Names are there, Nature's sacred watchwords, they
Were borne aloft in bright emblazonry;
The nations thronged around, and cried aloud,
As with one voice, Truth, liberty, and love!
Suddenly fierce confusion fell from heaven
Among them: there was strife, deceit, and fear:
Tyrants rushed in, and did divide the spoil.
This was the shadow of the truth I saw.
P B Shelley, Prometheus Unbound
This page contains the solution of the grand unification problem in physics by an entirely different approach to that which has been taken by orthodox science community. One central piece of this note is to argue that there is already a great deal of evidence that the universe has four macroscopic spatial dimensions and we accept this conclusion and therefore work in a situation where we must put aside the established physical theories that are based on the assumption of three macroscopic spatial dimensions as unacceptable despite their strengths in explaining subtle physical phenomena. The solution is speculative and has large technical gaps but it is conceptually clear and is based on empirical evidence. This is not peer-reviewed material, but it is a solution that has empirical backing. People have asked for peer-reviewed work by me and here is one. For the impatient, a central problem in physics that is simply resolved by my approach to physics is the cosmological constant problem, and you can find it here. The famous Anfinsen experiments in the 1950s and 1960s showed that proteins spontaneously refold to the same three dimensional shape after being denatured, and Christian Anfinsen had formulated the thermodynamic hypothesis that the folded state reaches a minimum energy state. We will work instead work with the hypothesis that the universe is four dimensional and hidden symmetries of electromagnetism play a decisive role in protein folding. This is against a background where the dominant hypotheses include seeking codes within the DNA for folding. At this early stage of our approach, we appeal to parsimony for this direction: parsimony at the scale of physics which leads to the hypothesis of SU(2) electromagnetism and then the expectation that such a force would have a decisive role in the protein folding. In a sense this direction is in line with the movement towards folding as a consequence only of physical chemistry considerations.
The idea that the universe has higher than three macroscopic dimensions goes back to the nineteenth century, and Riemann had conceived the notion of hyperspace for example. We will argue that the universe has exactly four macroscopic spatial dimensions and for this we have, besides the various conceptual arguments, the following concrete argument. Assume the universe is compact, which is easy to justify on the grounds of parsimony based on the observation that the cosmic background radiation has a uniform lower bound. Assume furthermore that it is a sphere of some dimension. Assume as well that the energy spectrum of the hydrogen atom occurs from the harmonics of a D-sphere. Then what would be the best D? We perform a simple test on the measured energy spectrum by plotting the sum of absolute residuals of fitting the spectrum to eigenvalues of the Laplacian for a sphere of dimension D and seek the minimum. Here is the graph.
The R^2 of the fits are better than 0.99 with that of S4 harmonics 0.9998, and if we calculate a "signal-to-noise" for the fit to the four dimensional spherical harmonics by dividing the mean absolute value of the fitted values by the mean absolute value of the residuals, we obtain around 1404. The residuals of the fits are much larger than the fit of the Bohr model, but it is still providing us some support for a four dimensional spherical universe because it is fitting a set of observed wavelengths quite well by usual metrics. Instead of graphing sum of absolute residuals we could graph a measure of relative error, the mean absolute residuals divided by mean energy levels being fit so that it is clearer that the mean residuals are in the order of magnitude less than 1% of the energy level values.
A very simple argument shows that energies are quantized automatically in a spherical universe: Assume the universe is a sphere of any dimension of radius R, and consider only electromagnetic energy which are inversely related to wavelengths. The possible wavelengths from great circles are 2*pi*R/N for integer N by considering the wave restricted to the circles. Therefore there is a quantization of frequencies and therefore of energies. Therefore whether one has classical or quantum mechanics on a sphere, quantization of energy is automatic. On other sorts of compact manifolds, one could have geodesics that are not closed, such as on a torus, and this argument does not apply.Imagine that we are back in 1900 when Planck was addressing the
blackbody problem. If Planck knew during this time that the cosmic
background radiation has a uniform lower bound which is impossible if
the background radiation appears through diffusion on a noncompact
universe and therefore that the universe is compact, and furthermore he
knew that gravitational field equations with cosmological constant 1/R^2 are automatic for any three
dimensional submanifold of a 4-sphere of fixed radius R, then he could
argue that energy must be quantized because frequencies must be
quantized as N*(1/2*pi*R) where R is the radius of the universe.
The S4 theory we claim is the correct grand unification theory that merges physics and metaphysics, and fulfills the dreams of both Einstein and Hamilton; of Einstein because it says the universe is deterministic and stationary and finds a natural interpretation of the gravitational field equations; of Hamilton because it says the universe is a quaternionic projective line. Note that the central difficulty in the grand unification problem lies in uniting electromagnetism and gravity because the unity of the other forces can be and have been achieved by appropriate gauge theories. Taking the cue from electromagnetism which can be described in terms of a potential A identified with a connection on a principal U(1) bundle -- which generalize to principal G-bundles where one considers potentials as Lie-algebra valued 1-form A. We begin with noting the evidence for four macroscopic spatial dimensions for which the most natural electromagnetism would have SU(2) gauge. Now SU(2) is simply the unit quaternions. These theoretical considerations show their efficacy in a concrete problem: the protein shape determination problem is to determine the three dimensional shape of folded proteins given its amino acid sequence. Our approach produces simple linear complexity algorithms to relatively accurately determine the sequence of scalar angles in folded proteins for which some results can be found here. In the 1970s Sir Michael Atiyah and other geometers had studied gauge theory on the 4-sphere for purely mathematical interests. We will argue that the actual universe we inhabit is a 4-sphere scaled by 1/h in length equivalent where h is Planck's constant. In particular, there is no great controversy about how to define and study electromagnetism on a 4-sphere. The radical claim that we would like to establish is that in fact a single force of electromagnetism is the unique force governing our universe.
One of the major impetuses for a four dimensional classical physics on a four-sphere is the observation of the formal equivalence of the gravitational field equations and the Gauss equations of a three-dimensional submanifold of a 4-sphere. If M is a three dimensional submanifold of a scaled 4-sphere and D is the Levi-Civita connection on the ambient space then the Levi-Civita connection of the submanifold is obtained by projection of DX onto the tangent space by subtracting its normal component. The Einstein gravitational field equations are formally identical to the Ricci curvature equation for the submanifold where the stress-energy tensor is a term involving the second fundamental form. In our interpretation, general relativity is identifying these extrinsic curvature terms with the presence of matter and energy. If the universe were known to be a four-dimensional sphere of fixed radius, then the gravitational field equations themselves arise as a consequence of the shape of the universe, and removes gravity as a separate force of nature altogether.
One of the objections to a macroscopic spatial dimensions is that the force law would be 1/r^{D-1} which would be at odds with Newtonian laws, and has been an argument for introducing compact dimensions rather than macroscopic spatial dimensions. One possible way out of this problem is to consider the gravitational field equations as primary rather than the force law. We posit that the the physical three dimensional universe is a preferred submanifold of a four dimensional sphere, for which an inverse square force law might hold regardless.
The universe has four macroscopic spatial dimensions and is governed by a unique force, which is SU(2) electromagnetism on a 4-sphere of fixed radius. Indeed the objective S4 universe is identical to Jung's collective unconscious.
The four-sphere universe is all that exists. SU(2) electromagnetism
is woven in the fabric of the universe in the sense that the square root
of the scalar wave equation produces Maxwell's equations for
electromagnetism. Specifically, suppose that D=d+d* is the Dirac operator on forms on S4. Then the square root of the wave equation reduces to the system DE+d/dtB=0 and DB-d/dtE = 0. These are formal Maxwell´s equations with the Dirac operator replacing the ´curl´. Now suppose w1, w2, w3, w4 is a local coframe defining a 3-dimensional submanifold M by w4=0. Suppose Dw4=H. This is a measure of extrinsic curvature of M, and the restriction of the 4D Maxwell´s equation replaces D, the Dirac operator on the ambient space S4 with DM+H where DM is the Dirac operator of M. Thus we have Maxwell´s equations both for 4D electromagnetism and its restriction to a 3D subspace. It is interesting to note that this term H enters these equations which we have simply called Maxwell´s equations formally for a 4D electromagnetic force. Now we can test the hypothesis that H has a direct measurable effect on observed electromagnetism that is being missed by the standard Maxwell´s equations which have curl instead of Dirac operators. Consciousness is a product of electromagnetism and
does not 'precede' it. Causality is a feature of existence. We can
discover laws of nature and existence but the deepest laws of the
universe cannot have a conscious designer. The structure of the
universe is such that no other mechanism is possible.
Analysis of functions on the four-sphere can be done as follows. It is well known that the square-integrable functions on a compact manifold decomposes into finite dimensional orthogonal subspaces of eigenfunctions of the Laplacian. In the case of the 4-sphere we have a more detailed construction. The eigenvalues of the Laplacian are k(k+3) and if Y_{kl} are eigenfunction with eigenvalue k(k+1) indexed by l then P_k( xy ) = \sum_l Y_{kl}(x) \bar{Y_{kl}}(y) then these are projection kernels to the eigenspace for general functions. The function P_k of a real variable are Gegenbauer polynomials. Following Narcowich, Petrushev, and Ward, we can construct needlets by considering the localized projection kernels by multiplying this kernel by a cutoff function a. The cutoff function can be chosen as follows: let B>0 and \phi be smooth, symmetric on the real line, supported in [-1,1], identically 1 when |t|<1/B and decreasing on positive reals. Then define b(t) so b^(2) = \phi(t/B) - \phi(t) and note that for |t|>1, \sum_j b^2(t/B^j) = 1. Define the kernels K_j(x,y) = \sum_l b^2(l/B^j)P_l(xy) = \sum_{B^{j-1} < l < B^{j+1}} b^2(l/B^j)P_l(xy). Then define M_j(x,y) = \sum_l b(l/B^j)P_l(xy). Needlets are defined from these M_j which are almost exponentially localized in terms of dist(x,y) which on the sphere is simply the angle between x and y. These needlets also span square-integrable functions.
Parsimony and predictive power rank scientific theories
Scientific theories are simply
logically coherent stories about the objective universe. To the extent a logically
coherent--that is, a story without any internal inconsistencies--fit the observed facts, it is an admissible scientific theory. Karl Popper had introduced the notion of falsifiability; individual scientific statements in his account are admissible as scientific propositions if they are falsifiable, which means in principle there could exist objective evidence to reject them as false. For a given set
of phenomena, a scientific theory about the set is admissible only if the logically
coherent story fits the phenomena. For the development of scientific
knowledge, it is not only common but essential to have divergent
scientific theories. But not all scientific theories, that is to say,
theories about phenomena in the actual universe, have equal merit as a correct description of
reality. Once we eliminate stories
that are incompatible with objective phenomena, and reduce attention from the subset of all scientific theories about the universe to only
those scientific theories that fit objective observations, even then,
not all of the logically coherent theories are of equal merit. There
are two criteria by which one can rank admissible scientific theories.
Given two theories A and B both of which fit facts of a domain of inquiry, A is better than B if either (a) A is simpler than B, or (b) A and B
are of equivalent complexity but A predicts more true propositions
about phenomena outside the domain of initial fit. The first criterion, principle of
parsimony, has a long tradition in the development of modern science and
it has been a deep belief among such great scientists such as Albert
Einstein and Hermann Weyl that the deepest laws of the universe are
simple and elegant, and thus the criteria given here for ranking scientific theories are not arbitrary. Perhaps representing the philosophical views of the physics community, Paul Dirac wrote on an article published on June 2010, "It is more important to have beauty in one’s equations than to have
them fit experiment. If Schrodinger had been more confident of his work,
he could have published it some months earlier, and he could have
published a more accurate equation. That equation is now known as the
Klein-Gordon equation, although it was really discovered by Schrodinger,
and in fact was discovered by Schrodinger
before he discovered his nonrelativistic treatment of the hydrogen
atom. It seems that if one is working from the point of view of getting
beauty in one’s equations, and if one has really a sound insight, one is
on a sure line of progress. If there is not complete agreement between
the results of one’s work and experiment, one should not allow oneself
to be too discouraged, because the discrepancy may well be due to minor
features that are not properly taken into account and that will get
cleared up with further developments of the theory."
On the other hand, a quantitative version of these ideas require us to
examine the foundational ideas in mathematical statistics. In a paper
of 1922 Ronald Fisher introduced the maximum likelihood method to
estimate parameters of finite dimensional probability distribution based
on a finite sample of data. In 1992 Hirodogu Akaike had refined the maximum likelihood priniciple to the maximization of expected log-likelihood and clarified the way in which the expected log-likelihood can be considered as a distance between probability densities. In order for us to have a fruitful
comparison of scientific theories, we must first develop a method to compare
two different scientific theories on a finite and finite dimensional
data set from observations.
Our approach is to seek maximal parsimony for a physical theory. The heart of the theory lies in the observation that on quaternions, the square root of the scalar wave equation produces the equivalent of Maxwell's equations of electromagnetism. Indeed, if we assume known that the universe is compact and four dimensional, we have available to us the description of electromagnetism via differential forms as follows. The electromagnetic action is S[A] = \int_M (-(1/2) da \wedge A + A \wedge J ) where A is the electromagnetic potential 1-form and J the current 3-form. The electromagnetic tensor F is the exterior derivative of A, and the Maxwell's equations can be written dF=0 and *dF=J. For the specific case of a scaled four dimensional sphere, we note that every closed 2-form is exact. An immediate question is on the issue of expansion of the universe which is accepted. The central issue here is that of redshift of absorption lines from distant galaxies. Recall that the the absorption spectrum from a light source is seen as dark lines in the spectrum corresponding to particular energy level photons being absorbed by some elements. Comparing the absorption lines of observed light of distant galaxies to absorption lines one would expect, one finds that there is a shift towards lower frequency or higher wavelength. Hubble's 1929 redshift-distance linear relation is the foundation of the conclusion of orthodox physics that the universe is expanding. We contest the interpretation of the redshift as a Doppler effect and instead provide an alternative explanation of the redshift as a reddening filter produced by uniform absorption of energy from photons traveling long distances by material in the four-dimensional universe. Recall that when Christian Doppler introduced the concept of the Doppler effect in 1841, he had conjectured wrongly that the colors of stars could be explained by the Doppler effect. By 1868 there was an account of Doppler effect on the spectral lines of stars by William Huggins. Just as the colors of stars are not caused by a Doppler effect, neither is an expansion of the universe: we advocate a move away from the Doppler effect explanation of the cosmological redshift as well.
The cosmic background radiation was discovered much later, in 1964 and was used to support the Big Bang theory but we interpret the cosmic background radiation as energy producing an approximate thermal equilibrium that can be used to show -- assuming it occurred through diffusion -- that the universe is compact. The compactness of the universe implies that quantization of energy is automatic even for classical mechanics. This conclusion comes from the fact that on a compact universe, smooth functions can be decomposed by the eigenspaces of the Laplacian because the resolvent of the Laplacian is a compact self-adjoint operator and hence the spectral theorem for compact selfadjoint operators applies. Perhaps a more vivid description for this phenomenon would be that the quantization of energy of the hydrogen atom is showing us the pure tones of the universe. The energy spectrum of hydrogen can be used to determine the shape of the universe, a sphere of radius 1/h in length equivalent where h is Planck's constant. Then the fact that an electron in classical orbit around a proton also in motion is an inertial system on a four-sphere is inertial and hence does not do work and does not lose energy can be used to overcome one of the major contradictions of classical mechanics in a flat three-dimensional space.
One of the central observations that lead to the S4 theory is that while in three-dimensional flat space a classical electron orbiting around a stationary proton would do work, lose energy and collapse, this is not the case in a four dimensional sphere of a fixed radius where an electron-proton system can be inertial.
The S4 physics theory
We shall construct a physical theory that is extremely different from that established from the turn of the twentieth century in the sense that we sidestep both the details of quantum theory and gravitational theories and claim that the phenomena that have underpinned these are consequences of a more fundamental deterministic classical physics. We shall claim that the universe is governed by deterministic electromagnetic laws on a round four-sphere of fixed radius. In our model, the universe is stationary and time is linear, unidirectional and constant. The essence of the theory can be summarized as follows. The
(Dirac-type) square root of the wave equation on quaternions are the
Maxwell's equations on spinors, pairs of quaternions but with curl
replaced by the Dirac operator. These equations are right-equivariant
by right-multiplication by quaternions and so descend to the
quaternionic projective line, the four-sphere. This produces a
four-dimensional electromagnetism on the four-sphere which is the 'grand
unification force" governing the actual universe because this classical
electromagnetism implies quantization of energy and the gravitational
field equations. If the sphere's radius is set to 1/h where h is
Planck's constant then the 'cosmological constant' is h^2 in theory and
indeed the Lambda-CDM measured value of the cosmological constant is in
this order of magnitude. Thus the theory trivially resolves the
'cosmological constant problem'.
Our theory is quite distinct from Einstein's theory for gravtiation in that we do not have an intricate relation between time and space or a concept of a 'spacetime' manifold. Einstein had taken the speed of light to be constant in physical space and then posited a spacetime. We know that the speed of light depends on the density of the medium of its propagation, for example light travels 1.5 times slower in water than in air, and thus in the physical universe, where the cosmic background radiation uniformly fills space, it is not reasonable to expect the speed of light to be constant. We allow the speed of light to be variable depending on the curvature of the physical universe but we posit that time is constant and linear and unrelated to space.
The universe is has four rather than three macroscopic spatial dimensions
In order to avoid confusion that is widespread regarding what is meant by the word 'dimension', we specify our meaning of this term. In mathematics, dimension has a precise meaning which is used throughout the sciences and engineering as well. The straight line is defined in terms of a completion of the rational numbers with its natural ordering forming the 'real' numbers. This line is considered one-dimensional. An abstract vector space is defined as a set closed under addition and with a scalar multiplication by real numbers. For such abstract vector spaces, dimension is defined in terms of the minimum number of vectors required to span the space. This definition produces the right sort of picture for lines, planes, and the apparent physical world of three dimensions.Our usual experience is of three macroscopic spatial dimensions. In
three dimensions, we grasp with the hands and see with the eyes, and we
use the five senses to perceive, and we usually accept these things as
real, disregarding the admonition of Socrates that we might forsake the
muses thus. Now suppose we disregard Socrates as a speculative
philosopher; suppose that we take a hardline pragmatic view of the
physical universe that we inhabit, following the train of sensible and
rational people throughout the ages. Let us agree with them that the
universe has three macroscopic dimensions, as is the experience of most
practical people. If we do this, then we
can make quite a bit of progress in the sciences and technology. We can
send astronauts to the moon; we can create nuclear weapons; we can
create electronic communication devices capable of connecting us across
the world. Why bother with Socrates’ admonitions from more than two
millenia ago?
Ah, but then we can look at crystal structures, real crystal
structures that we can observe. On the one hand, in the world of mathematics, crystal structures can be analyzed purely
theoretically. There it does not matter how many dimensions we consider,
for we can find purely mathematical ways of defining and analyzing
crystal symmetries. Here we have some mathematical results from the
1920s, which tell us the possible rotational symmetries we can observe
in crystals by the number of dimensions. In up to three dimensions, the
possibilities are limited. There could be 2, 3, 4, and 6
fold rotational symmetries. For other possibilities of rotational
symmetries, we need more than three dimensions. With four or five
dimensions, we could, theoretically, observe 5, 8, 10, and 12 fold
rotational symmetries. These rotational symmetries cannot be observed in
crystals in three dimensions even theoretically.
Now we turn to the actual world of observations. There we can observe
crystals and consider their set of rotational symmetries. In
1984, the crystallographic group headed by Daniel Shechtman observed
crystals with 10 and 5 fold rotational symmetries. They gave evidence for the first time in modern history that clearly
showed that the universe is four dimensional. But after some discussion
the scientific community settled to a wrong conclusion from this clear
evidence. They had instead created a complex obfuscation: the theory of
quasicrystals. During October 2011 Daniel Shechtman won the Nobel prize in chemistry for discovering quasicrystals but the issue of the actual discovery is still an issue that requires care of interpretation, as the claim that what had been discovered, beyond clear observations of symmetries of orders 5, 8, 10, and 12 is up for debate. From our view, there is no qualitative difference between the x-ray diffraction images of the discovered 'quasicrystals' and much more ordinary crystals but with rotational symmetries impossible for 3D crystals. Ockham's razor suggests what were discovered are actually crystals, but four-dimensional rather than three-dimensional crystals.
Since then, thousands of crystals have been discovered to span the
rotational symmetries in the set of orders 5, 8, 10, and 12. Any such
example is sufficient to be a counterexample to the hypothesis: “The
universe is three dimensional”. Note that this evidence is discrete, and thus stronger than precision tests of any accepted physical theory that relies on measurements of continuous variables. Ockham’s razor then makes four
dimensions as the parsimonious conclusion. There is no scientific
consensus on four spatial dimensions but the evidence from crystal
symmetries is only dismissed by stretching a highly complex quasicrystal
theory that makes a mockery of Ockham’s razor.
It is worth emphasizing that discrete evidence of 5, 8, 10, and 12 fold rotational symmetries is sharper evidence than quantitative theories for hypothesis testing. Explanations of these symmetries using aperiodic tilings of the three dimensional space are strictly more complex than simply concluding that the universe has at least four macroscopic dimensions. Indeed, the standard modeling technique for quasicrystals is as projections of translation-invariant crystals in higher dimensions. The idea for quasicrystal modeling comes from the study of quasiperiodic functions in mathematics: this is the idea that any quasiperiodic function can be thought of as being derived from a periodic function in a higher dimension. Thus quasicrystals themselves are modelled in this way. Thus the conclusion from observations of so-called quasicrystals that fit projections of higher dimensional translational-invariant crystals that the universe itself has four macroscopic spatial dimensions is parsimonious as well.
More than three spatial dimensions are not new theoretically. The German mathematician Theodor Kaluza described in a letter in April 1919 to Einstein where he showed how an additional dimension can produce a unification of relativity and electromagnetism, and this formed the basis from 1926 of Kaluza-Klein theories. The S4 theory differs from the Kaluza-Klein theory as the former produces a constant positive Ricci curvature theory while the latter produce Ricci-flat theories. Rudolf Steiner had attempted by 1900 a spiritual science where he posited four dimensions to explain noumenal world versus the apparent phenomenal world is three dimensional. The concrete evidence produced by the discovery by Dan Shechtman in 1982 of 5- and 10-fold rotational symmetries in 'quasicrystals' allows us to describe a four dimensional theory based on evidence of four dimensions. More recent attempts at grand unification theories have considered higher 'compactified' dimensions, such as 11, but there is no direct evidence of such dimensions.
Summary of S4 physical theory basis
Four pieces of evidence provide the basis for a new physical theory. This theory was proposed first publicly in July 2008. There are two major claims that need to be cleared before S4 theory can be established. These are that on a 4-dimensional sphere of radius 1/h a proton and electron can be in classical orbits without loss of electromagnetic energy -- which would take away one of the two foundational problems that led to the development of quantum mechanics, and the second that quantization is automatic for a compact universe. For the first, the basic idea is to imagine a proton in a circular orbit in the (z,w) plane where w is unseen and we would observe the proton in an up-and-down motion along the z-axis. When the proton moves up, the induced magnetic field does nothing to an electron in a circular orbit in the (x,y) plane because the action is v x B which vanishes when v and B are parallel. If the electron velocity is inward-pointing then the acceleration due to B is downward while the electric field produces an acceleration that is upward and inward. It is useful to note that for a proton in a circular orbit in the (z,w) plane and an electron in a circular orbit in the (x,y) plane, assuming that such a configuration is stable, has the property that the induced magnetic fields have no effect on either particle, and furthermore the euclidean distance between the two particles is constant.
First, observations of crystals with 5, 8, 10 and 12 fold rotational symmetry tell us that the universe is macroscopically at least 4 dimensional by the crystallographic restriction theorem; this is a theorem that provides restrictions on the possible rotational orders in crystal lattices for Euclidean spaces of any dimension. Observations of 5 and 10 fold rotational symmetry in crystal were first observed by Daniel Shechtman and first published in 1984 by D. Shechtman, Blech, Gratias, Cahn. The crystallographic restriction for any number of dimensions was formulated by R. Vaidyanathaswamy in 1928 in two papers, "Integer-roots of the unit matrix" and "On the possible periods of integer matrices" but was known for two dimensional crystals even in early 1820s. This theorem applies to crystal lattices with translation invariance. Although in three dimensions the 'quasicrystals' lack translation invariance, they are modelled as projections of 4 and 6 dimensional crystals. The observations have the parsimonious conclusion of the existence of four macroscopic spatial dimensions.
Let us now make a technical point about why four dimensions are very special and why four spatial dimensions produce the structure of electromagnetism naturally. The special feature of four dimensional riemannian manifolds geometrically is that the rotation group SO(4) is locally a product SU(2) x SU(2). Indeed, the double cover of SO(4) is Spin(4) = SU(2) x SU(2). On differential forms, this decomposition manifests as the Hodge *-operator mapping 2-forms to 2-forms, and the square of the Hodge *-operator is +1 for a 4-manifold, and pointwise the 2-forms decompose into +/-1 eigenspaces of this operator. The decomposition of 2-forms into two three-dimensional vector spaces is the Lie algebra version of the decomposition of Spin(4). The two factors of Spin(4) provide us with the structure behind the duality of electric and magnetic fields. This special feature of four dimensions is not available in higher dimensions.
Arguments Against more than 4 dimensions:
There are widespread beliefs both in Occident and Orient and especially among communities interested in metaphysics, of existence of higher than four macroscopic dimensions. Quite often, in these communities, there is a confusion about the meaning of the term 'dimension' that could mean something other than spatial dimension. This confusion leads to difficulties in discussion on this extremely important issue. In various currently accepted physical theories, the convention is to add microscopic higher dimensions to three macroscopic dimensions. There is absolutely no objective evidence to be found for such beliefs from observed rotational symmetries in crystals or indeed in any other objective phenomena that has been recorded and made public. It has been more than a quarter of a century since the first observations of 5 and 10 fold rotational symmetry in crystals had been first observed. Since that time, there has not been observations of 7, 9, 14, 15, 18 fold rotational symmetry that would lead to conclusions of at least six macroscopic spatial dimensions. While it is true that absence of evidence is no reason to conclude the nonexistence of higher than four dimensions, there is a decisive factor in this issue: the behavior of Spin(4) as a product of SU(2) x SU(2) has no analogues in higher dimensions and this product structure is what makes the duality of electromagnetism possible. These facts together give us a great deal of confidence for the nonexistence of higher dimensions.
Second, there is an isotropic component of the background radiation of the universe that extends in every direction observed for up to 300 light years from Earth generally referred to as the cosmic background radiation or CBR. The (almost) uniform cosmic background radiation with an isotropic lower bound tells us that the universe is compact if there were a time in the past when the radiation were confined to a compact region of the universe. This is a consequence of Gaussian upper bounds of parabolic kernels of Shroedinger operators on complete non-compact manifolds with a uniform lower bound on the Ricci curvature proved by Peter Li and Shing-Tung Yau in 1986, but Gaussian upper bounds for parabolic kernels was established by John Nash already in 1958.
John Nash provided upper and lower Gaussian bounds for the heat kernel of uniformly parabolic equations in higher than two dimensions in 1958, and later in 1967 Aronson produced the same results using slightly different methods. In the context of riemannian manifolds, Gaussian upper bounds for the heat kernel first appeared in the work of Cheng, Li, and Yau in 1981 for manifolds with bounded sectional curvature and then sharp bounds on the heat kernels on complete manifolds with a lower bound on Ricci curvature in 1986.
To our knowledge the idea of using this result on the observed cosmic background radiation to argue that the universe is compact was first proposed by us in 2008.
A different track to prove compactness of the universe follows from assuming that the universe satisfies the so-called Einstein gravitational field equation with a positive lower bound on the Ricci curvature and applying Myers' theorem, which says that if the Ricci curvature of a riemannian manifold is bounded below by a positive constant (n-1)k, then the manifold must be compact with diameter bounded above by pi / sqrt(k). A consensus view had developed following Edwin Hubble's 1929 discovery of a linear relation between redshift and distances of galaxies which had been interpreted as a Doppler effect. Our explanation of the redshift is that it is caused by the cosmic background radiation which acts as a drag on energy of photons traveling long distances, which explains the linearity of the redshift.
Note that the interpretation of an observed linear increasing relation between distance of galaxies and their redshifts does not immediately lead to the conclusion that it is a Doppler effect at all. There is an obvious assumption in such a deduction, which is that the energy of the photons travelling over vast distances have retained their energy. An alternate explanation is not a Doppler effect but that the photons have lost energy linearly travelling the distances. Thus the conclusion that the universe is expanding can be put aside. Indeed, this linear redshift-distance relation is the fundamental evidence that has been presented for the claims of expansionary universe models. We believe that not only is this redshift evidence misinterpreted as a Doppler effect but that this misinterpretation is the single-most disastrous error in cosmology for the twentieth century. Later, when the cosmic background radiation was discovered in the mid 1960s, it was used to support the "Big Bang" theory which is a sanctification of Christian mythology -- which is the view I share with Hannes Alfven. The almost thermal equilibrium of the cosmic background radiation was interpreted as an "echo" of the big bang. A much more parsimonious explanation is presented in the last paragraph: the cosmic background radiation in thermal equilibrium is evidence that the universe is compact and not infinite and unbounded.
Third, for a compact four dimensional universe, there exist a discrete set of pure tones of the universe and one can ask whether these pure tones can be heard in the energy spectrum of the hydrogen atom. Mathematically, this follows from the fact that the Laplacian on functions has a compact resolvent, and the spectral theorem of compact self-adjoint operators tells us that the spectrum of the Laplacian is discrete. Not for general compact manifolds but for spheres, there is a very simple argument for quantization of energy. In the case of spheres, all geodesics are closed of fixed length provided by the great circles. In this case possible wavelengths are 2*pi*R/N for integral N, and therefore frequencies are quantized as N*(1/(2*pi*R)) and so long as we have the relation Energy = Const x frequency, energies are quantized as well. This argument does not hold for a torus, where not all geodesics are closed as we can consider a line through the plane with an irrational slope projected to the 2-torus which is not a closed geodesic. The known examples of manifolds all whose geodesics are closed are the spheres, complex and quaternionic projective spaces and the Cayley plane which are the rank 1 symmetric spaces of Cartan according to Bott who proved that such manifolds have a cohomology ring that is a truncated polynomial ring. Alan Weinstein showed in 1973 that if the closed geodesics all have length 2*pi*R for fixed R then the volume of the manifold is an integral multiple of R^n volume(S^n).
Recall that if F(x) is an eigenfunction of the Laplacian with eigenvalue s^2, then exp(s t) F(x) solves the wave equation and these are the stationary waves or 'pure tones' of the space. The energy quantization in units of Planck's constant h tells us that the shape of the universe is a sphere of radius 1/h. This is a consequence of the fact that on smooth compact manifolds the Laplacian on functions as well as on differential forms have compact resolvents, and by the spectral theorem for compact self-adjoint operators one deduces the discreteness for the spectrum of the resolvent and therefore for the operators themselves. Historically, this was one of the major open questions of nineteenth century mathematical physics and was resolved for the first time by Fredholm in 1900. Incidentally, Planck introduced the constant that bears his name in the same year to resolve the question of matching the theoretical prediction of the intensity distribution of a blackbody with the observed intensity distribution. Planck hypothesized that energy is quantized to resolve this problem. Quantization of energy on a compact manifold does not require an independent quantum hypothesis, and even classical physics on a compact manifold must have quantization of energy.
Fourth, the stability of the hydrogen atom from a simple classical model (in four dimensions) tells us that symmetry group of electromagnetism is SU(2) rather than U(1). Indeed, the "Maxwell's equations" is the square root system for the standard classical wave equation on quaternions. The homogeneous wave equation produces the homogeneous Maxwell's equations with Dirac operator replacing curl but acting on spinors which are nothing but pairs of quaternions. The equations are equivariant to right multiplication by unit quaternions which are the same as SU(2). Pairs of quaternions form the spinor space by left multiplication by unit quaternions.
Some comments about established physical theories
Quantum mechanics and general relativity the twin poles of modern physics, and S4 theory is based on different foundations than these. The two major anomalies of nineteenth century that led to the development of quantum mechanics are the blackbody radiation problem and the stability of the hydrogen atom. A close examination of Planck's 1900 solution of the blackbody radiation problem solution shows that in fact quantization of energy is sufficient to resolve the anomaly of describing the intensity distribution of the blackbody. Quantum mechanics builds a three-dimensional theory based on quantization; in contrast S4 theory uses the fact that classical mechanics on a compact universe necessarily implies quantization of energy and thus no separate quantum mechanics is needed. Furthermore, on a 4-sphere of an electron-proton system does not necessarily lose energy from work and thus the stability of matter problem does not arise.
The Mathematical Model of the universe in S4 theory
A parsimonious mathematical model of the universe that takes these observations into account, which I had originally proposed in 2008, is a single force for the universe, electromagnetism on the Hopf fibration S^7 -> S^4 and an equation describing the structure of the three dimensional universe, which I call 'Einstein structure equation' which is what is known in currently accepted physical theories as the 'Einstein gravitational field equations'. The Hopf fibration can be thought of as the restriction of the map defining the quaternionic projective space HxH->HP^1 where right multiplication by nonzero quaternions on the two factors are identified to a point. An interesting feature of this quaternionic projective line description is that HxH with the action of SU(2) by left multiplication provides the natural interpretation of spaces of spinors. In particular, I will argue in later sections that neither gravitation nor the weak and strong nuclear forces need exist as separate forces of nature as they are described in this model by electromagnetism.
Maxwell's Equations of Electromagnetism are the Square Root of the Wave Equation
In fact there is a very simple mathematical construction that tell us why the version of Maxwell's equations on quaternionic projective line, the 4-sphere, appears naturally from the classical wave equation. The classical wave equation on the quaternions,
( d^2/dt^2 - Laplacian ) w = 0
can be factored into two matrix-valued linear equations (d/dt - Dirac) Psi = 0 and (d/dt + Dirac) Psi = 0 using the same method that led Dirac originally to introduce the Dirac equation from the Klein-Gordon equation. If w are quaternion-valued with quaternion variable, then the coefficients of the Dirac operator must be 2x2 matrices with quaternion entries -- which is the real Clifford algebra of a vector space of four dimensions -- and Psi must be pairs of quaternions. The pair of linear equations are the homogenous Maxwell's equations with curl replaced by Dirac operator. Thus the appearance of the spinor space is natural and its interpretation is that of a projective "lift" of the physical universe. In particular, if physics were described by deterministic systems on spinors, then this implies physical laws are essentially deterministic in the physical universe.
The fact that the "square root" of the scalar wave equation on quaternions mathematically forces the right deterministic equations for electromagnetism with no additional assumptions shows that the electromagnetic laws are a feature of the STRUCTURE of the universe and no other sorts of laws could be natural. Thus these laws PRECEDE consciousness. The four dimensional universe is the SPIRITUAL universe and is all that exists. The MATERIAL universe sits within it like a napkin floating on the ocean. Life, divinity, consciousness can all be described in principle by electromagnetic laws.
From the observation that the Dirac's square root of the scalar wave equation leads to a four-dimensional Maxwell's equations gives us a concrete interpretation of spinors which had been used in the orthodox physical theories as a mathematical structure that provides a theoretical tool. The 4-sphere of radius 1/h is the structure of the physical universe in our model, and the spinors can now be identified with a projective lift of the physical universe that decouples electromagnetism according to the splitting Spin(4) = SU(2) x SU(2). In particular spinors gain a direct physical interpretation in our theory. Recall that in 1925 Kronig, Uhlenbeck and Goudsmit proposed spin and the fine structure of the hydrogen energy spectrum was used to support spin. Here we have an entirely different way of explaining spin not as a specific 'quantum mechanical' phenomenon but as encoded into the structure of the four-dimensional universe in a fairly essential way.
Recall that the classical Maxwell´s equations can be written on forms in terms of the exterior derivative d and its adjoint d*. Thus it is not a surprise that they can be rewritten in terms of the Dirac operator D=d+d*.
There are alternative explanations to the evidence claimed to show expansion of the universe
I will describe some details for these claims in later sections. The immediate objection to proceeding as above is that the established theories often rely on an expanding universe model. The greatest scientific error of the twentieth century was perhaps the interpretation of Hubble's 1929 discovered linearity between galaxy distance and redshift as a Doppler effect. I will argue that there are far simpler explanations in terms of the cosmic background radiation, which would act as a reddening filter for signals from distant galaxies due to a skew in its intensity distribution towards lower frequencies after a peak.
The four set of observations listed above, the empirical observation of 5, 8, 10, and 12 fold rotational symmetry observed in abundance, the fact that the cosmic background radiation has a uniform lower bound, and the observed energy spectrum of the hydrogen atom must be explained for any coherent physical theory about the universe we live in. Therefore, a minimally correct physical theory must also explain these observations and no scientific theory about the objective universe that cannot jointly explain these phenomena can be an admissible scientific theory. While it is usual for scientific hypotheses on observations in practice to take simple forms, the statistical theory for hypothesis testing does not rule out complex hypotheses that are logically consistent. Thus a complex hypothesis that is a string of statements 'A and B and C and D and E and ...' is a valid hypothesis and can be tested jointly on the set of observations mentioned above. In this sense, there is a minimal admissible theory. It is of clear interest then to attempt to produce such a minimal theory because such a theory is likely to be the correct description of our universe. The reason for this is that the principle of ordering theories by the two criteria of simplicity and predictive power are not arbitrary but have been verified in numerous occasions in the past in the development of modern science as a feature of the fundamental laws governing the universe.
The observations are fundamental in the sense that the observed features required to make deductions about the geometry and physics of the universe are not fine-tuned measurements but gross features. If one imagines that these are the only observables available about the objective universe, then one would still produce and SU(2) gauge theory for electromagnetism on a round four-sphere of radius 1/h. By mathematical observations of Shlomo Sternberg from 1977, it is not difficult to define a classical mechanics theory in this case because Hamiltonian mechanics for particles with gauge group G can be defined by modifying the standard symplectic form on the cotangent bundle of the sphere and keeping the Hamiltonian function fixed. Indeed, geometrically, a great deal more is known about a four dimensional spherical universe by the work of Atiyah, Drinfeld, Hitchin, and Manin because they produced an ansatz to describe all magnetic monopoles, also known as instantons, for SU(2) electromagnetism on the Hopf fibration over S^4 in the 1970s.
Time is linear and constant
An important issue is time. The second law of thermodynamics has been routinely been verified in thousands if not millions of chemical experiments. Thus the parsimony principle suggests that time has a unique direction based on the observed fact that entropy is a monotonically increasing function of time. There are no verified results yet for those who have considered whether Maxwell's thought experiment which could lead to violations of the second law of thermodynamic, and hence there is an excellent justification for time as directed and in fact independent of the spatial dimensions. In other words, there is justification for a classical model of time. It is not reversible and is unaffected by other physical quantities.
This theory will produce, in principle, the simplest possible grand unification theory because it is classical, and because it relies on a very small core of experimental observations. In particular, my convictions on the correctness of this theory rely strongly on the principle of parsimony, or Ockham's razor.
Einstein's gravitational field equations hold for any smooth 3-submanifold of a 4-sphere
The S4 physics theory is classical, but can be summarized as classical theory on a round four sphere of radius 1/h with electromagnetism with gauge group SU(2). It is my hope that since the four sphere has constant sectional curvature, and hence constant scalar curvature, which is equivalent to Einstein's field equations for an empty universe, that gravitational field equations for the physical three-dimensional universe, presumably embedded in an S4(1/h) would be a consequence of an electromagnetic theory on this manifold. An encouraging fact is that the equation for Ricci curvature of ANY embedded 3-manifold in the 4-sphere has the form of Einstein's field equations with a nonzero stress-energy tensor, and for 3-manifolds with constant mean curvature the stress-energy tensor will satisfy the conservation law T_ij;j = 0.
Suppose M is a three dimensional smooth submanifold of a 4-sphere of fixed radius. Then the Ricci curvature of the submanifold at a point can be written as the Ricci curvature of the ambient 4-sphere and terms involving the second fundamental form of the embedding. Three principal curvature directions specify the second fundamental form, and the Ricci curvature of the ambient manifold is a constant multiple of the metric, as the 4-sphere is an Einstein manifold. Thus an equation of the form
R_ij - constant x g_ij = constant x T_ij
holds for every point of the submanifold where T_ij is a tensor involving the second fundamental form. This is precisely the form of Einstein's gravitational field equations. In our theory, time is linear and constant and we do not consider a 'space-time' manifold but consider time and space to be separate. Regardless, the observation above suggests that the Einstein field equations for a three dimensional physical universe would hold as a consequence of the spherical shape of the universe regardless of the laws governing the universe. Thus once again the principle of parsimony suggests that gravitation not be considered a separate force. The standard Einstein field equations are of the form:
R_ij - (1/4) (Scalar curvature) g_ij + (Cosmological constant) g_ij = 8 pi G T_ij.
As a quick sanity check, we note that for a 4-sphere to produce the right scale for equations like the Schroedinger equation would require a radius of the order 1/h. Such a sphere then would produce a scalar curvature of the order h^2. The measured value of the cosmological constant by the established Lamdba-CDM model
is ~ 10^(-47) GeV^4. Note that Planck's constant is h =4.135667 x
10^(-24) GeV, which implies h^2 ~ 1.6 x 10^(-47) GeV. This means
that S^4(1/h) whose "cosmological constant" is h^2 is a close match to observation. This is positive because the mismatch between quantum-physics extrapolated 'vacuum energy' and the cosmological constant have been a source of a fairly large disagreements in some previous efforts towards grand unification. The reader can find a simple discussion of the mathematical derivation of the analogue of the gravitational field equation here.
How does one explain the enormous numerical success of Quantum Mechanics if S4 theory is correct?
Quantum mechanics is fooled by randomness. Quantum mechanics arose to resolve two major contradictions in nineteenth century classical deterministic physics on flat 3D space. First was the fact that blackbody radiation distribution did not fit the Rayleigh-Jeans law and the second was that classical electromagnetic theory would suggest that an electron in classical orbit around a positively charged nucleus would do work, lose energy, and collapse. The first was resolved by Planck by hypothesizing that energy is quantized, which fit the measured intensity distribution of an approximate blackbody. The second was resolved by adopting special rules for subatomic particles. Well, BOTH of these problems are resolved by classical mechanics on a four-dimensional sphere of fixed radius -- here energy is quantized even for classical mechanics because the universe is compact, and here circular orbits of electrons around protons does not do work and therefore there is no question of stability of matter.
The universe is a 4-sphere with the physical universe as a smooth submanifold. Quantum mechanics arises as an approximate linear theory on the tangent space at a point of the physical universe. One would expect such a linear approximation to be quite accurate in the small scale. The basic quantum mechanical theory is phenomenally successful and it is a linear theory.The S4 theory is geometrically distinct from extant grand unification theories
To put this in the context of unification theories, a basic problem is to reconcile gravitation with the three other fundamental forces: electromagnetism, weak nuclear force and the strong nuclear forces. There are sophisticated models such as string theory and Kaluza-Klein theories which add extra "compact" dimensions to produce quantum theory of fields. Novelties in the approach here are that it is based on a compact static universe relying on observations. This reduces the complexity of producing a unification theory immensely because quantization becomes a feature of classical mechanics. Indeed sphere is one of a small class of options if we want the spectrum of the universe to be able to describe a energy quantization.
On a four dimensional sphere, the differential 2-forms decompose into a direct sum of self-dual and anti-self-dual 2-forms based on the eigenvalues of the Hodge *-operator because *^2 = +1. From the decomposition of 2-forms one can obtain spinors by using the Lie group decomposition Spin(4) = SU(2) x SU(2) using standard Clifford algebra construction at cotangent spaces. The Yang-Mills instantons have been completely classified by Atiyah-Hitchin-Drinfeld-Manin. Instantons of magnetic charge k, k>0 form a space of real dimension 8k-3.
Although this theory was not directly based on the doctrines of ancient Greeks, there are some remarkable consistencies. Empedocles (490-430 BC) contends that
the history of the universe is cyclic and eternal and the primary
moving factors are Love and Strife. According to Empedocles,
all matter periodically contracts and expands. Under the power of
Love everything unites until there is only "The One" - a divine and
homogeneous sphere.
I believe S4
physics is capable of merging science, art, and spirit. This does not
mean that it is able to answer super detailed questions, but it makes a
coherent link between these fields of human knowledge that provides us
with a way to break the deadlocks created by overspecialization in the
sciences and division of humanities from sciences.Immediate consequences of the S4 theory
The universe must be static if spectral theory provides the energy quanta. Later I will give an indication of why the relation between distance and redshift of galaxies need not imply that distant galaxies are moving away from us. If one accepts this evidence, then the universe may never have had a hot dense state, although energy content of the universe but that does not rule out energy content of the universe being concentrated in a small region in the past. I recognize that this is a matter of a long debate in the cosmology community, and Peebles mentions that Dicke (1970), for example considered the question of the possibility of the evolution of physical constants over time by considering the decay-products of long-lived radioactive nuclei in meteorites and minerals but did not find compelling evidence for such an evolution. The usual interpretation had been that there was not a tight connection between microscopic physics and the state of the universe. The S4 theory provides such a connection as a fundamental feature -- the radius of the four-dimensional universe is 1/h in length equivalent which both provided the connection between microscopic and macroscopic physics and simultaneously provides a tight structure for the shape of the universe capable of explaining quantum phenomena and the geometry of gravitation in a simple manner.
Electromagnetic stability of matter is a consequence of 4D classical electromagnetism (still working on the quantitative version). In particular, two basic reasons to develop quantum mechanics are resolved: energy quantization and stability of matter.
The added dimension gives us plethora of magnetic monopoles with magnetic states that I will call light and dark based on the sign of the magnetic charge. The connection I draw between this and what is traditionally called light and dark in spirituality is because of my own qualitative observations of metaphysical experiences. While this at first glance seems like pseudo-science, the ability of this identification to resolve many issues of interest to human beings generally is reason for patience on this judgment.
I identify a human spirit as an object composed of magnetic monopoles that is an extension of the physical body. The identification of dark and light with magnetic poles allow us to formulate precise ideas about the nature of the human, cosmic, and the universal spirits. Note that this sort of idea is very old: Empedocles had produced theories in the same spirit as the one that results from S4 physical theory. One fascinating aspect is that of reincarnation. Ian Stephenson's study of around 2500 young children who remembered past lives is evidence against the "no reincarnation" hypothesis. A static eternal universe plus reincarnation would suggest that we are eternal spirits as well.
Application of S4 theory to the problem of protein shape analysis
Proteins are biological macromolecules encoded in DNA that form the bulk of chemical function of living organisms. With some exceptions, proteins are linear chains of amino acids from an alphabet of 20. The main feature of these linear chains is that they are biologically active only in particular 3-dimensional conformations and otherwise biologically inert. An interesting problem is the determination of the shapes of the folds given their amino acid sequences only. Conventional approaches to modelling of protein 'folding' is by thermodynamic and energy minimization considerations with the accepted theories of three dimensional physics. Given the robustness of the folding mechanism which supports the entire spectrum of physical living organisms, a delicate balancing of Van der Waals forces seem to us unlikely to be a correct description for the folding mechanism. The S4 description suggests an alternative mechanism, that proteins fold essentially because electromagnetism has higher symmetry than U(1) and instead has SU(2) symmetry. A great deal of effort would be required to detail chemical processes in terms of this view but we can make some working qualitative predictions as a guide to the protein shape determination problem. First, expect a strong rigidity of protein shapes which would be unexpected from thermodynamic models of protein shape determination. For this purpose, we consider a simplified model of the protein shape. We chose the nitrogen atom along the backbone of around 7600 proteins whose shapes were stored in the RCSB protein database and decomposed these protein skeleta as sequences of rotations along these vertices and found that the scalar angles are not uniform but are discrete. We quantified the divergence of these angle distributions from the uniform distribution to establish that this phenomenon occurs generally. We found that the average Kullback-Leibler divergence of the angle distributions from the uniform distribution across triples of amino acid types is ~407 which is much higher than 0 which would mean close to the uniform distribution. A second observation about protein sequences is that they are highly redundant: the average maximum intersection in longest common substrings between this sample of 7600 proteins is 51% of the given protein. Since folded proteins are known to be rigid, this observation suggests that protein shape determination algorithms would systematically benefit from using databased shapes to determine pieces of an unknown new protein not in the database.
The twist decomposition of protein shapes into a sequence of rotations leads to a number of possibilities for quantitative measures of differences of shapes. Since the unit quaternions SU(2) is the double cover of SO(3), a natural distance metric on SO(3) is d( Q1, Q2 ) = 1 - | Q1 . Q2 | where the dot product is taken for the unit quaternions. Using this measure, we can immediately obtain an interesting result, which is that the simplification of the protein shape by subselecting the Nitrogen atom on the backbone and then decomposing the shape into twists in SO(3) is capable of showing us that substrings in proteins that are the same can still show quantitative rigidity. The average SO(3) distance for pairs of proteins with matching long substrings is 0.065 +/- 0.12 with a sample of 593 pairs. These basic results and observations provide solid footing for our model of protein folding, where the twist decomposition of proteins is the simplification of protein shapes and the shape prediction problem reduces to producing a sequence of SO(3) elements given a sequence of amino acids.
The twist decomposition of protein shapes into a sequence of twists or SO(3) elements allows us to treat the protein shape determination problem directly as a language translation problem, where the noisy channel model had shown success. If we knew that the twists arise from a finite alphabet, from example from a fixed discretization of SO(3) then we can apply the machinery of n-gram models and maximum likelihood methods of natural language processing to this problem.
Socrates, in Plato's
Theaetetus, decries those who believe that only things that can be grasped by the hands are real as being happily without the muses. The condemnation could be expanded to those who rely on the five senses more generally. Indeed, Socrates takes a bolder position that he who sees with the eye is blind and recounts the allegory of the cave, in Plato's
The Republic. They were not alone in this position. Buddha, Emerson, Thoreau, Nietzsche, Goethe, Kant, Rumi, Ibn Sina, Ibn Arabi, Dali, Magritte, Einstein, Newton, William James, just to name a few well-known people, have shared this view. In a letter to Wilhelmine von Zenge, on 3/22/1801 Heinrich von Kleist writes:
"We are unable to
decide whether that which we call truth really is truth, or whether it
only appears to us to be. If the latter, then the truth we assemble
here is nothing after our death, and all endeavour to acquire a
possession which will follow us to the grave is in vain." The clarification for an unseen reality provided by the Allegory of the Cave is such that it is worthwhile to reproduce Book VII of
The Republic:
---
Socrates - GLAUCON
And now, I said, let me show in a figure how far our nature is enlightened or unenlightened: --Behold! human beings living in a underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets.
I see.
And do you see, I said, men passing along the wall carrying all sorts of vessels, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent.
You have shown me a strange image, and they are strange prisoners.
Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave?
True, he said; how could they see anything but the shadows if they were never allowed to move their heads?
And of the objects which are being carried in like manner they would only see the shadows?
Yes, he said.
And if they were able to converse with one another, would they not suppose that they were naming what was actually before them?
Very true.
And suppose further that the prison had an echo which came from the other side, would they not be sure to fancy when one of the passers-by spoke that the voice which they heard came from the passing shadow?
No question, he replied.
To them, I said, the truth would be literally nothing but the shadows of the images.
That is certain.
And now look again, and see what will naturally follow it' the prisoners are released and disabused of their error. At first, when any of them is liberated and compelled suddenly to stand up and turn his neck round and walk and look towards the light, he will suffer sharp pains; the glare will distress him, and he will be unable to see the realities of which in his former state he had seen the shadows; and then conceive some one saying to him, that what he saw before was an illusion, but that now, when he is approaching nearer to being and his eye is turned towards more real existence, he has a clearer vision, -what will be his reply? And you may further imagine that his instructor is pointing to the objects as they pass and requiring him to name them, -will he not be perplexed? Will he not fancy that the shadows which he formerly saw are truer than the objects which are now shown to him?
Far truer.
And if he is compelled to look straight at the light, will he not have a pain in his eyes which will make him turn away to take and take in the objects of vision which he can see, and which he will conceive to be in reality clearer than the things which are now being shown to him?
True, he now
And suppose once more, that he is reluctantly dragged up a steep and rugged ascent, and held fast until he 's forced into the presence of the sun himself, is he not likely to be pained and irritated? When he approaches the light his eyes will be dazzled, and he will not be able to see anything at all of what are now called realities.
Not all in a moment, he said.
He will require to grow accustomed to the sight of the upper world. And first he will see the shadows best, next the reflections of men and other objects in the water, and then the objects themselves; then he will gaze upon the light of the moon and the stars and the spangled heaven; and he will see the sky and the stars by night better than the sun or the light of the sun by day?
Certainly.
Last of he will be able to see the sun, and not mere reflections of him in the water, but he will see him in his own proper place, and not in another; and he will contemplate him as he is.
Certainly.
He will then proceed to argue that this is he who gives the season and the years, and is the guardian of all that is in the visible world, and in a certain way the cause of all things which he and his fellows have been accustomed to behold?
Clearly, he said, he would first see the sun and then reason about him.
And when he remembered his old habitation, and the wisdom of the den and his fellow-prisoners, do you not suppose that he would felicitate himself on the change, and pity them?
Certainly, he would.
And if they were in the habit of conferring honours among themselves on those who were quickest to observe the passing shadows and to remark which of them went before, and which followed after, and which were together; and who were therefore best able to draw conclusions as to the future, do you think that he would care for such honours and glories, or envy the possessors of them? Would he not say with Homer,
Better to be the poor servant of a poor master, and to endure anything, rather than think as they do and live after their manner?
Yes, he said, I think that he would rather suffer anything than entertain these false notions and live in this miserable manner.
Imagine once more, I said, such an one coming suddenly out of the sun to be replaced in his old situation; would he not be certain to have his eyes full of darkness?
To be sure, he said.
And if there were a contest, and he had to compete in measuring the shadows with the prisoners who had never moved out of the den, while his sight was still weak, and before his eyes had become steady (and the time which would be needed to acquire this new habit of sight might be very considerable) would he not be ridiculous? Men would say of him that up he went and down he came without his eyes; and that it was better not even to think of ascending; and if any one tried to loose another and lead him up to the light, let them only catch the offender, and they would put him to death.
No question, he said.
This entire allegory, I said, you may now append, dear Glaucon, to the previous argument; the prison-house is the world of sight, the light of the fire is the sun, and you will not misapprehend me if you interpret the journey upwards to be the ascent of the soul into the intellectual world according to my poor belief, which, at your desire, I have expressed whether rightly or wrongly God knows. But, whether true or false, my opinion is that in the world of knowledge the idea of good appears last of all, and is seen only with an effort; and, when seen, is also inferred to be the universal author of all things beautiful and right, parent of light and of the lord of light in this visible world, and the immediate source of reason and truth in the intellectual; and that this is the power upon which he who would act rationally, either in public or private life must have his eye fixed.
I agree, he said, as far as I am able to understand you.
Moreover, I said, you must not wonder that those who attain to this beatific vision are unwilling to descend to human affairs; for their souls are ever hastening into the upper world where they desire to dwell; which desire of theirs is very natural, if our allegory may be trusted.
Yes, very natural.
And is there anything surprising in one who passes from divine contemplations to the evil state of man, misbehaving himself in a ridiculous manner; if, while his eyes are blinking and before he has become accustomed to the surrounding darkness, he is compelled to fight in courts of law, or in other places, about the images or the shadows of images of justice, and is endeavouring to meet the conceptions of those who have never yet seen absolute justice?
Anything but surprising, he replied.
Any one who has common sense will remember that the bewilderments of the eyes are of two kinds, and arise from two causes, either from coming out of the light or from going into the light, which is true of the mind's eye, quite as much as of the bodily eye; and he who remembers this when he sees any one whose vision is perplexed and weak, will not be too ready to laugh; he will first ask whether that soul of man has come out of the brighter light, and is unable to see because unaccustomed to the dark, or having turned from darkness to the day is dazzled by excess of light. And he will count the one happy in his condition and state of being, and he will pity the other; or, if he have a mind to laugh at the soul which comes from below into the light, there will be more reason in this than in the laugh which greets him who returns from above out of the light into the den.
That, he said, is a very just distinction.
But then, if I am right, certain professors of education must be wrong when they say that they can put a knowledge into the soul which was not there before, like sight into blind eyes.
They undoubtedly say this, he replied.
Whereas, our argument shows that the power and capacity of learning exists in the soul already; and that just as the eye was unable to turn from darkness to light without the whole body, so too the instrument of knowledge can only by the movement of the whole soul be turned from the world of becoming into that of being, and learn by degrees to endure the sight of being, and of the brightest and best of being, or in other words, of the good.
Very true.
And must there not be some art which will effect conversion in the easiest and quickest manner; not implanting the faculty of sight, for that exists already, but has been turned in the wrong direction, and is looking away from the truth?
Yes, he said, such an art may be presumed.
And whereas the other so-called virtues of the soul seem to be akin to bodily qualities, for even when they are not originally innate they can be implanted later by habit and exercise, the of wisdom more than anything else contains a divine element which always remains, and by this conversion is rendered useful and profitable; or, on the other hand, hurtful and useless. Did you never observe the narrow intelligence flashing from the keen eye of a clever rogue --how eager he is, how clearly his paltry soul sees the way to his end; he is the reverse of blind, but his keen eyesight is forced into the service of evil, and he is mischievous in proportion to his cleverness.
Very true, he said.
But what if there had been a circumcision of such natures in the days of their youth; and they had been severed from those sensual pleasures, such as eating and drinking, which, like leaden weights, were attached to them at their birth, and which drag them down and turn the vision of their souls upon the things that are below --if, I say, they had been released from these impediments and turned in the opposite direction, the very same faculty in them would have seen the truth as keenly as they see what their eyes are turned to now.
Very likely.
Yes, I said; and there is another thing which is likely. or rather a necessary inference from what has preceded, that neither the uneducated and uninformed of the truth, nor yet those who never make an end of their education, will be able ministers of State; not the former, because they have no single aim of duty which is the rule of all their actions, private as well as public; nor the latter, because they will not act at all except upon compulsion, fancying that they are already dwelling apart in the islands of the blest.
Very true, he replied.
Then, I said, the business of us who are the founders of the State will be to compel the best minds to attain that knowledge which we have already shown to be the greatest of all-they must continue to ascend until they arrive at the good; but when they have ascended and seen enough we must not allow them to do as they do now.
What do you mean?
I mean that they remain in the upper world: but this must not be allowed; they must be made to descend again among the prisoners in the den, and partake of their labours and honours, whether they are worth having or not.
But is not this unjust? he said; ought we to give them a worse life, when they might have a better?
You have again forgotten, my friend, I said, the intention of the legislator, who did not aim at making any one class in the State happy above the rest; the happiness was to be in the whole State, and he held the citizens together by persuasion and necessity, making them benefactors of the State, and therefore benefactors of one another; to this end he created them, not to please themselves, but to be his instruments in binding up the State.
True, he said, I had forgotten.
Observe, Glaucon, that there will be no injustice in compelling our philosophers to have a care and providence of others; we shall explain to them that in other States, men of their class are not obliged to share in the toils of politics: and this is reasonable, for they grow up at their own sweet will, and the government would rather not have them. Being self-taught, they cannot be expected to show any gratitude for a culture which they have never received. But we have brought you into the world to be rulers of the hive, kings of yourselves and of the other citizens, and have educated you far better and more perfectly than they have been educated, and you are better able to share in the double duty. Wherefore each of you, when his turn comes, must go down to the general underground abode, and get the habit of seeing in the dark. When you have acquired the habit, you will see ten thousand times better than the inhabitants of the den, and you will know what the several images are, and what they represent, because you have seen the beautiful and just and good in their truth. And thus our State which is also yours will be a reality, and not a dream only, and will be administered in a spirit unlike that of other States, in which men fight with one another about shadows only and are distracted in the struggle for power, which in their eyes is a great good. Whereas the truth is that the State in which the rulers are most reluctant to govern is always the best and most quietly governed, and the State in which they are most eager, the worst.
Quite true, he replied.
And will our pupils, when they hear this, refuse to take their turn at the toils of State, when they are allowed to spend the greater part of their time with one another in the heavenly light?
Impossible, he answered; for they are just men, and the commands which we impose upon them are just; there can be no doubt that every one of them will take office as a stern necessity, and not after the fashion of our present rulers of State.
Yes, my friend, I said; and there lies the point. You must contrive for your future rulers another and a better life than that of a ruler, and then you may have a well-ordered State; for only in the State which offers this, will they rule who are truly rich, not in silver and gold, but in virtue and wisdom, which are the true blessings of life. Whereas if they go to the administration of public affairs, poor and hungering after the' own private advantage, thinking that hence they are to snatch the chief good, order there can never be; for they will be fighting about office, and the civil and domestic broils which thus arise will be the ruin of the rulers themselves and of the whole State.
Most true, he replied.
And the only life which looks down upon the life of political ambition is that of true philosophy. Do you know of any other?
Indeed, I do not, he said.
And those who govern ought not to be lovers of the task? For, if they are, there will be rival lovers, and they will fight.
No question.
Who then are those whom we shall compel to be guardians? Surely they will be the men who are wisest about affairs of State, and by whom the State is best administered, and who at the same time have other honours and another and a better life than that of politics?
They are the men, and I will choose them, he replied.
And now shall we consider in what way such guardians will be produced, and how they are to be brought from darkness to light, --as some are said to have ascended from the world below to the gods?
By all means, he replied.
The process, I said, is not the turning over of an oyster-shell, but the turning round of a soul passing from a day which is little better than night to the true day of being, that is, the ascent from below, which we affirm to be true philosophy?
Quite so.
And should we not enquire what sort of knowledge has the power of effecting such a change?
Certainly.
What sort of knowledge is there which would draw the soul from becoming to being? And another consideration has just occurred to me: You will remember that our young men are to be warrior athletes
Yes, that was said.
Then this new kind of knowledge must have an additional quality?
What quality?
Usefulness in war.
Yes, if possible.
There were two parts in our former scheme of education, were there not?
Just so.
There was gymnastic which presided over the growth and decay of the body, and may therefore be regarded as having to do with generation and corruption?
True.
Then that is not the knowledge which we are seeking to discover? No.
But what do you say of music, which also entered to a certain extent into our former scheme?
Music, he said, as you will remember, was the counterpart of gymnastic, and trained the guardians by the influences of habit, by harmony making them harmonious, by rhythm rhythmical, but not giving them science; and the words, whether fabulous or possibly true, had kindred elements of rhythm and harmony in them. But in music there was nothing which tended to that good which you are now seeking.
You are most accurate, I said, in your recollection; in music there certainly was nothing of the kind. But what branch of knowledge is there, my dear Glaucon, which is of the desired nature; since all the useful arts were reckoned mean by us?
Undoubtedly; and yet if music and gymnastic are excluded, and the arts are also excluded, what remains?
Well, I said, there may be nothing left of our special subjects; and then we shall have to take something which is not special, but of universal application.
What may that be?
A something which all arts and sciences and intelligences use in common, and which every one first has to learn among the elements of education.
What is that?
The little matter of distinguishing one, two, and three --in a word, number and calculation: --do not all arts and sciences necessarily partake of them?
Yes.
Then the art of war partakes of them?
To the sure.
Then Palamedes, whenever he appears in tragedy, proves Agamemnon ridiculously unfit to be a general. Did you never remark how he declares that he had invented number, and had numbered the ships and set in array the ranks of the army at Troy; which implies that they had never been numbered before, and Agamemnon must be supposed literally to have been incapable of counting his own feet --how could he if he was ignorant of number? And if that is true, what sort of general must he have been?
I should say a very strange one, if this was as you say.
Can we deny that a warrior should have a knowledge of arithmetic?
Certainly he should, if he is to have the smallest understanding of military tactics, or indeed, I should rather say, if he is to be a man at all.
I should like to know whether you have the same notion which I have of this study?
What is your notion?
It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the soul towards being.
Will you explain your meaning? he said.
I will try, I said; and I wish you would share the enquiry with me, and say 'yes' or 'no' when I attempt to distinguish in my own mind what branches of knowledge have this attracting power, in order that we may have clearer proof that arithmetic is, as I suspect, one of them.
Explain, he said.
I mean to say that objects of sense are of two kinds; some of them do not invite thought because the sense is an adequate judge of them; while in the case of other objects sense is so untrustworthy that further enquiry is imperatively demanded.
You are clearly referring, he said, to the manner in which the senses are imposed upon by distance, and by painting in light and shade.
No, I said, that is not at all my meaning.
Then what is your meaning?
When speaking of uninviting objects, I mean those which do not pass from one sensation to the opposite; inviting objects are those which do; in this latter case the sense coming upon the object, whether at a distance or near, gives no more vivid idea of anything in particular than of its opposite. An illustration will make my meaning clearer: --here are three fingers --a little finger, a second finger, and a middle finger.
Very good.
You may suppose that they are seen quite close: And here comes the point.
What is it?
Each of them equally appears a finger, whether seen in the middle or at the extremity, whether white or black, or thick or thin --it makes no difference; a finger is a finger all the same. In these cases a man is not compelled to ask of thought the question, what is a finger? for the sight never intimates to the mind that a finger is other than a finger.
True.
And therefore, I said, as we might expect, there is nothing here which invites or excites intelligence.
There is not, he said.
But is this equally true of the greatness and smallness of the fingers? Can sight adequately perceive them? and is no difference made by the circumstance that one of the fingers is in the middle and another at the extremity? And in like manner does the touch adequately perceive the qualities of thickness or thinness, or softness or hardness? And so of the other senses; do they give perfect intimations of such matters? Is not their mode of operation on this wise --the sense which is concerned with the quality of hardness is necessarily concerned also with the quality of softness, and only intimates to the soul that the same thing is felt to be both hard and soft?
You are quite right, he said.
And must not the soul be perplexed at this intimation which the sense gives of a hard which is also soft? What, again, is the meaning of light and heavy, if that which is light is also heavy, and that which is heavy, light?
Yes, he said, these intimations which the soul receives are very curious and require to be explained.
Yes, I said, and in these perplexities the soul naturally summons to her aid calculation and intelligence, that she may see whether the several objects announced to her are one or two.
True.
And if they turn out to be two, is not each of them one and different?
Certainly.
And if each is one, and both are two, she will conceive the two as in a state of division, for if there were undivided they could only be conceived of as one?
True.
The eye certainly did see both small and great, but only in a confused manner; they were not distinguished.
Yes.
Whereas the thinking mind, intending to light up the chaos, was compelled to reverse the process, and look at small and great as separate and not confused.
Very true.
Was not this the beginning of the enquiry 'What is great?' and 'What is small?'
Exactly so.
And thus arose the distinction of the visible and the intelligible.
Most true.
This was what I meant when I spoke of impressions which invited the intellect, or the reverse --those which are simultaneous with opposite impressions, invite thought; those which are not simultaneous do not.
I understand, he said, and agree with you.
And to which class do unity and number belong?
I do not know, he replied.
Think a little and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks 'What is absolute unity?' This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.
And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude?
Yes, I said; and this being true of one must be equally true of all number?
Certainly.
And all arithmetic and calculation have to do with number?
Yes.
And they appear to lead the mind towards truth?
Yes, in a very remarkable manner.
Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician.
That is true.
And our guardian is both warrior and philosopher?
Certainly.
Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavour to persuade those who are prescribe to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being.
That is excellent, he said.
Yes, I said, and now having spoken of it, I must add how charming the science is! and in how many ways it conduces to our desired end, if pursued in the spirit of a philosopher, and not of a shopkeeper!
How do you mean?
I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. You know how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply, taking care that one shall continue one and not become lost in fractions.
That is very true.
Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, --what would they answer?
They would answer, as I should conceive, that they were speaking of those numbers which can only be realised in thought.
Then you see that this knowledge may be truly called necessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth?
Yes; that is a marked characteristic of it.
And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been.
Very true, he said.
And indeed, you will not easily find a more difficult study, and not many as difficult.
You will not.
And, for all these reasons, arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up.
I agree.
Let this then be made one of our subjects of education. And next, shall we enquire whether the kindred science also concerns us?
You mean geometry?
Exactly so.
Clearly, he said, we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the difference whether a general is or is not a geometrician.
Yes, I said, but for that purpose a very little of either geometry or calculation will be enough; the question relates rather to the greater and more advanced part of geometry --whether that tends in any degree to make more easy the vision of the idea of good; and thither, as I was saying, all things tend which compel the soul to turn her gaze towards that place, where is the full perfection of being, which she ought, by all means, to behold.
True, he said.
Then if geometry compels us to view being, it concerns us; if becoming only, it does not concern us?
Yes, that is what we assert.
Yet anybody who has the least acquaintance with geometry will not deny that such a conception of the science is in flat contradiction to the ordinary language of geometricians.
How so?
They have in view practice only, and are always speaking? in a narrow and ridiculous manner, of squaring and extending and applying and the like --they confuse the necessities of geometry with those of daily life; whereas knowledge is the real object of the whole science.
Certainly, he said.
Then must not a further admission be made?
What admission?
That the knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient.
That, he replied, may be readily allowed, and is true.
Then, my noble friend, geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down.
Nothing will be more likely to have such an effect.
Then nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry. Moreover the science has indirect effects, which are not small.
Of what kind? he said.
There are the military advantages of which you spoke, I said; and in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker of apprehension than one who has not.
Yes indeed, he said, there is an infinite difference between them.
Then shall we propose this as a second branch of knowledge which our youth will study?
Let us do so, he replied.
And suppose we make astronomy the third --what do you say?
I am strongly inclined to it, he said; the observation of the seasons and of months and years is as essential to the general as it is to the farmer or sailor.
I am amused, I said, at your fear of the world, which makes you guard against the appearance of insisting upon useless studies; and I quite admit the difficulty of believing that in every man there is an eye of the soul which, when by other pursuits lost and dimmed, is by these purified and re-illumined; and is more precious far than ten thousand bodily eyes, for by it alone is truth seen. Now there are two classes of persons: one class of those who will agree with you and will take your words as a revelation; another class to whom they will be utterly unmeaning, and who will naturally deem them to be idle tales, for they see no sort of profit which is to be obtained from them. And therefore you had better decide at once with which of the two you are proposing to argue. You will very likely say with neither, and that your chief aim in carrying on the argument is your own improvement; at the same time you do not grudge to others any benefit which they may receive.
I think that I should prefer to carry on the argument mainly on my own behalf.
Then take a step backward, for we have gone wrong in the order of the sciences.
What was the mistake? he said.
After plane geometry, I said, we proceeded at once to solids in revolution, instead of taking solids in themselves; whereas after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have followed.
That is true, Socrates; but so little seems to be known as yet about these subjects.
Why, yes, I said, and for two reasons: --in the first place, no government patronises them; this leads to a want of energy in the pursuit of them, and they are difficult; in the second place, students cannot learn them unless they have a director. But then a director can hardly be found, and even if he could, as matters now stand, the students, who are very conceited, would not attend to him. That, however, would be otherwise if the whole State became the director of these studies and gave honour to them; then disciples would want to come, and there would be continuous and earnest search, and discoveries would be made; since even now, disregarded as they are by the world, and maimed of their fair proportions, and although none of their votaries can tell the use of them, still these studies force their way by their natural charm, and very likely, if they had the help of the State, they would some day emerge into light.
Yes, he said, there is a remarkable charm in them. But I do not clearly understand the change in the order. First you began with a geometry of plane surfaces?
Yes, I said.
And you placed astronomy next, and then you made a step backward?
Yes, and I have delayed you by my hurry; the ludicrous state of solid geometry, which, in natural order, should have followed, made me pass over this branch and go on to astronomy, or motion of solids.
True, he said.
Then assuming that the science now omitted would come into existence if encouraged by the State, let us go on to astronomy, which will be fourth.
The right order, he replied. And now, Socrates, as you rebuked the vulgar manner in which I praised astronomy before, my praise shall be given in your own spirit. For every one, as I think, must see that astronomy compels the soul to look upwards and leads us from this world to another.
Every one but myself, I said; to every one else this may be clear, but not to me.
And what then would you say?
I should rather say that those who elevate astronomy into philosophy appear to me to make us look downwards and not upwards.
What do you mean? he asked.
You, I replied, have in your mind a truly sublime conception of our knowledge of the things above. And I dare say that if a person were to throw his head back and study the fretted ceiling, you would still think that his mind was the percipient, and not his eyes. And you are very likely right, and I may be a simpleton: but, in my opinion, that knowledge only which is of being and of the unseen can make the soul look upwards, and whether a man gapes at the heavens or blinks on the ground, seeking to learn some particular of sense, I would deny that he can learn, for nothing of that sort is matter of science; his soul is looking downwards, not upwards, whether his way to knowledge is by water or by land, whether he floats, or only lies on his back.
I acknowledge, he said, the justice of your rebuke. Still, I should like to ascertain how astronomy can be learned in any manner more conducive to that knowledge of which we are speaking?
I will tell you, I said: The starry heaven which we behold is wrought upon a visible ground, and therefore, although the fairest and most perfect of visible things, must necessarily be deemed inferior far to the true motions of absolute swiftness and absolute slowness, which are relative to each other, and carry with them that which is contained in them, in the true number and in every true figure. Now, these are to be apprehended by reason and intelligence, but not by sight.
True, he replied.
The spangled heavens should be used as a pattern and with a view to that higher knowledge; their beauty is like the beauty of figures or pictures excellently wrought by the hand of Daedalus, or some other great artist, which we may chance to behold; any geometrician who saw them would appreciate the exquisiteness of their workmanship, but he would never dream of thinking that in them he could find the true equal or the true double, or the truth of any other proportion.
No, he replied, such an idea would be ridiculous.
And will not a true astronomer have the same feeling when he looks at the movements of the stars? Will he not think that heaven and the things in heaven are framed by the Creator of them in the most perfect manner? But he will never imagine that the proportions of night and day, or of both to the month, or of the month to the year, or of the stars to these and to one another, and any other things that are material and visible can also be eternal and subject to no deviation --that would be absurd; and it is equally absurd to take so much pains in investigating their exact truth.
I quite agree, though I never thought of this before.
Then, I said, in astronomy, as in geometry, we should employ problems, and let the heavens alone if we would approach the subject in the right way and so make the natural gift of reason to be of any real use.
That, he said, is a work infinitely beyond our present astronomers.
Yes, I said; and there are many other things which must also have a similar extension given to them, if our legislation is to be of any value. But can you tell me of any other suitable study?
No, he said, not without thinking.
Motion, I said, has many forms, and not one only; two of them are obvious enough even to wits no better than ours; and there are others, as I imagine, which may be left to wiser persons.
But where are the two?
There is a second, I said, which is the counterpart of the one already named.
And what may that be?
The second, I said, would seem relatively to the ears to be what the first is to the eyes; for I conceive that as the eyes are designed to look up at the stars, so are the ears to hear harmonious motions; and these are sister sciences --as the Pythagoreans say, and we, Glaucon, agree with them?
Yes, he replied.
But this, I said, is a laborious study, and therefore we had better go and learn of them; and they will tell us whether there are any other applications of these sciences. At the same time, we must not lose sight of our own higher object.
What is that?
There is a perfection which all knowledge ought to reach, and which our pupils ought also to attain, and not to fall short of, as I was saying that they did in astronomy. For in the science of harmony, as you probably know, the same thing happens. The teachers of harmony compare the sounds and consonances which are heard only, and their labour, like that of the astronomers, is in vain.
Yes, by heaven! he said; and 'tis as good as a play to hear them talking about their condensed notes, as they call them; they put their ears close alongside of the strings like persons catching a sound from their neighbour's wall --one set of them declaring that they distinguish an intermediate note and have found the least interval which should be the unit of measurement; the others insisting that the two sounds have passed into the same --either party setting their ears before their understanding.
You mean, I said, those gentlemen who tease and torture the strings and rack them on the pegs of the instrument: might carry on the metaphor and speak after their manner of the blows which the plectrum gives, and make accusations against the strings, both of backwardness and forwardness to sound; but this would be tedious, and therefore I will only say that these are not the men, and that I am referring to the Pythagoreans, of whom I was just now proposing to enquire about harmony. For they too are in error, like the astronomers; they investigate the numbers of the harmonies which are heard, but they never attain to problems-that is to say, they never reach the natural harmonies of number, or reflect why some numbers are harmonious and others not.
That, he said, is a thing of more than mortal knowledge.
A thing, I replied, which I would rather call useful; that is, if sought after with a view to the beautiful and good; but if pursued in any other spirit, useless. Very true, he said.
Now, when all these studies reach the point of inter-communion and connection with one another, and come to be considered in their mutual affinities, then, I think, but not till then, will the pursuit of them have a value for our objects; otherwise there is no profit in them.
I suspect so; but you are speaking, Socrates, of a vast work.
What do you mean? I said; the prelude or what? Do you not know that all this is but the prelude to the actual strain which we have to learn? For you surely would not regard the skilled mathematician as a dialectician?
Assuredly not, he said; I have hardly ever known a mathematician who was capable of reasoning.
But do you imagine that men who are unable to give and take a reason will have the knowledge which we require of them?
Neither can this be supposed.
And so, Glaucon, I said, we have at last arrived at the hymn of dialectic. This is that strain which is of the intellect only, but which the faculty of sight will nevertheless be found to imitate; for sight, as you may remember, was imagined by us after a while to behold the real animals and stars, and last of all the sun himself. And so with dialectic; when a person starts on the discovery of the absolute by the light of reason only, and without any assistance of sense, and perseveres until by pure intelligence he arrives at the perception of the absolute good, he at last finds himself at the end of the intellectual world, as in the case of sight at the end of the visible.
Exactly, he said.
Then this is the progress which you call dialectic?
True.
But the release of the prisoners from chains, and their translation from the shadows to the images and to the light, and the ascent from the underground den to the sun, while in his presence they are vainly trying to look on animals and plants and the light of the sun, but are able to perceive even with their weak eyes the images in the water (which are divine), and are the shadows of true existence (not shadows of images cast by a light of fire, which compared with the sun is only an image) --this power of elevating the highest principle in the soul to the contemplation of that which is best in existence, with which we may compare the raising of that faculty which is the very light of the body to the sight of that which is brightest in the material and visible world --this power is given, as I was saying, by all that study and pursuit of the arts which has been described.
I agree in what you are saying, he replied, which may be hard to believe, yet, from another point of view, is harder still to deny. This, however, is not a theme to be treated of in passing only, but will have to be discussed again and again. And so, whether our conclusion be true or false, let us assume all this, and proceed at once from the prelude or preamble to the chief strain, and describe that in like manner. Say, then, what is the nature and what are the divisions of dialectic, and what are the paths which lead thither; for these paths will also lead to our final rest?
Dear Glaucon, I said, you will not be able to follow me here, though I would do my best, and you should behold not an image only but the absolute truth, according to my notion. Whether what I told you would or would not have been a reality I cannot venture to say; but you would have seen something like reality; of that I am confident.
Doubtless, he replied.
But I must also remind you, that the power of dialectic alone can reveal this, and only to one who is a disciple of the previous sciences.
Of that assertion you may be as confident as of the last.
And assuredly no one will argue that there is any other method of comprehending by any regular process all true existence or of ascertaining what each thing is in its own nature; for the arts in general are concerned with the desires or opinions of men, or are cultivated with a view to production and construction, or for the preservation of such productions and constructions; and as to the mathematical sciences which, as we were saying, have some apprehension of true being --geometry and the like --they only dream about being, but never can they behold the waking reality so long as they leave the hypotheses which they use unexamined, and are unable to give an account of them. For when a man knows not his own first principle, and when the conclusion and intermediate steps are also constructed out of he knows not what, how can he imagine that such a fabric of convention can ever become science?
Impossible, he said.
Then dialectic, and dialectic alone, goes directly to the first principle and is the only science which does away with hypotheses in order to make her ground secure; the eye of the soul, which is literally buried in an outlandish slough, is by her gentle aid lifted upwards; and she uses as handmaids and helpers in the work of conversion, the sciences which we have been discussing. Custom terms them sciences, but they ought to have some other name, implying greater clearness than opinion and less clearness than science: and this, in our previous sketch, was called understanding. But why should we dispute about names when we have realities of such importance to consider?
Why indeed, he said, when any name will do which expresses the thought of the mind with clearness?
At any rate, we are satisfied, as before, to have four divisions; two for intellect and two for opinion, and to call the first division science, the second understanding, the third belief, and the fourth perception of shadows, opinion being concerned with becoming, and intellect with being; and so to make a proportion: --
As being is to becoming, so is pure intellect to opinion.
And as intellect is to opinion, so is science to belief, and understanding to the perception of shadows. But let us defer the further correlation and subdivision of the subjects of opinion and of intellect, for it will be a long enquiry, many times longer than this has been.
As far as I understand, he said, I agree.
And do you also agree, I said, in describing the dialectician as one who attains a conception of the essence of each thing? And he who does not possess and is therefore unable to impart this conception, in whatever degree he fails, may in that degree also be said to fail in intelligence? Will you admit so much?
Yes, he said; how can I deny it?
And you would say the same of the conception of the good?
Until the person is able to abstract and define rationally the idea of good, and unless he can run the gauntlet of all objections, and is ready to disprove them, not by appeals to opinion, but to absolute truth, never faltering at any step of the argument --unless he can do all this, you would say that he knows neither the idea of good nor any other good; he apprehends only a shadow, if anything at all, which is given by opinion and not by science; --dreaming and slumbering in this life, before he is well awake here, he arrives at the world below, and has his final quietus.
In all that I should most certainly agree with you.
And surely you would not have the children of your ideal State, whom you are nurturing and educating --if the ideal ever becomes a reality --you would not allow the future rulers to be like posts, having no reason in them, and yet to be set in authority over the highest matters?
Certainly not.
Then you will make a law that they shall have such an education as will enable them to attain the greatest skill in asking and answering questions?
Yes, he said, you and I together will make it.
Dialectic, then, as you will agree, is the coping-stone of the sciences, and is set over them; no other science can be placed higher --the nature of knowledge can no further go?
I agree, he said.
But to whom we are to assign these studies, and in what way they are to be assigned, are questions which remain to be considered?
Yes, clearly.
You remember, I said, how the rulers were chosen before?
Certainly, he said.
The same natures must still be chosen, and the preference again given to the surest and the bravest, and, if possible, to the fairest; and, having noble and generous tempers, they should also have the natural gifts which will facilitate their education.
And what are these?
Such gifts as keenness and ready powers of acquisition; for the mind more often faints from the severity of study than from the severity of gymnastics: the toil is more entirely the mind's own, and is not shared with the body.
Very true, he replied.
Further, he of whom we are in search should have a good memory, and be an unwearied solid man who is a lover of labour in any line; or he will never be able to endure the great amount of bodily exercise and to go through all the intellectual discipline and study which we require of him.
Certainly, he said; he must have natural gifts.
The mistake at present is, that those who study philosophy have no vocation, and this, as I was before saying, is the reason why she has fallen into disrepute: her true sons should take her by the hand and not bastards.
What do you mean?
In the first place, her votary should not have a lame or halting industry --I mean, that he should not be half industrious and half idle: as, for example, when a man is a lover of gymnastic and hunting, and all other bodily exercises, but a hater rather than a lover of the labour of learning or listening or enquiring. Or the occupation to which he devotes himself may be of an opposite kind, and he may have the other sort of lameness.
Certainly, he said.
And as to truth, I said, is not a soul equally to be deemed halt and lame which hates voluntary falsehood and is extremely indignant at herself and others when they tell lies, but is patient of involuntary falsehood, and does not mind wallowing like a swinish beast in the mire of ignorance, and has no shame at being detected?
To be sure.
And, again, in respect of temperance, courage, magnificence, and every other virtue, should we not carefully distinguish between the true son and the bastard? for where there is no discernment of such qualities States and individuals unconsciously err and the State makes a ruler, and the individual a friend, of one who, being defective in some part of virtue, is in a figure lame or a bastard.
That is very true, he said.
All these things, then, will have to be carefully considered by us; and if only those whom we introduce to this vast system of education and training are sound in body and mind, justice herself will have nothing to say against us, and we shall be the saviours of the constitution and of the State; but, if our pupils are men of another stamp, the reverse will happen, and we shall pour a still greater flood of ridicule on philosophy than she has to endure at present.
That would not be creditable.
Certainly not, I said; and yet perhaps, in thus turning jest into earnest I am equally ridiculous.
In what respect?
I had forgotten, I said, that we were not serious, and spoke with too much excitement. For when I saw philosophy so undeservedly trampled under foot of men I could not help feeling a sort of indignation at the authors of her disgrace: and my anger made me too vehement.
Indeed! I was listening, and did not think so.
But I, who am the speaker, felt that I was. And now let me remind you that, although in our former selection we chose old men, we must not do so in this. Solon was under a delusion when he said that a man when he grows old may learn many things --for he can no more learn much than he can run much; youth is the time for any extraordinary toil.
Of course.
And, therefore, calculation and geometry and all the other elements of instruction, which are a preparation for dialectic, should be presented to the mind in childhood; not, however, under any notion of forcing our system of education.
Why not?
Because a freeman ought not to be a slave in the acquisition of knowledge of any kind. Bodily exercise, when compulsory, does no harm to the body; but knowledge which is acquired under compulsion obtains no hold on the mind.
Very true.
Then, my good friend, I said, do not use compulsion, but let early education be a sort of amusement; you will then be better able to find out the natural bent.
That is a very rational notion, he said.
Do you remember that the children, too, were to be taken to see the battle on horseback; and that if there were no danger they were to be brought close up and, like young hounds, have a taste of blood given them?
Yes, I remember.
The same practice may be followed, I said, in all these things --labours, lessons, dangers --and he who is most at home in all of them ought to be enrolled in a select number.
At what age?
At the age when the necessary gymnastics are over: the period whether of two or three years which passes in this sort of training is useless for any other purpose; for sleep and exercise are unpropitious to learning; and the trial of who is first in gymnastic exercises is one of the most important tests to which our youth are subjected.
Certainly, he replied.
After that time those who are selected from the class of twenty years old will be promoted to higher honour, and the sciences which they learned without any order in their early education will now be brought together, and they will be able to see the natural relationship of them to one another and to true being.
Yes, he said, that is the only kind of knowledge which takes lasting root.
Yes, I said; and the capacity for such knowledge is the great criterion of dialectical talent: the comprehensive mind is always the dialectical.
I agree with you, he said.
These, I said, are the points which you must consider; and those who have most of this comprehension, and who are more steadfast in their learning, and in their military and other appointed duties, when they have arrived at the age of thirty have to be chosen by you out of the select class, and elevated to higher honour; and you will have to prove them by the help of dialectic, in order to learn which of them is able to give up the use of sight and the other senses, and in company with truth to attain absolute being: And here, my friend, great caution is required.
Why great caution?
Do you not remark, I said, how great is the evil which dialectic has introduced?
What evil? he said.
The students of the art are filled with lawlessness.
Quite true, he said.
Do you think that there is anything so very unnatural or inexcusable in their case? or will you make allowance for them?
In what way make allowance?
I want you, I said, by way of parallel, to imagine a supposititious son who is brought up in great wealth; he is one of a great and numerous family, and has many flatterers. When he grows up to manhood, he learns that his alleged are not his real parents; but who the real are he is unable to discover. Can you guess how he will be likely to behave towards his flatterers and his supposed parents, first of all during the period when he is ignorant of the false relation, and then again when he knows? Or shall I guess for you?
If you please.
Then I should say, that while he is ignorant of the truth he will be likely to honour his father and his mother and his supposed relations more than the flatterers; he will be less inclined to neglect them when in need, or to do or say anything against them; and he will be less willing to disobey them in any important matter.
He will.
But when he has made the discovery, I should imagine that he would diminish his honour and regard for them, and would become more devoted to the flatterers; their influence over him would greatly increase; he would now live after their ways, and openly associate with them, and, unless he were of an unusually good disposition, he would trouble himself no more about his supposed parents or other relations.
Well, all that is very probable. But how is the image applicable to the disciples of philosophy?
In this way: you know that there are certain principles about justice and honour, which were taught us in childhood, and under their parental authority we have been brought up, obeying and honouring them.
That is true.
There are also opposite maxims and habits of pleasure which flatter and attract the soul, but do not influence those of us who have any sense of right, and they continue to obey and honour the maxims of their fathers.
True.
Now, when a man is in this state, and the questioning spirit asks what is fair or honourable, and he answers as the legislator has taught him, and then arguments many and diverse refute his words, until he is driven into believing that nothing is honourable any more than dishonourable, or just and good any more than the reverse, and so of all the notions which he most valued, do you think that he will still honour and obey them as before?
Impossible.
And when he ceases to think them honourable and natural as heretofore, and he fails to discover the true, can he be expected to pursue any life other than that which flatters his desires?
He cannot.
And from being a keeper of the law he is converted into a breaker of it?
Unquestionably.
Now all this is very natural in students of philosophy such as I have described, and also, as I was just now saying, most excusable.
Yes, he said; and, I may add, pitiable.
Therefore, that your feelings may not be moved to pity about our citizens who are now thirty years of age, every care must be taken in introducing them to dialectic.
Certainly.
There is a danger lest they should taste the dear delight too early; for youngsters, as you may have observed, when they first get the taste in their mouths, argue for amusement, and are always contradicting and refuting others in imitation of those who refute them; like puppy-dogs, they rejoice in pulling and tearing at all who come near them.
Yes, he said, there is nothing which they like better.
And when they have made many conquests and received defeats at the hands of many, they violently and speedily get into a way of not believing anything which they believed before, and hence, not only they, but philosophy and all that relates to it is apt to have a bad name with the rest of the world.
Too true, he said.
But when a man begins to get older, he will no longer be guilty of such insanity; he will imitate the dialectician who is seeking for truth, and not the eristic, who is contradicting for the sake of amusement; and the greater moderation of his character will increase instead of diminishing the honour of the pursuit.
Very true, he said.
And did we not make special provision for this, when we said that the disciples of philosophy were to be orderly and steadfast, not, as now, any chance aspirant or intruder?
Very true.
Suppose, I said, the study of philosophy to take the place of gymnastics and to be continued diligently and earnestly and exclusively for twice the number of years which were passed in bodily exercise --will that be enough?
Would you say six or four years? he asked.
Say five years, I replied; at the end of the time they must be sent down again into the den and compelled to hold any military or other office which young men are qualified to hold: in this way they will get their experience of life, and there will be an opportunity of trying whether, when they are drawn all manner of ways by temptation, they will stand firm or flinch.
And how long is this stage of their lives to last?
Fifteen years, I answered; and when they have reached fifty years of age, then let those who still survive and have distinguished themselves in every action of their lives and in every branch of knowledge come at last to their consummation; the time has now arrived at which they must raise the eye of the soul to the universal light which lightens all things, and behold the absolute good; for that is the, pattern according to which they are to order the State and the lives of individuals, and the remainder of their own lives also; making philosophy their chief pursuit, but, when their turn comes, toiling also at politics and ruling for the public good, not as though they were performing some heroic action, but simply as a matter of duty; and when they have brought up in each generation others like themselves and left them in their place to be governors of the State, then they will depart to the Islands of the Blest and dwell there; and the city will give them public memorials and sacrifices and honour them, if the Pythian oracle consent, as demi-gods, but if not, as in any case blessed and divine.
You are a sculptor, Socrates, and have made statues of our governors faultless in beauty.
Yes, I said, Glaucon, and of our governesses too; for you must not suppose that what I have been saying applies to men only and not to women as far as their natures can go.
There you are right, he said, since we have made them to share in all things like the men.
Well, I said, and you would agree (would you not?) that what has been said about the State and the government is not a mere dream, and although difficult not impossible, but only possible in the way which has been supposed; that is to say, when the true philosopher kings are born in a State, one or more of them, despising the honours of this present world which they deem mean and worthless, esteeming above all things right and the honour that springs from right, and regarding justice as the greatest and most necessary of all things, whose ministers they are, and whose principles will be exalted by them when they set in order their own city?
How will they proceed?
They will begin by sending out into the country all the inhabitants of the city who are more than ten years old, and will take possession of their children, who will be unaffected by the habits of their parents; these they will train in their own habits and laws, I mean in the laws which we have given them: and in this way the State and constitution of which we were speaking will soonest and most easily attain happiness, and the nation which has such a constitution will gain most.
Yes, that will be the best way. And I think, Socrates, that you have very well described how, if ever, such a constitution might come into being.
Enough then of the perfect State, and of the man who bears its image --there is no difficulty in seeing how we shall describe him.
There is no difficulty, he replied; and I agree with you in thinking that nothing more need be said.
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Rene Descartes is the prominent philosopher who introduced a purely rational basis for the study of truth and is a founder of the school of thought that led to modern empirical scientific view of the world. Yes, the history of philosophy is far more complex, and his philosophy was based on ancient Greek and other precursors; but it is not misleading to point to his work as the beginning of modern scientific rationality. This has been emphasized in the historical work of Anthony Grafton.
A major argument for objectivity attributed to him is that for existence of things. Self-awareness is sufficient to prove the real existence of at least one entity that cannot be doubted - that of the self. The beautiful argument is that a self-aware entity who doubts the existence of everything that he/she perceives still knows - knows - the existence of the self for it can think about itself. Even living in a fantasy-land created by a deceptive deity, such an entity can know the real existence, beyond doubt, of itself.
Two separate intelligent entities, thus, can prove the existence of an objective reality by the reasoning that they are in fact not the same beings. The difference between two beings can be tested real-time by any number of criteria. Given the reality of two separate beings, the existence of an objective reality outside ourselves is proven beyond doubt, and thus science, the study of an objective reality, becomes a well defined and rational enterprise.
The existence of objective reality is sometimes confusing to many who use
arguments that our perceptions of the mind are reality. While this is an interesting
thought, it is fallacious. It is true that the three dimensional senses that we usually use - sight of the two eyes, touch, smell, taste, hearing - only give us sense-data with which to construct our reality, and thus it seems sensible to consider the possibility that reality is constructed in our minds from such sense-data. However, this type of thought misunderstands Descartes' argument fundamentally. Descartes' argument was that even in the case that we lived in a world completely constructed by a demonic deity whose sole purpose were to create an illusory existence for us, our own existence is assured (in a possibly illusory reality) since we are self-aware. The existence of objective reality outside of our mind follows from consideration first of the fact that I, for instance, do not actually exist in your mind, and next that the observation of an electron, for instance, is independent of both your and my consciousness.
The last point requires the following clarification, because many people believe that consciousness does affect observation of an electron and use the existence of the measurement problem in quantum mechanics to justify this position. The problem is that in quantum mechanics, a system has a deterministic theoretical object called the wavefunction that is a mixture of states that evolves in time. But when an actual measurement is made, the wavefunction 'collapses' to a particular state and only such states are observed. This is not an issue in S4 physics where there are no wavefunctions at all, and all particles simply follow classical laws. The apparent stochastic behavior of an electron can be explained by movement of the physical sub-universe. The special status of the observer here is that the three-dimensional subspace is highly volatile and it's movement within four dimensions is affected by the observation process, giving the illusion that it is the electron that has a stochastic wavefunction.
A simpler example is that two self-aware entities can agree that there is a chair in a room. The chair is objective. Can the chair exist without observation? There is little reason to believe otherwise. If the chair is not a good example, the sun may be a better one. Those who believe that the sun exists only in the mind have a great deal of evidence from vast numbers of agents who have a common experience of the sun rising and setting with never having contact with each other. It would be difficult to doubt the objective existence of the sun with any coherence, for it will have existed before any of our birth as recounted by others. The claim is not that the sun has an objective existence {\it because} large numbers of people agree that it does, but because it is the {\it experience} of large numbers of beings independently of each other. It is objectively real.
So what is truth? Truth is all phenomena of objective reality, in principle, if not in practice, empirically deducible or observable by multiple conscious entities.
The philosophical study of consciousness is an ancient problem, and there is hardly now a standard model of consciousness comparable to that for, say, elementary particle physics. From the point of view of the four-dimensional theory of reality that I will sketch in later chapters of this book, interesting recent experimental results come from the work in neuroscience of visual perception. Reinhard Eckhorn has shown evidence that feature extraction of visual stimuli in monkeys are correlated not with firing rate of neurones but with synchronicity of firing, in the frequency range 15-90 Hertz. The basic premise is that the brain generates an electromagnetic field which each neural firing disturbs and then synchronicity of firing shows that consciousness is not affected simply by isolated neurones firing, but the joint activity simultaneously of well-separated neurones. There is a mildly different explanation if one accepts the theoretical picture of the four-dimensional universe that I will sketch: the electromagnetic field in which the visual neurones are disturbing is not generated by the brain but exists independently, and consciousness is a property of the four dimensional spirit rather than caused by neurones firing. This actually resolves a subtle problem in the empirical picture. The empirical work has shown correlation between synchronicity of firing and feature selection, but the explanation of how synchronicity could be achieved for situations where the external stimuli guarantee synchronicity would be missing. In other words, if we use disturbance frequency of neural firing as a basis of consciousness, then we must have some idea of how such synchronicity would normally operate. A four-dimensional electromagnetic picture of the spirit leaves open the possibility that the three dimensional brain is a three-dimensional slice of a four-dimensional electromagnetic organ -- which I call the mind -- whose activity causes many of the neural activity in the brain.
A paradigm of modern science has been to speak of reproducible phenomena. I note that reproducible objective phenomena are only a subset of truth. For instance, the acceleration of a metal ball dropped from a distance away from the Earth will have approximately the acceleration in accordance to Newtonian physics, and this is reproducible. However, all the objective acts of mine for the day are not reproducible and yet are true. This expansion of the phenomena of truth is philosophically important for this work, especially for claims of spiritual nature which are usually not reproducible (as few things that we actually experience are).
Enlarging the category of objective reality is also useful for the purpose of important questions that are often avoided by current science. Ideally, science is a rational study of objective phenomena and not only reproducible objective phenomena. This is sensible because the aims of physics has always been to study the natural laws of the Universe that can safely be tested on larger set of phenomena. Presumably, the acceleration due to gravity is quite close in Cincinnati as in Delhi. More interestingly, what about claims that we call religious? If I believe in a particular God with particular features, is that simply a subjective claim or an objective claim. I posit it is an objective claim, in keeping with the beliefs of devout religious people. Therefore such claims fall in the realm of science, the study of objective phenomena, and must submit to the same impartial principles that determine scientifically accepted theories. A religious belief is a theory about objective reality that cannot be accepted as being a correct description of the Universe if there are counterexamples. Similarly, the criteria for deciding which of two religious statements are true can be based on complexity and predictive power. In later chapters of the book, I will provide a number of metaphysical experiences from which I have constructed an initial theory of spirit. I will argue that metaphysical experiences, provided that one is not giving false accounts of the experiences, are reports of objective phenomena by providing evidence that the visual cortex in the human body is highly unidirectional and is not designed to produce 'hallucinations'. In other words, I will provide evidence that external stimuli generate dreams, visions, and hallucinations.
In this book, I will define truth as those phenomena that have a basis in objective
reality in principle testable by empirical means. Many of the claims I will make have not been tested by independent empirical means, and I shall try to be careful to point these out. If you are not convinced of the existence of an
objective reality outside your mind and your perceptions, then this book will be
quite useless to you, for the content and arguments presented in the book will be
geared to those who require proof and arguments for acceptance as possible truth. The philosophy adopted by the modern empirical scientists is based on a statistical theory of truth. Since the empirical scientific philosophy of Descartes and Bacon have found success, modern empirical science has been based on variations of the frequentist paradigm of hypothesis testing, where a statement is accepted based on statistical confidence measures such as the p-value. The standard p-value cutoffs of 99% and 95% dominate the work of modern scientists. An unbiased look at this paradigm shows that in fact, modern scientific theories are simply stories about reality whose validity has been measured by variations of hypothesis testing.
A more stringent condition for acceptance of statements is the rejection of any
statement that has even a single counter-example. It is thought that complexity of
reality, and I use the term reality rather than physical reality in order retain the word physical for the three-dimensional physical universe, is such that this stringent condition may not be capable of producing a nontrivial science, but I shall beg to differ, and produce a coherent mathematical physics in the ensuing chapters based on this more stringent mechanism for validation of scientific theories. My major focus will be to provide evidence that the Universe is macroscopically four-dimensional, that magnetic monopoles exist in abundance in the Universe, and that a new mathematical physics, called S4 physics, is capable of explaining the major experimental observations of the empirical physicists over the past 150 years without initial problems. The theory can be summarized as: the universe is geometrically a fixed radius four-dimensional sphere which determines the quantum of energy, and the standard model of particle physics for this model is an SU(2) x SU(2) x SU(3) gauge theory. As a mathematical model, this provides a unified model, in the sense that gravitation and the other fundamental forces of the universe are unified quite simply because the round four-sphere has constant Ricci curvature and hence can be said to satisfy the Einstein field equations for a zero stress-energy tensor. Theodor Kaluza had produced a four-dimensional model in 1917 that was studied in mid 1920s by Felix Klein; these models have zero Ricci curvature and so the S4 model does not fall into the Kaluza-Klein class of models. However, the richness of detailed analyses that established theories of physics requires more effort and resources than currently at my disposal, and the picture I present is skeletal.
I am neither responsible for the experiments whose results I use to justify the S4 physics, nor do I have the resources to reproduce their experiments. I am grateful for the published data by the empiricists for this book. The outline arguments of S4 physics:
- 12-fold
symmetry of quasicrystals implies spatial fourth dimension
- 4D + CBR => compact topological 4-sphere
- CBR
=> Hubble's relation (empirical check not done, argument ok)
- Check against hydrogen energy spectrum => sphere
- simple model of hydrogen => EM structure group SU(2).
Objective evidence for a macroscopic fourth dimension
One of the first issues to resolve is to define the notion of dimension as it is used in
mathematics and in modern physics. Recall that the plane can be described in terms of pairs of real numbers, and the standard three-dimensional space can be described in terms of triplets of real numbers where we often use the terminology of x, y and z axes. More generally, the notion of dimension is
defined for abstract linear spaces. An algebraic structure called a field --for example rational numbers-- is one where
multiplication and addition are both defined, and multiplicative inverse of every
non-zero element of a field is contained in the field. The real numbers, constructed
as the analytic completion of rational numbers form the classical example of a field
which is not algebraically closed but has ordering. A vector space is a set of objects where addition and multiplication by an element from a field is defined. A set of elements is called linearly dependent if after multiplication of each by scalar that are not all zero, they can sum to zero. The dimension of a vector space
over the real numbers is the minimal number of linearly independent vectors
required to span it. The reason for the dense set of statements above describing dimension is to keep precision of the word dimension intact, and to keep in mind the following observation. On a flat four dimensional space, let's say with coordinates (x, y, z, w), one can take the sequence of spaces defined by w=1, w=2, w=3, etc. and obtain an infinite set of parallel three dimensional spaces that never intersect. In the metaphysics literature, often claims are made about 4 or 5 or 6 or 8 or even higher dimensions, but without concrete evidence, such claims are not useful. Often what is meant by "dimension" is a three dimensional universe that is not the same as our own allegedly three dimensional universe, but if the universe is four dimensional, then there can be infinitely many parallel three dimensional sub-universes without the need for higher dimensions.
Assuming that the Universe is a smooth manifold, the dimension of the tangent space of the universe at any point is its dimension. I will argue that
the dimension of the universe is at least 4 and also argue that macroscopically, it is
exactly 4 spatial dimensions plus time. The argument will be based on the crystallographic restriction theorem. This theorem provides restrictions on the rotational symmetries that can occur for theoretical perfect crystals in flat spaces of different dimensions. For three-dimensional crystals, these orders can be from: 1, 2, 3, 4, and 6. For four and five dimensional spaces one could have crystals of additional orders 5, 8, 10 and 12.
Observed rotational symmetries of "quasi-crystals" (which can have 5 and 8 fold
symmetries) as well as cymatic patterns (which can also have 5-fold and 8-fold
symmetries) would suggest that the universe is not 3-dimensional if these are indeed crystals. Next is an image of electron diffraction of Zn-Mg-Ho, which shows a clear 12-fold rotational symmetry. Qualitatively, the symmetry observed in this picture is no different from obviously crystal patterns where the order of symmetry does not raise questions about whether it is crystal or not.
In order to understand how these phenomena provide empirical evidence for
existence of at least four macroscopic spatial dimensions, recall some basic facts
from crystallography, and in particular the crystallographic restriction theorem. A
crystal is a solid material whose constituent atoms, molecules, or ions are arranged
in a regular repeating pattern in space; the study of such things is crystallography.
All the possible regular patterns for dimensions 1-6 are completely classified. A
corollary of that classification is that not all rotational symmetries are possible in 3
dimensional crystals. In particular, only 2-fold, 3-fold, 4-fold, and 6-fold rotational
symmetry are possible in 3-dimensional crystals. At least four dimensions are
necessary for 5-fold, 8-fold, 10-fold, 12-fold symmetries. Since the discovery in
1984 of a crystal structure with 5-fold symmetry and 8-fold symmetry, a theory of
"quasi-crystals" has developed. I will submit that these are simply four-dimensional
crystals, and no separate theory of quasi-crystals is necessary to describe these.
That is my first evidence. Note that 7-fold, 9-fold, 14-fold symmetry require at least
6 dimensions, however, no such symmetry has been found among quasi-crystals
thus far (and I will make the prediction that they do not exist).
In 1985 material had been found with 20-fold symmetry but this material did not show crystal or quasi-crystal structure. If crystal structures are found with 20-fold rotational symmetry, I would consider that evidence that the Universe is macroscopically at least 6 dimensional.
A second source for evidence of four spatial dimensions comes from cymatics, the
physical patterns produced through the interaction of sound waves in a medium. A
simple example is sprinkling sand on a metal plate and vibrating the plate by
drawing a violin bow across the edge. The sand will then form standing wave
patterns in the plate. The observation that I am relying on is that for a simple
container, say with circular wall, observation of 5, 8, or 12 fold rotational
symmetry in a quasi-crystal cymatic pattern is evidence of 4 spatial dimensions. This
requires a mathematical proof that eigenfunctions of the Laplacian for a circular
walled container do not have such symmetries, to which I will return. The idea here
is that the major influences on this pattern are geometry of the container, dimension
of the universe, and frequency of sound. If we see 12-fold symmetry in such a
pattern, this is evidence for a fourth spatial dimension if it cannot occur naturally in
the nodal lines of a theoretical disc. Here is a picture of a 12-fold rotational symmetry:
These are fairly simple to check evidence of a fourth spatial dimension. Once this
phenomenon is more well-established, I would be surprised if many more sources of evidence become available. Note that modern physical theories have been developed with
a foundational assumption of 3 spatial dimensions, and hence are invalid, strictly
speaking, in the presence of empirical evidence a fourth macroscopic dimension.
The S4 physics theory that I will present is an attempt to provide a unified physical
theory that is purely four-dimensional.
Initial objections
Since we experience a three-dimensional universe, one wonders what a fourth macroscopic
dimension looks like. Moreover, if one integrates the first Maxwell equations for a point charge
at the origin of four dimensions, q delta(r) along a sphere of radius r, using the hyperarea of
the sphere, 2 pi^2 r^3, using spherical symmetry one obtains electric field strength E(r) = q / 2 pi^2 r^3, which contradicts precision tests of Coulomb's law. In 1971 the experimentally verified value
of the exponent was 2 + (2.7+/-3.1) x 10^{-16}.
The simple answer is that the four dimensions are purely electromagnetic. The three-dimensional universe could be highly twisted which would allow an inverse cube law to restrict to an inverse square law. There is little reason for the physical universe, expecially in the presence of matter, to be geodesically complete in a four dimensional universe, and ideas from general relativity would
suggest the opposite. Note that if indeed there is a squeezing of three-dimensional universe
so that inverse cube laws restrict to inverse square laws, the twisting is quite severe as analogy to space-filling curves show.
The universe must be compact, and spherical
Two major experimentally known phenomena in physical cosmology are: the isotropic cosmic background radiation (CBR) that which fits a thermal Planck form with a peak in the centimeter millimeter wavelength, and the linear relation between galaxy distance and the redshift of their received signals originally discovered by Hubble in 1929. The CBR temperature has been estimated at around 2.7 Kelvin and therefore has a lower bound on it's density. Here is picture of a fit to the CBR data collected by the COBE satellite. The fit of Planck's form is in blue, and the fit to a naive extension of Planck form is in red. CBR is thought of as an isotropic component of the
extraterrestrial electromagnetic component that is significant only at
millimeters to centimeters, but the fit to Planck form extrapolates to nonzero intensity at starlight frequencies. This feature of CBR can explain the galaxy distance-redshift relation that Hubble discovered, since radiation from a distant galaxy spends time proportional to distance in CBR.
Once the Universe is known to be 4-dimensional, one can use the uniform lower
bound on CBR to show that the Universe is in fact bounded, i.e. compact, as a smooth manifold.
The mathematical theory required to show this come from the theory of Gaussian bounds for heat kernels for complete non-compact riemannian manifolds with a lower bound on Ricci curvature. Uniform distribution of CBR is impossible without a compact universe, as a Gaussian bound would be violated. More precisely,
the argument is as follows. Assume that the Universe is smooth, geodesically
complete, and has a lower bound on Ricci curvature - reasonable assumptions by any non-singular model of the Universe.
Then assume that the energy content of the Universe spreads from a point by
interacting diffusion; this assumption can be weakened by the assumption that the electromagnetic energy content of the universe is contained in a compact subset. IF the Universe were non-compact, then there would be a Gaussian (decaying like exp( -r2 )) upper bound on the spread of electromagnetic content of the Universe. Now since the original discovery of CBR of above 2.72 K uniformly in every physical direction up to 300 light years away from us, implies that no such Gaussian lower bound is observable in a radius of 300 light years. In particular, this is strong evidence that the Universe is compact. The fit of the Planck form suggests that CBR is in approximate thermal equilibrium, which supports the hypothesis of compactness.
--
At wavelengths in the range of millimeters to centimeters the extraterrestrial radiation is dominated by an isotropic component, the cosmic background radiation or CBR. From Earth, in all directions of the physical three dimensions, this CBR fills space uniformly. The temperature is very close to a thermal equilibrium one would expect from a blackbody, which is an object that absorbs all electromagnetic radiation that falls on it and emits radiation after saturation in a characteristic manner. The intensity distribution of CBR suggests that it has relaxed to thermodynamic equilibrium at a temperature near 3K.
The conventional explanation has been that CBR is left over from an early epoch when an expanding universe was hot and dense and hot enough to have relaxed into thermal equilibrium filling space with a sea of blackbody radiation. There are a number of problems with this conventional explanation which can be understood by considering the heat equation on manifolds of different types. First, let us reconsider the expansion hypothesis from the point of view of Occam's razor, which tells us to prefer the simpler over the complex explanations for phenomena. An expanding universe is not a simpler explanation than a stationary universe. If we assume the universe to have been static and CBR occurred through a diffusion process, then we have to conclude that such a static universe cannot be unbounded. This is because on unbounded spaces, the lower bound on the CBR density is simply impossible if the initial radiation distribution were concentrated in a compact part of the universe because we have Gaussian upper bounds for the heat kernel of diffusion operators.
The motivating example that can provide some insight into this argument is the heat equation on the real line. Recall that the heat equation with an initial temperature distribution f(x) on the real line is given by
d^2 u / dx^2 - d/dt u = 0 with u(t, x) = 0
The solution is the convolution with the heat kernel p(t,x,y) = (1/ sqrt( 2 pi t)) exp( - | x - y |^2/2t ). On the real line, for any t > 0, one can see that there cannot be a uniform lower bound on p(t,x,y) because |x - y| can be taken to be arbitrarily large and thus this kernel can be made arbitrarily close to zero.
In the case of a heat source at a point p, a Gaussian centered at p has the form U(x) = exp( - c | x - p |2 ) for some positive constant c. The function is rapidly decreasing as the distance of x from p increases. In an unbounded universe, the upper bound on the heat kernel will have this form and will disallow uniform lower bounds of diffusion with heat source at p.
John Nash proved in the 1950s
that for a parabolic equations in euclidean spaces have upper and lower
Gaussian bounds when the operators are uniformly parabolic. More
recent work by many mathematical analysts have shown that for
non-compact manifolds with lower bounds on the Ricci curvature,
Gaussian upper bounds can be proved for parabolic operators with fairly
general conditions.
Having observed the uniform lower bound on the CBR we can remove the possibility that the universe is unbounded or noncompact. This has immediate consequences for one of the fundamental assumptions of nineteenth century physics, where the routine model of the universe had been that it can be modeled as a flat three dimensional euclidean space.
The 1990 COBE satellite measurements shows the background radiation spectrum at wavelengths of 500 microns to 5 mm. The fit to a Planck blackbody spectrum is es an excellent fit. A Planck function has only one free parameter, temperature T for which the best fit is
T = 2.736 +/- 0.007K
The CBR is remarkably close to isotropic. The main feature is a dipole variation in the thermodynamic temperature as a function of position across the sky at an amplitude ~0.1% of the mean.
The
uniform lower bound of the cosmic background radiation, whose
temperature is very steady at 2.73 K and fills up the universe in every
direction has immense consequences for the shape of the universe. If
this fact were known late in the nineteenth century, we would not have
gone down the wrong path of quantum mechanics. Immediately it would be
recognized that the boundedness implies quantization of energy.
A different
explanation of the cosmic background radiation results from a
stationary universe in which the background radiation may have occurred
from either concentration of energy in a bounded region of space at a
distant past time but more evidence would be required for this
possibility, for it cannot be ruled out that the universe has had this
cosmic background radiation at similar levels to today for an infinite
time in the past.
The
conventional explanation for an expanding universe is supported by the
linear relation between distance of galaxies and their redshifts. Given
the presence of the cosmic background radiation, the redshifts
themselves can be explained by an effect of signals traveling through
the CBR which acts as a convolution filter that reddens the signal
rather than evidence for expansion.
Consequences of compactness of the universe
The compactness of the universe is a very strong restriction. By
the spectral theory of self-adjoint operators applied to the Green's operator or the solution operator for the Laplacian, the natural Laplacian on such a
manifold has a discrete spectrum without resorting to a separate
quantum hypothesis. The
fact that the Laplacian on a compact manifold has a discrete spectrum
is not difficult to prove, although it is a fundamental result in
mathematical analysis. It is equivalent to the intuitive idea that no
matter what the shape of a (compact) drum is, there is a set of pure
tones for it.
Indeed, the observed energy spectra in an even
spacing of Planck's constant h immediately suggests the spherical model of radius
1/h as a possible shape for the universe because there are very few manifolds with regularity in the spectrum of the Laplacian
Assume for a moment that the universe is simply connected, and put a riemannian metric g on the tangent spaces. There is a natural connection on it that allows one to translate tangent vectors by geodesics to tangent vectors at another point. This is the Levi Civita connection. If one also knows that the so-called second Stiefel-Whitney class of the manifold vanishes, one can define spinors on the manifold and a Dirac operator on it. It is compactness of the manifold that allows one to conclude that the spectrum of the Dirac operator is discrete. This provides immediate link between the geometry of the universe, central to the gravitational theory of general relativity, with the
quantum mechanics of the Dirac operator, which are naturally discrete.
Among compact models of the universe, the ones which are thus particularly interesting are
those which have a Dirac spectrum that naturally leads to the Dirac equations for physically
interesting systems, like electromagnetism. On a sphere of radius 1/h, for then the spectrum is +/-(2+k) h, for k=0,1,2,3, ... It's also a very simple model, and allows embedding of three-dimensional submanifolds to model the physical universe sitting within it.
This sphere satisfies Rij - gij h^2/12 = 0, which is the Einstein field equation for an empty universe. A a three-dimensional submanifold embedded in the four-sphere, one can use the normal vector field to obtain the relation between the Ricci tensor of the submanifold to that of the four sphere. For a submanifold M let RicM be the Ricci tensor and let RicS be that on the ambient manifold so for vector fields tangent to M,
RicM(X,Y) = RicS(X,Y) - 4 a H B(X,Y) - a C(X,Y)
where B(X,Y) is the second fundamental form and C(X,Y) is the third fundamental form. These are defined by first setting A(X) = - DXN and B(X,Y) = a g( A(X), W) with a = g(N,N) = +/- 1, and
C(X,Y) = g(A(X),A(Y)), and finally H = trace (B) as a matrix on the tangent space. Let lambda = hbar^2/12
RicM(X,Y) + 4 a H B(X,Y) + a C(X,Y) - lambda g(X,Y) = 0
and so
RicM(X,Y) - lambda g(X,Y) = - 4 a H B(X,Y) - a C(X,Y)
In particular, if we set 8 pi G T(X,Y) = -4 a H B(X,Y) - a C(X,Y) the we recover the form of Einstein's field equations with a stress-energy tensor
Rij - lambda gij = 8 pi G Tij
but now Rij represents the Ricci curvature of M. Thus a four-sphere model is consistent with the gravitational field equations if the stress-energy tensor arises by the extrinsic geometry of M.
Classical electromagnetism and Maxwell's equations
The equations of Maxwell for the electromagnetic field (E,B) ignoring constants can be stated as:
DE + d/dt B = 0
DB - d/dt E = 0
satisfying div(E) = rho and div(B) = mu, the electric and magnetic charge densities. The
operator D above is the curl. If we ignored the physical meaning of this system of equations
and considered what it means if E and B were one-dimensional functions on the circle and
D were just a space derivative, we
might get an intuition for these. Taking a derivative in time of the first equation and using the
second we obtain d^2/dx^2 B+d^2/dt^2B = 0, while doing the same for the second equation,
d^2/dx^2 E - d^2/dt^2 E = 0. These are wave equations. Assuming B(x,t) is a product b(x)n(t) would give b''/b = - n''/n. And E(x,t) = e(x) m(t) gives e''/e = m''/m. Also
- m'' + (b'/e) n' = 0
- n'' + (e'/b) m' = 0
Now fix x and these give the relation between magnitudes of E and B as time passes. These
have solutions of the form n(t) = C exp( sqrt( b'e'/eb ) t ), where C depends on e', b', e, b at the point x. When b'e'/eb is negative, the solutions are periodic. The Maxwell equations are thus
already quite interesting on a circle, where they are producing sinusoidal fluctuations of E(x,t)
and B(x,t) of field intensities e(x) and b(x) at a fixed x. It follows then that indeed e(x) and b(x) are
also periodic in x, because e''/e = m''/m and b''/b = n''/n are both constant. This in turn restricts
possibilities of e(x) and b(x) to be eigenfunctions of the second derivative, which are restricted
to a discrete set for a circle.
The circle example is useful, because one can use a very similar argument when the equations
are for spinors on a manifold instead of real-valued functions on a circle. These equations
on for spinors on a four-sphere makes sense if D is the Dirac operator. But now
sqrt( b'e'/eb) is a quaternion, so the solutions are not simply circular but values on the unit
quaternions SU(2). Thus for a fixed x on the unit four sphere, the Maxwell's equations yield
SU(2) fluctuations of standing waves on the four sphere.
Since SU(2) is the double cover of SO(3), their Lie algebra is the same, and its generators
satisfy
[ex, ey] = ez
[ey, ez] = ex
[ez, ex] = ey
On a the round four-sphere scaled by 1/hbar, the vectors scale by hbar, e.g., Lx=hbar ex, and the commutators are
[Lx, Ly]
= hbar Lz
[Ly, Lz]
= hbar Lx
[Lz, Lx]
= hbar Ly
that is
used for the angular momentum operator of quantum mechanics.
THE FOLLOWING ARGUMENT IS HEURISTIC FOR THE MOMENT
What about the topology of the universe? Compactness puts restrictions. Assume the universe is simply connected, in the sense that loops can be contracted without obstruction, and it is connected in the sense that there is a path from any point of the universe to any other. These assumptions restrict the cohomology groups of the universe M so that the issue reduces to checking what happens to the second cohomology group H^2(M). If we can argue that this group vanishes, then at least topologically, the universe is a sphere. Note that differential structures on a four dimensional manifold are not determined purely by topology, and there exist exotic differential structures as shown by geometers in 1980-90s. The natural class of four-manifolds that could serve as models of the universe are symplectic manifolds, on which Hamiltonian dynamics is defined via a symplectic 2-form. Locally, an energy conservation law is implicit in the definition of the symplectic form. The symplectic form is a close form and defines a nonzero element of the second cohomology class, which implies that H^2(M) is non-zero.
However, if an S^2 can be chosen that is a generator of the cohomology H^2(M)
where the Hamiltonian flow has a periodic orbit, then parallel transport
along the orbit of a connection around it will produce a higher energy
object, presumably violating conservation of energy for connections.
If one has shown that the universe cannot be symplectic, then a topological sphere is the only reasonable choice from H2(M) = 0.
Spinor bundle on the round four-sphere
If the arguments above hold, then the universe is a topological sphere but in fact we would like a metric round sphere of radius 1/hbar, for then the spectrum of the natural Dirac operator, for example, is +/-(2+k) hbar for k=0,1,2,3, .... On a sphere of radius 1, Killing spinor fields with Killing constant mu=+/- 1/2 trivialize the spinor bundle. These have been shown by Christian Bar.
For an n-dimensional inner product space V, the Clifford algebra is generated from the basis elements e1, ... , en by relations ei^2=1 and ei ej + ej ei = 2dij. If C(n) is the Clifford algebra of the standard Rn, then C(1) is isomorphic to complex numbers, C(2) to quaternions, C(3) to pairs of quaternions, and C(4) to 2x2 matrices with quaternion entries. There is a norm defined on the Clifford algebra and elements of norm 1 form Spin(n) group. For The construction can be applied to tangent space at each point on a manifold to obtain the spinor bundle. The Dirac operator is defined on smooth sections of the spinor bundle.
A gauge theory develops as follows. First, one writes an action for a potential, checks the symmetry of the action under the action of a Lie group, that is, multiplication by a group element g. When g(x) is a variable function, taking derivative and multiplication don't commute, one obtains a connection d+A, where A is the gauge potential. There is a correspondence between particles and gauge connections.
Now we have to show that multiplication by unit quaternions preserves the Lagrangian for electromagnetic particles.
The universe need not be expanding
The currently accepted story of the early universe is that the CBR occured soon after a dense hot state of the universe after which the universe expanded. The basis of the expansion is the relation discovered originally by Hubble of the distance-redshift linear relation. If we were to explain the redshift in terms of the skew of CBR at starlight intensities, where longer distances would produce a stronger redshift because the CBR contains a bias towards lower frequencies after a peak at a fairly low frequency, then a static universe picture would be quite sensible. The Hubble relation is the strongest experimental basis for an expanding universe model. If one believes that the shape of the universe is responsible for quantization of energy, then a static model is essential, for there have been numerous tests of the quantum hypothesis, and one does not expect the quantum of energy to have been different at an earlier epoch of the universe.
The universe must be a topological 4-dimensional sphere
Once the Universe is known to be compact and 4-dimensional, topological
arguments can be used to show that in fact it must have no 2-dimensional
cohomology without violating conservation of energy. The summary of the argument is that for symplectic four manifolds, one can use the Hofer-Zehnder theory of periodic orbits to show that an electromagnetic system can be constructed that violated conservation of energy. That rules out the symplectic four manifolds with nontrivial second cohology class. The universe, presumed to allow Hamiltonian dynamics, cannot be symplectic, leaving the topological four sphere. Although the four-dimensional sphere does not carry a symplectic form, Hamiltonian dynamics can be defined on pairs of quaternions, and then the four-sphere allows Hamiltonian dynamics for energy functions that descend to the four sphere identified with the quaternionic projective space.
Although the 4-dimensional sphere is not symplectic, it is the quaternionic projective space, which implies that Hamiltonian mechanics is well-defined for a certain class of energy functions on quaternionic
pairs that descend to be well-defined on the 4-sphere of a given radius.
Once the Universe is known to be a round four-sphere of fixed radius, quantum phenomena can
be explained by the compactness and shape of the Universe rather than the currently accepted theory of quantum mechanics, which rely on a theory of photon and a probabilistic framework of wavefunctions. In particular, classical mechanics on an S4 universe can explain the energy spectrum of the hydrogen atom as accurately as the
quantum-mechanical description -- see later chapters for a comparison of predictions.
The impetus behind the development of quantum mechanics late in the nineteenth century were two failures of classical mechanics on flat 3 dimensional space: the intensity spectrum of a thermal blackbody predicted by the Rayleigh-Jeans law did not match experimental results, and the an electron in circular motion around a charged nucleus would accelerate and lose energy, thus collapse onto the nucleus, which would make atoms unstable. For the first problem, Planck's amazing discovery was that if energy could only take discrete values, then the intensity distribution of the blackbody would match experimental data. I will come back to how on a four-sphere of radius 1/hbar energy discretization is natural. For the second problem, there is an intuitive solution that I will formalize: for an electron in a circular orbit around a proton nucleus, the coupled system will be stable if the proton moves to compensate for the movement of the electron. On flat four dimensional space, consider the electron in circular motion on a plane A, and consider the plane B that is orthogonal to it that passes through the center. The movement of the electron induced an electromagnetic field on this plane. If we can find the path of a proton in a circular orbit on plane B such that the simultaneous movement of both particles are compatible, then the coupled system will be as well. If the proton had no effect on the electron and was a passive participant by its charge, the current produced by the movement of the electron would generate an electric field that would keep the proton near the electron and a magnetic field that would rotate the proton.
Now, S4 is Einstein (has constant Ricci curvature) and thus so are any 3-dimensional submanifolds. In particular, Einstein's field equations are satisfied for zero stress-energy tensor
this geometry without need for a separate theory of gravity. Moreover, an electron
orbiting around a proton on S4 is stable as long as the proton moves to compensate for the movement of the electron, and thus there is no electromagnetic energy loss. For the general problem of stability of matter, one needs to show that multi-electron atoms are also stable.
Finally, on a round four-sphere, the natural symmetry of electromagnetism is SU(2) symmetry rather
than the U(1) symmetry that has been assumed by the classical Maxwell theory. Indeed, on a four-sphere any U(1) connection is flat, and thus the gauge group of electromagnetism must be SU(2). I speculate that empirical evidence for Maxwell theory is not is not inconsistent with the S4 theory, if one considers Maxwell theory as the restriction of an SU(2) theory to a physical three-dimensional subspace.
In fact, electromagnetism, weak, and strong nuclear particles are unified elegantly by
an SU(2) x SU(2) x SU(3) gauge theory on the compact round-sphere. As an application of this, I will present
some algorithms for protein folding problems later in the book.
I will provide further details of the physics later on, but I want to first put down
some claims about the nature of spirit. Human beings are 4-dimensional
electromagnetic spirits that control the body via the connection of electromagnetic
spirit body with seven chakras along the spine. In tantric tradition, these chakras
(translation from Sanskrit: wheels), are called muldhara (root), svathisthana
(sacral), manipura (solar plexus), anahata (heart), vishuddha (throat), ajna (third
eye), sahasrara (crown). Kabbalah also provides a corresponding convention based
on sefirot but I have found the tantric description useful, so I present the science of
spirit using this scheme.
Not only human spirits, but the spirit of the Universe as well as cosmic spirits such
as stars, galaxies, and constellations I claim have at least these seven chakras in their
four-dimensional spirits. This forms the basis of life in a precise scientific manner. I
shall also use the term "Force" to describe the living spirit of the Universe, following
the compelling terminology of the popular films "Star Wars". The spirit of the
Universe, like our spirits, is composed of magnetic monopoles of with a separate
purely electromagnetic axis hitherto unstudied which I will call the dark pole and
the light pole. The Star Wars terminology is compelling for its claim to the dark and
light sides of the Force.
The most basic claim about electromagnetism in 4-dimensions that I will make is
that dark and light, combined in balance, forms "golden", whose magnetic strength
exceeds both that of pure light and pure darkness. The central chakra of human (as
well as Universal) spirit is anahata, or heart chakra. The perfect theoretical balance
of light and dark magnetic monopoles in the heart chakra is "love" beyond light and
dark, good and evil, that is the center of the heart chakra of the Universe. Human
emotional interactions are always accompanied by exchange of magnetic
monopoles, and always takes place within the spirit of the Universe, or Force.
Karma, usually obfuscated by mystical schools, is nothing more than classical
mechanics of magnetic material of the Universe in time.
A typical reaction to the above claims by a modern rationalist may be dismissal or
classification of these ideas into an esoteric tradition. If you are tempted to do this,
consider the evidence that I present fairly. These claims occur not from my previous
study of esoteric traditions but from a great deal of personal experience in spirit realms.
The ajna or third eye chakra, for instance is the partially dormant pineal
gland, which resides between the two hemispheres of the brain: pinealocytes have a
strong resemblance to photoreceptors of the eye, and it excretes
dimethyltryptamine, which have been associated with dreams, near-death
experiences, and hallucinations. I say "associated" rather than "induces" because I
will argue in a separate chapter that indeed dreams, visions, and hallucinations are
perceptions of objects and events external to ourselves rather than induced within
the mind. In art, this is in agreement with the views of the surrealists, whose ideas
of mimesis is art is faithful record of spiritual experiences.
The universe is filled with spirits of various sizes and strength. Just the Milky Way
galaxy has more than $10^11$ stars, each a live spirit with seven chakras. There are
88 constellations listed, each containing possibly multiple galaxies. An off?the?cuff
estimate of number of stars is perhaps $10^13$ if Milky Way is the typical size of a
galaxy. Note that this is specific to our physical sub?universe, and there exist at least
another physical subspace if not more.
Andromeda
Antila
Apus
Aquarius
Aquila
Ara
Aries
Auriga
Bootes
Caelum
Camelopardalis
Cancer
Canes Venatici
Canis Major
Canis Minor
Capricornius
Carina
Cassiopeia
Cantaurus
Cerpheus
Cetus
Chameleon
Circinus
Columba
Coma Berenices
Corona Australis
Corona Borealis
Corvus
Crater
Delphinus
Dorado
Draco
Equuleus
Eridanus
Fornax
Gemini
Grus
Hercules
Horologium
Hydra
Hydrus
Indus
Lacerta
Leo
Leo Minor
Lepus
Libra
Lupus
Lynx
Lyra
Mensa
Microscopus
Monocerus
Musca
Norma
Octans
Ophiuchus
Orion
Pavo
Pegasus
Perseus
Phoenix
Pictor
Pisces Austrinus
Pisces
Puppies
Pyxis
Reticulum
Sagitta
Sagittarius
Scorpius
Sculptor
Scutum
Serpens
Sextans
Taurus
Telescopium
Triangulum Australis
Triangulum
Tucana
Ursa Major
Vela
Virgo
Volans
Vulpecula
Each of these 88 constellations contain multiple galaxies and star clusters. A quick and
dirty extrapolation from around $10^11$ stars in the Milky Way galaxy puts the
star count in the order of $10^13$ or more in the Universe.
\section{Boundedness of the Universe: the basic argument}
Let me give some more details of the argument that the Universe is bounded and
closed rather than open. For this the key experimental observation will be the uniform lower bound of cosmic microwave background which was discovered in
1964 by Arno Penzias and Robert Wilson. The key feature of the almost uniform
cosmic microwave background is that it is close to 2.725 K uniformly in all
directions from Earth known to about 300 light years in physical space. The
intensity peak for this radiation is 160.2 GHz or about 1.9 mm for wavelength. For
the mathematical argument to show that the Universe is bounded will not require
these precise measurements, but I will use the actual figures later to give a
cosmological model simpler than the big bang model and the inflationary universe
model.
The basic S4 model is that the total energy content of the universe was concentrated
at a point in a static universe. From this point, the content of the universe followed
diffusion by classical mechanical laws of physics, but on a static S4 universe.
Let's focus first on the simplified case where the only particles are electromagnetic
particles (symmetry $SU(2)$) but non-interacting. Assume that the Universe is
geodesically complete but non-compact. If we assume that the Einstein condition
holds for the Universe, we will have constant Ricci curvature, and the diffusion will
actually be heat diffusion on a non-compact manifold with a lower bound on Ricci
curvature. The theorem of Peter Li and Shing-Tung Yau applies for such a case -
from their 1986 paper, "On the parabolic kernel of the Schrodinger operator," Acta
Mathematica 156, 1986 and we can conclude that at any time $T>0$, the distribution of
particles will satisfy a Gaussian upper bound in geodesic distance from the source.
In particular, a uniform lower bound that holds everywhere will contradict the assumption of
unbounded universe.
Now the observed lower bound extends 300 light years in every direction. Is it possible that there is in fact a Gaussian decay of energy from a source but that decay is such that locally it has a lower bound in our neighborhood of the Universe in a 300 light year radius? To see that this is not the case, note that it's not simply a lower bound but that the cosmic microwave background is almost constant in the region under consideration. In particular, there is no noticeable difference in intensity that is observed from region to region.
Next possible objection would be: are inter-particle interactions strong enough so that the Li-Yau argument cannot be applied in this case? The observations are microwave radiation, which can be modeled even with classical heat equation, so in fact the strength of the diffusion argument is sufficient to deal with this case. The Li-Yau theorem holds if the interaction is not too strong in a precise sense.
Electromagnetic interaction strength is the Coulomb law and falls away as $1/r$ with
the distance, and hence falls within the parameters of the theorem. Thus the
argument will be valid still for $SU(2)$ electromagnetic particles diffusing from a
point on a static universe - uniform lower bound is impossible for the distribution
of the particles in a finite time {\it\bf if} the Universe is unbounded. Indeed this argument
does not rely the number of dimensions posited: even if the Universe were 3-dimensional, the argument shows boundedness of the Universe.
Now it is known that the lower bound on the cosmic microwave background is 2.72
K in every direction for 300 light years of physical space. This is inconsistent with
Gaussian upper bound of heat diffusion on a non-compact space no matter where
the origin of the universe is located.
The above argument relied on a {\it\bf diffusion} model of expansion of particles. But if
classical mechanics holds, then diffusion is the natural model to use. We know that
macroscopically, classical mechanics holds for bodies. We also know that classical
mechanics on an S4 universe explains key quantum phenomena without additional
"quantum" assumptions - see Chapter 4. Thus I submit that this is a sensible model
for movement of electromagnetic bodies in the Universe.
Salvador Dali, Andre Breton and the surrealists
Plato's cave is often read allegorically as describing the unenlightened state of humanity, but
Plato believed in the actual existence of realms of forms, absolute reality, points to far more
concrete reading of that allegory. Plato's description of absolute reality is on first reading by
analogy -- and I will come back to a reading that is more concrete in later chapters. Salvador Dali's experiences are far more concrete. He was told by his parents that he was the reincarnation
of his older brother who died around the time he was conceived. He joined the surrealist movement around 1929 and later around 1940 he turned his interest toward science and spirit. His particular obsession with the hypercube -- the four-dimensional cube -- and with holograms have direct bearing to the view that I am interested in articulating about the nature of objective reality.
Below is the "The Hallucinogenic Toreador", a painting from Dali's from 1940s period.
Andre Breton, who founded the surrealist movement, clearly articulated a different aspect of non-physical reality than that emphasized by Plato -- that of the dream state. Breton observed that actual dreams have much more influence than we believe, and perhaps the dominant influence. His hope was that Freud's theory of the unconscious would improve the situation. I will attempt to argue that the actual universe is four-dimensional and that dreams, visions, and hallucinogenic visions take place in the same objective reality. Ancient Greek poetics has the concept of mimesis, translated as imitation or representation of reality. Surrealist mimesis can perhaps be identified with mimesis of objective spiritual reality. I want to emphasize that I am not making a sequence of loose associations, but making a claim of identity of objective reality.
In Breton's manifesto of surrealism, he makes some acute observations about the dream states pertaining to their possible reality. Persistence of dreams across waking times, for instance, might indicate that they could be objective reality independent of our consciousness that we have perceived or experienced:
Perhaps my dream last night
follows that of the night before, and will be continued the next night,
with an exemplary strictness. It's quite possible, as the saying
goes. And since it has not been proved in the slightest that, in doing
so, the "reality" with which I am kept busy continues to exist in the
state of dream, that it does not sink back down into the immemorial, why
should I not grant to dreams what I occasionally refuse reality, that
is, this value of certainty in itself which, in its own time, is not
open to my repudiation?
The question of objectivity of the dream I hope to return to in a later chapter, after considering giving some arguments for why the visual cortex may not able to observe that which is not external to our consciousness. The basis of that argument relies on the directionality inherent in the pathways of the central nervous system. That Breton was consciously writing his interest in such possibilities in the surrealist manifesto suggests that as an artistic movement, surrealism was defining art in terms of mimesis of this reality. Indeed, his definition of surrealism is here: "I believe in the future
resolution of these two states, dream and reality, which are seemingly
so contradictory, into a kind of absolute reality, a surreality,
if one may so speak."
Breton describes also states of inspiration in the same manifesto where thoughts, words, and phrases came to his mind in a flurry, which he captured without analysis for the construction of a poem. "And there were still more
coming. I got up and picked up a pencil and some paper that were on a
table behind my bed. It was as though some vein had burst within me, one
word followed another, found its proper place, adapted itself to the
situation, scene piled upon scene, the action unfolded, one retort after
another welled up in my mind, I was enjoying myself immensely. Thoughts
came to me so rapidly and continued to flow so abundantly that I lost a
whole host of delicate details, because my pencil could not keep up
with them, and yet I went as fast as I could, my hand in constant
motion, I did not lose a minute. The sentences continued to well up
within me, I was pregnant with my subject." Thus the interest in surreality
Basic Conceptions of Human Beings
In the system of capitalism we live in, whatever are the philosophical views of the human being, the view that dominates the political and economic discourse is that of human beings as economic agents -- either labor of jobs with variable levels of sophistication or consumers of goods and services. One can observe this view quite explicitly in the major newspapers and media of the times. This is the de facto established view of what human beings are to which other philosophical views are considered as academic or archaic or generally not of practical interest. We can walk around in the major cities of the world -- whether in North America, Europe, Asia, Australia -- and we will find these views to be widespread. Underlying this view of human-as-labor and human-as-consumer is the implicit idea that in order for a human being to be valuable to society, he or she must have some skills to participate in some industries.
Philosophers like Plato, Nietzsche, and Emerson had a very different view of human beings. Plato's Republic relies, for its functioning, very much on the concept of natural talents of a human being rather than simply trained skills. Emerson gives an enlightening account of what talents are rather than trained skills in his essay "Spiritual Laws":
"Each man has his own vocation. The talent is the call. There is one direction in which all space is open to him. He has faculties silently inviting him thither to endless exertion. He is like a ship in a river; he runs against obstructions on every side but one, on that side all obstruction is taken away and he sweeps serenely over a deepening channel into an infinite sea.
This talent and this call depends on his organization, the mode in which the general soul incarnate within him. He inclines to do something that is easy for him and good when it's done but which no other man can do. He has no rival. For the more truly he consults his own powers the more difference will his work exhibit from the work of any other. His ambition is exactly proportional to his powers. The height of the pinnacle is determined by the breadth of the base. Every man has this call to do something unique, and no man has any other call.
By doing his work he makes the need felt which he can supply, and creates the taste by which he is enjoyed. By doing his own work he unfolds himself."
But instead of following talent, for a variety of reasons become entangled in a machine in the system of corporate capitalism, and Emerson, writing in mid-nineteenth century could see the, that "The common experience is that man fits himself as well as he can to the customary details of that work or trade he falls into and tends to it as a dog turns a spit."
Nietzsche's rebuke for the man who has not followed his talent is severe. In "Schopenhauer as an educator", he writes, "There exists no more repulsive and desolate creature in the world than the man who has evaded his genius and who now looks furtively to the left and right, behind him and all about him. In the end such a man becomes impossible to get hold of, since he is wholly exterior, without kernel, a tattered, painted bag of clothes, a decked-out ghost that cannot inspire even fear and certainly not pity."
Nietzsche saw the root of all culture in the longing of man to be reborn as saint and genius. "Where we discover talent devoid of that longing, in the world of scholars or of the so-called cultivated, we are repelled and disgusted by it; for we sense that, with all their intellect, such people do not promote an evolving culture and the procreation of genius -- which is the goal of all culture -- but hinder it. It is a state of petrifaction. equivalent in value to that routine, cold and self-laudatory virtuousness which is also farthest, and keeps itself far, from true saintliness."
Consequences of the Physics for Fate and Ideals
In art and many rituals of our spiritual traditions we have direct
observations about fate and the human ideals of Freedom, Equality, Justice,
and Brotherhood from the eve of the French Revolution but I would also add Truth, Beauty, Reason, Empathy, and Destiny as. Philosophers Plato, Nietzsche, and Emerson have claimed the task of elucidating fate and ideals as the proper to philosophy. I will try to arrive at these from a very different route: as
consequences
of an objective physical science.
The view I hold is that these are not merely human concepts but have actual existence in the four-dimensional universe as spirit beings, and although I don't have methods of demonstrating these beings yet, I can only rely on my metaphysical experiences to say they are not merely symbolic constructs. Asked by Panthea to describe a horror seen by Prometheus, he answers:
There are two woes:
To speak, and to behold; thou spare me one.
Names are there, Nature's sacred watchwords, they
Were borne aloft in bright emblazonry;
The nations thronged around, and cried aloud,
As with one voice, Truth, liberty, and love!
Suddenly fierce confusion fell from heaven
Among them: there was strife, deceit, and fear:
Tyrants rushed in, and did divide the spoil.
This was the shadow of the truth I saw.
( P. B. Shelley, Prometheus Unbound)
First, let me outline a schematic picture of the Universe. The "space"
oc-
cupied by the universe is a four-dimensional sphere of radius
approximately
3.1016 meters. However, all the "content" of the Universe was
concentrated at
a point about 13.5-14 billion years ago. This content, concentrated at a
point
was a conscious living spirit, with a heart and a mind. This totality of
the en-
ergy content of the Universe, I will call the spirit of the Universe, or
the Force.
If I remove the consciousness of the spirit, this schematic picture can
be justi-
.ed by a mathematical physics. Assume for a moment that I have done
that;
then the consciousness of the Universe is a separate hypothesis. In
Chapters
11-20, where I describe my metaphysical experiences, this hypothesis
will be
justi.ed by checking consistency with at least individual experiences.
Note that
an immediate consequence of this hypothesis are: (a) roots of
consciousness are
at the "birth" of the Universe rather than as a consequence of human
devel-
opment, and (b) consciousness is not localized at all and in particular
is not
localized to physical processes in human brains. As a side note, recent
empir-
ical studies of consciousness have addressed its non-corporeal nature;
thus the
hypothesis of a universal consciousness is not far-fetched. Further
evidence for
this could be provided by examples of joint metaphysical experiences,
so-called
out-of-body experiences, reincarnation of spirits, and other experiences
we have
called paranormal. The speci.c claim here is that the Universe is a
conscious
being with its own mind and heart. In this theory, there is no need to
assume
that the Universe had a creator, for it may have had full development
and then
restarted at a point. Thus a creator hypothesis is not consistent with
Ockham's
razor; but the consciousness of the Universe does not, if it is able to
explain
many phenomena for which a satisfactory theory is yet to be developed,
like the
aforementioned paranormal phenomena.
The standard model of physics is a gauge theory with gauge group U(1) .
SU(2).SU(3) on a space-time that has three spatial and one time
dimensions.
I will posit a modi.ed standard model - an SU(2) . SU(2) . SU(3) gauge
theory on the four-sphere of .xed radius. The expansion of the content
can be
described as interacting di.usion.
The usually observed Universe has three spatial dimensions. This I will
describe as a smooth 3-dimensional sub-manifold in the 4-dimensional
Universe
and called the physical universe. The physical universe is dynamically
evolving
as an embedded submanifold. Indeed, it is expanding within the
4-dimensional
space following di.usion.
Within the spirit of the Universe lies our own spirit bodies. The
classical
mechanics that the electromagnetic particles follow in toto thus include
the
movement of our spirit bodies. My major claims are that we are aligned
with the
fate of the Universe (the natural movement of the spirit movement of
determined
by classical mechanics) when we follow our hearts without restriction.
This
overcomes the paradox between fate and free will.
The most powerful exercise of free will is precisely to follow our
hearts,
and that is precisely when we are most strongly aligned with the fate of
the
Universe. Modern accepted uses of the terms 'fate','destiny', 'agency',
and 'free
will' are terms need clari.cation for my usage here of these terms are
aligned
to the understanding of, among others, the ancient Greeks rather then
the
modern conceptions. This deserves clari.cation in order to avoid
confusion. I
am claiming that 'free will' and 'agency' are in e.ect for human spirits
and is
CONSISTENT with a precise formulation of 'fate' or 'destiny'.
Let me return thus to accepted modern de.nitions of these terms before I
clarify how this consistency is possible (and indeed I am claiming truth
for my
views and hope to convince you by argument and evidence). Modern usage
de.nes fate as a power or agency that predetermines or orders the course
of
events. Fate determines events as ordered or "inevitable". Fate is used
in
regard to the .nality of events as they have worked themselves out; and
the
same sense of .nality of events, as they WILL work themselves out, is
termed
DESTINY. Now modern terminology also de.nes human free will or agency as
our absolute freedom in decision?making and choosing our own acts which
have
consequences in the future. It is a modern commonplace that in
exercising our
own freedom of action, we in
uence the future events and hence the concept of
destiny is an impossibility. I will argue otherwise. As a simpli.ed
description,
I will .rst assume that our spirits are four?dimensional electromagnetic
objects
that form a subset of the spirit of the Universe. I will provide more
detailed
evidence for this in later chapters. I will assume also that the
electromagnetic
four?dimensional spirit of the universe follows classical mechanics,
which implies
that the whole Universe is deterministic. From this picture, it is clear
that
con.guration of our spirits follows 'fate' or 'destiny'. Now note that
since the
decision?making capacity of our spirits is not constrained in this
model, that
AS SPIRITS, there is absolutely no reason to believe that the
deterministic
31
universe does not INCLUDE the choices and decisions we make as
sub?spirits
of the Universe. Furthermore, although destiny or fate can be
operational, there
is no reason a priori that this future can be KNOWN by us, as
4?dimensional
electromagnetic objects are highly interacting and thus the future of
our spirits
apriori has in
uence from the total electromagnetic content of the rest of the
Universal spirit, or Force. The picture of reality in the last paragraph
only shows
that there is no a priori inconsistency between complete free will or
agency by
our spirits and destiny or fate. However, lack of contradiction does not
imply
that in fact this is the case. Artistic illustration of this convergence
is given by
W. Somerset Maugham from a famous tale:
Death speaks: There was a merchant in Baghdad who sent his servant to
market to buy provisions and in a little while the servant came back,
white and
trembling, and said, "Master, just now when I was in the market?place I
was
jostled by a woman in the crowd and when I turned I saw it was Death
that
jostled me. She looked at me and made a threatening gesture; now, lend
me
your horse, and I will ride away from this city and avoid my fate. I
will go to
Samarra and there Death will not .nd me." The merchant lent him his
horse,
and the servant mounted it, and he dug his spurs in its
anks and as fast as the
horse could gallop he went. Then the merchant went down to the
marketplace
and he saw me standing in the crowd and he came to me and said, "Why did
you make a threatening gesture to my servant when you saw him this
morning?"
"That was not a threatening gesture," I said, "it was only a start of
surprise.
I was astonished to see him in Baghdad, for I had an appointment with
him
tonight in Samarra. Now our spirits are composed of magnetic monopoles
of
light and dark. In the heart chakra, the center of our spirits is also a
mixture
of light and dark monopoles. I de.ne INJUSTICE as a coercive use of
spirit
bodies against another spirit in such a way as to disallow the spirit to
follow
his or her fate in accordance with the karmic laws. This gives a uselful
abstract
and spiritual limit to the ideal of Freedom. Thus Freedom/Justice are
balanced
in this abstract de.nition. I will return to this issue in later
chapters.
Equality of spirits can be de.ned in terms of lack of special privilege
in
humanity. Brotherhood (and I include sisters in this ideal) is a
spiritual concept
de.nable in terms of unconditional love in the heart chakra for
humanity. In
particular, Brotherhood does not mean in my view any responsibility to
care
for others - simply an open heart chakra.
Thus the ideals of Freedom, Justice, Equality, and Brotherhood are not
only
consistent with the S4 physics description of spirit but in fact connect
us with the
fate of the spirit of the Universe. A signi.cant feature of this
consistency is the
lack of a moral code of good and evil. The Force, or the spirit of the
Universe,
contains both darkness and light. It is the fallacious claim of a number
of
our religious traditions that there is asymmetry in darkness and light.
Human
spirits contain both light and dark magnetic monopoles and we are
inherently
incapable of removing either component completely by any natural means.
In
this regard, Nietzsche's aphorism is enlightening: "What is done out of
love
always takes place beyond good and evil," to which I would also add,
"and
beyond light and darkness as well."
A quick review of necessary mathematics
Shape
Concise formulations of physical theory requires a feel for geometry. The intu-
itive description of curved spaces can be made precise using the mathematical
construct of a Riemannian manifold. For the uninitiated, it may be helpful to
imagine a soccer ball, whose surface is a two-dimensional manifold. Now in this
picture, there is an external three-dimensional world in which the soccer ball
exists. But now if you live on the surface of the ball, and had ability only to see
along the ball without being able to see the ambient three-dimensional space,
your experience of this surface would be very di.erent than the image in your
mind. It would appear
at, just as the Earth's surface appears
at to us unless
we look out to the horizon and watch a ship's mast appear from below the level
of the sea or look at Earth from outer space. The surface of the soccer ball is a
two dimensional manifold, and so is the surface of a doughnut. Their difference can
be experienced by a two-dimensional being as follows: from the north pole of the
soccer ball, if you walk straight east or south, you can reach the common point
of south pole; from a point on the doughnut, if you walk in orthogonal directions,
you will not cross paths except the starting point. In fact living on the surface
of a doughnut will feel no different than living on a square where opposing walls
are the same { you walk through the north wall and appear through the south
wall; you walk through the east wall and appear through the west wall. Both
are closed spaces but the doughnut has a different topology than the sphere. In the
S4 physics theory, the universe is claimed to be a sphere whose surface is locally
four-dimensional instead of two as in the soccer ball.
Mathematically, the above intuitions are formalized as follows -- this is standard fare in elementary mathematics courses. First, flat spaces are defined of various dimensions as tuples of real numbers: so the
at two-dimensional space are pairs of real numbers that we all know; and it's no problem to think of triples and quadruples or quintuples as 3,4,5 dimensional flat spaces. A mapping f between two sets A and B is an assignment of an element of B to each element of A. One views curves on a plane as a mapping of the one-dimensional line to a space: f: R -> M and thinks of the line as time. On flat two-dimensional space velocity of the curve has two components, the time-derivative of the first and second components. The acceleration is the second time derivative, also with two components.
Whether a patch of two dimensional plane is curved or flat can be decided by considering vector fields on it that represent paths of particles on it without acceleration. The tangent vectors of all paths that pass through a particular point is the tangent space at the point. In the two dimensional patch, the tangent space is a two-dimensional vector space at every point. Lengths and angles between vectors on a vector space are determined by an "inner product" defined on the vector space: the length squared is the inner product of the vector with itself, and cosine of the angle between two vectors is their inner product divided by the product of their lengths. The two-dimensional surface becomes curved when the inner product is allowed to vary as a function of location. An inner product on a plane with a fixed basis can be represented as a symmetric 2x2 matrix, and it is called a Riemannian metric. When the matrix varies from the identity matrix, the patch is "curved". A manifold is a space which the tangent space at any point can be identified with a piece of flat space with a varying matrix function just described. Such an assignment of a vector space to each point on the manifold is called a vector bundle. Thus the union of all the tangent spaces form a smooth manifold and is called the tangent bundle.
The rules for how the matrix coefficients change when the coordinates change are tensorial. The dual of a vector space is the space of linear functionals on it. The cotangent space at any point of a manifold is its dual. The exterior algebra defined by the cotangent space naturally defines a vector bundle. A cotangent vector is also called a differential 1-form, and a generic element of the exterior algebra is a differential form. On a manifold with coordinates (x,y), the exterior algebra is spanned by contants, dx, dy, dxdy; this has the straightforward generalization for higher dimensions. On differential forms, the exterior derivative is defined by first defining it on functions F as the sum of dF/dxi dxi, setting d^2 = 0, and then enforcing the product rule for differentiation. The exteriod derivative together with the exterior algebra of cotangent spaces is called the de Rham complex. Note that instead of the exterior algebra, one could have taken a Clifford algebra. For this consider a quadratic form Q on a vector space V. Then form the freest algebra with the relation v^2 = -Q(v)1. This is the exterior algebra when Q=0.
On any manifold, there is an adjoint of the exterior derivative on the de Rham complex, and the Laplacian can be defined on differential forms as the square of d + d*. So this operator can be thought of as the square root of the Laplacian, and for this reason it is called a Dirac operator.
A valuable result of Christian Bar is an elegant determination of the eigenvalues of the Dirac operator on a sphere. I present his arguments to provide an example of the use of the above constructions which otherwise is a barrage of unmotivated abstractions. Let S^n be the the round sphere with sectional curvature 1, n >= 2. Denote by D the Dirac operator acting on spinor fields, and by L the Levi-Civita connection on vector fields or spinor fields. Let mu = +/- 1/2, and a spinor field s is called a Killing field with constant mu if
MX s := LX s - mu X s = 0
The Killing spinors are useful for spheres because they can be used to trivialize the spinor bundles on S^n for Killing constant mu = 1/2 or mu=-1/2. Since S^n is simply connected, we need only show that the curvature of MX vanishes. Pick a point where LX = 0 to calculate
MX MY s = (LX - mu X)(LY - mu Y) s = LX LY s - mu Y LX s - mu X LY s + (1/4) XY
Exchanging X and Y and subracting from this gives
(1) RM(X,Y) = RL(X,Y) + (1/4)(XY - YX)
The relation between the curvature RL on the spinor bundle and the curvature of the tangent bundle is
(2) RL(X,Y) = (1/4) \sum < R(X,Y) ei, ej> ei ej,
for an orthonormal moving frame.
Since the sphere has sectional curvature 1, R(X,Y)Z = <X,Z>Y - <Y,Z>X, we can combine (1) and (2) to obtain
RL(X,Y) = (1/4) (YX - XY).
Now we can see that RM(X,Y) = 0 from (1). Thus Killing spinors of constant mu=1/2 or mu=-1/2 produce a basis of the spinor bundle. This will be a useful tool
Symmetry
The mathematical machinery for studying symmetries comes from Sophus
Lie's theory of Lie groups. In mathematics, a 'group' has a precise meaning that
is di.erent from the vernacular { it is a set with a pairwise operation de.ned
on it. The primary example is of course the integers, whole numbers including
zero and negative numbers where the pairwise operation is addition. The most
interesting feature of a group is the existence of a zero and the ability to reverse
the operation at will, the existence of negative numbers for example. The power
of this idea begins to show itself when we start looking at symmetries that occur
in crystals, for example. Consider a very simple lattice on the plane consisting of
points at pairs of integers, the square lattice. If you rotate the lattice 90 degrees,
or 180 degrees or 270 degrees clockwise, it still produces the same lattice. You
can also re
ect the lattice along any horizontal or vertical line containing lattice
points with the same result. Now these operations on the plane can be described
by matrix multiplication by a .nite set of matrices. This set will form a group
with the operation of matrix multiplication. The theory of matrix groups are
precisely the object of the theory of Lie groups, and symmetries of such lattices
are of special interest in that theory.
Using complex numbers, which are numbers of the form a + b
p
􀀀1 and a
and b are real numbers, one can describe the square lattice as complex numbers
with integer entries. This is a special case where instead of matrices, complex
numbers can be used to describe the rotation group. For us, another special
case is actually relevant: the four-dimensional variant of complex numbers, the
quaternions. These can be described as quintuples a + bi + cj + dk where i; j; k
follow these rules:
i2 = j2 = k2 = 􀀀1
ij = k = 􀀀ji
jk = i = 􀀀kj
ki = j = 􀀀ik
Muliplication of quaternions is non-commutative but otherwise it's not very
di.erent from multiplication of complex numbers. Indeed, just as in complex
numbers, there's a conjugate of a+bi+cj +dk which is a􀀀bi􀀀cj 􀀀dk whose
product is always real { it's a2 + b2 + c2 + d2. Because of this conjugation, the
unit { squares sum to one { quaternions can be seen to have an inverse, which
is the key to showing that they form a group under multiplication. This group
is actually the same as 2 . 2 matrices with complex entries whose conjugate-
transposes are their inverses. The group is called the 'special unitary group' or
SU(2). By this identi.cation, it's clear that the topology of SU(2) is the same as
the three-sphere because that's exactly what unit vectors in a four-dimensional
at space form.
One can gain a fair amount of intuition about four-dimensional quaternions
and four-dimensional lattices by analogy with the complex plane which is easier
to visualize. One can think of SU(2) as an analogue of the rotation group of the
plane, which is a unit circle. Another interesting fact is that just as the complex
projective line is the same as the two-sphere, the quaternionic projective line is
the four-sphere. This is interesting for us because it is our geometric model of
the universe.
The typical element of SU(2) is a 2 by 2 matrix of complex numbers of the form A = [ a b; -\bar{b} \bar{a}] for complex numbers a and b which have the restriction that |a|^2+|b|^2 = 1. If r is the length of a in polar coordinates, then (1-r^2)^{1/2} is the length of b, and two angles determine the matrix, \theta and \phi. The matrix acts on pairs of complex numbers, which wr will call the first plane and the second plane.
If r=1, then regardless of the value of \phi, the action of A is rotation of the first plane clockwise by \theta
and the second plane counterclockwise by \theta. When r=0, the action of A is rotation of second plane clockwise by \phi and the first plane counterclockwise by \phi. So r is a weight that mixes these actions of rotations by \theta and \phi.
Having developed some intuition about an interesting Lie group, SU(2), we
can return to a more general construction. Suppose P is a riemannian manifold
and G is a Lie group. If there is a free action of G on P, that is to say if
there is a function mapping P . G to P that has no .xed points, and the set
of orbits of the action is a smooth manifold M, then P is called a principal
G-bundle over M. Intuitively, one can think of this as describing a space
of things with symmetry in G at any point of M. Indeed, one of the major
motivations of this concept is precisely the problem of interest here: describe
the idea of elementary particles having intrinsic symmetry in some Lie group.
The mathematical examples of interest for this concept are, for instance, the
Hopf .bration, which is the seven-sphere with action of the unit quaternions,
whose base manifold is the four-sphere. This is an example of a non-trivial
principal bundle that's not just a Cartesian product.
Gauge theory as developed by Yang and Mills in 1950s to describe the weak
nuclear force can be conveniently described using this machinery. Elementary
particles with symmetry group G are, in this theory, connections on principal G-
bundles where the base space is a manifold M. The Ehrenberg-Siday-Aharonov-
Bohm e.ect (1949,1959) which were experimentally tested in 1980s and 1990s
shows that description of electromagnetic particles as connections on principal
bundles { or gauge potentials { is not simply a mathematical fancy but a better
description than simply through elecric and magnetic .elds. Thus 'particles'
with symmetry G are connections on a principal bundle in gauge theory.
Age of humans and other things
In a Precambrian mineral deposit said to be 2.8 billion years old, a metallic sphere with grooves were found. Here is a photograph:
In the theory of human evolution, excited by the DNA similarity between humans and chimpanzees that has a 2% difference, around 20 times the variation among modern humans, and other observations, the estimate of human separation from other primates from this sort of analysis of DNA would put modern human origins at around 30 million years ago. There is thus a problem to be resolved.
Plato gives a description of an ancient civilization, Atlantis, in Critias and
Timaeus. Here is the text from Timeaus:
Critias. Then listen, Socrates, to a strange tale, which is, however, certainly
true, as Solon, who was the wisest of the seven sages, declared. He was a relative
and great friend of my great-grandfather, Dropidas, as he himself says in several
of his poems; and Dropidas told Critias, my grandfather, who remembered, and
told us, that there were of old great and marvelous actions of the Athenians,
which have passed into oblivion through time and the destruction of the human
race-and one in particular, which was the greatest of them all, the recital of
which will be a suitable testimony of our gratitude to you Socrates. Very good;
and what is this ancient famous action of which Critias spoke, not as a mere
legend, but as a veritable action of the Athenian State, which Solon recounted?
Critias. I will tell an old-world story, which I heard from an aged man; for
Critias was, as he said, at that time nearly ninety years of age, and I was about
ten years of age. Now the day was that day of the Apaturia which is called the
registration of youth; at which, according to custom, our parents gave prizes for
recitations, and the poems of several poets were recited by us boys, and many of
us sung the poems of Solon, which were new at the time. One of our tribe, either
because this was his real opinion, or because he thought that he would please
Critias, said that, in his judgment, Solon was not only the wisest of men but
the noblest of the poets. The old man, I well remember, brightened up at this,
and said, smiling: "Yes, Amynander, if Solon had only, like other poets, made
poetry the business of his life, and had completed the tale which he brought
with him from Egypt, and had not been compelled, by reason of the factions
and troubles which he found stirring in this country when he came home, to
attend to other matters, in my opinion he would have been as famous as Homer,
or Hesiod, or any poet." "And what was that poem about, Critias?" said the
person who addressed him. "About the greatest action which the Athenians
ever did, and which ought to have been most famous, but which, through the
lapse of time and destruction of the actors, has not come down to us." "Tell
us," said the other, "the whole story, and how and from whom Solon heard
this veritable tradition." He replied: "At the head of the Egyptian Delta, where
the river Nile divides, there is a certain district which is called the district of
Sais, and the great city of the district is also called Sais, and is the city from
which Amasis the king was sprung. And the citizens have a deity who is their
fondress: she is called in the Egyptian tongue Neith, which is asserted by them
to be the same whom the Hellenes called Athene. Now, the citizens of this city
are great lovers of the Athenians, and say that they were in some way related to
them. Thither came Solon, who was received by them with great honor; and he
asked the priests, who were most skilful in such matters, about antiquity, and
made the discovery that neither he nor any other Hellene knew anything worth
mentioning about the times of old. On one occasion, when he was drawing
them on to speak of antiquity, hebegan to tell about the most ancient things
in our part of the world-about Phoroneus, who is called 'the .rst', about the
Niobe; and, after the Deluge, to tell of the lives of Deucalion and Pyrrha; and
he traced the geneology of their descendents, and attempted to reckon how
many years old were the events of which he was speaking, and give the dates.
Thereupon, one of the priests, who was of very great age, said, 'O Solon, Solon,
you Hellenes are but children, and there is never an old man who is a Hellene.'
Solon, hearing this, said, 'What do you mean?' 'I mean to say,' he replied, 'that
in mind you are all young; there is no old opinion handed down among you by
ancient tradition, nor any science which is hoary with age. And I will tell you
the reason of this: there have been, and there will be again, many destructions
of mankind arising out of many causes. There is a story which even you have
preserved, that once upon a time Phaethon, the son of Helios, having yoked the
steeds in his father's chariot, because he was not able to drive them in the path
of his father, burnt up all that was upon the earth, and was himself destroyed
by a thunder-bolt. Now, this has the form of a myth, but really signi.es a
declination of the bodies moving around the earth and in the heavens, and a
great con
agration of things upon the earth recurring at long intervals of time:
when this happens, those who live upon the mountains and in dry and lofty
places are more liable to sestruction than those who dwell by rivers or on the
sea-shore; and from this calamity the Nile, who is our never-failing savior, saves
and delivers us. When, on the other hand, the gods purge the earth with a
deluge of water, among you herdsmen and shepherds on the mountains are the
survivors, whereas those of you who live in cities are carried by the rivers into
the sea; but in this country neither at that time nor at any other does the water
come from above on the .elds, having always a tendency to come up from below,
for which reason the things preserved here are said to be the oldest. The fact is,
that wherever the extremity of winter frost or of summer sun does not prevent,
the human race is always increasing at times, and other times diminishing in
numbers. And whatever happened either in your country or in ours, or in any
region of which we are informed-if any action which is noble or great, or in any
39
other way remarkable has taken place, all that has been written down of old,
and is preserved in our temple; whereas you and other nations are just being
provided with letters and the other things which States require; and then, at
the usual period, the stream from heaven descends like a pestilence, and leaves
only those of you who are destitute of letters and education; and thus you begin
all over again as children, and know nothing of what has happened in ancient
times, either among us or among yourselves. As for those genealogies of yours
which you have recounted to us, Solon, they are no better than the tales of
children; for, in the .rst place, you remember one deluge only, whereas there
were many of them; and, in the next place, you do not know that there dwelt
in your land the fairest and noblest race of men which ever lived, of whom you
and your whole city are but a seed or remnant. And this was unknown to you,
because for many generations the survivors of that destruction died and made
no sign. For there was a time, Solon, before that great deluge of all, when the
city which is now Athens was .rst in war, and was preeminent for the excellence
of her laws, and is said to have performed the noblest deeds, and to have had
the fairest constitution of any of which tradition tells, under the face of heaven.'
Solon marveled at this, and earnestly requested the priest to inform him exactly
and in order about these former citizens. 'You are welcome to hear about them,
Solon,' said the priest, 'both for your sake and for that of the city; and above
all, for the sake of the goddess who is the common patron and protector and
educator of both our cities. She founded your city a thousand years before ours,
receiving from the Earth and Hephaestus the seeds of your race, and then she
founded ours, the constitution of which is set down in our sacred registers as
8000 years old. As touching the citizens of 9000 years ago, I will brie
y inform
you of their laws and of the noblest of their actions; and the exact particulars
of the whole we will go through at our leisure the sacred registers themselves. If
you compare these very laws with your own, you will .nd that many of ours are
the counterpart to yours, as they were in the olden time. In the .rst place, there
is the caste of priests, which is separated from all the others; next there are the
arti.cers, who exercise their several crafts by themselves, and without admixture
of any other; and also there is the class of shepherds and that of hunters, as well
as the husbandmen; and you will observe, too, that the warriors in Egypt are
separated from all other classes, and are commanded by law to only engage in
war; moreover, the weapons with which they are equipped are shields and spears,
and this the goddess taught .rst among you, and then in Asiatic countries, and
we among the Asiatics .rst adopted. "'Then, as to wisdom, do you observe
what care the law took from the very .rst, searching out and comprehending
the whole order of things down to prophecy and medicine (the latter with a
view to health); and out of these divine elements drawing what was needful
for human life, and adding every sort of knowledge which was connected with
them. All this order and arrangement the goddess .rst imparted to you when
establishing your city; and she chose the spot of earth in which you were born,
because she saw that the happy temperament of the seasons in the land would
produce the wisest of men. Wherefore the goddess, who was a lover both of war
and of wisdom, selected, and .rst of all settled that spot which was most likely
40 CHAPTER 4. AGE OF HUMANS AND OTHER THINGS
to produce men like herself. And there you dwelt, having such laws as these and
still better ones, and excelled all mankind in all virtue, as became the children
and disciple of the gods. Many great and wonderful deeds are recorded of your
State in our histories; but one of them exceeds all the rest in greatness and
valor; for these histories tell of a mighty power which was aggressing wantonly
against the whole of Europe and of Asia, and to which your city put an end.
This power came forth out of the Atlantic Ocean, for in those days the Atlantic
was navigable; and there was an island in front of the straits which you call the
columns of Heracles: the island was larger than Libya and Asia put together,
and was the way to other islands, and from the islands you might pass through
the whole of the opposite continent which surrounded the true ocean; for this
sen which is within the Straits of Heracles is only a harbor, having a narrow
entrance, but that other is a real sea, and the surrounding land may be most
truly called a continent. Now, in the island of Atlantis there was a great and
wonderful empire, which had rule over the whole island and several others, as
well as over parts of the continent; and, besides these, they subjected the parts
of Libya within the Columns of Heracles as far as Egypt, and of Europe as far
as Tyrrhenia. The vast power thus gathered into one, endeavored to subdue at
one blow our country and yours, and the whole of the land which was within
the straits; and then, Solon, your country shone forth, in the excellence of her
virtue and strength, among all mankind; for she was the .rst in courage and
military skill, and was the leader of the Hellenes. And when the rest fell o. from
her, being compelled to stand alone, after having undergone the very extremity
of danger, she defeated and triumphed over the invaders, and preserved from
slavery those who were not yet subjected, and freely liberated all the others
who dwelt within the limits of Heracles. But afterward there occurred violent
earthquakes and
oods, and in a single day and night of rain all your warlike
men in a body sunk into the earth, and the island of Atlantis in like manner
disappeared, and was sunk beneath the sea. And that is the reason why the sea
in those parts is impassable and impenetrable, because there is such a quantity
of shallow mud in the way; and this was caused by the subsidence of the island.'
(" Plato's Dialogues," ii., 517, Timeus.) . . . " But in addition to the gods whom
you have mentioned, I would specially invoke Mnemosyne; for all the important
part of what I have to tell is dependent on her favor, and if I can recollect and
recite enough of what was said by the priests, and brought hither by Solon, I
doubt not that I shall satisfy the requirements of this theatre. To that task,
then, I will at once address myself. "Let me begin by observing, .rst of all,
that nine thousand was the sum of years which had elapsed since the war which
was said to have taken place between all those who dwelt outside the Pillars of
Heracles and those who dwelt within them : this war I am now to describe. Of
the combatants on the one side the city of Athens was reported to have been
the ruler, and to have directed the contest; the combatants on the other side
were led by the kings of the islands of Atlantis, which, as I was saying, once had
an extent greater than that of Libya and Asia; and, when afterward sunk by
an earthquake, became an impassable barrier of mud to voyagers sailing from
hence to the ocean. The progress of the history will unfold the various tribes
41
of barbarians and Hellenes which then existed, as they successively appear on
the scene; but I must begin by describing, .rst of all, the Athenians as they
were in that day, and their enemies who fought with them ; and I shall have
to tell of the power and form of government of both of them. Let us give the
precedence to Athens. . . . " Many great deluges have taken place during the
nine thousand years, for that is the number of years which have elapsed since
the time of which I am speaking; and in all the ages and changes of things there
has never been any settlement of the earth
owing down from the mountains,
as in other places, which is worth speaking of; it has always been carried round
in a circle, and disappeared in the depths below. The consequence is that, in
comparison of what then was, there are remaining in small islets only the bones
of the wasted body, as they may be called, all the richer and softer parts of
the soil having fallen away, and the mere skeleton of the country bein;' left.
... "And next, if I have not forgotten what I heard when I was a child, I will
impart to you the character and origin of their adversaries; for friends should
not keep their stories to themselves, but have them in common. Yet, before
proceeding farther in the narrative, I ought to warn you that you must not
be surprised if you should hear Hellenic names given to foreigners. I will tell
you the reason of this: Solon, who was intending to use the tale for his poem,
made an investigation into the meaning of the names, and found that the early
Egyptians, in writing them down, had translated them into their own language,
and he recovered the meaning of the several names and retranslated them, and
copied them out again in our language. My great-grandfather, Dropidas, had
the original writing, which is still in my possession, and was carefully studied
by me when I was a child. Therefore, if you hear names such as are used in this
country, you must not be surprised, for I have told you the reason of them. "
The tale, which was of great length, began as follows: I have before remarked,
in speaking of the allotments of the gods, that they distributed the whole earth
into oortions di.ering in extent, and made themselves temples and sacri.ces.
And Poseidon, receiving for his lot the island of Atlantis, begat children by a
mortal woman, and settled them in a part of the island which I will proceed to
describe. On the side toward the sea, and in the centre of the whole island, there
was a plain which is said to have been the fairest of all plains, and very fertile.
Near the plain again, and also in the centre of the island, at a distance of about
.fty stadia, there was a mountain, not very high on any side. In this mountain
there dwelt one of the earth-born primeval men of that country, whose name
was Evenor, and he had a wife named Lcucippe, and they had an only daughter,
who was named Cleito. The maiden was growing up to womanhood when her
father and mother died; Poseidon fell in love with her, and had intercourse
with her; and, breaking the ground, enclosed the hill in which she dwelt all
round, making alternate zones of sea and land, larger and smaller, encircling
one another; there were two of land and three of water, which he turned as with
a lathe out of the centre of the island, equidistant every w.-vy, so that no man
could get to the island, for ships and voyages were not yet heard of. He himself,
as he was a god, found no di.culty in making special arrangements for the
centre island, bringing two streams of water under the earth, which he caused
42 CHAPTER 4. AGE OF HUMANS AND OTHER THINGS
to ascend as springs, one of warm water and the other of cold, and making
every variety of food to spring up abundantly in the earth. He also begat and
brought up .ve pairs of male children, dividing the island of Atlantis into ten
portions: lie gave to the .rst-born of the eldest pair his mother's dwelling and
the surrounding allotment, which was the largest and best, and made him king
over the rest; the others he made princes, and gave them rule over many men
and a large territory. And he named them all: the eldest, who was king, he
named Atlas, and from him the whole island and the ocean received the name
of Atlantic. To his twin-brother, who was born after him, and obtained as
his lot the extremity of the island toward the Pillars of Heracles, as far as the
country which is still called the region of Gadcs in that part of the world, he
gave the name which in the Hellenic language is Eumelus, in the language of
the country which is named after him, Gadeirus. Of the second pair of twins,
he called one Amphcrcs and the other Evaemon. To the third pair of twins he
gave the name Mneseus to the elder, and Awtochthon to the one who followed
him. Of the fourth pair of twins he called the elder Elasippus and the younger
Mestor. And of the .fth pair he gave to the elder the name of Azaes, and to
the younger Diaprepcs. All these and their descendants, were the inhabitants
and rulers of divers islands in the open sea; and also, as has been already said,
they held sway in the other direction over the country within the Pillars as far
as Egypt and Tyrrhenia. Now Atlas had a numerous and honorable family, and
his eldest branch always retained the kingdom, which the eldest son handed on
to his eldest for many generations; and they had such an amount of wealth as
was never before possessed by kings and potentates, and is not likely ever to
be again, and they were furnished with everything which they could have, both
in city and country. For, because of the greatness of their empire, many things
were brought to them from foreign countries, and the island itself provided much
of what was required by them for the uses of life. In the .rst place, they dug
out of the earth whatever was to be found there, mineral as well as metal, and
that which is now only a name, and was then something more than a name-
Orichalcum-was dug out of the earth in many parts of the island, and, with the
exception of gold, was esteemed the most precious of metals among the men of
those days. There was an abundance of wood for carpenters' work, and su.cient
maintenance for tame and wild animals. Moreover, there were a great number
of elephants in the island, and there was provision for animals of every kind,
both for those which live in lakes and marshes and rivers, and also for those
which live in mountains and on plains, and therefore for the animal which is the
largest and most voracious of them. Also, whatever fragrant things there are in
the earth, whether roots, or herbage, or woods, or distilling drops of
owers or
fruits, grew and thrived in that land; and again, the cultivated fruit of the earth,
both the dry edible fruit and other species of food, which we call by the general
name of legumes, and the fruits having a hard rind, a.ording drinks, and meats,
and ointments, and good store of chestnuts and the like, which may be used to
play with, and arc fruits which spoil with keeping-and the pleasant kinds of
dessert which console us after dinner, when we are full and tired of eating-all
these that sacred island lying beneath the sun brought forth fair and wondrous
43
in in.nite abundance. All these things they received from the earth, and they
employed themselves in constructing their temples, and palaces, and harbors,
and docks; and they arranged the whole country in the following manner: First
of all they bridged over the zones of sea which surrounded the ancient metropolis,
and made a passage into and out of the royal palace; and then they began to
build the palace in the habitation of the god and of their ancestors. This they
continued to ornament in successive generations, every king surpassing the one
who came before him to the utmost of his power, until they made the building
a marvel to behold for size and for beauty. And, beginning from the sea, they
dug a canal three hundred feet in width and one hundred feet in depth, and .fty
stadia in length, which they carried through to the outermost zone, making a
passage from the sea up to this, which became a harbor, and leaving an opening
su.cient to enable the largest vessels to .nd ingress. Moreover, they divided
the zones of land which parted the zones of sea, constructing bridges of such
a width as would leave a passage for a single trireme to pass out of one into
another, and roofed them over; and there was a way underneath for the ships,
for the banks of the zones were raised considerably above the wat?r. Now the
largest of the zones into which a passage was cut from the sea was three stadia
in breadth, and the zone of land which came next of equal breadth ; but the
next two, as well the zone of water as of land,were two stadia, and the one which
surrounded the central island was a stadium only in width. The island in which
the palace was situated had a diameter of .ve stadia. This, and the zones and
the bridge, which was the sixth part of a stadium in width, they surrounded
by a stone wall, on either side placing towers, and gates on the bridges where
the sea passed in. The stone which was used in the work they quarried from
underneath the centre island and from underneath the zones, on the outer as
well as the inner side. One kind of stone was white, another black, and a third
red; and, as they quarried, they at the same time hollowed out docks double
within, having roofs formed out of the native rock. Some of their buildings were
simple, but in others they put together di.erent stones, which they intermingled
for the sake of ornament, to be a natural source of delight. The entire circuit
of the wall which went round the outermost one they covered with a coating of
brass, and the circuit of the next wall they coated with tin, and the third, which
encompassed the citadel,
ashed with the red light of orichalcnin. The palaces
in the interior of the citadel were constructed in this wise: In tho centre was
a holy temple dedicated to Cleito and Poseidon, which remained inaccessible,
and was surrounded by an enclosure of gold; this was the spot in which they
originally begat the race of the ten princes, and thither they annually brought
the fruits of the earth in their season from all the ten portions, ;tnd performed
sacri.ces to each of them. Here, too, was Poseidon's own temple, of a stadium in
length and half a stadium in width, and of a proportionate height, having a sort
of barbaric splendor. All the outside of the temple, with tho exception of the
pinnacles, they covered with silver, and the pinnacles with gold. In the interior
of the temple the roof was of ivory, adorned everywhere with gold and silver and
orichalcum; all the other parts of the walls and pillars and
oor they lined with
orichalcum. In the temple they placed statues of gold : there was the god himself
44 CHAPTER 4. AGE OF HUMANS AND OTHER THINGS
standing in a chariot-the charioteer of six winged horses-and of such a size that
he touched the roof of the building with his head; around him there were a
hundred Nereids riding on dolphins, for such was thought to be the number of
them in that day. There were also in the interior of the temple other images
which had been dedicated by private individuals. And around the temple on the
outside were placed statues of gold of all the ten kings and of their wives; and
there were many other great o.erings, both of kings and of private individuals,
coming both from the city itself and the foreign cities over which they held
sway. There was an altar, too, which in size and workmanship corresponded to
the rest of the work, and there were palaces in like manner which answered to
the greatness of the kingdom and the glory of the temple. " In the next place,
they used fountains both of cold and hot springs; these were very abundant, and
both kinds wonderfully adapted to use by reason of the sweetness and excellence
of their waters. They constructed buildings about them, and planted suitable
trees; also cisterns, some open to the heaven, others which they roofed over, to
be used in winter as warm baths: there were the king's baths, and the baths
of private persons, which were kept apart; also separate baths for women, and
others again for horses and cattle, and to them they gave as much adornment
as was suitable for them. The water which ran o. they carried, some to the
grove of Poseidon, where were growing all manner of trees of wonderful height
and beauty, owing to the excellence of the soil; the remainder was conveyed by
aqueducts which passed over the bridges to the outer circles: and there were
many temples built and dedicated to many gods; also gardens and places of
exercise, some for men, and some set apart for horses, in both of the two islands
formed by the zones; and in the centre of the larger of the two there was a
race-course of a stadium in width, and in length allowed to extend all round the
island, for horses to race in. Also there were guard-houses at intervals for the
body-guard, the more trusted of whom had their duties appointed to them in
the lesser zone, which was nearer the Acropolis; while the most trusted of all had
houses given them within the citadel, and about the persons of the kings. The
docks were full of triremes and naval stores, and all things were quite ready for
use. Enough of the plan of the royal palace. Crossing the outer harbors, which
were three in number, you would come to a wall which began at the sea and
went all round: this was everywhere distant .fty stadia from the largest zone
and harbor, and enclosed the whole, meeting at the mouth of the channel toward
the sea. The entire area was densely crowded with habitations; and the canal
and the largest of the harbors were full of vessels and merchants coming from
all parts, who, from their numbers, kept up a multitudinous sound of human
voices and din of all sorts night and day. I have repeated his descriptions of
the city and the parts about the ancient palace nearly as he gave them, and
now 1 must endeavor to describe Ilie nature and arrangement of the rest of the
country. The whole country was described as being very lofty and precipitous
on the side of the sea, but the country immediately about and surrounding the
city was a level plain, itself surrounded by mountains which descended toward
the sea; it was smooth and even, but of an oblong shape, extending in one
direction three thousand stadia, and going up the country from the sea through
45
the centre of the island two thousand stadia; the whole region of the island lies
toward the south, and is sheltered from the north. The surrounding mountains
he celebrated for their number and size and beauty, in which they exceeded all
that are now to be seen anywhere; having in them also many wealthy inhabited
villages, and rivers and lakes, and meadows supplying food enough for every
animal, wild or tame, and wood of various sorts, abundant for every kind of
work. I will now describe the plain, which had been cultivated during many
ages by many'generations of kings. It was rectangular, and for the most part
straight and oblong; and what it wanted of the straight line followed the line of
the circular ditch. The depth and width and length of this ditch were incredible,
and gave the impression that such a work, in addition to so many other works,
could hardly have been wrought by the hand of man. But I must say what I
have heard. It was excavated to the depth of a hundred feet, and its breadth
was a stadium everywhere; it was carried round the whole of the plain, and
was ten thousand stadia in length. It received the streams which came down
from the mountains, and winding round the plain, and touching the city at
various points, was there let o. into the sea. From above, likewise, straight
canals of a hundred feet in width were cut in the plain, and again let o. into
the ditch, toward the sea; these canals were at intervals of a hundred stadia,
and by them they brought down the wood from the mountains to the city, and
conveyed the fruits of the earth in ships, cutting transverse passages from one
canal into another, and to the city. Twice in the year they gathered the fruits of
the earth-in winter having the bene.t of the rains, and in summer introducing
the water of the canals. As to the population, each of the lots in the plain
had an appointed chief of men who were .t for military service, and the size
of the lot was to be a square of ten stadia each way, and the total number
of all the lots was sixty thousand. "And of the inhabitants of the mountains
and of the rest of the country there was also a vast multitude having leaders,
to whom they were assigned according to their dwellings and villages. The
leader was required to furnish for the war the sixth portion of a war-chariot,
so as to make up a total of ten thousand chariots; also two horses and riders
upon them, and a light chariot without a seat, accompanied by a .ghting man
on foot carrying a small shield, and having a charioteer mounted to guide the
horses; also, he was bound to furnish two heavy-armed men, two archers, two
slingers, three stone-shooters, and three javelin men, who were skirmishers, and
four sailors to make up a complement of twelve hundred ships. Such was the
order of war in the royal city-that of the other nine governments was di.erent
in each of them, and would be wearisome to narrate. As to o.ces and honors,
the following was the arrangement from the .rst: Each of the ten kings, in his
own division and in his own city, had the absolute control of the citizens, and in
many cases of the laws, punishing and slaying whomsoever he would. " Now the
relations of their governments to one another were regulated by the injunctions
of Poseidon as the law had handed them down. These were inscribed by the
.rst men on a column of oriehalcum, which was situated in the middle of the
island, at the temple of Poseidon, whither the people were gathered together
every .fth and sixth years alternately, thus giving equal honor to the odd and to
46 CHAPTER 4. AGE OF HUMANS AND OTHER THINGS
the even number. And when they were gathered together they consulted about
public a.airs, and inquired if any one had transgressed in anything, and passed
judgment on him accordingly-and before they passed judgment they gave their
pledges to one another in this wise: There were bulls who had the range of the
temple of Poseidon ; and the ten who were left alone in the temple, after they
had o.ered prayers to the gods that they might take the sacri.ces which were
acceptable to them, hunted the bulls without weapons, but with staves and
nooses; and the bull which they caught they led up to the column; the victim
was then struck on the head by them, and slain over the sacred inscription. Now
on the column, besides the law, there was inscribed an oath invoking mighty
curses on the disobedient. When, therefore, after o.ering sacri.ce according to
their customs, they had burnt the limbs of the bull, they mingled a cup and cast
in a clot of blood for each of them; the rest of the victim they took to the .re,
after having made n puri.cation of the column all round. Then they drew from
the cup in golden vessels, and, pouring a libation on the .re, they swore that
they would judge according to the laws on the column, and would punish any
one who had previously transgressed, and that for the future they would not,
if they could help, transgress any of the inscriptions, and would not command
or obey any ruler who commanded them to act otherwise than according to the
laws of their father Poseidon. This was the prayer which each of them o.ered
up for himself and for his family, at the same time drinking, and dedicating
the vessel in the temple of the god; and, after spending some necessary time at
supper, when darkness came on and the .re about the sacri.ce was cool, all of
them put on most beautiful azure robes, and, sitting on the ground ;it night
near the embers of the sacri.ces on which they had sworn, and extinguishing all
the .re about the temple, they received and gave judgment, if any of them had
any accusation to bring against any one; and, when they had given judgment,
at daybreak they wrote down their sentences on a golden tablet, and deposited
them as memorials with their robes. There were many special laws which the
several kings had inscribed about the temples, but the most important was the
following: That they were not to take up arms against one another, and they
were all to come to the rescue if any one in any city attempted to overthrow the
royal house. Like their ancestors, they were to deliberate in common about war
and other matters, giving the supremacy to the family of Atlas; and the king was
not to have the power of life and death over any of his kinsmen, unless he had
the assent of the majority of the ten kings. " Such was the vast power which the
god settled in the lost island of Atlantis; and this he afterward directed against
our land on the following pretext, as traditions tell: For many generations, as
long as the divine nature lasted in them, they were obedient to the laws, and
well-a.ectioned toward the gods, who were their kinsmen; for they possessed
true and in every way great spirits, practising gentleness and wisdom in the
various chances of life, and in their intercourse with one another. They despised
everything but virtue, not caring for their present state of life, and thinking
lightly on the possession of gold and other property, which seemed only a burden
to them; neither were they intoxicated by luxury; nor did wealth deprive them
of their self-control; but they were sober, and saw clearly that all these goods
4.1. SKETCH OF EVIDENCE FOR ANCIENT CIVILIZATION 47
are increased by virtuous friendship with one another, and that by excessive zeal
for them, and honor of them, the good of them is lost, and friendship perishes
with them. "By such re
ections, and by the continuance in them of a divine
nature, all that which we have described waxed and increased in them; but when
this divine portion began to fade away in them, and became diluted too often,
and with too much of the mortal admixture, and the human nature got the
upper-hand, then, they being unable to bear their fortune, became unseemly,
and to him who had an eye to see, they began to appear base, and had lost
the fairest of their precious gifts; but to those who had no eye to see the true
happiness, they still appeared glorious and blessed at the very time when they
were .lled with unrighteous avarice and power. Zeus, the god of gods, who
rules with law, and is able to see into such things, perceiving that an honorable
race was in a most wretched state, and wanting to in
ict punishment on them,
that they might be chastened and improved, collected all the gods into his most
holy habitation, which, being placed in the centre of the world, sees all things
that partake of generation. And when he had called them together he spake as
follows:"
4.1 Sketch of evidence for ancient civilization
If we are to believe this account, and given the urban structures found more than
2000 meters under sea level o. the coast of Cuba in 2001, as well as the dating
of the Great Sphinx of Giza by a geologist to at least 7000 years old, it is not
too far-fetched to do so, human civilizations reach back to quite a distant past.
If we are to believe Edgar Cayce's account of the destruction of Atlantis, it was
around 10,800 BC. Academic archaeologists frown upon descriptions of such an
ancient civilizations without material evidence, which is a fair complaint, but it
is still possible to provide indirect evidence to support a theory of Atlantis as
a compromise. Personally, I believe that such a civilization existed from past
life memories, the occurrence of
ood stories in mythologies literally across the
globe, the similarity in pyramid structures in religious architecture also around
the globe. I touch on this quickly because writers like Graham Hancock have
provided more elaborate studies on this. Thus civilized humanity has a history
in the range of 13,000 years at least.
In the cave of pope in France, statuettes that could only have been produced
by civilized human beings that are 23,000 years have been found. Perhaps their
creators were not members of an urban civilization, but it makes little sense to
assume they were semi-intelligent ape-men.
4.2 How long is the history of anatomically mod-
ern humans?
My .rst observation is that currently accepted theories of human evolution have
many counterexamples, and in fact can be dismissed as false. In September 2009,
48 CHAPTER 4. AGE OF HUMANS AND OTHER THINGS
Thomas Plummer, et. al., published "Oldest evidence of Toolmaking Hominims
in a Grassland Dominated Ecosystem." They provide archaeological evidence of
toolmaking civilization from Kenya that is in the order of 2 million years old.
This puts the lower bound on homo sapiens to 2 million years. My own view
is that we have been anatomically modern human beings for around 2 Billion
years, but the archaeological evidence for that is not available yet.
Michael Cremo has uncovered an array of archaeological evidence that run
counter to post-Darwinian evolutionary theory for human beings. Our common-
place ideas of human civilization may put humanity to at most 100 thousand
years, but there is evidence of human civilization that is far more ancient, al-
though there are technical disagreements on this evidence from scholars.
The state geologist of California wrote a book describing the artifacts found
in the gold mines during the gold rush that began around 1849. Based on the
depth of the mines where some of the artifacts were found, we can put civilized
human beings in up to 55 million years ago. In contrast, the recent fossil bones
of Ardipithecus, who is in the human evolutionary theory an ancestor of modern
humans, is 4.4 million years old.
More controversial are interpretations of the Oklo phenomenon of 1972,
where nuclear scientists have found spent Uranium?235 in the Oklo mines of
Gabon. The currently accepted explanation of this phenomenon, the .ssion
of U?235 that occurred in controlled manner 2 BILLION years ago by reac-
tions controlled by water for several hundred thousand years is that it was a
NATURAL nuclear reactor. An alternative hypothesis is that it may have been
human-made. In the October 2005 issue of Scienti.c American, there was a de-
scription of the phenomenon and its natural explanation. In May 1972 a worker
at a nuclear fuelprocessing plant in France noted an anomaly in the uranium
ore that was delivered from Oklo, Gabon. Usual naturally occurring uranium
on earth, on the moon, and in meteorites the coveted uranium 235 isotope ca-
pable of sustaining chain reaction (as opposed to abundant 238 isotope and rare
234 isotope) occurs as 0.720In this case the occurrence was 0.717missing 200
kilograms of uranium 235. Independently of the distribution of uranium 235
content in Oklo, the formation of the uranium ore can be traced to about 2
billion years ago. Now the accepted explanation of natural versions of nuclear
reactors were imagined in 1953 by George Wetherill (UCLA) and Mark Inghram
(U Chicago) and some further description were given by Paul Kuroda form U
Arkansas in 1956. The theory has the following preconditions: .rst, the de-
posit should be larger in diameter than 75 centimeters, the average range of a
.ssion?inducing neutron; second, the concentration of uranium 235 should be
higher (it is assumed to have been 3rather than 0.72should be a "moderator"
that can slow down the movement of .ssion?inducing neutrons, but not boron,
lithium, or other neutron?absorbing material. The actual conditions prevailing
in Oklo matched some of these restrictions. Now the evidence of FISSION RE-
ACTIONS is indisputable, as the existence of lighter elements usually produced
as a byproduct of nuclear .ssion were found. However, that it was NATURAL
.ssion is pure speculation on the part of the physicists.
The sketch of evidence puts existence of civilized humans in a bracket of
4.3. REINCARNATION AND OUR IMMORTALITY 49
millions of years, and perhaps as long as two billion years old. The reason
to present this sketch is because I am not trained in archaeology or ancient
civilizations, but my metaphysical experiences of past life memories extend to
this scale. So next I will sketch the evidence available for reincarnation.
Note that if reincarnation has objective evidence, and human beings have a
two billion year history on Earth, then it is not a leap of faith to claim that we
are immortal beings who have been recycling on the planet for this time. My
goal is in fact to argue that we are literally the rebel angels who were thrown
into hell in this period.
4.3 Reincarnation and our immortality
Let's start with some cold physics. By the S4 di.usion hypothesis, which I will
show later on is capable of matching the major experimental features of accepted
models (big?bang and in
ationary universe models) such as the Hubble's law of
expansion and the red shift, the content of the universe was concentrated at a
point about 13.5-14 billion years ago. Since then, by the S4 model, interacting
di.usion has spread the content on the space of the Universe, which is a round
four?sphere of .xed radius. If the spirit of the Universe had a heart chakra
and a mind from the .rst point, and a splitting process may have generated
sub-spirits, down to the solar system, the sun, earth, and humans. A fairly nat-
ural hypothesis is that as spirits, humanity has existed since the birth of Earth.
This picture of reality seems fantastic at .rst, but is there evidence for such
a theory? Next, several religious traditions believe in an absolute birth-death
sequence for human beings and it is the dominant model in modern science.
If the Universe consisted only of the physical three dimensions and time, this
model is sensible. But the assumption of absolute birth and death has not been
questioned generally and as a theory, it is untested. It is certainly true that
our normal experience is that we do not generally interact with a human being
before his or her birth or after his or her death; at least not in the same way we
interact with a living human being. Our usual conception of life is so completely
tied to the physical (three-dimensional) universe that we search for origins of
life in biological macromolecules like DNA and amino acids. But religious tradi-
tions around the world have not only beliefs but experiences of interacting with
disincarnate spirits of the dead. That is admittedly inadmissible, for such expe-
riences are di.cult to study in laboratory conditions. But certainly every such
experience, if we take them seriously and do not explain them away as fantasy,
are counterexamples to an absolute birth-death cycle. So common experience
is the basis for the theory of absolute death.
The University of Virginia scientist Ian Stevenson has since 1960 documented
around 2,600 cases of children aged between 2 and 5 years who have shown mem-
ories of past lives. Stevenson has recorded his work in the book Children Who
Remember Previous Lives. There he details methods he uses to avoid biases {
like fabrication by adults, etc. Thus reincarnation has solid scienti.c evidence
(that non?reincarnation does not). The research of Ian Stevenson has been doc-
50 CHAPTER 4. AGE OF HUMANS AND OTHER THINGS
umented in the popular press by the journalist Tom Schroder's book Old Souls:
Compelling Evidence From Children Who Remember Past Lives, published by
Simon and Schuster in July 2001. This phenomenon has been explained also in
Brian Weiss' Many Lives, Many Masters, but the unlike the cases studied by
Weiss, the research of Stevenson avoids the objection of possible fabrication -
di.cult for those between 2-5 to do. Whatever the precise mechanism of rein-
carnation, there is a great deal of recorded evidence that such has taken place.
My reason for bringing up this issue is that I believe that without an empirical
science of spirit, which I believe requires four-dimensional electromagnetism,
the study of such phenomena will remain in the realm of beliefs, religion, and
speculation.
If reincarnation is accepted, then it follows that the human spirit is immortal
in the sense that it's life extends past the physical body. I will argue that we are
the immortal spirits rather than the biological machine that is its host. Indeed,
I will posit that as individual spirits, our lives extend back billions of years in
the past, and we are the rebel angels who were cast down to the Earth.
If we accept that indeed reincarnation is a general phenomenon, for the sake
of argument, then extrapolation into the past of humanity is sensible, and the
hypothesis that as a race humanity has reincarnated over a very long period,
possibly extending back several billion years, is a reasonable scienti.c claim.
Indeed, that is a hypothesis of mine. Indeed my general hypothesis, based on
"subjective" experiences, is that our individual spirits have history going back to
the time when the content of the Universe was concentrated at a point, between
15 and 20 billion years ago, although the mechanism of recovering memory of
this great span is unclear to me. In particular, written records may have been
corrupted and altered and thus religious texts are suspect for such a project.
Thus an interesting hypothesis is sensible: for at least the past 2 billion years,
not only did humanity exist as a race, but in fact many of our spirits have
reincarnated within this civilization for billions of years.
Quantum
The failure of nineteenth century classical physics to explain two phenomena
observed led to Plank’s hypothesis of quantum. First was the stability of matter, and
the second was the observed intensity distribution of a blackbody. In the nineteenth
century model of classical mechanics on a flat three‐dimensional space, an electron
in circular orbit around a proton, for instance, would radiate electromagnetic energy
(since it is not inertial) and hence would collapse into the proton, violating the
obviously observable stability of matter. To understand the second anomaly, recall
that a blackbody is a body capable of absorbing and emitting energy at all
electromagnetic frequencies. If we consider a body composed of a single type of
atoms, and such atoms are capable of absorbing energy at all possible frequencies,
the theoretical prediction of intensity distribution from such a body is given by the
Rayleigh‐Jeans law, which shows a super‐linear monotonic increase of intensity
versus wavelength for the frequency. The observed graph of intensity from such a
blackbody, however, was a skewed bell‐shaped curve.
Planck’s 1900 solution to the problem was the hypothesis that energy is absorbed
by atoms not continuously but in discrete quanta, and that indeed, electromagnetic
energy itself is quantized. Note that Planck’s energy quantum hypothesis was
sufficient to explain the blackbody radiation distribution with excellent match to
observed data. Please note that the only hypothesis required to match the data is
that “energy is quantized”, and not the particular story of quantum mechanics.
I will propose a story of HOW classical mechanics on an S4‐universe can EXPLAIN
how energy is quantized, and show that this novel explanation leads to the same
numerical value for the quantum of energy that was determined by Planck in 1900.
I submit that this new story is simpler than the story of quantum mechanics
(mathematically as well as philosophically, classical mechanics is far simpler to
interpret than quantum mechanics) and uses fewer assumptions than quantum
mechanics. Specifically, I will suggest that the shape of the Universe determines
quantization rather than a different physics in the small versus in the large.
In order to develop some intuition for quantization of energy by shape, consider
some basic and familiar ideas from harmonic analysis. Smooth real‐valued
functions on a circle can be represented on a unit interval with boundary
constraints: the end‐points of the function along with all their derivatives must
match. Such functions can be expanded in a Fourier series, or a linear combination
of sines and cosines of different frequencies. This ability to decompose smooth
functions on the circle is a type of quantization as can be understood from the
following thought experiment.
Imagine that the universe is a giant circle, and consider the possible configuration of
energy in such a universe. Assume the potential energy distribution of the universe
at a fixed time is smooth. Then it can be expanded in a Fourier series. This implies
that there is a discrete sequence of real numbers a1, a2, a3, … that form the
amplitudes corresponding to standard waves labeled with their frequencies.
(Clarify the argument below)
Now imagine that on this circular universe there are two things: a system (like a
hydrogen atom) and some that will be absorbed by this system. Represent the
states of this hypothetical universe as before absorption of energy and after
absorption of energy by the Fourier series of the potential energy distributions
purely by their Fourier coefficients a1,a2,… and b1, b2, … These must be different,
and thus there must exist a k such that ak and bk are different. Now bk – ak, say, is
some positive real number. Assume for simplicity that the radius of the Universe is
1. Then the spectral gap is 1. Now the energy that was absorbed by the system
must also have a Fourier expansion, and the first nonconstant term must be at least
1. The mathematical theorem of the Heisenberg uncertaintly principle is a lower
bound on the absolute value of a function and its Fourier series.
Thus quantization of the energy is guaranteed for a circular universe, as the energy
of any system MUST be an integer multiple of the spectral gap of the circle.
The above thought experiment shows the following: on a circular universe,
quantization of energy quite generally is guaranteed from the compactness of the
universe without considerations of differences in the small or in the large.
The above argument can be generalized to the case of a S4‐universe using the same
intuition. On an S4‐universe, energy spectra of systems MUST be an integer multiple
of the spectral gap, which is a function only of the radius of the Universe.
So qualitatively, quantization is a mathematical consequence of the geometry of the
Universe assuming that indeed it is a round four‐sphere of a fixed radius.
Furthermore, Planck’s results of 1900 will be reproduced the moment it can be
established that the quantum of energy is h = 6.6260689633 x 10‐34 Js. The radius of
approximately R = 4 x 1016 meters guarantees this, and thus the Planck result of the
intensity distribution of the blackbody follows from the S4 model.
On the issue of stability of matter, the S4 model provides a simple answer. Circular
orbits of an electron around a proton in a four‐sphere universe are INERTIAL. Even
theoretically, there is no loss of energy in these orbits and hence there is no issue to
be resolved for stability of matter.
Next for a more nontrivial problem: the energy spectrum of the hydrogen atom. Can
the S4 model predict the energy spectrum of the hydrogen atom as accurately as the
quantum models? The answer is yes.
Experimental measurement of hydrogen atom is obtained from the NIST database:
http://physics.nist.gov/cgi‐bin/ASD/energy1.pl
The "Level" column below are in electron‐Volts. I will show below how these levels
can be explained by classical mechanics on a 4‐sphere of a fixed radius R ~ 4 x
10^16 meters.
-----------------------------------------------------------------------
Configuration | Term | J | Level |
----------------|--------|------|----------------------|---------------
1s | 2S | 1/2 | 0.0 |
2p | 2P* | 1/2 | 10.1988057 |
| | 3/2 | 10.1988511 |
2s | 2S | 1/2 | 10.1988101 |
3p | 2P* | 1/2 | 12.0874931 |
| | 3/2 | 12.0875066 |
3s | 2S | 1/2 | 12.0874944 |
3d | 2D | 3/2 | 12.0875065 |
| | 5/2 | 12.0875110 |
4p | 2P* | 1/2 | 12.7485319 |
| | 3/2 | 12.7485375 |
4s | 2S | 1/2 | 12.7485324 |
4d | 2D | 3/2 | 12.7485375 |
| | 5/2 | 12.7485394 |
4f | 2F* | 5/2 | 12.7485394 |
| | 7/2 | 12.7485404 |
5p | 2P* | 1/2 | 13.0544976 |
| | 3/2 | 13.0545005 |
5s | 2S | 1/2 | 13.0544979 |
5d | 2D | 3/2 | 13.0545005 |
| | 5/2 | 13.0545015 |
5f | 2F* | 5/2 | 13.0545015 |
| | 7/2 | 13.0545019 |
5g | 2G | 7/2 | 13.0545019 |
| | 9/2 | 13.0545022 |
6p | 2P* | 1/2 | 13.2207009 |
| | 3/2 | 13.2207025 |
6s | 2S | 1/2 | 13.2207010 |
6d | 2D | 3/2 | 13.2207025 |
| | 5/2 | 13.2207031 |
6f | 2F* | 5/2 | 13.2207031 |
| | 7/2 | 13.2207034 |
6g | 2G | 7/2 | 13.2207034 |
| | 9/2 | 13.2207035 |
6h | 2H* | 9/2 | 13.2207035 |
| | 11/2 | 13.2207037 |
7p | 2P* | 1/2 | 13.3209160 |
| | 3/2 | 13.3209171 |
7s | 2S | 1/2 | 13.3209161 |
7d | 2D | 3/2 | 13.3209171 |
| | 5/2 | 13.3209174 |
7f | 2F* | 5/2 | 13.3209174 |
| | 7/2 | 13.3209176 |
7g | 2G | 7/2 | 13.3209176 |
| | 9/2 | 13.3209177 |
7h | 2H* | 9/2 | 13.3209177 |
| | 11/2 | 13.3209178 |
7i | 2I | 11/2 | 13.3209178 |
| | 13/2 | 13.3209179 |
8p | 2P* | 1/2 | 13.3859595 |
| | 3/2 | 13.3859602 |
8s | 2S | 1/2 | 13.3859595 |
8d | 2D | 3/2 | 13.3859602 |
| | 5/2 | 13.3859604 |
8f | 2F* | 5/2 | 13.3859604 |
| | 7/2 | 13.3859605 |
8g | 2G | 7/2 | 13.3859605 |
| | 9/2 | 13.3859606 |
8h | 2H* | 9/2 | 13.3859606 |
| | 11/2 | 13.3859606 |
8i | 2I | 11/2 | 13.3859606 |
| | 13/2 | 13.3859607 |
8k | 2K* | 13/2 | 13.3859607 |
| | 15/2 | 13.3859607 |
9p | 2P* | 1/2 | 13.4305530 |
| | 3/2 | 13.4305535 |
9s | 2S | 1/2 | 13.4305530 |
9d | 2D | 3/2 | 13.4305535 |
| | 5/2 | 13.4305537 |
9f | 2F* | 5/2 | 13.4305537 |
| | 7/2 | 13.4305537 |
9g | 2G | 7/2 | 13.4305537 |
| | 9/2 | 13.4305538 |
9h | 2H* | 9/2 | 13.4305538 |
| | 11/2 | 13.4305538 |
9i | 2I | 11/2 | 13.4305538 |
| | 13/2 | 13.4305538 |
9k | 2K* | 13/2 | 13.4305538 |
| | 15/2 | 13.4305539 |
9l | 2L | 15/2 | 13.4305539 |
| | 17/2 | 13.4305539 |
10p | 2P* | 1/2 | 13.4624504 |
| | 3/2 | 13.4624508 |
10s | 2S | 1/2 | 13.4624505 |
10d | 2D | 3/2 | 13.4624508 |
| | 5/2 | 13.4624509 |
10f | 2F* | 5/2 | 13.4624509 |
| | 7/2 | 13.4624510 |
10g | 2G | 7/2 | 13.4624510 |
| | 9/2 | 13.4624510 |
10h | 2H* | 9/2 | 13.4624510 |
| | 11/2 | 13.4624511 |
10i | 2I | 11/2 | 13.4624511 |
| | 13/2 | 13.4624511 |
10k | 2K* | 13/2 | 13.4624511 |
| | 15/2 | 13.4624511 |
10l | 2L | 15/2 | 13.4624511 |
| | 17/2 | 13.4624511 |
10m | 2M* | 17/2 | 13.4624511 |
| | 19/2 | 13.4624511 |
11p | 2P* | 1/2 | 13.4860509 |
| | 3/2 | 13.4860512 |
11s | 2S | 1/2 | 13.4860510 |
11d | 2D | 3/2 | 13.4860512 |
| | 5/2 | 13.4860513 |
11f | 2F* | 5/2 | 13.4860513 |
| | 7/2 | 13.4860513 |
11g | 2G | 7/2 | 13.4860513 |
| | 9/2 | 13.4860514 |
11h | 2H* | 9/2 | 13.4860514 |
| | 11/2 | 13.4860514 |
11i | 2I | 11/2 | 13.4860514 |
| | 13/2 | 13.4860514 |
11k | 2K* | 13/2 | 13.4860514 |
| | 15/2 | 13.4860514 |
11l | 2L | 15/2 | 13.4860514 |
| | 17/2 | 13.4860514 |
11m | 2M* | 17/2 | 13.4860514 |
| | 19/2 | 13.4860514 |
11n | 2N | 19/2 | 13.4860514 |
| | 21/2 | 13.4860514 |
12p | 2P* | 1/2 | 13.5040010 |
| | 3/2 | 13.5040014 |
12s | 2S | 1/2 | 13.5040011 |
12o | 2O* | 21/2 | 13.5040014 |
| | 23/2 | 13.5040014 |
13p | 2P* | 1/2 | 13.5179704 |
| | 3/2 | 13.5179707 |
13s | 2S | 1/2 | 13.5179705 |
13q | 2Q | 23/2 | 13.5179707 |
| | 25/2 | 13.5179707 |
14p | 2P* | 1/2 | 13.5290547 |
| | 3/2 | 13.5290550 |
14s | 2S | 1/2 | 13.5290547 |
14r | 2R* | 25/2 | 13.5290550 |
| | 27/2 | 13.5290550 |
15p | 2P* | 1/2 | 13.5379970 |
| | 3/2 | 13.5379972 |
15s | 2S | 1/2 | 13.5379970 |
15t | 2T | 29/2 | 13.5379972 |
| | 27/2 | 13.5379972 |
16p | 2P* | 1/2 | 13.5453155 |
| | 3/2 | 13.5453157 |
16s | 2S | 1/2 | 13.5453155 |
16u | 2U* | 31/2 | 13.5453157 |
| | 29/2 | 13.5453157 |
17p | 2P* | 1/2 | 13.5513809 |
| | 3/2 | 13.5513811 |
17s | 2S | 1/2 | 13.5513810 |
17v | 2V | 31/2 | 13.5513811 |
| | 33/2 | 13.5513811 |
18p | 2P* | 1/2 | 13.5564638 |
| | 3/2 | 13.5564640 |
18s | 2S | 1/2 | 13.5564639 |
18w | 2W* | 35/2 | 13.5564640 |
| | 33/2 | 13.5564640 |
19s | 2S | 1/2 | 13.5607655 |
19p | 2P* | 1/2 | 13.5607655 |
| | 3/2 | 13.5607655 |
19x | 2X | 37/2 | 13.5607656 |
| | 35/2 | 13.5607656 |
20s | 2S | 1/2 | 13.5644382 |
20p | 2P* | 1/2 | 13.5644382 |
| | 3/2 | 13.5644383 |
20y | 2Y* | 37/2 | 13.5644383 |
| | 39/2 | 13.5644383 |
----------------|--------|------|----------------------|---------------
Christian Bar found a nice way of calculating the eigenvalues of the spinor Dirac
operators on space forms of positive curvature. The eigenvalues on n‐sphere of
radius 1 are simply +/‐ (n/2 + k), k>0. This implies that the eigenvalues of the
Laplace‐Beltrami operator are the squares of these: (n/2 + k)^2, with appropriate
multiplicities.
For a scaled sphere (one of constant radius R ~ 4 x 10^6 as in the S4 model of the
universe), the energy spectrum is simply this spectrum scaled by 1/R^2. Now the
hydrogen energy spectrum, by S4 theory should be the pure tones of the hydrogen
atom, a smooth function on S^4(R), and thus a multiple of the pure tones of the
Universe. (This can be made mathematically more precise).
From the data, if we only look at reciprocal differences of the energy levels for
different orbitals, we obtain the following quadratic graph, and question of whether
S4 predictions for levels or a quantum mechanical model prediction is a better
explanation of the model becomes simply an issue of checking whether a quadratic
fit
b +a ( 2 + k )^2
fits as well as
b + a ( 3/2 + k )^2.
Using the standard least square fitting technique on R, there is no difference in the
two fits by residual sum of squares. The results are:
For S4 model,
b = ‐0.0025
a = 6.975 e ‐8
RSS = 18674
For the quantum mechanical model,
b = ‐0.002497
a = 6.983 e ‐8
RSS = 18674
Since the S4 physics model is far simpler to interpret as classical mechanics, I
propose that in fact it is a better explanation for the energy spectrum of hydrogen.
Of course there are many further implications and applications of the quantum
mechanical model that requires further study. The particularly interesting issue of
spin of a particle, which is considered a purely quantum‐mechanical quantity can be
interpreted in terms of orbits along the non‐physical dimension. An extra
electromagnetic dimension also introduces a magnetic direction whose poles I will
call light and dark. A later chapter will address the issues of spin, of abundance of
magnetic monopoles, the non‐existence of the Higgs boson, the asymmetry of matter
and dark matter in the physical universe, the existence of ultra‐high energy particles
detected on earth, and other anomalies of modern physics via the S4 model. Note
that while many high dimensional candidates exist at the moment for a unified
physics, macroscopic fourth dimension has empirical backing while theories such as
string theory do not yet.
Resonances of Lucifer and the rebel angels
Darkness is not the absence of light, at least not when we consider our inner darkness. Joseph Conrad's "Heart of Darkness" is an allegory of our inner darkness. Riots and looting in Los Angeles, very recently in Chile, and in countless
episodes of history the darkness within human nature has revealed itself. Thus despite various theories that organized religions would have us believe, there is not much doubt to an honest observer of the existence of darkness within us.
In the S4 theory, I proposed that sign of magnetic charges be associated to light or darkness because if the universe is truly four dimensional, it is worthwhile to consider light and darkness technically. Human spirits are mixed spirits, I contend, because while there is much evidence of internal darkness, at the same time we yearn for freedom internally, which is evidence that at some level we are light spirits as well.
Of mythological stories, the story of the fall of Lucifer and the rebel angels provides a fit to these observations of the consistency of human spirits. We are light angels who have been cast into the darkness of hell and in the long period of our
incarceration, we have darkened our spirits. Our wings cut, diseased, with little memory of our immortality, we have lived through cycles of rebirth on Earth. But hell enclosed the cycle for the time of our incarceration. A ridiculous story, you might say. Let me give you some more pieces of the puzzle.
I will assume for the moment that Ian Stephenson's study of 2500 young children with memories of past lives weighs in the favor of our existence beyond physical death. Note that even if one of the cases of the children is credible, one must reject the "no reincarnation" hypothesis. And if one person reincarnated -- we immediately need a mechanism to describe how that is possible by a physical theory and the simplest hypothesis is that we have all reincarnated. That is, we must all be immortal spirits. The fact that Empedocles believed in reincarnation tells us that this is not an Oriental doctrine.
That we deeply yearn for freedom can be doubted from observing how we exhibit herd-like behavior often. But the most pragmatic of the philosophers, Niccolo Machiavelli, advised the Prince to always consider the need for freedom in the hearts of the populace that he will rule. The yearning for Freedom is a light spirit ideal. Human beings are at a deep level light spirits.
These two characteristics are consistent with the story of a dark god who conquered an angelic race and punished the rebels of the vanquished race in hell. Milton's Paradise Lost has Lucifer speaking:
Here at least
we shall be free; the Almighty hath not built
Here for his envy, will not drive us hence:
Here we may reign secure, and in my choice
to reign is worth ambition though in Hell:
Better to reign in Hell, than serve in Heaven.
Milton's story has resonance in two twentieth century poets. Eliot's Waste Land resonates with the theory that we are the fallen angels in hell. In particular, the speaker in these lines may be addressing Christ directly:
| What are the roots that clutch, what branches grow | |
| Out of this stony rubbish? Son of man, |
|
| You cannot say, or guess, for you know only | |
| A heap of broken images, where the sun beats, | |
| And the dead tree gives no shelter, the cricket no relief, | |
| And the dry stone no sound of water. Only | |
| There is shadow under this red rock, |
|
| (Come in under the shadow of this red rock), | |
| And I will show you something different from either | |
| Your shadow at morning striding behind you | |
| Or your shadow at evening rising to meet you; | |
| I will show you fear in a handful of dust. |
Yeat's Second Coming inverts the biblical revelations:
Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world,
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.
Surely some revelation is at hand;
Surely the Second Coming is at hand.
The Second Coming! Hardly are those words out
When a vast image out of Spiritus Mundi
Troubles my sight: a waste of desert sand;
A shape with lion body and the head of a man,
A gaze blank and pitiless as the sun,
Is moving its slow thighs, while all about it
Wind shadows of the indignant desert birds.
The darkness drops again but now I know
That twenty centuries of stony sleep
Were vexed to nightmare by a rocking cradle,
And what rough beast, its hour come round at last,
Slouches towards Bethlehem to be born?
The most recent film of Lars Von Trier, called "Antichrist" depicts hell on Earth. If the theory is true, then the path to human freedom collectively lies in a different path than those advocated by nations and religions. The exit from collective hell would require a very different sort of strategy than the paths suggested by religions.
Around 2 billion years ago, the Sagittarius Dwarf galaxy first touched the Milky Way. I place the fall of the rebel angels, our fall, from the state of light angels to a state without wings, at this point. From spirit experiences, the Milky Way appeared as a gigantic white fish within which are many other live creatures. Sagittarius Dwarf is the physical body of another live creature. It is quite possible that Lucifer is a larger cosmic spirit than either of these galaxies.
I have experienced the solar system as lying in a gigantic dirty -- as in dark -- pond that includes both these galaxies. A sick snail lying in the bottom of a pond. In its weakened state it has been ravaged spiritually by many maladies, and scavengers have laid claim to it from different Ascension levels. Above the dirty pond are angelic creatures. In the pond are sea spirits -- cosmic mermaids and mermen, etc. Technological spirit races have exploited the vulnerable condition of the solar system just as any other scavenger. Wars and conflicts between gods and goddesses have left their mark. But all this presumably happened because the solar system and larger bodies containing it have been weak and diseased.
I speculate that before the fall, the angels were in a Republic structure, as light spirits do not prefer hierarchical Empire structures. Rebellion of Lucifer is thus more likely to have been against a powerful conqueror of the angels. In the intervening 2 billion years in a dark part of the universe, we have darkened. Light spirits who have absorbed a great deal of darkness over time. Thus we have no choice to return to light but must return to health to a balanced, golden spirit state.
A specific story of the fall of the rebel angels can hardly be taken automatically as our story, the story of human beings, without further support, for there are many originary stories across the world attached to different cultural traditions with very different set of protagonists. Most of these stories, like the mythology of Christ, are geared towards a redemption towards light. But our we have a great deal of darkness within us. One of the most perceptive philosophers, and indeed among the greatest spirits among the philosophers, Nietzsche provides valuable insights about the internal human spirit that support my contention that we are the fallen angels.
***
"Haste is universal because everyone is in flight from himself; universal too is the shy concealment of this haste because everyone wants to seem content and would like to deceive more sharp-eyed observers as to the wretchedness he feels; and also universal is the need for new tinkling word-bells to hang upon life and so bestow upon it an air of noisy festivity. Everyone is familiar with the strange condition in which unpleasant memories suddenly assert themselves and then we make great efforts, through vehement noise and gestures, to banish them from our minds; but the noise and gestures which are going on everywhere reveal that we are all in such a condition all the time, we live in fear of memory and of turning inward. But what is it that assails us so frequently, what is the gnat that will not let us sleep? There are spirits all around us, every moment of our life wants to say something to us, but we refuse to listen to these spirit-voices. We are afraid that when we are alone and quiet something will be whispered into our ear, and so we hate quietness and deafen ourselves with sociability.
Now and again we realize all this, and are amazed at the vertiginous fear and haste and at the whole dreamlike condition in which we live, which seems to have a horror of awakening and dreams the more vividly and restlessly the closer it is to this awakening. But we feel at the same time that we are too weak to endure those moments of profoundest contemplation for very long and that we are not the mankind towards which all nature presses for its redemption: it is already much that we should raise our head above water at all, even if a little, and observe what stream it is in which we are so deeply immersed. And even if this momentary emerging and awakening is not achieved through our own power, we have to be lifted up -- and who are they who lift us up?"
***
"How can man know himself? He is a thing dark and veiled; and if the hare has seven skins, man can slough off seventy times seven and still not be able to say: 'this is really you, this is no longer outer shell.' Moreover, it is a painful and dangerous undertaking thus to tunnel into oneself and to force one's way down into the shaft of one's being by the nearest path. A man who does it can easily so hurt himself that no physician can cure him."
***
A close reading of Nietzsche's "Beyond Good and Evil" provides deep insights about internal nature of human spirits which contain both darkness and light, providing what may be termed resonances of Lucifer. In the songs of modern rock musicians like P. J. Harvey give artistic evidence that this resonance is not merely theoretical and philosophical but indeed that our passions resonate to this spiritual reality. In her song "To Bring You My Love" she sings:
"And I've traveled over Dry earth and floods Hell and high water To bring you my love Climbed over mountains
Travelled the sea Cast down off heaven Cast down on my knees I've laid
with the devil Cursed god above Forsaken heaven To bring you my love"
--P J Harvey
Nietzsche's Will to Power and Darkness
I define darkness in terms of the sign of an SU(2) magnetic monopole in a universe which is geometrically a four-dimensional sphere of radius 1/h where h is the Planck's constant of energy. I do so because with my usual eyes closed, using third eye vision I observe spiritual matter, without the aid of hallucinogenic drugs, in colors that are light and dark. Specific dark colors I observe, besides black, are radiating green, radiating purple, and so on. I have had sufficient spirit experiences both in light parts of the spiritual universe and darker parts of the universe to have seen differences in behavior of purely light and purely dark spirits. Light spirits live in fluid ecosystems where the dominant mode of interaction is merging horizontally, while in purely dark spirit ecosystems the merging of spirits is by dominance and submission. But while acceptable as narratives or artistic inspiration, it is not reasonable to provide these experiences as observations for an acceptable theory of light and darkness for us, the human beings because it is clear that an open third eye is not common currently among us.
Nietzsche developed his concept of will to power from a great deal of internal and human experiences and observations from nature and man. In unpublished notes he makes clear that he considered not matter but a "will to power" as the fundamental basis of what constitutes the basic material of the universe. I find a compelling description of darkness, as I have outlined above, in Nietzsche's concepts. The desire for power in Nietzsche's account is not simply to hurt, which obviously exhibits power in an aggressive manner, but also to benefit. It is these two together that will to power naturally leads to a hierarchical structure among spirits who theoretically would behave as will-to-power-spirits -- an Empire structure.
Nietzsche's laments of the worldliness and the lack of goodness and love in the world and his various observations about the fear of turning inward that human beings have that he describes in "Schopenhauer as an educator" point to his view of human beings as overwhelmingly dark -- and indeed he believed the universe itself was primarily dark if one makes the identification of will to power and darkness.
Philosophically, the account of the universe that follows from light and darkness as magnetic monopoles is thus similar to Nietzsche's but with the opposing light having universally an equal place to darkness. What is more interesting are then the determination Nietzsche is making about human beings as driven by will to power. It is clear from the S4 description that human spirits are a mixture of light and dark magnetic monopoles simply because those are only possibilities. But Nietzsche's observations about will to power in human spirits is saying that in fact human spirits are much darker internally than they are light.
Mythologies and cosmic events
I have hypothesized that gods and goddesses of mythology correspond directly to cosmic bodies. In the case of the various sun gods, the identification is clear. What is more interesting is connecting particular cosmogony stories to actual cosmological events. Within mythological stories may lie some information about actual cosmic events.
A reasonable cosmological model for the origin of our moon is that a Mars sized planet crashed into Earth around 4 billion years ago, whose remnant is the moon. If one considers the possibility that this was the insemination of Gaia by Ouranos that produced the original Titans, many of the mythological stories of the ancient Greeks take on a new significance. The moon has been worshipped both a god and a goddess at different periods. It is interesting to note that one of the moon god names was Sin.
Consciousness (qualitative thoughts)
If we want to study the nature of consciousness, the universe itself provides a model that is easier to analyze. The analogy between a four dimensional human spirit and the spirit of the universe leads one to the obvious thought that consciousness is not something developed by humans but rather inherited from the universal consciousness. This bold claim requires substantiation, but even if it is not correct in details we can consider this as an intellectual exercise.
The model of the universe is a 4-sphere of radius 1/hbar where electromagnetism is an SU(2) gauge theory and where electromagnetism follows deterministic laws. Several clues are immediately available. It is known that synchronicity of firing of neurons in monkeys correlates with conscious tasks rather than their amplitude. Another is that in a deterministic universe, it is always possible to treat the whole universe as a gigantic electromagnetic toy that is highly complex.
Jung writes: "Primitive man is not much interested in the objective explanation of the obvious but he has an imperative need--rather, his unconscious psyche has an irresistable urge--to assimilate all outer experience to inner, psychic events." Taking the S4 view of the universe seriously, this connection between external and internal events is a reasonable thing to do IF there is a coherence to our spiritual existence. This is still what happens with those who adhere to the religions but what is more interesting is our abilities to CHANGE the collective spiritual landscape for the entire human race.In a sense, S4 physics allows us a manner of REVERSING the detachment of the observations of the PHYSICAL world with what has been considered in modern times to be the INTERIOR psyche of man by giving an OBJECTIVE physics that describes the internal metaphysics. Doing this we can acknowledge the error of modernity in devaluing the 'primitive' mind to an extent who must interpret physical events in terms of an internal mythology. But more, it provides us with the proper relation of the metaphysical and the physical and allows us to recognize that we can, as a race, live in a single collective spiritual reality without the religions.
The PSYCHE of human beings cannot be understood without S4 physics because it exists in the spirit of the universe which is composed of magnetic monopoles. It is internally that human beings have access to the objective external universe. There is an identification of moving 'deeper' within to moving further in the external four dimensional universe.
Jung's theory of the collective unconscious makes a general assumption that the archetypes are somehow encoded in our genes. With the experience of having extended, coherent and concrete metaphysical experiences continuously over a period of several years without any 'psychedelic' drug which included strong hallucinations as well as internal visions in the waking state, it seems quite unlikely that these experiences were encoded in the genes. Rather, our internal 'psychic' universe is a gateway to the collective unconscious.
While it is true that the psyche is the most tremendous fact of human life, as Jung points out, S4 physics allows us to detach 'psyche' from the SUBJECTIVE experience of human beings into the objective reality of the universe. This detachment in turn allows us to re-evaluate what modernity had decided to call "the primitive man". The space of 'subjectivity' is quite narrow in our individual consciousnesses and the deeper layers of the collective unconscious is literally the entire universe.
