The structure of Physics

Abstract

The science of Physics is in transition today. What is the problem? There are actually many of them, but the most disputed one is the problem of unification. Scientists are looking for a general theory, one that relies on a coherent formalism to encompass all known empirical facts. It is a very old aspiration of man, a quest for omniscience, an effort to approach God. In Herman Dooyeweerd's terms, this is a ground motive, a spiritual driving force that impels each thinker to interpret reality under its influence (Ref: Wikipedia).  There are other problems inspired from philosophy and from new advances in mathematics (from our new understanding of chaos and complexity, for example), but they are all subsumed to the first one, considered as the most important. In this article, I break away from this ancient tradition of assigning to science the role of "bringing man closer to God". I propose a new project, one that is inspired by practical considerations.

Specialized intelligent machines will play an important role in the near future. Their intelligence is called artificial, which means that it is not naturally acquired by them, it is rather put in them by their creators, by us. But our general approach to give "intelligence" to these machines is highly anthropocentric. That is to say, we try to implement in them the way we see reality, we program them using our science. We are all amazed by what machines can do but in fact they are very limited.

My thesis is that we can greatly improve the efficiency of machines by tailoring the way they process information to their architecture, as well as to the particular tasks that they are built for. We need to create specific science, as opposed to what we wrongfully call universal science. 

Physics as we know it  

Before we analyze the profound structure of our physical science, let us make a quick review of some historical facts. In the middle of the seventeenth century, Galileo transformed the science of Physics by making use of mathematics to represent natural phenomena. Only about fifty years later, Newton built on Galileo's work and founded the base of modern mechanics. His calculus formalism (other contributors are Leibniz, Laplace, Euler, Lagrange) remained the language through which the "laws of nature" are expressed, in all the branches of modern Physics. But at an even deeper level, since Aristotle we continue to use the same old classical bivalent logic to reason about the physical world. The figure below represents the profound underlying structure of modern Physics.

This diagram has three levels that are represented vertically: the reality (Kant's noumenal world), theories constructed on a natural language (the narrative), and formal systems (the model). At the top, we have a brown rectangle that represents the world, or the reality. Below it, we have three circles containing the inscriptions: Quantum, Newtonian, and Relativistic. These three circles represent three different theories, which in turn constitute three different types of “theoretical worlds”. We must distinguish here between the real world RW and a world described by a certain theory TW, between a real object RO, whatever that is, and an object described by a theory TO. We say that TW-s are applied to RW. 

Any scientific theory is built on a specific ontology, which proposes a certain number of TO-s to be considered as RO-s. In physics, we find three main types of ontologies. One is used to speak about quantum phenomena and microscopic entities, such as they appear to us via our instruments, or directly to our senses. The other one is used to speak about effects and entities of another type, the ones with which we are accustomed in our daily life: the gravitational attraction, the collision between two billiard balls, the flow of water, etc. And finally, the third one is used to speak about relativistic effects, very heavy objects (like stars and black holes), or objects that move at very high speed (close to the speed of light). These three ontologies have a lot in common, but they can be considered as being distinct because of some fundamental differences between them.

Thus there are three major types of theoretical worlds, or TW-s, which are easily distinguishable. We possess three theories to describe these three types of “worlds”, namely quantum mechanics, classical mechanics, and relativistic mechanics. Each type of TO belonging to one of these TW-s is unambiguously described by the theory to which it belongs, which means that their fundamental properties are well defined, the relations between different TO-s and between their respective properties are also clearly given by the theory, as well as the rules of quantification. Even if the ranges of quantum mechanics and of relativistic mechanics cover the part of RW to which the Newtonian theoretical world (nTW) is normally applied, we are always able to distinguish between objects of Newtonian type (nTO) and objects of other types (rTO or qTO). Indeed, the characteristics of the first are obtained from the characteristics of the other two, by a process called limit. Speaking about real objects, or RO-s, sometimes we are faced with the problem of choosing a proper theoretical representation for them, i.e. choosing between a rTO or a nTO for example. Physicists face this kind of problem very often, for example, when an electron possesses a speed which is neither too slow to be considered as a classical particle, nor too high to be considered relativistic. It gets even more complicated than this. For example, when studying a gas, one can consider molecules as qTO-s (material systems with discrete energy levels / states), or as nTO (something like hard little balls), or even as a hybrid (hard balls moving in a well defined trajectory, with a quantified internal energy structure). Somehow physicists can make sense of this last option, and can predict the behavior of a gas within an acceptable margin of error. Their decision, whether to consider the molecule as a qTO, or a nTO, or as a hybrid  is based essentially on pragmatic reasons. The RO under study is the actual thing, and the physicist simply decides what theoretical representation to use to represent it. What is actually important, is the fact that the physicist is always conscious about his choice, and that he always makes the distinction between a qTO, a nTO, or some hybrid. See Appendix 1 below for more details.

Going back to our diagram, the oval containing the inscription Calculus represents the formal system used in physics to deal with quantities. A qualitative formal system, a logic, or a deductive system of inference, is grafted to this quantitative system. In all three cases, the logic used is the classical Sentential Logic. Without a logic the concept of mathematical proof wouldn't be defined. In another article, I will explore the connection between these two types of formal systems, which is represented on the diagram by the red arrow. The green arrows represent the semantics of the formal system. For every element belonging to the formal system there is an element within our theory. Usually the power of representation of a formal system is smaller than the power of representation of the corresponding theory.

According to the methods introduced by Galileo more than 400 years ago and, which still persist today in modern physics, there is a TO (or a property of a TO) for every relevant phenomenological entity, which in turn is represented by a string of symbols within a coherent formal system. The symbols within the formal system are of different types: operators, operands, relations, etc. The formal properties of these symbols are given in a coherent manner within the formal system. After a detailed observation of a RO, the physicist tries to formulate an explanation about it (the narrative), using natural language, and using parts of previously accepted knowledge. Thus, he develops a theory. He then adopts a quantitative and a qualitative formal system to build a model. Not any formalism will do the job. A string of symbols (operators, operations, relations, etc.) is chosen to represent the TO-s, which represent in turn all RO-s under study. As RO-s seem to maintain some sort of relations between them, these relations must also be represented within our theory (like relations of causality, for example), and must also be transposed within the formal system by taking advantage of its internal coherence, or of its internal "mechanics". Let's say, for example, that the event Y always follows the event X according to the observation of some RO-s. X and Y are represented within our formal system by two symbols, say x and y, and x must somehow lead to y following a finite number of operations on the formal model, respecting some strict predefined rules. See Appendix 2 below for more details.

About Newton's work in physics

What was Newton's contribution to science? Initially, Newton had to construct a theory (a narrative) to understand and to explain empirical observations of the movement of celestial objects, and of the free falling of objects, down here on Earth. A theory is knowledge that can be verbalized, can be put into language and transmitted to others. Thereafter, Newton had to design a formal system (a model) that could match his theory. Galileo had already established the mathematical basis of modeling, getting inspired from ancient science, which itself relied on geometry. Formalizing a theory is adding rigor to it, is constructing a "scaffold" that supports our deductive reasoning, clarifies the meaning of concepts, and helps with quantitative predictions. Newton designated symbolic expressions (strings of symbols) within his formal system to represent the TO-s belonging to his theory (see Appendix 3 below for an example). Its classical mechanics is in essence a set of formal expressions connected to each other by a strict set of rules and, based on the internal "mechanics" of this formal system, he could simulate empirical observations. A formal system is the type of thing that you put into your computer to simulate some aspects of reality. A theory is like a virtual world that exists in people's minds and represents something in reality, helps us navigate the world, allows us to pass information to each others about the world. The formal system is what computers need to construct yet another representation of that virtual world, and to display it somehow on the screen, is what we use to make accurate predictions.

About Einstein's work in physics  

Using Riemannian geometry and calculus, Einstein identified formal expressions that could represent TO-s, which he used to speak about relativistic entities and effects. In doing this, Einstein had to satisfy same constraints. Like Newton, he also had to transpose the relations between TO-s contained within his theory into formal representations. Let us call this type of constraint material adequacy. Moreover, it had to satisfy additional constraints of another type, which we will call of unification: some of these formal expressions must be reduced syntactically to formal expressions belonging to Newton's formal system. This is actually done by a limit process, considering the speed of the particle and the intensity of the gravitational field very small, and simply erasing certain formal expressions from a more complex formal expression, based on the argument that their quantitative value becomes negligibly small. In the case of quantum mechanics and classical mechanics this constraint was conceptualized by Neils Bohr, using the term of correspondence principle. Syntactically speaking, this process represents a rupture (a process that goes outside of the formal system, following a rule that is external to it), but it is justified in terms of relative contribution to the final quantity. It changes the form of the symbolic expressions in such a way that by manipulating them further, respecting the rules of operations, we can obtain precisely the formal expressions used in classical mechanics to represent corresponding TO-s. In other words, we believe that there is a RO out there, that we can represent by a nTO or by a rTO, depending on certain conditions. The nTO is represented by a formal expression within the formal system of classical mechanics, and the rTO is represented by another formal expression within the formal system of relativity. By applying the limit process, one can obtain the formal expression corresponding to the nTO from the one representing the rTO, and vice versa. We can say that the nTO is the limit of rTO for small values of speed and gravitational field. This connection is the formal basis of the constraint of unification.

The work of the fathers of quantum mechanics is very similar to Einstein's work. They also had to respect these two types of constraints. There is also a limit process that reduces formal expressions representing qTO-s to expressions representing correspondent nTO-s. The problem is that there are no formal processes that reduce formal expressions representing nTO-s to expressions representing their corresponding qTO-s, or vice versa. This is another constraint of unification that operates at a higher level of generality, which physicists are still struggling to satisfy. 

What does the underlying structure of modern physics allow us in terms of representations of the world? 

The formalism at the base of modern physics enables us to represent in a symbolic manner (by using a string of symbols) TO-s (theoretical events and objects, and their properties), which in turn represent phenomenological entities, or RO-s. Natural language finds itself between the experiment/phenomena and the formal representation (see the first figure). The various types of relations between RO-s (relations which are behind our perception of stability and continuity and help us make predictions) can also be transposed within the natural language in the form of causality, or another type of relation, and further within a formal system by direct syntactic connections, or by following some rules of inference/deduction. In other words, we use strings of symbols to represent TO-s, and the internal "mechanics" of the formal system to represent causality between these TO-s, or other types of relations, which hold between them.

The material adequacy of a formal representation is ensured by a rather lose isomorphism between a pattern perceived by us by experiencing the reality, and the structure of the formal system. TO-s, which are interpreted within our theories, maintain some type of relations between them, and these relations must be transposed within a formal system. By exploiting the internal "mechanics" of the formal system, we can reproduce the properties of TO-s, and emulate, to some degree, the pattern perceived by us, by experiencing the reality. (See Appendix 4 below for more details) This constitutes a set of constraints that must be respected, which means that we cannot choose any formal expression to represent a given theoretical entity. The choice is limited by the internal coherence of the formal system (its internal "mechanics"), and this limitation becomes more severe if the number of TO-s to be represented is high, and if the relations between them are complex.

The formalism must satisfy other more fundamental constraints that we did not consider yet. It allows us to represent differences, similarities, order, symmetries, etc. Let's call them instrumental constraints. The expressions which have the same form cannot represent different TO-s. Thus, within the context of Newton's theory, the expressions which represent the gravitational interaction between two bodies, or between three bodies don't have the same form, but they present some similarities. Also, the expressions which represent the gravitational interaction according to the theory of relativity, quantum mechanics, and classical mechanics don't have the same form either. In the last example, the focus is on the same RO, however, the theories which we use to interpret this RO are different. These formal expressions do not represent the same TO, but a nTO, a rTO, and a qTO, which are different from one another. For example, the concept of time does not behave in the same manner in classical mechanics and in relativity.

It is important to realize at this point that we use the same type of formal system for all three major theories discussed above, namely analytic geometry, calculus, and sentential logic. In other words, the same RO is represented by three different TO-s within three different theories (quantum, classical, relativistic), and these three types of TO-s are represented symbolically in three different ways, using the same type of formal system.

Formal constraints, theory, and empirical domain

Newton, Einstein, and the fathers of quantum mechanics used the same deductive system (bivalent sentential logic), and very similar quantitative formal systems. This is the reason why I connected on the diagram with the green arrows Calculus with the three theories. Einstein's equations contain tensors and operations on tensors, but these mathematical entities and these operations can be reduced to algebra and calculus. The same remark holds for quantum mechanics: the Hamiltonian formalism is only a very compact way to write complex expressions, which are ultimately based on algebra and calculus.

Let's stop for a moment and make the following observation: The formal expression in classical mechanics that represents the gravitational interaction between two bodies is syntactically simpler than the equivalent expressions in quantum mechanics and relativity. Why is that?

First, the complexity of the formal expressions obtained increases with the number of constraints. In all three cases, we use the same type of formal system. Thus in order to represent in a symbolic manner a rTO, a nTO, and a qTO we must find a way to respect not only material constraints, but also instrumental constraints. The formal expressions representing these three TO-s must be different from one another, otherwise we cannot distinguish them. Moreover, we must also respect the unification constraint, to connect all these three theories somehow. We saw that this is done by the limit process.

Second, we must realize that the formal arsenal used in modern physics was developed for classical mechanics, or to deal with entities at the Newtonian scale. Therefore, in some sense, it must be adequate to Newton's theory. In other words, Newtonian theoretical entities can be represented within this formalism by strings of symbols that appear to be syntactically simple, because there must be some sort of compatibility between the theory and the formalism. After Newton, we automatically adopted the same type of formalism to represent other types of theoretical entities, not being aware of other alternatives. Nothing guarantees that this particular formal system first adopted by Newton is also adequate for other theories, which are applied to different empirical domains.

Going back to our remark, gravity doesn't have the same meaning within these three theories, and must be represented differently within this same type of formal system, so that we can distinguish between these different interpretations. The three formal expressions which represent the gravitational interaction according to these three theories are, in fact, all different. It is also true that the formal expression in classical mechanics is syntactically simpler than its correspondents in the other two models. Thus, we can say that the formalism of modern physics is more adequate to treat Newtonian gravitation, and less adequate to treat other theoretical interpretations of gravitation. We can do nothing to reduce the complexity of the others two expressions if we decide to stay within the framework of the same formal system. This situation is imposed by the internal structure of the formalism which was adopted, and by the type of constraints that we must satisfy. If someone aspires to reduce the complexity of the formal expressions representing theoretical entities of relativistic mechanics, he must consider adopting a different type of formal system.

Here is one of the most important limitations of the fundamental structure of modern physics. Since it allows the use of only one type of formal system (the one adopted by Newton), and only one type of logic (classical bivalent logic), we cannot reduce the complexity of formal expressions representing TO-s from other physical theories, like quantum mechanics and relativity. The dream of physicists today is to find a unifying theory that accounts for all observed phenomena. If we carry out this project by using the same type of formal system, we must satisfy additional constraints, and the result will be an even more complex formalism. However, if we look at science from a pragmatic point of view, if we consider the technological applications of scientific knowledge, we understand that there is a growing interest for specialized models, that are to be applied on very restrained empirical domains, and, which could represent TO-s in a very simple form. Why engineers at Ford aren't using quantum mechanics to manufacture their car engines? They don't need this theory because their margin of error is higher than the precision offered by the Newtonian models. They continue to use classical mechanics for its simplicity. Do you think engineers at IBM working on optical and quantum devices would be happy to have at their disposal a theory tailored for their empirical context, for their specific needs, one, which would offer a formalism that represents relevant TO-s in a simple form? I bet they would be thrilled. Can physicists offer them these theoretical tools? Certainly! (See more on this here: L'Approche Scientifique) However, in order to do this physicists must adopt a formal system adequate to their particular empirical domain. Thus we must break with the tradition in physics. Here is the underlying structure of physics that I propose for the future.

For each empirical domain, we can formulate specific theories and adopt adequate formal systems, in order to build models that represent relevant theoretical entities in a syntactically simple way. Quantitative calculations will be in this case simplified, and the reasoning will be shortened.

If we return to the project of unification in physics, which we discussed previously, we can expose the whole physical theory in the following manner, connecting the classical, quantum, and relativist worlds. 

The first problem is to construct the formal systems adequate to each theory. These formal systems must satisfy the material adequacy constraint. Moreover, they must have a structure that can be considered as a substructure of a larger structure, the one of a more general formal system. The more general formal system doesn't necessarily need an interpretation (a semantics, or a theory associated with it). It can exist only at the formal level to make the bridge between these three pairs, and to ensure the unity of the physical theory.

Epistemological implications

Can we still call this thing that I have just described above physical science?

The changes to the underlying structure of physics that I proposed do not affect the characteristic of scientificity. We only allow the physicist to choose a formalism which is better suited for its theory. Nothing in the criteria of scientificity prohibits that, simply because we long believed that the formalism adopted by Newton was the only good one, the one that God used to construct the Universe, and that put in us to help us understand His creation. All widely accepted epistemological theories are not in contradiction with the ideas presented here, they are just extended by the addition of another dimension.

What will happen to Einstein's theory of relativity and to quantum mechanics in the future? I believe that if we manage to build a physical theory comprising the structure presented on the last diagram, the formalisms developed by Einstein and by the fathers of quantum mechanics will not be used any more in the industry. They will, nevertheless, retain their theoretical interest. 

Future research

Empirical work

• • Is there a way to extract an ontology from the structure of a trained neuronal network? One could use neural networks to solve a Newtonian problem and another neuronal network to solve the same problem but using quantum mechanics. The comparison between the two trained networks, in terms of their structure, must contain the difference between the two ontologies that emerge within these networks while training. If the experiment is repeated different structures might emerge, showing ontological degeneracy (i.e. more than one proper ontology for a single problem). [this idea was added on June 09 2012 after the CAMBAM meeting]

Formalisms

• Develop a theory and a formal system adequate for the relativistic empirical domain and for the microscopic empirical domain. 

• Develop a theory and a formal system adequate for the microscopic empirical domain and try to develop a more global formal system that includes as substructures both classical and microscopic models. Do the same for the relativistic and Newtonian empirical domains.

Philosophy 

• Develop further the philosophical aspect of this theory to guide at the formal level. Formulate a theory of knowledge, a theory of the process of theorization and formalization, show the impacts that these ideas have at the epistemological level, demonstrate the possibility of a physical science comprising the new structure, and show the advantages of the changes suggested.

Appendix 1

Nobody prevents us from representing a real entity that we call electron like a solid ball (classical representation), but if we apply this image to explain the phenomenon of superconductivity, as we observe it via our instruments, we realize that this simple representation doesn't hold. Within this empirical context, it is better to adopt for what we call an electron an image as described by quantum mechanics. On the other side, if we restrict our language at the theoretical level, we can say that, even if we have the possibility to explain the behavior of a rock in quantum mechanical terms, by analyzing the molecular composition of this object and the behavior of this cluster of molecules, arranged according to a certain structure, under the influence of gravity, and subjected to the bombardment of molecules composing the air, it remains that this behavior is of Newtonian type. A quantum particle individually will not behave like an aggregate of a few billion quantum particles coupled between them and coupled to other elements outside of this very complex system which they form. Moreover, two rocks which enter in collision (Newtonian objects) behave in a radically different manner compared with two helium atoms entering in collision (microscopic objects). And finally, to understend the collision between an electron and a rock, physicists treat the collision between this electron and one or more atoms or molecules composing this rock (described in microscopic terms). In this last example, the electron is a quantum object, the rock a Newtonian object, and the process of collision can be treated by the two theories, but not in the same form.

A phenomenological entity analyzed by a physicist is the real thing. The physicist assigns a certain theoretical image to it, to be able to treat this phenomenological entity within the framework of a scientific theory. Assigning an image means to decide to associate this phenomenological entity (WO) with a theoretical object (TO), for which there have been defined quantifiable properties, precise rules of quantification, and quantitative relations between various properties of this TO, and/or the properties of other TO-s resulting from the same theory. We can always distinguish between TO-s belonging to different theories. However, to decide to assign to a certain RO a TO' defined within a Theory 1, rather than another TO" defined within a Theory 2 is another question.

Appendix 2

Adopting a pair of quantitative-qualitative formal systems to be applied to a certain empirical domain.

What is the movement in scientific modeling? The scientist begins by observing some part of reality. This observation can be a direct one, or a mediated one, i.e. with the help of an instrument. The passive observation moves rapidly to a process of interaction-observation, where the scientist acts on the reality more or less in a random fashion, monitoring its behavior. A natural tendency pushes our scientist to verbalize his observations, to speak about them. From his observations, a pattern starts to emerge. The scientist realizes that he can reproduce some effects by acting on the reality in a certain manner. He sees similarities and he associates some of his observations to observations made in another context, of some other parts of reality. Progressively, his language becomes more accurate and more effective to describe what he observes. Little by little, an ontology emerges for the empirical domain at the center of his attention. Most of the language he uses to talk about his new observations is an old language, only a few new concepts emerge if the observed reality is entirely new to him. At the beginning, the scientist is tempted to apply old terms to the new reality, and to make analogies with other realities he knows of. He uses other theories he has, or other ontologies which he applies to other domains. Gradually, he realizes that certain theoretical objects he uses from other theories don't have the desired properties, they are not entirely adequate to describe the new reality. Other concepts might still hold to explain some other aspects of the observation. Therefore, our scientist is forced to expand his ontology, by creating new theoretical objects, new properties, new relations, and to cast these new theoretical entities into his old language. During this process, his language changes a little, our scientist creates a new technical language, but he doesn't loose his capacity to speak to the rest of the world. Along with this process of language refinement, concepts get better defined and the relations between concepts become clearer: a formalism starts to emerge. To build a formalism, the scientist adopts a pair of known quantitative-qualitative formal systems, and tries to find strings of operators, operands, relations, etc., which can represent the theoretical entities. The selected formal expressions cannot be anything. They are used to represent empirical entities. Therefore, they must respect some empirical constraints, i.e. we must be able to make some parallel between their formal "behavior", or structure, and the physical behavior of the reality under observation. They must also respect some formal constraints, which come from the fact that a formal expression must be connected to other formal expressions within a coherent formalism, the latter formal expressions representing other theoretical elements defined by the same theory. The trick is to exploit the internal coherence of the chosen formal system to extract some patterns, which by some strange process we map into relations between theoretical entities (our concepts, linguistic entities), which, by some other strange process, we map into relations between empirical entities (detected by observation). In other words, a relation of causality between event X and the event Y (X and Y being theoretical entities) will thus have as representation at the formal level, for example, in the form of a deductive chain between the formal expression which represents X and that which represents Y. The formal model and the language used to describe the empirical domain in question can continue to be refined by a recursive movement, from observation, to language, to formalism, and back.

Appendix 3

In classical mechanics the concept of force is represented by the symbol F. The mass, an inherent property of material objects, is represented by the symbol m. The kinetic property called acceleration is represented by the symbol a. The formal expression F = ma represents the force that acts on an object in terms of basic properties of the object. We can also have F = mdv/dt, which also represents the force in terms of the speed of the object, represented by the symbol v, and in terms of time, represented by the symbol t. To go from F = ma to F = mdv/dt we must have a = dv/dt. It is due to the nature of the relation of equality, represented by the symbol =, and to the nature of the derivative operation, represented by the symbol d, that the expression «to go from F = ma to F = mdv/dt we must have a = dv/dt. » has a meaning. Newton chose the operators and the relations between these operands/variables so that F could be expressed as a function of v or a, respecting the relation between a and v. 

Appendix 4

The theoretical entities, as well as the relations between them, are represented in natural language by terms, which become meaningful only within a theory. This theory, which is applied to a given empirical domain, contains elements that are specific to it, but also other elements that take their meaning from a more general theory. The formal model captures only a few aspects of the theory, and it connects to the theory via its semantics. But in itself the formal model acquires its proper identity, and its "existence" is given only by the description of its structure. For this reason, the same formal model can be used for other theories. The semantics of a formal system, constructed according to a given theory, consists of a description of the correspondence between elements of the formal system and elements of the theory. The semantics cannot capture the significance of all the theoretical terms, or all aspects related to the relations between theoretical elements. The relation of correspondence selects only part of the significance of the theoretical elements, certain aspects of the theory simply cannot be represented syntactically within the formal system, but must be added at the semantic level. To build a formal model one must refer to a proto-theory. The proto-theory preexists the formal model, although beyond of a certain level of development both can influence each other reciprocally, and become operational in the refinement process.