By Jim J. McCrea
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A line of thinking, following in the wake of Descartes (d. 1650), maintains that all of physical reality can be reduced to mathematical descriptions. This is because Descartes believed that the material world is nothing but a vast variation of shapes, sizes, and motions of undifferentiated matter he terms “res extensa” (Latin: extended substance). Much modern thinking lends to the idea that the essence of this may be captured mathematically. This also includes biological entities such as plant life, animal life, and human bodies.
Indeed, Descartes himself developed what is known as "analytical geometry" which has been proven to describe many physical attributes of things. Analytical geometry has been essential for the development of modern science. Modern science and technology could not exist without it. With analytical geometry, the shapes (or motions) of things are explained by mathematical formulae, as that shape is plotted on an x-y co-ordinate grid (or x-y-z for three dimensions, or x-y-z-t if the fourth dimension of time is included). For example, a circle is described by the formula:
x^2+y^2=r^2
An object thrown into the air inscribes an arc, which is a parabola, indicated by the formula:
y=x^2
(where '^' is to the power of).
The thesis of this paper is that only certain features of physical things can be described mathematically (an example of such is the circularity of a wheel). Physical things, in that even though they are ordered as God created them (or as man has fashioned them well), cannot in their entirety be captured mathematically but have an order that exceeds mathematics. This order, I shall call "suprageometric." I also apply the concept of suprageometricity to spiritual things to show that they too follow the order of God in their truth, beauty, and goodness, and these even less so can be captured geometrically or mathematically. Spiritual things are thus more suprageometric than physical things. Being more suprageometric they are a higher manifestation of reality, so suprageometricity far from being a deficiency (because it escapes a geometrical or mathematical description) is an indication of its superiority over pure geometry or mathematics. This, I will explain in this essay.
Some personal examples of mine can be given. I live[ed] in a small rural Ontario town. It has winding roads interspersed with small lakes. In the area out back, there are patches of meadows, in which each patch is creatively different from the others. The houses are in a rustic arrangement that has escaped the hands of urban planners. All in all, everything has a "haphazard" layout. However, the area is not disordered, but has a suprageometric order. That is what gives it its really good feel. It is unlike those city parks or estates of the rich which are manicured to a cold exactness.
A while back, I acquired a cat (named Kit Kat). There are certain things that cannot be explained or read about, but can only be known by experience. Observing the cat in action, observing the "personality" of the cat, I understood definitely that there is a creativity, spontaneity, and responsiveness that can not be reduced to cold mechanics - that is, capturable by mathematical formulae. There is a positive suprageometric behavior in the animal that points to levels of being far above the materialistic conception of the universe of a mechanism that can be described by finite laws.
Nature itself is suprageometric. Scientists have attempted to replicate nature with computer simulation, using fractals and other mathematical objects. Mountains, clouds, plants, wavy water, and animals in motion are the things that have been replicated. Such a simulation can come across as extremely beautiful and interesting, especially when those scenes of artificial reality are animated so that one thing changes to another in spectacular and creative ways. Superficially, it may even seem far better than nature. But man, with his science, has not bested nature. Looking deeper, there is always a coldness and lack of naturalness to such a thing. One quickly grows tired of it. It is what one would call "pseudo-suprageometric." It resembles the suprageometricity of nature, but the whole thing is reducible to cold mathematics.
As computer simulations are artificial, nature is natural. Nature is truly suprageometric. If one observes the flora and fauna, the lakes and streams, the blue dome of the sky, and the starry firmament, their essence cannot be captured by mathematics or geometry. Certain features of them can, as for example when we observe the various symmetries of flowers that have something of a geometrical form. As one always tires of computer simulations, one never tires of nature. It can continue to refresh, console, and inspire through one's life. Nature has a type of infinity to it, so the human understanding cannot exhaust it. Suprageometricity, specifically, has a property of infinity. More and more statements can be applied to a suprageometric thing and one never exhaustively explains it, for there is an infinite number of true statements that describes it. In contrast, something that is purely geometric has a finite number of statements that can describe it.
All real things are suprageometric to one degree or another (which means that suprageometricity is a transcendental property of being). Even a billiard ball, which appears so simple, is suprageometric which means that an infinite number of truths can be applied to it. Science with its finite methodology can investigate some of these truths. A billiard ball is spherical in shape. Spherically is one abstracted feature of the ball which can be described by the formula:
x^2+y^2+z^2=r^2.
It is a truth about the ball. It is a geometric truth of a suprageometric thing. If the ball is in motion, then we can apply the formula:
p=mv
to describe its momentum (where m is mass, v is velocity, and p is momentum). Momentum is another abstracted feature of the ball, which is another geometric truth about it.
We can get an idea that there is an infinite number of such truths that can be applied to the billiard ball by examining its responses to all possible outside influences that can be applied to it. It can be shot with a bullet, it can be bombarded with a beam of microwaves, it can be dropped in acid, and an infinity of other things can be done to it, each eliciting its own response from the ball . For each of these three examples of the bullet, the microwaves, and the acid, each can be applied in an infinite number of possible ways, each bringing forth a unique response (for example, various types of acids at various temperatures and pressures can be used and there is an infinite possible number of configurations of microwaves that can be used). Science can investigate this sort of thing, but it must do so by choosing specific features or truths from an infinite manifold. The scientist must set up the specific experiment he wishes to conduct on it (of which there are an infinite number possible). The infinite manifold itself, which is the entire truth about the billiard ball, science can never reach.
** Footnote - Possible experiments on the billiard ball may be divided into *destructive* and *non-destructive types.* The destructive types would be the bullet shattering the ball and acid dissolving the ball. A non-destructive type would be observing the scattering of microwaves from the ball which would not harm it.
The materialist argues that only the physical world is real with its measurable observable properties. Anything spiritual or supernatural is denied by such a materialist. Even things such as beauty, love, goodness, etc. are seen as purely subjective experiences of a conscious person. Their objective truth value is denied. Such a materialist basis that on the claim that he is only looking at the facts of reality objectively. However, such a viewpoint is the result of seeing reality in a very specific way. What one is doing in that case is looking down on things. Only that is accepted which one can dominate with the understanding *from above* - putting things under one's thumb, so to speak, so that they are understood with a comprehensive clarity (this is the approach that Descartes promoted in understanding reality. According to him, things are only accepted if they fit into a system of clear and distinct ideas).
As a result only those aspects of reality are recognized that are amenable to clear and comprehensive understanding. That is why so many today accept only the material sciences, with their mathematical and logical methodology as truth. In this, only the geometric aspects of being are recognized as being real. The suprageometric properties of being are missed, which are almost all the important attributes of being. To see these suprageometric attributes, one must *look up.* Rather than dominating reality from a superior vantage point, one must let one's understanding be drawn upwards towards being in contemplation, awe, and wonder, to abide in the mystery of it. This is the only condition of joy in life. In looking down on everything, there may be a form of pleasure that is the result of pride, but there can be no joy or happiness. Joy and happiness can only come from what is above us, never from what is below.
As a result of this stance of looking down, existence is severely truncated, and is not seen as it really is. Much modern secular university education promotes this and indoctrinates students in such a view. That explains why the higher the education that one has, the less likely that one is religious. Higher education causing a loss of faith, does not prove the falsity of faith as many claim, but demonstrates the defective and ideological nature of much higher education. Not only does this promote a sterile form of control over the material world, which shuts out true joy and happiness, but as knowledge (of that type) multiplies, so do world problems. This is because without a suprageometric view of things, there is a near infinity of material facts known, but little wisdom attained.
The elements of the Catholic faith are infinitely suprageometric. This is why modern culture with its need for control from above, may incapacitate one for receiving the faith. The Eucharist is where Christ is present body, blood, soul, and divinity, in all His power and glory, as substantially existing as He does at the right hand of the Father, under the appearances of bread and wine. The Eucharist is infinitely and supernaturally suprageometric. It is infinitely above what can be captured in any rationalistic way, through mathematics, geometry, or Aristotelian logic. A certain supernatural humility is required to accept it, because in its presence, we are drawn infinitely above ourselves to reach a mystery which is nearly absolute. The Eucharist is almost completely beyond the earthbound human intellect's ability to understand. To worship Christ in the Eucharist is eminently pleasing to God, because it is the complete antithesis of the proud attitude that wishes to dominate everything from above.
Supernatural faith, in general, is eminently pleasing to God for the same reason. This is because with that we are drawn infinitely upwards to make contact with God, as He is in Himself, with our intellects. Many atheists say that they would accept God's existence if they could find proof, but they claim that they cannot find any. This is because they are looking down to find God with the same methodology that they would use to do physics. God can only be found by looking up. God Himself is infinitely and supernaturally suprageometric. That is why dark faith is the means to reach Him. As Christ said, one must become like a little child to enter the kingdom of heaven (cf. Matt 18:2-4). Children, generally, have the attitude of looking up to things in wonder and awe, and they are open to the truth that is presented to their minds. Adults, on the other hand tend to be much more closed, accepting only what will agree with their prejudices and conform to their preconceived ideas. This is why it is always far easier to evangelize children than adults. Even though children generally have far less raw knowledge than adults, they most often take a proper suprageometric stance towards reality and thus see things as they really are.
** Footnote - Even the supernatural infinitely suprageometric things of God have geometrical aspects. Theology proper is the deduction of certain conclusions from the data of divine revelation via a process of finite logic. That branch of metaphysics known as natural theology, which brings us to the existence of God and many of His attributes by the unaided human reason, uses a logical chain of reasoning based on natural first principles. In this, we move "geometrically" to God. However, such knowledge is only a knowledge by analogy, in which we do not know God as He is in Himself. It is the supernatural light of faith and divine revelation that brings us to this intimate knowledge of God (although in an obscure manner. The face to face vision of Him is reserved for the next life). Natural knowledge via 'geometry' is necessary to form the basis of faith and revelation. For if we can naturally deduce that God exists and is all perfect, we then know that what is given in faith and divine revelation is perfectly reliable. For it is axiomatic that grace builds upon nature.
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Part II: Topics in Suprageometricity
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(1) Suprageometric Time: The Physics of Heaven
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Heaven completely transcends this earth. Mystics have said that the music in heaven has no beat. That is because heaven does not have time - or at least the type of time that we have on earth. In heaven, the inhabitants do not experience the absolute eternity of God, where He experiences all of time all in an instant in an everlasting "now." With God, there is no time absolutely, and no motion whatsoever. The infinite richness of God, in His eternity, does not require motion, whereas with us, each particular moment is so poor that we constantly need to be moving to the next one.
What the inhabitants of heaven experience, theologians tell us, is something between physical time and the absolute eternity of God. This type of time is known as "aeviternity." With physical time there is what is known as a "metric." The metric of time is the measuring rod of time. It ensures that any given one second interval, or one minute interval, or one hour interval is equal to any other one second, one minute, or one hour interval respectively. This is based on regular physical processes in the universe. In fact, one second is defined as 9,192,631,770 vibrations of a cesium atom, and this is consistent from one second interval to another. The metric of time in our universe is defined by that.
But in heaven, such a thing does not exist. Aeviternity is something which transcends this. It is based on the on thought processes or other events considered holistically. Each event is so unique that it cannot be measured against another. A clock cannot be applied to its duration, but simply the fullness of it is experienced as it happens. This is what we would call suprageometric time. Likewise rulers cannot be applied to the dimensions of things in heaven. They are likewise suprageometric.
(2) Suprageometric Music
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This is a quote from the book "Metamagical Themas" by Douglas R. Hofstadter. It is a large tome on the essence of pattern in reality. He has many fascinating things to say. I quote what he says on the topic of music:
"...With Chopin though, preoccupation with strict pattern never took precedence over the expression of heartfelt emotions. One must distinguish, it seems to me, between 'head pattern' and 'heart pattern', or, in more objective-sounding terms, between *syntactic* pattern and *semantic* pattern. The notion of a syntactic pattern in music corresponds to the formal structural devices used in poetry: alliteration, rhyme, meter, repetition of sounds, and so on. The notion of a semantic pattern is analogous to the pattern of logic that underlies a poem and gives it reason to exist: the inspiration, in short. [that is, the meaning] That there are such semantic patterns [meaning patterns] in music is as undeniable as that there are courses in the theory of harmony. Yet harmony theory has no more succeeded in explaining such [semantic meaning] patterns than any set of rules has yet succeeded in capturing the essence of artistic creativity. To be sure, there are words to describe well-formed patterns and progressions, but no theory yet invented has come close to creating a semantic sieve so fine as to let all bad compositions fall through and to retain all good ones. Theories of musical quality are still descriptive and *not* generative; to some extent, they can explain in hindsight why a piece seems good, but they are not sufficient to allow someone to create new pieces of quality and interest. It is nonetheless fascinating, if not downright compelling, to try to find certain earmarks of greatness, to try to understand why it is that one composer's music can reach in and touch you innermost core while another composer's music leaves you cold and unmoved. It is a mystery."
Section III. Chapter 9. Page 181.
Here Hofstadter is talking about the suprageometric quality of good music (referring in particular to Chopin). With something that is merely geometric, each element in a pattern can be deduced from a general formula. Many materialists today think that all of reality is measurable and reducible to formulae. They believe that when all of the laws of nature, mathematically described, are discovered, reality will be comprehensively understood. That was Descartes' dream. However, reality is largely suprageometric. It is ordered, but transcends formulae. Nature is like that. Personality is far more suprageometric still, and the realities which are the object of supernatural faith are infinitely still more. Good music is suprageometric. That is why a mere computer program cannot suffice to compose good music. The human soul, made in the image of God, with intellect, will, and spiritual affectivity has that creative ability which transcends the generative power of mere algorithms (automatic processes). True artistic work cannot be generated mechanically. As Hofstadter says, certain features of music can be analyzed in hindsight to give insight as to why it is good, but analysis cannot explain the entire pattern. Only the human soul with its intuitive faculty can judge it as good and as having meaning.
(3) The Trinity and Reality as "Revelation"
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It is a teaching of the Catholic Faith that the truth of the Trinity cannot be derived by reason, but must be revealed by God. This is because the doctrine of the Trinity is a suprageometric doctrine. No analysis of language or concepts can derive it. Remember, that a suprageometric form cannot be derived by something prior to it, but must simply be given.
Reality itself, being suprageometric, must be "revealed." When we are "thrown into" reality, what we encounter there (in the main) cannot be derived by any logical or computational process. This is why experience is absolutely essential to learning. Even an infinite computing mind, if it were in a void, could not deduce the nature of the least being that a child would know who lives in this world.
However, certain *features* of reality are geometric, as opposed to suprageometric. A form of logical or mathematical analysis can deduce them. This is what makes physics or engineering possible, which heavily utilizes mathematics. For example, a tire has circularity which can be described by the formula:
x^2+y^2=r^2.
The circularity of a tire is a geometric truth of it. Circularity is a truth of it, but it is not it. A wheel is not a mere circle. The wheel itself is suprageometric, which transcends any geometric formula. The wheel has quality, texture, thickness, pattern etc, which as a whole cannot be derived mathematically. The essence and existence of the wheel - in short its being - is suprageometric and cannot be derived mathematically. This is why, even though mathematics is essential to the work of an engineer, empirical experiments are also essential in his work. For the mathematics gives the initial outline of the device to be developed, but as the device is developed, empirical experimentation is required in the development process to ensure that the specified device materializes. An empirical experiment is simply doing something and experiencing what happens. That experiencing of what happens is a "revelation" that cannot be derived either logically or mathematically.
(4) Suprageometric Living
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Life can be frustrating. We are constantly making plans and having them upset. The true Christian knows that all things happen only by divine providence. How does divine providence fit into the upsetting of our plans? It fits in, simply because as fallen creatures, our need to plan is often inordinate. A certain amount of planning is necessary in life. For example, we have to ensure that we keep our proper appointments with other people. But the thing is, we tend to take planning too far. This destroys spontaneity in life. Spontaneity is the key to joy in life.
All in all, the properly balanced life is a combination of planning and spontaneity. In other words, the properly balanced life is a combination of geometric and suprageometric behavior. What the upsetting of our plans in life does, is tend to cut down and humiliate our tendency to excessively plan. It is this type of cross, which if patiently endured, sets us free over time.
What we are being set free from is being a "control freak" which makes ours and other people's lives miserable. One we become free of that, we learn to "go with the flow" and we and other people are much happier.
A perfect human society would have the divergent qualities of order (planning) and freedom (spontaneity) in perfect balance - or geometric (planning) and suprageometric (spontaneous) behavior in perfect balance.