Wednesday, February 5, 2014

Is Mathematics the Real Reality?

Mathematics is the language with which God wrote the universe.—Galileo Galilei1

I like mathematics largely because it is not human and has nothing particular to do with this planet or with the whole accidental universe - because, like Spinoza's God, it won't love us in return.—Bertrand Russell2

When we ask the question seen in the title of this chapter, what are we really trying to ascertain? We are asking whether mathematics is ultimately that which is true, actual, real in the sense that it would be evident even to a being possessing Absolute Perspective (which we, of course, cannot know with any certainty). We are asking whether mathematical realities would still exist even if there were no space-time or energy-matter. We are asking whether what humans interpret as mathematics is a stand-alone phenomenon, something that requires the existence of nothing else to serve as a point of reference. We are trying to imagine ourselves on an expedition to the heart of things, so to speak, stripping away all definitions that depend solely on human perception, penetrating layer after layer of intellectual underbrush, going for that which would be true at any time, in any place, whether it was being perceived or not. We are seeking something that is permanent, invariant, Universal in the true sense of the word, and unquestionably real. That’s what we’re looking for, and that’s what mathematicians devote their lives to.

And that to which mathematicians devote their professional lives seems to be the area of human study most removed from the vagaries of human judgment, an area which seems to exemplify “objectivity” itself. Of course, some figures in the history of mathematics have seen it as a mystical door into the world of metaphysics. Pythagoras is the example that most readily comes to mind. But for the most part, mathematicians are a level-headed group of individuals not given to making mystical pronouncements about the magical qualities of numbers.

Mathematics is a tool that can seemingly lay bare the nature of the physical world down to its deepest levels. Mathematics, in the words of Eugene P. Wigner, possesses an “unreasonable effectiveness” in scientific disciplines. It was Wigner’s opinion that the principal task of mathematics is the formulation of concepts, for which the genius and ingenuity of mathematicians can discover uses and applications. Wigner pointed out that mathematics is especially important in physics and that the deeper into physical reality physicists push, the more these scientists rely on the concepts devised by mathematicians to both understand and express that reality.3 It would seem inarguable that the Universe’s nature can best be described through the application of mathematical thought.

But what is mathematical thought?  And what is mathematics itself? Is it pure logic? Is it an act of mental construction (intuition)? A formal system for demonstrating propositions, the goal of which is complete consistency? The study of number, relationship, change, and quantity?  It is telling that there is no real consensus, even among the philosophers of mathematics, about the exact subject matter of the discipline itself. This makes an analysis of its ultimate nature problematic.

Are mathematicians inventing mathematical forms? If this is true, the study of mathematics is simply another emergent aspect of human consciousness, a particularly elaborate psychological construct, no different in nature from any other produced by our brains. Alternatively, are mathematicians engaged not in “creating” mathematical principles but rather in discovering principles which seem to be inherent in the fabric of reality itself (although the discovery of such principles may demand creative and imaginative thinking of the kind Wigner described)? If this is the case, it might be said that when we are engaged in mathematical thought we are closer to touching the “true essence” than in any other human endeavor. More generally, when we are thinking mathematically, we are perhaps engaged in a type of thought, a method of analysis which, although expressed in our own unique cultural symbols, might be one that would be familiar to any intelligent life form anywhere in the Universe. Those who designed and installed the famous plaque that went on the Pioneer 10 spaceship in 1972 thought so; the relative distances between the planets in our solar system are depicted in binary.4

The chief issue in the philosophy of mathematics (as it appears to this layman) is: Do numbers, and objects which are solely described by mathematics, objectively exist? Do mathematical objects exist outside space and time? Are mathematical objects real in the sense that Plato’s ideal forms are purported to be real? (As we will see later, Plato postulated that there are ideal forms of all things and these forms are the true reality rather than the reality presented to us by our senses.) Indeed, it might fairly be said that many mathematicians tend toward Platonism in their thinking. The concepts and objects with which they work are so reliable, so consistent, and so applicable to a wide range of problems, that they often appear to be real in the ontological sense.

Bertrand Russell had very definite opinions on this matter. In an essay from 1904 he wrote:

The truth is that, throughout logic and mathematics, the existence of the human or any other mind is totally irrelevant; mental processes are studied by means of logic, but the subject-matter of logic does not presuppose mental processes, and would be equally true if there were no mental processes. It is true that, in that case, we should not know logic; but our knowledge must not be confounded with the truths which we know, and in the case of logic, although our knowledge of course involves mental processes, that which we know does not involve them. Logic will never acquire its proper place among the sciences until it is recognized that a truth and the knowledge of it are as distinct as an apple and the eating of it.5

Kurt Gödel, one of the giants of twentieth century mathematics, was a mathematical Platonist of the first order. In a 1995 paper describing the evolution of his thought, Gödel’s position is expressed as follows:

Speaking quite generally, philosophers often talk as if we all know what it is to be a realist, or a realist about a particular domain of discourse: realism holds that the objects the discourse talks about exist, and are as they are, independently of our thought about them and knowledge of them, and similarly truths in the domain hold independently of our knowledge. One meaning of the term “platonism” which is applied to Gödel (even by himself) is simply realism about abstract objects and particularly the objects of mathematics.

The inadequacy of this formulaic characterization of realism is widely attested, and the question what realism is is itself a subject of philosophical examination and debate. One does find Gödel using the standard formulae. For example in his Gibbs lecture of 1951, he characterizes as “Platonism or ‘Realism’ ” the view that “mathematical objects and facts (or at least something in them) exist independently of our mental acts and decisions”… and that “the objects and theorems of mathematics are as objective and independent of our free choice and our creative acts as is the physical world”6

Gödel was adamant in his insistence that mathematical objects were not created by humans and exist fully independently of any human intuition of them. This mode of thought even takes on a theological dimension for some people. To them, mathematical axioms and propositions are to be found in the “Mind of God”, from whence they find their way into our world.7

In response to such thinking, Mario Livio has posed an interesting conjecture. It is his contention that the question of whether mathematics is discovered or created is misleading. He argues that it is, in part, both—that some of it has been discovered and some of it created. Livio uses Euclid’s definition of the Golden Ratio, which is (1 + √5) /2 = 1.6180339887… , as an example of something that mathematicians created rather than discovered. Livio recognizes that the Golden Ratio pops up in several places—the nature of pentagrams, the construction of pentagons, aspects of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.)—but he argues that “Euclid’s inventive act singled out this ratio and attracted the attention of mathematicians to it.”8  He adds that in China and India no such ratio was invented and it had no effect on mathematical research in those regions. He contends that triangles, parallelograms, prime numbers, and many other examples were concepts that were invented, but that the theorems that are used to describe them were discovered.9

More broadly, philosophers of mathematics have posed key questions aimed at addressing the issue of whether mathematical entities exist in some timeless, dimensionless reality. Some have asked, do all mathematical entities correspond to, describe, or apply to actual entities in the known Universe? Can we say that if a thing can be described mathematically that it must exist, by necessity? In a related manner, doubters of mathematical Platonism have asked, is the physical world contained within mathematics, or is mathematics contained within the physical world? In other words, does the ultimate nature of actual, finite physical reality describe the limits of mathematics? Is the ability to manipulate mathematical concepts and symbols, made possible by the evolution of the more advanced regions of the cerebral cortex, the final and definitive method by which physical reality may be apprehended?

And in relation to this there is the view of mathematician Michael Atiyah (quoted by Mario Livio) that our mathematics reflects our unique experience of the physical world. Atiyah points out that had the Universe been a one-dimensional entity that no concept of geometry could have developed, and he further points out that an animal that never encountered any other objects would have no concept of number (for example one living in isolation in the deep ocean). So if the human perception of physical reality were different, would the human understanding of mathematics be different?10

Others who doubt that mathematical objects exist in their own realm have asked, if mathematical objects have their own independent existence, and do not need to be instantiated (represented in some concrete form) in any way, how can they be known to mathematicians, who are instantiated, and moreover, how can these objects be manipulated by mathematicians?11

Is it right to say that mathematical entities exist in a Platonic sense, or is it more accurate to say that they represent ways of seeing reality that any rational creature would have to perceive? In Gödel, Escher, Bach Douglas Hofstadter asks whether every world that we can imagine would have to have an identical mathematics to ours. Hofstadter answers this question by saying that although not all conceivable worlds would necessarily use Euclidean geometry, for example (since non-Euclidean geometries do exist), any conceivable world would have to have a mathematics based on a foundation which at the very least included logic. And of course, in any world we can imagine, Hofstadter says, such propositions as “1+1=2” and “the number of prime numbers that exists is infinite” must, by logical necessity, be true.12 

Why does mathematics work? Why can it be said that even God himself (if such an entity exists) cannot make 2+2 equal anything but four? Why are mathematical theorems, once demonstrated, beyond all possible contradiction? These propositions, once uncovered, are regarded as eternal verities, expressions which have always been true, which will always be true, and which must be true in any conceivable reality. Indeed, why does it seem that mathematical reality is the ultimate boundary between the logically possible and the logically impossible? We are reduced, ultimately, to saying that mathematical truths simply are. They appear to be the most fundamentally true things which exist, the foundation upon which everything else must rest. And mathematics as a discipline does not depend on empiricism. Mathematical principles are deduced through the operations of logic and what may be called the mastering of chains of reasoning by means of the correct ordering of propositions, propositions which can only take one correct form. Moreover, it would appear that mathematics can even utilize numbers which do not exist on the number line, and can in fact use them to describe aspects of physical reality. Mathematician (and investigator of the nature of consciousness) Peter Russell expresses it like this:

Mathematics… is purely a creation of the mind. Mathematics is that body of knowledge that is arrived at by pure reason, and does not rely upon any observations of the phenomenal world. It is free from the limitations imposed by the particular way human minds create their experience of the underlying. As such it is probably the closest the human mind can come to understanding the thing-in-itself. 

Russell goes on to describe the tremendous utility of a completely imaginary number: the square root of -1, designated simply as i.

From this arose a new and even larger set of numbers, the so-called "complex" numbers, that were a combination of real and imaginary numbers. And these, it turned out were invaluable in helping mathematicians solve equations that had no solution in the realm of real numbers. Moreover the solutions applied to the real world.

Out of this panoply of numbers a most remarkable and intriguing relationship appeared. The irrational number "pi", the irrational number "e", and the imaginary number "i", come together in one of simplest equations ever; "e to the power of i times pi = -1".

This simple equation is the basic equation of any wave motion. Every wave from a wave on water, the air waves coming from a violin string, to light waves, can be expressed as a combination of simple equations of this form. It also expresses the orbits of the planets, the swing of a pendulum and the oscillation of an atom. In fact, every single motion in the cosmos can ultimately be reduced to an equation of this form. The whole of quantum physics depends upon it. If mathematicians had not discovered this most remarkable relationship, the strange story of the quantum would never have been told.

And all of this without a single empirical observation. No wonder then, that in the end all science comes down to mathematics. The very fact that it is not based upon phenomena, is why it is probably the best approximation to the underlying reality we have. 13

The argument made by Peter Russell is certainly a compelling one, but the heart of the matter is, to my way of thinking, this: scientists can describe the most fundamental levels of physical reality only through the use of mathematics. But is there a difference between a thing in itself and the description of that thing? To say that mathematics is an approximation of the underlying reality borders on saying that the underlying reality is somehow  fundamentally a perspective or a method. To paraphrase something which I believe Bertrand Russell said about electricity, mathematics isn't a thing; it is a way in which things behave. Things behave in a mathematically-describable way. But does that really get us closer to what things ultimately are? If by saying mathematics is an approximation of the underlying reality, does Dr. Russell mean that the utility of mathematics argues for its existence as a stand-alone entity?  I obviously don’t have the background to make these arguments in a mathematically sophisticated manner. But I wonder if we are conflating reality itself with its description.

Still, it must be admitted that the number of places i, e, and pi (π) can be used is amazing. Euler’s number, e, (2.718281828459045… ), is the most important mathematical constant in existence.14 Calculus, one of the most powerful methods by which the world can be examined, uses it constantly. And the fact that π, an irrational number (meaning that it can be carried out to an infinite number of places) is used to calculate the area of a finite space (the area of a circle) seems to say something about the essential mystery of numbers. The very fact that there are constants, and many of them, in mathematics, and that their application is completely invariant, would seem to demonstrate that these constants represent fundamental aspects of the real. (But again—are the constants invented concepts described by discovered axioms?) Further, there are countless mathematically regular patterns in nature (see the chapter entitled Patterns,  Cycles and Shapes) and it would seem as if physical reality is saturated with such patterns. All of this, combined with the extraordinary usefulness of mathematics, would seem to argue for the primacy of mathematics as the real reality.

But even if we were to grant that mathematics is more than the description of reality, can we ask if mathematics is reality itself, or is it rather the only non-emergent level of reality? The two are not the same thing, in my opinion. The non-emergent level of reality is necessary, but is it sufficient? And what of the limits of mathematical description? If Gödel, the über-Platonist, has proven the incompleteness of mathematics, which is to say the inability of mathematics to either prove or disprove every possible statement in number theory, does that undermine in some way the claim that mathematics constitutes its own world? How could any ultimate reality be incomplete in any way, or be vulnerable to any uncertainty? (It could be replied, I suppose, that uncertainty is an inevitable and inherent feature of ultimate reality.) And what of the limitations of mathematics in other areas? Since only the collective behavior of particles at the quantum level can be predicted, and the behavior of individual particles cannot be predicted, does this demonstrate the limits of mathematical description, and therefore weaken the claim that mathematics is “the thing in itself”?

Even granting that mathematical language is more precise than ordinary human language, it still must be grasped within the confines of the human animal’s brain. Since that brain is inherently limited, can we know with certainty that we are perceiving mathematical propositions with complete accuracy? Even more, can we know with certainty that the mathematical propositions flowing from the variety of specialties within mathematics are all congruent with each other, and are all describing, ultimately, the same reality? There are now several thousand categories and subcategories within mathematics. How can it possibly be known with certainty that none of the theorems of some arcane sub-specialty contradict those of another? (A mathematician might respond that if those studying these fields use the methods of reasoning taught by mathematics, that there should be no inconsistencies, incongruities, or contradictions at all. But can this be known?) Further, since mathematics can be used to describe things that do not exist (such as planetary epicycles or the ether), what does that say about claims that mathematics rests on fundamental truth? (Here a mathematician might argue that were there actually such things as planetary epicycles they would have to behave in the ways described by mathematics.) And then there is the world of the intangible.

If we assume certain emotional states are real in the sense that they are experienced in consciousness, even if we can define: A. the specific region of the brain from which the emotion is emanating (biological origin) B. the chemical composition of the neurotransmitters involved in the perception of the emotion (chemical origin) and the collective action of the subatomic particles composing the neurotransmitters (physical origin), have we described the emotional state? Or could an observer of  the emotional state only understand it by referencing his or her own experience of it? If a person has never been angry, for example, he or she might be able to look at the pattern of subatomic particle behavior, neurotransmitter action, and neuronal activity in a person and say, “The person experiencing these things is currently displaying an affect which I am told is anger.” But such an observer cannot know the experience of the angry person, any more than a color blind individual could know the experience of color perception merely by analyzing the light waves by which it is conveyed and the cones of the retina by which those waves are perceived. Mathematics cannot describe the whole of human reality because much of that reality is clearly emergent, is clearly more than the sum of its parts. In a sense, breaking things down to their mathematical descriptions is severely reductionistic in nature. The quantum probability waves we encountered in the previous chapter by themselves tell us nothing. It is only when they undergo decoherence that they have meaning for us, and it is only when many stages of emergence have been “piled up on top of each other” that human reality emerges. Mathematical analysis by itself is insufficient to the understanding of reality.

And the attempt to “mathematize” all aspects of the human experience could lead to disaster. Philip J. Davis and Reuben Hersh explain it like this:

Whenever anyone writes down an equation that explicitly or implicitly alludes to an individual or a group of individuals, whether this be in economics, sociology, psychology, medicine, politics, demography, or military affairs, the possibility of dehumanization exists. Whenever we use computerization to proceed from formulas and algorithms to policy and to actions affecting humans, we stand open to good and evil on a massive scale. What is often not pointed out is that this dehumanization is intrinsic to the fundamental intellectual processes that are inherent in mathematics. [Emphasis in the original]15

So what, in summary can we say? The methodology of mathematics is clearly indispensable to an examination of physical reality, but not all (or even most) of human social or cultural reality is amenable to its methods. It would be possible, perhaps, to quantize all the variables that affect human history, but any attempt to predict the future course that such variables might lead to could only be described in the “soft” terms of probability, where the boundaries are hazy and indefinite. Mathematics cannot be used to decide aesthetic questions (although it can explain, in conjunction with evolutionary psychology, why humans seem to like certain shapes and ratios better than others), nor can it be used to explain the appeal of literature. Mathematics cannot, in isolation, deal with moral or ethical questions, even if its methods can be used to help lay out the consequences of human decisions. (“If you do x, then a, b, and c are more probable.”) Mathematics cannot explain love, cannot quantify despair, cannot measure devotion, cannot explain the unfolding of human history, and cannot lay bare all the mysteries of human psychology. All of these phenomena could, perhaps, be analyzed mathematically, but they cannot be experienced in that way. Having said that, therefore, let us return to the original question: is mathematics the real reality, that which would exist after all else was stripped away? I can only give my naïve, non-mathematician’s response.

Perhaps it would be best to say that reality isn’t mathematics itself. It is, rather, a set of interrelationships and space-time phenomena that are explicable in mathematical terms.   It might be that the number 2 does not exist in any Platonic, metaphysical realm, possessing an independent reality. Rather, it may be that the real reality has aspects that when perceived and analyzed by conscious minds (from any planet in any solar system in any galaxy), cause the number 2 and all the relationships it has to other points on the number line to manifest themselves. This is not philosophical idealism—I am not saying that the fundamental nature of reality does not exist except when we are attempting to perceive it—but rather that numbers, geometric figures, etc., are interpretations of the real reality’s links to us, in much the same way that hearing is an interpretation of energies which exist independently but are not interpreted as sound until they reach the physical structures found in ears. There is something about the real reality, therefore, that when we look at it, can only be interpreted as the number 2.

Bearing this in mind, therefore, let us turn to the realms of randomness and probability, both aspects of mathematical reality, and both features more ubiquitous than humans ordinarily perceive.

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