Abstract
The source of human applications of mathematics to natural phenomena is very likely to be found in the perceptual processes necessary for most animals to survive, either as prey, predator, or forager. Much of the resources of the brain are used to transform the patterns of energy that impinge on sensory nerve endings into representations more or less isomorphic to the structure of some portion of the world. It is one of the intellectual triumphs of Ancient Greek thought, especially by Aristotle, to have recognized for the first time systematically the centrality of such isomorphisms for perception. The move from visual perception to intuitive geometry, and then to systematic Euclidean geometry seems psychologically natural. The Greek application of this geometric viewpoint to the motion of the heavens constitutes the first great work in mathematical physics. The progress from the Stone Age to Ptolemy is surely a very much bigger step than the move from Ptolemy to Newton and Einstein.
Stone Age paintings
Humans have been representing the structure of the external world in remarkable paintings and drawings almost from the beginning of any kind of civilized activity. It is very likely that this beautiful work represents in itself a very long tradition that goes back still many more thousands of years. What I take these early developments to show is that humans have long had the capacity to represent the geometrical structure of the visual world they experience. Moreover all the evidence seems to point to the naturalness and the widespread use of the representational skills that do not require any elaborate mathematical justification or development for the recognition of their correctness. An ancient drawing of a lion or some other large animal is now, and I am sure, was then, immediately recognized by children as well as adults as a representation of a particular kind of animal. Contrary to Wigner, I find this early use of intuitive mathematical skills of perceptual representation as natural as the early development of language in humans. It is a nice question, to which we can now give no satisfactory answer, as to what came first, the ability to draw perceptual representations, or the ability to speak a language that could refer to the objects or processes of such representations. In any case, these two unparalleled abilities are to be found in no animals at the level already exhibited by early humans.
Animal capacities
My second point is a more general one about the ability of higher animals as well as humans to represent internally features of the world significant to them as prey, predator or forager. What is the mechanism that permits birds to fly several thousands of miles to reach an appropriate destination? What is the mechanism that permits horses and dogs to return home when released miles away? Even this restriction to higher animals is mistaken. We can ask the same sort of question of fish, the great variety of insects, especially ants, and many other species we have yet to investigate with any thoroughness.
The general point is clear. Not only humans but most animals have some mechanism of representing the external world. Without such a mechanism, there seems to be no scientific explanation possible of their remarkable ability to choose and follow a given trail or route. Of course, as any neuroscientist knows, we are very far from understanding in any complete detail most of these internal representations. In many instances, the mathematics and physics required, to represent them in a language of concepts we understand explicitly, are surprisingly subtle.
Internal perceptual representations
In the case of visual perception, we do have good evidence that the visual cortex has a natural and relatively simple geometric representation of what is visually perceived, and even an internal representation in memory that preserves to a surprising degree the intuitive geometry of the perceptual representation. Given the richness and power of this natural perceptual processing in the brain, it is scarcely surprising that early humans, with brains very similar to ours, found it easy to draw or paint visual representations of objects that were important in their lives. In the case of the auditory perception of the world, used even in the case of many mammals as much as visual perception, the internal representation problem is more complicated and harder for us to understand in as complete detail, but at least there seems to be strong evidence of a systematic organization according to frequency of the sounds heard. Of course this is the thinnest of possible descriptions I am giving. Much more can be said and can be found in the scientific literature. When we move on to the haptic space of touch, also much used by many mammals and other animals, the representation data are more scarce. At least this is my appraisal of the literature I know. In the case of humans there has been a continual useful tradition of experiments on haptic perception, often with blind persons as the participants (Lederman and Klatzky 2009). In the case of smell, the olfactory representation seems to be of great importance to almost all mammals, least of all humans, but also to hundreds of thousands of insect species where mechanisms for using any of the senses are as yet not well understood (Steuernagel et al. 1994). Our knowledge of these subtle representations seems to be still rather primitive. More generally, it is also important to emphasize that when we turn to the lower species, it is still difficult even to assess what their sensory capabilities are. Much is yet to be learned about their internal representations of the world.
Perceptual motor control
I have primarily emphasized the relatively passive activity of visual perception. In many ways this is a mistake. Perception is really needed to help all animals guide their movements, often under severe pressure to make a quick and accurate assessment, depending on perceptual input, on where and how to move. A distinct physical theory of movement seems to be needed for most species — from the flight of birds, the coordinated leg movements of centipedes, to the fast vehicle-driving of humans.
The emergence of geometry
One of the great feats of ancient times was the gradual progression from intuitive visual geometry to the systematic geometry of Euclid, the creation of many minds. The early history of this development is probably forever hidden from us, except for a few rather untrustworthy reports. But certainly by the fourth century BCE, Eudoxus was leading the way to new concepts and new proofs. By the second century BCE Apollonius was proving theorems sufficiently sophisticated about conic sections that Kepler, almost 2000 years later, referred to them in his work on elliptic orbits in the solar system. For those Greek mathematicians responsible for these remarkable developments I conjecture that the increasingly sophisticated use of geometric concepts seemed very natural and obvious to these insiders as they learned from one another. What kind of evidence supports my conjecture? Proclus’ (fifth century CE, 1970) many remarks about the beginning and the foundation of geometry contain some suggestive hints. I also think we can learn something pertinent to these early foundations from the ways students learn such geometric ideas today. Then as now, the same rich perceptual visual data on shapes and sizes, which the Stone Age painters handled so easily, fully supported the Greek conceptual effort. This depended all the way on such data-driven intuitions, needed to develop their early mathematical system of geometry.
Ancient astronomy
Trailing not far behind the geometers was the application of their rich ideas to the motion of the heavens. By the time of Eudoxus, a geometric model of the heavens was widely accepted by Greek astronomers (Neugebauer 1975). Here it seems that the big intuitive leap was the application of geometric concepts to the motion of the Sun, Moon and inner planets. It seems extremely likely that once the idea was put out there, Greek astronomers found a geometric model of the heavenly movements the only possible way to think about the subject. But it is important to note that something perceptually very much more sophisticated and difficult had to occur. Only the Sun traced a simple arc across the sky each day. This easily suggested a circle as the orbit, but this was especially less obvious in the case of the planets whose directly observable movements are very complicated. The pieces were put together using especially the hypothesis that natural and eternal motion must be circular. Patterns of motion that did not seem to be simple circular orbits around the centre of the Earth must be composed from several circles, giving rise to the epicycles and epicentres so central to Ptolemaic astronomy during the 1500 years of its dominance. The phrase ‘Ptolemaic astronomy’ is well justified, for it was Claudius Ptolemy above all who systematized and organized the observational data and the geometric theorems that established the structure of the theory. Of course, I do not mean to suggest that this was all his work. Living and writing toward the end of the first century and the beginning of the second century CE, Ptolemy used data from the Babylonians that was more than 1000 years old, and theoretical ideas from his fellow Greek mathematicians that were at least 500 years old in some cases (Neugebauer 1975).
In many different ways Ptolemy's great work The Almagest (1984) was the pinnacle of Greek science. The conceptual mountain climbing it describes so well took many centuries. But as each plateau was reached, and a conceptual place was cleared for further work, the geometric view soon seemed obvious and inevitable, surely in a way close to how those early Stone-Age painters thought about the naturalness of their visual representational creations.
All the same, the conceptual distance covered from the time of those early painters to Ptolemy seems very much greater than the distance from Ptolemy to Newton and Einstein. The real triumph, to be celebrated forever in the long history of human thinking, is how we got to Ptolemy from such intuitive but primitive beginnings.
In one respect I may have given the wrong impression of these early developments. I may have made it seem that the only way was a geometric way. But as in many great intellectual developments, pluralism not unique-ness is the order of the day. So it is in astronomy. More than 2000 years before Ptolemy, astronomical observations were systematically made and recorded by the Sumerians, who were the ancestors of the Babylonians. As mathematics developed in these early days, slowly and gradually a math-ematical theory of the motion of the heavens, quite different from that of the Greeks, starting around the eighth century BCE reached a comparable level of scientific sophistication in all respects really, except that of providing explicit mathematical proofs (Neugebauer 1975). The non-geometric arithmetic methods of calculation of the Babylonians were clearly earlier than those of the Greeks. Still another interesting example is a development around the fourth century BCE in China of the flat-earth hypothesis as a basis of astronomical calculations (Cullen 1996). Other civilizations of ancient times such as that of the Mayas or the Incas had original astronomical theories. But rather than explore these alternatives I want to move to a more detailed psychological view of how perception works. The ancient Greeks had a lot to say about this, and there is one work that, like The Almagest, consolidates and dominates Greek thinking on these matters. It is Aristotle's treatise On the Soul (1975).
Aristotle's on the soul
This is undoubtedly the most important systematic statement in ancient times, both scientifically and philosophically, of a theory, not only of the mind, but of the main characteristics of biological life. Probably the only serious competitor is the extensive and often sophisticated body of remarks about such matters in similar writings of Plato. Other philosophers, both before and after Aristotle, of course, had a great deal of relevant things to say. (A good survey of other philosophies of mind, such as that of Stoic philosophers, can be found in (Annas 1992).) All the same, in terms of later influence and, equally, in terms of detailed development of ideas of how the mind perceives and thinks, there is no real equal to Aristotle in ancient times. Here are his opening words at the beginning of Book I of On the Soul (Aristotle 1975).
‘We regard all knowledge as beautiful and valuable, but one kind more so than another, either in virtue of its accuracy, or because it relates to higher and more wonderful things. On both these counts it is reasonable to regard the inquiry concerning the soul as of the first importance. Moreover this investigation seems likely to make a substantial contribution to the whole body of truth, and particularly to the study of nature; for the soul is in a sense the principle of animal life. So we seek to examine and investigate first the nature and essence of the soul, and then its 〈essential〉 attributes. Of the latter some seem to be affections peculiar to the soul, and others seem to belong to living things also, by virtue of the soul. But to attain any sure belief on the subject is hedged with difficulties on every side’. 402a, I 1–12, p. 9.1
Form and matter
A central tenet of Aristotle's general theory is that we must distinguish between the form and the matter of something. In general, this distinction is a relative one. For example, a brick has a certain obvious geometrical form — forms are not always geometrical, but it is easy to convey the idea when they are — and the matter of the brick is the clay of which it is composed. On the other hand, if we focus on a brick house, the form of the house is its plan and the matter of the house consists of the bricks from which it is built. So, bricks in this case become matter and the form dictates how the bricks are to be assembled. Everything in nature has its form, animate or inanimate.
Potentiality and actuality
I skip the rest of Book I and go to Book II. Aristotle turns to his own theory of the soul. In doing so, he opens with one of his most original and important contributions, namely, applying his general distinction between potentiality and actuality. I quote the opening paragraph to give you a sense of this and how his doctrine is developed.
‘The theories of the soul handed down by our predecessors have been sufficiently discussed; now let us start afresh, as it were, and try to determine what the soul is, and what definition of it will be most comprehensive. We describe one class of existing things as substance; and this we subdivide into three: (1) matter, which in itself is not an individual thing; (2) shape or form, in virtue of which individuality is directly attributed, and (3) the compound of the two. Matter is potentiality, while form is realization or actuality, and the word actuality is used in two senses, illustrated by the possession of knowledge and the exercise of it. Bodies seem to be pre-eminently substances, and most particularly those which are of natural origin; for these are the sources from which the rest are derived. But of natural bodies some have life and some have not; by life we mean the capacity for self-sustenance, growth, and decay. Every natural body, then, which possesses life must be substance, and substance of the compound type. But since it is a body of a definite kind, viz., having life, the body cannot be soul, for the body is not something predicated of a subject, but rather is itself to be regarded as a subject, i.e., as matter. So the soul must be substance in the sense of being the form of a natural body, which potentially has life. And substance in this sense is actuality. The soul, then, is the actuality of the kind of body we have described. But actuality has two senses, analogous to the possession of knowledge and the exercise of it. Clearly actuality in our present sense is analogous to the possession of knowledge; for both sleep and waking depend upon the presence of soul, and waking is analogous to the exercise of knowledge, sleep to its possession but not its exercise’. pp. 67–69.
As you can easily see, there are two important major distinctions here. First, the fundamental one, between what is potential and that which is actual. Pure matter, in Aristotle's view, is pure potential and pure actuality is pure form. (Do not take my use of the adjective ‘pure’ as Aristotle's. I have used it here just to state matters in a certain slightly extreme way.) The second is that there are two senses of actuality, for example, possessing knowledge and exercising it. I can have the knowledge of someone's date of birth but not be thinking about it at the present moment. If I search for it and find it, I am then exercising and using that knowledge. This is an easy and common distinction. Aristotle wants, of course, to say that the soul is actual primarily in the sense of possession. He says in the next sentence, after the quotation I have given,
Now in one and the same person the possession of knowledge comes first. The soul may therefore be defined as the first actuality of a natural body potentially possessing life; and such will be any body which possesses organs’ p. 69.
Summing up again in somewhat different words, with what I regard as a rather poor translation using the word formula, which I have noted here seems best to mean function, we get the following passage:
‘We have, then, given a general definition of what the soul is: it is substance in the sense of formula [function]; i.e., the essence of such-and-such a body’. pp. 69–71.
Here is a further passage amplifying this general definition, in the same paragraph:
‘… We must, however, investigate our definition in relation to the parts of the body. If the eye were a living creature, its soul would be its vision; for this is the substance in the sense of formula [function] of the eye. But the eye is the matter of vision, and if vision fails there is no eye, except in an equivocal sense, as for instance a stone or painted eye. Now we must apply what we have found true of the part to the whole living body. For the same relation must hold good of the whole of sensation to the whole sentient body qua sentient as obtains between their respective parts. That which has the capacity to live is not the body which has lost its soul, but that which possesses its soul; so seed and fruit are potentially bodies of this kind’. p. 71.
To repeat what I said above, because form suggests too geometrical an image, spatial form or shape, it is often better to translate formula not as formula but as function. So we could now say the function of the eye is to see. If vision fails, then the eye is no longer functioning, ‘except in an equivocal sense’, to quote Aristotle's phrase.
I now move on, also skipping over Chapter V of Book II, in which Aristotle begins his discussion of sensation, or, as we would say, perception.
Perceiving and thinking
In Book III, I concentrate only on the parts dealing with perceiving and thinking. Aristotle asks how can we perceive or judge the difference between objects of different senses, e.g., how do we perceive that sweet and white are different? Notice that this problem is a dual of the famous binding problem in neuroscience. When the brain perceives something white and something sweet, how does it put these sensations together to judge that what is being perceived is a white peach? Aristotle continues,
‘Evidently, therefore, it is impossible to pass judgement on separate objects by separate faculties; … in so far as the judging faculty is indivisible, it is one and instantaneous in action; but in so far as it is divisible, it uses the same symbol twice at the same time. In so far, then, as it treats the limit as two, it passes judgement on two distinct things, as being itself in a sense distinct; but in so far as it judges of it as only one, it judges by one faculty and at one time’. pp. 153–155.
Discussion so far in this chapter applies to all animals that can sense something, an important point in arguing for the naturalness of mathematical thinking about the world. This ends the first two chapters of Book III.
More on thinking
Aristotle begins Chapter III with the remark that older philosophers, such as Empedocles, identified thinking with perceiving, for in both cases we have an object that exists. He has a quick response to this view.
‘Now it is quite clear that perceiving and practical thinking are not the same; for all living creatures have a share in the former, but only a few in the latter. Nor again is speculative thinking, which involves being right or wrong — ‘being right’ corresponding to intelligence and knowledge and true opinion, and ‘being wrong’ to their contraries — the same thing as perceiving; for the perception of proper objects is always true, and is a characteristic of all living creatures, but it is possible to think falsely, and thought belongs to no animal which has not reasoning power; for imagination is different from both perception and thought; imagination always implies perception, and is itself implied by judgement. But clearly imagination and judgement are different modes of thought’. p. 157.
Imagination
Aristotle now turns to imagining, saying that thinking is mainly comprised of imagining and judging. He first gives a clear argument as to why imagination is not sensation.
‘If imagination is (apart from any metaphorical sense of the word) the process by which we say that an image is presented to us, it is one of those faculties or states of mind by which we judge and are either right or wrong. Such are sensation, opinion, knowledge and intelligence. It is clear from the following considerations that imagination is not sensation. Sensation is either potential or actual, e.g., either sight or seeing, but imagination occurs when neither of these is present, as when objects are seen in dreams. Secondly, sensation is always present but imagination is not. If sensation and imagination were identical in actuality, then imagination would be possible for all creatures; but this appears not to be the case; for instance it is not true of the ant, the bee, or the grub. Again, all sensations are true, but most imaginations are false. Nor do we say ‘I imagine that it is a man’ when our sense is functioning accurately with regard to its object, but only when we do not perceive distinctly’. p. 159.
This clear characterization of imagination is followed by further useful distinctions.
Thinking and mental images
Table I move on to Chapter IV of Book III, which is mainly about thinking.
‘Concerning that part of the soul (whether it is separable in extended space, or only in thought) with which the soul knows and thinks, we have to consider what is its distinguishing characteristic, and how thinking comes about. If it is analogous to perceiving, it must be either a process in which the soul is acted upon by what is thinkable, or something else of a similar kind. This part, then, must (although impassive) be receptive of the form of an object, i.e., must be potentially the same as its object, although not identical with it: as the sensitive is to the sensible, so must mind be to the thinkable’. pp. 163–165.
Delaying for now any comments, I go directly to a clarifying passage in Chapter VII on how the mind thinks.
‘So the thinking faculty thinks the forms in mental images, and just as in the sphere of sense what is to be pursued and avoided is defined for it, so also outside sensation, when it is occupied with mental images, is moved. For instance in perceiving a beacon …’, pp. 177–179.
Aristotle's summary
Finally, I cite the entire Chapter VIII which sums up nicely how sensing and thinking are postulated to work.
‘Now summing up what we have said about the soul, let us assert once more that in a sense the soul is all existing things. What exists is either sensible or intelligible; and in a sense knowledge is the knowable and sensation the sensible. We must consider in what sense this is so. Both knowledge and sensation are divided to correspond to their objects, the potential to the potential, and the actual to the actual. The sensitive and cognitive faculties of the soul are potentially these objects, viz., the sensible and the knowable. These faculties, then, must be identical either with the objects themselves or with their forms. Now they are not identical with the objects; for the stone does not exist in the soul, but only the form of the stone. The soul, then, acts like a hand; for the hand is an instrument which employs instruments, and in the same way the mind is a form which employs forms, and sense is a form which employs the forms of sensible objects. But since apparently nothing has a separate existence, except sensible magnitudes, the objects of thought — both the so-called abstractions of mathematics and all states and affections of sensible things — reside in the sensible forms. And for this reason as no one could ever learn or understand anything without the exercise of perception, so even when we think speculatively, we must have some mental picture of which to think; for mental images are similar to objects perceived except that they are without matter. But imagination is not the same thing as assertion and denial; for truth and falsehood involve a combination of notions. How then will the simplest notions differ from mental pictures? Surely neither these simple notions nor any others are mental pictures, but they cannot occur without such mental pictures’. pp. 179–181.
Comments on Aristotle and Wigner
My most important comment on Wigner's analysis of the effectiveness of mathematics in developing natural laws and theories is to note his neglect of any psychological concepts or theories relevant to thinking systematically about the world. It is for this reason that I concentrated on Aristotle's famous treatise. My purpose was to show that sophisticated ideas about the nature of perceiving and thinking were as old as comparable ideas about mathematics and astronomy. Of course, just as now and then as well, psychological theories are not as advanced as physical ones. But this does not mean there are not serious and useful psychological ideas about how the mind or brain works in doing mathematics or science. Aristotle's doctrine of form as central to both perceiving and thinking provides already the key idea undoubtedly behind even the earliest developments of geometric representations, and is just as present in thinking about modern quantum mechanics or astrophysics. The key notion, put in modern terms, is that of structural isomorphism between, in one case a perception and an object or process in the world, and in another, between a mental image and an ‘abstract’ structure. Such isomorphisms are indeed as important, or perhaps even more important, to modern pure mathematics as they are to theoretical physics. How such isomorphisms are physically implemented in perceiving and thinking was beyond Aristotle, and they have also been beyond most of modern psychology. But in his concept of form, Aristotle had the right conception of how such a physical process would have to work in perceiving. Just as significant was his extension of the same ideas to thinking, and thereby to the realization of the critical role of mental images in all of thought.
Moreover, his passage about the ‘abstract’ concepts of mathematics in the final quotation I gave from On The Soul reflects, in my judgment, the correct attitude toward mathematical thinking. It is a natural and continuing extension over thousands of years of our stock of perceptual and cognitive concepts. But there is no agreed-upon cognitive metric that convincingly shows that today's Hilbert-space computations in quantum mechanics, for example, are conceptually more difficult than those of spherical trigonometry in the time of Menelaus and Ptolemy.
Finally, although having the greatest respect for the extraordinary developments of physics in the past 300 years, I am not at all persuaded that physics occupies a unique position among theories of nature, including human nature, or that we can expect in the future, what we do not have now, anything like proofs of the uniqueness of the most important physical theories. The level of controversy and uncertainty in modern astrophysics should certainly give anyone pause in making strong claims about such uniqueness any time in the near future. What we can expect, in another direction, is ever better scientific theories of how the brain thinks, even though it will surely take more than the present century to bring neuroscience to the level now evident in the theories of modern physics.
Note
1 Hereafter I cite only the page numbers in the translation listed in the references.
Notes on contributor
Patrick Suppes is the Lucie Stern Professor Emeritus of Philosophy at Stanford University. He has published widely in philosophy of science and the social sciences, especially psychology. He is currently doing research on the brain, with emphasis on language and emotion, and the physical mechanisms of brain computations. He is a member of the US National Academy of Sciences. His last book appeared in 2002, Representation and Invariance of Scientific Structures.