Wednesday, February 11, 2009


... the map is not the territory...(Alfred Korzybski)


One day, Alfred Korzybski the inventor of the theory of general semantics was giving a lecture to a group of students, and he suddenly interrupted the lesson in order to retrieve a packet of biscuits, wrapped in white paper, from his briefcase. He muttered that he just had to eat something, and he asked the students on the seats in the front row, if they would also like a biscuit. A few students took a biscuit. “Nice biscuit, don't you think”, said Korzybski, while he took a second one. The students were chewing vigorously. Then he tore the white paper from the biscuits, in order to reveal the original packaging. On it was a big picture of a dog's head and the words “Dog Cookies”. The students looked at the package, and were shocked. Two of them wanted to throw up, put their hands in front of their mouths, and ran out of the lecture hall to the toilet. “You see, ladies and gentlemen”, Korzybski remarked, “I have just demonstrated that people don't just eat food, but also words, and that the taste of the former is often outdone by the taste of the latter.” Apparently his prank aimed to illustrate how some human suffering originates from the confusion or conflation of linguistic representations of reality and reality itself.

For Bohm this confusion was not limited to words but extended also to mathematical models.  He referred to Korzybski’s thesis that maths is a limited linguistic scheme and that necessarily what we say about a thing in this language is always less, than the thing itself, the thing is always different from what we say, always more than what we say. If reality stopped exhibiting features that are not in our thought, then the distinction between reality and our thought would cease, but the notion of objective reality, which proposes that reality has an existence apart from us precludes that possibility. Despite the declarations of some physicists, mathematics is an abstract system that cannot cover the whole of reality. Different kinds of thought and different kinds of abstraction may give us a better understanding of reality than the exclusively mathematical type. To borrow David Peat’s image, what Bohm was talking about is reminiscent of René Magritte’s painting Ceci n’est pas une Pipe (a painting designed to question our tendency to confuse reality and representation) insofar as he believed that every theory of the universe should have the caption ceci n’est pas un univers. It is not a universe because it is just a model. There is no possibility that a model of the universe could equal or replace the universe: model and reality are eternally distinct. And this disjunction, moreover, goes for every use of a mathematical model whether in science, economics or whatever. It almost seems fatuous to make these points, but they are made because the confusion of model and reality is an ever-present danger in our ‘scientific’ approach to our affairs.

These words are being written as one of the worst recessions in modern history begins to bite and begins to look like a 1930s-style Depression. It is a sobering fact that this economic crash was caused, in major part, by the rash confidence of investment bankers who were using exotic mathematical models derived from rocket-science. These so-called ‘masters of the universe’ were using their models to construct exotic financial instruments for the creation of wealth out of its opposites, debt and poverty. The mathematical models proved to be all too fallible precisely because they aroused the confidence of investors in wholly illusory possibilities of risk-free speculation. As these mathematical models were applied more and more and on a large scale by computers without the intervention of human beings, their inadequacies – the things they left out – compounded  their damaging effects and the result was a near total collapse of the world’s financial system. This is just one example of the possible disasters lurking behind the seductions of bewitching  mathematical models. There are many others, but the point is this: in the urge to mathematise reality as far as possible and in a culture for which mathematics is almost the only intellectual authority, the dangers are evident: they are dangers of a wholly misleading precision fostering decisions made on the basis of imperfect information. The problems of the planet are in large part caused by this kind of decision. More of the same kinds of decision can hardly be expected to help. I what follows a modest attempt will be made to put our passion for mathematical reasoning in its place.

Those with a mathematical bent should not expect to find any interesting mathematics here, for there will be none; and those who are allergic to mathematical squiggles need not worry, for the same reason. The point here is simply to show firstly, that everyone is a mathematician, just as everyone is a philosopher (all things being equal), it’s simply a matter of following the natural bent of the mind; and secondly that our dependence upon mathematics indicates in many ways that it is almost an essential image of fallible human understanding as such. The philosophically interesting issue here is why this should be the case. Why should we owe so much to mathematics?


Models and Pictures: Our Urge to ‘Represent’


Whenever we say of something that it is ‘round’ or ‘square’, or whenever we say of a group of some things that there are ‘a couple’ or ‘one or two’ of them, we are showing our everyday dependence upon mathematical modes of thought. We do it from our very earliest age. Little children, when they first start to draw trees or people, do so in a very stylised manner which is substantially mathematical, involving the simple ‘things’ of mathematics, points and lines. They represent a face as a circle with three or four points in it. They represent a tree as two vertical parallel lines with a circle on top – a kind of lollipop. They represent a human being as an inverted ‘V’ with a straight line rising from its apex to a circle and a horizontal line drawn across the vertical one – the ubiquitous ‘stick’-person. The resemblance between these drawings and what they are supposed to represent is not very great, but they are instantly recognisable, nevertheless; and no-one hesitates to use the word ‘is’ in statements such as that is a man’ whereas it is no more than a few lines. We are clearly simplifying the resemblance between the things we see before us to a very great extent, reducing things that are very complex shapes to the simplest of geometrical patterns. We have been doing this sort of thing for a long time as the abstract patterns of ancient rock art show, but why we do it is in itself a very deep and interesting question. Answering it would give us insight into the origins of mathematics.

We might ask, ‘where does mathematics come from?’ One ‘obvious’ answer is, ‘from our minds.’ That doesn’t get us very far. But it’s a start. Another possible answer is ‘from the world’; but no-one has ever found an equation lying under a hedge. If mathematics does come from our minds, then we are naturally disposed to count similar things, things that appear to us to exist in numbers greater than one. The repetition of similar appendages of our own bodies could well have been among the first similar things we spotted in early childhood. The realisation that we possess the same number of fingers on our hands as we do toes on our feet must have been an illuminating discovery to our early ancestors. The fact that the word ‘digit’ means both ‘number’ and ‘finger’ is revealing and there is obviously a connection between our fingers and our use of a number system to the base ten. We must have developed the practical need for simple geometrical and numerical ideas as soon as we began to cultivate land, keep animals, make tools and construct shelter; but the most obvious origin of the use we make of more complex geometrical forms is probably our fascination with the sky and the objects we find in it both during the day and at night.

The sun and the moon are discs that move in apparently repetitive, curved paths across the sky with great regularity, with a periodicity that invites us to count. The ‘stars’ do the same, though some of them are not stars but planets and do more complicated things than simply describe arcs. The sun and the moon themselves are the most prominent natural circles in our world. At all events, one of the first insights of the earliest major civilisations was mathematical abstraction from the regular movements of heavenly bodies and the imposition of the idea of circles on their movement, in a directly analogous way to that in which children represent faces as disks and trees as lollipops. The earliest astronomers, simply by observation, decided that the heavens are organised on the basis of circles: when one completes the visible arc of the sun, one must conclude that wherever it goes when it disappears, the simplest explanation is that it moves in a circle. Ditto for the moon, and the same then goes for the stars, or at least most of them.

Thus it became ‘obvious’ to these early observers that the entire universe was organised on the basis of things moving in circles, except the earth which obviously did not move. It’s not surprising therefore, that the first comprehensive model of the universe was that of a spherical set-up with concentric spheres containing all the various bodies we espy in the sky above our heads. For the Ancient Greeks, the fact that the heavenly bodies moved in circles, in a perfectly orderly manner, and didn’t just tumble to the ground like earthly objects, proved that they were made of more noble stuff than the things on earth. Now we know that these early astronomers were ‘wrong’, that the heavenly bodies do not move in perfect circles, and that some of the circular movements are only apparent. Moreover, we know that the ‘falling’ of earthly bodies is directly similar to the ‘falling’ of heavenly bodies. But were the ancients so wrong? Their insight, that the world could be modelled by means of geometrical shapes and arithmetical concepts was the important thing and successive models of the universe have largely become more ‘right’ because they have refined the original insights of those pioneers, replacing circles with ellipses, for example, expressing geometrical notions algebraically, showing which bodies appear to move, relative to a particular frame, and which do not, and so on.

The essential character of the process of understanding in this way, however, has always remained the same. It has to do with comparison. We not only compare natural objects and events to each other and to mathematical abstractions thought to capture their similarities, we also compare them to things that we invent and that are not real in the sense that objects are real, but to which we attribute a higher sort of reality; and then we compare natural events and processes to these things – as in myths, for example, and in mathematical models. What precisely is going on in all this? Why do we tend to attribute more reality to the repetitions of our abstractions than to what we actually experience as, for example, when we consider the abstract, mathematical ‘laws of nature’ as having more reality than the changing events they govern? This is a complex question, but it certainly has to do with the ‘reliability’ of mathematical objects: they are predictable and one knows that two mathematical abstractions that are identical are truly identical without any possibility of hidden differences. This dual problem of apparent similarity and hidden difference is the great bugbear of our cognitive approach to the world. At least what is on either side of the ‘=’ sign in an equation is really the same and not just similar.

What is going on in the eternal tendency of our imagination to try and compare different things and to say that this is ‘like’ that? In our comparisons, we get the impression that we have latched on to some invariant feature of different processes, some unchanging feature of the world. Most frequently we even drop the ‘like’ and just say, “this is that,” – just as a child, pointing to its lollipop may say, “this is a tree.” Our fundamental desire to understand is closely related to our desire to find natural correspondences to find occasions for detecting essential similarities between quite disparate items of our experience. In mythology, we likened the thunder of the storm to the bellowing of a bull, we likened the wind to the chords of a harp, we likened the curling rollers that crash on the beach to the charging chariots of war, we likened the constellations of the night sky to the protagonists of our legends and so on. But we became dissatisfied by these likenesses: though there was perhaps a satisfying moral truth in many mythologems, there was always something deeply unsatisfying in the clear differences involved in the specific comparison. The profound similarities in the comparisons were insufficient to keep us happy. We were always able to see where our comparisons succeeded and where they failed. In our geometrical and arithmetical analogies, however, we seemed to be able to see directly where the comparison succeeded; and the failure (e.g. the lack of a moral dimension to the comparison) was less important. In conceiving of the movement of the heavenly bodies as circles, we could see directly that we were on to something deep.

The most satisfying feature of these comparisons more geometrico, however, was the precision, rigour and sense of control that we were able to bring to our analogies. Once we got the habit of inventing abstractions, we began to get interested in the internal properties of these abstractions themselves in the patterns that our imagination espied within them. We were able to ‘prove’ certain things by demonstrating that the mathematics gave rise to constant relations, ‘necessary truths’, as some optimists called them, that were true simply by virtue of our thinking about them. We didn’t have to do the legwork and go scouting around for the evidence; we simply extracted it by thought. The necessary truths were obviously true in an irrefutable sense. They just ‘had to’ be true; and if people denied this they contradicted themselves and demonstrated their ignorance. The properties of a circle were demonstrable simply by pure reflection on what was obvious and we did not have to adduce examples to make the case. The fact that many different items could be seen as moving in circles, for example, and the fact that circles could be demonstrated mathematically to have the same properties in all possible circumstances gave us a powerful tool for seeing regularities in the processes of the world around us.

By comparing natural phenomena with mathematical models, we invented a most productive method of talking about the ‘likenesses’ between various phenomena and their common ‘likeness’ to our mathematical conceptions. The reason why this kind of comparison was so much more satisfying than the mythological comparisons was that we could demonstrate with convincing rigour the exactitude of our conception without any fuzzy bits hanging out. The fuzzy bits were just chopped off. And moreover, we could remain in complete control of the images. The mathematics worked out with complete irrefutability and anyone could see the force of the analogy between the observed phenomenon and the mathematical model. The mathematical model could therefore be put in place of the observed phenomenon and more mathematical deductions could be made from the model in order to come to conclusions about the universe as a whole. We could begin to make predictions about devices that we made – such as the antikythera mechanism and the later mechanical timepieces that in many senses were miniature universes working on the principle of circular movement – and once we had mastered that trick, not only was our technological Pandora’s Box open, the delicious dream of controlling our environment by means of abstract models seemed close to realisation.

Once we had developed the skill of modelling reality in mathematics, we were off on the passionate quest that we call ‘precise science’ and the modern world of amazing technology had begun. But the method that has permitted this modern world certainly goes back at least to the ancient civilisations of Babylon and Greece and probably further back to a long gone civilisation of the Indian sub-continent. The essence of the method is closely similar to the child’s modelling of complicated items of its experience, faces as disks, trees as lollipops, mountains as triangles, houses as cubes and so on, with simple geometrical shapes. The method proved fantastically successful in classical physics and is still operating, though with far more exotic geometries, in the world of quantum physics, where the mathematics involved is vastly more complicated than the simple plane and solid geometry of Euclid.

The really interesting questions concerning our use of mathematical models are questions such as these: 1) why do we have this urge to compare different things and to say ‘x is like y’ or even ‘x is y’?  2) why do we instantly reach for mathematical concepts and models in our urge to compare? 3) since the mathematics by means of which we refine our conceptions of reality is thought up, in many cases, for its own sake and long before the observations to which we find we can apply it, where does it originate?

There is a deep issue here that concerns the very nature of our intellect and our irresistible urge to apply our minds to our environment in more than routine ways. The animals around us do incredibly clever things and show staggering abilities. But they are to a large extent both limited in and to a definite range of behaviour and cannot adapt outside of the environment in which their skills have sense. We, by contrast, have fewer such pre-programmed patterns of behaviour and apply our eternal tendency to find correspondences and similarities to every aspect of our experience. In this we discover the regularities of our natural environment, become able to predict its behaviour and thereby acquire the ability to manipulate it. By means of our comparisons we become masters of our environment in a broader sense than is available to any of the other animals with which we share the planet. Yet this ability seems capable both of setting us apart from nature and also of establishing deeper links between us and it than could ever have been provided by a purely instinctive life.

But where does this ability come from that we possess not only to compare the regularities of the environment with mathematical models, but also to create entirely new mathematics and new models that both follow and drive the expanding range of our experience?

Why mathematical models? We can simply throw up our hands and say, ‘well, our minds just work like that!’ and give up the urge to understand. Philosophy, however, wishes to understand everything, even our methods of understanding. The puzzle gets even deeper when you realise that mathematics is almost miraculously successful at modelling the processes of the universe that we can experience and also in uncovering some we cannot – at least in physics which describes the behaviour of matter. There is a deep truth in our discovery of mathematical regularities in the universe that we have not yet got to the bottom of. The physicist Dirac believed that by following the principle of what he called ‘beauty’ in our equations, we could be confident that we were moving towards deeper understanding of the world. Mathematics is by far the best method of describing the physical processes of the world, since these processes seem in themselves, at least to the extent that we can observe them, to work in many ways on deep mathematical principles. We ought to be cautious of such a perfect fit between the world and our minds, however, and at least entertain the idea that we might only be seeing what we are programmed to see. Our tendency to interpret our experience in accordance to what we wish to find in it is notorious. The chief disadvantage with our mathematical way of understanding is that often we confuse the precision and rigour of the maths, that we happen to be using, with the processes that we are observing. We have, as observed, a deep tendency to confuse the model with the reality, description with thing described, midworld with foreworld. Once we realise that we have this tendency, however, we are liberated anew to generate yet more mathematical models. The way in which we do this is profoundly wonderful and clear demonstration that our minds are in some sense always above and beyond any formal language that we may use to express our thoughts. Our minds are in some almost miraculous way superior to all the forms in which we think. Herein lies a conundrum of the most fascinating kind.

The origin of mathematics would seem to be both in the enumeration of similar items and in the exploration of shapes and solids by the infant, in the discovery of the relationships and differences between open, closed disjoint and intersecting figures, between squares, triangles and circles. It would seem to lie in the detection of patterns of resemblance between 3D objects. This is not an abstract intellectual process, but a process of active sensory experience and physical, muscular discovery. The manipulation of objects, internal visualisations, the ability to distinguish between various types of sizes, areas, volumes etc. involves not abstract reasoning, at the outset, but much more simple physical sensation. The perception of an object is almost one with the intention to grasp and manipulate it. Imagine the mathematical principles encoded in the motor skills of the gibbon that flings itself with such astonishing athletic ability and precision through the trees, swinging at enormous speed through the branches and rarely making mistakes in its estimation of trajectory and force of launch or distance of landing sites. There is no point in saying that maths has no role to play in a gibbon’s gymnastic skill because that forces one to say something like, ‘that’s just what gibbons do, it’s instinct’ or some such lame nonsense. That fact is, if we wanted to construct a gibbon or even to reverse-engineer a real one, we would have a very complicated mathematical task on our hands. Imagine then the gibbon’s discovering intellectually through the invention of an abstract mathematical formalism these depths of mathematical potential in its own brain.

Something analogous to such a discovery must have gone on in the development of mathematics from the simple activities of our prehistoric forefathers: the counting of animals and measuring areas of fields, of quantities of grain and the best dimensions of tools. The discovery of mathematics extends conscious thought down through the musculature of the body and beyond into the very structure of matter itself. This may be the origin of the celebrated mathematical ‘intuition’ that is at the heart of all truly creative mathematical thinking. The algorithmic, formalised part of mathematical thought is only its public, consensual part. The body of knowledge known as ‘mathematics’ is no more than the frozen traces of countless spasms of creative frenzy that still continue alongside the discipline unabated. The real work of generating mathematical insight goes on in the conscious and unconscious uncovering of principles that are encoded in our bodies, in the memory-traces of the interaction of these bodies with the material world and in the structure of the stuff out of which both we and the world are made. Perhaps situating the origin of maths vaguely in our minds is totally misguided: perhaps maths is inherent in the structure of the physical world of which our bodies and our brains are a part. Here again, the spurious distinction between subject and object is abolished.

The self-similarity of the universe from top to bottom guarantees that mathematical principles of fundamental similarity and applicability will be found at all levels. Small wonder then that the mathematical principles that we invent are applicable to all parts of the world of our experience. Like the self-similarity (with rich, subtle differences) of all levels of a piece of fractal geometry, the universe must give evidence of its fundamental unity in the reappearance of similar mathematical regularities in systems of great diversity, from sub-atomic particles to the behaviour of large numbers of people.  The science of chaos-theory has taken mathematics beyond the world of three dimensions as boldly as it has taken the discovery of mathematics beyond the deductions of formal reasoning into realms of empirical investigation. It may be that now, with the strange non-Euclidean and fractal geometries, mathematics has finally been cut loose from our experience of the 3D world and our weddedness to it. It may be that now mathematics, thus cut loose will open up vistas of possibility unimagined in the past. If this is so, then our mathematical ability would seem to be an ability that takes us in principle beyond the limitations of the body, not only by prosthetically extending the body, but also by transforming the mind and enabling us to overcome the mental habits of millennia of our forefathers. This suggests that there is some agency in the mind that from a vantage beyond the physical is constantly transforming and extending us and taking us beyond the evidence of our senses. What then could be going on?


The Limits of Our Mathematical Thinking


There is a very deep connection between 1) the human tendency to go beyond the evidence in any theory of the world, 2) the so-called mind-body problem, and 3) the famous Incompleteness Theorems of Kurt Gödel.

Obviously, as one convinced of the imperfections of language and of its inability to provide a structure corresponding to our conception of what truth should be, I’m not about to elaborate a neat and tidy theory with all the ends tied up about how these three things fit together. Language is only good at the description of three-dimensional objects and their everyday disposition or movement in space. We apply concepts derived from the objects of sense-experience to entities of which we could never have had experience, and we could never have had experience of them simply because such entities have no presence in the world of three-dimensional objects. Thus language is strained and begins to crack up. Yet we do it nevertheless, and do it with much greater enthusiasm and persistence than ever we applied to the description of objects in front of our noses. The whole world of abstractions, the world of universals, the world of so-called a priori concepts is manipulated by us according to logical concepts and rules that are derived from the world of three-dimensional objects. In mathematics, we have the freedom to vary the rules and can use that very logic and those very rules to go beyond the world from which it and they are derived. We can think up geometries which are not-three-dimensional simply by dropping this or that assumption which seems intuitively to belong to logic because logic too is derived from the three-dimensional world. We drop the fifth axiom of Euclidian geometry (the one about parallel lines meeting at infinity, which is the weakest of the five) and hey presto! we can dream up abstract worlds in which parallel lines meet in all sorts of circumstances, where space is curved, where triangles have internal angles adding up to more or less than two right-angles and so on. On the basis of such uncommon-sense geometries, we can then construct models of the physical universe which stack up mathematically and which fit the observational evidence, but which correspond not at all to our pre-programmed way of imagining reality that still handicap our logic. Once one masters the significance of imaginary numbers, and other exotic mathematical notions, the maths gets ever further from thelogic of everyday sense-experience. This is why the most modern theories of physics seem unacceptable or unsettling to some: they appear illogical.

The state of modern physics is such that its best theory of everything seems immune from experimental validation. Critics of String- or M-Theory say that this lack of experimental corroboration means that the theory is not physics at all but philosophy. But it was ever thus. Theory has very often led the way in science. Remember Dirac! The theory is dreamt up to account for the facts as they are known, sure, but every theory validated by experiment always throws up new and inconvenient facts which destroy the integrity of the theory. For this reason, theory will always be speculative and go beyond the evidence of the senses. The question is: how do we perform the trick of imagining and then discovering depths to reality that our sensory experience could never have opened up to us? From what position in the world do we dream up these things, if not in some sense from outside of it? We certainly do not dream them up from within any closed system of deductions: not from within a closed system of physically limited experience and not from within a closed system of mathematical possibility. How can that be?

One can imagine the whole of mathematics as a system which validates itself and which is then both internally self-consistent and complete, i.e. safe from new discovery. The basic axioms would be demonstrated from the fundamental logic of the system and all its theorems would then be derived from the basic axioms. Such a beautifully tidy and definitive system is still the Holy Grail of some pure mathematicians. When such a system is presented to certain physicists – who are completely dependent upon mathematicians – they are likely to be encouraged in their ambition to write the fundamental algorithm of the universe. The alliance between mathematician and theoretical physicist would then seem to hold out the possibility of definitive and absolute truth being attained, truth which would be self-validating, since the language in which it is written is a self-validating system.  Unfortunately, the great mathematician Kurt Gödel demonstrated by his famous theorems that such a dream is a pipe-dream. He pointed out that in any system of mathematical formulations more complex than arithmetic, there are bound to be some axioms which are used and understood, but which can not be proven from within the system. The upshot of this is the truth that for any mathematical language more complex than arithmetic, the user is always in some sense beyond or above the system, because he is using knowledge which is not derived from within the system. He is always going beyond the evidence within the system; and he is thus superior to the system in the sense that he is beyond it. And there is no purpose served in saying that he is simply in a super-ordinate and higher system, which is complete and internally self-consistent. The reason for this is that in order to manipulate such a higher system, one has to be beyond it in exactly the same fashion. So we are always beyond and above our logic and our maths. If we describe our experience in the language of mathematics, we are always beyond and above the resultant system of ‘truth’. This probably means that we are inevitably beyond and above all our truth.

What does this ‘beyond’ and this ‘above’ mean?

Well this brings me back to the statement used earlier: there is a deep connection between the mind-body problem, Gödel’s theorems and the human tendency to go beyond the evidence in any picture of the world. In Descartes’ view, the mind, since it had properties different from and opposite to those of the body, was a different substance from the body. Being a different substance, it did different things. One of the different things it did was to think of the experience of the body in terms of the eternal and necessary truths of God. In a sense, then, the mind was always above and beyond the body in that it had access, in a way that the limited body could never have, to a vantage-point from which it could contemplate reality without the constraints of special and temporal finitude. I see no reason at all why the mind should not be conceived of as logically utterly different from the body as 3D object, in the manner suggested by Descartes, despite the notorious difficulties concerning the interaction of the two, which the grand man simply ignored. (The contemptuous speculation about weird transactions in the pineal gland is no more than a provocation.) If the mind is always above and beyond any system of description of physical objects, there is a real sense in which the mind is in different dimensions from the body, if the body is conceived of as a 3D object.

One has only to think of the fact that human beings are judged when all is said and done not by their physique, nor by their possessions, nor by their power, nor by their accomplishments, but by their moral qualities. To take a homely example, “the spirit is willing but the flesh is weak” we say; and we thereby state the schism between mind and body. But if the spirit is after all only body, how can a thing be both weak and strong in the same sense? If the body is strong and the sprit weak there is patently a distinction to be made between the two. We can describe the human mind in terms of 3D objects located in the three dimensional space inside our skulls until the cows some home and we will never eradicate this distinction. We simply cannot shake off this unshakeable intuition that the mind is different from the body in the face of the best arguments that it is not. Gödel’s theorem and our innate (and justified) tendency to go beyond the evidence of our body-mediated experience seem to suggest that there is something in our belief. What then is the point of sticking dogmatically to the belief of the eliminative materialists that ‘essentially’ the mind is just an object?

But let us return to the flipside of this freedom for a moment, to our dogmatising tendency to confuse our models with what is modelled. Take the concept of ‘necessity’, i.e. the idea that some truths just ‘have to’ be true and cannot not be true. This is a feature of mathematics, not of the world. If you disagree with a mathematical equation that is done correctly, then for a mathematician that is because you are stupid or ignorant or both. The correct calculation cannot be wrong. Now when we apply that calculation to a natural process, we have the impression that the rigour and necessity of the calculation is the same as the necessity of the process that we observe. At that point we say of this or that natural process “it just has to be that way!” and we think that we have found an example of natural necessity, something in nature that just has to come about in the way that we think it comes about. We confuse the necessity of the calculation with the properties of the phenomenon observed. This can make us dogmatic and certain where we have no right to be certain. The certainty of the mathematics misleads us into thinking that we are certain about the natural phenomenon. This sometimes makes us arrogant and likely to embark on ill-advised courses of action, like the ‘masters of the universe’ mentioned earlier. But this unfortunate tendency arises precisely because maths is so staggeringly successful at modelling the world and because we can come up with ‘right’ answers to our questions rather than with hit-and-miss analogies. So there are huge advantages and real disadvantages in our method. But all of this simply evokes anew the question “where do we get the mathematics from?” Now we are not going to answer that question in these few pages. But we can continue to ask the question and raise some of the interesting issues that surround it.

It almost seems as if our tendency to use mathematics as models for reality and the tendency of the world to conform, at least apparently, to our mathematical models, indicates a kind of pre-existent harmony between the world and our minds. We don’t simply find the mathematics in the world, as we simply find rocks and trees, even though some things such as crystals, orbiting moons, traces in the bubble-chamber and so on, look pretty mathematical in themselves. We have more frequently invented the mathematics first and then found that we could apply it to the world. The most complicated mathematics arises often from the purely mental operations of gifted and creative mathematicians who dream these things up, spin them out of their minds like weirdly creative spiders, simply because they follow the implications of a particular line of thought. The biography of the philosopher Blaise Pascal is very instructive in this respect. His father thought it wise to keep him away from mathematics until he was older, so the little boy just went ahead and invented a lot of it for himself. We could also cite the case of the weirdly talented Indian mathematical genius, Ramanujan who had little education but who amazed the Cambridge establishment with his staggering mathematical creativity.

We have a passion for mathematical invention and we often discover, after the mathematics has been invented, that we can use it to model ever more complicated and subtle processes of nature. This process seems endless. The more complicated the mathematics we invent (or is it ‘discover’?), the more the reality we apply it to seems to us to be subtle and complicated. There seems to be a feedback loop involved: as the maths gets more complex, so does the world. We are now at a frontier in our understanding of the complicated world of ‘string-theory’ because we lack a sufficiently powerful mathematics to deal with the problems it poses. But that mathematics will doubtless be discovered (or ‘we’ will invent it) and the process of understanding by means of mathematical models will take off again and lead us into an understanding of new, unsuspected depths to the natural world – as long, that is as we can make our hypotheses conform to experiment. There seems no possible end to this process of discovery.

Why do we do this? Where does the mathematics come from? If it comes from our minds, then how can we be sure that it really does apply to the world? How can we be sure that we are not deluding ourselves? If it comes from the world, how do we discover it? We do not simply ‘read it off’ from nature, because nature does not appear mathematical to our eyes, it’s too messy. We have to impose simple mathematical models that distort reality before we discover more complicated and ‘truer’ models by a process of refinement. This is certainly not ‘reading off’ mathematics from what we see around us. Take pi for example. Everywhere in physics where rotations, i.e. circles, are of importance in the modelling of physical process, pi is a vital number to our understanding. And yet we do not understand it. Perhaps if we were creatures with eight fingers and eight toes we might see some pattern in its extension. But for our base ten numbers system, pi is irrational and we can’t get a handle on it. Since we can’t get a handle on it, we work at the problem in purely theoretical ways and that theoretical work illuminates our exploration of nature. The point here is that it doesn’t work the other way around: we don’t discover pi in natural circles, we discover circles, we discover pi as an important number to understand them and then we extend our understanding by mere calculation, not by observation (the discoveries made by observing a computer that is working through the complex implications of a calculation too long for us to do is a different matter). This nicely illustrates the manner in which we apply maths to reality without understanding how the two relate, but feeling that in some wonderful way they do. We just ‘know’ that there is an intimate connection between our delight in doing maths and our delight in finding that the world responds to mathematical description. The origin of maths would then seem not to be in the world, nor in our minds, nor in our bodies but in the confluence of all three.

So if we want an answer to the question where does maths come from? We can only say that it comes from the creative imagination of the mathematicians who over the ages have dreamt up the ‘formalisms’, as they are called, in which mathematical ideas are expressed. It is useless to say that maths is a property of either mind or matter: it demonstrates that the two are one. These creative individuals dreamt up counting, maybe starting from the invention of the tally-stick. Then they dreamt up measurement, giving quantity to shapes. Then came plane geometry. Then solid geometry. Once these things were formalised, the stage was set for the explosion of mathematical creativity that characterised first ancient civilisations of the Indian sub-continent, then Greek civilisation, then the Islamic world and then the European civilisation after the Renaissance, when Indian, Greek and Arabic inventions in mathematics came together and cross-fertilized each other. Mathematical invention is always a matter, in the end, of rigorous ‘proof’. But mathematical creativity is not primarily to do with proof. It has more to do with the spontaneous working of the creative fantasy, of imagination, that espies, perhaps without understanding all the details, that further patterns, further mathematical depths can be opened up by altering the assumptions that govern existing knowledge. This suggests again that the creative mathematical imagination is always outside of any system of axioms and theorems that it might be using, always ‘above’ and ‘beyond’ such a system in some indefinable region of thought where entirely new insights are sparked off from the contemplation of old problems.

Gödel’s theorems show that the mind using such a system will always ‘know’ certain things that cannot be demonstrated from within the system. The mind, itself is always creatively beyond its mathematical inventions and able to survey its mathematical systems from a vantage-point outside them. The really weird thing is that the world itself seems to possess this property, too: it too is somehow always ‘beyond’ any mathematical models that we impose upon it, in the sense of being more complex than the model. Nature is almost but not quite mathematical. We can capture it to a certain extent in our net of conceptual thought, but something will always slip through. That’s why Plato, the great admirer of maths, though that dialectical reasoning was above mathematical reasoning. That’s why our mathematical models get progressively more complicated. Mathematical creativity is a matter firstly of conceiving in imaginative terms an abstract possibility and then of generating ever more powerful formalisms and ever more powerful models to deal with the world’s slipperiness. It may well be that this new mathematics arises from the same indeterminate essence of the world as is at the heart of what we call ‘matter’. It may be that at the heart of foreworld, at the heart of hindworld and at the heart of midworld, there lies the same indeterminate fund of creative and meaningful novelty. After all, the world is a co-ordinated totality and not a bag of marbles. So it is entirely possible that its indeterminate core is everywhere the same, whether it is in the world we observe (foreworld) in the subjective reactions consequent upon that observation (hindworld) or in the publicly accessible thoughts we generate in consensus-building language (midworld). It is for reasons such as this that the fourth term ‘hyperworld’ is here considered necessary. This is only a speculation, but beware of simply rejecting it out of hand, for that way lies dogmatism.

Maths as established formalism is a world in itself. It is pure midworld. It has properties that can be discovered. But it is not the world of sense. This demonstrates once more that the concept ‘world’ has more to it than we generally think. So the problem remains; and a very deep problem it is. The power of mathematics is in its ‘obviousness’, in our conviction that the truths of maths are obvious and that they obviously apply to the world. That we invent (or discover) them and that they do apply to the world are two almost miraculous facts in themselves. The comprehensibility of the world in terms of mathematics is a very suggestive link between mind and world, a link that seems to suggest that the observer is not just a featureless subject facing a monolithic and alien object, but that there is a much more intimate connection between the self and the world that it encounters. Self and world and language seem to flow together in mathematics and seem all three to be harmonised by some mysterious fourth term. There seems to be an intimate connection between our mathematical ability and the world itself. We seem to be more deeply enmeshed in the world than we ever suspected. This ‘enmeshedness’ seems to speak to us of an intelligence in nature that our own intelligence constantly pursues but never catches. But there is a connection between our intelligence and the intelligence we devine and that some call ‘divine’. We are at home in the universe. We are part of the universe and our mathematical ability is an indication of how as parts we reflect the whole.

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