Is Mathematics a Free Creation of the Human Mind?
In these posts I defend the commonsense view that mathematics originates in our sense perceptions and is neither a free invention of the human mind nor a window on the eternal. Although a certain part of mathematics ─ essentially that based on the properties of the ‘natural numbers’ ─ is empirically based, this is by no means true of all branches of mathematics. Modern society has been unwise in accepting at face value the exaggerated claims made by late nineteenth-cnetury and twentieth-century mathematicians about the origin and nature of mathematics and which have now become unassailable dogma.
1.Introduction
Prior to the nineteenth century, practically all mathematicians in the West thought mathematics dealt with the ‘real world’, was indeed the surest way of getting a handle on it. In consequence, no very great distinction was made between pure and applied mathematics and the greatest names, Newton, Gauss, Euler, worked indifferently in both spheres. The important theorems were ‘ideas in the mind of God’ : some of these ideas the great Geometer had employed in the natural world, others He had seemingly kept up his sleeve.
Conversely, what made little sense in physical terms was treated with scepticism. It was a long time before negative numbers were accepted, let alone the square root of (−1) and Newton seems to have had serious doubts about his own greatest invention, the Calculus, which is why he returned to more cumbersome geometrical methods in his Principia.
All this changed dramatically during the second half of the nineteenth century : more or less at the same time as Gautier in France and Oscar Wilde in England launched the ‘Art of Art’s sake’ movement. A handful of analysts, especially in Germany, decided that mathematics was a law unto itself and could be developed along ‘pure’ or abstract lines, with no reference to material reality whatsoever.
The so-called ‘Formalist’ approach remains orthodoxy today. Ask any pure mathematician and he or she will tell you that “mathematics is a free creation of the human mind” (Dedekind) : any attempt to tie mathematics down to physical reality is met with incredulity and insdignation. The weakness of this position is, of course, that it makes the predictive power of mathematics utterly mysterious, indeed incredible. As one author put it disingenuously, “Mathematics is an abstract construction of the human mind, and it is really quite miraculous that it should have an immediate and practical application to the real world” (Backus, The Acoustical Foundations of Music).
Actually, this is not the whole story. As Davis and Hersch shrewdly point out (Davis & Hersch, 1983), the typical contemporary mathematician is two things at once, for he or she is both a Formalist and a Platonist. In the quiet of the study, mesmerised by the dancing symbols, he feels he is looking in on ultimate reality. This was indeed what Kepler and Leibnitz thought when they stumbled upon their greatest formulae and theorems, and, within the context of rational deism, it made quite a lot of sense. But today? In the present agnostic or fiercely anti-religious climate of the West, such feelings hardly pass muster. So the contemporary mathematician takes the easy way out : if challenged to give some sort of rationalisation of his or her transcendental view of mathematics, he puts on the Formalist mask which has at least the merit of keeping the Empiricist at bay.
I believe that both these positions, the Formalist and the Platonist, have very little to commend them ─ except to persons who are specialist mathematicians (and even then). If we take the commonsense view that mathematics is rooted in our experience of the real world there is nothing mysterious about its success as a symbolic model and prediction system, quite the reverse. As to the more fanciful inventions of modern mathematicians, they should be classed more as art than as science, and judged accordingly, i.e. on aesthetic, not empirical, criteria. To be sure what starts as science fiction can sometimes turn out to be little short of the truth ─ but there is certainly no guarantee that this will come about.
Euclidian Geometry
Euclid’s Elements is not a Formalist work. The author always has his eye on the actual construction of shapes and figures : the very first Proposition of Book I is “[how] To construct an equilateral triangle on a given straight line”. The famous joke proof that every triangle is isosceles is obviously fallacious if you actually try drawing the figure (as Greek geometers would have done) since the two lines cannot possibly meet inside the triangle as the ‘proof’ requires.
Should one, then, view Euclid as a compendium of verifiably correct statements about the physical world? This would be going too far. If you actually measure the angles in a triangle you will almost certainly find that they do not add up to 180°. An eleven year old girl once came up to me after a lesson to tell me this with deep indignation, accusing me of having told an untruth. I asked her what she had got and she replied, “179 and a half”3. And, as a Sceptic philosopher objected even in Plato’s time, if you draw a tangent to a circle you will find that it certainly touches it at more than one ‘point’.
Absolutely straight lines do not exist in Nature and raindrops are far from being true spheres. Euclidian geometry is a grid that we impose on the real world : most of the time Nature does not bother with it. The twisted tangles of branches on a tree haven’t the slightest resemblance to the clearcut shapes of school geometry and even fractals are far too regular. All this is a little worrisome since the basic propositions of Euclid must be true ─ after all, we can prove them! It is here that Platonism provides a very influential and, at first sight, satisfying way of resolving the problem. Euclidian geometry deals with an ideal world which exists independently of the actual world and is ‘more real’ than it. In such a world straight lines really are straight and tangents touch the circumference of a circle at one point only. Everything down here is an imperfect copy of these timeless Forms, or ‘Ideas’ 4.
Personally, I consider Platonism to be a delusion, though admittedly a seductive and historically very important one : the ‘real’ is best defined as what actually occurs not what is supposed to occur or exists in a mathematical Fairyland. So how do I view Euclidian geometry? As an ensemble of true, i.e. empirically verifiable propositions, but only in a statistical sense. Speaking rather pedantically, one could put it this way:
“If you take the angle sum of any triangle drawn on a flat surface, the mean will be 180° (better, half a full turn) approaching 180° in the limit as n, the number of trials, increases without bound and p, the resolving power of the measuring device likewise increases.”
So at any rate, I believe ─ I have not put the proposition to the test. Gauss, the foremost mathematician of his time, was sufficiently bothered by the question that he took the trouble to work out the sum of the three angles of a triangle formed by the peaks of three mountains in the State of Hanover (using surveying data he had collected himself). He did not obtain 180° but was relieved to find that the discrepancy was “within the limits of experimental error”. It is rather pathetic to consider that the same concern with ‘reality’ exhibited by the ‘Prince of Mathematicians’ would be considered ludicrous today and, indeed, the contemporary tutors of such a person would probably advise him or her to drop mathematics in favour of biology or mechanical engineering. (I very much doubt that my eleven-year old empirical philosopher actually went on to study mathematics, or even physics.)
Clearly, it would be insufferable to have to translate all the theorems of Euclid and, for that matter, Newton’s Principia, into the language of statistics : it is not only convenient, but, practically speaking, mandatory to formulate geometry and mechanics in ‘absolute’, not relative, terms. This does not make beautiful ‘perfect spheres’ or absolutely frictionless pulleys real, however, any more than Shakespeare’s definitive study of a reluctant revenge hero makes his Hamlet a real person. If there is a ‘hierarchy of realism’, it works the other way round : by my book it is the actual people, events and imperfect shapes that rate higher in the scale of what is, and the actual gives rise, via human invention, to the ideal, certainly not the other way round. In chemistry, we need the eminently Platonic concept of an ‘ideal gas’ (one that obeys Boyle’s Law exactly), but there are no such gases, nor are there likely to be any.
On the other hand, Euclidian geometry is not a free creation of the human mind : if it were, it would hardly be much use in industry. The subject matter of geometry is certain spatial properties which actual objects, to a high degree of approximation, are found to possess and, subject to certain conditions, seemingly must possess. There is thus nothing mysterious about the successful application of Euclidian geometry to technology, nor anything paradoxical about Euclidian geometry not being the only pebble on the beach. If there is only one Set of Ideal Forms and only one prophet, Euclid, then there is a problem for the true believer. But Euclidian geometry is only one — though by far the simplest and most useful — out of several ‘geometries’, each of which have their appropriate spheres of application such as spherical geometry, hyperbolic geometry, fractal geometry and so on. The key proposition of hyperbolic geometry, that the angle sum of a ‘triangle in space’ is less than 180°, started off in Riemann’s imagination, but is currently taken perfectly seriously by the practising astronomer because it gives rise to slightly better predictions than Euclidian geometry. This is not a matter of matrhmatical fashion but of empirical fact : I believe that laser beams have been used to demonstrate that the sum of the three angles of a triangle in space is somewhat less than 180° as Einstein’s Theory requires.
To be continued S.H. 23/10/12
Note: The image is Vivacity by Jane Maitland
Origins of Mathematics 
Mathematics may be grounded in reality to a greater or lesser extent, but our thinking when doing mathematics should not be limited to reality — or more precisely, reality as we know it, since unknown reality is by definition not accessed by us. The non-existence of a 4-D hypercube in “reality” should not dissuade me from pursuing it as an object of investigation, even though we may simultaneously acknowledge 4-D space’s roots in a generalization of 3-D space which is our reality, or at least an approximation (String Theory or the like notwithstanding, should it turn out to be true).
You mention that mathematics divorced from reality would be addressed on grounds more like art than on “empirical” grounds, but I’m not sure who exactly considers, say large cardinals in infinite set theory to be “empirical” in the sense of a physical scientists, so the complaint does not make much sense to me.
Yes, what you say is basically OK but there are two points that need to be made. Firstly, mathematics, even the most abstruse, is far more grounded in our physiology, prejudices and specific human culture than most mathematicians are prepared to admit. Lakoff in “Where does Mathematics come from?” demonstrates this very effectively. A different intelligent species would develop a different mathematics with some overlap.
Secondly, pure mathematicians are very good at evading, precisely the questions you raise about whether certain branches of mathematics are in any sense reality based. Essentially, they believe in their own inventions however fantastic and are outraged if anyone calls them to task. Tell them that Cantor’s Theory of the Transfinite is just a piece of intellectual embroidery and you will meet with a rude reaction. I will never forget the outrage on the face of a pure mathematician when he blurted out, “Good God, this guy does not even believe in ordinary infinity let alone the transfinite!” It was like a Victorian bishop meeting someone who said he had doubts about the divinity of Christ. It has been well said that a pure mathematician is both a Platonist and a Formalist, a Platonist in his study and a Formalist when mingling with the hoi polloi and asked to justify his belief. I view this attitude as cowardly and deceitful, what we call “having it both ways”. Yes, in reply to your question, what I said to ‘these people’ was that Cantor’s theory was simply a mental construct with no more reality or relevance than a pattern in embroidery. But strangely enough Cantor’s definition of ‘cardinal number’ as a set of actual or imaginary objects is very useful and sensible. SH
Yes, I’ve heard the idea that aliens may have a different mathematics: e.g. someone considered that hypothetical kinds of aliens that lived in the atmosphere of a gas giant (were such things possible), may not develop a notion of “number” as such, but instead their maths would look primarily more like topology, because the notion of “number” comes from having discrete objects you can put together and assort, whereas in the atmosphere of a gas giant you would not have that. An alien civilization living in the ocean might also develop different mathematics, e.g. they may find fluid physics to be more important and so would develop mathematics and metaphors conducive to describing that first. And also, as you mention, the culture plays a role too, e.g. formal proof only appeared in Greece, I believe, whereas other human cultures did not use it (but despite this were nevertheless able to produce some interesting and sophisticated mathematics of their own, like India).
What I think with regards to mathematics is that the axioms are invented, but the truths, e.g. if you assume this you get that, aren’t. But that doesn’t mean an alien will assume the same mathematics as us due to its different development. Likely, an alien won’t even THINK the same way we do, for its brains (or whatever) will have evolved along a fundamentally different evolutionary route. That would certainly frustrate any attempts to “communicate” with aliens.
What do you think is a non-“cowardly” attitude? Can you accept that, say, the Cantor’s infinity does not “exist physically”, yet still consider it a legitimate mental exercise? Or do these people you talk to actually believe the Cantor infinity exists “physically”? What exactly did you say to them? That it’s “just” a mental construct and not a physical reality?
Also, the question of whether or not “ordinary” infinity exists physically is unresolved: in particular, we cannot see whether the universe goes on beyond the cosmic horizon or not. It may or may not, we do not know. Yet, that doesn’t mean I cannot “believe” in such infinity as a useful construct, e.g. I can imagine taking a limit “as x goes to infinity” or “approaches infinitely close to some value”, and the entailed “complete” Cantor-Dedekind real numbers, as a mental construct, and use the mental construct as a model and reason from the model and get useful results, and also find it interesting to study the mathematics of the model itself in its own right.
Dear Mike, Yes, thanks for your very insightful comments. I will think a bit before I reply about Cantor’s theory. But, in brief I believe the infinite should be banished entirely from mathematics. Instead we could (and I sometimes do) use an arrow instructing the reader to “increase to any appropriate value” (or maybe decrease) : infinity is not a thing, not a reality, but essentially just an ‘operator’. “If there were no limits there would be nothing” as Parmnenides said. I believe that there is such a thing as the ‘parafinite’ if you like, but this is not amenable to mathematical/physical treatment and should be left to mystics : mathematics deals with the measurable or at least ‘conformable’ and should stick to this. Yes, we can talk about “going to the limit” but most readers even mathematical students do not realize that this limit is practically never attained, it is like an ideal gas, a useful concept but not a reality. In brief this is my position. I am an empiricist when it comes to mathematics and physics. SH