What is mathematics about? We know what biology is about; it’s about living things. Or more exactly, the living aspects of living things – the motion of a cat thrown out of a window is a matter for physics, but its physiology is a topic for biology. Oceanography is about oceans; sociology is about human behaviour in the mass longterm; and so on. When all the sciences and their subject matters are laid out, is there any aspect of reality left over for mathematics to be about? That is the basic question in the philosophy of mathematics.
People care about the philosophy of mathematics in a way they do not care about, say, the philosophy of accountancy. Perhaps the reason is that the certainty and objectivity of mathematics, its onceandforall establishment of rocksolid truths, stands as a challenge to many common philosophical positions. It is not just extreme sceptical views such as postmodernism that have a problem with it. So do all empiricist and naturalist views that hope for a fully ‘scientific’ explanation of reality and our knowledge of it. The problem is not so much that mathematics is true, but that its truths are absolutely necessary, and that the human mind can establish those necessities and understand why they must be so. It is very difficult to explain how a physical brain could do that.
One famous philosopher who finds mathematical necessity an inconvenience is Peter Singer. In one of his bestselling books on ethics, he argues that we cannot rely on intuiting ethical truths, since the most convincing case of intuition, in mathematics, is not correct. ‘The selfevidence of the basic truths of mathematics,’ he says, ‘could be explained… by seeing mathematics as a system of tautologies… true by virtue of the meanings of the terms used.’ Singer is wrong to claim that this philosophy of mathematics, called logicism, is ‘widely, if not universally accepted’. It has not been accepted by any serious philosopher of mathematics for 100 years. But it is clear why anyone who, like Singer, wishes to explain away the strange power of human intuition might want a deflationary philosophy of mathematics to be true.
To the question: ‘Is mathematics about something?’ there are two answers: ‘Yes’ and ‘No’. Both are profoundly unsatisfying.
The ‘No’ answer, whose champions are known as nominalists, says that mathematics is just a language. On this view, it is just a way of talking about other things, or a collection of logical trivialities (as Singer claims), or a formal manipulation of symbols according to rules. However you cut it, it is not really about anything. Those whose encounter with mathematics at school was less than happy (‘Minus times minus equals plus/The reason for this we need not discuss’) might feel some sympathy with the nominalist picture. Then again, it is also a view that appeals to physicists and engineers who regard serious propositions about reality as their business. They look on tables of Laplace transforms and other such mathematical paraphernalia as, in the words of the German philosopher Carl Hempel, ‘theoretical juice extractors’: useful for getting extra sense out of meaty physical propositions, but not contentful in themselves.
Nominalism might have a certain downtoearth appeal, but further reflection suggests that it can’t be right. Although manipulation of symbols is useful as a technique, we also have a strong sense that mathematics makes objective discoveries about a terrain that is in some sense ‘out there’. Take the subtleties of the distribution of primes. Some numbers are prime, some not. A dozen eggs can be arranged in cartons of 6 × 2 or 3 × 4, but eggs are not sold in lots of 11 or 13 because there is no neat way of organising 11 or 13 of them into an eggbox: 11 and 13, unlike 12, are prime, and primes cannot be formed by multiplying two smaller numbers. The idea is very easy to grasp. But this doesn’t mean there’s nothing to discover about it.
It turns out that the way in which the primes are distributed among numbers involves a complex interplay of pattern and irregularity. On the small scale, the latter is most evident: there are long stretches without any primes at all – indefinitely long stretches, in fact. At the same time, it is widely believed that there are infinitely many ‘prime pairs’; that is, pairs of numbers only two apart that are both prime, such as 41 and 43.
When we turn to the large scale, the impression of disorder fades and a pattern starts to emerge after all. Primes become gradually less dense as one counts up: the density of primes around a large number is inversely proportional to its order of magnitude. The density of primes around a trillion (10^{12}), for example, is about half what it is around a million (10^{6}). More exact information on the intricacies of the distribution of primes is contained in the Riemann Hypothesis, currently the most famous unproved conjecture of mathematics.
It seems as if pure mathematics reveals the topography of a region whose truths preexisted investigatio, even language
This is typical of the results of pure mathematics, from simple school facts such as the divisibility of numbers by 9 if the sum of their digits is divisible by 9, up to the higher reaches of abstract algebra. It is impossible to escape the conclusion that pure mathematics reveals to us the topography of a region whose truths preexisted our investigations and even our language.
Inspired by that thought, Platonism proposes a philosophy of mathematics opposite to nominalism. It says that mathematics is about a realm of nonphysical objects such as numbers and sets, abstracta that exist in a mysterious realm of forms beyond space and time. If that sounds farfetched, note that pure mathematicians certainly speak and often think that way about their subject. Platonism also fits well with the apparent success of mathematical proof, which seems to demonstrate how things must be in all possible worlds, irrespective of what the laws of nature might be in any particular world. The proof that the square root of 2 is an irrational number does not rely on any observationally established laws. It shows how things must be, suggesting that the square root of 2 is an entity beyond our changeable world of space and time.
Still, despite its clean lines and long history, Platonism cannot be right either. Since the time of Plato himself, nominalists have been urging very convincing objections. Here’s one: if abstracta float somewhere outside our own universe of space and time, it’s hard to imagine how can we see them or have any other perceptual contact with them. So how do we know they’re there? Some contemporary Platonists claim that we infer them, much as we infer the existence of atoms to explain the results of chemistry experiments. But that seems not to be how we know about numbers. Fiveyearolds learning to count don’t perform sophisticated inferences about abstractions; their contact with the numerical aspect of reality is somehow more perceptual and direct. Even animals can count, up to a point.
In any case, the problem with Platonism is not so much about knowledge as about its view of mathematical entities. Surely when we measure, or calculate, or model the weather mathematically, we are dealing with mathematical properties of real things in this world, such as their quantities. Such properties are not abstracta: like colours, they have causal powers that result in our seeing them. The visual system easily detects such properties as the ratio of your height to mine (if we stand next to each other). There is no room for abstracta in other worlds to enter the story, even if they did exist.
Nominalists and Platonists have fought each other to a standstill, each convincingly revealing the fatal flaws in their opponents’ views, each unable to establish their own position. Let’s start again.
Imagine the Earth before there were humans to think mathematics and write formulas. There were dinosaurs large and small, trees, volcanoes, flowing rivers and winds… Were there, in that world, any properties of a mathematical nature (to speak as noncommittally as possible)? That is, were there, among the properties of the real things in that world (not some abstract world), some that we would have to recognise as mathematical?
There were many such properties. Symmetry, for one. Like most animals, the dinosaurs had approximate bilateral symmetry. The trees and volcanoes had an approximate circular symmetry with random elements – seen from above, they look much the same when rotated around their axis. The same goes for the eggs. But symmetry, whether exact or approximate, is a property that is not exactly physical. Nonphysical things can have symmetry; arguments, for example, have symmetry if the last half repeats the first half in the opposite order. Symmetry is an uncontroversially mathematical property, and a major branch of pure mathematics – group theory – is devoted to classifying its kinds. When symmetry is realised in physical things, it is often very obvious to perception; if you have an asymmetrical face, don’t go into politics, because it makes an immediate bad impression on TV. Symmetry, like other mathematical properties, can have causal powers, unlike abstracta as conceived by Platonists.
Another mathematical property, which like symmetry is realisable in many sorts of physical things, is ratio. The height of a big dinosaur stands in a certain ratio to the height of a small dinosaur. The ratio of their volumes is different – in fact, the ratio of their volumes is much greater than the ratio of their heights, which is what makes big dinosaurs ungainly and small ones sprightly. A given ratio is something that can be the relation between two heights, or two volumes, or two time intervals; a ratio is just what those relations between different kinds of physical entities share, and is thus a more mathematical property than the physical lengths, volumes and so on. Ratio is what we measure when we determine how a length (or volume, or time, etc) relates to an arbitrarily chosen unit. It is one of the basic kinds of number. As Isaac Newton put it in his uniquely magisterial language: ‘By Number we understand not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same kind, which we take for Unity.’
Any digression into applied mathematics – rarely undertaken by philosophers of mathematics, who prefer the familiar ground of numbers and logic – will turn up, for the alert observer, many other quantitative and structural properties that are not themselves physical but can be realised in the physical world (and any other worlds there might be): flows, order relations, continuity and discreteness, alternation, linearity, feedback, network topology, and many others.
There is a name for a philosophy of mathematics that emphasises the way in which mathematical properties crop up in the actual world. It is called Aristotelian realism. It is based on Aristotle’s view, opposed to that of his teacher Plato, that the properties of things are real and in the things themselves, not in another world of abstracta. A version of it, holding that mathematics was the ‘science of quantity’, was actually the leading philosophy of mathematics up to the time of Newton, but the idea has been largely off the agenda since then.
Infants and animals demonstrably do have the ability to recognise pattern and estimate number, shape and symmetry
Because Aristotelian realism insists on the realisability of mathematical properties in the world, it can give a straightforward account of how basic mathematical facts are known: by perception, the same as other simple facts. Ratios of heights are visible (to a degree of approximation, of course). Infants and animals demonstrably do have the ability to recognise pattern and estimate number, shape and symmetry.
Our developed human intellectual abilities add two things to those simple perceptions. The first is visualisation, which allows us to understand necessary relations between mathematical facts. Try this easy mental exercise: imagine six crosses arranged in two rows of three crosses each, one row directly above the other. I can equally imagine the same six crosses as three columns of two each. Therefore 2 × 3 = 3 × 2. I not only notice that 2 × 3 is in fact equal to 3 × 2, I understand why 2 × 3 must equal 3 × 2. So the Platonists were right to call attention to the ability of the human mind to grasp mathematical necessities; they just failed to notice that those necessities are often realised in this world. The second intellectual ability by which the human mind extends the results of perception is proof. Mathematical proofs chain together a series of insights, individually similar to ‘2 × 3 = 3 × 2’, to demonstrate necessities that cannot be understood at a glance, such as how the density of primes tails off for large numbers.
Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience. If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds? The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.
The standard alternatives in the philosophy of mathematics have failed to account for the simplest facts about how mathematics tells us about the world we live in – nominalism by reducing mathematics to trivialities, and Platonism by divorcing it from the world, the real world of which mathematical truths form a necessary skeleton. Aristotelian realism is a new beginning. It connects the philosophy of mathematics back to the applications that have always been the fertile ground from which mathematics grows. It has a message both for philosophy and for mathematics and its teaching: don’t get blinded by shuffling symbols, don’t disappear into a realm of abstractions, just keep an eye fixed on the mathematical structure of the real world.
7 April 2014
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