Integrationist Notes and Papers. No.19 © Roy Harris 2008

The grammar of numbers

   •Roy Harris

Reification takes many curious forms in human thinking. INP No.14 (‘Integrationism and the Foundations of Mathematics’) draws attention to what is there called the ‘doctrine of Natural units’ as one of the foundations of mathematical thought. But it says nothing about the reification of numbers as such, which goes back in the Western tradition at least as far as Pythagoras.

Circularity and infinite regress are the Scylla and Charybdis of advanced mathematical thinking. For the modern period, we see a classic example in Bertrand Russell’s famous attempt to answer the question ‘What is a number?’ (Russell 1919: 11-19). According to Russell, this is a question that had been asked for centuries, but it ‘has only been correctly answered in our own time’. (The philosophical arrogance of the word correctly is worthy of note, and should alert readers to the reificatory presuppositions that underlie Russell’s treatment of the topic.)

Russell’s claim is that the ‘correct’ answer was first given in 1884 by Frege, but lay unnoticed until ‘rediscovered’ by Russell himself in 1901. The odd implications of this claim can hardly fail to raise eyebrows: i.e. that all the great mathematicians of the past, from Euclid down to Euler, really did not understand what they were dealing with; and that even when Frege came up with the ‘correct’ answer, no one paid much attention. All of which in turn throws doubts on whether ‘What is a number?’ is actually such an important question for mathematics as Russell seemed to think. (There is nowadays a school of thought which says that how you answer the question makes no difference anyway. But that was not Russell’s view.)

The first thing we need to be clear about, according to Russell, is ‘the grammar of our inquiry’ (Russell 1919: 11). Those who have not got their ‘grammar’ straight, says Russell, are easily misled into trying to define plurality instead of trying to define number. Number is quite different from plurality. So what is it? ‘Number is what is characteristic of numbers, as man is what is characteristic of men.’

The first thing to note is the sleight of hand by which the grammar of the question has straight away been changed. The indefinite article has dropped out of sight. What is now being asked is not ‘What is a number?’ but ‘What is number?’. (Cf. the difference between ‘What is a man?’ and ‘What is man?’) This grammatical trick is necessary in order to give Russell’s own account any semblance of plausibility. In other words, it is now being taken for granted that there are numbers, just as there are men, and all that remains is to specify what is ‘characteristic’ of them. Whatever it may turn out to be, that is what number is. More basic questions such as ‘Are there any numbers?’, ‘Where are they?’ and ‘How do we recognize them?’ are not even allowed to arise. By means of this subterfuge Russell appears to be giving an informative answer when all that he is offering is a mere tautology. (Cf. ‘What is a colour?’ Replace by ‘What is colour?’. Then reply ‘Colour is what is characteristic of colours.’ We are no wiser about what a colour is than we were in the first place. This is not an answer but a verbal smokescreen. In effect, the question has been thrown back at the inquirer, who is presumed to be already familiar with colours, and is invited to inspect them, and draw the conclusion that whatever they have in common – even though it may not be at all obvious that there is anything – is what colour is.)

What has gone astray in Russell’s thinking here? From an integrationist perspective, the flaw is evident. Like so many champions of reocentric semantics, Russell has been misled by the language myth into confusing lexical definition with real definition. (For other examples of this pervasive confusion in the Western tradition, see Harris and Hutton 2007.) That is, he has treated the question ‘How do we define the term number?’ as if it were a question about the actual existence of certain presupposed arithmetical objects. In short, the same philosopher who is so quick to tell others how important it is to get ‘the grammar of their inquiry’ clear has himself committed one of the most basic ‘grammatical’ howlers.

But the confusion does not end there. Russell thought that his ‘solution’ was immune from any charge of circularity or tautology because in practice it was always possible to establish the numerical identity or non-identity of two classes of items empirically by putting the members of each class in one-one correspondence. But this will not do either. For, as Wittgenstein pointed out, putting the members of two sets in one-one correspondence is not somehow prior to or independent of our grasping any concept of number, but, on the contrary, presupposes that we have already grasped some such concept. In the end, therefore, Russell’s account of number is a muddle from beginning to end. It is rather like an account of colour which claims that there is no circularity in defining colour as the property characteristic of all colours because in the end you can always tell whether something is coloured just by inspecting it.

One might have hoped that what is wrong with Russell’s reasoning about numbers would be by now well understood in academic mathematics. Far from it. The reification of numbers is still thriving today, to judge by a much acclaimed book, The Music of the Primes, by Marcus du Sautoy, Professor of Mathematics at Oxford and a Fellow of All Souls. He declares his Platonistic faith in ‘an absolute and eternal reality beyond human existence’ and quotes with approval the assertion of a Cambridge mathematician, G.H. Hardy, that ‘317 is a prime not because we think so, or because our minds are shaped in one way or another, but because it is so, because mathematical reality is built that way’ (du Sautoy 2003: 7. Italics in the original.). But he admits it is a perfectly reasonable question to ask whether there might not be, for all we know, numbers that are neither primes nor multiples of primes. ‘How can we be really sure that there aren’t some rogue numbers out there which can’t actually be built by multiplying together prime numbers?’ (du Sautoy 2003: 35).

The form of the question is itself revealing. Numbers are ‘out there’. Evidently, du Sautoy’s grasp of the grammar of number questions is every bit as naive as Russell’s. Checking for these rogue numbers is implicitly likened to an astonomer making sure that there aren’t any rogue planets in the solar system that are not actually orbiting the sun like the others. So for du Sautoy it would seem reasonable to ask, say, ‘Are there prime numbers in the Andromeda galaxy?’ And one assumes his answer would be ‘yes’: in fact, he claims that ‘prime numbers will remain prime whichever galaxy you are counting in’.

The problem is to make sense of the claim – as Wittgenstein remarked of the suspiciously similar statement ‘It’s 5 o’clock on the sun’ (Philosophical Investigations §350). As Wittgenstein pointed out, it is not enough to know what it is 5 o’clock means, and where the sun is in relation to the earth. The assertion ‘It’s 5 o’clock on the sun’ is still semantically problematic (unlike, say, ‘It’s raining on the sun’). Similarly, in the case of arithmetic in Andromeda, it is not enough to know what prime number means or where Andromeda is.

According to du Sautoy, our ignorance about what conditions are like in Andromeda need not bother us too much, because we can be sure back on Earth that there are no rogue numbers anywhere in the universe (presumably not even in black holes). Mathematicians can prove it. Here du Sautoy out-Russells Russell: no mean feat. His proof runs as follows. Let N be a candidate rogue number somewhere in the universe. Since ex hypothesi N is not prime, there must be two smaller numbers (say A and B) of which it is the product. ‘Since A and B are smaller than N, our choice of N implies that A and B can be written as products of primes. So if we multiply together all the primes coming from A and all the primes coming from B, then we must get the original number, N.’ That shows ‘that N can be written as prime numbers multiplied together, which contradicts our original choice of N.’ Apply this to any N, and we get the same result. ‘Hence every number must be prime or built by multiplying primes’ (du Sautoy 2003: 35-6). It does not seem to worry du Sautoy that if this is a typical example of what counts as a ‘proof’ in mathematics, then mathematical arguments are characteristically circular. The ‘proof’ reveals nothing about what is ‘out there’ in the universe, but only about what was contained in one (terrestrial) definition of the term number. (It is rather like ‘proving’ that if there were a large body in the solar system not orbiting the sun it couldn’t by definition fall under the term planet.)

The popular alternative among mathematicians is to treat numbers as ‘concepts’, i.e. to substitute a psychocentric for a reocentric semantics of numeration. Thus Graham Flegg, in his book Numbers, warns us: ‘There is nothing in the physical world that is two’, and ‘Since the symbols clearly exist in the physical world, we tend to grant the same status of existence to the concepts which they represent’. Numbers, he declares, are ‘idealizations in the mind of particular experiences encountered in the world’ (Flegg 1983: 3). This would not do at all for du Sautoy, because we have no experiences of what life is like in Andromeda, and thus no basis for saying what Andromedan numbers are like, or whether there are any at all.

Numbers, according to Flegg, are like colours. Just as the number two ‘does not have an independent existence of its own except as a concept, neither does redness have an existence except as a concept’ (Flegg 1983: 3). So isn’t blood ‘really’ red after all? Flegg never tells us. But two or twoness is nevertheless, he claims, a property of real objects. What objects? Groups of two objects, of course. Provided there are ‘really’ just two of them. (It is interesting that seeing that there is nothing out there that is ‘really’ two does not automatically lead people to see there is nothing out there that is ‘really’ a group of two, either. It is even more interesting to speculate whether any research council in the future would ever fund a programme to count how many observable groups of two there really are the universe. That would surely be a contribution to scientific knowledge.)

For an integrationist answer to the question ‘What is a number?’, see Harris 2005: 106-128, where there is more on the futility of debates in philosophy of mathematics. As for numerals, their meaning depends on context. They integrate human activities in a complex variety of ways in the world we know. Whether numerals will ‘work’ just as well on the far side of the Milky Way is something most integrationists will be happy to wait and see when we get there.

Despite the claims of many mathematicians from Pythagoras onwards, numerical signs have no special status in the universe. Their grammar is no more obscure than the grammar of road signs or cooking recipes (in both of which they commonly appear). So why have ‘numbers’ been conjured up as the occult referents that supposedly underpin the universal semiology of numeration?

One reason why mathematicians are so touchy about the ontological status of numbers is presumably their fear that ‘pure’ mathematics might be seen as an idle intellectual pursuit far removed from any concern with the common problems of humanity, and justified only when ‘applied’ to enterprises like running banks and building bridges. It is indeed difficult to see that the world is any better off, or ever will be, for Andrew Wiles’ demonstration of Fermat’s Last Theorem, as great an intellectual achievement as it doubtless is. But another reason is that mathematicians are genuinely mesmerized by the doctrine that the ‘language’ of mathematics has a special relationship with matter that somehow guarantees its predictions in advance. How else, it is sometimes asked, could we trust mathematics when sending astronauts to Mars, or bombers to put down unruly tribesmen in Afghanistan? How else could numbers be the foundation of science in all its forms? How else could we be sure that our mathematics was perfectly valid for galaxies on the other side of the universe? Reflexion upon these and similar ‘how else?’ questions is supposed to leave us with no alternative but to admit that ‘2+2=4’ is not just a handy rule of thumb for everyday arithmetic, but in some ‘deep’ sense captures the fact that two plus two ‘really does’ universally and sempiternally make four. This piece of mathematical mystique has come to supplant the Book of Genesis as the basis for believing in the possibility of a pre-lapsarian language, i.e. a language that is not based on arbitrary human signs, but on signs that directly and reliably reflect the ultimate nature of reality (or, in some versions, the way God decreed it should be).

du Sautoy, M. (2003), The Music of the Primes, London, Fourth Estate.
Flegg, G. (1983), Numbers. Their History and Meaning, London, Deutsch.
Harris, R. (2005), The Semantics of Science, London, Continuum.
Harris, R. and Hutton, C. (2007), Definition in Theory and Practice. Language, Lexicography and the Law, London, Continuum.
Russell, B. (1919), Introduction to Mathematical Philosophy, London, Allen & Unwin.
Wittgenstein, L. (2001), Philosophical Investigations, trans. G.E.M. Anscombe, 3rd edn., Oxford, Blackwell.

© Roy Harris, Emeritus Professor of General Linguistics, Oxford, 2010