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What is the Philosophy of Mathematics?

Stephen Ferguson asks whether mathematical objects are real.

I work on the philosophy of mathematics – in fact, I’m just finishing off my PhD dissertation. When I meet people and they ask what I do, they always look puzzled by my reply. Typical responses include surprise that mathematics has a philosophy, complete bewilderment, or else they conclude that I must do some form of esoteric mathematics and confess that they could never get their head round maths when they were at school. The situation is usually slightly better when I meet other philosophers – they know that it has something to do with Bertrand Russell; sometimes they will mention Wittgenstein or even Frege or Gödel. But no-one ever seems to be able to put their finger on just what it is that I do.

In a nutshell, the philosophy of mathematics deals with the special problems that arise from our possession of mathematical knowledge. Therefore it is a branch of epistemology, the study of how we know things, just as philosophy of science and philosophy of perception are. Unlike other forms of knowledge, where we learn by experience, mathematical knowledge seems to be purely concerned with the realm of thought. In addition to specific questions about mathematics, discussion also concerns how mathematical knowledge fits into the broader scheme of things, and more general accounts of our cognitive capacities.

Truth is generally considered to be a matter of correspondence between our thoughts and language on the one hand and reality on the other. As mathematical statements are (hopefully) true, this suggests that the objects referred to in these statements – numbers, sets, functions, etc. – must exist. In the past twenty years, two questions generally attributed to American philosopher Paul Benacerraf have become paramount. They are:

1) given that mathematical objects don’t have causes or effects, how can we refer to them?

2) given that mathematical objects don’t have causes or effects, how do we have any knowledge of them?

These have been called Benacerraf’s twin puzzles of Referential and Epistemic Access. As you can imagine, there has been much debate about different ways to explain these puzzles.

In what follows, I would like to outline some of the main responses to these puzzles, to pass on some of the feel for the philosophy of mathematics, after which I’d also like to outline the position that I advocate myself.

I. Mathematical Realism

There are three, not entirely separate, main lines of response to these puzzles, which are usually adopted depending upon whether the philosopher thinks that there really are numbers, in the same way as there are tables, chairs and beermugs – this is what is called a realist or platonist response. Alternatively, they might deny that there really are numbers, and explain our apparent commitment to them as a convenient fiction, claiming roughly that mathematics is not about some abstract realm of mathematical objects, but about certain concepts which we employ: this might be called an anti-realist strategy. Thirdly, there are those who argue that to concentrate upon individual objects – such as natural numbers or sets – is the cause of the problems; instead we should be concerned with mathematical structures. Typically structuralists also espouse some form of realism about structures, but this is not always the case.

There are three main arguments for mathematical realism. The first, put forward by German mathematician Gottlob Frege (1848- 1925) relies on the reality of language. He argued that where our language does genuinely engage with the world, singular terms (what he called Proper Names) refer to, or stand for, objects. So the proper name ‘chair’ refers to an actual object. He suggested that true statements such as those of numerical identity (e.g. 2+2=4) supply the requisite contexts to conclude that numerals too refer to objects, i.e. that numbers are objects. This is sometimes called linguistic or semantic realism, because of the prominence of language in this account.

The second argument is due to Kurt Gödel, the logician who had the office next door to Einstein at the Institute of Advanced Studies at Princeton. Gödel argued that mathematicians are able to perceive mathematical objects using a special faculty of intuition. This is Platonism proper (with a capital ‘p’) because, as with Plato’s theory of the forms, an abstract realm is postulated as well as a means of being acquainted with that realm.

Thirdly, Hilary Putnam – one of America’s most influential philosophers today – has put forward an argument based on Quine’s arguments for epistemological holism. Quine contends that our knowledge is a unity, and that it is not possible to isolate one aspect of our knowledge from all others. Putnam’s argument runs like this – if we take our scientific theories seriously, we are committed to believing in the entities postulated by those theories, even when it isn’t possible to experimentally verify their existence. However, if our knowledge really is holistic, as Quine claims, then the theoretical entities aren’t the only objects to which we commit ourselves by accepting a theory: as mathematics is involved in expressing physical theory, we are also committed to accepting the existence of mathematical objects. This is the Indispensability Argument – so called because mathematics is indispensable in science.

II. Anti-realism about mathematics

Traditionally two positions oppose platonism: Intuitionism and Formalism. Recently a variety of new anti-realist positions have also been articulated, notably Quasi-realism and Irrealism. More of them later.

Intuitionism was first put forward as a philosophical account of mathematics by Dutch mathematician Jan Brouwer (1881-1966) as an alternative to Platonism. According to him, mathematics is not about an abstract realm of mind-independent objects, but rather about the creation of mathematical objects by the human mind. Rather than a statement being true or false of some preexisting mathematical reality, he argued that we create that reality as we go. Mathematical statements are true or false when they deal with objects already created, but neither true nor false when they break new ground. So what logicians call the Law of Excluded Middle – that for every statement, either it or its negation is true – must be rejected, if Brouwer is correct. But as the proofs of certain results in ‘classical’ mathematics rest essentially on the use of this Law, denying it requires us to rethink much of the subject.

Over the past 30 years, Michael Dummett, sometime professor of logic at Oxford, has argued for Intuitionism by attacking Frege’s semantic arguments for realism. On the basis of the Law of Excluded Middle, classical mathematics is committed to the existence of statements which are true, but which we cannot prove are true; these are known as evidence transcendent truths. A good example is Goldbach’s conjecture that every even number is the sum of two primes. Most mathematicians are reasonably certain that this is true, but as proving it would involve finding the prime components of infinitely many even numbers, such a proof could never be completed. Dummett’s arguments concern how we learn mathematics; someone has to teach us, and moreover, we have to be able to show that we’ve understood what we’ve been taught. He argues that if knowing the meaning of a statement is knowing what the terms within it stand for, as Frege claimed, then we would never be able to learn the meaning of evidence transcendent statements, as no-one could show us the truth of them. This has become known as semantic anti-realism, and like the Intuitionism that inspired it, if correct, will involve the rejection of the Law of Excluded Middle.

Like Intuitionism, Formalism opposes the Realist conception of truth which supports Platonism. The usual interpretation of Formalism is that it treats mathematics as being fictional or like a game; but this would be a misinterpretation of at least one Formalist – the most famous of all: David Hilbert (1862-1943). His attempts to deflate the metaphysical worries about truth might be put in modern terms by saying that there is nothing more to truth than is entailed by a principle of correspondence so weak as to be a platitude:

‘P’ is true if and only if P

More recent attacks on Realism have come in the form of Quasi-realism and Irrealism. Quasi-realism is a term coined by Simon Blackburn, in connection with debates in the philosophy of language and in ethics. He suggests that in certain discourses, although the surface grammar has a certain shape, it misrepresents the underlying logical structure of the syntax. This enables us to carry on our various linguistic practices, without being bogged down with burdensome philosophical commitments – such as to the existence of non-physical objects. Geoff Hellman has used this strategy to suggest that mathematical statements are not straightforward indicative claims, but instead that they are subjunctive claims about what the consequences would be if there were such objects as numbers or sets.

Quasi-realists accept that mathematical statements are true, but deny that there are any numbers, sets or functions corresponding to the names occurring in the surface grammar. lrrealists on the other hand, such as Hartry Field, have argued that mathematical language should be taken at face value, but as there are no such objects as those named in the statements of mathematical theories, these statements must be false. Field takes his task as two-fold: to show that what he takes to be the main argument for mathematical realism – the Indispensability Argument – is misguided, and to show that despite mathematical statements being false, they can nevertheless be useful. He accounts for the usefulness of mathematics by demonstrating that all mathematics satisfies a certain principle of normativity which he calls conservativeness; Inputting true information into the mathematical machinery will result in output which is also true. He argues that this principle of conservativeness shows that mathematics is ultimately no more than a convenient shortcut, and that science can be conducted without express mention of mathematics.

III. Structuralism

Unlike most of the mathematics studied by professional mathematicians, the surface grammar of arithmetical practice suggests that particular objects are at stake. But when it comes to the items studied by working mathematicians, it is not objects but structures which are of prime importance. One typical mathematical structure is the group : given a set g and a binary operation +, ⟨g, +⟩ is a group if the following hold:

(G-i) closure – for any a, b in the set, there is some c in the set such that a+b=c;

(G-ii) identity – there is an element, e, so that for any element a, a+e=e+a=a;

(G-iii) inverses – each a element has another element, b, related to it, so that a+b=b+a=e. This is the inverse of a, written a^-1;

(G-iv) associativity – for any a, b, c in the set (a+b)+c=a+(b+c).

Rather than think of any individual object in the structure as having an important mathematical role, the key insight when dealing with structures is that the whole structure is mathematically important: no part of it can perform in isolation. Perhaps the best way to grasp the mathematical concept of structure is to consider physical structures. In his forthcoming book, Stewart Shapiro describes structures as being similar to sports teams. Take a typical football team: there is a goalkeeper, players in central defence, midfield and with the strikers up front. Some teams may play with three strikers, others with two: these then have a different structure.

Now consider the problems that faced platonism: like numbers, positions in a football team are not physical things, so how do we have knowledge of them? The obvious answer is that we come to know what role a goalkeeper plays by watching several instances of goalkeepers, and abstracting what is common to all of them. We would be missing something if we thought that all goalkeepers had brown hair: what is important is the role that they play within the structure – within the team. The structuralist argues that if all of mathematics is considered structurally (not just the obvious algebraic structures such as groups, but also areas such as arithmetic) then this perspective offers simple solutions to various philosophical worries, such as Benacerraf’s twin puzzles.

We learn mathematics by spotting patterns – by what is called pattern recognition. In fact, Resnik argues that we would be better off talking generally in terms of patterns than structures, but the differences is merely terminological. Both Resnik and Shapiro, two of the most influential structuralists, have argued that mathematical knowledge comes by first experiencing various ‘concrete’ or physical patterns, and then abstracting to the underlying structure.

I’ve already briefly mentioned Geoff Hellman’s modal structuralism, as an example of a quasi-realist strategy; like Shapiro and Resnik, he takes the philosophical problems with platonism to stem from its focus on mathematical objects, and suggests that a structurebased account would remedy this. Unlike Shapiro and Resnik however, he bases his structuralism not on pattern recognition, but on the thought that structures describe the possible combination of objects, and so takes statements of mathematics to be concerned with our grasp of what is possible and what is necessary.

Structuralism is attractive because it offers an account of the subject matter of most of modern mathematics – unlike platonism which focuses almost exclusively upon the mathematics with which the man in the street is familiar. However, the exact shape of the structuralist’s arguments are worth considering in closer detail. First there is this obvious desire to account for mathematical practice. Secondly, the claim that ‘All mathematics is structural’ is made for strategic rather than philosophical reasons. By this I mean that it gives the structuralist certain advantages. For example, without this claim, the structuralist’s theories would only apply to the obviously structural areas of mathematics, such as the theory of groups mentioned above. In order for structuralism to be an alternative to platonism, the structuralist needs to show that this view also applies to ordinary mathematics, i.e. to arithmetic.

The usual way to do this is to show the adequacy of the structural view of number. Rather than think of the natural numbers as being a collection of individual objects, it is possible to think of them as a structure, with each element of the structure being the successor of another element of the structure. Nothing can be said about any of these elements, except for the relationship it holds with the others. What distinguishes such structures from those of algebra, is that the underlying pattern of the natural numbers is categorical – for any given cardinality, all the models of the structure are the same. In the algebraic case, as with football teams, there may be different models of the structure, exhibiting differences similar to the differences between one team with three strikers, and another having only two.

IV. Modest structuralism

I’ve never been convinced that the structuralist is correct about arithmetic – I’ve always thought that the intuitive differences between, on the one hand, systems such as the natural numbers, real and complex numbers, and on the other, structures such as groups, was more than just a matter of the number of models of a theory.

I do think that the structuralist is basically correct about abstract algebra – which has led me to try to produce an account which retains all of the intuitive appeal of Frege’s platonist account of ordinary mathematics, along with the structuralist’s account of professional mathematics. Recall that Frege offered linguistic arguments for his conception of numbers as objects: to offer a two-fold account such as I’ve just proposed, entails giving reasons based on the features of mathematical language, to suggest why structures and systems might differ.

As well as considering arithmetic, Frege also wrote about the properties of ordinary language. He argued that in most sentences, terms which stood for the same object could be exchanged without changing the truth value of the sentence. For example, if it is true that:

(A) Lois Lane loves Clark Kent

then it is also true that:

(B) Lois Lane loves Superman.

However, in certain contexts, this breaks down. We all know that for the longest time, Lois was unaware of Superman’s secret identity, so it is false to say that:

but true to say that:

(D) Lois Lane believes that Superman is Superman.

Frege concluded that in non-indicative statements, proper names do not in fact stand for the objects they usually refer to, but instead proper names refer to the way in which we usually work out what the reference is, i.e. what he called the sense of the name.

Like statements of belief, statements of modality also cause problems for reference. Recall that Hellman argued that structures deal with the possible combinations of objects; this suggests that mathematical statements are not straightforward indicative claims, but instead that they are subjunctive claims about the consequences were there such objects as numbers or sets.

It is possible to combine these two approaches: I claim that statements of arithmetic are indicative propositions, and so the proper names that they contain make essential reference to objects: so numbers are objects. But statements of structural mathematics are not indicative – they are subjunctive statements about the consequences were there objects arranged in such-andsuch a fashion and so do not refer in the same way that arithmetical statements do.

V. Conclusions

I’ve tried to convey some of the main issues in the philosophy of mathematics, and hopefully have managed to make it appear accessible and interesting. Just about everything on the bibliography below is readable, although it varies in technical difficulty; for example, Shapiro’s first book is full of logical detail, while his second book contains virtually no complicated logic at all.

I have hinted that the philosophy of mathematics deals with whether there really are numbers, sets and functions. Different positions such as platonism, intuitionism and formalism offer different ways of tackling these questions; structuralism on the other hand, offers a radically different approach, providing a new perspective on the debates.

Rather than accept the structuralist slogan ‘All mathematics is structural’, I prefer to think of there being structural and non-structural areas of mathematics, and give separate though not independent accounts of each area, based on the differences in the context of statements of those discourses.

Bibliography
Benacerraf, P (1965) ‘What numbers could not be’, Philosophical Review 74, pp47-73; (1973) ‘Mathematical Truth’, Journal of Philosophy 70, pp661-80 – Putnam, H (1983) Philosophy of Mathematics: Selected Readings 2nd edition, Cambridge University Press

Blackburn, S (1984) Spreading the word. Oxford: Clarendon Press

Brouwer, J (1949) ‘Consciousness, Philosophy and Mathematics’, in Benacerraf & Putnam (1983), pp90-6

Dummett, M (1973) ‘The philosophical basis of Intuitionistic Logic’, in Benacerraf & Putnam (1983), pp97-130

Field, H (1980) Science without numbers. Oxford: Blackwell

Frege, G (1879) Die Grundlagen der Arithmetik; trans. Austin (1950) as Foundations of Arithmetic. Oxford:Blackwell; (1893) Die Grundgesetze der Arithmetik. Vol I, Olms: Hildesheim

Gödel, K (1947) ‘What is Cantor’s Continuum Problem?’, American Mathematical Monthly 54, pp515-25, reprinted in Benacerraf & Putnam (1983), pp470-86

Hellman, G (1989) Mathematics without number. Oxford; Clarendon Press

Putnam, H (1971) Philosophy of Logic. New York: Harper

Resnik, M (1981) ‘Mathematics as a science of patterns: Ontology and Reference’, Noûs 15, pp529-49; (1982),‘Mathematics as a science of patterns- Epistemology’, Noûs 16, pp95-105

Shapiro, S (1991) Foundations without foundationalism. Oxford Logic Guides 17, Oxford University Press; (forthcoming) Philosophy of Mathematics: Structure and Ontology. Oxford University Press

Wright, C (1983) Frege’s conception of numbers as objects. Scots Philosophical Monographs, Aberdeen University Press

Stephen Ferguson is a PhD student at the University of St Andrews