Mathematics

When we talk about mathematics, we are talking about what Wittgenstein might call a 'grammar'.

I want to say something a bit more specific, but consistent with Wittgenstein's usage, about grammars.

If we find that a statement - of whatever kind - threatens the intelligibility (as opposed to the veracity) of a conversation, then we are dealing with an element of the grammar of that conversation. Clearly, logics and literal (linguistic) grammars largely fall within this characterisation. However, so do some more peculiar things - Moore's paradox and its variants being one.

Under some circumstances, empirical statements can have a grammatical quality. If you and I appear to agree about almost all the contents of a room we share, but you insist on adding a pink elephant to the list, I will find the intelligibility of our whole discourse threatened, not just the obviously difficult bit. I realise that this challenges some principles of empiricist epistemology, but that seems more conducive to insight than to confusion, as far as I am concerned.

Sometimes we find that groups of these grammatical statements can be generalised over - we can articulate grammatical rules and, even, grammatical theories. (These, of course, cannot be compounded into a theory of intelligibility in general, because this would generate an OQ paradox.)

Some of these theories are like literal (linguistic) grammars, in so far as they do not adjudicate on content. Others seem to include explicit or tacit statements about the world. (Kant, but without the metaphysics).

Mathematics falls into this latter group. It is a set of instructions about how to speak, on the one hand, but it allows us to make reliable measurements of, and predictions about, the physical universe.

As I have argued elsewhere, this is not an accident - the fact that we can talk about the world in the way that we do (an apparently slightly weaker epistemological claim than that it is the way we talk about it - although the stronger claim may not be intelligibly distinguishable from the weaker one), means that the world must have some 'machine-like' qualities that permit this reliable measurement and prediction. Another way of putting this is to say that if  our predictions were radically unreliable, our  'language' would be unintelligible - we would have no language.

There are, in short, some grammars that tell us about the way we can speak, and about the way the world is, as though these were the same thing. And, of course, they are. Any world we can have a conversation about must conform to the grammar of that conversation.

We cannot bring a literally indescribable world into our conversations. We may try to imagine one outside the net of words - perhaps in some private phenomenological space. But even to say we are doing this is to describe it to the extent required for it to be theoretically relevant.

(It is important that to bring apparently inarticulable - wordless - experiences into a space where they can be seen and empathised with is an important therapeutic process.)

Since we can never have a 'complete' grammatical theory, there must be an element of experiment as well as calculation in the way we progress. The discovery that some radical move - zero, negative numbers, complex numbers - does not result in chaos is  characteristic of mathematics. As is the discovery that some 'obviously true' principles - e.g. mathematical induction - can be resistant to direct proof. Gödel wanted to slip a theory of forms in here, I would prefer to say that for our language to be presently intelligible, some future discoveries must come out the right way ...

Hilbert, of course, believed that generally intelligibility would turn out to depend on mathematics being formalisable - and he turned out, not surprisingly, perhaps - to be exactly wrong. It is because it is not completely formalisable that it is mathematics. We cannot predict how our language will develop, except to say that the future must preserve - perhaps as a special case - the intelligibility of how we speak now. (To say otherwise is to claim that the way we speak now is potentially unintelligible, which is incoherent.)

We will discover new ways of talking. We will discover what rules we must follow to remain intelligible.