6 - Metaphysics, Necessity and Grammar

Summary

Description and Necessary Truth

It is a part of what Wittgenstein called ‘Augustine's picture of language’ that sentences are ordered combinations of names. A corollary of that idea is the thought that it is by the combination of names in accordance with the rules of syntax that sentences can represent how things are. It is but one step more, in the developed forms of this conception of language, to suggest that just as it is of the essence of words to stand for things, so too it is of the essence of sentences to describe how things stand. And one may well go on to argue, as Wittgenstein did when he wrote the Tractatus, that the general form of a proposition – a sentence in use – is: such-and-such is thus-and-so. And that is clearly the general form of a description.

If we think thus, then we may be inclined to go on to say that different kinds of proposition describe different kinds of subject matter. We distinguish the subject matter of physics from that of chemistry, for example, and we distinguish the domains of physics and chemistry from that of biology. Propositions of physics describe how things are in the domain of physics. Propositions of chemistry describe how things stand in the domain of chemistry. And propositions of biology describe how things are in the realm of the biological. But if we go down this road, what are we going to say about propositions of mathematics?

Int. Well, surely arithmetical propositions such as ‘25 x 25 = 625’ are true descriptions of relations between numbers. They state mathematical facts. Just as there are physical facts in the spatio-temporal domain of physics, so too there are mathematical facts in the non-spatial and atemporal domain of numbers. Isn't that precisely what mathematicians study?

PMSH. Good, that is indeed how most great mathematicians have thought of themselves and of the propositions they investigate. Mathematicians conceived of themselves as explorers and discoverers of the non-spatial and atemporal domain of number – precisely analogous to physicists in the spatio-temporal domain of physical objects.