Why does mathematics get treated so poorly by analytic philosophy?
Here is an exciting challenge, which calls for close cooperation between physicists and logicians — better still, for the work of younger men who have studied both physics and logic. The application of modern logic and the axiomatic method to physics will, I believe, do much more than just improve communication among physicists and between physicists and other scientists. It will accomplish something of far greater importance: it will make it easier to create new concepts, to formulate fresh assumptions. (Carnap, quoted in Friedman’s Wissenschaftslogik).
Why not mathematics!
We see in Nagel 1939 paper ‘The formation of modern conceptions of formal logic in the development of geometry’, Osiris 7:142-224, what has gone wrong. He claimed that, as projective geometry suggested, geometry is no longer about geometric things, just logical relations of entities satisfying certain properties. Yes, but we still need to talk about ideational content. Mathematics cannot be reduced to logical if-thens. Nagel is providing a philosophical gloss decades after the event on one aspect of an important change, but contrast this with Lautman’s insights of the same time into the contemporary ideas governing mathematics (see my Philosophiques article), where logic is drawn within the scope of mathematics, a notion later wonderfully realised by Lawvere’s idea of quantifiers as adjoints.
Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth. (Russell)
…contrary to Quine’s well-known portrayal of Carnap’s position – as presented, for example, in Quine (1963) – Carnap’s own emphasis on the importance of the analytic/synthetic distinction is by no means derived from a foundational epistemological program aiming to explain how logical and mathematical certainty is possible in terms of truth-by-convention or truth-in-virtue-of-meaning. Rather,according to precisely the principle of tolerance, the point of regarding the statements of logic and mathematics as analytic lies in our freedom to choose which system of logic and mathematics best serves the formal deductive needs of empirical science. Classical mathematics, for example, is much easier to apply, especially in physics, than intuitionistic mathematics, while the latter, being logically weaker, is less likely to result in contradiction. The choice between the two systems is therefore purely practical or pragmatic, and it should thus be sharply separated, in particular, from all traditional philosophical disputes about what mathematical entities “really are” (independent “Platonic” objects or mental constructions, for example) or which such entities “really exist” (only natural numbers, for example, or also real numbers – that is, sets of natural numbers). Carnap aims to use the new tools of metamathematics definitively to dissolve all such metaphysical disputes and to replace them, instead, with the much more rigorous and fruitful project of language planning, language engineering – a project which, as Carnap understands it, has no involvement whatsoever with any traditional epistemological program. Indeed, as Carnap clearly and emphatically states in Logical Syntax, the new discipline he here calls Wissenschaftslogik (the logic of science) “_takes the place of the inextricable tangle of problems one calls philosophy_” (1934, Sect. 72). (Wissenschaftlogik, 392-393)