Thursday, 22 December 2011

A Revolution in Mathematics?

Via the FOM list, I link to a forthcoming (Jan 2012) article by Frank Quinn in Notices of the AMS, entitled "A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today". Rather interesting.

My main comment would be that, although Quinn is defending mathematical logic, the rigour it has brought to understanding mathematical reasoning, and in particular the work of Hilbert and Gödel, Quinn does not mention the semantic definition of truth used in mathematical logic, and which Gödel recognized and which played a role in his discovery of the incompleteness results. (Solomon Feferman's 1984 article "Kurt Gödel: Conviction and Caution", which is also Ch. 7 of Feferman's In The Light of Logic, has a discussion of this.)

I'm not sure of this, but I think Quinn's own view is some kind of formalism, as, more specifically, Quinn writes,
Ironically, it had the same practical consequences because it established “impossible to contradict” as the precise mathematical meaning of “true”.
The suggestion that "$\phi$ is true" be analysed as "$\phi$ is consistent" is not workable, because "$\phi$ is consistent" is not a truth predicate in the technical sense: it does not commute with the connectives and quantifiers. For example, "$\phi$ is consistent" and "$\theta$ is consistent" do not jointly imply "$\phi \wedge \theta$ is consistent". Consistency of a formula $\phi$ or a (recursively axiomatizable) theory $T$ is expressible using a $\Pi_1$-formula ("there is no derivation from $\phi$ (or $T$) ending with $\bot$"), whereas truth of a formula (or soundness of a theory) usually requires a higher level (in arithmetic, for $n>0$, the truth predicate for $\Pi_n$ statements is itself a $\Pi_n$-formula). Consistency and truth are not the same thing. A mathematical anti-realist is better off just saying straight out that the relevant theories are not true: i.e., they're useful-and-consistent-but-not-true. Quinn also doesn't mention the status of mixed sentences, such as "the mass-in-kg of $x$ is $3.21$", "the divergence of the magnetic field $\mathbf{B}$ is zero everywhere", or "the gauge group of the electromagnetic field is $U(1)$", etc., which are at the centre of science. Are these merely consistent? How is truth defined for them?

Suppose I define truth for sentences in the language of arithmetic in the usual way, as follows:
(i) An equation $t = u$ is true iff the values of $t$ and $u$ are identical.
(ii) A negation $\neg \phi$ is true iff $\phi$ is not true.
(iii) A conjunction $\phi \wedge \theta$ is true iff both $\phi$ and $\theta$ are true.
(iv) A universal quantification $\forall x \phi$ is true iff, for all $n$, $\phi(x/\underline{n})$ is true.
As Paul McCartney put it, what's wrong with that? (Ok, aside from the conceivably gratuitous mention of Paul McCartney ...)