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 ...)


  1. I just ran across this after reading Quinn's article. I think you miss the point. I do not see Quinn as making any philosophical points regarding the meaning of "truth" which remains quite vague even after reading Tarski and Godel. Quinn is a classical mathematician, hence his misguided defense of proof by contradiction which itself misses the very point he would make. He seems to mistake proof by contradiction with rigor, and he also seems to mistakenly equate intuitionism with lack of rigor. That is a mistake, to be certain. Intuitionism became a form of rigor, leading to questions about what can done without certain assumptions regarding infinity. What the methods of topoi showed is that mathematical truth is relative, hence in a sense justifying formalism but in a more comprehensive sense than Hilbert would have imagined (with a hierarchy of what might be "true". I suggest reading Toposes and Local Set Theory by J. L. Bell, available from Dover).

    Logic as cast in topoi has clearly shown that regardless of embracing proof or model theory, there is a hierarchy of logical methods that lead to more theorems as one moves up the hierarchy. Classical theorems in mathematics, including much of calculus, become less possible as one moves down the hierarchy.

    In essence the argument of what is "really" true in mathematics is nonsense. Realism versus formalism is dead, especially as almost any reality can be supported by a formal logic down the hierarchy (not sure about ultrafinitism though). Classical mathematics is an art form, not a scientific endeavor to determine what is "really" true. Ultrafinitists are much like the tea party, warning of imminent collapse in a bubble of contradiction (instead of debt).

    Quinn's concern that a lack of respect for rigor is the norm much of "mathematical science" and high school education, not to mention engineering, is quite well taken. Trained as a pure mathematician, I worked in industry with engineers and physicists and made a very good living fixing all the stuff that didn't work because they didn't respect hypotheses and seldom understood the theories they tried to apply (all mathematical). I merely needed to carefully adjust their algorithmic contraptions to account for hypotheses and such (while learning that certain mathematical assumptions like the axiom of choice make no difference in application other than it was generally impossible to build contraptions from theorems where the axiom of choice was necessary to posit existence). Meeting aerospace engineers with advanced degrees who did not know how to multiply matrices was awakening.

    If you are convinced of the "reality" of mathematical objects, I suggest you build a Klein bottle or even a Mobius band (not the paper model which as you know is three, not two, dimensional). Or write down the square root of two in all its digital glory in some base (not simply giving an algorithm for potentially computing them), say two or ten or sixteen so it will "require less space". Or build me an exotic 7 sphere (not mathematically, but physically).

    I don't think Hilbert or Quinn confused "semantic truth" with lack of contradiction. And I am sure they would be quite comfortable with your simple example at the end of the article, but be aware there is a problem with that final step, the universal quantification over a possibly infinite set. That is the heart of the intuitionistic argument that Quinn confuses with lack of rigor and that you seem to not understand at all. Mathematical realism is nonsense. The arithmetic you fall back on here is in dispute. That is the point. There is a formalism in which those integers do not exist. Can you show them (all of them) to me as an object?

  2. Many thanks for the comment!

    If you are convinced of the "reality" of mathematical objects, I suggest you build a Klein bottle or even a Mobius band (not the paper model which as you know is three, not two, dimensional).

    Why do you think that a Mobius band is a physical object? Do you think that SU(3) is a physical object?



  3. Did you bother to read what I wrote? From the question I would guess no. I can describe with great precision a Mobius band or a Klein bottle or the group SU(3), but that has nothing to with physical existence. Describing things that exist in physical reality with that sort of precision is impossible. Can you prove the existence of fish?

    Can't recall how I got on last time, so am posting anonymously this time.

  4. Thanks, Anonymous,

    This is good. You are now saying that there is a Lie group, e.g., SU(3), that you "can describe with great precision", and you also now concede that SU(3) does not have physical properties.
    To summarize your assertions:

    1. There is a Lie group.
    2. No Lie group has physical properties.

    What conclusion would you then draw from your assertions?



  5. Long time I have not visited here. In reply to that statement about Lie groups, you seem to believe that because you can describe something in great detail it must exist in some sense. As an idea that can be communicated, sure. That is using what might be termed operational definitions which is the heart of mathematics: you describe the class of things so that one can test whether or not some idea fits within that class. That is how it is that people who take the time to study mathematics can understand what another mathematician means and can determine whether or not some statement's proof is correct and hence true within a given framework of logic. (Even intuitionists and ultra-finitists can tell if a statement is true in a framework of logic other than the one they choose to accept.) That is not something that happens in everyday language much. Like, for example, the word God. Seems to be no operational definition for the word, which does not keep people from thinking they understand what someone might mean when he or she uses the word. Of course, they are wrong in an operational sense,. unlike say a Lie group or root two.

    Not just Lie groups don't have physical properties, neither do circles as described in mathematics. The idea that somehow those things had to have an existence outside the physical universe because they could be thought gave rise to the nonsense of Plato whose idealism was among the stupidest philosophies of all time (Russell does a good job of skewering it in his History of Western Philosophy). Neither do numbers.

    What is the physical reality of the square root of two which "exists" a a solution to a a quadratic equation? We know you can't represent it in any radix, though you can approximate it. But operationally it is that number whose square is two. Simple. Pi is more complex, not being given by a polynomial with rational coefficients, but it is based on the ratio pf two quantities that cannot be measured in the physical world and depend on some mathematical precision which can be thought only.

    My conclusion, in short, is that because one can think something does not mean it is any sense physically real. So what? Anyone who reads horror fiction knows that. Philosophers and mathematicians tend to be confused easily.