Friday, January 22, 2010


Have you seen this article by Willfred Hodges about reading people's "refutations" of Cantor's diagonal argument?

It's pretty funny, and he even raises some interesting philosophical issues about logic, how to think about logical mistakes, etc.


  1. Hi Sharon,

    I'm writing a paper on a posteriori mathematics.

    Are you ok with my citing:

    I note that it's listed as a paper in progress.

  2. OK with it? I'd be flattered!

    And maybe you could mail me when you find a devastating objection :).

  3. Truth be told, I'm still digesting it.

    I think that the axioms you've assumed are overly restrictive.

    I'm questioning whether any of mathematics is truly a priori knowledge.

    Wittgenstein's argument that we'd reject Principia if it conflicted with everyday counting comes to mind.

  4. btw: I think those that argue against Cantor's Diag would also accept zeno's paradox as showing there is no infinity. They would also claim the Axiom of Infinity in ZF to be inconsisent? (or simply false?)

  5. It's funny you should mention this because, my thesis is about exactly about this issue of revising mathematical beliefs when they conflict with certain (observational or mathematical or logical) applications.

    I think we totally do revise like this, and that this fact is crucial to understanding how we can learn about mathematics without having some kind of spooky vision of mathematical objects. BUT to say that conflict with experience is a crucial *causal mechanism* which helps correct our mathematical and logical beliefs, doesn't yet commit one to saying that it figures in the justification of those beliefs.

    So I think experiences like that do help causally lead us to have correct intuitions about math, but they don't then get cited in the justification of the true beliefs which we form using these good intuitions, and hence don't count as depending on experience for their justification.

    In slogan form - Revision of mathematical beliefs is like revision of scientific beliefs with Stockholm Syndrom; In both cases experience slaps you down and makes you revise your beliefs. But, in the science case you say you were initially justified in believing that P, and now experience justifies you in changing your mind. WHEREAS when experience prompts revision to *mathematical beliefs* you say 'oh I made a mistake, I should never have accepted that fallacious argument in the first place!' :)

    soo...IMO that's the sense in which mathematical beliefs are still a priori, but there's also a closer relationship between a priori and posteriori knowledge than you might think/ the a priori vs. a posteriori distinction is less 'deep' than you might think.

  6. That is very funny. I'm working pretty much the same angle.

    My slogan version is that all a priori knowledge is simply that which has not met its counterexample. I'm exploring the idea that what we are calling a proiri is really the sum of the millions of a posteriori experiences of our ancestors (i.e. we are not likely the descendents of the people who couldn't figure out that one apple and another two apples is the same as two apples plus another apple.)

    This gives basic counting and adding with small numbers an instinctive "rightness" in our minds. This could also explain why things like quantum mechanics are so confusing.

    My handicap is that I'm just returning to academics. I have an undergrad in math, a Master of Science in Business Admin (whatever that means) and I took a multi-year break. I only decided recently to apply for a Ph.D. in Philosophy. It is reassuring that I'm interested in something that those in the know are interested in too.

    Your paper was on of the first I found when I first thought of the concept of a posteriori mathematics. I too was thinking about Godel undecidable statements as the example of such.

    However, have you ever considered this: Take a theorem such as Goldbach's Conjecture. What if I told you it was undecidable? i.e. I somehow proved that there is no number theoretical argument that would prove it.

    Surprisingly: it would then have to be True. You could rest easy knowing that a counter example would never be found by ω-consistency. And that implies that there is no even number that is not the sum of two primes.

    I'm ultimately hoping that I can show that anything that is Godel Undecidable with respect to PA has unknowable truth value iff it has no empirical analog.

    And I think that is why Godel's Theorems hold no water with respect to things such as Euclidean Geometry, but the logical connection...I have not made that yet (other than the obvious Euclidean Geometry doesn't have PA as a subset).

    Please excuse the sloppiness of the language...I'm still learnin'

  7. Corrections/Clarifications:

    "multi-year break" was from academics.

    Sorry for the double negative. "That would imply all even numbers are the sum of two primes"

    PA = Peano Arith

    Empirical Analog: By this I mean a arithmetically testable statement. I'm still working on formalizing this concept.

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