Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.

Saturday, August 22, 2009

The mathematician on the street...

That last post was a bit of frustration about an ongoing discussion of AC/CH on the FOM mailing list. Not everything about the discussion has been quite as frustrating as the whole discussion-- namely, some fantastic quotes have come from it. Here are some of my favorite in posting order (I think they slowly get less and less technical...):

"All that we have here in this quasi-paradox is confirmation that reals are not a perfect model of dart throwing and vice versa." -Thomas Lord

"It would be useful to provide some rationale why the continuum having cardinality aleph_1 leads to more unusual results than, say, the Banach-Tarski paradox. Furthermore, I would like to know why you think these results should lead us to reject the continuum hypothesis but not the axiom of choice. Finally, I would be interested to know what has led you to conclude that most 'mainstream mathematicians' find your arguments convincing." -Lasse Rempe

"They are not claiming to have an argument formalizable in ZFC; they are merely claiming that mathematicians have overreacted to the results of Banach-Tarski, Godel, and Cohen by throwing out too much of their intuition about assigning measures to subsets of R^n." -Joe Shipman

"If you think it's an interesting question to investigate plausible extensions of ZFC that settle CH, then you're already a dyed-in-the-wool" -Tim Chow

"But in my experience, if you pick a random mathematician who is not already interested in f.o.m., there's at least a 50% chance that you'll have to remind them of the definition of a well-ordering of the reals and of its relationship to the axiom of choice." -Tim Chow

"So you do not accept AC in the same way you accept the other ZF axioms? That's fine, but it's not the position of the mathematician in the street." -Joe Shipman

" I don't think the mathematician in the street will respond, 'Gee, since I accept AC as gospel, I am forced to blame these pathologies entirely on CH!'" -Tim Chow

"For starters, [the mathematician on the street] is unlikely to be able even to list the axioms of ZF, but he or she will know AC explicitly, precisely because it is known to have some strange consequences." -Tim Chow

(Included because of how wrong it is): "I think we are in danger of forgetting that not only do most mathematicians-in-the-street not believe AC, most of them have no intuitions about it and cannot state it even roughly, let alone have any idea how to use it." -T Foster

"Someone who does not know such basic material cannot be called 'a mathematician' (neither in the street nor anywhere else)." -Arnon Avron

"I am reminded of a time in graduate school [...] when I delivered my self of the opinion that cardinal trichotomy was, intuitively, OBVIOUSLY true and that the Well-Ordering Theorem was, intuitively, OBVIOUSY very fishy." -Allen Hazen

"This certainly circumvents the use of AC, but I submit that it is somewhat contrary to the mathematical practice of *not* equipping structures with non-canonical stuff that is extraneous to their essence. You could define a vector space as something that comes equipped with a basis, or a manifold as something that comes equipped with an embedding in R^n, or a group as something that comes equipped with a homomorphism to an automorphism group of something, etc." -Tim Chow

"mathematicians tend to replace the use of existential statements by the introduction of skolem functions. This is such a common procedure that they do not even notice that they are using AC when they do so." -Arnon Avron

"In particular we agree that these street mathematicians (one pictures them performing Hilbert's Nullstellensatz while passers-by drop coins in their hat) are enumerating witnesses to countability rather than countable sets." -Vaughn Pratt

"Depriving the street mathematician of her witnesses is like depriving a boxer of his fists. [...] Why should she care that foundationalists make things harder by killing off her witnesses?" -Vaughn Pratt

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