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 f.o.m.er." -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

## Saturday, August 22, 2009

## Sunday, August 16, 2009

### A little rant about AC...

So. Before we get started. Math is based on logic. The most important result of this fact is that there must be some axioms-- a starting point for the logical "gears". So, there must be a few things which are accepted as true, without argument, in order to "prove" anything. The really nice thing about this is you can use different axiom sets for different purposes; for example, the Peano Axioms are the axioms for number theory; any set theory capable of producing arithmetic will have axioms that imply the Peano Axioms.

The problem with this forced reliance on axioms is that mathematical truth is based to some small extent on human intuition. And human intuition of mathematical concepts is notoriously fickle.

One point that is still a hot topic amongst foundational researchers, and amongst those who spend their time discussing the philosophy of math, is called the Axiom of Choice. The "normal" statement (which gives the axiom its name) is a bit technical, but there's a completely equivalent statement: A non-empty Cartesian product of non-empty sets is non-empty. In other words, if we have a bunch of non-empty sets, and we take the set of all tuples (ordered lists) of these sets, we have a non-empty set. An example: X={1,2}, Y={1,3} Z={3}, X×Y×Z={(1,1,3),(1,3,3),(2,1,3),(2,3,3)}. (For the more interested reader, the first statement of AC in the wikipedia article mentions a choice function. Any point in the cartesian product encodes such a choice function. If we have a non-empty cartesian product, we have a choice function.)

The debate is mostly about whether or not this statement is intuitively true-- can we say it can be placed with the "obvious" axioms? It seems to make sense to do so, but it leads to a lot of counterintuitive results. The most famous such result is the Banach-Tarski "paradox", which says it is possible to take apart a sphere to create 2 spheres whose sizes are each equal to that of the first. Let's repeat that: Start with one sphere of a certain volume. Split it into 2 in a very clever way. Now you have two spheres, each with volume equal to the first. The "clever way" of splitting the sphere requires the axiom of choice (in a way I'm not sure I have the background to understand.)

The problem, of course, is that counter-intuitive starts at infinity, not at AC. We can split the set of all even numbers into two copies without choice (pull out 2,6,10,... then divide those by two and add 1, and simply divide the rest by 2), so why is a sphere less intuitive? How is it intuitive that there are as many rationals as there are integers? How is it intuitive that you have 0 probability of selecting an algebraic number from the reals, despite the fact that they are dense?

I don't mind people rejecting choice for certain work: constructive logic is incredibly useful for CS, but it explicitly contradicts choice. What I mind is people bringing up "counter-intuitive" results that are no more counter-intuitive than results that have long been taken for granted, because we're so used to seeing them.

My last sentence reminds me of another problem with the whole discussion: Human intuition is so fickle! The results I mentioned are not considered counter-intuitive to most working mathematicians, because the results are so fundamental. In addition, in the same breath that they say "AC leads to counter-intuitive results", they talk about how certain people haven't built up an intuition for these sorts of foundational results. Perhaps none of us have built up an intuition for certain results?

The problem with this forced reliance on axioms is that mathematical truth is based to some small extent on human intuition. And human intuition of mathematical concepts is notoriously fickle.

One point that is still a hot topic amongst foundational researchers, and amongst those who spend their time discussing the philosophy of math, is called the Axiom of Choice. The "normal" statement (which gives the axiom its name) is a bit technical, but there's a completely equivalent statement: A non-empty Cartesian product of non-empty sets is non-empty. In other words, if we have a bunch of non-empty sets, and we take the set of all tuples (ordered lists) of these sets, we have a non-empty set. An example: X={1,2}, Y={1,3} Z={3}, X×Y×Z={(1,1,3),(1,3,3),(2,1,3),(2,3,3)}. (For the more interested reader, the first statement of AC in the wikipedia article mentions a choice function. Any point in the cartesian product encodes such a choice function. If we have a non-empty cartesian product, we have a choice function.)

The debate is mostly about whether or not this statement is intuitively true-- can we say it can be placed with the "obvious" axioms? It seems to make sense to do so, but it leads to a lot of counterintuitive results. The most famous such result is the Banach-Tarski "paradox", which says it is possible to take apart a sphere to create 2 spheres whose sizes are each equal to that of the first. Let's repeat that: Start with one sphere of a certain volume. Split it into 2 in a very clever way. Now you have two spheres, each with volume equal to the first. The "clever way" of splitting the sphere requires the axiom of choice (in a way I'm not sure I have the background to understand.)

The problem, of course, is that counter-intuitive starts at infinity, not at AC. We can split the set of all even numbers into two copies without choice (pull out 2,6,10,... then divide those by two and add 1, and simply divide the rest by 2), so why is a sphere less intuitive? How is it intuitive that there are as many rationals as there are integers? How is it intuitive that you have 0 probability of selecting an algebraic number from the reals, despite the fact that they are dense?

I don't mind people rejecting choice for certain work: constructive logic is incredibly useful for CS, but it explicitly contradicts choice. What I mind is people bringing up "counter-intuitive" results that are no more counter-intuitive than results that have long been taken for granted, because we're so used to seeing them.

My last sentence reminds me of another problem with the whole discussion: Human intuition is so fickle! The results I mentioned are not considered counter-intuitive to most working mathematicians, because the results are so fundamental. In addition, in the same breath that they say "AC leads to counter-intuitive results", they talk about how certain people haven't built up an intuition for these sorts of foundational results. Perhaps none of us have built up an intuition for certain results?

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