I recently read and reviewed The Pea and The Sun by Leonard Wapner, which is about the Banach-Tarski Paradox. Recall that the Banach-Tarski Paradox is actually a theorem (not a paradox) that says you can take apart a ball of volume 1 into a finite number of pieces (some of the pieces will be VERY funky) and reassemble to get a ball of volume 2. By repeating this process you can take a pea and get a ball the size of the sun. (My review is here).
When I told my wife this theorem (she has a Masters in Comp Sci and knows some math)
she said Math is Broken!!!. She later wondered why mathematicians
didn't just toss out The Axiom of Choice (for uncountable sets)
since it leads to such an obviously false theorem.
By coincidence, Leonard Wapner had just emailed me to thank me for my review (and to agree that
MacArthur Park is the worst song ever written, see
on my website).
SO I put the question to him- why didn't mathematicians just toss out AC
since it leads to BT?
What follows is an edited version of our back and fourth emails.
BILL: I told my wife the Banach-Tarski Paradox
and she wonders why mathematicians didn't just IMMEDIATELY toss out
the Axiom of choice for uncountable sets since BT is so obviously false.
LEONARD: I do not believe BT is obviously false. I believe it to be true, based
on the axioms of set theory, including AC.
Your wife is claiming there are no real world manifestations of the
phenomenon. She is not alone with this feeling. Personally, I am
highly skeptical of there being any real world models of the BT result.
(Those given in the book were clearly labeled as speculations at best.
But, I also believe we can't rule out the possibility, based on
mathematical and physical discoveries, to date. BT does not contradict
any theorem or axiom. And, it's no more counterintuitive than some Baby
BTs (mini version of BT that proceeded it that are also counter-intuitive)
and some physical phenomena (relativity, quantum mechanics, etc.)
BILL: By the time BT was proven AC was embedded into the
math culture. Many things had already been build on it.
Is that why Math folks didn't just toss it out?
Would anything important be lost if we tossed uncountable AC out?
LEONARD: I believe it wasn't tossed out because there was no mathematical
justification (in the way of a mathematical contradiction) to do so. If
AC were to produce a mathematical contradiction, then, being a
questionable axiom, it might be tossed. But, there is no mathematical
contradiction and I think most mathematicians accept AC. In any case,
it would be even more counterintuitive for some to drop it than accept
the BT result. AC allows for surprises, but no mathematical contradictions.
I suspect that there
are useful theorems, relying on AC, which would be lost if AC were to be
tossed. I can't give a specific example now, but I do know that there
have been proofs relying on AC which have encouraged others to prove
those same theorems without AC. BT, in its usual form, requires AC.
Some variations of it do not.
BILL: BT is obv false in the real world. Is this enough of a reason to toss
it out (my wife things so).
LEONARD: Coincidentally, my wife asks me the exact same question. And, this is
precisely what intrigues me about BT and other paradoxes. As I write
above, I prefer to say it is "apparently" false in the real world,
rather than obviously false. I can't rule out that which I've not seen
simply because I've not seen it.