Counterintuitions and the well-ordering theorem
in [Logic]
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From: Aatu Koskensilta on 18 Oct 2009 10:39 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > I don't think what you are saying makes any sense. Bringing up > non-measurable sets is not a way to explain the counter-intuitiveness > of Banach-Tarski. I didn't say it was. I said that in order for the Banach-Tarski theorem to strike anyone as counter-intuitive we must implicitly depend on some identification of arbitrary and wild sets of points with physical volumes etc. > It's a way of arguing that it *ISN'T* counter-intuitive (because > people don't have good intuitions about non-measurable sets). Most people have no intuitions whatever about non-measurable sets. Of course, in order to illustrate that mathematical constructions needn't have anything to do with balls and pieces in the ordinary sense we can just observe that e.g. removing all points with rational coordinates from the unit ball doesn't correspond to any physical operation. > The situation with Banach-Tarski to me is that (1) There is an > informal, intuitively true claim, (something along the lines of: If > you cut up a solid object into pieces, and you rearrange the pieces, > you'll get a new object that has the same volume as the original. (2) > There is an attempt to formalize the informal claim, by defining what > a "piece" might be, what "rearrange" means, etc. (3) Then > Banach-Tarski shows that the formal version is false. This is not a very accurate account of the situation. There are usually no attempts to formalise anything. Rather, some people simply mistakenly think a mathematical result means something it doesn't mean. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 18 Oct 2009 10:46 David C. Ullrich <dullrich(a)sprynet.com> writes: > Come on. Surely it's clear that we need someone with a _less_ keen > intuition to explain this. So it appears. Regarding the evidence you provided for the evidence of the axiom of choice (or at least dependent choice or countable choice), well, there is a reason the axiom of choice is an axiom and the well-ordering theorem a theorem. (Cantor did at one point think the well-ordering principle was a "law of thought", but later came around the view it's something that needs to be proven.) My favourite example of a result where it's often difficult even for people with some experience in such matters to spot the invocation of (countable) choice is the theorem that omega_1 is regular. > Otoh one sees the same people giving the impression that the existence > of an uncountable well-ordered set requires AC. This idea is not uncommon in news... -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 18 Oct 2009 11:30 Aatu Koskensilta says... > >stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > >> I don't think what you are saying makes any sense. Bringing up >> non-measurable sets is not a way to explain the counter-intuitiveness >> of Banach-Tarski. > >I didn't say it was. I said that in order for the Banach-Tarski theorem >to strike anyone as counter-intuitive we must implicitly depend on some >identification of arbitrary and wild sets of points with physical >volumes etc. Yes, and I'm saying that that is completely wrong. As I explained, there are two claims at work: (1) a formal statement, and (2) an informal paraphrase of that statement. The informal paraphrase is counterintuitive. It is not counter-intuitive because someone has implicitly accepted some identification of arbitrary and wild sets of points with physical volumes. You can argue that the informal paraphrase is *inaccurate*, that it doesn't really capture the meaning of the formal statement (or that the formal statement doesn't capture the meaning of the informal statement). >> The situation with Banach-Tarski to me is that (1) There is an >> informal, intuitively true claim, (something along the lines of: If >> you cut up a solid object into pieces, and you rearrange the pieces, >> you'll get a new object that has the same volume as the original. (2) >> There is an attempt to formalize the informal claim, by defining what >> a "piece" might be, what "rearrange" means, etc. (3) Then >> Banach-Tarski shows that the formal version is false. > >This is not a very accurate account of the situation. Sure it is. There are two claims, one is informal and intuitively true, and the other is formal and provably false. They are claimed to be paraphrases of each other. >There are usually no attempts to formalise anything. Sure there is. You have to formalize what a rearrangement means, and you have to formalize what a "piece" of a solid object is. >Rather, some people simply mistakenly think a mathematical result >means something it doesn't mean. I don't know what point you are making, unless you are saying that the order is: formal result --> inaccurate informal paraphrase, instead of informal statement --> inaccurate formal paraphrase. What difference does that make? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 18 Oct 2009 11:44 Aatu Koskensilta says... Anyway, your claim about the counterintuitveness of the Banach Tarski result being dependent on wild assumptions about non-measurable sets is patently false. Maybe you don't care about whether your observations are true or not, but that particular observation is certainly not true. It's provably false because I am a counter-example. The Banach Tarski theorem is counter-intuitive to ME, and I believe I understood perfectly well that the "pieces" into which they cut a sphere are non-measurable, and that therefore they do not correspond to anything like a physical object. So what you are saying is false. Not that you necessarily care. -- Daryl McCullough Ithaca, NY
From: Rupert on 18 Oct 2009 16:51
On Oct 19, 12:39 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > stevendaryl3...(a)yahoo.com (Daryl McCullough) writes: > > I don't think what you are saying makes any sense. Bringing up > > non-measurable sets is not a way to explain the counter-intuitiveness > > of Banach-Tarski. > > I didn't say it was. I said that in order for the Banach-Tarski theorem > to strike anyone as counter-intuitive we must implicitly depend on some > identification of arbitrary and wild sets of points with physical > volumes etc. > I would think when you tell the man in the street about the Banach- Tarski paradox, he doesn't explicitly formulate to himself some kind of argument involving volumes, he just has the strong intuition based on his experience with the physical world that that couldn't possibly be the case. |