Obections to Cantor's Theory (Wikipedia article)
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From: David Kastrup on 21 Jul 2005 03:49 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Daryl McCullough wrote: > >> Nothing of any importance about mathematics would change if we >> substituted different words for the basic concepts. In contrast, >> your arguments are about nothing *but* terminology. To me, that >> shows that there is no actual content to your arguments. An actual >> mathematical argument does not depend on word choice. As a >> challenge, see if you can express your claims about infinite sets, >> or infinite naturals, or set size, or whatever, *without* using the >> words "infinite", "larger", "size", etc. > > This clearly represents the formalist (Hilbertian) view on > mathematics as a "senseless game with symbols". So, it represents the view that it does not make sense to talk about different things using the same words. > The problem is that pro-Cantorians think that it is the only > possible view. Wrong. The problem is that _if_ you want to talk about the state of mathematics, you have to do it using the definitions that it is built upon. If you want to talk about something different, that is fine too, but then you need to redefine the terms. If you don't do that, you are not free to attach different meanings to them, and if you do, you are not free to criticize the conclusions from them. If you don't like counterintuitive results, you have to replace their foundations. Failure to do so renders your case ridiculous. > Mathematicians like Brouwer, on the other hand, have repeatadly > emphasized that mathematics should have a _meaning_. And thus they changed the definitions, and built their own systems based on the changed definitions. This is the legitimate way to proceed against counterintuitive results: replace the foundations that led to them. But you can't just protest the results alone. > But such a meaning can only be attached if it is _outside_ the > formalism. The desired outside meaning steers the direction your formalism is taking, but it can't replace it. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Han de Bruijn on 21 Jul 2005 03:51 stephen(a)nomail.com wrote: > In sci.math Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>Nah, nah. First comes the theorem. And then comes the proof. > > You cannot prove something without axioms. So if you > start with your conclusions, you have to create some > axioms before you can find a proof. Unfortunately they > never seem to actually define any axioms. Look at Tony Orlow > in this thread for a fine example of this sort of behavior. You cannot prove something without having a clue what to prove. So the conjecture comes first. Then we have to find out, eventually, which set of axioms fits our conjecture. Some people arrive at wrong conclusions because they create the _wrong axioms_. Look at mainstream mathematics for a fine example of this sort of behavior. Nah, nah. Han de Bruijn
From: Han de Bruijn on 21 Jul 2005 03:57 Virgil wrote: > Every bit of "Cantorianism" has been well enough defined for the > understanding of thousands upon thousands of people. That TO fails where > so many have succeeded says more about TO than about the adequacy of > "Cantorianism's" explanations. The fact that a faith has millions of adherants doesn't say anything about its validity. It says something about the society wherein it is accepted, though. Han de Bruijn
From: David Kastrup on 21 Jul 2005 04:09 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > stephen(a)nomail.com wrote: > >> In sci.math Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> >>>Nah, nah. First comes the theorem. And then comes the proof. >> You cannot prove something without axioms. So if you >> start with your conclusions, you have to create some >> axioms before you can find a proof. Unfortunately they >> never seem to actually define any axioms. Look at Tony Orlow >> in this thread for a fine example of this sort of behavior. > > You cannot prove something without having a clue what to prove. So > the conjecture comes first. Then we have to find out, eventually, > which set of axioms fits our conjecture. Some people arrive at wrong > conclusions because they create the _wrong axioms_. That does not invalidate the conclusions as conclusions. It invalidates the model. The problem is that our cranks here don't bother replacing the axioms while protesting some of their conclusion as unintuitive, and wanting to keep others. And that's not going anywhere. The rules are: if you don't like some conclusions, you have to change the axioms, and then you lose all other conclusions (many of them might be easy to reacquire, but that process is not automatic). -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Han de Bruijn on 21 Jul 2005 05:44
Tony Orlow (aeo6) wrote: > Well, that's about as close to a lie as one can get, eh? I asked for a > definition of infinite, and no one could give me a definition of that word. The > best I could get was that an infinite set can have a bijection with a proper > subset, which is hardly a definition of the word "infinite". In fact I went to > the etymology, which literally means "without end". Finite means with a known > end or bound, and infinite means without end. Of course, I got all sorts of > flack for my definition, from those that couldn't even suggest one outside of > the set theory they were regurgitating. let's try to be straight here, and no > more insulting than necessary, so it doesn't come back to bite us, why don't > we? The infinite they define as "that an infinite set can have a bijection with a proper subset" of itself is known as "_actual_ infinite", which is rejected by most anti-Cantorians as sheer nonsense. Apparently, you are rather talking about "_potential_" infinity: something finite that becomes larger and larger. The indisputably useful concept of a "limit" falls within the latter category. Everything that involves the infinite and cannot be handled with limits is rather suspect IMHO. Han de Bruijn |