Obections to Cantor's Theory (Wikipedia article)
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From: Martin Shobe on 23 Jul 2005 08:57 On Fri, 22 Jul 2005 14:14:51 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Martin Shobe wrote: > >> On Fri, 22 Jul 2005 10:22:58 +0200, Han de Bruijn >> <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >> >>>Robert Kolker wrote: >>> >>>>Han de Bruijn wrote: >>>> >>>>>It's the standard definition of the "actual infinite", but it is not >>>>>"perfectly good". Worse. It's not good at all. >>>> >>>>What is wrong with it? No one has shown it leads to a contradiction. >>> >>>No contradiction. That's perhaps the only good thing about it. If that >>>is the only thing you care about, let me tell you that most of us care >>>about other things, such as a physics that has a reliable mathematical >>>machinery at its disposal. >> >> And physics does have reliable mathematical machinery at its disposal. >> You are blaming mathematics for the choices made by physicists. It's >> the physicist who are choosing to continue using the mathematics that >> involve infinity. > >I am not blaming mathematics. This is what I said somewhere else in this >thread: Now I ... Your response to what is wrong with a definition of infinity was that "Most of us care about other things, such as a physics that has a reliable mathematical machinery at its disposal". The only connection between physics and that definition of infinity, is that the physicist choose to use that portion of mathematics that makes use of it. If that portion of mathematics does not reliably model reality, the *physicist* should be using a different portion of mathematics. This may result in the mathematicians having to develop a new branch (see Calculus, Dirac's delta, etc.), but that doesn't invalidate anything that was there before. >> don't say that mathematicians should be blamed for this, if it happens. >> But the fact is that most physicists have a blind faith in mathematics >> and can be easily deluded by the fact that infinities actually exists, >> within mainstream mathematics, and think that they do exist in physics >> as well. Yet your response is that *mathematics* is what must change. Not the physicists. Most mathematicians do not believe that mathematics has more than an accidental connection to reality. If the physicists believe otherwise, then the phsycists need to have their training adjusted to emphisize that mathematics is not reality. Martin
From: malbrain on 23 Jul 2005 10:11 Peter Webb wrote: > > > > e.g. we agree on the basis of our experience with the axiom's veracity > > and viability. > > > > > You seem to think that somehow mathematics is a physical science, and the > axioms are like physical laws, which can be true or false. You think that > you can observe that zero does not have a suucessor just as you observe that > every action has an equal and opposite reaction. When we talk about natural numbers we AGREE that zero has no predecessor. We get to say with complete VERACITY that minus one is not a natural number. karl m
From: malbrain on 23 Jul 2005 10:15 David Kastrup wrote: > malbrain(a)yahoo.com writes: > > > David Kastrup wrote: > >> malbrain(a)yahoo.com writes: > >> > >> > David Kastrup wrote: > >> >> Robert Kolker <nowhere(a)nowhere.com> writes: > >> >> > >> >> > Peter Webb wrote: > >> >> >> How do you disagree with an axiom? > >> >> > > >> >> > By assuming a contrary axiom, as is done in non-Euclidean > >> >> > geometry. > >> >> > >> >> Oh, but that is not disagreeing with it. In fact, it is expressing > >> >> faith that the axiom indeed _is_ an axiom. > >> > > >> > Axioms are agreements -- an shared expression of faith. > >> > >> Uh, no. Axioms have nothing to with faith at all. If you are playing > >> chess, you don't have _faith_ that a knight moves always two squares > >> and then one perpendicular. If it moves differently, that does not > >> cause you to lose faith in the knight, but rather in your opponent's > >> mental sanity. > > > >>From webster (1913): > > > > "Faith (?), n. [OE. feith, fayth, fay, OF. feid, feit, fei, F. foi, fr. > > L. fides; akin to fidere to trust, Gr. to persuade." > > > > "1. Belief; the assent of the mind to the truth of what is declared by > > another, resting solely and implicitly on his authority and veracity; > > reliance on testimony." > > > > e.g. we agree on the basis of our experience with the axiom's veracity > > and viability. > > Axioms don't have "authority" outside of the game, and certainly not > "veracity". And I don't see why you drag in the dictionary here. The > meaning of the word "faith" was not at all in question. Natural numbers are not a game. They are a part of our language. We have a set of agreements that define them and their properties. karl m
From: klaus.schmid on 23 Jul 2005 13:02 david petry wrote: > For those who missed it, the key sentence is: > > "But, at least, there should exist a path from the abstractions > back to the observable objects." Wouldn't conceivable be better than observable? Anyway, I think it is difficult to express it in one sentence. Mathematics reminds me to a laboratory without any physical objects except a blackboard, symbols written on paper and things like that. I think it is a very nice, economic approach to have a separate room for this. But as with any spezialization, persons not so familiar with this room, may doubt about what is going on here, is it useful? I see some benefit and applications here and there, but some things look very strange, and I cannot see any relation to my work and experience. Therefor I think this laboratory needs more popular explanations about its work, results and basics. Popular should mean: explain a topic as simply as possible, but not more simply. I would wish this mainly for basic topics, topics which at least seem to be easy to understand -- and to attack. Less historical reviews, which may lead away from current mathematics and may end up in dumb associations, e.g. insane Cantor, insane ideas. More discussions about possible applications and interpretations, e.g. what could the different sized infinities mean philosophically. Just my 2 cents. -- Klaus
From: Han.deBruijn on 23 Jul 2005 15:56
Martin Shobe wrote: > Your response to what is wrong with a definition of infinity was that > "Most of us care about other things, such as a physics that has a > reliable mathematical machinery at its disposal". The only connection > between physics and that definition of infinity, is that the physicist > choose to use that portion of mathematics that makes use of it. If > that portion of mathematics does not reliably model reality, the > *physicist* should be using a different portion of mathematics. This > may result in the mathematicians having to develop a new branch (see > Calculus, Dirac's delta, etc.), but that doesn't invalidate anything > that was there before. > > Yet your response is that *mathematics* is what must change. Not the > physicists. Most mathematicians do not believe that mathematics has > more than an accidental connection to reality. If the physicists > believe otherwise, then the phsycists need to have their training > adjusted to emphisize that mathematics is not reality. That sounds reasonable. Alas. Seems that we have become such separatists that almost nobody of us still believes in that great ideal: One World or No World Doesn't anybody agree with me that a Unified Science would be desirable? If it only where for the sake of efficiency. Not to mention the need to prevent schizophrenia. What's the beef in having all those little truths and little beliefs dispersed all over the planet. Sigh ! Han de Bruijn |