Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 26 Jul 2005 13:08 Robert Kolker said: > Han de Bruijn wrote: > > > > True. That's why: > > > > A little bit of Physics would be NO Idleness in Mathematics > > You are pissing and moaning that a peach is not a pear. Mathematics is > deductive. Physics is empirical. Mathmematics is about the relation of > ideas. Physicis is about how the real world works. Two completely > different things. > > Bob Kolker > Mathematics is NOT purely deductive. Before you can apply your rules to the facts at hand, you have to develop the rules. Without an inductive process of establishing axioms, you have no tools with which to deduce anything. So, dismissing the question of whether certain axioms are correct, and ignoring inductive arguments concerning them, is a real mistake that occurs too often in mathematics. It's largely at the root of this particular problem. -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 13:12 Han de Bruijn said: > Peter Webb wrote: > > > 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. > > That's true. But mathematical axioms start to behave _as if_ they were > physical laws, as soon as they become being _applied_ to i.e. physics. > > > The axioms of set theory (ZFC) make a really, really interesting game. > > Set theory doesn't deserve such a predominant place in mathematics. > After the discovery of Russell's paradox et all, everybody should have > become most reluctant. > > Han de Bruijn > > Not to mention the Banach-Tarski spheres. Doesn't that derivation constitute a disproof by contradiction? Isn't the result absolutely nonsensical? And yet, it is accepted, somehow, as truth that one can chop a ball into a finite number of pieces and reassemble them into two solid balls, each the same size as the original, despite all evidence and logic to the contrary. It's a clear sign of something wrong in the system, when it produces results like that. -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 13:18 Peter Webb said: > > > > 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. > > Well, they are not like that. As David said, they are just the rules of the > game. Mathematicians just pick the rules in order to make interesting games. > The example I gave before - the axioms of group theory - make for a really > interesting game. The axioms of set theory (ZFC) make a really, really > interesting game. There is no question of "veracity". Is it true that there > is an element of the group e such that for all g, g*e=g ? Yes, because that > is an axiom of group theory. It is true *by definition*. It is part of the > rules of the game called group theory. As David said, arguing about whether > axioms are true makes about as much sense as arguing about how knights move > in chess - if you think you can invent axioms which make better games, go > for it. But you can't say - "that isn't how a knight moves" because we > defined a knight in chess as moving that way. > > > > Well, if this is how mathematicians feel about their study, and claim it has no connection to reality, then Cantorian set theorists really have no reason to claim that anything they say is more correct than any competing claims. In fact, if they are not at all concerned with understanding numbers as they pertain to reality, then I am not sure what their motivation is besides playing games, and it is not surprising that their solution to such a complex subject as infinity is as nonsensical as it is. If axioms are unquestionable, then I can prove anything true, just by making it an axiom. But, where does that lead us? Certainly not towards any deeper understanding of anything. -- Smiles, Tony
From: malbrain on 26 Jul 2005 13:22 aeo6 Tony Orlow wrote: > Peter Webb said: > > > > > > 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. > > > > Well, they are not like that. As David said, they are just the rules of the > > game. Mathematicians just pick the rules in order to make interesting games. > > The example I gave before - the axioms of group theory - make for a really > > interesting game. The axioms of set theory (ZFC) make a really, really > > interesting game. There is no question of "veracity". Is it true that there > > is an element of the group e such that for all g, g*e=g ? Yes, because that > > is an axiom of group theory. It is true *by definition*. It is part of the > > rules of the game called group theory. As David said, arguing about whether > > axioms are true makes about as much sense as arguing about how knights move > > in chess - if you think you can invent axioms which make better games, go > > for it. But you can't say - "that isn't how a knight moves" because we > > defined a knight in chess as moving that way. > > > > > > > > > Well, if this is how mathematicians feel about their study, and claim it has no > connection to reality, then Cantorian set theorists really have no reason to > claim that anything they say is more correct than any competing claims. In > fact, if they are not at all concerned with understanding numbers as they > pertain to reality, then I am not sure what their motivation is besides playing > games, and it is not surprising that their solution to such a complex subject > as infinity is as nonsensical as it is. > > If axioms are unquestionable, then I can prove anything true, just by making it > an axiom. But, where does that lead us? Certainly not towards any deeper > understanding of anything. As part of a system they are unquestionable authorities when relied upon in that system. They exist FOR the system. We AGREE to them as things-for-themselves. I'm a programmer by trade. Perhaps a mathematician on this list (sci.logic) can enlighten us as to the history the sticking point--the axiom of infinity v. the axiom of induction. karl m
From: Randy Poe on 26 Jul 2005 13:27
Tony Orlow (aeo6) wrote: > Well, if this is how mathematicians feel about their study, and claim it has no > connection to reality, then Cantorian set theorists really have no reason to > claim that anything they say is more correct than any competing claims. As usual, you are confused. Your arguments deal with conclusions which you draw FROM THE SAME AXIOMS as the competing, mainstream theorems you denigrate. Mathematicians have every right to say that theorems deduced from a given set of axioms by rigorous logic are more correct UNDER THOSE AXIOMS than a competing, contradicting set of conclusions someone claims arise UNDER THE SAME AXIOMS. Some outside "reality" is not the standard. The basis of argument in your eternal threads is the validity of your conclusions UNDER A SPECIFIC SET OF AXIOMS AND THE RULES OF DEDUCTION. - Randy |