Obections to Cantor's Theory (Wikipedia article)
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From: Daryl McCullough on 24 Jul 2005 13:33 David McAnally says... >pratt(a)boole.Stanford.EDU (Email me at CS not Boole) writes: >>Point taken. I guess my (narrower) point would have to be that it >>seems a shame for physicists to have to surrender the term "Dirac delta >>function" merely because the Cantorian perspective disallows it as a >>function. > >Not strictly accurate. The Dirac delta function can be treated as a >function. The domain is S, the set of Schwartz test functions. The range >is C. The Dirac delta function maps the Schwartz function f to f(0). What Vaughn means is that the delta function is treated by physicists as a function on *reals*, not as a function on functions. They take derivatives of it, they compose it with other functions on the reals, etc. -- Daryl McCullough Ithaca, NY
From: Jiri Lebl on 25 Jul 2005 01:19 david petry wrote: > My claim is that all the results of analysis which have observable > implications, an hence have the potential to be applied as models > of the real world, necessarily do not need Cantor's Theory. I assume "cantor's theory" means "set theory." As far as I know set theory was invented pretty much precisely to arithmetize analysis. To make it possible to make rigorous rather then just handwavy arguments. > Mathematicians were solving real world problems long before Cantor's > Theory came along. Yes, and they're now able to solve much more complicated problems because of it. Set theory made it possible to get much more profound mathematical results that were then applied to real systems where such predictions agreed with observation. That's your connection to reality. > Applied mathematicians never need to think about > completeness, and I suspect many don't know what it is. That's the whole point of completeness, that you can just take limits willy nilly and assume you get a real number back as long as the sequence is cauchy (most people then just equate this with convergence precisely BECAUSE the reals are complete). > My argument is that the mathematics that is relevant to real > world problems necessarily has a property of "observability" - that is, > mathematical statements must have observable implications, where > we think of the computer as the mathematician's microscope which > lets us make observations. Before Cantor came along, mathematicians > would have listened to that argument. Now they don't. What is this obsession with "computer as a microscope"? Computer is a very limitted device capable of making some limited formal manipulations on strings of meaningless characters. Further there are observable facts that agree with mathematical models created by deductive reasoning and not possible by brute force calculation by a computer. Computers are but a tool that can help you with tedious calculations but most definately no microscope for observations to base all "relevant" problems on. Observation for applied mathematics is done in whatever the system is that you are observing, not in a computer. Jiri
From: Jiri Lebl on 25 Jul 2005 01:39 Vaughan Pratt wrote: > Daryl McCullough wrote: > > Give an example of a calculation in physics in which using modern > > ("Cantorian") mathematics gives you the wrong answer, and using > > some other kind of mathematics gives you the right answer. > > That's pretty easy, isn't it? The Dirac delta function pervades > physics, but if you used "Cantorian mathematics", which defines a > function as a set of ordered pairs, you'd get either the wrong answer, > an overflow error of some kind, or a type check error. Not even Dirac would call the "delta function" a set theoretic function. It is NOT a set theoretic function. It is a measure! And totaly fine within the cantorian picture of the world. Or you can think of delta function as a distribution acting on say the continuous integrable functions by giving the value of the function at 0. The reason for calling it a "function" is purely from sloppiness reasons. You want to just treat it as a function which you can AS LONG AS YOU INTEGRATE IT, and not take it's value as it is not a function. BTW from the formulation of your answer I assume you're a programmer and not a mathematician. Jiri
From: Han de Bruijn on 25 Jul 2005 03:32 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
From: Han de Bruijn on 25 Jul 2005 03:50
klaus.schmid(a)wtal.de wrote: > 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. Suggesting that mathematics is good in its overall content. But it needs better "sales", the problem is in the packaging. (Like with the European Constitution ...) Han de Bruijn |