Obections to Cantor's Theory (Wikipedia article)
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From: David Kastrup on 25 Jul 2005 04:42 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > 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. Oh nonsense. It is not the mathematics that draws the connection between its axioms and the physical world, but the physicists. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: David Kastrup on 25 Jul 2005 04:45 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > David Kastrup wrote: > >> Uh, no. I don't want to recreate the whole building of mathematics >> whenever physics discovers something new. The whole point of >> mathematics is that it is not subject to external influences. New >> math models don't invalidate previous ones, new physics models _do_ >> invalidate previous ones. > > It's not that bad. The Theory of Relativity didn't invalidate the > use of Newtonian Mechanics with designing bridges and cars. There is no room for a bit of error in the tenth place if you are factoring primes. > And what would be wrong with a mathematics which evolves in time, > like everything else in life ? That's like saying what is wrong with a building that evolved in time, like everything else in life? Let's use sand instead of bricks. You don't get anywhere serious if your structures keep crumbling down. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: David Kastrup on 25 Jul 2005 04:46 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > david petry wrote: > >> Cantor's set theory has been intertwined with modern analysis to >> such an extent that it sometimes appears that analysis would fall >> apart without the glue of set theory to hold it together. >> 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. > > Affirmative. After I discovered (in 1973) that Set Theory was just a > burden on analysis (Lie groups), I've managed to do analysis without > set theory for another 30 years. That's like a person that gets along without burning any coal since he does everything with electricity. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Han de Bruijn on 25 Jul 2005 05:04 David Kastrup wrote: > If you want to work with things like Dirac delta, the way to do this > is not to use functions (which can't represent them), but > distributions. Those are well-defined, too. Nah, nah. Despite of the fact that mainstream mathematics knows so well what at Dirac delta is, they were highly _surprised_ when I pointed out my 'Crossing Number = Winding Number' theorem to them. As Robin Chapman says in http://groups.google.nl/groups?q=sci.math+%22new+alas%22&hl=nl : > Your awkward integral on the left may be new alas Where I used Green's Theorem and a Dirac delta to arrive at the result: http://huizen.dto.tudelft.nl/deBruijn/grondig/crossing.htm Clearly, it has never crossed their minds that such a nice relationship between topology and calculus could possibly esists. Han de Bruijn
From: Han de Bruijn on 25 Jul 2005 05:28
Jiri Lebl wrote: > 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. So maybe you should broaden your view on the "laboratory", David. I tend to agree with Jiri on this issue a great deal. As it is said in my .sig: "A little bit of Physics would be NO Idleness in Mathematics": there's a little bit of Physics in there, not a computer. A computer is a physical device, undeniably, but it's not the only thing available. Maybe Jiri's formulation is the best: whatever the system is that you are observing. Under the condition that it's a system in the outside world, not within mathematics itself, of course. Han de Bruijn |