Cantor Confusion
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From: stephen on 30 Oct 2006 10:30 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> >>>I'm DOING mathematics. Mathematics is NOT independent of Physics. >> >> 5*10^8 m/s + 5*10^8 m/s = 10*10^8 m/s > There are 10 men in a room. Each has a body temperature of 37 Celcius. > This means that the temperature in the room is: 10 x 37 = 370 Celcius. > Satisfied? Or do you rather prefer it in Kelvin? > Han de Bruijn Satisfied? I have no idea what you mean. Either you are agreeing with me or you totally missed my point. Mathematics can easily describe things that are physically impossible or physically inaccurate. If Mathematics were not independent of physics, this would not be the case. If Mathematics were truly dependent on physics, then it would be impossible to mathematically describe something that is physically impossible. It is a good thing that mathematics is capable of describing things other than what is physically possible, because we do not know all that is physically possible. Demanding that we limit our knowledge to what is currently known will guarantee that we never learn anything. Stephen
From: Randy Poe on 30 Oct 2006 10:38 Han.deBruijn(a)DTO.TUDelft.NL wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > stephen(a)nomail.com schrieb: > > > > > Can you describe a continuous version of the problem where each > > > "unit" of water has a well defined exit time? A key part of > > > the original problem is that the time at which each ball is > > > removed is defined and reached. This is crucial to the problem. It is > > > not just a matter of rates. If you added balls 1-10, then 2-20, > > > 3-30, ... but you removed balls 2,4,6,8, ... then the vase is > > > not empty at noon, even though the rates of insertions and removals > > > are the same as in the original problem. So you cannot just > > > say the rate is 10 in and 1 out and base an answer on that. > > > > > The answer for any time t *before noon* is independent of the chosen > > enumeration of the balls. Doesn't that fact make you think a bit > > deeper? > > And add this to the fact that noon and beyond cannot exist > in this problem. I'll ask you the same question I've been asking Tony about this "existence". Did noon exist yesterday? Is there anything that prevents me from defining a set of times t_n, n=1, 2, ... where t_n = 1/n minutes before noon yesterday? As soon as I define those times, does noon yesterday cease to exist? - Randy
From: William Hughes on 30 Oct 2006 10:44 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > If there is a limit to how many numbers we can define there > > > > is also a limit to how large a number we can define. > > > > > > Why. Why does "there is a limited number of numbers" imply > > > that "every number is limited"? > > > > > > > Not by simply reversing the quantifier. > > > > The reason for believing that there are only a finite > > number of integers is that there are only a finite number > > of bits in the lifetime of the universe, and therefore only > > a finite number of different integers will be defined. > > Correct. > > > > But note, to define an integer, the definition must be > > communicated. We have only a finite number of bits > > to use for this communication (this includes comunicating > > the details of any compression scheme). So there are > > only a limited number of integers that it is possible to define. > > Among these is the largest integer that it is possible to > > define. > > Sorry, I can't see that conclusion. You can always define a new and > larger base., a new and more efficient way to use the bits. To communicate your new and larger base you need to use bits. There are only a limited number of bits. Therefore there are only a limited number of new and and larger bases that can be communicated. To communicate your new and more efficient way to use the bits you need bits. There are only a limited number of bits. Therefore there are only a limited number of new and efficient ways to use these bits that can be communicated. (Combining compression schemes does not change things (using one compression scheme to compress the description of another). There are still only a limited number of ways to do this.) > > > > If we claim that the only integers that exist are the ones > > that have been or will be defined, > > then there is a largest possible integer. > > There is, or better there will be, a largest integer ever defined or > communicated. But this integer will only be fixed after all life has > been gone. Therefore it will never be fixed. So I don't think that it > does exist. > Whether it exists or not is beside the point. If we assume that there are only a limited number of bits in the lifefime of the universe, then whatever this number will be, it will be smaller than or equal to the largest number that it is possible to define. - William Hughes
From: Lester Zick on 30 Oct 2006 12:38 On Sun, 29 Oct 2006 20:56:16 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On 29 Oct 2006 13:35:46 -0800, mueckenh(a)rz.fh-augsburg.de wrote: >> >Dik T. Winter schrieb: >> > >> >> "positive numbers" with "numbers larger than 0". Because I understand the >> >> main domain is Anglo-Saxon mathematics. This is in contradiction to what >> >> I did learn at university (0 is both positive and negative). >> > >> >Really? I never heard of that. Is here anybody who learned that too? It >> >would interest me. No polemic intended. >> >> I have also heard that. In the early days of computing it was a >> significant issue as to whether zero was only positive or there was >> also a negative zero. These days zero is only regarded as positive and >> the so called negative zero is considered an overflow situation to the >> best of my knowledge. > >You've jumped from Mathematics to Computer Science. The subject jumped. I'm sure something of epistemological interest lies therein. >IEEE floating point does provide minus zero. It is useful for dealing >with underflow (not overflow). Technically correct. I misspoke. I once coded floating point conversion routines in assembler to no particular point since compilers do it for you. Rather interesting. ~v~~
From: Lester Zick on 30 Oct 2006 12:42
On Sun, 29 Oct 2006 19:45:35 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> Dik T. Winter schrieb: >> >> > Euclides was apparently smarter than Cantor in this. He used the >> > parallel axiom (postulate) for something he could not prove from the >> > other axioms. >> >> But which is obviously possible and correct under special >> circumstances, while Zermelo's AC is obviously false under any >> circumstances. > >What do you mean an axiom is "false"? Are you discussing philosophy or >mathematics? The same way any assumption or definition can be false: self contradiction. ~v~~ |