Cantor Confusion
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From: Han.deBruijn on 10 Dec 2006 09:22 David Marcus schreef: > Duh! Anyone could have told you that functions are a plausible > foundation to build mathematics on. However, I'm not sure if this has > been worked out in detail. In truth, most working mathematicians don't > really care what foundation is used. If you do not really care, why are you so offensive then? Han de Bruijn
From: Han.deBruijn on 10 Dec 2006 09:42 stephen(a)nomail.com schreef: > Han.deBruijn(a)dto.tudelft.nl wrote: > > David Marcus schreef: > > >> Han.deBruijn(a)DTO.TUDelft.NL wrote: > > >> > To those who are unable to see it, there is no evidence that there is > >> > any sort of mathematics not describable by set theory. But this is a > >> > vicious circle. Mainstream mathematics simply DOES NOT ALLOW any sort > >> > of mathematics that violates set theory. > >> > >> Quite a silly thing to say. Do you have any evidence for such an absurd > >> statement? [ ... snipped "counter evidence" with Category Theory ... ] > > > Ah, now don't act as if you didn't have those heated debates with some > > manifest opponents of set theory, i.e. Wolfgang Mueckenheim. I'm not > > talking about any possibilities to replace set theory by look-alikes. > > > I'm talking about rejecting any monolithic foundation for mathematics, > > any "foundation" that narrows down my freedom of thinking. I hate any > > form of NewSpeak, whether it is called Set Theory, Category Theory or > > Object Oriented Programming. I've seen too many of these. > > > Han de Bruijn > > You are the one who is trying to restrict peoples' freedoms. You > cannot name a single piece of mathematics that is "forbidden" > by set theory. Oh yes, I can: Let P(a) be the probability that an arbitrary natural is divisible by a fixed natural a. Then P(a) = 1/a . Forbidden by set theory. http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962 > You on the other hand want to abolish any mathematics > that involves infinity, which is a severe restriction on others > freedom of thinking. In short, you are a hypocrite. No. I am a truth seeker, against all odds. > I would imagine your understanding of Object Oriented Programming > is as woefully garbled as you understanding of set theory. I've done quite some of it for a living. And I'm still alive and well. Han de Bruijn
From: stephen on 10 Dec 2006 09:44 Han.deBruijn(a)dto.tudelft.nl wrote: > stephen(a)nomail.com schreef: > [ ... Thank you, Stephen, for this quite detailed explanation ... ] > But then: >> This whole calculation, including time, can be modelled >> in set theory. > It can't. Don't we have a debate with Wolfgang Mueckenhein about sets > that change in time? With a negative outcome? Yes it can. There is no reason a set itself has to change in order to model change. Just as there is no reason any individual natural number has to change in order to model change. Or are you in the camp that believes that 0 is going to turn into 1 some day? > Let us say that set theory is half the truth. Within set theory, any > function is a special relation between commodities and products, i.e. > domain and range. But the production process itself involves _labour_, > hence time, and this aspect is not covered by the static set theoretic > framework, where functions are reduced to "mappings" between sets. > Han de Bruijn You can include it. You are just talking about a cost being associated with a function. That is trivial to do. Can you give an example of a production process that you can describe mathematically that cannot be described using set theory? If your description just consists of functions, then how does it model time? Stephen
From: Tony Orlow on 10 Dec 2006 09:57 Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > >> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>> Ah, now don't act as if you didn't have those heated debates with some >>> manifest opponents of set theory, i.e. Wolfgang Mueckenheim. >> If you've been reading the threads, then you should know that WM has so >> far failed to present any mathematics at all. All he does is present >> incorrect arguments that he insists follow from the standard axioms and >> then proclaim: "Behold, standard mathematics is inconsistent." Big deal. > > Nonsense. The arguments presented by WM are quite reasonable. > I agree. He perceives the same problem with the von Neumann ordinals as I do, he just has a different answer. His objection is valid, however. >> Anyone can prove 2 = 1, if they bend a rule here and there. >> >>> I'm not >>> talking about any possibilities to replace set theory by look-alikes. >>> >>> I'm talking about rejecting any monolithic foundation for mathematics, >>> any "foundation" that narrows down my freedom of thinking. I hate any >>> form of NewSpeak, whether it is called Set Theory, Category Theory or >>> Object Oriented Programming. I've seen too many of these. >> Fine. Give some evidence that it "narrows your thinking". I.e., present >> some mathematics that can't be done using ZFC as a foundation. > > �'m pretty sure you can twist and bend any thought in such a way that > it fits into your set theoretical paradigm. And as soon it doesn't fit, > you'd simply say that it is not mathematics. Wolfgang Mueckenheim has > come up with several of such examples. You'd not be better off with me. > See e.g. the "Probability XOR Calculus" thread, which was initiated by > this author: > > http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962 > > Han de Bruijn > Yes, that was a very good thread, pointing out the need for infinitesimal probabilities within infinite sets of alternatives, as well as a number of other anomalies that came up. Is that where we discussed the "staircase"? Tony
From: stephen on 10 Dec 2006 09:55
Han.deBruijn(a)dto.tudelft.nl wrote: > stephen(a)nomail.com schreef: >> Han.deBruijn(a)dto.tudelft.nl wrote: >> > David Marcus schreef: >> >> >> Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >> >> > To those who are unable to see it, there is no evidence that there is >> >> > any sort of mathematics not describable by set theory. But this is a >> >> > vicious circle. Mainstream mathematics simply DOES NOT ALLOW any sort >> >> > of mathematics that violates set theory. >> >> >> >> Quite a silly thing to say. Do you have any evidence for such an absurd >> >> statement? [ ... snipped "counter evidence" with Category Theory ... ] >> >> > Ah, now don't act as if you didn't have those heated debates with some >> > manifest opponents of set theory, i.e. Wolfgang Mueckenheim. I'm not >> > talking about any possibilities to replace set theory by look-alikes. >> >> > I'm talking about rejecting any monolithic foundation for mathematics, >> > any "foundation" that narrows down my freedom of thinking. I hate any >> > form of NewSpeak, whether it is called Set Theory, Category Theory or >> > Object Oriented Programming. I've seen too many of these. >> >> > Han de Bruijn >> >> You are the one who is trying to restrict peoples' freedoms. You >> cannot name a single piece of mathematics that is "forbidden" >> by set theory. > Oh yes, I can: > Let P(a) be the probability that an arbitrary natural is divisible by > a fixed natural a. Then P(a) = 1/a . Forbidden by set theory. No. It is forbidden by the definitions used in probability theory. Your argument that P(a)=1/a is simply using the density of a set of naturals, which is describably by set theory. Of course this is more hypocrisy from you, because you do not even believe that the set of all natural numbers exists, and do not believe that you can choose a natural at random from the actually infinite set of all natural numbers, yet you get all in a huff when mathematicians say that P(a) does not exist. So your one example of mathematics that you believe is forbidden by set theory is something that you do not even think is mathematics. > http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962 >> You on the other hand want to abolish any mathematics >> that involves infinity, which is a severe restriction on others >> freedom of thinking. In short, you are a hypocrite. > No. I am a truth seeker, against all odds. Do you or do you not wish to abolish any mathematics that involves infinity? If you are perfectly content to let others freely explore whatever they wish, then why are you so aggressive? >> I would imagine your understanding of Object Oriented Programming >> is as woefully garbled as you understanding of set theory. > I've done quite some of it for a living. And I'm still alive and well. > Han de Bruijn There are a lot of people out there who think they are doing object oriented programing who really do not have a clue. I suppose you think 'while' loops restrict your freedom. Stephen |