Cantor Confusion
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From: Han.deBruijn on 10 Dec 2006 10:00 step...(a)nomail.com schreef: > Han.deBruijn(a)dto.tudelft.nl wrote: > > > 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. > > You can include it. You are just talking about a cost being > associated with a function. That is trivial to do. Exactly! In the eye of the capitalist beholder labour is identical with "cost". And that cost is an easy thing to incorporate. But a fact is that labour is not cost and not static and it involves time. And time is not a set. Constructive mathematics is different from mainstream mathematics, precisely for that sole reason: time. Time excludes actual infinities as well, because they cannot be "done". Han de Bruijn
From: Han.deBruijn on 10 Dec 2006 10:08 step...(a)nomail.com schreef: > 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? _What_ infinity. That's the question. Mainstream mathematics has mixed up infinity so much that it's not a sensible notion anymore. Han de Bruijn
From: Tony Orlow on 10 Dec 2006 11:04 stephen(a)nomail.com wrote: > 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 believe Han agreed that, if he saw a more satisfying treatment of infinite sets, he'd be open to it. Of course, he's been rather resistant to my alternatives, but it seems everyone is, for one reason or another. In any case, I think his objections are specific enough to warrant some attention. "Calculus XOR Probability" was about the loss of additive probability measure, when you have an infinite set of equally likely possibilities, as a result of the notion of aleph_0 elements, and its standard inverse, if there is such a thing, of 0% probability each. No sum of 0's can be anything but 0. Is it unreasonable to want to preserve additive measure within probability over an infinite set? I don't think so, and the answer to that issue was obviously to allow some infinitesimal probability for each natural. In that case it can easily follow that the probability the n/3 e N is 1/3. Set theory "disagrees". Interpret that as you wish. >>> 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 > What do "while" loops have to do with object-oriented programming? That doesn't mean everything has to be recursive. It means each method or event is associated with an object. It takes a while to wrap your head the other way after doing iterative programming, handling events and including functions as elements of objects, but it makes a lot of sense in the end.
From: stephen on 10 Dec 2006 12:39 Han.deBruijn(a)dto.tudelft.nl wrote: > step...(a)nomail.com schreef: >> 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? > _What_ infinity. That's the question. Mainstream mathematics has mixed > up infinity so much that it's not a sensible notion anymore. > Han de Bruijn What is mixed up about infinity? Care to cite an example where mainstream mathematics has mixed up infinity? And shouldn't people be free to explore whatever infinities they wish? You were the one complaining that about your freedoms being restricted, yet you seem perfectly content to deny those freedoms to others. Stephen
From: Lester Zick on 10 Dec 2006 12:44
On Sat, 9 Dec 2006 17:30:18 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Tony Orlow wrote: [. . .] >Of course I don't agree. You seem to be saying that infinite sequences >can't be handled in ZFC. Since ZFC has no trouble modeling the natural >numbers and defining functions, it clearly has no trouble acting as a >foundation for all of calculus and analysis. It might be nice however if the Peano axioms could produce straight lines for the natural numbers except by assumption. ~v~~ |