Cantor Confusion
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From: Lester Zick on 19 Nov 2006 17:23 On Sat, 18 Nov 2006 13:37:07 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> What is it modern zen mathematikers do instead of thinking about the >> truth of what they say? Sit around all day massaging their middle >> legs? I mean really what is it they expect they get paid for? > >Proving theorems, of course. Fess up: you really knew that, didn't you? I already knew they don't prove the truth of their theorems. ~v~~
From: Lester Zick on 19 Nov 2006 17:23 On Sat, 18 Nov 2006 12:50:30 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Thu, 16 Nov 2006 01:35:12 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >Virgil wrote: >> >> In article <1163426009.651510.237050(a)h48g2000cwc.googlegroups.com>, >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > It is difficult to answer this question, because the expression "set" >> >> > is occupied in modern mathematics by collections of elements which are >> >> > actually there (you don't know what that means, imagine just a set as >> >> > you know it). Such infinite sets do not exist. >> >> >> >> While infinite collections in any physical sense are not possible, why >> >> are imaginary infinities, such as sets of numbers must be, unimaginable? >> >> Why are square circles unimaginable? > >Depends on what you mean by "unimaginable". Not my term, slick. Ask whatsitsface. >> >For that matter, we can always switch from Platonism to formalism and >> >declare the question of whether sets really exist to be a philosophical >> >question. >> >> So is the switch from platonism to formalism a philosophical question? > >Yes. Platonism and formalism are philosophies of mathematics. Untrue. They're philosophies of daydreaming. > Regardless >of which you prefer (or if you prefer something else), it doesn't change >which theorems are provable in which axiom systems. Fortunately it also doesn't change which theorems are demonstrably true and which are not demonstrably true except by assumptions of truth in modern mathematics. ~v~~
From: Lester Zick on 19 Nov 2006 17:23 On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> I'm saying that you don't understand what a mathematical definition is >> but nonetheless want to pretend you do. If a mathematical definition >> were "just" an abbreviation as you claim you wouldn't have any way to >> tell one mathematical definition from another. > >Why not? Suppose I make the following definitions. > > Let N denote the set of natural numbers. > Let R denote the set of real numbers. > >Then I can tell N and R are different because their defintions are >different. If I write > > 0.5 is not in N, > >then this means the same as > > 0.5 is not in the set of natural numbers. > >And, it means something different from > > 0.5 is not in R. But the problem, sport, is you claim mathematical definitions are "only abbreviations". Granted I suppose even mathematikers can tell the difference between N and R in typographical terms. I mean they may be too lazy and stupid to demonstrate the truth of what they say but even they can see differences in typography. But in terms of abbreviations alone we can't really say what the difference is between N and R because you insist their definitions are "only abbreviations" and not their conceptual content. ~v~~
From: Lester Zick on 19 Nov 2006 17:43 On Sun, 19 Nov 2006 11:16:14 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >imaginatorium(a)despammed.com wrote: >> >> David Marcus wrote: >> > imaginatorium(a)despammed.com wrote: >> > > David Marcus wrote: >> > > > imaginatorium(a)despammed.com wrote: >> > > > > Dik T. Winter wrote: >> > > > > > In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >> > > > > > > Dik T. Winter schrieb: >> > > > > > ... >> > > > > > > > In >> > > > > > > > principle no axiom is necessary. But you need a few to have some start >> > > > > > > > to work with. >> > > > > > > >> > > > > > > That's the question. By means of axioms you can produce conditional >> > > > > > > truth at most. I am interested in absolute truth. Axioms will not help >> > > > > > > us to find it. I don't think we need any axioms. >> > > > > > >> > > > > > If you want to find absolute truth you should not look at mathematics. >> > > > > >> > > > > Really? There are two groups of order 4; could any truth be more >> > > > > absolute than that? >> > > > >> > > > I think it depends on what the words mean. If the axioms are correct in >> > > > your model, then the theorems are correct in your model. >> > > >> > > Well, model-schmodel, really. (This stuff is a bit beyond me, >> > > actually...) >> > > >> > > It's not entirely clear what the notion of "absolute truth" refers to. >> > > Suppose you think it is a matter of absolute truth that all men are >> > > created equal. Then you go to Venus and discover that in their language >> > > the word 'All' means flying, 'men' means pigs, 'are' means eat, >> > > 'created' means chocolate, and 'equal' means icecream.* Moreover the >> > > atmosphere of Venus turns out to be full of flying pigs, but is of such >> > > chemical composition that icecream of any flavour self-combusts >> > > explosively. Well, has absolute truth varied? I think the reasonable >> > > answer is 'No', because a truth is _about_ something, not merely a >> > > string of formal symbols. >> > > >> > > : * Language doesn't work like this - I know, but I haven't time to >> > > assemble grammars and whatnot >> > > : just to make the same point. Anyway, see the Hilary Putnam stuff >> > > about horses and schmorses >> > > : (which I have only read secondhand in Dennett). >> > > >> > > > Why did you pick the statement you did, rather than something like 2 + 2 >> > > > = 4? >> > > >> > > Because as far as I know there is no (normal, sane) interpretation of >> > > the _words_ of my statement about groups of order 4 other than the >> > > standard one. Whereas, for example, in other contexts 2 + 2 = 1, so >> > > while the truth to which "2+2 = 4" refers is absolute, it takes longer >> > > to write, because you have to spell out the full context, and in >> > > present crank company even saying "integers" may take 2-3 lines. >> > > >> > > You say this depends on my axioms and my model; but are there such that >> > > make my claim about groups of order 4 untrue? >> > >> > I don't know. >> >> You claim to have a PhD in mathematics, and you "don't know"? What a >> feeble answer. So disappointed was I when I saw it, that I was tempted >> to say I begin to understand where Lester gets his "ideas" from, though >> mercifully I overcame that temptation, seeing it would probably start >> him off again. > >Are you making a joke? Somehow, David, I rather suspect not. Occupational hazard I expect when one begins to wonder what truth means and gets answers from modern mathematikers that when axioms are correct in your model theorems are correct in your model. So very enlightening since we don't quite know when that is and modern mathematikers can't really enlighten us. >> I think I see that there could be a set of rather weak axioms that >> formed something called groupette theorino, which were simply powerless >> to prove the existence of two groups of order 4, but my suggestion is >> that no-one would accept such a miniature as being grownup group >> theory. > >Obviously, if we throw out all the axioms, we can't do anything. Then obviously you can't do anything. > But, if >that was your question, you could answer it yourself. I don't know if >there is a more interesting answer. I was also thinking that it depends >on what you mean by "absolutely true". Apparently you're too lazy or stupid to mean anything by it. > This seems to relate to your >statement that "no one would accept" such a theory. It is hard to say >what people will accept. Since they accept modern math I certainly agree. ~v~~
From: Lester Zick on 19 Nov 2006 17:47
On 18 Nov 2006 21:33:39 -0800, imaginatorium(a)despammed.com wrote: > >David Marcus wrote: >> imaginatorium(a)despammed.com wrote: >> > >> > Dik T. Winter wrote: >> > > In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >> > > > Dik T. Winter schrieb: >> > > ... >> > > > > In >> > > > > principle no axiom is necessary. But you need a few to have some start >> > > > > to work with. >> > > > >> > > > That's the question. By means of axioms you can produce conditional >> > > > truth at most. I am interested in absolute truth. Axioms will not help >> > > > us to find it. I don't think we need any axioms. >> > > >> > > If you want to find absolute truth you should not look at mathematics. >> > >> > Really? There are two groups of order 4; could any truth be more >> > absolute than that? >> >> I think it depends on what the words mean. If the axioms are correct in >> your model, then the theorems are correct in your model. > >Well, model-schmodel, really. (This stuff is a bit beyond me, >actually...) > >It's not entirely clear what the notion of "absolute truth" refers to. So why not start off with a simpler question, Brian? What does it mean to be absolutely false? >Suppose you think it is a matter of absolute truth that all men are >created equal. Then you go to Venus and discover that in their language >the word 'All' means flying, 'men' means pigs, 'are' means eat, >'created' means chocolate, and 'equal' means icecream.* Moreover the >atmosphere of Venus turns out to be full of flying pigs, but is of such >chemical composition that icecream of any flavour self-combusts >explosively. Well, has absolute truth varied? I think the reasonable >answer is 'No', because a truth is _about_ something, not merely a >string of formal symbols. The difficulty with categorical assumptions of truth of this kind, Brian, is that they're only assumptions. You're still trying to get at the idea of truth by assumption. You need to be able to demonstrate "truth" and not just assume it. >: * Language doesn't work like this - I know, but I haven't time to >assemble grammars and whatnot >: just to make the same point. Anyway, see the Hilary Putnam stuff >about horses and schmorses >: (which I have only read secondhand in Dennett). > >> Why did you pick the statement you did, rather than something like 2 + 2 >> = 4? > >Because as far as I know there is no (normal, sane) interpretation of >the _words_ of my statement about groups of order 4 other than the >standard one. Whereas, for example, in other contexts 2 + 2 = 1, so >while the truth to which "2+2 = 4" refers is absolute, it takes longer >to write, because you have to spell out the full context, and in >present crank company even saying "integers" may take 2-3 lines. > >You say this depends on my axioms and my model; but are there such that >make my claim about groups of order 4 untrue? > >Brian Chandler >http://imaginatorium.org ~v~~ |