An Ultrafinite Set Theory
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From: Gc on 4 Mar 2010 01:53 On 4 maalis, 08:33, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >As > noted, as regards the philosophy of science and philosophy of > mathematics Feferman's claim has direct relevance to the Quinean idea > that the justification for our mathematical theories lies in their use > in our best scientific theories. OK. PS. In my view the justification lies only in that most of these strong theories make sense to me. In harmonic analysis people investigate(d) a lot sets of divergence of Fourier series, even if you sum the series in some other way they knew that the series do converge a.e. I read the essay by the great Antoni Zygmund who said the reason being just "I like it". How to spread resources in mathematics is another thing. I hope that also people who investigate large cardinals can make a career.
From: Aatu Koskensilta on 4 Mar 2010 02:11 Gc <gcut667(a)hotmail.com> writes: > PS. In my view the justification lies only in that most of these > strong theories make sense to me. An eminently sensible view! All this justification business, when taken literally, is mostly just hot air. In the end mathematicians, left to their own devices, will simply study stuff they find interesting, using principles congenial to their way of thinking, whatever is pleasing to their intellect. In so far as we're not dealing with real justification at all -- that is, something that is actually intended to convince real people of the correctness or worthwhileness of something -- I think it's better to phrase our observations (and logical results) about what rests on what, what concepts can be reduced to what concepts, what is in a strictly logical sense needed for this and that, what appears evident on this or that conception, what "can be argued", and so on, as just observations. Otherwise our waffling is, or at least appears to be, just feeble rationalization. There's nothing wrong in justifying one's interest in higher set theory, say, by a cheery: "I just like it!". > I hope that also people who investigate large cardinals can make a > career. Well, they evidently can. In order to do research level work in set theory one must essentially be pretty sharp logically. It is no accident that most people with a theorem to their name in set theory are also known for all sorts of results outside abstract set theory. (And, perhaps not surprisingly, a sizable portion of set theorists originally came from other areas of mathematics, Solovay and Cohen being obvious examples.) I think logic is not a bad area to choose for a young aspiring mathematician -- logic being a rather marginal field of mathematics there's lots of stuff one can make a substantial contribution to even if one is not at the avant-garde mathematical frontier; absorbing mind-numbing amount of mathematical theory and trivia is not a necessary prerequisite for fruitful work; and since the community is small as mathematical communities go, the atmosphere is perhaps more friendly, and even though logicians are as competitive as they come, there's usually enough interesting stuff for everyone. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 4 Mar 2010 04:37 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > You'll have to check, of course, that you *can* write down the axioms > for well-ordering. I don't see any issues, but I haven't thought it > through. There's nothing in the least problematic in formalizing the assertion that < is a well-ordering. I had either forgotten or not read your post before presenting an ultrafinite theory, as a boon for TransferPrinciple, a theory that is, I now find, essentially just what you outlined. I imagine TransferPrinciple is salivating vigorously in anticipation of all the exciting non-standard mathematics bound to follow now we've got the details sorted out. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Jesse F. Hughes on 4 Mar 2010 07:55 Transfer Principle <lwalke3(a)lausd.net> writes: > On Mar 3, 8:21 am, MoeBlee <jazzm...(a)hotmail.com> wrote: >> On Mar 2, 9:46 pm, Transfer Principle <lwal...(a)lausd.net> wrote: >> First, Transfer Prinicple, you blowhard, you lied again about me in >> your previous posts. And, so far, you've not responded to my latest >> requests that you stop doing that. > > The reason that I don't promise to stop "lying" is that after I do, > I'd > inevitably see a MoeBlee post that I'll consider to be representative > of what I call standard theorist/anti-"crank" behavior, and then I'd > use that post to make a generalization about standard theorists or > anti-"cranks," and that generalization would be considered a lie. So > I'd be making a promise that I know I wouldn't be able to keep. > > (Of course, the easiest way to stop posting "lies" about MoeBlee > on Usenet is just to stop posting on Usenet, period. But calling > me a "liar" isn't going to make me disappear that easily, any more > than calling someone a "crank" makes "cranks" stop posting.) An alternative is to stop generalizing. When Moe says something, he speaks for Moe. >> You have my contempt for that. > > We're opponents, and so I expect nothing less. Because you have odd ideas about intellectual disagreements. Newberry and I disagree on whether his intuitions are reasonable. I've never seen any real sense to his ideas, but I have no contempt for him. (This is not true for certain others, Andrew Usher, for example, who do behave in a truly contemptible manner.) -- Jesse F. Hughes "I have put all the information that you need at [a Yahoo! group] where you'll notice a significantly better signal to noise ratio, as I'm just about the only person posting." -- James S. Harris on noise
From: Jesse F. Hughes on 4 Mar 2010 08:00
Transfer Principle <lwalke3(a)lausd.net> writes: > Here's what I'm getting at -- if M is a subset of Z, then the elements > of M ought to be called "integers" -- even if M is a finite subset of > Z such as {neZ|-65536<=n<=65535} or {neZ|-2^31<=n<=2^31-1}. Well, duh. It does not follow that M is *the* integers. In any case, this is a dull and odd battle you're waging. Others suggested that Russell change his terminology in order to avoid confusion. Your immediate inclination is defend his right of confusion. -- Jesse F. Hughes "I just define real numbers to be all those on the number line, as they were defined before Dedekind and Cauchy." -- Ross Finlayson's simple definition. |