Give me ONE digit position of any real that isn't computable up
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From: Transfer Principle on 16 Jun 2010 20:44 On Jun 16, 7:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Leland McInnes says... > >I expect that relative newness of SIA is a big part. To make robust > >foundations for SIA possible you need to ground things in topos theory > >with its more flexible logics. That meant that SIA wasn't developed as > >a theory until the 1980s. Compare that to classical calculus which has > >more then a centurey of established history. > I think that there is another reason that most mathematicians prefer > standard analysis (or, for that matter, even nonstandard analysis): > because they feel more comfortable using classical logic. Yes. Greene writes something similar. > Anyway, I do think that it is a mistake to view people's > preferences for this framework or that framework in terms > of ideology and personal "belief". Herc refers to ZFC as a "religion." MoeBlee counters by calling Herc "dogmatic." And of course WM refers to classical mathematics as "Matheology." My references to "adherents," "beliefs," etc., reflects Herc's, MoeBlee's, and WM's use of similar words to describe the posters with whom they don't agree.
From: Transfer Principle on 16 Jun 2010 21:02 On Jun 16, 9:49 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 16, 9:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: > >I do think that it is a mistake to view people's > > preferences for this framework or that framework in terms > > of ideology and personal "belief". For practical purposes, > > you don't have to any metaphysical commitment to use a > > logic or a mathematical theory. You just need to understand > > the criteria for rigorous mathematics in whatever field. > Also, I wonder whether the poster Transfer Principle has the > impression that mathematicians in general even think very much about > such foundational matters as alternative logics, alternative theories, > etc. They sure think about it very much when posting in threads started by Herc, TO, WM, and other posters. This is also another way to understand what exactly I mean by an "adherent" of ZFC. An "adherent" of ZFC is someone who uses ZFC when arguing that someone else is wrong. Thus, when Herc attacks Cantor diagonalization and other posters respond by calling Herc "wrong," then giving the steps of the proof in the theory ZFC, I call such posters "adherents" of ZFC. No one is using Smooth Infinitesimal Analysis or any other theory to prove Herc "wrong." ZFC is the theory being used against Herc. > One may argue (I don't opine myself here) that this tradition-bound > acceptance deserves to be shaken up; that mathematics would be better > if we "leveled the playing field" for a variety of alternative > foundations. So that would be a call for changes in the OVERALL > institutions of mathematics. I have nothing against accepting the traditions of mathematics. What I am against is calling someone like Herc, who doesn't accept these traditions (deriding them as a "religion"), a five-letter insult. Those mathematics who don't think about the foundations of mathematics would have no reason to use such foundations against _Herc_.
From: Jesse F. Hughes on 16 Jun 2010 22:20
Transfer Principle <lwalke3(a)lausd.net> writes: > This is also another way to understand what exactly I mean by an > "adherent" of ZFC. An "adherent" of ZFC is someone who uses > ZFC when arguing that someone else is wrong. Thus, when Herc > attacks Cantor diagonalization and other posters respond by > calling Herc "wrong," then giving the steps of the proof in the > theory ZFC, I call such posters "adherents" of ZFC. > > No one is using Smooth Infinitesimal Analysis or any other theory > to prove Herc "wrong." ZFC is the theory being used against Herc. Yes, that's very insightful. Now let's see if we can figure out why people use ZFC when refuting Herc's claims. Oh, yeah! I remember! Herc's claim is that a particular theorem of ZFC is false or that its proof is invalid! Herc didn't say anything about smooth infinitesimals. I wonder if that's why people respond by posting the ZFC proof?[1] Footnotes: [1] Yes, of course, the same statement is a theorem in countless theories and also refuted by countless other theories, but it is the theorem of ZFC which is *obviously* at issue here, not some hypothetical theory that Herc might have in mind, although he forgot to say so. -- "I liked the world a lot better over ten years ago. I believed in a lot more things. Hell, most people believed in a lot more things. Back then the United States was still, well, known as most people used to know the United States." -- James S. Harris in a nostalgic mood |