Give me ONE digit position of any real that isn't computable up to that digit position!
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From: Marshall on 12 Jun 2010 14:33 On Jun 12, 11:11 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jun 9, 6:50 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > >> Nam > >> Nguyen > > > is overwhelmingly confused about some of the most basic matters in > > mathematical logic, and worse, (with rare exception) he will not allow > > himself to be corrected, educated, or enlightened by the many quite > > informed posters who try. It is usually a dead end trying to have a > > rational discussion with Nguyen. This is shown over and over again in > > thread after thread. And it's definitely not a matter of "a new > > theory" but rather that he purports to represent a correct > > understanding of standard, ordinary mathematical logic while yet he's > > horribly mixed up about it. > > The voice of a _pathetic_ incarnation of the Inquisition. > I had some technical issues to discuss about a new ZF axiom, > about FOL based definition of AI and Moeblee has nothing to > respond or to post except very pathetic trashing! Are you sure about that? If you aren't sure about whether you are a potato chip or not, you definitely can't be sure of that. Your appraisal of MoeBlee might be true in certain contexts, but it's not *true* true; that's for sure. It's false in some contexts as well. Reality, for one. Marshall
From: Leland McInnes on 12 Jun 2010 16:51 On Jun 10, 4:43 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 10, 11:33 am, Leland McInnes <leland.mcin...(a)gmail.com> wrote: > > > > > > > On Jun 9, 7:50 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > I want to see a peaceful discussion in which the OP > > > discusses why they work in the alternate theory, while > > > the others post why they prefer ZFC to the theory, and > > > so on. > > Just in case I'll get the ball rolling on a discussion of synthetic > > differential geometry/smooth infinitesimal analysis as compared to > > classical calculus. The prime difference here is that SDG supposes a > > non-punctiform continuum -- you can't pick out individual points from > > it -- which squares nicely with the intuitive notion of a continuum as > > an indivisible whole that it ultimately incapable of being described > > in terms of discrete/distinguishable points. This is, of course, > > incompatible with classical calculus and the classical real line which > > is nothing but distinguishable points. Now, to have such a notion of a > > continuum, you need to make some sacrifices, like, for instance, > > forgoing the law of excluded middle. On the upside you can develop a > > purely synthetic geometric calculus in which standard results fall out > > naturally as simple algebra involving infinitesimals. Indeed, you can > > build up a very natural differential geometry in which, for example, > > thinking about a vector field on a manifold as infinitesimal flow > > across the manifold versus an infinitesimal element of a symmetry > > group of the manifold is just a matter of lambda abstraction. > > So there we go; some positive comments about a theory that is not > > formulatable in ZFC (I suppose it might be, but it would not be > > pretty). I doubt you'll see many comments though, because I doubt I've > > said much that is especially controversial. > > For one thing, the vast majority of posters who do > mention infinitesimals are criticized for doing so. I think it depends on how they mention infinitesimals. If they mention rigorously defined infinitesimals (such as in Robinson's non-standard analysis, or the rather different nilpotent infinitesimals of smooth infinitesimal analysis) I doubt you'll see much controversy or complaint. If they mention ill-defined infinitesimals the exact meaning of which changes moment to moment upon the whims of the poster they might get called on that. If they mention poorly defined infinitesimals whose obvious self contradictions are ignored by the poster I expect they may get called on that too. I'm in the first camp, and can readily direct people to appropriate resources giving full formal definitions for smooth infinitesimals. Moreover I'm simply saying that such a thing can be defined and that this results in a rather nice theory. I'm not claiming that everyone is else is wrong about everything. As a result, I don't expect much controversy. > MR is the most well-known infinitesimalist here on > sci.math (though currently, MR is posting against > negative numbers, rather in favor of nonzero > infinitesimals like his "smallest quantity"). AP > is another infinitesimalist. TO mentions such > numbers from time to time. RF has his "iota." I think you'll find these posters fall into the latter cases -- ill defined and changing exact meaning at the whim of the poster, or poorly defined and self contradictory. I think you'll also find that, far more than simply mentioning infinitesimals, it is the attitude of the posters that tends to draw ire and blunt responses. > Would any of these posters be open to the smooth > infinitesimals described by McInnes? I can't be > sure of this. For one thing, these infinitesimals > are said to be nilpotent. It's hard to say whether > RF, TO, MR would accept them. AP is unlikely to > accept any infinitesimals that don't have digits, > unless there's way to define digits for these > smooth infinitesimals. Also, I'm not sure whether > any of them want to give up Excluded Middle. > > Still, McInnes has started the type of discussion > that I'd like to see. The response I'd like to > see is one which defends classical analysis > against these smooth infinitesimals -- and I mean > something more like "Smooth infinitesimals are bad > because they contradict LEM" than "There are no > nonzero infinitesimals, and anyone who thinks so > is a --" (five-letter insult). I think you've already found that such a discussion isn't going to happen, because no-one who understands ZFC well has any need to "attack" smooth infinitesimal analysis since it is a perfectly self consistent (if not compatible with classical analysis) theory that develops a lot of interesting math. At worst I suspect anyone with decent understanding of mathematics will simply not be interested in SIA or SDG. Que sera sera. But no argument really.
From: FredJeffries on 12 Jun 2010 19:04 On Jun 12, 1:51 pm, Leland McInnes <leland.mcin...(a)gmail.com> wrote: > I'm in the first > camp, and can readily direct people to appropriate resources giving > full formal definitions for smooth infinitesimals. Do you know of any beach-reading level references on the subject?
From: Leland McInnes on 12 Jun 2010 22:12 On Jun 12, 7:04 pm, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jun 12, 1:51 pm, Leland McInnes <leland.mcin...(a)gmail.com> wrote: > > > I'm in the first > > camp, and can readily direct people to appropriate resources giving > > full formal definitions for smooth infinitesimals. > > Do you know of any beach-reading level references on the subject? That depends on what you consider beach reading. "A Primer of Infinitesimal Analysis" by J.L. Bell makes for fairly easy reading and covers that basics well. You can also try http://arxiv4.library.cornell.edu/abs/0805.3307 if you prefer a little instant gratification from freely available web sources. Then there's "Synthetic Differential Geometry" by A. Kock (available for download from http://home.imf.au.dk/kock/sdg99.pdf); I would consider the first part of that beach reading (if your willing to stretch yourself a little bit while reading at the beach). The second half, which provides all the full rigorous justification via topos theory is really not beach reading unfortunately.
From: FredJeffries on 15 Jun 2010 10:53
On Jun 10, 1:43 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Still, McInnes has started the type of discussion > that I'd like to see. The response I'd like to > see is one which defends classical analysis > against these smooth infinitesimals -- and I mean > something more like "Smooth infinitesimals are bad > because they contradict LEM" than "There are no > nonzero infinitesimals, and anyone who thinks so > is a --" (five-letter insult). Everything is C-infinity. Where's the singularities? Where's the everywhere continuous nowhere differentiable functions? How can George Lucas make those planets with coastlines and mountains in Star Wars? What about Velcro? |