Give me ONE digit position of any real that isn't computable up to that digit position!
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From: Transfer Principle on 15 Jun 2010 17:00 On Jun 12, 8:24 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > The response I'd like to see is one which defends classical analysis > > against these smooth infinitesimals > This idea, that classical analysis needs defending against smooth > infinitesimals, is bizarre. But there has to be a reason why most mathematicians use classical analysis and not smooth infinitesimal analysis. I thought the fact that the latter contradicts the Law of the Excluded Middle was one reason to reject it. If not, then I'd like to see some of the real reasons that the classical analyis is more prevalent -- and once again, without the use of five-letter insults.
From: George Greene on 15 Jun 2010 17:52 On Jun 15, 5:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > But there has to be a reason why most mathematicians use > classical analysis and not smooth infinitesimal analysis. No, really, there doesn't. At least not beyond "It's simpler and it was discovered first". If you want more support for THAT reason, read "The Structure of Scientific Revolutions" by Thomas Kuhn. This makes almost as little sense as asking why people use the standard (rather than a non-standard) model of PA; why are they being so unfair to supernatural numbers?
From: George Greene on 15 Jun 2010 18:08 On Jun 15, 5:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > But there has to be a reason why most mathematicians use > classical analysis and not smooth infinitesimal analysis. > > I thought the fact that the latter contradicts the Law of > the Excluded Middle was one reason to reject it. You can't believe everything you read in wikipedia. It is not possible for an axiom system to accept or reject the law of the excluded middle. Changing something like that means changing THE UNDERLYING LOGIC from standard classical first-order logic TO SOMETHING ELSE. It is THAT change AND NOT a change in "analysis" that is relevant to your point here. Traditionally, first-order model theory is done in ZFC. In what you are talking about, the model theory has to be category theoretic instead. Category theory and doing logic THAT way really is a separate can of worms. There will be threads on topoi before this is over, if you don't stop now. The wikipedia article begins with such tomfoolery as: > In particular, for all infinitesimals ε, neither ε = 0 nor NOT (ε = 0) is provable Which one would have to rebut with "So What?? In standard classical first-order ZFC, neitehr G nor ~G is provable, and THAT doesn't require suspending or losing excluded middle!" The problem, rather, is that for certain kinds of infinitesimals, In particular, for all infinitesimals ε, neither ε = 0 nor NOT (ε = 0) is TRUE! THAT is a problem for the MODEL theory in question. In standard classical first-order model theory, the model has to be "semantically" complete in the sense that EVERY MODEL (even if not every recursive axiomatization of the model) actually PROVIDES A TRUTH VALUE for the application of EVERY predicate to EVERY element (or appropriate n-tuple of elements) of the domain. A "partial" or "partially defined" model isn't merely disallowed: it's disallowed BY DEFINITION; it's a CONTRADICTION IN TERMS. The wikipedia article gets better as it goes along, changing from "provable" to > there are plenty of x, namely the infinitesimals, such that > neither x = 0 nor x â 0 hold That's right, HOLD. That's almost as good as "is True". But only almost. In any case, the point, as the wikipedia article belatedly concedes, is that YOU HAVE TO USE A DIFFERENT *LOGIC*. This is not merely a different kind of analysis; the article correctly points out that > Smooth infinitesimal analysis is like non-standard analysis in that > (1) it is meant to serve as a foundation for analysis, and > (2) the infinitesimal quantities do not have concrete sizes > (as opposed to the surreals, in which a typical infinitesimal is 1/Ï, > where Ï is the von Neumann ordinal). However, smooth infinitesimal > analysis differs from non-standard analysis in its use of > nonclassical logic, and in lacking the transfer principle. That last line makes the whole discussion ironic. One would expect any contributor using THAT handle to be a supporter of the principle, not an advocate of a logic that doesn't use it. In any case, the question you need to be asking is why are people sticking with classical logic, not why aren't they using smooth infinitesimal analysis. Your question is not actually about analysis at all. And why doesn't the wikipedia article name THE LOGIC that this analysis IS using?? Cart, meet horse.
From: Tim Little on 16 Jun 2010 04:45 On 2010-06-13, Leland McInnes <leland.mcinnes(a)gmail.com> wrote: > You can also try http://arxiv4.library.cornell.edu/abs/0805.3307 if > you prefer a little instant gratification from freely available web > sources. That is really quite interesting material. The idea may be old hat to some readers here, but to me that's the first time I've seen the weaker logic used as an enhancement in allowing stronger sets of axioms without contradiction. In hindsight it is obvious. > Then there's "Synthetic Differential Geometry" by A. Kock (available > for download from http://home.imf.au.dk/kock/sdg99.pdf); I haven't got into this one yet, but looking forward to it. - Tim
From: Jesse F. Hughes on 16 Jun 2010 08:14
Tim Little <tim(a)little-possums.net> writes: > On 2010-06-13, Leland McInnes <leland.mcinnes(a)gmail.com> wrote: >> You can also try http://arxiv4.library.cornell.edu/abs/0805.3307 if >> you prefer a little instant gratification from freely available web >> sources. > > That is really quite interesting material. The idea may be old hat to > some readers here, but to me that's the first time I've seen the > weaker logic used as an enhancement in allowing stronger sets of > axioms without contradiction. In hindsight it is obvious. No! No no no no NO! You're supposed to defend ZFC by rejecting Smooth Infinitesimals because they are different than ZFC. Has Walker taught you *nothing* about what it means to be an adherent? Geez. -- Jesse F. Hughes "I think the problem for some of you is that you think you are very smart. I AM very smart. I am smarter on a scale you cannot really comprehend and there is the problem." -- James S. Harris |