Cantor's Diagonal?
in [Math]
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From: Aatu Koskensilta on 10 Feb 2010 09:32 Gc <gcut667(a)hotmail.com> writes: > On 10 helmi, 16:04, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >> Setting aside the rather fantastic notion that the axiom of choice, >> say, could have any physical consequences, we may note that as >> regards logical strength the mathematics used in physics is rather >> wimpy -- PA, and even PRA, suffices just fine, with some tedious >> coding and massaging of the mathematical machinery. Of course, we >> can't draw from this purely logical observation any conclusion about >> whether it would be humanly possible to do mathematical physics using >> only weak mathematical principles. > > Are you sure about this? Well, yes. I've been told so by people who should know! Feferman -- who often likes to point out that in physics we don't really need anything beyond predicative mathematics -- is again whom you should turn to if you're interested in the details. > I think you need at least borel measure to define a Hilbert space > which is used in Quantum mechanics, but in the other hand Hilbert > space probably contains functions which have no physical meaning, not > differentiable anywhere etc. Anyway you probably need some axiom of > choice like stuff in functional analysis, operator algebras etc. This is where the tedious coding and massaging, the chiseling off of spurious generality, comes in. There's loads of stuff on this in the literature on reverse mathematics. As to the axiom of choice, its use in physical predictions and such like is always eliminable. > I agree (if that is what you are saying) that doing things rigorously > as in mathematical physics you probably will need more stronger > mathematics. Implicitly ordinary physicist also us those notions. Physicist as a rule couldn't care less about the logical strength of their mathematical machinations, and will in blissful logical ignorance apply any piece of mathematics that strikes their fancy or they find useful, even if, as some logician may find out, what actual use they make of the mathematical machinery only requires (in the logical sense) vastly weaker mathematical principles. Similarly, although number theorists do have a healthy interest in elementary proofs they won't shy from analytic methods, stuff from algebraic geometry, and so on, if it helps them in their work -- this even though probably[1] all known number theoretic theorems can be proved in very weak fragments of arithmetic, the proofs of course being possibly utterly beyond human comprehension. Footnotes: [1] "Probably" in the sense that those who make it their business to know about these things would be rather surprised if this wasn't the case. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Gc on 10 Feb 2010 10:06 On 10 helmi, 16:32, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Gc <gcut...(a)hotmail.com> writes: > > On 10 helmi, 16:04, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > >> Setting aside the rather fantastic notion that the axiom of choice, > >> say, could have any physical consequences, we may note that as > >> regards logical strength the mathematics used in physics is rather > >> wimpy -- PA, and even PRA, suffices just fine, with some tedious > >> coding and massaging of the mathematical machinery. Of course, we > >> can't draw from this purely logical observation any conclusion about > >> whether it would be humanly possible to do mathematical physics using > >> only weak mathematical principles. > > > Are you sure about this? > > Well, yes. I've been told so by people who should know! Feferman -- who > often likes to point out that in physics we don't really need anything > beyond predicative mathematics -- is again whom you should turn to if > you're interested in the details. Well, I`m not that interested, but isn`t predicativism a lot less restrictive than PA? Anyway I would like more to read some expert views which think of nonpredicativist mathematics more highly than Feferman. > > I think you need at least borel measure to define a Hilbert space > > which is used in Quantum mechanics, but in the other hand Hilbert > > space probably contains functions which have no physical meaning, not > > differentiable anywhere etc. Anyway you probably need some axiom of > > choice like stuff in functional analysis, operator algebras etc. > > This is where the tedious coding and massaging, the chiseling off of > spurious generality, comes in. There's loads of stuff on this in the > literature on reverse mathematics. As to the axiom of choice, its use in > physical predictions and such like is always eliminable. You may need to use axiom of Choice in proofs, when you are doing mathematical physics rigorously. Actually applicable mathematics use surprisingly advanced mathematics, and the basics they teach does not of course differ from pure mathematics. If someone believes that everything could be done some other way, fine, but if that were nontrivial we would have more of those using only weaker principles in their work, because there are always people who don`t like current strong principles. > > > I agree (if that is what you are saying) that doing things rigorously > > as in mathematical physics you probably will need more stronger > > mathematics. Implicitly ordinary physicist also us those notions. > > Physicist as a rule couldn't care less about the logical strength of > their mathematical machinations, and will in blissful logical ignorance > apply any piece of mathematics that strikes their fancy or they find > useful, even if, as some logician may find out, what actual use they > make of the mathematical machinery only requires (in the logical sense) > vastly weaker mathematical principles. In the end it`s is hard to say because the mathematics they use lack rigour. I have studied physics and very often infinitesimals are used ,but not in any kind of rigorous way (like in nonstandard analysis).. >Similarly, although number > theorists do have a healthy interest in elementary proofs they won't shy > from analytic methods, stuff from algebraic geometry, and so on, if it > helps them in their work -- this even though probably[1] all known > number theoretic theorems can be proved in very weak fragments of > arithmetic, Well ,this is just a guess. Mathematics of physics uses analysis, sets of real numbers, functional analysis, algebraic geometry, fractal geometry etc. IMHO It is a real possibility that you need something more than PA to actually build their mathematics from the ground up. Anyway it is clear that restrictions would cripple the subject. >the proofs of course being possibly utterly beyond human > comprehension. > > Footnotes: > [1] "Probably" in the sense that those who make it their business to > know about these things would be rather surprised if this wasn't the case.. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 10 Feb 2010 11:20 On Feb 9, 10:52 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > standard theorist MoeBlee I'm a standard theorist now? This should give my parents such naches! > Typical of the standard theorists, MoeBlee desires a theory > in which calculus can be axiomatized. It matters little to > such standard theorists what "extra" can be axiomatized in > theory I NEVER said that it matters little what else is proven. In fact, I've posted that being ontologically (if you will) conservative may be a desideratum. In fact, IF you ASKED me about this instead of PUTTING WORDS IN MY MOUTH, you'd find that I even have some (I stress 'some') sympathy for "V=L" as an axiom. > The standard theorist MoeBlee > says to include calculus in the foundational theory (no > matter what else gets included). I NEVER said the parenthetical portion or implied it. You're plain lying about me again. Stop it. Also, I've allowed that the calculus doesn't necessarily have to be classical but could be constructivist instead. MoeBlee
From: William Hughes on 10 Feb 2010 12:38 On Feb 9, 6:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Feb 9, 8:14 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > On Feb 7, 10:55 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > Therefore, _ultra_finitists such as WM and > > > Yessenin-Volpin will now be considered cranks (and that's > > > _without_ the scare quotes, since the label is deserved). > > That you consider Alexander Yessenin-Volpin a crank does not say much > > for your method of classification. > > I believe that Yessenin-Volpin is a "crank" if and only if WM > is a "crank." (For this post, I return to scare quotes until > the "crank"-hood of WM and Y-V is settled.) I do not. Indeed my view of WM as a crank has very little to do with his views on ultrafinitism. > > Recall that Yessenin-Volpin is the mathematician famous for, > upon being asked "Is n a natural number?" (for the standard > or Peano natural n), waiting until a time proportional to the > magnitude of n has passed before answering "yes." In other > words, he waits until time nt has passed (for some fixed > constant time t) before answering the question. > > There are many similarities between Yessenin-Volpin and WM > that lead me to believe that each deserves to be a "crank" if > the other so deserves. These similarities include: > > 1. Both are ultrafinitists. Except for WM who actively denies the existence of a largest integer. > 2. Neither actually has a fixed upper bound on the magnitude > of a permissible natural number. Thus Y-V can't answer the > question "What is the largest number?" He may, however, be able to answer the question "Is there a largest number" The answers "yes" (but it is not possible to know what it is", "no" (I have not constructed such a beast) and "this question is meaningless" would all be consistent with what I do know. > WM has also stated > that although some Peano natural numbers "don't exist," there > is no largest permissible natural number. And I consider this inconsistent with his arguments that there are a limited number of natural numbers. (Though I think it is possible to make this consistent by saying that the question "Are there a limited number of natural numbers" has meaning and "Is there a largest natural number" does not.) However, in of itself, I do not think that espousing a view I find to be inconsistent makes someone a crank. Y-V may or may not be a crank. I do not know (my limited exposure would lead me to think not). In my opinion WM is a crank because of the way he argues. E.g. - he mostly refuses to give definitions, and many of the definitions he does use (generally implicitly) are idiosyncratic in the extreme. - he changes the subject frequently. If you challenge a "proof" that X is true, you will not get a reply to your challenge, but another "proof" of X. - many discussions with him follow the pattern WM: A is true and A implies B. O: A is true but A does not imply B. WM: Clearly you do not understand. Here is another proof that A is true. [Note that your categorization of crank only applies to construction of mathematics. E.g. your stuff does not apply to JSH.] - William Hughes
From: Tonico on 10 Feb 2010 14:08
On Feb 10, 12:01 am, Transfer Principle <lwal...(a)lausd.net> wrote: > On Feb 9, 8:14 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > On Feb 7, 10:55 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > Therefore, _ultra_finitists such as WM and > > > Yessenin-Volpin will now be considered cranks (and that's > > > _without_ the scare quotes, since the label is deserved). > > That you consider Alexander Yessenin-Volpin a crank does not say much > > for your method of classification. > > I believe that Yessenin-Volpin is a "crank" if and only if WM > is a "crank." (For this post, I return to scare quotes until > the "crank"-hood of WM and Y-V is settled.) > ****Sigh**** Of all the nonsenses you've written in this forum, either "deffending" (against what or whom?) people like Tommy or speaking in the name of others about stuff you don't know, this last one must surely be one of the highest peaks of them all: you actually dare to compare in any sense an actual mathematician, and a great logician, thinker, poet and even philosopher, as Essenin-Volpin, with a megacrank like WM...do you have some nerve or what! You better check yourself...with a shrink, preferably. Tonio Ps. Unbelievable...! As you've |