Cantor's Diagonal?
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From: Aatu Koskensilta on 10 Feb 2010 08:58 Transfer Principle <lwalke3(a)lausd.net> writes: > Consider the thread in the link above. In this thread, Nguyen used > mostly symbolic language, which according to criterion III above, > suggests that he is standard. You do realize that in passages such as this you come off as extremely silly? -- 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 10 Feb 2010 09:04 Transfer Principle <lwalke3(a)lausd.net> writes: > I agree with Webb here -- in the end, it's a wash as to which side > physics really favors, so perhaps it's best just to avoid physics in > any discussion of set theory. The point that I was making is that > members of both sides _do_ make physical-based arguments, not that > such arguments are actually valid. 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. -- 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 09:17 On 10 helmi, 16:04, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > I agree with Webb here -- in the end, it's a wash as to which side > > physics really favors, so perhaps it's best just to avoid physics in > > any discussion of set theory. The point that I was making is that > > members of both sides _do_ make physical-based arguments, not that > > such arguments are actually valid. > > 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? 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. 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.
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 |