On Ultrafinitism
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From: Bill Taylor on 27 Oct 2006 00:37 Is the following a reasonable point of view, do people think? I'm still kind of wondering where Yessenin-Volpin, Edward Nelson, and other ultrafinitists are coming from. They purport to find, or rather take the public stance of finding, that the concept of "all the naturals" is confusing and vague, whereas it is indeed *crystal-clear* to the rest of us. I'm sure it was once crystal clear to them too. It is NOT necessarily crystal clear, initially, to the non-mathematician - sometimes I've had CS or business students (amazingly) wonder "do the numbers go on for ever", though they are always happy with the simple answer "yes". But by and large, it might almost be considered a criterion to be a "natural mathematician", that the idea of N, the naturals, is crystal clear. (As opposed to, say, an intelligent doctor or lawyer who may have doubts about it.) Now, given this, what are we to make of ultrafinitists, who purport to find vagueness or ambiguity in this basic crystalline abstract jewell of ours, but who nevertheless seem to be reputable mathematicians. At least it seems so, judging from the fact that they get quite a bit of air time. My take is this, and I wonder if it is a reasonable view? """ Some time ago, back in the late seventies to early eighties, there was a brief flurry of interest from fringe mathematicians in "fuzzy math". It was never quite clear what this was, but it still has a small amount of library shelf space, though perhaps little or no presence in math departments in academia. It seemed to be (AFAICT), basically, that joke that used to go around about "Generalized Mathematics" - * "In Orthodox math we derive true results by valid means; * in Generalized math both these restrictions are dropped!" Anyway, one can hardly say that Fuzzy math even died - it was practically still-born... math departments gave it very short shrift. """ So finally, my question is this:- is it a fair point of view to regard ultrafinitism as essentially, fuzzy mathematical logic? They insist on keeping a fuzzy view on what is the largest feasible number, and similarly with the largest feasible derivation; indeed feasible anything - the very concept of feasibility seems to be the ultimate in fuzzy concepts. This viewpoint is *not necessarily* a negative one, I must point out. It may be that (unknown to me) there IS a lot of value in fuzzy math, whatever FM may be. This being so, there could easily be value in ultrafinitist math logic, also. So without necessarily making any approbation or disapprobation of either, is it fair to regard ultrafinitism as "fuzzy mathematical logic"? wfct Christian rationality: A god killing himself to save his own creation from his own wrath.
From: MoeBlee on 27 Oct 2006 15:28 Bill Taylor wrote: > They insist on keeping a fuzzy view on what is the largest > feasible number Thank you for your post and all the questions in it. I hope it gets a lot of response. A related question that I have, which you or perphas others can help with, is that of the putative axiomatizations of utltrafinitist mathematics. Do these axiomatizations really hold up as formal (recursively axiomatized) systems? Do they really provide systems such that there is an effective method to check whether a string of formulas is a proof and still have only finitely many natural numbers and still also allow for proofs of theorems that serve to provide the basic mathematics (or some version of) calculus? MoeBlee
From: galathaea on 28 Oct 2006 22:04 In article <1161923879.394890.236700(a)k70g2000cwa.googlegroups.com>, "Bill Taylor" <w.taylor(a)math.canterbury.ac.nz> wrote: !! Is the following a reasonable point of view, do people think? !! !! I'm still kind of wondering where Yessenin-Volpin, Edward Nelson, !! and other ultrafinitists are coming from. !! !! They purport to find, or rather take the public stance of finding, !! that the concept of "all the naturals" is confusing and vague, !! whereas it is indeed *crystal-clear* to the rest of us. I'm sure !! it was once crystal clear to them too. It is NOT necessarily !! crystal clear, initially, to the non-mathematician - sometimes !! I've had CS or business students (amazingly) wonder !! "do the numbers go on for ever", though they are !! always happy with the simple answer "yes". !! !! But by and large, it might almost be considered a criterion to !! be a "natural mathematician", that the idea of N, the naturals, !! is crystal clear. (As opposed to, say, an intelligent doctor !! or lawyer who may have doubts about it.) !! !! Now, given this, what are we to make of ultrafinitists, !! who purport to find vagueness or ambiguity in this basic !! crystalline abstract jewell of ours, but who nevertheless seem !! to be reputable mathematicians. At least it seems so, judging !! from the fact that they get quite a bit of air time. !! !! My take is this, and I wonder if it is a reasonable view? !! !! """ !! !! Some time ago, back in the late seventies to early eighties, !! there was a brief flurry of interest from fringe mathematicians !! in "fuzzy math". It was never quite clear what this was, but it !! still has a small amount of library shelf space, though perhaps !! little or no presence in math departments in academia. !! It seemed to be (AFAICT), basically, that joke that !! used to go around about "Generalized Mathematics" - !! !! * "In Orthodox math we derive true results by valid means; !! * in Generalized math both these restrictions are dropped!" !! !! Anyway, one can hardly say that Fuzzy math even died - !! it was practically still-born... math departments !! gave it very short shrift. !! !! """ !! !! So finally, my question is this:- is it a fair point of view !! to regard ultrafinitism as essentially, fuzzy mathematical logic? !! !! They insist on keeping a fuzzy view on what is the largest !! feasible number, and similarly with the largest feasible !! derivation; indeed feasible anything - the very concept !! of feasibility seems to be the ultimate in fuzzy concepts. !! This viewpoint is *not necessarily* a negative one, !! I must point out. It may be that (unknown to me) there IS !! a lot of value in fuzzy math, whatever FM may be. !! This being so, there could easily be value in !! ultrafinitist math logic, also. !! !! So without necessarily making any approbation or !! disapprobation of either, is it fair to regard ultrafinitism !! as "fuzzy mathematical logic"? my biggest exposure to finitists of various persuasions comes from philosophy departments that is an indicator of the type of arguments i have seen the approaches i have seen might be better called "ultra-engineer" the claim behind many approaches is that mathematics does not model mathematics very accurately that mathematicians are finite manipulators that the number of steps they will ever take is finite and the number of symbols manipulated also finite and so it is not meaningful at least in modelling mathematics to speak of infinite processes they are counterfactual and in fact anti-factual can never be factual now of course this can be formalised in various ways and one that is particularly well done can be found at http://arxiv.org/abs/quant-ph/0108121 the point of ultrafinitists is that this changes the types of objects that are allowable in the model's ontology esenin-volpin describes some of the structural limitations one must place on the models of natural numbers in a modal logical representation but that is only a small illustration of the many constraints finite models obey ebbinghaus has a classic book on finite model theory ultrafinitsm takes a conceptual step beyond finitism by stressing that not only is mathematics a finite process but there exist hard limits there is no potential infinity or legitimate means to assume a process can be continued indefinitely in any derivation it is necessary to question for any process specification whether that process can complete "within the limits of resources" available to mathematics because they insist mathematics is a physical process and one day too may suffer the entropic decay -+-+-+-+- i have not seen this done in "fuzzifying" the truth characteristic (by for instance making it a more general subobject classifier) but you could possibly develop such a theory if you study the formalisation of ultrafinitsm and follow the debates on appropriate model constraints papers like "characterising finite kripke structures in propositional temporal logic" by browne, clarke, and grumberg start standing out !! Christian rationality: !! A god killing himself to save his own creation from his own wrath. -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar
From: Rupert on 30 Oct 2006 00:23 MoeBlee wrote: > Bill Taylor wrote: > > They insist on keeping a fuzzy view on what is the largest > > feasible number > > Thank you for your post and all the questions in it. I hope it gets a > lot of response. > > A related question that I have, which you or perphas others can help > with, is that of the putative axiomatizations of utltrafinitist > mathematics. Yessenin-Volpin said explicitly that he didn't want to commit to a specific axiomatic theory. Edward Nelson, on the other hand, has done some interesting research about certain weak axiomatic theories in arithmetic, which may embody his stance. See his book "Predicative Arithmetic". Edward Nelson has put active effort into proving certain results in these weak arithmetics. Success would imply that Peano Arithmetic is inconsistent. > Do these axiomatizations really hold up as formal > (recursively axiomatized) systems? Do they really provide systems such > that there is an effective method to check whether a string of formulas > is a proof and still have only finitely many natural numbers and still > also allow for proofs of theorems that serve to provide the basic > mathematics (or some version of) calculus? > > MoeBlee
From: Eckard Blumschein on 30 Oct 2006 08:07
On 10/29/2006 3:04 AM, galathaea wrote: > ultrafinitsm takes a conceptual step beyond finitism > by stressing that > not only is mathematics a finite process > but there exist hard limits > > there is no potential infinity Mueckenheim was blamed an ultrafinitist. However, he denies the actual infinity while the potential infinity seems to be obvious to anybody. > or legitimate means to assume > a process can be continued indefinitely > > in any derivation > > it is necessary to question > for any process specification > whether that process can complete > "within the limits of resources" > available to mathematics > > because > they insist > mathematics is a physical process > and one day too may suffer the entropic decay Mueckenheim obviously shares this view. I do not understand why he cannot accept mathematics like dealing with the two abstract ideas number and continuum. One alternative after the other failed to unmask Cantors paradise as what I consider the Dedekind-Cantor Utopia. Kronecker even called the natural numbers given by the Lord. Brouwer even intended to improve set theory. Weyl suggested an atomist continuum. Even Cantor and Hilbert started at some sound finitist views. Now ultrafinitism is rumored to be the most silly counterpoint to formalism. Tell me please whether or not there is a drawer you may put me in? I consider the world of (countable) numbers quite different from the complementing world of (uncountable) continuum. In principle Cantor was conjecturing almost the same when he believed that there is nothing between aleph_0 and aleph_1. Eckard Blumschein > > -+-+-+-+- > > i have not seen this done in "fuzzifying" the truth characteristic > (by > for instance > making it a more general subobject classifier) > but you could possibly develop such a theory > > > if you study the formalisation of ultrafinitsm > and follow the debates on appropriate model constraints > papers like > "characterising finite kripke structures > in propositional temporal logic" > by browne, clarke, and grumberg > start standing out > > !! Christian rationality: > !! A god killing himself to save his own creation from his own wrath. > > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > galathaea: prankster, fablist, magician, liar |