On Ultrafinitism
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From: Rupert on 2 Nov 2006 18:31 MoeBlee wrote: > Rupert wrote: > > > At least with Nelson > > that's made clear. Nelson's work indisputably has some serious > > mathematical content which is of interest to mathematicians regardless > > of their foundational stance. It's really quite startling how serious > > his skepticism is: he is actually making serious efforts to prove the > > inconsistency of EFA. > > EFA? Is that PA without induction schema but with two axioms for > exponentiation? > It has the axioms for Robinson Arithmetic, and a symbol for exponentiation with axioms giving the recursive definition of this symbol, and induction for all formulae where the quantifiers are bounded by a term in the language not including the quantified variable. It can be interpreted in a quantifier-free version which is a subtheory of Primitive Recursive Arithmetic, with only symbols for the elementary functions (those whose computing time is O(2^n), or O(2^2^n), etc.). > Thanks also for the rest of your remarks, which will require though > some time for me to get caught up on more set theory for me to fully > understand what you wrote. > > MoeBlee
From: MoeBlee on 2 Nov 2006 19:50 Rupert wrote: > It has the axioms for Robinson Arithmetic, and a symbol for > exponentiation with axioms giving the recursive definition of this > symbol, and induction for all formulae where the quantifiers are > bounded by a term in the language not including the quantified > variable. It can be interpreted in a quantifier-free version which is a > subtheory of Primitive Recursive Arithmetic, with only symbols for the > elementary functions (those whose computing time is O(2^n), or > O(2^2^n), etc.). Okay, thanks. MoeBlee
From: R. Srinivasan on 3 Nov 2006 03:28 On Oct 27, 9:37 am, "Bill Taylor" <w.tay...(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"? > I believe that the logic NAFL (see <http://arxiv.org/abs/math.LO/0506475>) has an element of ultrafinitism in it, and my hunch is that this is the correct way to go about it. I claim this even though the infinite class of all natural numbers (N) provably exists in the NAFL version of PA. The reason is that in NAFL, the claim that there is a maximum element m in N, but we cannot know what m is, is the *same* as saying that N is an infinite class. Since m is unspecified, it must be in a superposed state of all possible values and this makes N an infinite class. Thus the notion of "all" or "infinitely many" is already fuzzy enough to capture ultrafinitism in NAFL. For example, let us say we set up a quantum algorithm in NAFL (at the moment this is a distant dream) which has the ability to access random natural numbers. So the algorithm will access m. Since m is unknowable, the quantum algorithm has theoretically exhibited infinite parallelism, i.e., has accessed infinitely many natural numbers at the same time. One might be tempted to dub this as "physically impossible". But in practice, when the algorithm is actually implemented on a physical quantum computer, what has been accessed may be thought of as a theoretically unpredictable, large natural number m. Regards, RS
From: Keith Ramsay on 10 Nov 2006 03:32
Bill Taylor wrote: |I'm still kind of wondering where Yessenin-Volpin, Edward Nelson, |and other ultrafinitists are coming from. I'd suggest reading their books and papers carefully, and trying to figure out what they mean by what they're writing, piecemeal. Trying to take it all in in one grand sweep isn't probably going to give you a good grasp of what they're trying to say. |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 don't know of any reason to think that they're being insincere in any of their "public stances", so it seems a little unfair to write as if this were more of an issue for them than it is for anybody. I think it would be good to try to pose the question in a more neutral way. To say that it *is indeed* clear to "us" while they just "find" it not so clear is basically to start out by assuming they're wrong. I realize you've come to that conclusion and are simply describing the situation as you see it, but I think it still is easier to grasp someone else's position if you describe it more neutrally. There are plenty of people who'd say that "it *is* crystal clear to us that Jesus loves us" whereas there are those like us who merely "find" it doubtful... and around here there are more of them than of us. And I think part of why many of them find it hard to understand us is that they approach it as a matter of explaining how, *given* that it is *so* breathtakingly clear, some people don't find it so. So start by taking it only as a sense of clarity, knowing that there can be a spurious sense of clarity as well as a spurious sense of unclarity. I'm unsure whether "confusing" is the right descriptive terms here. That's not the way I remember either Nelson or Essenin-Volpin describing it, although perhaps I don't remember. "Confusing" is a subjective property that often says more about the person being confused than the thing they're confused about. I don't remember either of them claiming to be confused. Both of them have something to say about the natural numbers that we "can" count to, and I don't know of anybody who disputes that this is vague because of the fuzziness of the concept of ability. I think both of them are well aware that this is different from the usual notion of "N" the set of all natural numbers. It seems more pertinent to say that each of them has doubts about the existence of "N", the thing that mathematicians assume exists The way one usually defines "finite" and the definition of "N" are very closely related. I don't think Nelson has any trouble at all describing the theory that mathematicians usually work with, describing "N", nor does he seem to say that it's described in a nonrigorous way. It's just that he doubts it's factually correct. |I'm sure |it was once crystal clear to them too. I don't know how Essenin-Volpin's views changed with time. Nelson describes having had a kind of epiphany in which he lost the belief in N. Perhaps it's fair to say he came to believe there is no "N". |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". I don't think it's so amazing that people are puzzled by the idea that in some sense there "exist" more numbers than have ever been thought of. In what sense do they "exist", many have asked. |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.) You can point to lots of cases of students progressing from finding something unclear to finding something clear. I think once a person has the understanding of a Ph.D. in mathematics of such a concept as "natural number" and they stop finding it obviously sound, it's very rarely a matter of their regressing in understanding. I think it's more a question of trust. Certainly there appear to be people who fail to understand the critique of the natural numbers as impredicative, not at all because they can see more clearly, but because they see the issue less clearly. I think we may not be in a good position to address the question of what it's reasonable to doubt, whether for example it's reasonable to consider the consistency of PA to be seriously uncertain. On what objective basis could one assign it a probability near 1, say? The kind of rationality that comes with science, almost required by science, mainly calls for not being overly certain of things without a reason. If one wants to address the reasonableness of these people, I think one has mainly to set aside such questions as "is he sure that N exists?" as being too personal and subjective to be relevant. To understand Essenin-Volpin, I think it helps to remember that he was a Soviet dissident. He took the approach of calling on the authorities to obey the law. He would go to trials of other dissidents, which were often conducted in places that were difficult to go to, taking with him a copy of the Soviet legal code requiring that trials be open, asserting that they were therefore REQUIRED to let him in. One of Essenin-Volpin's claims is that mathematical argument should be noncoercive. When assumptions are invoked, they should be stated openly, and open to questioning, just like laws. A lot of what I've seen him write amounts to laying out various issues to be investigated this way. I think for the most part the real issue is not whether there's a valid point being made by these guys, but whether there's much to be done about it. I think to a large extent assumptions like mathematical induction are accepted because their consequences are interesting from various points of view: intellectually, for the sake of applications, and so on. Investigations of what follows from weaker assumptions is not considered as interesting, and it's mainly done by logicians who aren't considered ultrafinitists. (Studies of fragments of arithmetic are not to be sneezed at, however! It's just that it's not on the same scale as classical mathematics as a whole.) |Now, given this, what ar |