From W.Taylor at math.canterbury.ac.nz Wed Nov 1 00:32:46 2006 From: W.Taylor at math.canterbury.ac.nz (Bill Taylor) Date: Wed, 01 Nov 2006 18:32:46 +1300 (NZDT) Subject: [FOM] Concerning Ultrafinitism. Message-ID: <200611010532.kA15Wkjn015939@math.canterbury.ac.nz> [Note to moderator: I sent this several days ago, and have seen/had no reply. Is it still in the pipeline? ] ============================================================================= 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 From aatu.koskensilta at xortec.fi Wed Nov 1 06:30:21 2006 From: aatu.koskensilta at xortec.fi (Aatu Koskensilta) Date: Wed, 01 Nov 2006 13:30:21 +0200 Subject: [FOM] On "ruling out" non-standard models of first order arithmetic Message-ID: <4548854D.8000603@xortec.fi> One of the perennial "puzzles" concerning certain results about first order logic that I personally regard as totally wrongheaded has been subject of much discussion on this list lately, namely how come we are able to refer to the structure of natural numbers unambiguously, given that it cannot be characterised by saying that it satisfies some set of first order sentences. This question pops up every now and then in different guises, in whatever forum logic, foundations or philosophy of mathematics are discussed. For what it's worth here are my thoughts and observations on the subject, as trivial and unoriginal as they might be. As is obvious, much of what I say below I have shamelessly stolen from various remarks by Torkel Franz?n, both in the news and in his books. Now, how do we come to know what the natural numbers are? Certainly by some very complex process of indoctrination, innate cognitive inclinations, mechanical practice, reflection, education and so forth. Of that process I have very little to say, developmental psychology and such like not being my forte. A plausible "rational reconstruction" can be offered, though. The basic idea or conceptual picture we have in mind is a never ending sequence of things obtained from 0 by repeatedly adding 1 - perhaps pictorially in the form of appending a stroke to a sequence of strokes - or, in formal mode should we wish to sound professional, obtained from 0 by repeatedly applying the successor function. For some strange reason a qualifier "for a finite number of times" is often added to "repeatedly applying the successor function", as if applying any operation an infinite number of times to anything made some sense in this context. Reflecting on this - not that is something a single individual does, but rather a long historical development - we come to realize that the fundamental idea behind this picture is captured by the induction principle Whatever determinate property of natural numbers P is, if 0 has P, and x+1 has P whenever x has P, all natural numbers have the property P I have intentionally avoided using any logical symbolism, because this is an informal principle, not a sentence in some first order language, or in second order language, or what you have; it is just a piece of ordinary language, as vague or crystal-clear as any such piece. Of course, what applications this principle has depends on what properties we regard as determinate - in the way "if a person has x hairs he's bald" is not - and intelligible. In any case, it seems that properties that are expressible using similar concepts as those used in framing the theorems we prove, or conjectures we wonder about count. Similarly, for any interpreted formal language we regard as meaningful, the principle immediately yields induction for that language, which we might express schematically, or by going second order and postulating comprehension for the relevant language, or by any such device. So far so good - I don't expect the above to be particularly controversial. Now, enter the basic results about first order logic - incompleteness, the true theory of naturals having non-isomorphic models, L?wenheim-Skolem, pick your favourite. By any of these, the structure of natural numbers can not be characterized by any set of first order sentences (axiomatizable or not). The conundrum then is, supposedly, how can we successfully and unambiguously refer to the structure of natural numbers? Must it be that we're using second order logic? Because all the non-standard models are non-recursive? Or perhaps we can't, and it's all just an illusion? And so forth. Before trying to pick the answer, let's pause for a moment and have a closer look at what the supposed conundrum is. The obvious question to ask is why should a mathematical result about first order logic be at all relevant to how mathematical language works, how we refer to mathematical structures, how we come to learn what naturals are - or what sets are, for that matter. It appears that the picture (or a model, if you like that word better) giving rise to the conundrum is of us somehow being presented with an array of structures, one of which we must pick by providing some sort of description for it, the way we might be presented with 100 chairs one of which we want to identify. If all we know of a given structure is what first order sentences it satisfies, the mathematical results tell us that we simply cannot pick the correct structure, expect by chance, if all we are allowed to know about it is some set of first order sentences it satisfies. Ok. But the picture is totally implausible - there is simply no reason to suppose our understanding of anything, let alone mathematical structures, is obtained or mediated that way. We simply aren't "given" any mathematical structures, be it ones where non-standard numbers might secretly be hiding or not. Rather, if we consider some array of structures, it is by means of some set of concepts, some set of mathematical ideas and ways of describing structures. And then we can unproblematically say that the standard ones are those for which induction holds - in the form of the informal principle - and in particular that if induction fails for some structure living in the mathematical world envisioned in terms of those concepts and ideas, it is non-standard. As our only access to these structures are as parts of the mathematical pictures or stories we come up, we simply cannot "accidentally" think some structure is standard while, in fact, it is not - for we can't directly pick a structure x and assert of it that it is or is not standard, we can only refer to them using our mathematical concepts and ideas, as parts of mathematical worlds we fantasize, discover or invent, however that works. (Of course, one might refer to some structure in this way of which we simply don't know whether it is standard or not, e.g. by saying that we're considering a structure in which the axioms of PA hold but the Goldbach conjecture doesn't). It is sometimes suggested that we manage to "rule out non-standard models" by using second order logic. I'm not sure what that means exactly; second order logic is a mathematical system, and it's not obvious how to "use" it in any interesting sense. Of course, it's possible to use second order logic for all sorts of purposed in the ordinary mathematical sense, e.g. specifying which structure we're talking about - by saying that it's the structure satisfying this or that set of second order sentences, not in any sense relevant to the conundrums here - and so forth. But it seems something more substantial is meant, and I haven't really seen that spelled out anywhere, and as Tim Chow already pointed out any worries we might have about the naturals extend to pretty much all mathematics - without falling prey to circularity there seems to be no way to use any piece of mathematics to support the idea that we really do understand what we mean when we talk about naturals, as if that needed *any* support to begin with. Now, there is a *honest* way to question the standard picture of naturals, which is to question its coherence, or the induction principle applied to logically complex properties, or something on those lines. Such questions are not, however, based on any result of mathematical logic, but rather on conceptual considerations that might or might not lead to interesting mathematics e.g. in the hands of competent ultra-finitists and ultra-intuitionists. Who knows, perhaps some conception of ultra-finitistic naturals or several systems of natural numbers proves to be as rich as the hierarchy of large cardinals, say? Not that I'm holding my breath... As to first order logic and its role in foundations, I'm in total agreement with Harvey's assertion that it is fundamental. The relevant observation here is not that we "use" first order logic in contrast to second order logic, but rather that the completeness theorem and a few conceptual reflections a la Kreisel show that the first order consequence relation correctly captures our informal notion of something being provable on basis of something else. Thus to determine what follows from what, what does not follow from what, what is the relation of this sets of principles to that set of principles, and so forth, first order logic is the tool of choice. For all sorts of mathematical purposes higher-order logics, infinitary languages, exotic subsystems of first order logic, all are useful and interesting, but they do not have the same sort of a fundamental relation to our basic conception of mathematics as first order logic. That said, most of the results about first order logic are purely mathematical and do not necessarily have any philosophical significance, at least with some special argument to that effect. Aatu Koskensilta (aatu.koskensilta at xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus From rupertmccallum at yahoo.com Wed Nov 1 17:56:50 2006 From: rupertmccallum at yahoo.com (Rupert McCallum) Date: Wed, 1 Nov 2006 14:56:50 -0800 (PST) Subject: [FOM] Is there a "naturally occurring" theorem provable in EFA but not in predicative arithmetic? Message-ID: <20061101225650.99269.qmail@web51903.mail.yahoo.com> My friend Norman Wildberger thinks there is a part of mathematical reasoning which is self-evident and doesn't need to be analyzed by the axiomatic method. I want to cast doubt on this belief of his by giving him an example of a "naturally occurring" theorem, of the kind mainstream mathematicians would be interested in, which can be proved in EFA but not in predicative arithmetic as developed in Edward Nelson's book "Predicative Arithmetic". Of course one example is "exponentiation is total", but that is not the sort of theorem which would be of interest to a mainstream mathematician. ____________________________________________________________________________________ Access over 1 million songs - Yahoo! Music Unlimited (http://music.yahoo.com/unlimited) From rupertmccallum at yahoo.com Wed Nov 1 18:19:25 2006 From: rupertmccallum at yahoo.com (Rupert McCallum) Date: Wed, 1 Nov 2006 15:19:25 -0800 (PST) Subject: [FOM] Concerning Ultrafinitism. In-Reply-To: <200611010532.kA15Wkjn015939@math.canterbury.ac.nz> Message-ID: <20061101231925.19259.qmail@web51915.mail.yahoo.com> Have you read Edward Nelson's book "Predicative Arithmetic"? If you read the details of when he regards induction as justifiable and when he doesn't, you might get a better idea of where he's coming from. Actually, Edward Nelson is a formalist, he even regards the consistency of Robinson Arithmetic as an open problem. But let's pretend that the theory he develops in "Predicative Arithmetic" embodies his stance. An example of a sentence which this theory can't prove is one which might be paraphrased as "exponentiation is total". Nelson gives a way of interpreting a theory of finite sets in his theory, using a scheme of coding which doesn't appeal to the totality of any function except just a shade beyond polynomial. Then the theorem that says exponentiation is total can be stated as: For every a and b, there exists a function f defined on {0,...,b} such that f(0)=1 and f(i+1)=a.f(i). This theorem cannot be proved because in Nelson's theory induction is only allowed for formulas where the quantifiers are "bounded" using functions which are either polynomial or slightly beyond, and in this context that cannot be done, the code for the function grows too fast. His justification for this stance is discussed in the opening chapter called "The impredicativity of induction". Take a model M for Robinson Arithmetic. Take an inductive formula phi. Then Nelson shows that we can construct a formula phi', such that, if M' is the set of x in M satisfying phi', then M' is a model for Robinson Arithmetic as well, and every x in M' satisfies phi in the original model M. Now, if we can also prove that every x in M' satisfies phi in M', then we can interpret Q+"(Ax)phi(x)" in Q. Nelson regards induction as justified in such an instance, since assuming induction in this case gives us a theory which can be interpreted in Q. But when we cannot prove this, we cannot give a syntactic proof that Q+"(Ax)phi(x)" can be interpreted in Q. We can give a semantic proof that there is a submodel of our original model which satisfies Q+"(Ax)phi(x)" as follows. Let M' be the submodel of M consisting of all x which satisfy phi' in M, then let M'' be the submodel of M' consisting of all x which satisfy phi' in M', and keep on going in this way, and take the intersection of all these models. This might be thought to justify induction in general. But this involves impredicative second-order reasoning, which Nelson rejects. All the theories Nelson works with can be interpreted in Q. So, Nelson rejects the idea that we have a clear understanding of the distinction between the finite and the infinite. Every time we apply a new instance of the principle of induction, we refine our understanding of what a number is. We cannot assume that the natural number system is given as something fixed once and for all. So we cannot justify induction in general. __________________________________________________________________________________________ Check out the New Yahoo! Mail - Fire up a more powerful email and get things done faster. (http://advision.webevents.yahoo.com/mailbeta) From neilt at mercutio.cohums.ohio-state.edu Wed Nov 1 18:28:33 2006 From: neilt at mercutio.cohums.ohio-state.edu (Neil Tennant) Date: Wed, 1 Nov 2006 18:28:33 -0500 (EST) Subject: [FOM] On "ruling out" non-standard models of first order arithmetic In-Reply-To: <4548854D.8000603@xortec.fi> Message-ID: On Wed, 1 Nov 2006, Aatu Koskensilta wrote: > ... if we > consider some array of structures, it is by means of some set of > concepts, some set of mathematical ideas and ways of describing > structures. And then we can unproblematically say that the standard ones > are those for which induction holds - in the form of the informal > principle - and in particular that if induction fails for some structure > living in the mathematical world envisioned in terms of those concepts > and ideas, it is non-standard. Isn't part of the problem that mathematical induction holds (for all first-order instances) in the non-standard models? Neil Tennant From examachine at gmail.com Wed Nov 1 17:26:26 2006 From: examachine at gmail.com (Eray Ozkural) Date: Thu, 2 Nov 2006 00:26:26 +0200 Subject: [FOM] Concerning Ultrafinitism. In-Reply-To: <200611010532.kA15Wkjn015939@math.canterbury.ac.nz> References: <200611010532.kA15Wkjn015939@math.canterbury.ac.nz> Message-ID: <320e992a0611011426v3447b58fpe5f843278e3ebe80@mail.gmail.com> On 11/1/06, Bill Taylor wrote: > So without necessarily making any approbation or disapprobation of either, > is it fair to regard ultrafinitism as "fuzzy mathematical logic"? I would be grateful if you would consider my inexpert opinion. >From what I have read, my impression on finitism has been that it arises from philosophical preferences. However, your conclusion is also philosophically motivated. In particular, you seem to assume the truth of (a particular form of) realism, and thus you seem to think that there is something that is _really_ false about ultrafinitism. I beg to differ. I think we can sort of enjoy a flexible view in which different philosophical foundations are considered as mathematics proper. If you would ask my own view instead, I would say that both assertions are equally imaginery (I don't say false): 1) there are an infinite number of integers 2) there is a largest integer Because, in my own view, integers do not exist in any real sense of the word. On the other hand, both seem to be equally valid from a flexible philosophical point of view (as above). I can also offer a third interpretation, which is computationalism, this would seem to say something like, a non-halting computation can generate an ordered list of integers, however, there is a largest integer that can be generated in any finite *physical* universe. I take this point of view to be equivalent to what Godel himself dubbed as Aristotelian Realism. Unfortunately, I haven't heard any other author use the same phrase. As an additional claim, I will suggest that there is no mathematical way of deciding the truth of your claim. Regards, PS: Many cosmologists seem to think that our universe is finite indeed, therefore the considerations of ultrafinitists actually make physical sense, even if they do not designate the only admissible metaphysics (or mathematics). (I do recognize that physics itself is beyond the scope of FOM list) -- Eray Ozkural, PhD candidate. Comp. Sci. Dept., Bilkent University, Ankara From dana.scott at cs.cmu.edu Wed Nov 1 23:57:28 2006 From: dana.scott at cs.cmu.edu (Dana Scott) Date: Wed, 1 Nov 2006 20:57:28 -0800 Subject: [FOM] Fwd: 303: PA Completeness (restatement) (Andrej Bauer) References: <668D7565-E6E1-4B0F-928D-1D96EF7DE82F@cs.cmu.edu> Message-ID: <3DB63A99-3312-4F84-95C1-6F14C3B20D2B@cs.cmu.edu> In FOM Digest, Vol 46, Issue 40 (October 31), Andrej Bauer asked about the provability in PA of two quite short statements: (1) exists x, exists y, x^4 = 1 + x + y^2 (2) exists x, exists y, x^2 (1 + y) = 1 + x + y^2 I concluded that both can be REFUTED. I outline two arguments which can surely be done formally in PA. (Ad 1) Look, for x = 0, 1, or 2 the euqation is impossible for any natural number. Right? Consider x^4 - 1 - x for x >= 3. That is a little bit less than the square of x^2. Now, if we think of any n^2, the square right BELOW it is n^2 - 2 n +1 = (n - 1)^2. Therefore, the square just below x^4 has to be x^4 - 2 x^2 + 1. However, (x^4 - 1 - x) - (x^4 - 2 x^2 + 1) = 2 x^2 - x - 2. That quantity is POSITIVE for x >= 2. In other words, x^4 - 1 - x is never a square. Q.E.D. (2) I checked that the equation has no integer solutions up to x = 2,000,000. But that is not a proof. So, solve the equation for y. The positive root is: y = (x^2 + Sqrt(- 4 - 4 x + 4 x^2 + x^4))/2. Note that this is in general a fraction. For y to be an integer we should find out something about what is under the square root sign. If x is even, then the stuff under the root is even, so the square root is even, so the numerator is even, so y is an integer (if the root is an integer). If x is odd, then x^4 is odd, so the stuff under the root is odd, so the square root is odd, so the numerator is even, so y is again an integer. HENCE, the problem comes down to whether the stuff under the root can ever be a SQUARE. Write it as x^4 + 4(x^2 - x - 1). Remark that for large x, the term x^4 is much larger than 4(x^2 - x - 1). Now 4(x^2 - x - 1) is larger than 2 x^2 + 1, so x^4 + 4(x^2 - x - 1) might exceed the NEXT square after x^4. But the SECOND square after x^4 is x^4 + 4 x^2 + 4, which is LARGER than x^4 + 4(x^2 - x - 1). Q.E.D. The point of these simple arguments is that there are bigger and bigger gaps between squares. From friedman at math.ohio-state.edu Thu Nov 2 05:40:37 2006 From: friedman at math.ohio-state.edu (Harvey Friedman) Date: Thu, 02 Nov 2006 05:40:37 -0500 Subject: [FOM] Is there a "naturally occurring" theorem provable in EFA but not in predicative arithmetic? In-Reply-To: <20061101225650.99269.qmail@web51903.mail.yahoo.com> Message-ID: On 11/1/06 5:56 PM, "Rupert McCallum" wrote: > My friend Norman Wildberger thinks there is a part of mathematical > reasoning which is self-evident and doesn't need to be analyzed by the > axiomatic method. I want to cast doubt on this belief of his by giving > him an example of a "naturally occurring" theorem, of the kind > mainstream mathematicians would be interested in, which can be proved > in EFA but not in predicative arithmetic as developed in Edward > Nelson's book "Predicative Arithmetic". Of course one example is > "exponentiation is total", but that is not the sort of theorem which > would be of interest to a mainstream mathematician. > 1. For all n, the Pell equation x^2 = dy^2 + 1 where d > 0 and not a square, has at least n solutions. This can be proved in EFA but not in, say, PFA = Idelta0. There is known exponential behavior here, even for fixed d = 2. 2. Consider the statement "every Pell equation has a solution". If I recall properly, D'Aquino proved that EFA is equivalent to PFA + "every Pell equation has a solution". In particular, it is provable in EFA but not in PFA. 3. For all 0 < p < 1, there exists n such that the probability that a positive integer < n is prime is less than p. 4. Burnside problems in group theory (3rd ursl below). Also van der Waerden's theorem, although the known lower bounds are very low, even the Gowers proof seriously uses exponentials (I think it works in EFA). 5. Falting's theorem should be looked at from this point of view. 6. People tend to believe that FLT can be proved in EFA but not in PFA. One reason is that an appropriate statement in PFA is awkward. 7. Various Ramsey theory statements. http://www.cs.nyu.edu/pipermail/fom/2006-April/010367.html http://www.cs.nyu.edu/pipermail/fom/2006-April/010366.html http://www.cs.nyu.edu/pipermail/fom/2006-April/010379.html Harvey Friedman From aa at tau.ac.il Thu Nov 2 06:39:48 2006 From: aa at tau.ac.il (Arnon Avron) Date: Thu, 2 Nov 2006 13:39:48 +0200 Subject: [FOM] 303: PA Completeness (restatement) In-Reply-To: References: Message-ID: <20061102113948.GA946@nova.cs.tau.ac.il> Even if Friedman's stronger conjecture concerning PA is true, I have strong doubts concerning the value of such a result and its possible philosophical implications. The reasons are: 1) It seems to me somewhat misleading to talk about the induction schema as mentioning only one variable (or two, if the metavariable R is counted too - is it?). In the actual language of PA instances of it might involve a huge number of variables. It is quite likely that the only proofs in PA of some (2,2) theorems require instances of the induction scheme involving more (and maybe much more) variables. In such a case what is really hapening is that a system with axioms of high complexity can decide some fragment of lower complexity. So what? 2) The (2,2) fragment of PA seems to me too weak to be significat or interesting. Even the most elementary properties of the structure N (like associativity of + or the transitivity of <) need at least 3 quantifiers for their formulation. Indeed, it seems to me that only corresponding results about sentences with at least 3 variables might be really interesting. I strongly doubt that such results may be obtained. As an example: suppose that instead of using as primitives 0,S,+, and dot we take as primitives 0,1,+,<, and | (where the meaning of x|y is "x devides y"). The two languages are equivalent in their expressive power (concerning N), and it is not difficult to produce a system equivalent to PA in the second language, which has the same syntactic properties as those noted by Friedman in PA. Now in this language Goldbach's conjecture can be formulated using a sentence of complexity (3,1) (and involving 4 quantifiers). This seems to indicate that already the (3,1 )-level may be very difficult. Arnon Avron From gstolzen at math.bu.edu Thu Nov 2 13:48:08 2006 From: gstolzen at math.bu.edu (Gabriel Stolzenberg) Date: Thu, 2 Nov 2006 13:48:08 -0500 (EST) Subject: [FOM] Yessenin-Volpin Message-ID: Alik Volpin's "ultra-finitist" program had the aim of proving the consistency of mathematics. Gabriel Stolzenberg From V.Sazonov at csc.liv.ac.uk Thu Nov 2 15:39:57 2006 From: V.Sazonov at csc.liv.ac.uk (V.Sazonov@csc.liv.ac.uk) Date: Thu, 02 Nov 2006 20:39:57 +0000 Subject: [FOM] Concerning Ultrafinitism. In-Reply-To: <200611010532.kA15Wkjn015939@math.canterbury.ac.nz> References: <200611010532.kA15Wkjn015939@math.canterbury.ac.nz> Message-ID: <20061102203957.6o7i88hegco0o84o@cgi.csc.liv.ac.uk> Quoting Bill Taylor Wed, 01 Nov 2006: > I'm still kind of wondering where Yessenin-Volpin, Edward Nelson, > and other ultrafinitists are coming from. One of the first papers of Yessenin-Volpin on ultrafinitism was called Analysis of Potential Feasibility (Logical Investigations, Moscow, AN USSR, 1959, 218-262 (in Russian). I think, the title says for itself. > > 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. Until they have not tried to analyse. It is NOT necessarily > crystal clear, initially, to the non-mathematician > But by and large, it might almost be considered a criterion to > be a "natural mathematician", ...appropriately educated one [all of us are somewhat "damaged" by the education system] and knowing (either explicitly or implicitly just by the training provided by the "school") which formal rules and axioms to use. When you have learned how to reason and have no further questions then, of course, you will start to believe that the idea of N, the naturals, > is crystal clear. see also below about ideas and fantasies and mathematics. (As opposed to, say, an intelligent doctor > or lawyer who may have doubts about it.) or an educated mathematician who starts asking some "stupid" questions. > > 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, a "Sacred cow"? > 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" - an awful idea on mathematics, by my opinion. > * "In Orthodox math we derive true results by valid means; > * in Generalized math both these restrictions are dropped!" Mathematics, "by definition", deals with our (inevitably vague) imagination and fantasies governed/restricted/strengthened by FORMAL axioms and rules. This definition is very general, but it does not change the nature of mathematics. It only changes some traditional angle of view on mathematics - a wrong angle assuming beliefs in fictions having no scientific grounds. Of course this definition is not about true results (which ever truth if we are dealing with fantasies and imaginations; recall, e.g. Imaginary Geometry of Lobatshevsky). But what you call valid methods is, in fact, formal axioms and rules (governing our fantasies). > > Anyway, one can hardly say that Fuzzy math even died - it was > practically still-born... math departments gave it very short shrift. I am not an advocate of Fuzzy math, but as I know Peter Hajek presented a quite nice mathematical (rigorous) approach to fuzzy logic. Any problems with this? Or any pretensions to probability theory, etc? The ideas may be as arbitrary as our fantasies. The only problem is whether they are governed by formal rules. Then it is just a normal mathematics. > So finally, my question is this:- is it a fair point of view > to regard ultrafinitism as essentially, fuzzy mathematical logic? I think not. The point of ultrafinitism, as I understand it, is the analysis of potential feasibility. Fuzzy math has some other starting point. They might be related in some way, but this is a different story which does not exist yet. See also some citations from Troelstra and van Dalen presented, e.g., in http://cs.nyu.edu/pipermail/fom/2003-June/006716.html The point is that constructivism requires that existence should mean the possibility to construct an object. The question is whether we can guarantee that the construction is real(istic). Thus, the question is how the ordinary abstract natural numbers are related with "real", or feasible numbers (some initial part of N). Either we reduce everything to complexity theory based mainly on asymptotical considerations (which, being based on classical mathematics, also ignores or has no non-asymptotic approaches to the "real" feasibility) or we could try to do something on the level of foundations of mathematics trying to find a reasonable formal approach to the very concept of feasible numbers. > > So without necessarily making any approbation or disapprobation of either, > is it fair to regard ultrafinitism as "fuzzy mathematical logic"? As I told, I do not believe so. The "set" of feasible numbers looks a vague or fuzzy one, but the question is rather about how, starting with naively understood feasible numbers we arrive to highly idealised (still vague AS ANY OTHER IDEA at all) of mathematical numbers. Can we approach to all of this rigorously, as it is required in mathematics so that the formalisation will include the idea of feasibility in an appropriate way, or the only thing we could do is dully speculations? This could extend mathematics by new concept of feasible number, but the main feature of mathematics - its rigorous character - will remain at least at the same level if not higher. Thus, as I understand, the question is not in introducing in mathematics something foreign to its nature. Mathematical rigour is not intended to be relaxed in any way! It is about extension of its ability to formalise new kind of ideas (or fantasies; see the definition above). Quoting Gabriel Stolzenberg Thu, 02 Nov 2006: > Alik Volpin's "ultra-finitist" program had the aim of proving > the consistency of mathematics. I remember his further paper with this "attempt", but, by my opinion, this was not a real proof and too far to generate a proof. To prove anything from "ultra-finitism" it was necessary to have it first formalised. I like the general ideas and fantasies of Yesenin-Volpin as stimulating ones, but I have not seen anything written by him what could be considered as a sound formalisation of *these* his ideas. More precisely, I was never been able to understand his writings on this subject (even when he used some formulas) as truely mathematical. (See again the above definition of mathematics.) Is anybody able to understand his "proofs" concerning ultrafinitism? As I know, the first person who indeed formalised something similar was Rohit Parikh. Vladimir Sazonov ---------------------------------------------------------------- This message was sent using IMP, the Internet Messaging Program. From rupertmccallum at yahoo.com Thu Nov 2 16:36:22 2006 From: rupertmccallum at yahoo.com (Rupert McCallum) Date: Thu, 2 Nov 2006 13:36:22 -0800 (PST) Subject: [FOM] Yessenin-Volpin In-Reply-To: Message-ID: <20061102213622.40941.qmail@web51903.mail.yahoo.com> Yes, he claimed to have a proof of the consistency of ZF with any finite number of inaccessible cardinals. Unfortunately it seems to be hard to get hold of a copy of this proof. I would really like to know in which axiomatic theory he claimed it could be done. --- Gabriel Stolzenberg wrote: > > Alik Volpin's "ultra-finitist" program had the aim of proving > the consistency of mathematics. > > Gabriel Stolzenberg > _______________________________________________ > FOM mailing list > FOM at cs.nyu.edu > http://www.cs.nyu.edu/mailman/listinfo/fom > ____________________________________________________________________________________ Access over 1 million songs - Yahoo! Music Unlimited (http://music.yahoo.com/unlimited) From pietro.kc at gmail.com Thu Nov 2 14:57:35 2006 From: pietro.kc at gmail.com (Pietro Kreitlon Carolino) Date: Thu, 2 Nov 2006 17:57:35 -0200 Subject: [FOM] 303: PA Completeness (restatement) In-Reply-To: References: <454682C0.1090509@fmf.uni-lj.si> Message-ID: [Note to moderator: professor Scott's recent post has brought it to my attention that my original message was bounced. However, I've become interested in the (admittedly vague) question asked in its first paragraph, and would like to hear from subscribers. Sorry for the mix-up.] ---------- Forwarded message ---------- From: Pietro Kreitlon Carolino Date: Nov 1, 2006 12:10 AM Subject: Re: [FOM] 303: PA Completeness (restatement) To: Andrej.Bauer at andrej.com, Foundations of Mathematics [Original note to moderator: I'm not sure how interesting this is to subscribers, or even how obvious, but for completeness I submit these results to your judgement.] On 10/30/06, Andrej Bauer wrote: > I have played with Mathematica to see which sentences of the forms > > exists x, exists y, e1 = e2, > forall x, forall y, e1 = e2, > > with e1 and e2 terms of complexity at most 2, we can decide. Mathematica > easily gets them all, except two Diophantine equations: > > exists x, exists y, x^4 = 1 + x + y^2 > > exists x, exists y, x^2 (1 + y) = 1 + x + y^2 I would like to know the proportion of ExEy sentences that came up true in professor Bauer's Mathematica run. It might be interesting to see how a "random sentence" of PA behaves truth-wise, maybe fixing parameters like quantifier complexity, number of variables etc. Has something like this been done, akin to random graph theory maybe? Anyway, the answer to both equations above is negative. Below are technical details. They are a bit different in flavor from professor Scott's proof, which gets the result with fewer resources (though both are obviously formalizable in PA). It is interesting that we both solved equations 1 and 2 in similar ways, though our methods differ from each other. I wonder what the peculiarity is that prevented Mathematica solving them. The second one is a bit easier, so I start off with it. Working mod 2, we see that x = 1 --> 1+y = y^2 --> 1 = 0 (contradict.) x = 0 --> y^2+1 = 0 --> y = 1 So x is even and y is odd. Mod 8, that means x^2 = 0 or 4, and y^2 = 1. So: x^2 = 0 --> 0 = x+2 --> x = 6 --> x^2 = 4 != 0 (contradict.) x^2 = 4 --> 4(1+y) = x+2 --> x+2 = 0 (as 1+y is even) --> x=6. Therefore x = 6 (mod 8). But then x/2 = 3 (mod 4). Hence there exists a prime q dividing x/2, with q=3(mod 4). Then, taking the original equation mod q, we have: x^2 (1 + y) = 1 + x + y^2 ---(mod q)---> y^2+1 = 0 (mod q) which is well-known to be unsolvable for q=3 (mod 4). Now for the first equation. Rewrite x^4 = 1 + x + y^2 as x^4 - x = y^2 + 1 <--> x(x-1)(x^2+x+1) = y^2+1 Reduce it mod x: then y^2+1 = 0 (mod x). As we reasoned above, x can have no prime factors congruent to 3 mod 4. Also, the highest power of 2 to divide x is 0 or 1, otherwise y^2+1 = 0 (mod 4). If it were 0, x would be =1 (mod 4) (as all its prime factors apart from 2 are), and by the factorization above on the right, y^2+1 = 0 (mod 4) again. Therefore: x= 2*pq...r where p,q,...,r are primes = 1 (mod 4) In particular, x=2 (mod 4) (this will save some labor just ahead). On the other hand, working mod 2 on the original equation, y is easily seen to be odd, and therefore y^2 = 1 (mod 8). So: x(x-1)(x^2+x+1) = 2 (mod 8) Trying x=2 and x=6 (mod 8), we see that only x=6 (mod 8) works. But this gives x/2=3 (mod 4), contrary to the factorization obtained above. Hence there is no solution. Cheers, --Pietro From robblin at thetip.org Thu Nov 2 17:52:38 2006 From: robblin at thetip.org (Robbie Lindauer) Date: Thu, 2 Nov 2006 14:52:38 -0800 Subject: [FOM] Concerning Ultrafinitism. In-Reply-To: <320e992a0611011426v3447b58fpe5f843278e3ebe80@mail.gmail.com> References: <200611010532.kA15Wkjn015939@math.canterbury.ac.nz> <320e992a0611011426v3447b58fpe5f843278e3ebe80@mail.gmail.com> Message-ID: Mr. Ozkural, the dichotomy between: 1) There are an infinite number of integers and 2) there is a largest integer is a false dichotomy since there are at least these three other options: a) There may be no number of integers at all. There is no reason to suspect that there is a number of Cardinal Numbers, why should we expect that there is a number of integers? b) The integers AS A WHOLE may not themselves exist, just as the Cardinal Numbers AS A WHOLE may not exist. The integers as defined by successors in counting of "1" where counting is the practice of identifying the succeeding integer by adding 1 to the given integer may or may not identify any REAL THINGS. That is, it is a practice of generating words which themselves may not have referents. There is no reason, in general, to assume that just because there is a word for something that the thing it is supposed to refer to exists. So the fact that we have a word-structure "All the Cardinal Numbers" does not lead us to deduce that there is any such thing, neither should it in the case of "All the Integers". c) The integers may be fundamentally incomplete. Any given set of integers can be show to NOT be the complete set of integers. The "complete set of integers" then may not in fact exist - that is, may not in fact be giveable in a complete intuition. For if it were so given, it would be possible to produce an integer not in the given set, thus contradicting the theory that it is the whole set of integers. Finally a not-very-related mathematical question: is this a proved theorem (false or true): Given a unique and infinite decimal expansion of two real numbers X and Y, there is a non-finite intersection of those expansions of X and Y, e.g. given a series of integers in the decimal expansion of x: ...21837283940....274829304... there is a non-finite series of integers in Y identical to it. Robbie Lindauer From rupertmccallum at yahoo.com Thu Nov 2 18:49:45 2006 From: rupertmccallum at yahoo.com (Rupert McCallum) Date: Thu, 2 Nov 2006 15:49:45 -0800 (PST) Subject: [FOM] Concerning Ultrafinitism. In-Reply-To: Message-ID: <20061102234945.69946.qmail@web51912.mail.yahoo.com> --- Robbie Lindauer wrote: > Finally a not-very-related mathematical question: > > > is this a proved theorem (false or true): > > Given a unique and infinite decimal expansion of two real numbers X > and > Y, there is a non-finite intersection of those expansions of X and Y, > > e.g. given a series of integers in the decimal expansion of x: > > ...21837283940....274829304... there is a non-finite series of > integers > in Y identical to it. > I'm not sure I understand what your question is. How about 0.3333333333333.......... and 0.444444444444........., would that be a counterexample? > > > Robbie Lindauer > > _______________________________________________ > FOM mailing list > FOM at cs.nyu.edu > http://www.cs.nyu.edu/mailman/listinfo/fom > ____________________________________________________________________________________ Low, Low, Low Rates! Check out Yahoo! Messenger's cheap PC-to-Phone call rates (http://voice.yahoo.com) From W.Taylor at math.canterbury.ac.nz Thu Nov 2 21:35:20 2006 From: W.Taylor at math.canterbury.ac.nz (Bill Taylor) Date: Fri, 03 Nov 2006 15:35:20 +1300 (NZDT) Subject: [FOM] Concerning Ancestral Logic Message-ID: <200611030235.kA32ZKC0000215@math.canterbury.ac.nz> This query is mainly directed to Arnon Avron, but no doubt others may feel like joining in. Arnon - you have been extolling the virtues of AL as an alternative, (an extension?), of FOL. It sounds good, but I don't yet know what it is. You mention it was given a treatment in Shapiro's F-without-F, but I don't recall noticing it from when I read that. So I wonder if you could give us all a brief rundown on it. Nothing too massively technical, but just enough to get the flavour and main ideas. I was thinking of, say, 2 or 3 standard paragraphs. Could you do that please? Thanks muchly, Bill Taylor From rupertmccallum at yahoo.com Fri Nov 3 04:00:56 2006 From: rupertmccallum at yahoo.com (Rupert McCallum) Date: Fri, 3 Nov 2006 01:00:56 -0800 (PST) Subject: [FOM] formalism Message-ID: <20061103090056.55275.qmail@web51912.mail.yahoo.com> I'm interested in understanding the formalist position better. One version of it is that mathematics is "the science of formal systems", but then the question arises in what metatheory do we study these formal systems. Edward Nelson says he is a formalist and that he thinks the consistency of Robinson Arithmetic is an open problem. This makes me wonder what metatheory he accepts. (I have sent him an email asking him about this). Is it just the set of true Sigma-1 sentences, for example? Or the set of Sigma-1 sentences which we can feasibly prove? Have any formalists expressed a position on this issue? ____________________________________________________________________________________ We have the perfect Group for you. Check out the handy changes to Yahoo! Groups (http://groups.yahoo.com) From mmannucc at cs.gmu.edu Fri Nov 3 07:44:22 2006 From: mmannucc at cs.gmu.edu (Mirco Mannucci) Date: Fri, 3 Nov 2006 07:44:22 -0500 (EST) Subject: [FOM] Concerning Ultrafinitism Message-ID: --Hay un concepto que es el corruptor y el desatinador de los otros. No hablo del Mal, cuyo limitado imperio es la Etica; hablo del infinito-- (There is a concept which corrupts and upsets all others. I refer not to Evil,whose limited realm is that of Ethics; I refer to the infinite) Jorge Luis Borges ---- Bill Taylor wrote: > So without necessarily making any approbation or disapprobation of either, > is it fair to regard ultrafinitism as "fuzzy mathematical logic"? > The answer to your question is simply NO. Ultrafinitism is not easy to pin down and strait-jacket, as it is still a FOM Program in its very infancy (but NOT still-born!), and people who have an interest in it hold quite different views, as this list has amply borne out...... However, I think there IS a unifying theme: ----> Ultrafinitism = Constructivism MINUS Potential Infinity In other words, one could say that Ultrafinitism is any serious attempt to establish a coherent fundation of mathematics on constructive basis WITHOUT giving a free ticket to ANY form of infinity, not excluded POTENTIAL INFINITY (which incarnates in the infamous 1,2, 3 ..... to an ultrafinitist the dots ARE THE PROBLEM!!!), in all its various forms and disguises (for instance, using unbridled and unchecked induction at the meta-level). This goes as far (see Rupert's excellent reply to you, chiefly focused on Nelson) as to cast doubts on whether the long-standing dichotomy FINITE-INFINITE has an absolute, uncontroversial character. To this effect, notions such as feasibility are investigated, involving a degree of "fuzziness". The Model Theory and, as I suggested in my two postings on TTP, the Proof Theory of Ultrafinitism, may (I say may, not must) use methods and techniques from Fuzzy Set Theory and other tools developed to deal with vagueness, in the attempt to create CLASSICAL models of ultrafinitistic mathematics. This is exactly the same situation as, say, the use of Kripke models/realizability models/topos models for standard intuitionism: these are classical mathematics structures that provide a REPRESENTATION of intuitionism WITHIN the (supposedly solid) framework of classical mathematics. My personal goal is NOT to evangelize anybody on the absence of N (I could not care less); instead, I intend to provide a concrete Model Theory and a Proof Theory for Ultrafinitistic Mathematics that can be understood, studied and played with by ANYONE within the FOM community (as long as he/she knows some math, and has no irrational apriori prejudices). ----> Do you want to believe in your "crystal-clear jewel"? I say, no problem at all. Just do not try to suggest that your faith (and faith it is) should be necessarily accepted by everybody and do not assume that you can read other's minds (I can assure you that if you try to scan mine, you will get a very BIG head-ache!!!). As for myself, I have other plans: I am striving to show that it is at least conceivable to think coherently of a mathematical universe where the very notion of infinite is blurred, where the entire chain of large cardinals can be reproduced in the finite realm, where the essential theorems of FOL (completeness, incompleteness, Lowenheim-Skolem, etc, etc, etc.) can be reproduced at the "below-omega" scale, where.... In other words, my ultimate objective is to show that there is NO NEED whatever to banish anyone from Cantor's Paradise, because it can be miniaturized down to the FINITE REALM (this Program will be the topic of a future posting of mine, titled: Cantorian Nanotechnology). If such a program can be carried out, even in part, I honestly think that it would be of some substantial value to the FOM community at large, as it would force everybody to re-think WHERE the widespread belief in (or presumed knowledge of) INFINITY comes FROM. Sincerely Mirco A. Mannucci P.S. On Nelson: Edward Nelson is one of the prominent US mathematicians, with fundamental contributions to Analysis, Set Theory, Stochastic Calculus, Foundations of Quantum Mechanics, and God only knows what else (see Nelson' s recent celebration in the book- Diffusion, Quantum Theory, and Radically Elementary Mathematics- Princeton University Press, I believe). In my modest belief though, the single most important thing Nelson ever wrote is not his magnificient mathematics, not even his predicative arithmetics (not radical enough for ultrafinitism, by the way), but a small confession (available at http://www.math.princeton.edu/~nelson/papers/s.pdf), where he candidly tells the world when and where he lost his faith in N. I invite you and everyone on this list to read it (incidentally, according to your unfortunate definition, he is an "un-natural mathematician", at least since his apostasy (1976). Well, apparently he can still do excellent math, in spite of his avowed lack of belief in N. And, last time I checked, he did not turn into a lawyer or a doctor...) From friedman at math.ohio-state.edu Fri Nov 3 11:30:00 2006 From: friedman at math.ohio-state.edu (Harvey Friedman) Date: Fri, 03 Nov 2006 11:30:00 -0500 Subject: [FOM] PA Completeness/progress Message-ID: I have been able to prove the following. THEOREM. Let A be a sentence in 0,S,+,dot, with two quantifiers, in which all terms use at most two operation symbols. Then A is true iff A is provable in (a weak fragment of) PA. This class of sentences contains the axioms of PA except for the axiom scheme of induction. I believe that the following is within reach: CONJECTURE. Let A be a sentence in 0,S,+,dot, with three quantifiers, in which all terms use at most two operation symbols. Then A is true iff A is provable in (a weak fragment of) PA. This class of sentences also includes the associative law of addition. To include the distributive law, we make the following conjecture. CONJECTURE. Let A be a sentence in 0,S,+,dot, with three quantifiers, in which all terms use at most three operation symbols. Then A is true iff A is provable in (a weak fragment of) PA. This, also, will be proved with some effort. Harvey Friedman From friedman at math.ohio-state.edu Fri Nov 3 12:36:36 2006 From: friedman at math.ohio-state.edu (Harvey Friedman) Date: Fri, 03 Nov 2006 12:36:36 -0500 Subject: [FOM] PA Completeness/more Message-ID: The following adds to the growing completeness phenomena for PA. It goes in a somewhat different direction than the earlier recent postings of mine on this topic. THEOREM. Any universal sentence in 0,S,+,dot that is true in the reals is provable in PA. Any scheme in 0,S and unary schematic letters is true for all set interpretations of those letters if and only if every substitution by formulas in 0,S,+,dot is true in the natural numbers if and only if every substitution by formulas in 0,S,+,dot is provable in PA. THEOREM. PA can be axiomatized as follows. 1. All universal sentences in 0,S,+,dot true in the reals. 2. All schemes using only 0,S and unary schematic letters all of whose substitution instances in 0,S,+,dot are true. Note that these results do not treat sentences and schemes the same way. The completeness phenomena for PA will become deeper and deeper through sustained effort over many years, and will vindicate the idea that "PA does not miss any fundamental axioms and axiom schemes in its language". The same efforts will also prove dramatically fruitful for many other axiom systems that we use in f.o.m. This kind of paradigm shift is one of many other paradigm shifts that are needed to make real progress in f.o.m. Harvey Friedman From gstolzen at math.bu.edu Fri Nov 3 16:13:43 2006 From: gstolzen at math.bu.edu (Gabriel Stolzenberg) Date: Fri, 3 Nov 2006 16:13:43 -0500 (EST) Subject: [FOM] Reply to Rupert MCallum on Voplin and consistency In-Reply-To: <20061102213622.40941.qmail@web51903.mail.yahoo.com> References: <20061102213622.40941.qmail@web51903.mail.yahoo.com> Message-ID: Re a copy of Volpin's consistency proof manuscript, I would first try David Isles in the math department at Tufts. Also, if you know enough to do so (I don't), it might be helpful to explain to the members of the fom list how Alik thought he could "prove the unprovable" by making finer distinctions (of a certain kind) than traditionally is done. (E.g., distinguishing between the "near" and "distant" future, which I recall him claiming is a distinction that exists in Russian.) I myself never had more than a very sketchy idea of what Alik was up to. Or against. For someone seemingly (and, in some ways, truly) unworldly, Alik seemed very knowledgable. A long time ago, I asked both Alik and Kleene (not at the same time) for the simplest theorem that came to mind in the proof of which the law of excluded middle was really doing some work. Alik gave a reasonable answer. (I think he said the Intermediate Value Theorem.) Kleene did not. Which suggested to me that he hadn't thought about this before, whereas Alik had. In the proof of the theorem he chose, the law of excluded middle wasn't really doing any work. (I don't remember which theorem he offered but the proof was a fake proof by contradiction, like standard proof of the infinitude of primes or the traditional proof that square root of 2 is irrational.) Gabriel Stolzenberg P.S. My all-time favorite "proof" of the consistency of mathematics, which I learned from Fred Richman, goes like this: Suppose not. Then we have a contradiction. On Thu, 2 Nov 2006, Rupert McCallum wrote: > Yes, he claimed to have a proof of the consistency of ZF with any > finite number of inaccessible cardinals. Unfortunately it seems to be > hard to get hold of a copy of this proof. I would really like to know > in which axiomatic theory he claimed it could be done. > > --- Gabriel Stolzenberg wrote: > > Alik Volpin's "ultra-finitist" program had the aim of proving > > the consistency of mathematics. From tchow at alum.mit.edu Fri Nov 3 17:10:26 2006 From: tchow at alum.mit.edu (Timothy Y. Chow) Date: Fri, 3 Nov 2006 17:10:26 -0500 (EST) Subject: [FOM] Concerning Ultrafinitism. In-Reply-To: References: Message-ID: Bill Taylor wrote: > 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. This seems to be an unnecessarily disparaging view of fuzzy logic. It may not be particularly deep from a theoretical point of view, but this does not prevent the idea from being successful in engineering applications. The Wikipedia article is a pretty good introduction to the subject, listing some "practical applications" as well as clearing up certain misconceptions. In particular, until I see some more historical evidence, I'm quite skeptical about any alleged historical connection between fuzzy logic and ultrafinitism. http://en.wikipedia.org/wiki/Fuzzy_logic Tim Chow From V.Sazonov at csc.liv.ac.uk Fri Nov 3 19:41:37 2006 From: V.Sazonov at csc.liv.ac.uk (V.Sazonov@csc.liv.ac.uk) Date: Sat, 04 Nov 2006 00:41:37 +0000 Subject: [FOM] formalism In-Reply-To: <20061103090056.55275.qmail@web51912.mail.yahoo.com> References: <20061103090056.55275.qmail@web51912.mail.yahoo.com> Message-ID: <20061104004137.obrrwhh61w4skkc4@cgi.csc.liv.ac.uk> Quoting Rupert McCallum Fri, 03 Nov 2006: > I'm interested in understanding the formalist position better. One > version of it is that mathematics is "the science of formal systems", I consider myself to be a formalist. My understanding of formalism differs from the traditional caricature view assuming just a play with symbols and nothing else. Moreover, I have no idea about any real mathematician having a formalist view who would have such a caricature view. I believe that this is rather "invention" of platonists or so called "realists" (an awfully misleading term) to show that formalism is something stupid. Let me just repeat the following definition of mathematics from my last posting (I have presented in FOM various versions of this) which describes my formalist view: Mathematics, "by definition", deals with our (inevitably vague) imagination and fantasies governed/restricted/strengthened by FORMAL axioms and rules. This definition is very general, but it does not change the nature of mathematics. It only changes some traditional angle of view on mathematics - a wrong angle assuming beliefs in fictions having no scientific grounds. Of course this definition is not about the truth of mathematical results (which ever truth if we are dealing with fantasies and imaginations? recall, e.g. Imaginary Geometry of Lobatshevsky). > but then the question arises in what metatheory do we study these > formal systems. The above definition does not require any metatheory. Formal systems are assumed to be considered in a naive manner (to avoid the evident vicious circle) as I described in another recent posting answering to Timothy Y. Chow. Of course, any metatheoretical considerations are not excluded by this formalist view as it does not restrict mathematics in any way, just vice versa! Vladimir Sazonov ---------------------------------------------------------------- This message was sent using IMP, the Internet Messaging Program. From V.Sazonov at csc.liv.ac.uk Sat Nov 4 08:31:02 2006 From: V.Sazonov at csc.liv.ac.uk (V.Sazonov@csc.liv.ac.uk) Date: Sat, 04 Nov 2006 13:31:02 +0000 Subject: [FOM] Feasible consistency--Truth Transfer Policy In-Reply-To: References: Message-ID: <20061104133102.4kj3rd5gkcg0cg4g@cgi.csc.liv.ac.uk> Quoting Mirco Mannucci Sun, 29 Oct 2006: Dear Mirco, Feasible consistency means that the derivation of the contradiction is impossible if it has feasible number of symbols. (I would call this also the naive or even genuine understanding of formal consistency.) It could have some other versions, for example, that any derivation with feasible number of symbols in each participating term does not lead to a contradiction, or the like. When we discussed some measures like complexity of cut elimination or elimination of term abbreviations, I understood that this was about whether this measure will be of feasible value, just in correspondence with the above simple idea of feasible consistency. Now you suggest rather a measure which is a real number between 0 and 1 and that some Truth Transfer Principle (TTP) would "govern" this measure. This seems to me going far from the very idea of feasible consistency. Your examples of using such a TTP are non-convincing to me. The first example with the TTP for modus ponens based on taking max ( 0, min( p_(A-->B), p_A ) - E) for E=1/100 seems to me just ad hoc. Note that the predicate M which I suggested to consider as satisfying M(0), forall x (M(x) => M(x+1)) and not M(100) was not a primitive one (for any interested reader, it makes sense to look at http://cs.nyu.edu/pipermail/fom/2006-February/009746.html for the precise definitions). The point was that the trivial proofs of M(0), M(1), M(2),..., M(100) by using modus ponens are, in fact too complex in the sense of the measure mentioned above (of complexity of eliminations of abbreviations of terms), just *impossible*, for sufficiently large numbers even smaller than 100. The problem with your TTP in this case is that it is not related with the proof-theoretical measure I am considering. It is, in fact, related with my example only very superficially. You simply consider a quite different example and idea which I cannot relate with the idea of feasible consistency as I understand it. In your case p_M(100) = 0, whereas in my case the complexity measure of proving M(100) is "feasible infinity", and this is not so adequate in this context to consider 0 as 1/"feasible infinity". Say, in your example p_M(99) = 1/100 is quite different from 0 (this is quite far from to be considered as an "infinitesimal") whereas in my case M(99), M(98), etc are still "feasibly non-provable". The point is that neither the above form of your TTP nor some other its version you hinted at in your posting has anything general with (and cannot imitate) the proof-theoretic measure I consider. If you have a proof of the length like 100 above and apply any version of TTP to each derivation step, you will need to accumulate in some way the "credibility values" for about 100 times by some, basically, "uniform" and locally applied TTP. I do not see how anything like cut elimination can be grasped by such a procedure. What you suggest is just a quite different story. It is more alike to fuzzy logic theory (dealing with measures like you suggest in a local style) than to the idea of feasible consistency or may be posed somewhere in between of these two ideas but more close to fuzzy logic. As I remember, fuzzy logic deals with a special real number valued interpretation in [0,1] of logical operators whereas you suggest doing something similar with proof rules. This might be interesting, but, I repeat, this is rather not about feasible consistency - even in the case if nothing like cut elimination is assumed as a measure of (non)feasibility (i.e. just feasible length of proofs is required). Of course, you could suggest to consider also natural numbers as the measures of complexity instead of [0,1], but I still do not see how TTP (applied locally)would imitate the complexity of cut elimination what is essential for the effects like in the above example with M and 100 and for formalising the "true" feasibility. I repeat, what you suggest is rather a renewed version of fuzzy logic with the accent on proof rules instead (or alongside with??) that on logical operators. Just quite a different story in comparison with the approach via feasible consistency. Any analogy seems to me superficial. Best wishes, Vladimir Sazonov ---------------------------------------------------------------- This message was sent using IMP, the Internet Messaging Program. From friedman at math.ohio-state.edu Sat Nov 4 10:57:10 2006 From: friedman at math.ohio-state.edu (Harvey Friedman) Date: Sat, 04 Nov 2006 10:57:10 -0500 Subject: [FOM] 304: PA Completeness/strategy Message-ID: We begin with some general remarks concerning the paradigm shift represented by investigations in the style of the PA Completeness investigation. (This is not to say that there is anything illegitimate about traditional and ongoing PA Incompleteness investigations). We begin with the observation that many of our usual formal systems for f.o.m. are obtained by first specifying a language and then formulating as many "obvious natural fundamental axioms" as one can come with up within the specified language. F.o.m. is generally speaking absorbed by the various incompleteness phenomena. Of course, an exception is where one has completeness in the strong sense - e.g., the theory of real closed fields. But even here, there is substantial incompleteness phenomena of a more subtle nature connected with lengths of proof and computational complexity. In this paradigm shift, we concentrate instead on the various senses in which PA (and other systems) are complete. At the outset, we can expect to establish completeness in an important but special sense: *That any true fact in the language of 0,S,+,dot that is no more complicated than the usual axioms and axiom schemes for PA, is already provable in PA.* This is well within reach for a variety of very reasonable notions of "no more complicated". What is harder to obtain in a truly convincing way is **That any serious axiom or axiom (scheme) candidate in the language of 0,S,+,dot is already provable in PA.** Complexity measures on schemes are entirely natural and appropriate and fit in well with basic mathematical logic. For example, the axiom scheme A implies A is very simple, and is appropriately measured by the complexity of what is written. The latter will be obtained incrementally over long periods of time with results that are more and more compelling. But I don't want to rule out the stunning discovery of a truly convincing notion of "axiom (scheme) candidate" for which one can prove that any such is already provable in PA. Let me mention a fundamental point that is easily missed. Suppose that at some stage we consider a natural (finite) family of sentences in 0,S,+,dot that is sufficiently rich to contain well known open number theoretic conjectures. E.g., the twin prime conjecture. Of course, this will prevent us from actually showing that every true element of that family is already provable in PA (at least for a long time). HOWEVER, this is not really an obstacle to **. For it is completely obvious that ***The twin prime conjecture is not a serious axiom candidate***. Let me illustrate this point as follows. Suppose we can show that all true sentences in this rich family are provable in PA with a list of possible exceptions that we don't know the truth value of. We can then hope to argue in some convincing way that none of these exceptions are axiom candidates. If the only exception is the twin prime conjecture, then of course we are done. More realistically, we might show that these exceptions are all equivalent to well known open number theoretic problems and variants thereof. We would then want to argue that none of them are axiom candidates - and various arguments will be given, including "it it obvious that...". The above discussion lays the groundwork for the importance and potential importance of such completeness investigations for f.o.m. in a SUFFICIENTLY CONVINCING way. SUFFICIENTLY CONVINCING for what? Every such foundational program immediately leads to incremental formal investigations of great depth, ****THAT TAKE ON A LIFE OF THEIR OWN.**** E.g., Reverse Mathematics, Concrete Independence Results, Concept Calculus, and several others that have set up over the years. SUFFICIENTLY CONVINCING to appropriately motivate the pursuit of the associated formal investigations on the part of those with sufficient interest in f.o.m. and sufficient abilities. 1. A NEW FORMULATION. It is a challenge to come up with a complexity measure for sentences and sentence schemes in 0,S,+,dot which lead to powerful results of the kind we want than can actually be obtained. Currently we are most optimistic about the following complexity measure. *The number of occurrences of general operations*. In the case of sentences, the general operations are not, and, or, implies, iff, S,+,dot,=, forall, therexists. In the case of axiom schemes, we add the schematic symbols to this list. Let us look at the counts for the various axioms of PA. 1. (forall x)(not Sx = 0). 4. 2. (forall x)(forall y)(Sx = Sy implies x = y). 7. 3. (forall x)(x + 0 = x). 3. 4. (forall x)(forall y)(x+Sy = S(x+y)). 7. 5. (forall x)(x dot 0 = 0). 3. 6. (forall x)(forall y)(x dot Sy = (x dot y) + x). 7. 7. (forall x)(R(x) implies R(Sx)) implies (forall x)(R(x)). 8. Also (forall x)(forall y)(forall z)(x+(y+z) = (x+y)+z). 8. (forall x)(forall y)(forall z)(x dot y+z = x dot y + x dot z). 9. CONJECTURE. Every true sentence in 0,S,+,dot with general operation complexity at most 7 is provable in PA. This is also true for 8. This is also true for 9,10,11. The Twin Prime Conjecture appears to be the smallest under general operation complexity of the open problems, coming in at 12. For reasons given earlier we should not be automatically frightened of 12. There is another closely related complexity measure. Count the number of symbols, except for parentheses. This is the same as general operation complexity except that we add the number occurrences of variables and 0. Under this measure you get penalized 2 for each quantifier. The shape of the results ought to be the same, but the numbers are higher. CONJECTURE. Every instance of every true sentence scheme in 0,S,+,dot with general operation complexity at most 8 is provable in PA. This is also true for 9,10,11. 2. SOME PA COMPLETENESS. THEOREM 2.1. Any universal sentence in 0,S,+,dot that is true in the reals is provable in PA. Any scheme in 0,S and unary schematic letters is true for all set interpretations of those letters if and only if every substitution by formulas in 0,S,+,dot is true in the natural numbers if and only if every substitution by formulas in 0,S,+,dot is provable in PA. THEOREM 2.2. PA can be axiomatized as follows. 1. All universal sentences in 0,S,+,dot true in the reals. 2. All schemes using only 0,S and unary schematic letters all of whose substitution instances in 0,S,+,dot are true. Note that these results do not treat sentences and schemes the same way. In terms of the original motivation, this is not good. THEOREM 2.3. Let A be a sentence in 0,S,+,dot, with two quantifiers, in which all terms use at most two operation symbols. Then A is true iff A is provable in (a weak fragment of) PA. This class of sentences contains the axioms of PA except for the axiom scheme of induction. CONJECTURE. Let A be a sentence in 0,S,+,dot, with three quantifiers, in which all terms use at most two operation symbols. Then A is true iff A is provable in (a weak fragment of) PA. This class of sentences also includes the associative law of addition. To include the distributive law, we make the following conjecture. CONJECTURE. Let A be a sentence in 0,S,+,dot, with three quantifiers, in which all terms use at most three operation symbols. Then A is true iff A is provable in (a weak fragment of) PA. ********************************** I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable manuscripts. This is the 304th in a series of self contained numbered postings to FOM covering a wide range of topics in f.o.m. The list of previous numbered postings #1-249 can be found at http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM archives, 6/15/05, 9:18PM. NOTE: The title of #269 has been corrected from the original. 250. Extreme Cardinals/Pi01 7/31/05 8:34PM 251. Embedding Axioms 8/1/05 10:40AM 252. Pi01 Revisited 10/25/05 10:35PM 253. Pi01 Progress 10/26/05 6:32AM 254. Pi01 Progress/more 11/10/05 4:37AM 255. Controlling Pi01 11/12 5:10PM 256. NAME:finite inclusion theory 11/21/05 2:34AM 257. FIT/more 11/22/05 5:34AM 258. Pi01/Simplification/Restatement 11/27/05 2:12AM 259. Pi01 pointer 11/30/05 10:36AM 260. Pi01/simplification 12/3/05 3:11PM 261. Pi01/nicer 12/5/05 2:26AM 262. Correction/Restatement 12/9/05 10:13AM 263. Pi01/digraphs 1 1/13/06 1:11AM 264. Pi01/digraphs 2 1/27/06 11:34AM 265. Pi01/digraphs 2/more 1/28/06 2:46PM 266. Pi01/digraphs/unifying 2/4/06 5:27AM 267. Pi01/digraphs/progress 2/8/06 2:44AM 268. Finite to Infinite 1 2/22/06 9:01AM 269. Pi01,Pi00/digraphs 2/25/06 3:09AM 270. Finite to Infinite/Restatement 2/25/06 8:25PM 271. Clarification of Smith Article 3/22/06 5:58PM 272. Sigma01/optimal 3/24/06 1:45PM 273: Sigma01/optimal/size 3/28/06 12:57PM 274: Subcubic Graph Numbers 4/1/06 11:23AM 275: Kruskal Theorem/Impredicativity 4/2/06 12:16PM 276: Higman/Kruskal/impredicativity 4/4/06 6:31AM 277: Strict Predicativity 4/5/06 1:58PM 278: Ultra/Strict/Predicativity/Higman 4/8/06 1:33AM 279: Subcubic graph numbers/restated 4/8/06 3:14AN 280: Generating large caridnals/self embedding axioms 5/2/06 4:55AM 281: Linear Self Embedding Axioms 5/5/06 2:32AM 282: Adventures in Pi01 Independence 5/7/06 283: A theory of indiscernibles 5/7/06 6:42PM 284: Godel's Second 5/9/06 10:02AM 285: Godel's Second/more 5/10/06 5:55PM 286: Godel's Second/still more 5/11/06 2:05PM 287: More Pi01 adventures 5/18/06 9:19AM 288: Discrete ordered rings and large cardinals 6/1/06 11:28AM 289: Integer Thresholds in FFF 6/6/06 10:23PM 290: Independently Free Minds/Collectively Random Agents 6/12/06 11:01AM 291: Independently Free Minds/Collectively Random Agents (more) 6/13/06 5:01PM 292: Concept Calculus 1 6/17/06 5:26PM 293: Concept Calculus 2 6/20/06 6:27PM 294: Concept Calculus 3 6/25/06 5:15PM 295: Concept Calculus 4 7/3/06 2:34AM 296: Order Calculus 7/7/06 12:13PM 297: Order Calculus/restatement 7/11/06 12:16PM 298: Concept Calculus 5 7/14/06 5:40AM 299: Order Calculus/simplification 7/23/06 7:38PM 300: Exotic Prefix Theory 9/14/06 7:11AM 301: Exotic Prefix Theory (correction) 9/14/06 6:09PM 302: PA Completeness 10/29/06 2:38AM 303: PA Completeness (restatement) 10/30/06 11:53AM Harvey Friedman From rgheck at brown.edu Sat Nov 4 12:24:38 2006 From: rgheck at brown.edu (Richard Heck) Date: Sat, 04 Nov 2006 12:24:38 -0500 Subject: [FOM] Concerning Ancestral Logic In-Reply-To: <200611030235.kA32ZKC0000215@math.canterbury.ac.nz> References: <200611030235.kA32ZKC0000215@math.canterbury.ac.nz> Message-ID: <454CCCD6.7020901@brown.edu> The question was mainly for Arnon, but let me add a bit. Shapiro does discuss AL in his book, around p. 227. The characterization he gives there is purely semantical. Here's one way to do it. Introduce a new symbol *. Given a formula \phi, *xy(\phi(x,y))(a,b) is a formula, where x and y are variables and a and b are terms. It is true iff there is a finite sequence x_1, ..., x_n such that \phi(x_i,x_{i+1}). (Of course, satisfaction is what we really need.) Note that this is Frege's `strong' ancestral: We do not, in general, have *xy(\phi(x,y))(a,a). Since the ancestral is definable by a \Pi^1_1 formula, AL is a sub-system of \Pi^1_1 second-order logic. I believe it's a proper sub-system but I'm not sure I'm remembering that corrrectly. In any event, it follows that AL is both incomplete and non-compact, since arithmetic therefore has a categorical formulation in AL. Define the `weak' ancestral thus: *=xy(\phi(x,y))(a,b) iff *xy(\phi(x,y))(a,b) \vel a = b Then induction takes the form: \forall n[*=xy(y=Sx)(0,n)] Just as Dedekind and Frege taught. That AL is incomplete does not, of course, imply that there are not nice partial axiomatizations of it, just as there are nice partial axiomatizations of second-order logic. I discuss the question how to axiomatize it in my paper "The Logic of Frege's Theorem", available from my web site. Richard Heck Bill Taylor wrote: > This query is mainly directed to Arnon Avron, but no doubt others > may feel like joining in. > > Arnon - you have been extolling the virtues of AL as an alternative, > (an extension?), of FOL. It sounds good, but I don't yet know what it is. > You mention it was given a treatment in Shapiro's F-without-F, but I don't > recall noticing it from when I read that. > > So I wonder if you could give us all a brief rundown on it. > Nothing too massively technical, but just enough to get the flavour > and main ideas. I was thinking of, say, 2 or 3 standard paragraphs. > > Could you do that please? Thanks muchly, > > Bill Taylor > > _______________________________________________ > FOM mailing list > FOM at cs.nyu.edu > http://www.cs.nyu.edu/mailman/listinfo/fom > -- ================================================================== Richard G Heck, Jr Professor of Philosophy Brown University http://bobjweil.com/heck/ ================================================================== Get my public key from http://sks.keyserver.penguin.de Hash: 0x1DE91F1E66FFBDEC Learn how to sign your email using Thunderbird and GnuPG at: http://dudu.dyn.2-h.org/nist/gpg-enigmail-howto From T.Forster at dpmms.cam.ac.uk Sat Nov 4 15:15:49 2006 From: T.Forster at dpmms.cam.ac.uk (Thomas Forster) Date: Sat, 4 Nov 2006 20:15:49 +0000 (GMT) Subject: [FOM] Concerning Ultrafinitism In-Reply-To: References: Message-ID: I've often thought that Borges was a mathematician manqu\'e. He even has his own version of the Liar Paradox. "In Sumatra, someone wishes to receive a doctorate in Prophecy. The master seer who administers his exam asks if he will fail or pass. The candidate replies that he will fail.." Review of Kasner & Newman *Mathematics and the Imagination* (1940) Can anyone in this list tell us how much Mathematics Borges actually knew? On Fri, 3 Nov 2006, Mirco Mannucci wrote: > > --Hay un concepto que es el corruptor y el desatinador de los otros. No > hablo del Mal, cuyo limitado imperio es la Etica; hablo del infinito-- > > (There is a concept which corrupts and upsets all others. I refer not to > Evil,whose limited realm is that of Ethics; I refer to the infinite) > > Jorge Luis Borges > -- www.dpmms.cam.ac.uk/~tf; Home phone +44-1223-704452. work: +44-1223-337981. Mobile in UK +44-7887-701-562 From Andrej.Bauer at fmf.uni-lj.si Sat Nov 4 16:58:34 2006 From: Andrej.Bauer at fmf.uni-lj.si (Andrej Bauer) Date: Sat, 04 Nov 2006 22:58:34 +0100 Subject: [FOM] 303: PA Completeness (restatement) In-Reply-To: References: <454682C0.1090509@fmf.uni-lj.si> Message-ID: <454D0D0A.1030805@fmf.uni-lj.si> Pietro Kreitlon Carolino wrote: > I would like to know the proportion of ExEy sentences that came up > true in professor Bauer's Mathematica run. I did not actually count how many were true and how many false. For example, we can throw out anything that has just one variable or is within Presburger arithmetic without deciding whether it's true or false. So, I made another computation which does what you asked for. You can see the results in the Mathematica notebook (also available in PDF), which I published at http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/ in order not do send attachments to FOM. I should point out that there are many different ways of counting here. For example, is "x + S 0" the same thing as "S x"? Should we put equations in canonical form before we start counting them? I indicate some possibilities in the Mathematica notebook. I hope other FOM readers find this interesting, too. Andrej Bauer From rupertmccallum at yahoo.com Sat Nov 4 19:38:59 2006 From: rupertmccallum at yahoo.com (Rupert McCallum) Date: Sat, 4 Nov 2006 16:38:59 -0800 (PST) Subject: [FOM] formalism In-Reply-To: <20061104004137.obrrwhh61w4skkc4@cgi.csc.liv.ac.uk> Message-ID: <20061105003900.57469.qmail@web51909.mail.yahoo.com> --- V.Sazonov at csc.liv.ac.uk wrote: > > The above definition does not require any metatheory. Formal systems > are assumed to be considered in a naive manner (to avoid the evident > vicious circle) as I described in another recent posting answering to > But what statements are accepted as known from this naive point of view? Edward Nelson considers the consistency of Robinson Arithmetic to be an open problem. Do you? Supposing a dispute arose between two formalists about that issue, wouldn't it have to be settled by a choice of metatheory? ____________________________________________________________________________________ Access over 1 million songs - Yahoo! Music Unlimited (http://music.yahoo.com/unlimited) From rgheck at brown.edu Sat Nov 4 22:49:32 2006 From: rgheck at brown.edu (Richard Heck) Date: Sat, 04 Nov 2006 22:49:32 -0500 Subject: [FOM] Concerning Ancestral Logic (CORRECTION) In-Reply-To: <200611030235.kA32ZKC0000215@math.canterbury.ac.nz> References: <200611030235.kA32ZKC0000215@math.canterbury.ac.nz> Message-ID: <454D5F4C.8040505@brown.edu> Silly omission in my previous note, one people could probably see and fix themselves, but.... ===== Given a formula \phi, *xy(\phi(x,y))(a,b) is a formula, where x and y are variables and a and b are terms. It is true iff there is a finite sequence x_1, ..., x_n such that \phi(x_i,x_{i+1}) AND a=x_1 AND b=x_n. Etc. -- ================================================================== Richard G Heck, Jr Professor of Philosophy Brown University http://bobjweil.com/heck/ ================================================================== Get my public key from http://sks.keyserver.penguin.de Hash: 0x1DE91F1E66FFBDEC Learn how to sign your email using Thunderbird and GnuPG at: http://dudu.dyn.2-h.org/nist/gpg-enigmail-howto _______________________________________________ FOM mailing list FOM at cs.nyu.edu http://www.cs.nyu.edu/mailman/listinfo/fom From slaterbh at cyllene.uwa.edu.au Sun Nov 5 05:24:43 2006 From: slaterbh at cyllene.uwa.edu.au (Hartley Slater) Date: Sun, 5 Nov 2006 18:24:43 +0800 Subject: [FOM] Concerning Ultrafinitism Message-ID: Mirco Mannucci writes >In my modest belief though, the single most important thing Nelson >ever wrote is not his >magnificient mathematics, not even his predicative arithmetics (not >radical enough >for ultrafinitism, by the way), but a small confession (available at >http://www.math.princeton.edu/~nelson/papers/s.pdf), where he candidly tells >the world when and where he lost his faith in N. I have read this paper ('Syntax and Semantics'), and can readily see why it should be thought important. Having found that the procedures Hilbert put in place cannot prove that 2+3=5, for instance, (because that would involve giving a semantics to the metamathematical syntax) the novel idea is to deny that there is such a semantically based fact, enabling one to continue guiltlessly with Hilbert's line of research into what are then regarded as the only proper facts - syntactic ones. Gone is the thought that there might be a conclusive proof that 2+3=5 some other way, that might be a more appropriate object for the Foundations of Mathematics to locate and investigate. So 'Hilbert's plane geometry is flawless' (Nelson, p3)?? Certainly it meticulously avoided any statement about lines and planes and was scrupulously concerned merely with the words 'lines' and 'planes', in a certain accurately deductive context. But that shows that it was not in the same business as Euclid, not that it improved on that business. At one time I thought that a use-mention confusion was endemic in foundational studies, when metamathical investigations about uninterpreted words were thought to be establishing, on a firmer basis, the older, mathematical truths, which involved words that were interpreted. And maybe that was the case in the early part of the 20th century. But the more modern trend, illustrated in this paper by Nelson, and some other messages to FOM recently, clearly involves, instead, just the total abnegation of use. -- Barry Hartley Slater Honorary Senior Research Fellow Philosophy, M207 School of Humanities University of Western Australia 35 Stirling Highway Crawley WA 6009, Australia Ph: (08) 6488 1246 (W), 9386 4812 (H) Fax: (08) 6488 1057 Url: http://www.philosophy.uwa.edu.au/staff/slater From jmarcos at dimap.ufrn.br Sun Nov 5 10:16:51 2006 From: jmarcos at dimap.ufrn.br (Joao Marcos) Date: Sun, 5 Nov 2006 12:16:51 -0300 Subject: [FOM] Borges and Mathematics Message-ID: > Can anyone in this list tell us how much Mathematics Borges actually knew? Borges was certainly not a "mathematician" stricto senso, but it seems he knew more than enough about modern mathematics so as to produce several stories in which mathematical concepts are anything but central ("The Aleph", "The Library of Babel", "The Book of Sand", "The Garden of Forking Paths"). A nice book on the matter called "Borges y las Matem?ticas" was written not long ago by Guillermo Mart?nez (http://guillermomartinez.8m.net/CV/publicacionesingles.htm), a competent Argentinian academic logician and best-selling novelist. It can be bought through Amazon. There is also a book called "Unthinking Thinking. Jorges Luis Borges, Mathematics, and the New Physics", by Floyd Merrell, but that one I haven't read. There is probably as much of modern mathematics and philosophy to be found in Borges as there is of kabbala and mythology. His whole oeuvre, full of mirrors, labyrinths and infinity, is an exploration into paradox. Best, Joao Marcos -- My homepages: http://geocities.com/jm_logica/ http://www.dimap.ufrn.br/~jmarcos/ http://slc.math.ist.utl.pt/jmarcos.html From mmannucc at cs.gmu.edu Sun Nov 5 12:31:52 2006 From: mmannucc at cs.gmu.edu (Mirco Mannucci) Date: Sun, 5 Nov 2006 12:31:52 -0500 (EST) Subject: [FOM] Nelson's ultraformalism --to Slater Message-ID: ---- Hartley Slater wrote: > I have read this paper ('Syntax and Semantics'), and can readily see > why it should be thought important. Actually, I ought to apologize, I am guilty of pointing everyone to the WRONG paper: the one I had in mind is called "Mathematics and Faith" http://www.math.princeton.edu/~nelson/papers/faith.pdf and the amazing "confession" I was referring to is at page 7, beginning with the sentence: --I must relate how I lost my faith in pythagorean numbers--. Anyway, the one you are discussing is along the same lines: Nelson, if I understand him at all, essentially proposes a rigorous ultra-formalism. ---> Ultra-formalism = everything (in math) is syntax, syntax and nothing but syntax I think that this is a FOM challenge for everybody, and I take it very seriously (to the point of taking Nelson's logic itself to task, when he proposes his own brand of finitistic arithmetics and finitistic set theory). I also think that there are a few interesting topics to ponder here. For instance: 1) can we keep a formalist approach in a coherent way, not just at the mathematical level, but also at the meta-mathematical one? (stopping finally the meta-delusion that there is a meta-level "above" math) 2) can we give a positive interpretation of the incompleteness phenomena in the ultra-formalist approach? In other words, can we account for Godel's incompleteness as saying something ABOUT our syntactical games, as opposed to referring to some "intended" pre-built platonic structures? (the negative interpretation, saying--they are great theorems, but they mean nothing---, is the cheap way out) 3) how does the math world looks to a ultra-formalist that does not allow himself/herself the (expensive!) luxury of naively using infinitary arguments of any type and shape? I have my own view (which I shall present in due time), but I would like to hear from others on 1)--> 3) Best Wishes Mirco A. Mannucci From corcoran at buffalo.edu Sun Nov 5 14:35:12 2006 From: corcoran at buffalo.edu (John Corcoran) Date: Sun, 5 Nov 2006 14:35:12 -0500 Subject: [FOM] FOM: BUFFALO LOGIC COLLOQUIUM 4TH FALL ANNOUNCEMENT Message-ID: <000701c70111$887b1cf0$cff7cd80@Alfred> BUFFALO LOGIC COLLOQUIUM http://www.philosophy.buffalo.edu/EVENTS/blc.htm 2006-7 THIRTY-SEVENTH YEAR FOURTH FALL ANNOUNCEMENT QUOTES OF THE MONTH: LEWIS ON MATHEMATICAL SYSTEMS: A mathematical system is any set of strings of recognizable marks in which some of the strings are taken initially and the remainder derived from these by operations performed according to rules which are independent of any meaning assigned to the marks. C. I. Lewis, 1918, 355. TARSKI ON FORMAL LANGUGES: We are not interested here in ?formal? languages to the expressions of which no meaning is attached. For such languages the problem here discussed has no relevance, it is not even meaningful. Tarski 1933/1956, 166. CORCORAN ON LEWIS, TARSKI, AND ZOMBIES. Is it an exquisite irony that 15 years later Tarski published a completely meaningful axiomatization of string theory, a theory about possibly meaningless strings? And in the same paper he made the concept of truth respectable and saved logic from the skeptics and the zombies. John Corcoran 2006. NOTE TIME CHANGE THIRD MEETING Friday, November 10, 2006 1:30-3:30 P.M. 141 Park Hall SPEAKER: Daniel Merrill, Philosophy, Oberlin College. COMMENTATOR: John Corcoran, Philosophy, University of Buffalo. TITLE: De Morgan?s Ways of Construing the Syllogism. ABSTRACT: Augustus De Morgan's logical work seems to have been constrained by a fixation on tinkering with the traditional syllogism. Nevertheless, he introduced three logical innovations which go far beyond the syllogism. What is notable is that the syllogism emerges as a special case of each approach and that each ends up construing the syllogism in a different way. The three innovations are: the logic of complex terms (Boolean algebra), the numerically definite syllogism, and the logic of relations. All are found in his FORMAL LOGIC (1847), though the logic of relations is only developed fully later on. This talk will outline the innovations, and discuss critically the ways in which De Morgan embeds the traditional syllogism within them. Dutch treat supper 6:30 pm at the Family Tree. Please try to let Corcoran know you are coming. Future Speakers: George Boger (Canisius College), William Demopoulos (University of Western Ontario and UC-Irvine), William Rapaport (University of Buffalo), Jos? Miguel Sag?illo (University of Santiago de Compostela), Stewart Shapiro (Ohio State University), Barry Smith (University of Buffalo). Sponsors: Some meetings of the Buffalo Logic Colloquium are sponsored in part by the C. S. Peirce Professorship in American Philosophy and by other institutions. All are welcome. To receive this via email, please send your full name and email address to John Corcoran. For further information, to report glitches, suggest a talk, unsubscribe or make other suggestions, please email: John Corcoran: corcoran at buffalo.edu - From zahidi at logique.jussieu.fr Mon Nov 6 04:52:02 2006 From: zahidi at logique.jussieu.fr (zahidi@logique.jussieu.fr) Date: Mon, 6 Nov 2006 10:52:02 +0100 (CET) Subject: [FOM] Consistency Message-ID: <1161.146.175.207.172.1162806722.squirrel@webmail.logique.jussieu.fr> Dear FOM-members I was asked the following question: "PA does not prove its own consistency. Are there other formal systems which do prove their own consistency?" I had to admit that I did not know the answer to this question, but that it seemed unlikely to me. It seems that just to formalize the sentence "i'm consistent" one would need a lot of coding techniques, which in their turn would require the fact that the formal system under scrutiny has elementary arithmetic inside it. Do any of the FOM-members know more about this question and/or the validity of the above sketched argument. Best regards Karim Zahidi From tchow at alum.mit.edu Sun Nov 5 17:57:15 2006 From: tchow at alum.mit.edu (Timothy Y. Chow) Date: Sun, 5 Nov 2006 17:57:15 -0500 (EST) Subject: [FOM] First-order arithmetical truth In-Reply-To: References: Message-ID: mario wrote: > 1. Do you argue for [1]->[3] above along the following (one line longer) > sketch: > > if (1) somebody lacks the ability to distinguish the intended model > of PA from another model, > then (2) (s)he lacks the ability to distinguish the intended model of > meta-PM from another model, > then (3) (s)he lacks the ability to distinguish the intended meaning of > the term "formal system" from some other meaning. Not really. The argument is just that any skeptical argument about the natural numbers translates easily into a skeptical argument about formal systems. One particular skeptical argument about the natural numbers might go as follows: "I lack the ability to understand precisely what some alleged mathematical entity is unless you can show me a first-order language for it and a set of first-order axioms with the property that the alleged mathematical entity is the unique structure satisfying the axioms. There is no such set of first-order axioms for N in the language of arithmetic. So I don't know what N is." The translation is: "I lack the ability to understand precisely what some alleged mathematical entity is unless you can show me a first-order language for it and a set of first-order axioms with the property that the alleged mathematical entity is the unique structure satisfying the axioms. There is no such set of first-order axioms for PA in the language of syntax. So I don't know what PA is." Here "language of syntax" is something like your "meta-PM." Maybe you don't think much of the above skeptical argument about the natural numbers, and think that there is some other skeptical argument that is more convincing. Fine. I'm sure that your favorite argument can also be translated, since it's very straightforward to pass between assertions about numbers and assertions about strings. Tim From examachine at gmail.com Sun Nov 5 17:46:35 2006 From: examachine at gmail.com (Eray Ozkural) Date: Mon, 6 Nov 2006 00:46:35 +0200 Subject: [FOM] Nelson's ultraformalism --to Slater In-Reply-To: References: Message-ID: <320e992a0611051446j1fd566f4v4c3348f9e18749ea@mail.gmail.com> On 11/5/06, Mirco Mannucci wrote: > 1) can we keep a formalist approach in a coherent way, not just > at the mathematical level, but also at the meta-mathematical one? (stopping > finally the meta-delusion that there is a meta-level "above" math) [In the below, I drop "I think that" in some places, but all of the below must be read as my ideas.] I think that a computationalist approach may fare better at this ambitious goal. Here is why. Abstract concepts help us predict physical events better. However, all abstract concepts are computational. There is no use for any abstract concept that does not lend to computation. Therefore, we imagine mathematics to form by self-reflective processes which have started finding out about facts of computation. Proving 2+3=5 is a shortcut, short of counting, which is a computation. In AI community, shortcuts have sometimes been called "chunking". However, those approaches are unfortunately not capable of inventing the ordinary kind of arithmetic (PA). Computational basis for such capability on the other hand might arise from the application of inductive inference (Reference available upon request). It is unknown at the present if this is the case. Note also the relation to the halting problem, which seems to correspond to the "golden standard" for computational problems. The theorem 2+3=5 is an easy instance of the halting problem, we can show a program that halts iff that is the case. On the other hand, a little longer program can turn out to be very hard (like twin primes conjecture). However, I think the (ultra) formalist position does not necessitate getting rid of potential infinity, because in my view the concept of potential infinity has a crystal clear formalization which is a non-terminating program, that everybody can objectively examine. I see no magic with that, nor any assumption of Pythagorean philosophy (as well as Platonism). It is false that numbers existed in some realm before we thought them up. However, it is true that they exist in a real sense of the word, as representations in our heads. This is a formalist position, but it does not strip numbers of their intended meaning (which is explained somewhat satisfactorily by the above use-theory of meaning). Thus, potential infinity seems to remain as an idealization that is often encountered in physical theory. Moreover, there are no gaps in our reasoning when we make this idealization. In particular, if we return to the twin primes conjecture, it seems innocent enough to introduce an open ended loop to our program. Why should we limit ourselves to only certain kinds of programs? They are all the same to a computer. And furthermore, how are we going to know which instance of the halting problem is admissible and which is not? (This may be a stronger objection than it seems, since instances of the halting problem do not necessarily refer to such abstract concepts as integers. They are simply strings of symbols to a computer.) Also, how are we going to know these upper bounds? Thus, what I mean is, we need not look at what lies above, but what lies beneath, in our heads. On the other hand, ultimately, all meta-theory must be formalized accordingly. I believe some formal meta-mathematical approaches comply with such restriction. I am leaving the matter of actual infinity untouched, because that is a topic that attracts too much debate and too few conclusions. Finally: > 2) can we give a positive interpretation of the incompleteness phenomena in > the ultra-formalist approach? In other words, can we account for Godel's > incompleteness as saying something ABOUT our syntactical games, as opposed > to referring to some "intended" pre-built platonic structures? > (the negative interpretation, saying--they are great theorems, but > they mean nothing---, is the cheap way out) Is not this already achieved by Kolmogorov complexity theory? I know that there has been widespread underrating of the theory, in favor of saving the day for Platonic forms, which do not exist at any rate. In particular, some criticisms contain seriously flawed assumptions about computation. More explanation available upon request, as that subject is beyond the scope of the present post. However, I will just state that there are incompleteness theorems that are strictly about computational facts. I believe that counts as syntactical enough. What do you have in your mind when you say that? Do you want to state Godel's theorem in another way? Or do you want to prove a new kind of incompleteness theorem? Regards, PS: Also, what do you think of Godel's statements when he said that the second incompleteness theorem is also valid for finite systems. This seems to weaken the finitist position a little, if we listen to Godel. PS2: I am hoping that this will finally make some sense to the moderators. I had expressed that it is best to understand set theory in terms of computations. I can and will explain in much finer detail any of the above points which hope to reconcile formalism and psychologism (the latter termn has been used by Godel). PS3: I apologize in advance if my ideas offend proponents of the Pythagorean view. -- Eray Ozkural, PhD candidate. Comp. Sci. Dept., Bilkent University, Ankara ai-philosophy: http://groups.yahoo.com/group/ai-philosophy From slaterbh at cyllene.uwa.edu.au Sun Nov 5 22:02:56 2006 From: slaterbh at cyllene.uwa.edu.au (Hartley Slater) Date: Mon, 6 Nov 2006 11:02:56 +0800 Subject: [FOM] Concerning Ultraformalism Message-ID: Mirco Mannucci points me/us towards another paper by Nelson, 'Mathematics and Faith' in his 'Concerning Ultraformalism --to Slater', but I am afraid I find Nelson's dream of an 'overwhelming presence' in that paper less worthy of addressing. Mannucci goes on to ask the crucial question, however: >1) can we keep a formalist approach in a coherent way, not just >at the mathematical level, but also at the meta-mathematical one? It is its formalism that has led the tradition to the, now very pressing problems it has had with Truth, and particularly The Liar and associated paradoxes, as I have pointed out in several previous FOM postings both this year and before. The recent books by McGee, Soames, etc, and the lifetime devotion of Priest to problems in this area, all arise, on my understanding of the matter, because the logical language that has been employed since Frege has lacked (primarily) any symbolism for 'that' clauses, in terms of which semantic remarks like 's means that p' can be formalised. So formalised treatments of issues in the area have ended up being entirely 'syntactic'. Frege, of course, fought with Hilbert expressly on the need for Hilbert's symbols to express thoughts - and Frege's 'horizontal', in at least one of its uses, could be taken to be a formalisation of the needed sentence nominaliser 'that'. But the formal tradition from Frege has tried to do without any such operator, and now has to swear blind that none is needed, to preserve its professional standing. But it can only do so at the cost of admitting that the problems with Truth and The Liar are intractable. "After all the work that has been done, there cannot be a solution to the Liar and Strengthened Liar in just two paragraphs!", has to be the thought. Indeed, there cannot be a formal solution to these problems at all, if one does not incorporate into one's formal language the ability to say what it means. -- Barry Hartley Slater Honorary Senior Research Fellow Philosophy, M207 School of Humanities University of Western Australia 35 Stirling Highway Crawley WA 6009, Australia Ph: (08) 6488 1246 (W), 9386 4812 (H) Fax: (08) 6488 1057 Url: http://www.philosophy.uwa.edu.au/staff/slater From W.Taylor at math.canterbury.ac.nz Sun Nov 5 23:20:02 2006 From: W.Taylor at math.canterbury.ac.nz (Bill Taylor) Date: Mon, 06 Nov 2006 17:20:02 +1300 (NZDT) Subject: [FOM] Concerning Ultrafinitism. Message-ID: <200611060420.kA64K2m7012478@math.canterbury.ac.nz> galathea wrote: > because they insist mathematics is a physical process > and one day too may suffer the entropic decay This is as good a place as any to make an observation. Ultrafinitists place a lot of significance on *what can physically be done* in the way of numbers, proofs, etc. The above quote gives another view of basically the same idea. The whole tradition of math for over 2500 years has been to abstract away from physical matters, whether entropy, the fine structure constant, the state of human biology and brain power, the state of calculator technology, and so on. Sure, all these things are relevant to what we may ever achieve in math; but not at all to what's IN math itself. Or so almost all have viewed it, for (as I say) so long a time. These other matters are highly relevant to other disciplines, such as sociology, psychology, the history of math, physics, engineering, medicine and (especially!) computer science. But they hardly seem relevant to MATH ITSELF. The above view (not necessarily Platonist) is almost completely standard among mathematicians of most stripes. Math is about abstractions, and abstraction, NOT physicality. What, (other than ideological correctness), can possibly be gained (for math) by these physicalist views? I can see gain to physics, CS, etc... but to MATH? W F C Taylor From V.Sazonov at csc.liv.ac.uk Mon Nov 6 13:03:24 2006 From: V.Sazonov at csc.liv.ac.uk (V.Sazonov@csc.liv.ac.uk) Date: Mon, 06 Nov 2006 18:03:24 +0000 Subject: [FOM] formalism In-Reply-To: <20061105003900.57469.qmail@web51909.mail.yahoo.com> References: <20061105003900.57469.qmail@web51909.mail.yahoo.com> Message-ID: <20061106180324.lg27ldd28g8gcgg0@cgi.csc.liv.ac.uk> Quoting Rupert McCallum Sun, 05 Nov 2006: >> The above definition does not require any metatheory. Formal systems >> are assumed to be considered in a naive manner (to avoid the evident >> vicious circle) as I described in another recent posting answering to I think, I should clarify, that no metatheory is necessary to play with formal rules. Just like children play with Lego, domino, like we use key to unlock the door etc. ? quite naively. > But what statements are accepted as known from this naive point of > view? If you succeeded to derive formally a theorem you definitely know about this using only a quite naive understanding what your formal system is. Additionally, you might have some idea what this formal system is about: you are also able to see (quite informally) that the axiom and proof rules (roughly) correspond to your imagination on the "world" this theory is "describing". You will probably conclude that derived theorems fit well in your picture of this imaginary "world". Otherwise, if something will go not according to your intuition and expectations you will think that it is an exception, like well-known counterexamples in analysis. Most probably, you will not consider these exceptions as sufficient reason to change the formal system and will learn how to co-exist with these exceptions. May be you even will eventually find these exception also natural in a sense. May be the formal system will somewhat change your intuition and vision of this imaginary "world". That is normal when your intuition is governed by formal rules. That is basically all. In this sense formalism is even not about consistency of the formal systems you are working with. You just see that it agrees with your imagination (at least to some degree). You can discuss this with other people by appealing to their intuition. Probably you will find that you understand one another. Probably the illusion will appear that you are discussing about "the same" imaginary world (because you ground your discussion and intuition on the same formal system or a similar range of formal systems) and because your fantasies are somewhat related with (or are natural extrapolations from) our joint REAL WORLD. If you want to prove consistency of one formal system in another one - do this as usually. The full existing mathematical practice (and probably much more) is included in this picture. Edward Nelson considers the consistency of Robinson Arithmetic to > be an open problem. Do you? In principle, any sufficiently nontrivial formal system can be suspected to be contradictory, even if we have some imaginary world which this system seemingly describes. But our imagination is, in general, so vague. In some cases it seems solid enough and we are inclined to believe that the formal system is therefore consistent. But nobody can give a full, absolute guarantee. Any proofs of consistency are based on some other, stronger formalisms. This gives only a relative guarantee and depends on some (may be subjective) preferences to consider some theories as having a better intuitive background and believed as more reliable. For me personally, it is sufficient to prove consistency of a new formal system in ZFC. If it seems impossible, the intuition plays the role of a guarantee. Another thing, if by some reason I am interested in fantasies of a special kind such as feasibility. Supposing a dispute arose between two > formalists about that issue, wouldn't it have to be settled by a choice > of metatheory? If both like the same metatheory. . . But formalist point of view, in my understanding, is not and should not be related with some specific preferences to such or other formalisms and intuitions. It is quite a general view - a full freedom, except always being based on formal systems (once it is called mathematics, once it is not just a free art). It seems to me that Nelson's preferences to some formal systems and and his intuitions on arithmetic are not a direct consequence of his formalist view on mathematics. Rather formalist view is a necessary (or desirable) condition for taking his preferences (related with some doubts in natural numbers). Formalist is not necessarily an ultrafinitist, or the like. It is just a general view on mathematics in its widest sense. Vladimir Sazonov ---------------------------------------------------------------- This message was sent using IMP, the Internet Messaging Program. From Helene.Boucher at wanadoo.fr Mon Nov 6 16:13:22 2006 From: Helene.Boucher at wanadoo.fr (Andrew Boucher) Date: Mon, 6 Nov 2006 22:13:22 +0100 Subject: [FOM] Consistency In-Reply-To: <1161.146.175.207.172.1162806722.squirrel@webmail.logique.jussieu.fr> References: <1161.146.175.207.172.1162806722.squirrel@webmail.logique.jussieu.fr> Message-ID: <0B25D9ED-7818-4813-8B2A-3778D3F30A48@wanadoo.fr> On 6 Nov 2006, at 10:52 AM, zahidi at logique.jussieu.fr wrote: > Dear FOM-members > > I was asked the following question: > "PA does not prove its own consistency. Are there other formal systems > which do prove their own consistency?" The answer is "yes". 1/ Jeroslow ("Consistency Statements in Formal Mathematics", Fundamenta Mathematicae, vol. 72, 1971, pp. 17-40) used a particular fixed point extension of Q v (x)(y)(x = y). 2/ Dan Willard ("Self-verifying axioms systems, the Incompleteness Theorem and Related Reflection Principles, JSL, Vol 66, 2001, pp. 536-596) has presented several cases, based on seven "grounding" functions. He also has a more recent article in the JSL, as I recall, also on this topic. 3/ Yvon Gauthier ("The internal Consistency of Arithmetic with Infinite Descent", Modern Logic, Vol 8, no 1/2, Jan 1998-Apr 2000, pp. 47-86) presented a system with a special quantifier, called "effinite". There is also a system that I have looked at, which is second-order Peano Arithmetic without the Successor Axiom, for instance with this axiomatization: (a) (n)(m)(m')(Nn & Sn,m & Sn,m' => m = m') (b) (n)(m)(n')(Nn & Nn' & Sn,m & Sn',m => n = n') (c) (n)(Nn => not Sn,0) (d) Induction schema. See http://www.andrewboucher.com/papers/arith-succ.pdf. > > I had to admit that I did not know the answer to this question, but > that > it seemed unlikely to me. It seems that just to formalize the sentence > "i'm consistent" one would need a lot of coding techniques, which > in their > turn would require the fact that the formal system under scrutiny has > elementary arithmetic inside it. > I think you are in good company with this intuitive argument. Here's why it does not hold. Predicates like Proof(x,y) can be defined by referencing only numbers less than or equal to x and y. One therefore does not need the Successor Axiom which says that every number has a successor, which ensures that for every number there exist greater numbers; one only needs the fact that, given any number, there exist smaller numbers. So one does not need all of the Peano Axioms in order to set up coding. In brief, while elementary arithmetic may be needed for coding, Peano Arithmetic is more than the elementary required. From cfranks at nd.edu Mon Nov 6 16:55:04 2006 From: cfranks at nd.edu (Curtis Franks) Date: Mon, 06 Nov 2006 16:55:04 -0500 Subject: [FOM] Zahidi's question on consistency Message-ID: <454FAF38.6020806@nd.edu> Karim Zahidi asked if there are any formal systems that prove their own consistency, presumably intending consistent such theories. This question came up on this list in the Spring of 2005. The answers proposed then involved reformulations either of the theory's own representation of its own axioms or of the provability predicate used. Michael Detlefsen mentioned a weak extension of Robinson Arithmetic and the self-consistency proof Jeroslow gave for that theory in Fundamenta Mathematica 72 (1971). Andrew Boucher (in his June 1 2005 post) mentioned his own self-consistency proof for a subsystem of second order arithmetic. I mentioned some similar results. The discussion then was sensitive to the following question: Since these examples all seem somewhat contrived, which representations of a system's consistency are "intensionally adequate?" This question was first raised by Georg Kreisel and first studied systematically by Solomon Feferman. Curtis From rupertmccallum at yahoo.com Sun Nov 5 17:15:28 2006 From: rupertmccallum at yahoo.com (Rupert McCallum) Date: Sun, 5 Nov 2006 14:15:28 -0800 (PST) Subject: [FOM] Question Message-ID: <20061105221528.57435.qmail@web51904.mail.yahoo.com> Woodin defines a large cardinal axiom as follows. "(exists x) phi" is a _large_cardinal_axiom_ if 1. phi(x) is a Sigma_2 formula 2. (As a theorem of ZFC) if kappa is a cardinal such that V |= phi[kappa] then kappa is strongly inaccessible, and for all partial orders P \in V_kappa V^P |= phi[kappa]." Let T be the set of all sentences in the language of second-order arithmetic which are either provable in first-order logic or which are provable in an extension of ZFC by finitely many large-cardinal axioms which are all true in V_kappa for some inaccessible kappa. T is a consistent theory, and assuming two inaccessibles, it contains every true sentence in the first-order language of arithmetic. Does T contain every true sentence in the second-order language of arithmetic? ____________________________________________________________________________________ Get your email and see which of your friends are online - Right on the New Yahoo.com (http://www.yahoo.com/preview) From zahidi at logique.jussieu.fr Tue Nov 7 06:38:25 2006 From: zahidi at logique.jussieu.fr (zahidi@logique.jussieu.fr) Date: Tue, 7 Nov 2006 12:38:25 +0100 (CET) Subject: [FOM] Consistency In-Reply-To: <0B25D9ED-7818-4813-8B2A-3778D3F30A48@wanadoo.fr> References: <1161.146.175.207.172.1162806722.squirrel@webmail.logique.jussieu.fr> <0B25D9ED-7818-4813-8B2A-3778D3F30A48@wanadoo.fr> Message-ID: <1230.146.175.207.172.1162899505.squirrel@webmail.logique.jussieu.fr> Thanks for all who replied to my query on consistency. I'm sorry that I asked a question that had already been discussed before. Best regards Karim Zahidi Dept of Mathematics, statistics and actuarial science University of Antwerp Prinsenstraat 13 B-2000 Antwerpen Belgium From vladik at utep.edu Mon Nov 6 20:05:34 2006 From: vladik at utep.edu (Kreinovich, Vladik) Date: Mon, 6 Nov 2006 18:05:34 -0700 Subject: [FOM] Call for papers: Logic and Information - From Logic to Message-ID: <77B4C8824930004AAC10E1B01576254AE23E4C@itdsrvmail01.utep.edu> This may be of intereset to many researchers in our community. -----Original Message----- From: Vasco Brattka [mailto:BrattkaV at maths.uct.ac.za] Special Issue on Logic and Information: From Logic to Constructive Reasoning of the Journal of Logic and Algebraic Programming Submission Deadline: March 12, 2007 ______________________________________________________________________ Call for papers Following the Swiss-South African (SNSF-NRF) joint seminar Logic and Information: From Logic to Constructive Reasoning University of Berne, Switzerland, January 22-25, 2007 it is planned to publish a special issue of the Journal of Logic and Algebraic Programming (JLAP) http://www.elsevier.com/locate/jlap This issue is supposed to contain papers related to the seminar but it is also open to other submissions that meet the standards of JLAP and the scope of the seminar. Scope The purpose of the seminar is to bring together researchers working on combinatorial, probabilistic, logical and topological aspects of reasoning in Mathematics and Theoretical Computer Science in order to discuss problems of common interest related to various approaches to constructive mathematics. On the one hand, combinatorial, probabilistic and logical methods have many applications in theoretical computer science and a direct impact on algorithms in general and on reasoning in the area of artificial intelligence. Certain modal fixpoint logics, for example, are widely used for specification and verification of algorithms. On the other hand, proof theory has revealed a deep relationship between logic and computer science: the famous "proofs as computations" paradigm and the Curry-Howard interpretations are just examples of a flourishing area. Keywords - Logic and Proof Theory - Theoretical Computer Science - Computability and Constructivity - Topological Methods - Combinatorial and Probabilistic Methods - Fixpoint Logics - Specification and Verification - Artificial Intelligence Guest Editors Vasco Brattka (Cape Town) Gerhard J?ger (Berne) Hans-Peter K?nzi (Cape Town) Submissions Authors are invited to submit PDF versions of papers to: topcs at maths.uct.ac.za Submission deadline: March 12, 2007 Notification: July 2, 2007 Camera-ready versions: August 6, 2007 Papers have to be prepared using LaTeX2e and the LaTeX templates available for download at: http://www.authors.elsevier.com/latex ______________________________________________________________________ From slaterbh at cyllene.uwa.edu.au Mon Nov 6 21:07:52 2006 From: slaterbh at cyllene.uwa.edu.au (Hartley Slater) Date: Tue, 7 Nov 2006 10:07:52 +0800 Subject: [FOM] Nelson's ultraformalism --to Slater Message-ID: At 5:55 PM -0500 6/11/06, Eray Ozkural wrote: >Proving 2+3=5 is a shortcut, short of counting, which is a computation. >In AI community, shortcuts have sometimes been called "chunking". >However, those approaches are unfortunately not capable of inventing >the ordinary kind of arithmetic (PA). Computational basis for such capability >on the other hand might arise from the application of inductive inference >(Reference available upon request). It is unknown at the present if this >is the case. > >Note also the relation to the halting problem, which seems to >correspond to the "golden standard" for computational >problems. The theorem 2+3=5 is an easy instance of the halting >problem, we can show a program that halts iff that is the case. The extraordinary thing about these remarks will, I suspect, be hidden from most members of this list. For they demonstrate only too well my previous point about the absence of 'that' from current formal languages - with the resultant professional unconsciousness about the identity of mathematical truths. The phenomenon is quite general. Thus, when I sent the abstract of my paper, 'Proving that 2+3=5', to the organisers of my seminar at Monash University recently, the same thing occurred, and at my own institution, here at UWA, the same thing had happened the week before. Each time the title of my talk was advertised simply as 'Proving 2+3=5'. If you do not insert the 'that' you miss so much! Why is it that no Turing Machine can prove that 2+3=5? Because 'that 2+3=5' is not a sentence and so, a fortiori, it is not a sentence at the end of any rule governed sequence of sentences. Generating sentences, as Turing Machines do, is not in the right business even; a 'category mistake' is involved in thinking it is. And the same category mistake is what makes Avron, amongst many others, think he is, indeed must be a Turing Machine. Before, I gave a solution of The Liar in two paragraphs, relying on this point; but here is the proof, now in just two lines, that anything that can prove that 2+3=5 is not a Turing Machine. Can anyone else on FOM see these things? They have little chance of doing so, if they casually elide 'that' in the manner above. Nelson, in the second defense-of-Formalism paper that Mannucci previously referenced ('Mathematics and Faith') said (p7) that he had felt an 'overwhelming presence' which had removed from him the 'arrogance' of his previous belief that there was a real world numbers. '... the theorems [of Mathematics] are not about anything' he said (p4). Of course, if one does not have a referential phrase like 'that 2+3=5' in one's formal language then one's language cannot make reference to facts or putative facts; so if one chooses one's language appropriately there will be no such reference. But is there some 'arrogance' in incorporating referential phrases of the form 'that p'? Certainly all I take myself to be doing is reminding theorists of the things we all, both professionals and ordinary people, commonly and repeatedly say. Every day, continuously, everybody, whether educated or untutored, is saying that such and such, or that such and such is so. Whether or not that means there is some 'real world' of facts we can grant philosophers another two and a half millenia to decide. The main point for the moment, though, is: how many FOMers, reading this message, even noticed the 'that's in the 'that such and such, or that such and such' just then? -- Barry Hartley Slater Honorary Senior Research Fellow Philosophy, M207 School of Humanities University of Western Australia 35 Stirling Highway Crawley WA 6009, Australia Ph: (08) 6488 1246 (W), 9386 4812 (H) Fax: (08) 6488 1057 Url: http://www.philosophy.uwa.edu.au/staff/slater From mmannucc at cs.gmu.edu Tue Nov 7 10:51:12 2006 From: mmannucc at cs.gmu.edu (Mirco Mannucci) Date: Tue, 7 Nov 2006 10:51:12 -0500 (EST) Subject: [FOM] Concerning Ultraformalism-to Slater&Ozkural Message-ID: ---- Hartley Slater wrote: > Mirco Mannucci points me/us towards another paper by Nelson, > 'Mathematics and Faith' in his 'Concerning Ultraformalism --to > Slater', but I am afraid I find Nelson's dream of an 'overwhelming > presence' in that paper less worthy of addressing. I guess I have to apologize once again: my goal was not to endorse in any way the "overwhelming presence": this is a (perhaps worthy) topic for Psychology, Anthropology,or Theology, depending on one's interests, beliefs and biases. At any rate, I do not think that it fits the present FOM thread (or even this list's scope). Incidentally, I note in passing that your labelling it as a "dream" is also a bit biased, though I respect your opinion. The only thing I wish to add here, is that I entertain absolutely NO DOUBTS whatsoever about Nelson's integrity: he simply related things as he deemed fit, without any hidden agenda, and I will leave it at that. The reason why I love that paragraph instead, and why I pointed you and the other FOM fellows there, is that it hints at a view that I find fascinating, and, it seems to me, (as yet) not fully explored in all its implications: -----> from the (rigorous) formalist standpoint, THERE ARE NO NUMBERS whatsoever. What IS there is simply an "arithmetical game", and "numbers" are just the "characters" of such a game. Here is an important point,that goes against the grain, not only in the "platonic" camp, but in the "constructivist" one as well: -----> it is not true that numbers are constructed/known once and for all. Quite to the contrary, ------> each "number" progressively unfolds as new "facts" become known about it. Trivial (and a bit silly) example. I ask everyone here: ------> do you know the number 5? You will probably answer: of course, ARE YOU KIDDING ME ?????? Answer: 5 = SSSSS0. True, I say, but wait a minute: Answer2: 5= SS0 + SSS0 as well Answer3: 5 = S( SS0^SS0 ) Answer4: 5 = SSS(0 + 0 + S0) + SS0^SSS0 - SSSSSS0 -S0 ........ Answer2^100000: ??? ....... You got my point (there is an indefinite number of answers, some even beyond anyones' current imagination. It may turn out that 5 is the UNIQUE number satisfying some incredibly sophisticated number theory conjecture, or something along similar lines). To say that 5 IS SSSSS0 would be exactly the same as saying that a vector in R^2 is a list [x1, x2]. All right, what if I change coordinates? What if I choose a completely different basis? Or even I choose to represent it not in a basis, but using a over-complete frame? SSSSS0 is just the CANONICAL representative term in the (temporary) similarity classes of available terms denoting 5. Indeed is the simplest & most rudimentary way of denoting 5, but also an extremely expensive and clumsy one. Following the vector space analogy, one could say that SSSSS0 is the representation of 5 in the standard basis (i.e. the standard denotation system). The simple (and a bit puzzling) truth is that NOBODY knows 5 once and for all. As we further and further play the "Peano Game", potentially meaningful new "facts" about 5, and its relations to other "numbers", may unfold. So, I ask again: what IS 5? To me, 5 is a "character in the arithmetical game" that gets constantly re-constructed and re-assessed, as our grasp of the arithmetical game itself increases. To borrow a wondrous idea from quantum physicist David Bohm, I would say that PA (I mean the formal rules and axioms) is the arithmetical world (completely) folded, whereas the bulk of arithmetics that is based on PA is the same world unfolding. ----> BEGINS MINI-NOTE FOR DAVID ISLES To my knowledge, thus far only David Isles hinted as something along similar lines (or, to be more precise, and fair to Isles, that is the way I read him. I may be way off. I hope he will comment on the above himself). -----> ENDS MINI-NOTE FOR DAVID ISLES ************************************************ Meanwhile, here is another question/proposal: ------> Can we develop formal ways of seeing basic arithmetics from an "invariant standpoint"? In other words, can we develop a general framework of different arithmetical denotation systems, and mappings of one denotation system into another (coordinate transformations), together with measures of their computational advantages-disadvantages? Note: this GENERAL THEORY OF DENOTATION SYSTEMS is needed to develop and rigorously formalize a notion I introduced on this list a few months ago: utterable and unatterable numbers (see postings on Utterable Numbers). Meta-Conjecture: given ANY reasonable definition of utterability, MOST numbers will be unutterable (in palin words, most numbers are not only unfeasible, but they cannot even be NAMED!!!!) . *************************************************** Before I leave the section dedicated to you (Hartley Slater), I would like to say that I found your sentence >if one does not incorporate into one's formal language the ability to say >what it means. very intriguing and a bit mysterious. I assume that you would like a language that is both ground-language and meta-language at the same time. But PA does that already, via Godelization. What am I missing??? Can you either elaborate on that one, or point me somewhere for refs? Thanks ************************************************************** ---- Eray Ozkural wrote: > > I think that a computationalist approach may fare better at > this ambitious goal. Here is why. Abstract concepts help us predict > physical events better. However, all abstract concepts are computational. > There is no use for any abstract concept that does not lend to > computation. I basically agree, but... how can you develop a computational approach OUTSIDE formalism? I hope you do not mean PHYSICAL machines. Physical machines were built, as you know, because people like Turing had FORMALIZED an abstract notion of computability. > > However, I think the (ultra) formalist position does not necessitate getting > rid of potential infinity, because in my view the concept of potential > infinity has a crystal clear formalization which is a non-terminating > program, that everybody can objectively examine. I am afraid you are wrong. What does it MEAN to say that a computation is not terminating? Answer: that its length can get arbitrarily large without ending up in a terminating state. And what does it mean arbitrarily large? Potential infinity (and circularity) again... Not to mention that there are programs for which you do NOT know a priori whether they will terminate. By the way, I do NOT advocate killing potential infinity. My own view is quite more subtle than that. ------> I claim that the very distinction of finite-infinite is not ABSOLUTE, but CONTEXTUAL. In order to substantiate my claim, I intend to build mathematical structures, let us call them *ultrafinitistic universes*, such that what looks like infinity from inside is very small from outside... > Is not this already achieved by Kolmogorov complexity theory? It Would be nice indeed, but the claims made by Chaitin that his "complexity approach" EXPLAINS Godel are so far (alas!) unsubstantiated (which does not mean there is nothing in his view). What Chaitin did was to CREATE ANOTHER undecidable sentence, NOT showing how all incompleteness phenomena are in fact derived from unmanageable complexity issue. I personally think that Chaitin's overarching project is meaningful and promising; however, not there yet... > What do you have in your mind when you say that? Do you want to > state Godel's theorem in another way? No. I am asking the following question: is it possible to interpret Godel's incompleteness as saying something about the arithmetical game ITSELF (i.e. the way it works, the way it is played, etc.), as opposed to some (perhaps fictional) "intended " standard model of arithmetics? My answer to my own question is yes (it will be posted under the header "The Grand Peano Game"). But, as I said, I would like first to see if someone else has a view to put on the table... > > PS: Also, what do you think of Godel's statements when he said that > the second incompleteness theorem is also valid for finite systems. This > seems to weaken the finitist position a little, if we listen to Godel. > I am not sure I understand you here. Please elaborate this point. Best Wishes Mirco Mannucci P.S. Apparently, a subset of strictly positive Lebesgue measure on this list has *crystal clear* ideas on a number of issues... well, I do not. The only crystal clear idea I have is: ---I AM--- and sometimes I feel like doubting that one too... From aa at tau.ac.il Wed Nov 8 07:16:21 2006 From: aa at tau.ac.il (Arnon Avron) Date: Wed, 8 Nov 2006 14:16:21 +0200 Subject: [FOM] Concerning Ancestral Logic In-Reply-To: <200611030235.kA32ZKC0000215@math.canterbury.ac.nz> References: <200611030235.kA32ZKC0000215@math.canterbury.ac.nz> Message-ID: <20061108121621.GA16381@soul.cs.tau.ac.il> On Fri, Nov 03, 2006 at 03:35:20PM +1300, Bill Taylor wrote: > This query is mainly directed to Arnon Avron, but no doubt others > may feel like joining in. > > Arnon - you have been extolling the virtues of AL as an alternative, > (an extension?), of FOL. It sounds good, but I don't yet > know what it is. You mention it was given a treatment > in Shapiro's F-without-F, but I don't > recall noticing it from when I read that. > > So I wonder if you could give us all a brief rundown on it. > Nothing too massively technical, but just enough to get the flavour > and main ideas. Richard Heck has already partially replied to the question what is ancestral logic. Let me add to what he wrote the following: 1) Actually, for full generality, one should consider not a single TC operator, but for every n an operator TC^n, which applied to an 2n-ary predicate produces a new 2n-ary predicate (its transitive closure). Thus the intuitive meaning of (TC^2_{x_1,x_2,y_1,y_2}A)(s_1,s_2,t_1,t_2) is: A(S_1,S_2,t_1,t_2) or \exists z_1,w_1. A(s_1,s_2,z_1,w_1) and A(z_1,w_1,t_1,t_2) or \exists z_1,w_1,z_2,w_2. A(s_1,s_2,z_1,w_1) and A(z_1,w_1,z_2,w_2) and A(z_2,w_2,t_1,t_2) or ... (As I wrote in a previous posting, the main idea in using TC is indeed to incorporate into the language a formal counterpart of "...", without introducing new objects). In practice, it seems, one does not need more than TC^1 and TC^2. Of course, if the language contains the means to introduce ordered pairs than TC^1 suffices for defining TC^n for every n. However, this is not the case without the presence of ordered pairs. Thus + is not definable in the the language of 0 and S using only TC_1. However, it is easily definable if we use TC_2. So although the natural numbers can be characterize categorically in FOL(0,S)+TC^1, the expressive power of this language is too weak. The exact situation is as follows: all recursive functions and relations are definable in either of: FOL(=,0,S) plus TC^2 FOL(=,0,S,+) plus TC_1 FOL(=,0,S,+,x) [instead of FOL(=,0,S,+,x) one can use FOL(=,0,S,+,|) (where | is "divides")] 2) By generalizing a particular case which has been used by Gentzen in his proof of the consistency of PA, mathematical induction can be presented as a *logical* rule of languages with $TC$. Indeed, Using a Gentzen-type format, a general form of this principle in the case of $TC^1$ can be formulated as follows: \Gamma, A(x), B(x,y) => A(y), \Delta ------------------------------------- \Gamma, A(s), (TC^1_{x,y})B(s,t) => A(t), \Delta where $x$ and $y$ are not free in $\Gamma,\Delta$, and $y$ is not free in B. This is an introduction rule on the left. There are also obvious rules for introducing TC on the right. In this way one gets a very natural logical system in which, I believe (but have not tried to prove yet), cuts are eliminable. I also believe that this system is "complete" in some intuitive, imprecise sense, (or at least is strong enough for most needs). 3) In his paper "Finitary Inductively Presented Logics" (in Logic Colloquium 1988, Amsterdam, North-Holland, pp. 191-220. Reprinted in the collection "What is a Logical System?", edited by D. Gabbay, Oxford Science Publications, Clarendon Press, Oxford, 1994), Feferman developed a general framework (called FS_0) for introducing and investigating formal systems. The main feature of FS_0 is that it provides a "comprehensive definition of a finitary inductive system which includes both languages and logics, and which covers all examples met in practice". The main observation on which FS_0 is based is that all major concepts needed to understand and use formal systems (like terms, formulas, and formal proofs) are inductively defined. Now the intended mathematical structure on which Feferman's framework is based is V_0, which is the smallest set including $0$ and closed under the operation of pairing (as Feferman has shown, it is much more natural to work within V_0 than within the natural numbers, the use of which for the same goal requires heavy machinery of unnatural coding). Now FS_0 can most naturally be developed within a language which includes first-order variables, the constant 0, a binary function symbol P (for a pairing function), equality, conjunction, disjunction, negation and TC^1 (the existential quantifier can easily be defined using TC_1). This is shown in my paper "Transitive Closure and the mechanization of Mathematics" (available from my homepage: http://www.math.tau.ac.il/~aa). In that paper I proved also a result I have mentioned in a previous posting: that the relations on V_0 which are definable by the negation-free formulas of the above language are exactly the r.e. relations on V_0. These facts show that TC (together with the ability to introduce ordered pairs) is exactly what one needs (and I claim: should) add to FOL in order to introduce and investigate formal systems. In other words: ONE CANNOT UNDERSTAND WHAT IS FIRST-ORDER LOGIC WITHOUT UNDERSTANDING TC and assuming the ability to use it. 4) Significant attention to logics with TC have been given within the area of Finite Model Theory (FMT). The idea there is to give purely logical characterizations to various complexity classes. For this goal various "fix-point" logics have been introduced, and ancestral logic belongs to this class (or is a close relative of it). Details and references can be found in one of the several textbooks on FMT which are now available (e.g. the book of Ebbinghous and Flum, or the book of Libkin). However, I don't think that the work on FMT has much relevance to FOM (but maybe other people on this list think otherwise). Arnon Avron From tchow at alum.mit.edu Wed Nov 8 11:27:31 2006 From: tchow at alum.mit.edu (Timothy Y. Chow) Date: Wed, 8 Nov 2006 11:27:31 -0500 (EST) Subject: [FOM] Concerning Ultraformalism-to Slater&Ozkural Message-ID: Hartley Slater wrote: >The main point for the moment, though, is: how many FOMers, reading >this message, even noticed the 'that's in the 'that such and such, or >that such and such' just then? This question is almost equivalent to asking how many FOMers care about this particular philosophical question enough to have studied it and developed a sensitivity to this distinction. Working mathematicians don't particularly care about the Liar or about Truth. They don't particularly care if they can't prove that 2+3=5, as long as they can produce proofs of "2+3=5". If you want to convince them that they *should* care, then you need to demonstrate what difference it would make to the daily practice of mathematics. Tim From slaterbh at cyllene.uwa.edu.au Wed Nov 8 21:37:07 2006 From: slaterbh at cyllene.uwa.edu.au (Hartley Slater) Date: Thu, 9 Nov 2006 10:37:07 +0800 Subject: [FOM] Concerning Ultraformalism-to Slater&Ozkural Message-ID: Mirco Mannucci first says >The reason why I love that paragraph instead, and why I pointed you >and the other FOM >fellows there, is that it hints at a view that I find fascinating, >and, it seems to me, >(as yet) not fully explored in all its implications: > > >-----> from the (rigorous) formalist standpoint, > THERE ARE NO NUMBERS whatsoever. > > >What IS there is simply an "arithmetical game", and "numbers" are just the >"characters" of such a game. but later: >SSSSS0 is just the CANONICAL representative term in the (temporary) >similarity classes of available terms denoting 5. Indeed is the simplest & >most rudimentary way of denoting 5, but also an extremely >expensive and clumsy one. Following the vector space analogy, one could say >that SSSSS0 is the representation of 5 in the standard basis (i.e. >the standard denotation system). So here he has something *denoting* the number 5. Maybe the difficulty is that he does not put quotation marks around terms like 'SSSSS0'. Without then made explicit then even the '5' might look like it is a mentioned term, and so just a 'character in a game'. In short what is true is that 'SSSSS0' denotes 5, (which presumes there are numbers, or at least the number 5), but one cannot say either that SSSSS0 denotes '5', or that 'SSSSS0' denotes '5'. A similar point is involved when he goes on to say: >Before I leave the section dedicated to you (Hartley Slater), I would like >to say that I found your sentence > >>if one does not incorporate into one's formal language the ability to say > >what it means. > >very intriguing and a bit mysterious. I assume that you would >like a language that is both ground-language and meta-language at the >same time. > >But PA does that already, via Godelization. What am I missing??? Godelisation enables one to say things like 'n' represents 'p', i.e. its sets up a 1-1 correlation between certain numerals and certain syntactic formulae. That has nothing to do with saying what 'p' means, or 'n' refers to. To say that one has to say things like: '2+3=5' means that 2+3=5, 'SSO' refers to the number 2. -- Barry Hartley Slater Honorary Senior Research Fellow Philosophy, M207 School of Humanities University of Western Australia 35 Stirling Highway Crawley WA 6009, Australia Ph: (08) 6488 1246 (W), 9386 4812 (H) Fax: (08) 6488 1057 Url: http://www.philosophy.uwa.edu.au/staff/slater From vladik at utep.edu Thu Nov 9 11:22:03 2006 From: vladik at utep.edu (Kreinovich, Vladik) Date: Thu, 9 Nov 2006 09:22:03 -0700 Subject: [FOM] book announcement Message-ID: <77B4C8824930004AAC10E1B01576254AE23EC0@itdsrvmail01.utep.edu> Dear Friends, This book may be of interest to many of us: Recently published: Techniques of Constructive Analysis, by Douglas Bridges and Luminita Simona Vita, Universitext, Springer-New-York, 2006. For more information see these websites: http://www.springer.com/east/home/generic/search/results?SGWID=5-40109-2 2-158902589-0 http://www.math.canterbury.ac.nz/~dsb35/techniques_preface.pdf From sartemov at gc.cuny.edu Thu Nov 9 16:59:19 2006 From: sartemov at gc.cuny.edu (Sergei Artemov) Date: Thu, 09 Nov 2006 16:59:19 -0500 Subject: [FOM] SYMPOSIUM ON LOGICAL FOUNDATIONS OF COMPUTER SCIENCE (LFCS'07) Message-ID: <4553A4B7.5000908@gc.cuny.edu> SYMPOSIUM ON LOGICAL FOUNDATIONS OF COMPUTER SCIENCE (LFCS'07) Revised Call for papers NOTE THE REVISED SUBMISSION DEADLINE New York City, June 4 - 7, 2007 URL: www.cs.gc.cuny.edu/lfcs07 Email: lfcs07 at gmail.com * Purpose. The LFCS series provides an outlet for the fast-growing body of work in the logical foundations of computer science, e.g., areas of fundamental theoretical logic related to computer science. The LFCS schedule is consistent with LICS and CSL timelines. * Theme. Constructive mathematics and type theory; logical foundations of programming; logical aspects of computational complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logics; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple agent system logics; logics of proof and justification; non-monotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; system design logics; other logics in computer science. * All submissions must be done electronically (15 pages, pdf, 12pt) via http://www.easychair.org/LFCS07/ * Submission deadline: December 4, 2006 (Revised from December 18) * Notification: January 11, 2007 * Steering Committee. Anil Nerode (Cornell, General Chair); Stephen Cook (Toronto); Dirk van Dalen (Utrecht); Yuri Matiyasevich (St.Petersburg); John McCarthy (Stanford); J. Alan Robinson (Syracuse); Gerald Sacks (Harvard); Dana Scott (Carnegie-Mellon). * Program Committee. Samson Abramsky (Oxford); Sergei Artemov (New York City, PC Chair); Matthias Baaz (Vienna); Lev Beklemishev (Moscow); Andreas Blass (Ann Arbor); Lenore Blum (CMU); Samuel Buss (San Diego); Thierry Coquand (Go"teborg); Ruy de Queiroz (Recife, Brazil); Denis Hirschfeldt (Chicago); Bakhadyr Khoussainov (Auckland); Yves Lafont (Marseille); Joachim Lambek (McGill); Daniel Leivant (Indiana); Victor Marek (Kentucky); Anil Nerode (Cornell, General LFCS Chair); Philip Scott (Ottawa); Anatol Slissenko (Paris); Alex Simpson (Edinburgh); V.S. Subrahmanian (Maryland); Michael Rathjen (Columbus); Alasdair Urquhart (Toronto). From slaterbh at cyllene.uwa.edu.au Thu Nov 9 22:44:25 2006 From: slaterbh at cyllene.uwa.edu.au (Hartley Slater) Date: Fri, 10 Nov 2006 11:44:25 +0800 Subject: [FOM] Concerning Ultraformalism-to Slater&Ozkural Message-ID: At 12:00 PM -0500 9/11/06, Timothy Chow wrote: >Hartley Slater wrote: >>The main point for the moment, though, is: how many FOMers, reading >>this message, even noticed the 'that's in the 'that such and such, or >>that such and such' just then? > >This question is almost equivalent to asking how many FOMers care about >this particular philosophical question enough to have studied it and >developed a sensitivity to this distinction. > >Working mathematicians don't particularly care about the Liar or about >Truth. They don't particularly care if they can't prove that 2+3=5, as >long as they can produce proofs of "2+3=5". If you want to convince them >that they *should* care, then you need to demonstrate what difference it >would make to the daily practice of mathematics. I should think that, in the 'daily practice of mathematics', the formula '2+3=5' is written and uttered with its standard interpretation in mind, whereas in the debate here it has been the theoretical difficulty of locating the standard interpretation of such formulae, and removing non-standard interpretations, that has been of concern. The issue about the identity of mathematical truths - and more generally the identity of facts of any description, and the crucial categorical difference between facts and formulae - has arisen in a debate about Formalism, the status of semantic interpretations of meta-mathematical syntax, and specifically whether Hilbert's programme could even prove that 2+3=5 rather than just organise systems of (uninterpreted) formulae, like '2+3=5'. The debate on this matter goes back to Frege. It seems to me, therefore, to be entirely appropriate for discussion of the FOM list, as it stands. -- Barry Hartley Slater Honorary Senior Research Fellow Philosophy, M207 School of Humanities University of Western Australia 35 Stirling Highway Crawley WA 6009, Australia Ph: (08) 6488 1246 (W), 9386 4812 (H) Fax: (08) 6488 1057 Url: http://www.philosophy.uwa.edu.au/staff/slater From friedman at math.ohio-state.edu Fri Nov 10 07:24:08 2006 From: friedman at math.ohio-state.edu (Harvey Friedman) Date: Fri, 10 Nov 2006 07:24:08 -0500 Subject: [FOM] Connections Seminar Message-ID: This Fall, I started a new type of Seminar called the Connections Seminar, at the OSU mathematics department, jointly with Ovidiu Costin. The url for its website is http://www.math.ohio-state.edu/Research/seminars/connections where you will currently find the Manifesto, and five papers. The Connections Seminar is devoted, partly, to issues in the foundations of mathematics, as indicated in the Manifesto. Harvey Friedman From jpvbende at vub.ac.be Tue Nov 14 05:15:06 2006 From: jpvbende at vub.ac.be (jean paul van bendegem) Date: Tue, 14 Nov 2006 11:15:06 +0100 Subject: [FOM] Yessenin-Volpin References: <20061102213622.40941.qmail@web51903.mail.yahoo.com> Message-ID: <132401c707d5$c2bc5760$0301a8c0@lan> > Yes, he claimed to have a proof of the consistency of ZF with any > finite number of inaccessible cardinals. Unfortunately it seems to be > hard to get hold of a copy of this proof. I would really like to know > in which axiomatic theory he claimed it could be done. About a year ago, I sent in this reply to a similar question: I know of three papers by Volpin, not in Russian. There is supposed to be a typescript in Russian containing the full proof, but I have never seen it. I did write to him many years ago, but what I received was a list of publications, not the papers. YESSENIN-VOLPIN, A. S. : "Le programme ultra-intuitioniste des fondements des math?matiques". In: Infinitistic Methods, Proceedings Symposium on Foundations of Mathematics, Pergamon Press, Oxford, 1961, pp. 201-223. YESSENIN-VOLPIN, A. S. : "The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics". In: KINO, MYHILL & VESLEY (eds.), Intuitionism & proof theory. North-Holland, Amsterdam, 1970, pp. 3-45. YESSENIN-VOLPIN, A. S. : "About infinity, finiteness and finitization". In RICHMAN, F. (ed.), 1981, pp. 274-313. I have paper copies of these papers. Jean Paul Van Bendegem From martin at eipye.com Tue Nov 14 13:52:32 2006 From: martin at eipye.com (Martin Davis) Date: Tue, 14 Nov 2006 10:52:32 -0800 Subject: [FOM] New Journal Message-ID: <200611141852.kAEIqfSZ024536@nlpi043.sbcis.sbc.com> An advance announcment of a new journal from Springer entitled LOGIC AND ANALYSIS, and an invitation to submit papers for possible publication. Aims and Scope of Logic and Analysis: "Logic and Analysis" publishes papers of high quality involving interaction between ideas or techniques from mathematical logic and other areas of mathematics (especially - but not limited to - pure and applied analysis). The journal welcomes - papers in nonstandard analysis and related areas of applied model theory; - papers involving interplay between mathematics and logic (including foundational aspects of such interplay); - mathematical papers using or developing analytical methods having connections to any area of mathematical logic. "Logic and Analysis" is intended to be a natural home for papers with an essential interaction between mathematical logic and other areas of mathematics, rather than for papers purely in logic or analysis. URL of website: http://www.springer.com/uk/home/generic/search/results?SGWID=3-40109-70-173679804-0 Papers should be sent to Nigel Cutland Editor-in-Chief, Logic and Analysis at math502 at york.ac.uk in PDF form, and if possible citing the member of the editorial board whose interest is closest to that of the paper. From Mark.vanAtten at univ-paris1.fr Tue Nov 14 14:35:56 2006 From: Mark.vanAtten at univ-paris1.fr (mark van atten) Date: Tue, 14 Nov 2006 20:35:56 +0100 Subject: [FOM] book announcement: Brouwer meets Husserl Message-ID: <20061114203556.hme9f5gzakcgs404@courrier.univ-paris1.fr> A revised and somewhat expanded version of my dissertation has just been published. Details below. Best wishes, Mark. Brouwer meets Husserl. On the Phenomenology of Choice Sequences. Mark van Atten (CNRS, Paris) Synthese Library, Volume 335. Berlin: Springer. November 2006. Hardcover, xiii+206 pp. List price EUR 99.95 ISBN 978-1-4020-5086-2 Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. "It is almost as if one could hear the two rebels arguing their case in a European caf? or on a terrace, and coming to a common understanding, with both men taking their hat off to the other, in admiration and gratitude. Dr. van Atten has convincingly applied Husserl's method to Brouwer's program, and has equally convincingly applied Brouwer's intuition to Husserl's program. Both programs have come out the better." Piet Hut, professor of Interdisciplinary Studies, Institute for Advanced Study, Princeton, U.S.A. Contents: Preface. Acknowledgements.- 1 An Informal Introduction.- 2 Introduction.- 3 The Argument.- 4 The Original Positions.- 5 The Phenomenological Incorrectness of the Original Arguments.- 6 The Constitution of Choice Sequences.- 7 Application: an Argument for Weak Continuity.- 8 Concluding Remarks.- Appendix: Intuitionistic Remarks on Husserl's Analysis of Finite Number in the Philosophy of Arithmetic. Notes. References. Name and citation index.- Subject index. -- Ce message a ete verifie par MailScanner pour des virus ou des polluriels et rien de suspect n'a ete trouve. From till at informatik.uni-bremen.de Thu Nov 16 05:28:11 2006 From: till at informatik.uni-bremen.de (Till Mossakowski) Date: Thu, 16 Nov 2006 11:28:11 +0100 Subject: [FOM] 9 research assistant positions available Message-ID: <455C3D3B.802@informatik.uni-bremen.de> 9 research assistant positions (most of them TVL 13, approx. ? 35,000 to ? 50,000 p.a. gross) available, at Transregional Collaborative Research Center SFB/TR 8 Spatial Cognition: Reasoning, Action, Interaction at the Universities of Bremen and Freiburg, Germany The positions are in general concerned with interdisciplinary long-term research in Spatial Cognition. Some of the positions may be of interest to the readers of this list, because formal methods, logic and category theory are used. For details, see http://www.sfbtr8.uni-bremen.de/openpositions.html (in particular, projects I1, I3 and I4) -- Till Mossakowski Office: Phone +49-421-218-64226 DFKI Lab Bremen Cartesium Fax +49-421-218-9864226 Robert-Hooke-Str. 5 Enrique-Schmidt-Str. 5 till at tzi.de D-28359 Bremen Room 2.051 http://www.tzi.de/~till From joeshipman at aol.com Sat Nov 18 13:59:39 2006 From: joeshipman at aol.com (joeshipman@aol.com) Date: Sat, 18 Nov 2006 13:59:39 -0500 Subject: [FOM] Hard problems in the theory of addition Message-ID: <8C8D95AD5769B73-824-6EA3@FWM-M25.sysops.aol.com> Some recent discussion here has focused on how simple a statement in the language of arithmetic can be and still be either unprovable in PA, or "hard" (having no short proof). Friedman has suggested that Goldbach's conjecture and the twin prime conjecture are near the boundary and that statements significantly simpler than them are decidable in PA (I don't recall whether he ventured an opinion on whether they are decidable with short proofs). But ever since the work of Fischer and Rabin in 1974, it has been known that there are hard statements even in decidable theories involving only addition. (For simplicity, assume the constants 0 and 1 are in the language, though this is not necessary.) Their techniques are of very general applicability and deserve to be better known. In particular, they showed that there exist formulas in the language of addition of length O(n) which represent real multiplication "up to 2^(2^n)", and which represent integer multiplication "up to 2^(2^(2^n)))". Thus, there is a formula An(x,y,z) in the language of addition of length O(n) with a very reasonable constant, such that An(x,y,z) is true in the structure of the real numbers iff xy=z and x,y, and z are all less than 2^2^n. This formula is constructed by an easy induction, using the fact that x < 2^(2^(k+1)) iff there exist x1,x2,x3,x4 < 2^(2^k) with x = x1x2 + x3 + x4. There is another formula Bn(x,y,z) in the language of addition of length O(n) such that Bn(x,y,z) is true in the structure of the natural numbers iff xy=z and x,y, and z are all less than 2^2^2^n. This formula is constructed by an induction which involves the assertion that the previous formula B_(n-1) is true modulo all primes up to 2^2^n, since the earlier definitions allow correspondingly limited definitions of residues. Correspondingly limited versions of exponentiation can be defined by pretty much the same trick. By a straightforward argument involving the representation of Turing machine computations by integers whose base 2 representation consists of the concatenation of their instantaneous descriptions, Fischer and Rabin show that the theory of real addition has exponentially long proofs in any proof system whose axioms can be recognized in polynomial time, and Presburger arithmetic has double-exponentially long proofs. An additional argument shows that Skolem arithmetic, the theory of integer multiplication without addition, has triple-exponentially long proofs. Although these constructions are quite straightforward, and the size of the assertions with (super-)exponentially long proofs is a small constant times the size of the description of the Turing machine recognizing the axioms, these assertions are still much longer than, for example, Goldbach's conjecture or the twin prime conjecture. My question for this list is: Can anyone state a SHORT sentence in the language of addition (OK to use 0,1,< as well as +), where the intended model is EITHER the real numbers or the natural numbers, which is a candidate for "hardness" (that is, has only long known proofs, or is an open question)? (If anyone can come up with such a sentence in the language of multiplication, where it is OK to use constants for 0,1,2,... , that would also be quite interesting.) -- Joe Shipman ________________________________________________________________________ Check out the new AOL. Most comprehensive set of free safety and security tools, free access to millions of high-quality videos from across the web, free AOL Mail and more. From pmt6sbc at maths.leeds.ac.uk Mon Nov 20 10:55:04 2006 From: pmt6sbc at maths.leeds.ac.uk (S B Cooper) Date: Mon, 20 Nov 2006 15:55:04 +0000 (GMT) Subject: [FOM] TAMC 2007 Message-ID: CALL FOR PAPERS Theory and Applications of Models of Computation (TAMC) http://www.tamc2007.fudan.edu.cn/ May 22--25, 2007 Shanghai, China The 4th Annual Conference on Theory and Applications of Models of Computation (TAMC07) will be held in Shanghai, China, May 22 to 25, 2007. Three previous annual meetings were held in 2004, 2005 and 2006, with enthusiastic participation from researchers all around the world. (The web site for TAMC06 can be found at http://gcl.iscas.ac.cn/accl06/TAMC06_Home.htm.) The three main themes of the conference TAMC07 will continue to be Computability, Complexity, and Algorithms. It aims to bring together researchers with an interest in theoretical computer science, algorithmic mathematics, and applications to the physical sciences. Typical but not exclusive topics of interest include: algorithms and data structures, computational complexity, cryptography, computational geometry, computational game theory, algorithmic graph theory and combinatorics, algorithmic algebra, number theory and coding theory, learning theory, computational biology, theoretical problems in networks and security, quantum computing, randomness, on-line algorithms, and parallel algorithms, natural computation, models of computation, automata and neural networks, continuous and real computation, computable mathematics, relative computability and degree structures, Turing definability, generalised and higher type computation, proofs and computation, physical computability, decidability and undecidability. More information about the conference is available on the TAMC 07 web site. Submission and publication: Authors should submit an extended abstract (not a full paper). The submission should contain a scholarly exposition of ideas, techniques, and results, including motivation and a clear comparison with related work. The length of the extended abstract should not exceed ten (10) letter-sized pages (not including bibliography, appendices and figures.) Submitted papers must describe work not previously published. They must not be submitted simultaneously to another conference with refereed proceedings. Research that is already submitted to a journal may be submitted to TAMC07, provided that (a) the PC chair is notified in advance that this is the case, and (b) it is not scheduled for journal publication before the conference. Accepted papers will be published in the conference proceedings, in the LNCS Series by Springer. Authors of accepted papers are expected to present their work at the conference. Special issues of Theoretical Computer Science and Mathematical Structures in Computer Science devoted to a selected set of accepted papers of the conference are planned. Important dates: Submission deadline: Papers must be received electronically by 11:59 pm EST Dec. 18, 2006. Notification: Acceptance or rejection decisions will be sent by Feb. 12, 2007. Final versions: Final versions of accepted papers are due on March 7, 2007. Abstract submission: Authors are required to submit their extended abstracts electronically. A detailed description of the electronic submission process is available at the conference web site. Plenary speakers: Miklos Ajtai (IBM Research) Juris Hartmanis (Cornell University) Program committee PC Chair: Jin-Yi Cai (University of Wisconsin, Madison) jyc at cs.wisc.edu PC co-Chairs: Barry Cooper (University of Leeds, Leeds) pmt6sbc at maths.leeds.ac.uk Hong Zhu (Fudan University, Shanghai) hzhu at fudan.edu.cn PC members: Giorgio Ausiello (Rome, Italy) Eric Bach (UW Madison) Nicolo Cesa-Bianchi (Milano, Italy) Jianer Chen (Texas AM University) Yijia Chen (Shanghai Jiaotong University) Francis Chin (Hong Kong) C.T. Chong (Singapore) Kyung-Yong Chwa (KAIST, Korea) Decheng Ding (Nanjing University) Rod Downey (Wellington) Martin Dyer (Leeds) Rudolf Fleischer (Fudan University) Oscar Ibarra (UC Santa Barbara) Hiroshi Imai (University of Tokyo) Kazuo Iwama (Kyoto University) Tao Jiang (University of California-Riverside/Tsinghua, Beijing) Satyanarayana Lokam (Microsoft Research-India) D T Lee (Academia Sinica, Taipei) Angsheng Li (Institute of Software, CAS) Giuseppe Longo (Paris, France) Tian Liu (Beijing University) Rudiger Reischuk (Universitat zu Lubeck) Rocco Servedio (Columbia University) Alexander Shen (Institute for Information Transmission Problems, Moscow) Yaoyun Shi (University of Michigan, Ann Arbor) Ted Slaman (UC Berkeley) Xiaoming Sun (Tsinghua University) Shanghua Teng (Boston University) Luca Trevisan (UC Berkeley) Christopher Umans (Cal Tech) Alasdair Urquhart (University of Toronto) Hanpin Wang (Beijing University) Osamu Watanabe (Tokyo Institute of Technology) Zhiwei Xu (Institute of Computing Technology, CAS) Frances Yao (City University of Hong Kong) Mingsheng Ying (Tsinghua University, Beijing) For TAMC07, we gratefully acknowledge the generous support of the Information School of Fudan University. From gbezhani at nmsu.edu Mon Nov 20 19:07:36 2006 From: gbezhani at nmsu.edu (Guram Bezhanishvili) Date: Mon, 20 Nov 2006 16:07:36 -0800 Subject: [FOM] Teaching with original historical sources Message-ID: <200611210007.kAL07r1e030420@nlpi015.sbcis.sbc.com> A team of mathematicians and computer scientists at New Mexico State University and Colorado State University at Pueblo has developed an innovative pedagogical technique for teaching material in discrete mathematics, combinatorics, logic, and computer science, with National Science Foundation support for a pilot project. Topics are introduced and studied via primary historical sources, allowing students to participate in the sense of discovery, and to appreciate and gain motivation from the context in which concepts were developed. For example, we have authored classroom modules in which students learn mathematical induction from Pascal's "Treatise on the Arithmetical Triangle," written in the 1660's. Another module develops the short recursion relation for the Catalan numbers from a seminal paper of G. Lame in 1838 (based on a start by Euler!!) We also have authored modules on binary arithmetic, based on the original historical sources by Leibniz and von Neumann; on infinite sets, based on original historical sources by Cantor; and on Turing machines, and Church's Thesis, based on original historical sources by Goedel, Church, Turing, and Kleene. We have authored 18 modules so far; all these modules and more information can be found at www.math.nmsu.edu/hist_projects/. The modules will appear in a chapter of a forthcoming MAA resource book for teaching discrete mathematics. We found that 65% of the students who completed a course with these historical projects performed equally well or better than the mean GPA in subsequent mathematics and computer science courses. We are seeking to expand our pilot program with further major support from the National Science Foundation to create a full book with a comprehensive collection of classroom projects based on historical sources. We would like to invite any instructors of mathematics or computer science courses to agree to site test future projects in related courses in discrete mathematics, combinatorics, logic, or computer science, or perhaps even to design your own projects. We hope to be able to provide a little NSF support as travel and/or consulting for site testers. If you think that you (or a colleague) would be interested in teaching with a project during 2008-2011, we would like to hear from you. We plan to finalize our new NSF proposal by mid-December, and would like to attach a brief letter of support from you if you are interested. It would be nice if it indicated the institution, the course, nature of students, rough timeframe, why you think it would be good for your students, and possible choice of projects for your testing. Contact persons: Guram Bezhanishvili (gbezhani at nmsu.edu) Jerry Lodder (jlodder at nmsu.edu) David Pengelley (davidp at nmsu.edu) From joeshipman at aol.com Tue Nov 21 14:44:50 2006 From: joeshipman at aol.com (by way of Martin Davis ) Date: Tue, 21 Nov 2006 11:44:50 -0800 Subject: [FOM] What's so hard about addition? Message-ID: <200611211944.kALJiV8f005140@nlpi043.sbcis.sbc.com> Not having gotten any response to my previous post, let me simplify it and get right to the point. We know there are sentences of Presburger arithmetic (the theory of addition in the natural numbers) that require double-exponentially long proofs (this is true even if you allow constants for natural numbers, the successor function, and the < relation). Can anyone state a proposition about addition which is easy to understand and does not appear to have any short proofs or disproofs? -- JS ________________________________________________________________________ Check out the new AOL. Most comprehensive set of free safety and security tools, free access to millions of high-quality videos from across the web, free AOL Mail and more. From jdh at hamkins.org Wed Nov 22 01:05:43 2006 From: jdh at hamkins.org (Joel David Hamkins) Date: Tue, 21 Nov 2006 22:05:43 -0800 Subject: [FOM] Conference Announcement: in commemoration of Rohit Parikh's 70th birthday Message-ID: <200611220605.kAM65ocG027163@nlpi029.sbcis.sbc.com> Conference Announcement, December 1-2, 2006 "Logical methods in exact and social sciences: a conference to commemorate 70th birthday of Rohit Parikh" to be held at the CUNY Graduate Center 365 Fifth Avenue, New York Sponsored by the MidAtlantic Mathematical Logic Seminar, New York Logic Colloquium, and CUNY Computer Science Colloquium. http://nylogic.org/Colloquium/ParikhFest Schedule: December 1, Friday, Session 1, Recital Hall 9:00 - 9:10 Conference opening 9:10 - 10:00 Dexter Kozen (Cornell) "Parikh's Theorem in Commutative Kleene Algebra" 10:00 - 10:10 Coffee break 10:10 - 11:00 Steven Brams (NYU) "New Results on Fair Division" 11:00 - 11:10 Coffee Break 11:10 - noon Sam Buss (UC San Diego) "Randomized computability and bounded arithmetic" Noon - 2:00 Lunch Session 2. Science Center 4102 2:00 - 2:50 Saul Kripke (CUNY) Title: TBA 2:50 - 3:00 Coffee Break 3:00-3:50 Melvin Fitting (CUNY) "Reasoning with Justifications" 3:50 - 4:00 Coffee Break 4:00-4:50 Horacio Arlo Costa (Carnegie Mellon University) "First order classical modal logic" 6:00 - 10-00 Conference dinner December 2, Saturday Session 3. Science Center 4102 10:00 - 10:50 John Horty (University of Maryland) Title: TBA 10:50 - 11:00 Coffee Break 11:00 - 11:50 Amy Greenwald (Brown University) Title: TBA Noon - 2:00 Lunch Session 4. Science Center 4102 2:00 - 2:50 Ali Khan (The Johns Hopkins University) "Perfect Competition: Mathematical and Epistemological Foundations." 2:50 - 3:00 Coffee Break 3:00-3:50 Juliet Floyd (Boston University) From Klaus.Weihrauch at FernUni-Hagen.de Wed Nov 22 11:40:55 2006 From: Klaus.Weihrauch at FernUni-Hagen.de (Klaus Weihrauch) Date: Wed, 22 Nov 2006 08:40:55 -0800 Subject: [FOM] conference call for papers Message-ID: <200611221641.kAMGfKtr003666@nlpi012.sbcis.sbc.com> Call for Papers CCA 2007 Fourth International Conference on COMPUTABILITY AND COMPLEXITY IN ANALYSIS June 16-18, 2007, Siena, Italy http://cca-net.de/cca2007/ Authors are invited to submit a PostScript or PDF version of a paper to Submission deadline: March 15, 2007 Notification of authors: April 15, 2007 Final versions: May 5, 2007 Conference: June 16-18, 2007 CCA 2007 is co-located with the conference CiE 2007, Computability in Europe 2007: Computation and Logic in the Real World, University of Siena, June 18-23, 2007. http://www.amsta.leeds.ac.uk/~pmt6sbc/cie07.html SCOPE of CCA 2007 The conference is concerned with the theory of computability and complexity over real-valued data. Computable Analysis combines concepts from Analysis/Numerical Analysis and Computability/Computational Complexity and studies those functions over real-valued data, which can be realized by digital computers. =========================================================================== From artyom.chernikov at gmail.com Thu Nov 23 11:45:42 2006 From: artyom.chernikov at gmail.com (Artyom Chernikov) Date: Thu, 23 Nov 2006 19:45:42 +0300 Subject: [FOM] Question on ultraproducts Message-ID: <6efdd4fc0611230845j77fabe3rd7f319da1d47d96c@mail.gmail.com> Suppose $A_i, i \in \omega$ is an indexed family of structures, S is a permutation of naturals and U is an ultrafilter on $\omega$. Obviously $\prod_{ i\in\omega }A_i/U$ is not isomorphic to $\prod_{ i\in\omega }A_{S(i)}/U$ in general. Can we somehow specify (without complete trivialization) type of U, or S, or maybe complexity of A_i to keep such ultraproducts isomorphic. I mean to specify respective parameters in mutually independent fashion, not along lines of "S has infinite set of fixed points, wich is in U". Thanks in advance, Artyom Chernikov From g.sica at polimetrica.org Sat Nov 25 07:55:02 2006 From: g.sica at polimetrica.org (nico) Date: Sat, 25 Nov 2006 13:55:02 +0100 Subject: [FOM] 2nd World Congress and School on Universal Logic - Call for papers Message-ID: <20061125125402.M91945@polimetrica.it> 2nd World Congress and School on Universal Logic - Call for papers Xi'an, China, August 16--22, 2007 This event is the second in a series of events whose objective is to gather logicians from all orientations (philosophy, mathematics, computer science, linguistics, artificial intelligence, etc) - people not focusing only on some specific systems of logic or some particular problems, but inquiring the fundamental concepts of logic. There will be a four days school with about 20 tutorials followed by a 3 days congress. Among the participants there will be Walter Carnielli, Hartry Field, Valentin Goranko, Vincent Hendricks, Wilfrid Hodges, Istvan N?meti, Gabriel Sandu, Stan Surma, Heinrich Wansing and many others. The deadline for submission of contributed papers is March 15, 2007. There will also be a contest with subject: how to translate a logic into another one? This event will take place in Xi'an, the ancient capital of China, just after the 13th LMPS to happen in Beijing. For further information, visit the website below: http://www.uni-log.org -- This message has been scanned for viruses and dangerous content by MailScanner, and is believed to be clean. From mr30 at st-andrews.ac.uk Mon Nov 27 11:06:39 2006 From: mr30 at st-andrews.ac.uk (Marcus Rossberg) Date: Mon, 27 Nov 2006 16:06:39 +0000 Subject: [FOM] =?iso-8859-1?q?Arch=E9_Abstraction_Workshop_XI=2C_St_Andrew?= =?iso-8859-1?q?s?= Message-ID: <24A79014-04BB-490B-8AE9-C91EF3A3415F@st-andrews.ac.uk> The eleventh and final Abstraction Workshop of Arch? research project concerning The Logical and Metaphysical Foundations of Classical Mathematics is taking place in St Andrews, 8-10 December 2006, under the title: Status belli: Neo-Fregeans and Their Critics. The workshop focuses on topics concerning the very heart of the Neo- Fregean programme. It revisits a variety of specific criticisms and trouble-spots, and evaluates what progress has been made on these issues, or might yet be made. Speakers include Roy T. Cook, Philip Ebert, Bob Hale, Jeffrey Ketland, Hannes Leitgeb, ?ystein Linnebo, Paul McCallion, Friedericke Moltmann, Nikolaj Jang Pedersen, Agust?n Rayo, Marcus Rossberg, Stewart Shapiro, Peter Simons, William Stirton, Alan Weir, Robert Williams, and Crispin Wright. The programme of the workshop and more details can be found here: http://www.st-andrews.ac.uk/~arche/pages/workshops/ abstractionwrks11.html Please email us if you would like to register for the workshop. --------------------------------------------------------------- Arch? - The AHRC Research Centre for the Philosophy of Logic, Language, Mathematics, and Mind School of Philosophical, Anthropological and Film Studies University of St Andrews St Andrews, Fife KY16 9AL Scotland, U.K http://www.st-andrews.ac.uk/~arche/ --------------------------------------------------------------- From jean-yves.beziau at unine.ch Tue Nov 28 03:59:57 2006 From: jean-yves.beziau at unine.ch (BEZIAU Jean-Yves) Date: Tue, 28 Nov 2006 09:59:57 +0100 Subject: [FOM] =?iso-8859-1?q?Logica_Universalis=3A_new_journal_of_logic_b?= =?iso-8859-1?q?eing_launched_by_Birkh=E4user?= References: Message-ID: Logica Universalis, New journal of logic being launched by Birkh?user. A new journal is now being launched by Birkh?user. The first issue will be on-line next week and in print by beginning of 2007. The journal will publish papers related to universal features of logics. Topics include general tools and techniques for studying already existing logics and building new ones, the study of classes of logics, the scope of validity and the domain of application of fundamental theorems, and also the philosophical and historical aspects of general concepts of logic. For further information, visit the website below: http://www.birkhauser.ch/LU From vladik at utep.edu Tue Nov 28 10:27:16 2006 From: vladik at utep.edu (Kreinovich, Vladik) Date: Tue, 28 Nov 2006 08:27:16 -0700 Subject: [FOM] LFCS'07 = Nerode 75 Message-ID: <77B4C8824930004AAC10E1B01576254AE2412F@itdsrvmail01.utep.edu> Forwarding. -----Original Message----- From: Sergei Artemov [mailto:sartemov at gc.cuny.edu] ************************************************************************ SYMPOSIUM ON LOGICAL FOUNDATIONS OF COMPUTER SCIENCE (LFCS'07) Revised Call for papers NOTE THE REVISED SUBMISSION DEADLINE New York City, June 4 - 7, 2007 URL: www.cs.gc.cuny.edu/lfcs07 Email: lfcs07 at gmail.com * Purpose. The LFCS series provides an outlet for the fast-growing body of work in the logical foundations of computer science, e.g., areas of fundamental theoretical logic related to computer science. The LFCS schedule is consistent with LICS and CSL timelines. * Theme. Constructive mathematics and type theory; logical foundations of programming; logical aspects of computational complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logics; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple agent system logics; logics of proof and justification; non-monotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; system design logics; other logics in computer science. * All submissions must be done electronically (15 pages, pdf, 12pt) via http://www.easychair.org/LFCS07/ * Submission deadline: December 11, 2006 * Notification: January 11, 2007 * Steering Committee. Anil Nerode (Cornell, General Chair); Stephen Cook (Toronto); Dirk van Dalen (Utrecht); Yuri Matiyasevich (St.Petersburg); John McCarthy (Stanford); J. Alan Robinson (Syracuse); Gerald Sacks (Harvard); Dana Scott (Carnegie-Mellon). * Program Committee. Samson Abramsky (Oxford); Sergei Artemov (New York City, PC Chair); Matthias Baaz (Vienna); Lev Beklemishev (Moscow); Andreas Blass (Ann Arbor); Lenore Blum (CMU); Samuel Buss (San Diego); Thierry Coquand (Go"teborg); Ruy de Queiroz (Recife, Brazil); Denis Hirschfeldt (Chicago); Bakhadyr Khoussainov (Auckland); Yves Lafont (Marseille); Joachim Lambek (McGill); Daniel Leivant (Indiana); Victor Marek (Kentucky); Anil Nerode (Cornell, General LFCS Chair); Philip Scott (Ottawa); Anatol Slissenko (Paris); Alex Simpson (Edinburgh); V.S. Subrahmanian (Maryland); Michael Rathjen (Leeds); Alasdair Urquhart (Toronto). From ruy at cin.ufpe.br Wed Nov 29 12:53:34 2006 From: ruy at cin.ufpe.br (Ruy de Queiroz) Date: Wed, 29 Nov 2006 09:53:34 -0800 Subject: [FOM] WoLLIC'2007 - Second Call for Papers Message-ID: <200611291753.kATHrQVJ030852@nlpi001.sbcis.sbc.com> Call for Papers 14th Workshop on Logic, Language, Information and Computation (WoLLIC'2007) Rio de Janeiro, Brazil July 2-5, 2007 WoLLIC is an annual international forum on inter-disciplinary research involving formal logic, computing and programming theory, and natural language and reasoning. Each meeting includes invited talks and tutorials as well as contributed papers. The Fourteenth WoLLIC will be held in Rio de Janeiro, Brazil, from July 2 to July 5, 2007, and sponsored by the Association for Symbolic Logic (ASL), the Interest Group in Pure and Applied Logics (IGPL), the European Association for Logic, Language and Information (FoLLI), the European Association for Theoretical Computer Science (EATCS), the Sociedade Brasileira de Computacao (SBC), and the Sociedade Brasileira de Logica (SBL). PAPER SUBMISSION Contributions are invited on all pertinent subjects, with particular interest in cross-disciplinary topics. Typical but not exclusive areas of interest are: foundations of computing and programming; novel computation models and paradigms; broad notions of proof and belief; formal methods in software and hardware development; logical approach to natural language and reasoning; logics of programs, actions and resources; foundational aspects of information organization, search, flow, sharing, and protection. Proposed contributions should be in English, and consist of a scholarly exposition accessible to the non-specialist, including motivation, background, and comparison with related works. They must not exceed 10 pages (in font 10 or higher), with up to 5 additional pages for references and technical appendices. The paper's main results must not be published or submitted for publication in refereed venues, including journals and other scientific meetings. It is expected that each accepted paper be presented at the meeting by one of its authors. Papers must be submitted electronically at www.cin.ufpe.br/~wollic/wollic2007/instructions.html A title and single-paragraph abstract should be submitted by February 23, and the full paper by March 2 (firm date). Notifications are expected by April 13, and final papers for the proceedings will be due by April 27 (firm date). PROCEEDINGS Proceedings, including both invited and contributed papers, will be published in advance of the meeting. Publication venue: Springer's Lecture Notes in Computer Science. INVITED TALKS: Foundations of Security by Veronique Cortier (LORIA Nancy) Topological methods in denotational demantics by Martin Escardo (Birmingham) Database theory, FMT, logic programming by Georg Gottlob (Oxford) Domain theory by Achim Jung (Birmingham) New directions (quantum algebras, biological systems) by Louis Kauffman (U Illinois Chicago) Quantum Computing by Sam Lomonaco (U Maryland Baltimore) Proof Theory, Proof Complexity by Paulo Oliva (London/QM) Biological computing (DNA, biological systems) by John Reif (Duke) Modal logic, logic of action by Yde Venema (Amsterdam) STUDENT GRANTS ASL sponsorship of WoLLIC'2007 will permit ASL student members to apply for a modest travel grant (deadline: April 1, 2007). See www.aslonline.org/studenttravelawards.html for details. IMPORTANT DATES February 23, 2007: Paper title and abstract deadline March 2, 2007: Full paper deadline (firm) April 12, 2007: Author notification April 26, 2007: Final version deadline (firm) PROGRAM COMMITTEE Samson Abramsky (U Oxford) Michael Benedikt (Bell Labs) Lars Birkedal (ITU Copenhagen) Andreas Blass (U Michigan) Thierry Coquand (Chalmers U, Goteborg) Jan van Eijck (CWI, Amsterdam) Marcelo Finger (U Sao Paulo) Rob Goldblatt (Victoria U, Wellington) Yuri Gurevich (Microsoft Redmond) Hermann Haeusler (PUC Rio) Masami Hagiya (Tokyo U) Joseph Halpern (Cornell U) John Harrison (Intel UK) Wilfrid Hodges (U London/QM) Phokion Kolaitis (IBM Almaden Research Center) Marta Kwiatkowska (U Birmingham) Daniel Leivant (Indiana U) (Chair) Maurizio Lenzerini (U Rome) Jean-Yves Marion (LORIA Nancy) Dale Miller (Polytechnique Paris) John Mitchell (Stanford U) Lawrence Moss (Indiana U) Peter O'Hearn (U London/QM) Prakash Panangaden (McGill, Montreal) Christine Paulin-Mohring (Paris-Sud, Orsay) Alexander Razborov (Steklov, Moscow) Helmut Schwichtenberg (Munich U) Jouko Vaananen (U Helsinki) ORGANISING COMMITTEE Marcelo da Silva Correa (U Fed Fluminense) Renata P. de Freitas (U Fed Fluminense) Ana Teresa Martins (U Fed Ceara') Anjolina de Oliveira (U Fed Pernambuco) Ruy de Queiroz (U Fed Pernambuco, co-chair) Petrucio Viana (U Fed Fluminense, co-chair) WEB PAGE www.cin.ufpe.br/~wollic/wollic2007 ---