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*To*: fom@math.psu.edu, csilver@sophia.smith.edu, rhersh@math.unm.edu*Subject*: FOM: Dialogue with Hersh re Silver's "Wagging Dogs"*From*: Joe Shipman <shipman@savera.com>*Date*: Thu, 17 Dec 1998 14:27:00 -0500*Organization*: Savera systems*Sender*: owner-fom@math.psu.edu

Subject: wagging dogs Date: Wed, 16 Dec 1998 17:26:24 -0700 (MST) From: Reuben Hersh <rhersh@math.unm.edu> To: shipman@savera.com, Charles Silver <csilver@sophia.smith.edu> Aristotle is said to have defined "man" as "featherless biped." Later philosophers pointed out that removing its feathers won't turn a chicken into a man. A modern definition might, as you suggest, be in terms of DNA. Of course that would be backwards, for by knowing what mammal's DNA to single out requires that we already can tell a man from a German shepherd. Going back to your tail-wagging dog, we ought to try a little harder to answer, how do we recognize a dog when we meet one? Tail-wagging isn't enough, a dog could be from a tailless breed, or had its tail chopped off by accident or malice. We might find that we simply can't explain in a complete and precise manner what is a dog. Nevertheless, we do know a dog from a dinosaur. This is a question of tacit knowledge, expounded by Michael Polanyi, and religiously overlooked and ignored by analytic philosophy. We know more than we can say. Since you have been kind enough to resurrect my name from fom'ish oblivion, let me say what I meant to say when I was trying to say something about these matters, with reference to mathematics and mathematicians. To me it seems clear that there are numbers, circles, triangles, and all sorts of mathematical objects. It also seems clear that there is only one universe--the physical universe to begin with, and then the mental and social universes rooted in and growing out of the physical universe. It seems clear to me that mathematical objects are not physical, for we do not detect them with our sense organs or with scientific instruments. It's even clearer that they are not mental, in the sense of the individual mind of a single person. But then, observing mathematics in real life, I could see that it was comprised under the heading of the social universe. At this point one meets the question, how is mathematics distinguished from other part of the social universe? I concluded that the answer was not in terms of mathematical subject matter--"math is the science of number and figure", as Noah Webster would have had it. There is no limit to the possible kinds of objects and ideas that mathematics can include. Neither would I accept a definition in terms of deductive logic, for deductive logic is important only in the last phase of mathematical work. Look under the heading of "Riemann" in my book for evidence that deductive proof is not the whole story. I concluded that the distinctive mark of mathematics is that it maintains virtual unanimity through all its manifold growths and transformations. Martin Davis argued that this distinction didn't work, because priests and rabbis also maintain unanimity. As to rabbis, it's simply false, as you can check any week with the Weekly Forward. As to priests. the history of the catholic church belies any such unanimity. Look up under heresies and heretics, of whom Maratin Luther was the most successful but by no means the only one. It seems to me, so far as I know, that mathematics is the only non-physical activity that maintains unanimity. Some of the physical sciences maintain virtual unanimity, that's why the definition of mathematics requires both the non-physical and the unanimous features. Now, isn't that just like Aristotle's featherless biped? Even if we never pluck a chicken, we know that featherlessness and bipedalism are not of the essence in defining a man. This would be proved, of course, if we come up with a plucked chicken. But even without physically exhibiting a plucked chicken, we know the definition is unsatisfactory. I understand and agree with the feeling that my definition of mathematics is unsatisfactory. Even if you grant my claim that any-thing non-physical and maintaining unanimity would necessarily be a branch of mathematics, that would not abate your dissatisfaction. The truth is that there are two kinds of definitions here, or two uses of definitions. One is classificatory--distinguishing the definiens from other definienda. The other is essentialist. Finding those qualities which make the definiens the definiens. In biological classification, the passage from ancient to modern is well known. The ancient classifications used the most obvious traits of an animal, or those traits of use to man, to make the definition. The comparative study of skeletons showed which animals are related to which, and which are not; the position in the scheme of relatedness became the definition. Saying math does not deal with physical objects does serve in part to locate it with respect to other disciplines. How to demarcate it from law, lit crit, theology, economics, sociology, anthropology? It seems to me its unanimity does the job. Still granting that just unanimity and non-physicality do not do complete justice to the essence of mathematics. Reuben Hersh Subject: Re: wagging dogs Date: Thu, 17 Dec 1998 10:25:38 -0500 From: Joe Shipman <shipman@savera.com> Organization: Savera systems To: Reuben Hersh <rhersh@math.unm.edu>, csilver@sophia.smith.edu References: 1 Dear Reuben, That is a very good summary. Would you object if I posted it on the f.o.m. forum? I agree that your definition of mathematics is classificatory rather than essentialist. However, you are also trafficking in essentialism when you talk about what mathematicians are "really" doing. This is still all right -- your classificatory definition (non-physicality and unanimity/consensus) is not incompatible with your essentialist conclusions (mathematics is "really" social). The most important disagreement many mathematicians have with you is that your account does not allow for an absolute notion of mathematical truth. Most important theorems are expressible as sentences in first-order number theory, and practically all the remaining important theorems are expressible as sentences in second-order number theory, and almost all mathematicians feel that a sentence in first-order number theory has an absolute truth value independent of anything physical, mental, or social, and most mathematicians feel the same way about sentences in second-order number theory. This is a stance you reject as "Platonistic". Charlie argues that unanimity is not enough, that it is possible for mathematicians to be unanimously WRONG and that your account does not allow for this possibility. This is relevant to both the classificatory and essentialist aspects of your work. If your classificatory definition is correct, then it is only possible to talk about a mathematical statement being "true" if the kind of unanimous recognition you require is never "overturned"; otherwise you do violence to the common meanings attached to the concept "truth". (Some people objected that your account did not allow the four-squares theorem to be called "true" until Lagrange proved it and his proof was generally accepted, but a temporal unfolding or expansion of the collection of statements deemed "true" is not incompatible with ordinary language, while the possibility of a "mathematically true" statement having its status changed to "not established as true" or even "false" is.) If you do not want to say that there is no such thing as mathematical truth, then you can only deal with the possibility of unanimity being overturned by either denying that this occurs in a serious way or by admitting that it is possible for mathematicians to be unanimously wrong. To eliminate the first of these alternatives, I called for examples, placing the restrictions that the problem have been considered important prior to the announcement of the "proof" (to ensure sufficient scrutiny), that the proof be readable by one person (to ensure a sort of reproducibility), that the "proof" was generally accepted for at least 5 years before the consensus changed (again to ensure that sufficient efforts were made), and that the example occurred in the 20th Century (because standards of rigor have been high in this century). Two examples, both from the theory of simple groups, have been suggested to me privately by f.o.m. subscribers and I am investigating them (one involves a serious gap in a proof, the other involves a result that actually turned out to be not only unproven but false). If such examples establish that mathematicians can be unanimously wrong, this poses a serious problem for your essentialist conclusions, because absent an absolute notion of truth, how is one supposed to be able to say that they are wrong? The first person to find that a generally accepted result is incorrect will know that they are wrong even *before* he manages to persuade them and change the consensus, and your account does not seem to allow for this. Thus you would have difficulty upholding ANY notion of "truth", even a non-platonic, socially-oriented one, for mathematical statements. (And you would also have to explain what happened when consensus is overturned, how a "mistake" could be possible, which would require you to start talking about logic and validity and all that other stuff that Charlie and I and others claim cannot be left out of any "essentialist" account of mathematics.) Best Regards, Joe Subject: Re: wagging dogs Date: Thu, 17 Dec 1998 10:34:27 -0700 (MST) From: Reuben Hersh <rhersh@math.unm.edu> To: Joe Shipman <shipman@savera.com> CC: Reuben Hersh <rhersh@math.math.unm.edu>, csilver@sophia.smith.edu Thank you for a courteous, thoughtful message. I don't mind if you transmit my previous message to the FOM list. Your concretization and testing of my ideas is interesting. I wish you well in this investigation. I think G.-C. Rota might be a good person to ask for counterexamples to my views. One of my unstated assumptions is that the present century is not the culmination of all centuries with respect to standards of mathematical rigor. A favorite example of mine is the Pasch axiom of betweenness in 2-d Euclidean geometry. From the 19th and 20th century vantage point, Euclid's omission of any axiom of betweenness made a significant part of his Elements "wrong," in the sense that the proofs were incomplete. But a more historical view would say that Euclid was "right", even though his proofs had gaps and he used implicit, unstated axioms. After all, we don't reject any of his theorems as "false." He was right in his terms, wrong in ours. I take it as not unlikely that future centuries will use different standards of rigor than we. Perhaps stricter, for instance rejecting our ambiguous use of "exist." Perhaps looser, for instance accepting as definitive computer proofs that leave us dubious. That would mean that the notion of "proof" would be historically dependent. If so, we should think of our own notion of proof as proof a la 20th-21st century. Citing what most mathematicians think would also be historically conditioned. By what right do we claim our opinions to be final and eternal? This doesn't touch the absolute notion of truth. I did not attempt and would not attempt to disprove that notion, any more than I would attempt to disprove any other transcendental, absolute belief. I would put the burden of proof on the other side. Why should we believe in absolute mathematical truth? Has anyone proved there is such a thing? The arguments for it are thought to be plausible, comforting, "obvious" but certainly not rigorous. I think a scientific attitude is to be skeptical about unseen realities, especially if belief in them is comforting, perhaps wishful thinking. If you want to broadcast this letter, go ahead. If not, not. Reuben Hersh **************************** Dear Reuben, Thanks for your response. I will send this correspondence to FOM. Whether or not the 20th Century is a "culmination", the level of rigor is certainly higher then in previous centuries; it would have been easy to find examples of unanimous wrongness before 1900 but I felt that would be less than fair. You raise an interesting point about future standards of rigor. Would anything we say now is a "gap" in a proof still be considered so by a future mathematician? In the case where the claimed result is actually false, obviously so. But if the proof is merely incomplete the gap may not be considered essential by a future mathematician. (For a historical example, Cantor's proof of the well-ordering theorem was incomplete but a modern mathematician who is well-versed in the Axiom of Choice might fill the "gap" immediately and regard it as not serious.) I am informed that one important case of the Feit-Thompson Odd Order Theorem was overlooked for many years; I need to check this out because while this is an extremely important theorem the gap may not be serious enough to count (I think the mathematician who found the gap also filled it in; if he had not been able to then it clearly would have been serious enough to count, while if any good finite group theorist would have been able to quickly fill it in had he found it one could argue that it was simply an unimportant omission of detail). I am also informed that a result of Suzuki's about finite simple groups turned out to be not only unproved but false after standing unchallenged for more than a decade; this would count as an example if the problem was important enough, but I can't venture an opinion until I track down the details. Euclid's proofs were "wrong" in the sense that he did not make all his axioms explicit; it is not as easy as one might expect to distinguish this case from, say, the false 19th-century proofs of FLT which wrongly assumed unique factorization in number fields. By reinterpreting Euclid's theorems as only being "about" the structures to which his implicit axioms applied and saying Euclid never intended to say anything about nonstandard models of his axioms one can defend him, but there was definitely a sense in which he was wrong when he chose his axiom set (some of the axioms he did make explicit are more obvious, and some less obvious, than the Pasch axiom, and I suspect that if it had been pointed out to him he would have recognized that it needed to be added). The reason we have a right to regard some our own notions of proof as having some eternal validity is that we have managed to formalize them so that they are themselves mathematical concepts. We can define an algorithm that outputs all and only the theorems provable in ZFC; this formalizable notion of proof is NOT, as you have clearly pointed out, the notion that mathematicians actually use professionally, but it serves as such a good "classificatory definition" (because nothing commonly regarded as proved is omitted by this algorithm and any output of this algorithm would [assuming feasible size] be regarded as proved) that we have the strong feeling that an "essentialist definition" must involve something similar. You are free to reject the notion of "absolute" truth; the question is whether your account of mathematics allows for ANY kind of "truth", because it seems that an essential feature of any kind of truth is incorrigibility. If mathematics is "really" social and mathematicians are unanimous that something is true for many years, how it is possible for them to be mistaken unless truth is independent of what they do? It is a defensible position to say that, for example, the twin prime conjecture may have no "absolute" truth value, while still maintaining that theorems of ZFC are "true"; but if mathematics is purely social then I cannot see how we can ever say any theorem is "true" in a sense that implies permanence. -- Joe Shipman

**Follow-Ups**:**Vladimir Sazonov**- Re: FOM: Dialogue with Hersh re Silver's "Wagging Dogs"

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