Dear Lee, YOU ASKED: Did you read the article by Lorenz?
YOU COMPLAINED: (I wish *someone* would; But did you actually SEND the link to the Lorenz article? It wasnt attached to the message I got. N Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([email protected]) http://home.earthlink.net/~nickthompson/naturaldesigns/ http://www.cusf.org [City University of Santa Fe] > [Original Message] > From: <[email protected]> > To: The Friday Morning Applied Complexity Coffee Group <[email protected]> > Date: 4/25/2010 2:26:49 PM > Subject: Re: [FRIAM] Why are there theorems? > > On 25 Apr 2010 at 10:51, Russ Abbott wrote: > > > In answer to Eric and lrudolph, the answer I'm looking for is not related to > > epistemology. It is related to the domains to which mathematical thinking is > > successfully applied, where successfully means something like produces > > "interesting' theorems. (Please don't quibble with me about what *interesting > > *mean -- at least not in this thread. I expect that *interesting *can be > > defined so that we will be comfortable with the definition.) What is it > > about those domains that enables that. > > Did you read the article by Lorenz? (I wish *someone* would; so > far I haven't had any takers closer to home, which is one reason > I sent it to Friam. Content aside, it's a fun article!) It > does suggest an answer to your question, I think: humans' capacity > for "mathematical thinking" evolved to be useful for human survival > in the world; so did humans' capacity for attributing different > degrees of "being interesting" to different things and structures > in the world; so thinking effectively (i.e., mathematically) about > "interesting" things Builds Better Bodies^WSpecies Two Ways. > > Yes, that answer (or anything along its lines) does leave open that > other species might evolve so as to have "minds" that engage in > "mathematical thinking" that is quite different from human > "mathematical thinking". Lorenz suggests as much (with a > rather far-fetched imaginary example of how "counting numbers" > might not be "interesting" had things been otherwise). > > I'm not a philosopher (I'm a mathematician, who has proved quite a > few interesting theorems in my day [1]) so I probably shouldn't allow > myself to use a word like "epistemology", whose definition I am > never quite sure of--much less a coinage like "Evolutionary > Epistemology". Let's just take that word off the table for now. > > Like you, I am interested in "the domains to which mathematical > thinking is successfully applied", and I would like to know "what is > it about those domains that enables that". It was through my pursuit > of a satisfying answer (satisfying to me, of course) that I recently > (last week or so) found Lorenz's article. I will now describe the > path that got me there. > > First, I had been (probably since college) vaguely aware of > the famous title of the 1960 article by the mathematical > physicist Eugene Wigner, "The Unreasonable Effectiveness of > Mathematics in the Physical Sciences". Then, in 1978, I read > (in an endnote to a review in the Bulletin of the AMS of a book > on Wittgenstein) Georg Kreisel's off-hand one-liner in response > to Wigner: "(Would it be \textit{obviously} more `reasonable' > if we were not effective in thinking about the external world > in which we have evolved?)" That response, of course, > conflates "effective thinking" with "mathematical thinking", > but I can live with that; and it strongly suggests an answer > to your question about finding a characterization of "the > domains to which mathematical thinking is successfully > applied", namely, that they are (or at least necessarily > include) the domains for which "effective [i.e., mathematical] > thinking" promotes species survival in "the external world in > which we have evolved". (If there are also domains full of > *interesting* theorems that don't, and never will, lead > to "effective thinking" about any aspect of our world-- > and there may be--they can be treated as "spandrels".) > > I didn't think much more about the subject until four or > five years ago, when I was commissioned to write an article > on non-standard mathematical models of time that might be > useful to psychologists. While doing the (non-mathematical) > research necessary for that article, including a lot of > observations of psychologists in their native habitats, > I noted that no one has ever made a claim for the > "unreasonable effectiveness of mathematics in the social > sciences", and that anyone who did would be rightly > laughed at (except, possibly, in an economics department). > > Furthermore, most attempts, including attempts by some > *very* good mathematicians (like Rene Thom), as well as by > a fair number of fraudsters, hacks, and mystagogues, > to apply (much) mathematics to (much) social science, > have come to nothing (except, in some cases, to pseudoNobel > prizes in economics). What is it about the domain of > "social science" that seemingly *disenables* any serious > use of theorem-thinking? > > A few weeks ago, I found that Kreisl's point had been made > by Lorenz already in 1941--37 years before Kreisl made it, > and 19 years before Wigner's article! It was really, really > hard to get my hands on Lorenz's paper (for some reason, > not a lot of US libraries have German philosophical journals > from 1941...), so when I did get it, I wanted to spread it > around. > > As to what it might be about social science that makes it > resistant to mathematical thought, maybe it's because life > on earth has had a lot longer to adapt to the physical world > than to the (human) social world (for all I know, ants have > a well-developed mathematics of ant social science). > > Lee Rudolph > Professor of Mathematics and Computer Science > Clark University, Worcester MA > > [1] Leaving aside the precise number of theorems I've proved, > interesting or otherwise, I can quite accurately say that I've > had at least 4 good ideas, all of which continue to generate > new (and--to me!--interesting) theorems (proved and published > by others, none of whom belong to the empty set of my own > graduate students), in the case of the three oldest of the > ideas ("links at infinity", "quasipositivity", and "braided > surfaces") over 30 years since I had them--which is *many*, > *many* half-lives of the typical Modern Theorem. So I can > be interesting to mathematicians, anyway. > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
