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.
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