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Re: [HM] Magnitudes
Posted:
Nov 9, 2003 1:08 PM
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In the 1960s, motivated by exposure to foundations of geometry and various
"nontraditional" applications of logical and algebraic methods, I absorbed
some of the material covered in volumes edited by Luce: Readings in and
Handbook of Mathematical Psychology. They suggested this general question:
what conditions on a set M and ternary and binary relations S and L on
M suffice for the existence of a function f from M to the set of
nonnegative real numbers, such that for all a, b, and c in M, Sabc
iff f(a) + f(b) = f(c) and Lab iff f(a) < f(b). There is a lot of
literature on this subject, which I think is often called "measurement
theory". It goes back to Cantor, to foundations of geometry work around
1900 and to an extension of that, soon after, by Hölder. The literature up
to 1960 can probably be detected by following references in Luce's works. I
think there is considerable work in the area since then and even now. For
example, there is a research group at UC Irvine studying discrete
mathematics applied to social sciences.
--------------------------------------
James T. Smith
Professor of Mathematics
San Francisco State University
smith@math.sfsu.edu
http://math.sfsu.edu/smith
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