FOM: Objectivity and Truth in Maths
Edwin Mares
Edwin.Mares at vuw.ac.nz
Fri Jan 16 22:08:21 EST 1998
Ruben Hersh says:
"Is the Pythagorean theorem about the material world or about
some ideal immaterial things? Euclid already knew it was about ideal
things. But the meaning of it, the interest of it, the use of it, all
have to do with material things. For instance, right triangles
carefully drawn on the ground. No matter how big. But if a triangle
is big enough, the curvature of the Earth will have a measurable
effect, and the Pythagorean theorem becomes false. Moreover, there
is no reasonable way of deciding when it becomes false. To a high
enough degree of precision, it is false even for the most carefully
drawn right triangle on the ground."
It is true that we can't draw perfect Euclidean triangles on a curved (and
bumpy) surface like that of the earth. But what are we to conclude from
this? Consider the laws of nature. If they have mathematical form, as modern
physics would suggest, it would seem that at least some of maths is about an
extramental world. Now, you might want to say that the "laws" are themselves
human contrivances that only approximate physical phenomena. This is
suggested by the passage quoted above. The meaning of "approximate" here,
however, is hard to understand. If phenomena have real measurements, then
there is at least a mapping from mathematical structures like the real
numbers onto physical phenomena. Thus, it would seem that at some parts of
maths are (perhaps indirectly) about phenomena that we didn't create. If you
want to deny that real things have magnitudes that are independent of our
measuring them, then you are buying into a very strong form of anti-realism
about maths and science. If you accept that they have magnitudes, it would
seem that your position is much less radical than you make out. It
resembles, at least at first glance, Hartry Field's fictionalism.
Ed Mares
Ed Mares
Department of Philosophy
Victoria University of Wellington
P.O. Box 600
Wellington, New Zealand
Ph: 64-4-471-5368
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