FOM: Reals and reality -- reply to Machover
JSHIPMAN@bloomberg.net
JSHIPMAN at bloomberg.net
Wed Dec 17 10:06:21 EST 1997
Your point that to give an objective validity to, e.g., the proof of the
infinitude of primes there must be a process which causes the socially
constructed notion of mathematical truth to converge to something objective is
an excellent one, and interaction with an external world plus selection is a
fair guess for this process. For statements about integers I would be willing
to admit this as a hypothesis. I'm afraid our intuitions about "continua" may
not lead to the same type of objectively valid proof--though I won't rule out
the possibility that our intuition somehow accesses a Platonic world of
mathematical forms, I think the intuitive properties of continua we "see" which
lead us to model them with the real numbers don't correspond to anything in the
physical external world. These key properties are density (between two points
there is another) and connectedness (if you chop the line anywhere a point is
clinging to one of the edges). Strengthening or weakening these postulates leads
to less tractable systems of analysis (e.g. recursive analysis, Conway's
surreals, a finite grid). Where, really, do the "real" numbers exist?-JShipman
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