[FOM] Platonism and Formalism

Torkel Franzen torkel at sm.luth.se
Fri Sep 19 00:02:06 EDT 2003

Karl Podnieks says:

  >b) Advanced Formalism - on weekends - when I'm thinking "about" mathematics
  >(otherwise, I will end up in mysticism).

  I tried this, but it did not, alas, work out. On a weekday, I was
explaining how we use "probable primes" in cryptography. I found
myself saying that it was very likely indeed that a number obtained in
this way was in fact a prime, when my Sunday self suddenly made me
brutally aware that I was talking nonsense - I was talking as though
whether or not a particular number - some kind of abstraction
associated with a string of digits - is a prime is a question of fact,
a question concerning the intrinsic properties of the number,
independent of any calculations that may or may not be carried
out. Thus reproached by my philosophical self, I made the mistake
of trying to explain the concept and use of "probable primes" in
philosophically acceptable terms. Having had a computer carry out
certain calculations, conventionally described as "showing that k
is a prime with probability p", we use the number k in a certain way,
and tend to assume that certain other things will not happen in
the future, in particular we do not expect any other computer calculation
to produce a result conventionally described as "showing that k is
not a prime", at least not if that calculation is "respectable" in
a sense not easily explained. Now, why do we have these expectations?
I found I couldn't say. Having discarded the meaningless notion of
k "in fact" being a prime, and of an objective probability that this
supposed fact obtains, I simply didn't see any connection between
calculation and expectation, or any theory or principle to explain
what I was doing. Making feeble excuses, I hurriedly left the room.

  >Which properties of structures and methods used in mathematics and
  >metamathematics are leading to the illusion that the natural number
  >system is a stable and unique mathematical structure that exists
  >independently of any axioms and cannot be defined by using axioms?

  I think the more interesting question is how the (Sunday) conviction
arises that the properties of the natural numbers are somehow
indeterminate or ineffable! I hope to arrive at a better understanding
of this (but not of course to convince anybody of anything contrary
to their Sunday meditations).

Torkel Franzen

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