[FOM] The boundary of objective mathematics
Timothy Y. Chow
tchow at alum.mit.edu
Thu Mar 12 11:49:21 EDT 2009
Paul Budnik wrote:
>The Continuum Hypothesis is an obvious example of a statement that does
>not meet this condition. Intuitionists are more conservative (at least
>in terms of what proofs they accept) but most mathematicians are, I
>suspect, far more liberal in the mathematics they believe is objective.
>Yet there is increasing scepticism about the objective truth of the
>Continuum Hypothesis and similar statements.
I'm curious about this "increasing scepticism" that you speak of. Do you
have any statistical evidence of increasing scepticism? Or it is just a
rhetorical flourish?
>For those that question the objectivity of the Continuum Hypothesis,
>what do you think of this proposal for objective mathematics. If the
>answer is not much, where would you draw the boundary and why?
If you're looking to physics for objectivity, then it doesn't seem to me
that your boundary is the obvious choice. In one direction, I could argue
that as far as we know, the continuum hypothesis might play a key role in
the physical world.
Your reaction to this suggestion might be that the problem is, even if the
continuum hypothesis does play a key role in the physical world, there
isn't any way we could *know* it, given the limitations of our biological
existence as human beings. That is, when you speak of physics, you're
really thinking *epistemologically* rather than *metaphysically*. In that
case, I could argue in the other direction: The natural place to draw the
boundary is where the ultrafinitists draw it. Potential infinity, the
argument goes, is no less a human mental construct than uncountable sets
are. There's no direct counterpart to potential infinity in our physical
experience; we have to *imagine* some scenario that might go on forever.
So if you choose to draw the line at recursive enumerability, it seems you
will need to defend it on grounds other than "physics"---that is, until
you develop a more careful philosophy of physics.
Tim
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