[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.


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