[FOM] Certificates are fully practical
Nik Weaver
nweaver at math.wustl.edu
Sat Sep 28 15:57:28 EDT 2013
Alan Weir wrote:
> In general, I remain to be convinced that there is a sense of 'in
> principle possible' which is at all helpful in philosophy of
> mathematics, one in which, for example, all finite proofs are 'in
> principle' graspable but no countably infinite one is.
Alan, this passage makes it seem like you feel the goal of the
philosophy of mathematics is to justify finitary mathematics and
discredit infinitary mathematics ... and the notion of "possibility
in principle" is not helpful toward that goal. Surely this is not
what you meant?
It seems to me that if we don't begin with a preferred outcome in
mind, then the notion of possibility in principle could be very
valuable in helping us to answer foundational questions.
For instance, E. B. Davies has described a persuasive scenario in which
a machine performs one step of a computation, then builds a smaller,
faster version of itself and instructs it to perform the next step.
The envisioned result is a cascade of ever-smaller and -faster machines
which collectively perform an infinite computation in a finite amount
of time. Thought experiments like this could help convince us that
countable computations are mathematically legitimate, regardless of
whether or not they are physically possible in our universe.
(E. B. Davies, Br J Philos Sci (2001) 52 (4): 671-682.)
Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
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