[FOM] Certificates are fully practical
Timothy Y. Chow
tchow at alum.mit.edu
Sat Sep 28 14:49:13 EDT 2013
Alan Weir wrote:
> I prefer the following story: a crazed gunman enters your class and says
> he'll kill you unless you count from one to 10 to the power 35- with no
> gaps- in the next minute. A student helpfully remarks 'well, it could be
> worse: he could be asking you to count to epsilon_0' (and then starts
> whistling 'always look on the bright side of life' from Life of Brian).
What I believe your parable shows is the following: We know already that
the continuum hypothesis is neither provable nor disprovable in ZFC. It
would be interesting to strengthen this result by showing that there is no
infinitary proof or disproof either, in some suitable sense of "infinitary
proof."
That is, "infinite" is not a perfect stand-in for "infeasible"; perhaps a
stronger notion of infinity is a better one. But this doesn't contradict
my point that "infinite" is a worse stand-in for "feasible" than "finite"
is.
> 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.
Like most radically skeptical positions, yours is unassailable if you
insist on digging in your heels. I guess you would also argue for the
following positions:
1. "Polynomial time" is a problematic model of feasible computation.
So hypercomputation is a better model.
2. "Natural numbers" are a problematic model of ordinary counting.
So infinite ordinals are a better model.
Tim
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