[FOM] Certificates are fully practical
Alan Weir
Alan.Weir at glasgow.ac.uk
Thu Sep 26 19:07:22 EDT 2013
Thanks Tim:
>>Now if I understand you correctly, you're asking, why can't we take
consolation in the fact that every assertion has an infinitary proof or
disproof?
My reply would be that that's pretty small consolation, if any. "Finite"
is admittedly not a perfect stand-in for "feasible," but "infinite" is an
even worse stand-in.
Here's an analogy. A genie offers to make me rich beyond my wildest
dreams. "Hooray!" I say. The genie then adds that actually, it's really
only my posthumous estate that will be worth billions, and I won't see any
of the money during my lifetime. I am disappointed, but at least happy
for my children. Do you think I will be cheered up if the genie says,
"Just kidding---you'll be extremely rich only after an infinite amount of
time passes"?
Tim<< (FOM Digest, Vol 129, Issue 25)
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).
Note that it is not just that there are proofs which no single generation of mathematicians (and allowing for the use of computers) could concretely realise, but we can pass the baton of the task to our children and future generations (like the building of a medieval church). There are proofs with more steps than the estimated length of the universe from big bang to heat death divided by the Planck time (or some other gargantuan number along those lines.)
Godel's speed-up results seem relevant here, even if we confine ourselves to very simple recursive functions like multiplication by a scalar: if k is the highest-order logic which will ever be used by any mathematicians in the universe (so I am presupposing this is finite) then there will be theorems whose shortest proof in k is c.n, where c is some giant constant, such as the number of Planck periods in the universe supposing a heat death but whose shortest proof in k+1 logic is n. If n is less than c then there could be a token of the theorem which is universe-sized but no token of a proof in any system ever used that can 'fit' in the universe. This isn't a fully rigorous thought I admit, but isn't it very plausible that there will be sentences we could grasp whose proofs (or disproofs) we will not be able, even in a wide sense of 'able' which is nonetheless consonant with physical law, to grasp?
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. And my point is not, of course, that infinitary proofs are unproblematic, from a naturalistic, anti-platonist standpoint. Rather it is that all but a tiny fragment of finitary proofs are problematic from that perspective; the suggestion is that it is unclear why a satisfactory resolution of that problem (if there is one) will apply only to finitary idealisations and not infinitary ones. Hope that's clearer.
Alan
Professor Alan Weir
Roinn na Feallsanachd/Philosophy
Sgoil nan Daonnachdan/School of Humanities
Oilthigh Ghlaschu/University of Glasgow
GLASGOW G12 8QQ
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