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Matheology § 221
Posted:
Feb 13, 2013 11:46 AM
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Matheology § 221
As I get older I seem to be getting more and more relaxed about
foundational issues. I?m happy to see people formalize and explore all
imaginable attitudes toward the foundations of mathematics. I feel
confident that the more interesting axiom systems will eventually
attract more researchers, while the less interesting ones will remain
marginal. I am not eager for one system to prevail over all others ?
nor do I feel any desire for systems I dislike to go completely
extinct. It?s a lot like my fondness for biodiversity. I enjoy the
diversity of life, and am very happy there are tigers, and would be
sad for them to go extinct, even though I wouldn?t want a bunch
running around in my back yard.
In particular, I?m glad there are ultrafinitists, because I suspect
that only someone with views like that could be motivated to prove the
inconsistency of (say) Peano arithmetic, and seek plausible strategies
for doing it.
If everyone believes Peano arithmetic is consistent, and it?s not,
we?re in big trouble because it?ll take us a long time to discover it.
So we need a few lonely people working on the other side of this
issue. I don?t think they?ll succeed, but I?m glad they?re trying.
Even if they don?t succeed, there could be some interesting concepts
and theorems that only they are likely to find.
Finally, I don?t expect these people to take the same ?relaxed,
balanced? attitude that I have. I suspect that only someone with
strong opinions could possibly be motivated to spend a lot of time
developing ultrafinitism, or trying to prove the inconsistency of
Peano arithmetic. Expecting them to share my relaxed attitude is a bit
like expecting a tiger to be an environmentalist.
[John Baez: "The Inconsistency of Arithmetic", n-Category-Cafe, Sept.
30, 2011]
Regards, WM
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