[FOM] Mathematical Certainty?
David Corfield
david.corfield at philosophy.oxford.ac.uk
Mon Jun 23 12:22:05 EDT 2003
Dan Goodman wrote:
> Ideally we want to make our proofs such that isolated small
> mistakes don't ruin the entire proof. What's interesting is
> that informal proofs are often more stable than purely formal
> proofs because they are driven by ideas rather than purely
> symbolic calculations, and if the ideas are right then a
> mistake in the formalism should be easily corrected.
Which is part of the reason why, when asked what provides
the main evidence for the Riemann Hypothesis, mathematicians
point to a not-quite-worked out analogy to another field
rather than to the 1.5 billion verifications. Even a gappy network
of ideas can be quite convincing.
>The
> slightly counterintuitive consequence of this is that maybe
> we should actually prefer a slightly informal proof driven by
> strong ideas to a formal proof checked by a computer.
An excellent way of putting it.
Taking things a stage further, one could take proofs in a field to be
checks that we've got the strong ideas right and linked properly.
Just one further step and:
"The quest for ultimate triviality is characteristic of
the mathematical enterprise." (Rota, Indiscrete Thoughts, 93)
David Corfield
Faculty of Philosophy
10 Merton St.
Oxford OX1 4JJ
http://users.ox.ac.uk/~sfop0076
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