[FOM] Re: Disaster?

[Dan Goodman]

Timothy Chow wrote:
> In that case, I strongly suspect that the contradiction would
> *not* be shrugged off by most mathematicians.  There would be
> only finitely many instances of the first-order induction
> axiom used, and the arguments would use standard principles
> of classical logic, leading to any desired (first-order
> number-theoretic) conclusion whatsoever.  What would people do?

A number of ideas occur to me. I agree that the first response would be
to look at things like quasiconsistent logic, but if one encountered
endemic and systematic inconsistencies, you'd need another approach.

How about: switching attention from theorems to proofs. The problem with
an inconsistency is that you can never say "such and such a theorem is
true" because it would also be false. However, you could still say "such
and such a proof is valid" whether or not the system was inconsistent.
Of course, that on its own would be of little interest, you'd have to
also find a way of taking such a viewpoint and making it interesting to
study. But it has one immediate advantage: if you 'prove', say, that
Poincare's conjecture is true by saying "Suppose it were false: we
already know that p&~p, a contradiction. Hence it must be true" (where
we imagine p&~p to be some rather artificially constructed inconsistency
we already know about), this is obviously a very different thing to
proving it using geometric or analytic methods. Perhaps this idea can be
formalised, perhaps not.

Another idea along a similar vein would be to switch our attention from
theorems to calculations. We already know that many types of calculation
'work' (e.g. the computers seem to work most of the time, buildings
don't collapse, etc.), so perhaps we can develop the notion of
calculation further without falling into inconsistencies. It might need
a great deal of ingenuity to find a way to approach studying things like
analysis and geometry from this perspective, but perhaps it could be
done. (Incidentally, this would be akin to the sort of approach to the
foundations of mathematics that Wittgenstein took.)

These are the only two ideas that have occurred to me that might lead to
a mathematical system that is as precise and formal as the current one.
There are lots of other approaches which would mean making mathematics a
bit more wishy washy and informal, but these would presumably be last
resorts.

In general, one would try to retain features of mathematics as we know
it. The examples I gave above both try to preserve the features of
formality, precision and definiteness. Others might try to preserve
different features, like 'usefulness in science' or 'beauty'.

Dan Goodman