FOM: Reply to Wallen on "proof"

[JOE SHIPMAN, BLOOMBERG/ SKILLMAN]

This does need further explication on FOM.  The reason everyone (well, many
people anyway) took issue with Hersh was that according to his sociological
conditions proof was nothing more than the process by which mathematicians
arrive at a reproducible consensus about abstract statements.  Hersh was
misunderstood as failing to recognize the uniquely objective character of
mathematical arguments as opposed to other types, but in fact he not only
recognized this, he used it as his definition of mathematics. What he declined
to do is EXPLAIN what was special about mathematical proof despite proddings by
Machover and myself.  I used an example from chess to show that his purely
external definition of mathematics was insufficient to characterize it; even if
there were a better external definition (I suggested one based on the nature of
communicability of proofs) we still need a characterization of mathematical
proof that is based on what a proof is, internally, rather than the
sociological effect it has of reliably and reproducibly inducing consensus
about the statement to be proven.  -- Joe Shipman