In conversationally questioning the mathematical interest of the
Dedekind-Peano axioms [and formal logical systems generally],
Gian-Carlo Rota once remarked that number theorists he knew consider
using mathematical induction in a proof to be "tasteless" [I think
that was his term -- it may well have been "childish"]. He either
then conjectured or asserted that mathematicians generally consider
mentioning or explicitly mentioning/drawing on a formal logical
principle "tasteless" [again it may have been "childish"]. I think
that the point in both cases was that the need to do so implied a
weakness of one's mathematical insight.
I am curious about whether this is a true observation and if so how it
should be understood in terms of mathematical practice; at the time I
thought Rota was just being impish or provocative, but I've come to
have second thoughts.
Robert Tragesser
(Dry)West[running]brook, Connecticut