FOM: rigor and intuition

[Gordon Fisher]

Matthew Frank wrote:

>
> We--and especially we who are interested in foundations of math--should
> nurture our intuitions.  This could be an important function for
> foundations of math:  to help articulate various mathematical intuitions
> in such a way as to make them more useful in mathematical practice.
>
> --Matt

Which reminds me of a saying attributed to L E J Brouwer:

Logic creates nothing --  logic destroys.

I interpret the first clause to indicate mean that new things
in, say, geometry, don't arise from logical analysis, but
rather from intuition in the sense of Anschauung, which I
take to be close to some sort of internal visualization
or imaging (internal to a mathematician, that is).  If
memory serves me well, I think a statement of this sort
appears in the book Anschauliche Geometrie by Hilbert
and Cohn-Vossen.

As to the second clause, I can only guess that Brouwer
had in mind some sort of reference to analysis versus
synthesis in mathematics, or perhaps (more romantically)
the transition from geometric (or topological) insight
to strings of marks which constitute a proof of a theorem
based on that insight, or more elaborately an axiomatic
structure of some sort.

Gordon Fisher    gfisher at shentel.net