FOM: geometry

[Martin Davis]

At 03:59 PM 10/8/98 +0100, Robert Black wrote:


>So my (twofold) question is:
>
>1.	Should we identify and give separate treatment to 'geometrical
>structures' as basic to modern mathematics, and
>2.	Why is the geometrical mode of thought - a mode abstracted from our
>thought about the very special example of 3-dimensional physical space - so
>pervasive in abstract mathematics?
>

It seems to me that most mathematical progress has leaned on a small number
of intuitions derived from real-world experience: counting, space, chance,
force, motion.

It is striking how useful some of these intuitions have been in mathemtical
domains far removed from direct experience. Take the experience of the
effect of chanciness on us all. This leads to games of chance and then to
probability theory. Lo and behold! Probabilistic reasoning turns out to be
very important in number theory where presumably everything is entirely
deterministic and chance is irrelevant. Similarly on our limited experience
with physical space rests the myriad instances of geometrical thinking in
vast areas of mathematical research.

The finding that much of classical analysis is conservative over PA may well
reflect the paucity of underlying intuitions. The challenge that I like to
call "G"odel's Legacy" of learning how to use the higher transfinite methods
that we know will have arithmetic consequences, by developing appropriate
new intuitions to guide us is, I continue to believe, the most important
issue of our time for f.o.m.

Martin