FOM: geometry

Robert Black Robert.Black at
Thu Oct 8 10:59:01 EDT 1998

The historical picture given in Mic Detlefsen's long and interesting
posting of 2 October seems to me to be pretty accurate, though of course
one could quibble over certain details.  I'm a bit puzzled though by the
questions he thinks it would be profitable to discuss, in that it seems to
me he's posed these questions in a way which might have seemed natural to
Frege/Hilbert/Poincare but doesn't seem very natural today.  Let me just
take the first one:

1: Is or should the asymmetry between arithmetic and geometry that Gauss
and nearly all other 19th century foundational thinkers believed in still
be treated as a fundamental 'datum' of the foundations of mathematics today?

Kant's view that we know a priori that euclidean geometry is true of
physical space is now dead as a dodo, so we can leave applied geometry,
'geometry as a branch of physics' as Steve puts it, to one side.  So far as
pure geometry is concerned, ever since Bolyai and Lobachevsky we have had a
plurality of geometries, and now we have euclidean, non-euclidean, affine,
projective, riemannian, pseudoriemannian and God knows what else, each in
as many dimensions as you might happen to want.  Particularly in the case
of projective and affine geometries we have synthetic axiomatizations in
terms of incidence etc.  and analytic coordinatizations, i.e. models of
these axiomatizations in algebraic structures, together with theorems
relating the geometric to the algebraic point of view (e.g. Pappus' theorem
holds in a projective geometry iff the field underlying the
coordinatization is commutative).

As a result of all this, I'd have thought that *everybody* would agree that
the modern approach to (pure) geometry is structuralist, the subject matter
of geometry being a *plurality* of abstract structures.  Hilbert's
'axiomatic method' is paradigmatic for this way of viewing things, though
of course it goes back to Dedekind and Riemann, perhaps even to Gauss.  It
would have been totally foreign to Kant, however, and Frege had difficulty
with it, as is clear from his exchange of letters with Hilbert.

If one also takes a structuralist attitude to arithmetic, as many of us do,
then it would seem that there is no asymmetry between arithmetic and
geometry left.  So it seems to me that Mic's question boils down to:
should we be structuralist about arithmetic?  And the arguments on both
sides of that question are pretty familiar.

I'd like to ask another question about geometry though, roughly, just how
does it fit into the overall structure of modern mathematics?  Geometrical
thinking is all-pervasive - e.g. every time one uses linear algebra one is
in effect thinking geometrically.  Further:  at least differential geometry
and algebraic geometry are major research areas.  But Bourbaki, for
example, identifies the major structure-types of modern mathematics as
algebraic or topological:  there's no volume called 'Geometrie Generale'.
Indeed Bourbaki clearly regards synthetic geometry as dead, or at best as
no more than an occasionally useful language for expressing pieces of
algebra - see in particular in his 'Elements d'histoire des mathematiques'
the chapter 'Formes quadratiques: geometrie elementaire'.

The most general definition of geometry that I'm aware of is: a geometry is
a set with a symmetric and reflexive 'incidence' relation.  I suppose some
idea like that would come at the beginning of 'general geometry' the way
the definition of a magma comes at the beginning of Bourbaki's algebra.

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?

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845

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