FOM: geometry

[Neil Tennant]

Robert Black wrote (Thu Oct  8 12:23 EDT 1998)
 
 > 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.
 
Surely the kind of 'structuralism' at issue in the case of arithmetic is
different from that in the case of geometry?  As I understand it,
structuralist philosophers of mathematics (my colleague Stu Shapiro comes
to mind here; but presumably one could include also Resnik and Hellman)
propose 'stucturalism' as a response to the Benacerraf problem---namely,
that no *particular* sequence of sets has any privileged claim to 'be'
'the' natural numbers. Rather, all that there "is" to the natural number
sequence is its pattern, or structure, which could be instantiated by
many different kinds of things. This kind of structuralism, therefore,
can be advanced even in the case of a mathematical structure of which we
take ourselves to have a *categorical* conception (as we do for the natural
number sequence, even if it means advancing a second-order characterization).

But the structuralism that arises in the case of geometries has more to
do with there not being any correspondingly categorical conception of 
structured  geometrical space. Metrics can vary (Euclidean v. Riemannian, say), 
as can incidence relations (affine v. projective, say). Thus, geometries
may perhaps be better likened to groups, rings, fields etc., whose 
theories do not aspire to characterize any of them categorically.

So, to the extent that one can still hang on to a *categorical* conception
of the natural numbers, but cannot do so for the spaces treated in geometry,
perhaps this use of 'structuralism' hides an important ambiguity.

Neil Tennant