FOM: Reply to Shipman on arithmetic v. geometry

[Neil Tennant]

On Fri Oct  9 09:51 EDT 1998, Joe Shipman wrote:

> For both [arithmetic and geometry], there are two modes of
> investigation, the pure-mathematical and the empirico-physical.  You
> are contrasting these two modes rather than looking at the differences
> between arithmetic and geometry within each mode.
> 
> In the pure-mathematical mode, since the time of Descartes both
> arithmetic and geometry have had a unique intended structure (N and
> R^n).  

This misses out, I think, an important contrast even in the
pure-mathematical mode. What is the "structure" of R^n in the
geometric sense? Surely we have to specify what subsets of R^n
constitute its geodesics (straight lines)? Given just R^2, for
example, I need to know whether it's the projective plane, or the
affine plane, or the Euclidean plane, ... etc. that you are talking
about. Admittedly, mathematicians in Descartes's day might have just
assumed that the straight lines would be those subsets of R^2
answering to linear equations of the form ax+by=0 (a,b in R); but
today we would realize the arbitrariness of this further assumption,
even in the pure-mathematical mode.
 
> The problem was to discover the true first-order sentences of
> these structures (since we are talking about classical arithmetic and
> geometry; modern arithmetic and geometry deal with second-order
> sentences as well).  Categoricity is not the issue here--the point of
> finding good axioms was not so much to single out a unique structure
> that could satisfy them as to determine whether propositions were true
> or not.  

Agreed. But categoricity can be a very good regulative ideal as well!
For one thing, if one is successful in getting a categorical
axiomatization (at first order) then its completeness (hence also: its
decidability) is ensured. Moreover, the "cast of mind" in the search
for axioms (in order "to discover the true first-order sentences of
these structures") was that of focusing on a unique intended
structure, rather than, like a group theorist, say, trying to
characterize a class of non-isomorphic structures that sustain
interesting homomorphisms.

>The true first-order sentences of the intended structure were
> successfully logically captured by rigorous axioms in the case of
> geometry (with Hilbert taking the final steps a century ago, though I
> believe it was Tarski a bit later who proved that this axiomatization
> was complete and hence successful).  In the case of arithmetic Godel
> showed we cannot do this.  In both cases earlier axiomatizations were
> found incomplete and augmented, the important difference is that
> arithmetic turned out to be too complicated to be axiomatically
> understood by us in the same way geometry has been.

Agreed; I said as much in the posting to which the quote above is a
response. 

> The empirico-physical mode was not understood to be distinct from the
> pure-mathematical mode until Kant, and Gauss may have been the first
> to realize that (for geometry) the intended pure-mathematical
> structure was not necessarily instantiated by the physical universe.

I note with interest Joe's restriction "(for geometry)". Presumably he
would not enter the same claim for arithmetic. And that is the very
nub of the issue, it seems to me. (See my earlier posting in response
to Kanovei on counting.)

> This mode led to great advances in understanding in pure mathematics
> (Riemann) as well as physics (Einstein); but there is a disconnect
> between mathematics and physics because it is not at all clear
> (especially since the development of quantum mechanics) whether the
> uncountable and rigid mathematical real numbers are directly relevant
> to the physical universe ("officially" they are, but when you try to
> make this absolutely rigorous all sorts of problems arise).

This is an interesting point, but not within the immediate purview of
my earlier concerns. But the point bolsters my earlier view. For, if
physical space (or spacetime), as Joe is suggesting, is not in the
fullest sense a *continuum*, then the point about the contrast between
arithmetic and physical geometry is even sharper. For now, it would
seem, even the great range of currently available "pure geometries"
would not be adequate to the minutiae of the task of describing
physical space. One would be looking instead for some kind of
discretization of them. Fascinating!---but all the more indicative of
the *a posteriori* status of physical geometry.

> It is perhaps underappreciated that the empirico-physical mode of
> investigation is relevant to arithmetic as well.  It all started with
> counting pebbles and the like; ...

What started? Presumably, given the context, our hominid ancestors' slow
ascent to full conceptual grasp of the role that natural numbers can
and must play in the mature thought of all rational creatures.

> and although the principles we have
> abstracted from this mode and rendered in pure mathematics are quite
> compelling *to those of us whose intuition has been developed by a
> modern mathematical education*,...

I'm afraid, intellectual elitist that I am, that I am not much
interested in the intuitions of those who happen to *lack* a proper (I
hesitate to say: "modern") mathematical education. Indeed, what one
really yearns for is someone who has not only got the mathematics
under their belt, but the necessary *philosophical* education as
well. (No ad hominem implicatures intended!)

> we should remember that the physical
> universe may be finite and Friedman's theorem "there exists n such
> that all sequences from {0,1,2} of length n have i < j <= n/2 with
> s(i)...s(2i) a subsequence of s(j)...s(2j)" may be FALSE in some
> reasonably defined physical interpretation.  

I confess you have lost me here. What would a "reasonably defined
physical interpretation" BE? How do *theorems* about the natural
numbers get to be *falsified* in "reasonably physical defined
interpretations" within an ontologically challenged physical universe?

> I am sure Profesor
> Sazonov will have something more cogent to say about this possibility.
> We don't have alternative models of arithmetic as nice as the
> alternative models of geometry, but I don't see much epistomological
> difference here either.

I see a HUGE epistemological difference: one so great, in fact, that
anything that Professor Sazonov might say to try to disabuse me of
this view would have to be put under the analytic microscope even
before the printout ink had dried!


Neil Tennant