FOM: Arithmetic vs Geometry : reply to Tennant

[Joe Shipman]

Neil Tennant wrote:

> The *geometrical* problem has this flavor:  Out there is, presumably, a unique
> geometrical structure that we call SPACE (or SPACETIME). Let's conduct empirical
> investigations to find which (from among many possible competing theories) truly
> describes it.
>
> The *arithmetical* problem has this flavor: "Out there" (metaphorically, now!)
> is a unique structure that we call THE NATURAL NUMBERS. Let's conduct purely a
> priori investigations to axiomatize successively stronger and more inclusive
> subtheories of its theory, knowing that the job will never be complete.
>
> Isn't that still a substantial contrast between the two kinds of theorizing?

Neil, the epistemological difference between arithmetic and geometry is more
subtle than this.  For both, 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).  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.  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.

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.  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).

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; 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*, 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 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.

A final remark: if you take the position that addition is a "feasible" operation
and multiplication is not, and try to develop a theory of arithmetic based on
addition without multiplication, something nice happens in both modes of
investigation: the physical universe may be a better model of the theory, and the
theory itself is mathematically complete (Presburger Arithmetic)!  Presburger
Arithmetic (the first-order-theory of addition) is a highly nontrivial theory; the
decision procedure is superexponential.  (Can anyone give a precise comparison
between the computational complexities of Presburger arithmetic and elementary
geometry?  I don't remember which is more infeasible.)

-- Joe Shipman