FOM: Reply to Tennant on arithmetic vs. geometry

Joe Shipman shipman at
Fri Oct 9 15:31:47 EDT 1998

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.

Of course I meant what the mathematicians in Descartes's day meant.  And
the true first-order sentences about this structure (using Euclid's
primitive terms and a standard Cartesian translation between terms like
"right angle" and real arithmetic) are not just an "arbitrary" theory.
They are the propositions Euclid, Descartes, Kant, etc. had in mind all
along!  That some other geometries model a subset of these sentences is
no more significant than that other rings than the integers model a
subset of the true sentences of arithmetic.

> 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.
My restriction was because I was talking about Gauss.  See below.

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.
The contrast is not sharper because I go on to question the "a priority"
of physical arithmetic as well.  Again--I'm contrasting mathematical
arithmetic and mathematical geometry (and finding the epistomological
difference that geometry is decidable and arithmetic isn't), and then
contrasting physical arithmetic and physical geometry (and finding the
epistomological difference that physical arithmetic seems to correspond
more closely to mathematical arithmetic than physical geometry
corresponds to mathematical geometry--but this second contrast is a
matter of degree not of kind).

I think you inappropriately compare *mathematical* arithmetic with
*physical* geometry because you don't recognize that there is no
identity between the mathematical theory of arithmetic and the
arithmetic that is "true in the physical world".  Of course I don't have
a precise definition for such a "physical arithmetic", but there is no
precise definition of "physical geometry" either.  In both cases we
abstract principles from regularities in sense-impressions,  and the
difference is again a matter of degree.

I am NOT arguing that arithmetic isn't "a priori"!  I accept the
logicist thesis as far as arithmetic is concerned (and somewhat further,
as I have explained in a recent posting), and agree with you about
this.  Rather, I'm pointing out that the applicability of the rules of
arithmetic to objects in the physical universe is an empirically
observed phenomenon, and the possible finitude of the universe suggests
that an alternative "feasible arithmetic" or "bounded arithmetic" might
be applicable.

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?

One way they get *falsified* is by being used to predict results that
experiment doesn't support.  Wiles's proof tells me (if I make
"reasonable" assumptions about how my technology works etc.) that if I
program a computer to search for a counterexample to FLT it will fail to
find one.  But I'm not suggesting that real theorems of arithmetic can
be falsified, just that some theorems can't be verified (and hence their
negations can't be falsified)!  Friedman's number "n(4)" has been proven
to exist, but he has shown that in certain weak theories there is no
feasible proof of this, so adding "n(4) does not exist" to such a theory
gives an inconsistent but *feasibly consistent* theory.  Of course one
doesn't need to go so far as an inconsistent theory -- the point is that
we can't give the statement "n(4) exists" any meaningful physical
interpretation and it is conceivable that all the true sentences of
arithmetic that CAN be given a meaningful interpretation are also true
sentences of an alternate "bounded arithmetic" or "feasible arithmetic"
that is nonetheless different from classical arithmetic for other

So there is no "huge" epistemological difference as far as the
empirico-physical mode of investigation is concerned, just the smaller
ones I have mentioned.  For the pure-mathematical mode of investigation,
you could claim a significant difference in the a prioriness of
arithmetic and geometry because logic leads you to arithmetic pretty
directly but only to geometry via an elaborate detour through the
development of the real number system; I'll grant this because this
detour is far removed from the geometry qua geometry practiced by
Euclid.  On the other hand, for a mathematical realist (rather than a
logicist) the more important epistomological distinction is that
geometry is decidable and arithmetic isn't.  You're the
intuitionist--tell us what the intuitionist position is on this
arithmetic vs. geometry question.

-- Joe

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