FOM: Reply to Shipman on arithmetic v geometry

[Neil Tennant]

Joe Shipman makes the following clarification concerning what he meant
by the geometry of R^n:

> 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! 

OK, so we're dealing here with the Euclidean geometry of R^n.
The pure mathematical exercise of deriving its theorems will be a
priori. But it is an empirical question, as everyone concedes, whether
it correctly describes physical space, or spacetime, upon the most
convenient identification of what, in physical space, count as the
'straight lines' etc. of the geometry.

As I understand it, the application of geometry for the description of
physical space involves *determining what geometrical theory to use*
in our description of the structure of physical space. That is, we
have to choose between Euclidean v. Riemannian v. ... [continue this
list with the names of all those geometries that have ever been put
forward]. 

Now let's contrast this with arithmetic. There's the "pure" theory of
arithmetic---say, PA, in order to be specific (even though, as we all
know, it is incomplete). [There is still a deep and interesting
question, however, as to what is so special/significant/privileged
about this incomplete fragment of arithmetic; but that would deflect
the present discussion.] Proving results in PA is an a priori
matter. So also is proving that certain things cannot be proved in PA.

Thus far we have covered three boxes in the following four-box
diagram:

		Pure		Applied/Physical

Geometry	Eucidean	Euclidean/Riemannian/...

Arithmetic	PA		??

The symbol ?? is my way of asking what, in the arithmetical case,
would be analogous to the project of determining which geometry best
describes physical space.  Joe Shipman wrote

> ...I question the "a priority" of physical arithmetic as well.
> Again--I'm contrasting mathematical arithmetic and mathematical
> geometry (and finding the epist[e]mological difference that geometry
> is decidable and arithmetic isn't), and then contrasting physical
> arithmetic and physical geometry (and finding the epist[e]mological
> 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'm still in the dark, though, as to what could possibly be meant by
'physical arithmetic'!  No matter how unstable physical continuants
might be---no matter how often they pop in and out of existence---and
no matter whether there are only finitely many of them---arithmetic
can be applied when counting them.  But there is ONLY ONE arithmetical
theory, namely PA, that gets so applied. The notion of natural number
and the laws of arithmetic arise out of the working of *any concepts
whatsoever that divide their reference*, that is, under which
re-identifiable individuals fall. It is so deeply meshed in the
conceptual system of any sufficiently mature intellect that, once
discovered and appreciated, it's there to stay. 

Is 'physical arithmetic' some imagined alternative to PA---that is, a
theory that actually conflicts with PA? This is the sense of the
phrase that would be needed in order to fill in the space marked '??'
in the diagram above---that is, in order to undermine the essential
contrast that I claim between the epistemological status of arithmetic
(even when applied) and the epistemological status of physical
geometry. Until Shipman or Sazonov can come up with a plausible
theoretical alternative to PA *for the purposes of applications* their
argument by analogy (for arithmetic and geometry being in the same
epistemological boat) remains unconvincing. 

Shipman continued:

> 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.

This strongly suggests that THERE IS NO precise definition of
'physical arithmetic'. The most it can be is PA (or some consistent
extension thereof) in application to counted, finite, physical collections.
Shipman again:

> 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.

Please clarify the phrase that I have asterisked. Or at least tell me
whether I'm on the right track in thinking that it would cover the
following sort of thing:

	Take the rule of arithmetic (x)(y)(x+y=y+x). For the purposes
	of application, we have the following schema:

		#xFx=n
		#xGx=m
		Nothing is both F and G
		Therefore, #x(Fx v Gx) = n+m

	Having determined what number k the number n+m actually is,
	we can now without further ado deal with this case:

		#xFx=m
		#xGx=n
		Nothing is both F and G
		Therefore, #x(Fx v Gx) = m+n = k

	It's the law of commutativity that allows us to give k as
	the answer here, without laboriously computing m+n afresh.

No matter how unstable physical continuants might be, how could their
physical behavior infirm this sort of reasoning?

Shipman writes

> I'm not suggesting that real theorems of arithmetic can
> be falsified, just that some theorems can't be verified...

What puzzles me is why anyone would seek a physical verification of a
theorem already proved.

> 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 ...

OK then: don't use those weak theories! Since Friedman's proof is
itself feasible (otherwise he would not have claimed that n(4)
exists), I don't see what the problem is here.

> so adding "n(4) does not exist" to such a theory
> gives an inconsistent but *feasibly consistent* theory.  

Once again: don't use weak theories! If you know a theory is
inconsistent but 'feasibly consistent', don't use it.

> we can't give the statement "n(4) exists" any meaningful physical
> interpretation ...

This is not surprising. The claim "5 exists" does not have any
'meaningful physical interpretation' either! It's a claim of pure
mathematics! It speaks only of an abstract object!

> 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
> sentences.

Again, please clarify the phrase that I have asterisked. (I'm assuming
you meant to write "meaningful physical interpretation".)

The application of numbers to count finite collections of physical
objects is NOT a matter of giving *numbers* a "meaningful physical
interpretation". The numbers themselves don't need
interpretations. Only numerals do. The interpretation of a numeral is
that it denotes a number. If someone finds that there are at most N
fundamental particles in the universe, that does not strip *numerals*
denoting numbers greater than N of their proper interpretation. Their
proper interpretation is simply to stand for numbers greater than N!

Application is not interpretation, let's be clear about that.

Neil Tennant