FOM: Reply to Shipman on "feasible arithmetic"

[Neil Tennant]

Joe Shipman writes:

> Neil, I repeat that I don't disagree that PA is here to stay for intrinsic
> reasons.  

Good! We're closer to agreement then. I'd go further: PA is, for
intrinsic reasons, here to stay *for any physical applications
whatsoever*.

> All I am saying is that it is *possible* (and therefore the alternative is 
> not *necessary*) that PA contains numbers that are "too big" to have any
> reasonable interpretation in the physical universe because it is possible 
> that the universe is in some sense completely finite.  

Again, the question arises: What is it for numbers---or, perhaps more
felicitously, (the language of) any arithmetical theory---to "have a
reasonable interpretation in the physical universe"? My original
question is being avoided, and there is no engagement with my point
that *application is not a matter of interpretation*. Nor do you
engage my point that numbers don't need any physical
interpretation. *Numerals*, as expressions in language, need
interpretation; but that interpretation (on a denotational model) is
provided precisely by the numbers, which are abstract objects that the
numerals denote. Alternatively, on an inferential semantics, the
interpretation would be provided by certain meaning-constituting rules
for the use of numberals and of abstraction terms of the form
#xFx. But in this case also, reference to physical objects is
irrelevant---in the sense that the existence of physical objects can
be accommodated, but is in no way essential to the conceptual
foundations of arithmetic.

> The point is that this supposed alternative theory will conflict with PA only
> for infeasibly verifiable statements which we cannot rule out ... by
> any sort of direct falsfication.  

Again, why should the need for "feasible verification" of theorems
ever arise? Can one seriously expect a "direct falsification" of a
theorem of PA? Isn't that simply ruled out by its status as a theorem?

> an alternative to PA is possible as long as the theorems
> of PA it denies cannot be feasibly settled.  

"Possible" in what sense? And why should the inability feasibly to
settle the points of disagreement between PA and the alternative
theory matter one bit? Which theorems of PA might be denied in such a
scenario? Why should we not say that the theorist who denies theorems
of PA cannot be talking about the natural numbers?

> If a theorem is proved in PA I
> don't think it will be physically falsifiable 

Good!

> but its negation may also not be
> physically falsifiable.

So what? Isn't a priori proof of P better than so-called "physical
falsification" of ~P? And what exactly *is* a "physical falsification"
of a theorem of PA?

> An ambitious way to realize this would be in the context of a finitary "theory
> of everything" (e.g. the universe is a giant cellular automaton as Fredkin has
> argued) where some notions of "object" and "counting" can be reasonably
> defined; but much simpler, and sufficient for the purposes of this argument,
> is if you can define a physical experiment (possibly but not necessarily a
> computation) whose outcome is causally related to the truth value of the
> sentence (e.g. searching for a counterexample to Wiles's theorem or the
> Riemann Hypothesis below a certain bound), and which can in principle be
> carried out because the universe is "large enough".
> 
> I hope this clarifies the distinctions I was trying to make ...

I'm afraid not; and I trust you will believe that I'm not just trying
to be perverse.  What would be these notions of "objects" and
"counting" that can be "reasonably defined"?  How would they differ
from the present (logical) notions of object and counting that
undergirds PA?  And if they do genuinely differ, how can the deviant
notions lay claim to be the true subject-matter of a *revised
arithmetic* that is somehow *better than* PA, and which should
*displace* PA for these "applications"?

Now,

> if you can define a physical experiment (possibly but not necessarily a
> computation) whose outcome is causally related to the truth value of the
> sentence (e.g. searching for a counterexample to Wiles's theorem or the
> Riemann Hypothesis below a certain bound), and which can in principle be
> carried out because the universe is "large enough"

then my response would be "Oh dear, the causal relations didn't
stretch far enough for this physical experiment to tell us anything
about the truth value of the sentence." Moreover, if they *did*
stretch far enough, and indeed a long way, and gave an answer at odds
with a theorem of PA, I would sooner reject the result of the
experiment as most likely involving a computer malfunction, or whatever.

> -- I don't have a
> particular alternative to PA in mind, but others on this forum have discussed
> the possibility.

I've followed these discussions closely, and find them completely
unconvincing. But from my point of view it's no surprise that you
don't have a particular alternative to PA in mind---for there *could
be no such thing*, if the intention is to talk about those abstract
objects that we call the natural numbers. (I would regard an
"alternative" to PA, for the purposes of the present discussion, as
any theory in the language of arithmetic containing the negation of
any sentence that is knowable as true in the standard model.)

> If you still don't "get" my point that the theory PA is not
> merely an abstraction from observed empirical regularities in the way physical
> objects behave, but an extrapolation whose *full* power is justified on
> intrinsic rather than empirical grounds, please say so ...

I have no hesitation in confessing that I still don't "get" it. But
it's strange that you think PA is "not merely an abstraction from
observed empirical regularities...".  It's not the "not merely" part
that I dispute---it's the extraordinary thought that PA might be (even
in part) "an abstraction from observed empirical
regularities". Empirical regularities, observed or unobserved, have
nothing whatsoever to do with the grounds for the truth of statements
of arithmetic. Those grounds are *entirely conceptual*. And even if
someone were to claim (which claim I don't accept) that PA needs, in
addition to its purely a priori justifications for its claims, some
further corroboration or confirmation stemming from "abstraction from
observed empirical regularities", there would be a host of difficult
questions to face:  What sort of "abstraction" would this be? Why
should observation play any role at all? How do you know that it is
possible to describe all or even some empirical regularities without
presupposing the very notion of natural number, and relying on
antecedent truths of arithmetic? 

> -- maybe someone else
> can explain it better than I have.

I doubt it!  You have done better than anyone else, and I'm grateful
to you for taking this trouble on behalf of people whom you might
regard as an embattled minority (i.e. finitists and "feasible
arithmeticians"). That there is no better explanation is perhaps a
consequence of the fact that the view is ultimately untenable, and
this *for conceptual reasons* that can be established by 
philosophical and foundational debate dealing with fundamentally
logical and conceptual issues.

Neil Tennant