FOM: Reply to Tennant on arithmetic v geometry JoeShipman at
Sun Oct 11 02:49:00 EDT 1998

In a message dated 10/11/98 12:49:07 AM Eastern Daylight Time,
neilt at writes:

 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? 

Neil, I repeat that I don't disagree that PA is here to stay for intrinsic
reasons.  But I think Zermelo set theory (for example) is also here to stay.
The question is whether they can be reasonably interpreted in the physical
universe.  Apparently Z cannot be, there are sets in it that are "too big".
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.  In that case a "feasible
arithmetic" or "bounded arithmetic" of the type that has been discussed on the
FOM (most recently in reference to Parikh's work) may lay a competing claim.
The point is that this supposed alternative theory will conflict with PA only
for infeasibly verifiable statements which we cannot rule out (for example the
negation of Friedman's "n(4) exists") by any sort of direct falsfication.  The
alternative theory may be consistent or it may be only "feasibly consistent",
but it will agree with PA for feasibly verifiable statements (e.g. "Wiles's
theorem has no counterexamples with x,y,z, and n all less than 10000" which is
verifiable on a modern computer in a few hours).

> 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.

No, my point was that an alternative to PA is possible as long as the theorems
of PA it denies cannot be feasibly settled.  If a theorem is proved in PA I
don't think it will be physically falsifiable but its negation may also not be
physically falsifiable.

> and it is conceivable that all the true sentences of
> arithmetic that ***CAN be given a meaningful physical 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.

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 don't have a
particular alternative to PA in mind, but others on this forum have discussed
the possibility.  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 -- maybe someone else
can explain it better than I have.  You say "don't use weak theories"--but
surely the question here is how weak a subtheory of PA is truly *necessitated*
on empirico-physical grounds alone.

To repeat:  I find PA more compelling than alternatives even for describing
the physical universe -- but I *don't* find that it has no alternatives;
therefore the epistemological difference between arithmetic and geometry (for
the empirico-physical mode of investigation) is one of degree and not of kind.
For the pure-mathematical mode of investigation the difference is also one of
degree, as you recognize that the Euclidean geometry of R^n is also a priori
(though maybe not as primordially as arithmetic).

-- Joe

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