[FOM] Indispensability of the natural numbers

[Timothy Y. Chow]

After some further reflection, I think I can state more precisely some 
thoughts that I have partially expressed in my recent articles regarding
the consistency of PA.  These ideas are a variation on an argument I heard 
from Torkel Franzen 10+ years ago, that doubts about the natural numbers
carry over to doubts about syntactic entities such as PA itself.

If someone exhibits an explicit contradiction in PA, then I would argue
that this indicates that there is something fundamentally incoherent about
the mental concept of the natural numbers that I, along with presumably
every sufficiently educated person, carry around in my head.

An interesting property of the concept of the natural numbers is that it
is rather "rigid"; that is, if you try to create a "small perturbation"  
of the concept (e.g., by taking a nonstandard model of PA), you typically
get something that is clearly *more* complicated than the natural numbers
(e.g., nonstandard models of PA contain the naturals as an initial
segment).  I would argue that this indicates that if our concept of the
natural numbers is incoherent, then so is any halfway plausible infinitary
substitute that we might propose.

Next, I want to argue that if our concept of the natural numbers is 
incoherent, then so is our concept of a formal system.  In a previous
article, I suggested two processes, Process A and Process B.  Process A
starts with "1" and applies the rule "append 11 to the previously
generated string," thus producing 1, 111, 11111, etc.  Process B is the
same but starts with "11," thus producing 11, 1111, 111111, etc.  Now
it might appear that I have clearly defined these processes, but I claim
that appearances are deceiving; these processes seem clearly defined only
because we are used to assuming that our concept of the natural numbers
is coherent.  If our concept of the natural numbers is incoherent, then
I would argue that the concept of a *syntactic rule* that potentially
applies to infinitely many situations is also suspect.

One might try to counter my argument by claiming that in any given
situation where a syntactic rule is to be applied, the string in question
will be finite and one will be able to check that String 1 turns into
String 2 when Rule X is correctly applied to String 1.  But I say: What
does the word "correctly" mean here?  I cannot check that Rule X is being
*correctly* applied unless I have a clear notion *ahead of time* as to
what Rule X *is*.  And Rule X encompasses, implicitly, an infinitude of
instances.  If the concept of the natural numbers is ill-defined, then so
is the concept of Rule X; I cannot confidently speak of Rule X as
something that applies to an *arbitrary* string.  At best, I might be able
to specify rules that apply in finitely many situations, by listing all
the relevant situations explicitly.  But this won't help me know whether I
have the "correct" concept of Rule X in any new situation that is not
already in my table of known instances.  (This is what I take Kripkenstein
to be arguing, but if you disagree with my reading of Kripkenstein, just
ignore this parenthetical remark.)

I therefore argue that if an explicit PA-proof of 0=1 is found, then
moving to something like I_Delta_0(exp) does not really salvage much,
even if one is unable to find a proof of 0=1 in that system.  The first
reason is that a PA-proof of 0=1 demonstrates a fundamental problem with
our concept of the natural numbers, and so we won't be able to assign a
coherent *meaning* to the sentences of first-order arithmetic any more.
(Recall my remark about halfway plausible substitutes.)  Formal languages
are interesting mostly because they capture some features of an entity
that we're studying, and if the entity vanishes, then why bother with the
formal language any more?  The second reason is that I_Delta_0(exp) is a
formal system, and it won't be clear any more exactly what I_Delta_0(exp)
*is* (although I guess we could generate a finite fragment of it and check
with each other that we agree on that finite fragment, or we could cross
our fingers and hope that discrepancies between different human beings
won't show up in practice).

Coming back to Process A and Process B, I believe that if one seriously
maintains that we "don't know" if PA is consistent, then one should also
maintain that we "don't know" if there is a string that both Process A and
Process B will generate.  This might seem like a surprising claim, since
the relevant fact about Processes A and B can be formalized in systems
much weaker than PA.  However, what I claim is that the very act of (1)
setting up a weak formal system, including rules (rules!) of inference,
and (2) asserting that theorems in this weak system "say" something about
what Processes A and B "really do," presupposes the coherence of the
concept of the natural numbers.  The inconsistency of PA undermines this
presupposition.  In fact, it even undermines the presupposition that we
know what Processes A and B *are*.

Finally, let me turn this argument around and argue against the point of
view that "we don't know whether PA is consistent because we don't have a
proof of it."  This sounds convincing on the surface because it sounds 
"scientific" (or "mathematical"?) to demand proof.  However, the whole
framework of formal systems and the whole concept of consistency 
implicitly rely on the coherence of the concept of the natural numbers,
as I have argued.  Therefore, assuming that one has a clear notion of
what "PA is consistent" *means*, and of the notion of a "proof" of such
a thing (as in something that would compel assent to such a meaningful
proposition), already presupposes a clear enough notion of the natural
numbers to make the consistency of PA obvious.

Tim