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From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Re: Intuition and doubt (was Re: Minds, machines, and Godel)
Message-ID: <1991Feb5.191130.18793@sics.se>
Date: 5 Feb 91 19:11:30 GMT
References: <4361.9102051326@s4.sys.uea.ac.uk>
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Organization: Swedish Institute of Computer Science, Kista
Lines: 166
In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 5 Feb 91 13:26:08 GMT


In article <4361.9102051326@s4.sys.uea.ac.uk> jrk@information-systems.east-
anglia.ac.uk (Richard Kennaway CMP RA) writes:

   >If you feel that the changeableness of my views makes them impossible 
   >to discuss, that's up to you.

  By no means, but it helps to know what views one is discussing at a
given point. In these exchanges, it is customary to assume that what
is claimed in message X+1 is a continuation of what is claimed in
message X, unless there is an explicit explanation to the contrary in
message X+1. If I could make no assumption whatever that your earlier
messages can be taken in conjunction with the latest one, I would have
to preface each of my contributions to this enlightening debate with a
long statement of what I assume concerning the position I am arguing
against.

   >So I ask you once more: are you claiming that there is better evidence than
   >the empirical for the consistency of arithmetic?

    Certainly; of the same kind as the evidence for theorems of
ordinary mathematics in general - whatever may be the nature of that
evidence.

  Allow me to recapitulate a bit. This discussion started with your
remarking, not without a certain amount of finger-wagging, that you
saw nothing in the notion of "intuiting" the consistency of
arithmetic. "Dreams require interpretation; fiction requires facts;
and mathematical intuition requires proofs." To this I responded that
the consistency of arithmetic is as much or as little a matter for
"intuition" as the validity of mathematical theorems in general. For,
after all, that elementary arithmetic is consistent is a theorem of
elementary analysis.  In saying that it is *obviously* consistent I
meant that there is an *obvious* consistency proof for arithmetic.
That is, we don't need to drag in any fancy proof-theoretical stuff
like Gentzen's proof. The trivial proof is mathematically less
informative, but that doesn't render it any less a proof.

  And, you will recall, this was my main point in my first articles on
the topic: why make a hullabaloo about consistency in particular? If we
are unsatisfied with the mathematical evidence for the consistency of
arithmetic, this can only serve as a starting point for wondering about
what mathematical proofs in ordinary mathematics prove in general.

  Now my objections to your present standpoint concern two aspects.
First, the idea that there is good empirical evidence for the consistency
of arithmetic (and perhaps even for that of ZFC?), and second the idea
that the consistency of (say) elementary analysis is sufficient to justify
it, from the point of ordinary mathematics.

  As to empirical evidence, this is a rather delicate matter, since there
can't be any clearcut refutation of claims of this kind. However, I'll
explain why I reject the claim that there is good empirical evidence for
the consistency of arithmetic.

  First note that "arithmetic is inconsistent" is a purely existential
assertion. What it means is that it is *possible* to derive a
contradiction using the formal rules of first order arithmetic. Here
"possible" means "theoretically possible": it has nothing to do with
what can in fact be achieved. If, say, the shortest derivation in
elementary arithmetic of a contradiction is, when written out in
primitive notation (using a standard X font, but a very large
VDU), a billion light years long, there is no question of our
producing even an extremely compressed version of this derivation.
Rather, if we are to be able to realize that arithmetic is
inconsistent under such circumstances, we must do so on the basis of
general mathematical considerations, presumably of a considerable
degree of abstraction. And the claim that we must be able to prove by
such means that an inconsistent theory is inconsistent is not one whit
more justifiable than the claim that we must be able to prove the
consistency of any consistent theory.

  Now the only empirical evidence you have pointed to consists in the
fact that people have done a lot of work in elementary arithmetic (or
done work representable in elementary arithmetic) and have never come
across any contradiction. By what ordinary empirical criteria is this
good evidence? I am not aware of any such criterion. We have checked a
pretty large (or, if you like, extremely small) finite subset of a
potentially infinite set of derivations without finding a
contradiction: that seems to be the sum of the supposed empirical
evidence. For some reason you call this "the same sort of evidence we
have that the sun will rise tomorrow." Since you make no attempt to
suggest the existence of anything corresponding to physical theory in
the case of the consistency statement, the only obvious point of your
comparison seems to be an appeal to a general principle of the form
"what has been, will be".  In mathematics at least, I submit, this is
a rotten principle.

  We might of course attempt to argue that it may reasonably (and
perhaps "empirically"?) be assumed that a contradiction, if it exists,
should be accessible in length rather than astronomical, and should have been
discovered by now, since (for example) the axioms of arithmetic after
all encompass only a few basic principles: the definitions of addition
and multiplication, equality, the induction principle.

  This is where it is very easy to delude oneself into regarding
evidence that is in fact based on our abstract understanding of the
axioms as somehow empirical, in the sense that one need not appeal to any
such abstract understanding. For the induction schema in elementary
arithmetic is indeed very easily understood in its abstract
formulation - "every set of (or property of) natural numbers which
contains..." etc.  This abstract understanding of the principle is quite
sufficient to convince me that the theory is consistent. Given that the
natural numbers are 0, s(0), s(s(0)), and so on, the induction
principle is true of the natural numbers, whatever the nature of those
numbers - Platonic objects or pure figments of our imagination. 

 But suppose we reject such appeals to our "abstract understanding"
and stay strictly within the realm of the supposedly objective and empirical.
Then we must note the following: the induction axioms of elementary
arithmetic have an extraordinary degree of complexity.  We know that
the proof-theoretical strength of the schema

            A(0) & (x)(A(x)->A(x+1)) -> (x)A(x)

depends very sensitively on the allowable logical complexity of the
formula A(x). We also know that the complexity of those A(x) that are actually
used in ordinary arithmetical proofs is very low. In particular, most or all
classical theorems of the form (x)p(x) with p(x) primitive recursive
are provable in primitive recursive arithmetic, in which A(x) is
constrained to be a primitive recursive predicate. And the consistency of
*that* schema is easily provable in elementary arithmetic using a slightly more
complicated condition A(x). In fact the empirical evidence for the
consistency of the induction schema (in combination with the remaining
axioms) only supports, at best, the consistency of a highly restricted
version of the schema. When A(x) contains, say, 100 quantifier
alterations followed by a primitive recursive relation, there simply
isn't anything in ordinary mathematics that can be represented as
testing the validity of the corresponding schema.

  So, to sum up: I claim that those who regard the consistency of arithmetic
as well supported on empirical grounds haven't taken their own idea
seriously.

  Now it is of course perfectly legitimate to ask: if the evidence for
the consistency of arithmetic or analysis is not empirical, what is
it? And does it exist? These are classical questions in the philosophy
of mathematics. I don't believe any very good answers have been
arrived at; I do believe that references to "empirical" evidence of
the kind considered above are worthless. It would take us too far, and
involve too much work, to go more deeply into this question, and I'll
just assert here that we will get nowhere unless we consider the
*actual* use, meaning, teaching, and understanding of mathematical
statements and structures in and outside mathematics, rather than our
preconceived (and almost inevitably simpleminded) ideas about what
mathematical evidence must be like.

   >Can you clarify your question?  If A is a theorem of T, then A is
   >satisfied by every model of T.  This is so even (trivially) for
   >inconsistent T.  If you can be clearer about what you mean by "if T is
   >consistent, then A", then perhaps I can answer your question more clearly.

  My question is simple enough. Suppose A is an ordinary mathematical
statement. If (a formalized version of) A is proved in a formalized
mathematical theory T, what does this tell us? What do we know when we
know that A has been proved in T? The answer "we know that A is true
in every model of T" is the model-theoretical version of the answer
"we know that there is a formal derivation in T of A." If this is all
there is to it, there is no other knowledge to be obtained from (formalized)
mathematics than this purely formal information, which doesn't even
depend on our associating any meaning with the statement A. However,
obviously this can't be your view, if only because you said earlier
that a proof in an inconsistent system proves nothing - now what do
you mean by this? After all, as you yourself point out, a proof of A
in a theory T proves that A is true in every model of T, whether or
not T is consistent.