[FOM] the debate with Professor Kanovei

[Martin Davis]

Dear Subscribers,

This discussion has been going in circles and I would propose we call a 
halt. But first I'll exercise my privilege as moderator to make some 
comments that I hope will be clarifying.

My teacher E.L. Post commented on the conclusions to be drawn from 
G\"odel's incompleteness theorem as follows:

"The conclusion is inescapable
 mathematical thinking is, and must remain, 
essentially creative. 
 this conclusion must inevitably result in at least 
partial reversal of the entire axiomatic trend of the late nineteenth and 
early twentieth centuries, with a return to meaning and truth as being of 
the essence of mathematics."

In 1941, he wrote:

"It is to the writer's continuing amazement that ten years after G\"odel's 
remarkable achievement current views on the nature of mathematics are 
thereby affected only to the point of seeing the need of many formal 
systems, instead of a universal one."

This is where Kanovei is stuck: he can only see "many formal systems"; the 
possibility of speaking meaningfully of some assertion being "true" 
independent of provability in a specific formal system, he dismisses as a 
kind of mysticism, incompatible with science.  His position is perfectly 
consistent and there is no arguing him out of it. He is like one devoted to 
the Ptolemaic system in which the sun and planets revolve about the earth: 
his account will have to put up with baroque complexities (the famous 
epicycles) but he cannot be dislodged from his view by pure reason.  He 
will have to maintain that when Lagrange proved that every positive integer 
is the sum of four squares or when Gauss produced his several proofs of the 
law of quadratic reciprocity, they were mistaken in their belief that they 
had actually demonstrated some facts about the natural numbers. He would 
have to maintain that nothing had been established until their proofs had 
been formalized in a system like PA or ZF. The Paris-Harrington form of 
Ramsey's theorem, he would have to say, is not a fact of elementary 
combinatorics not demonstrable in PA.  It is just that this particular 
sentence in the language of arithmetic is not provable in PA but is 
provable in stronger systems.

Here is a form of G\"odel's theorem: I can produce a specific polynomial 
with integer coefficients
p(a,x_1,...,x_n) such that given any consistent theory T on a language that 
includes that of arithmetic, there is a number a_0 (computable from a 
description of T) such that the equation
                                  p(a_0,x_1,...,x_n) = 0
has no solutions in natural numbers but the formula of T that formalizes 
that fact is not provable in T.

I would conclude: "There is a polynomial equation that has no solutions, 
but that fact can't be proved in T." Kanovei would understand what had been 
done perfectly well, but would maintain that there is no absolute sense to 
the assertion that the equation has no solutions. All that one can say is 
that in a formal system strong enough to prove the consistency of T, one 
can prove that the equation has no solutions, because the bare assertion 
talks about the infinite set of natural numbers, and such talk is 
meaningless. It may be worth considering what this position (unassailable 
as it is) entails. An assertion that some particular tuple of numbers 
satisfies the equation involves nothing more complicated than addition and 
multiplication of integers. To claim that it is meaningless to say that 
there is no solution is really remarkable.

It is particularly remarkable given that Kanovei somehow regards formal 
syntax as immune from his philosophical objections. The assertion that a 
particular theory is consistent is no simpler than what we have been 
discussing - in fact it is equivalent to just such an assertion about a 
polynomial equation. Yet Kanovei doesn't seem to regard such a statement as 
meaningless. If he did, he would find himself in an infinite regress, in 
which assertions about a given formal system only makes sense relative to 
another more inclusive system.

One final remark: Noam Chomsky maintains that the grammatical sense that 
makes it possible for children to learn languages leads to their absorbing 
rules that encompass as grammatically correct, an infinite set of 
locutions. Perhaps it is this same sense that we use in coping with the 
infinitude of the natural numbers as well as the infinitude of the 
expressions of a formal language.

Martin