[FOM] 461: Reflections on Vienna Meeting
Timothy Y. Chow
tchow at alum.mit.edu
Sat May 14 19:47:37 EDT 2011
Carl Mummert wrote:
> Of course one can argue that the consistency question is ill posed, for
> example by adopting some form of finitism. I am more interested in
> contemporary arguments (preferably from those who make them, if they
> have put them in writing) that accept the existence of the usual set of
> natural numbers, and accept the usual methods of contemporary
> mathematics, but doubt the consistency of PA or HA. Would someone be
> willing to summarize these for the FOM list, or provide references?
In my experience, the suspicion that the "mathematician in the street"
feels towards consistency statements stems from something like this.
Mathematicians have an instinctive feeling that when you're trying to
prove something, you must avoid circular reasoning, and not "assume what
you're trying to prove." So when trying to prove something like the
consistency of PA, the feeling goes, one should start from principles that
are more "evident" than the consistency of PA itself. Now you can see why
we're headed for trouble. Inevitably, to prove the consistency of PA, one
has to use some kind of induction. But if I have preset my brain to a
state where I am at least nominally *skeptical* of the consistency of PA,
I am likely to be skeptical of induction as well.
The mathematician in the street is still locked into a pre-Goedel view of
Hilbert's program, and wants to be able to prove the consistency of a
strong system using only assumptions that are manifestly (or at least
intuitively) weaker. If this can't be done, then the conclusion is that
the consistency remains an "open question."
F.o.m. experts sometimes try to characterize "ordinary mathematical
reasoning" as being "reasoning formalizable in some specific system X."
"Ordinary mathematicians" may unwittingly aid and abet this
characterization by giving lip service to it. However, I think that if we
more carefully examine how mathematicians think, we find that the question
of what assumptions are acceptable is *context-dependent*. That is, an
assumption that would be perfectly admissible in one context might not be
admissible in another context. Here's another example. If I ask for a
proof of the Pythagorean theorem, I will not be satisfied if you say that
the distance between (a,0) and (0,b) in R^2 is sqrt(a^2 + b^2) by
definition, even though such an argument might be unobjectionable in some
context where we're trying to prove something much more complicated.
Similarly, an argument by induction that would be fine in some other
context might still be ruled unacceptable in the context of proving the
consistency of PA.
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