[FOM] 461: Reflections on Vienna Meeting

[Timothy Y. Chow]

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.

Tim