[FOM] What does Peano arithmetic have to offer?

Timothy Y. Chow tchow at alum.mit.edu
Mon May 3 22:29:59 EDT 2010

Harvey Friedman wrote:
>6. I am sure that we agree that there are incomparably more consumers  
>of elementary numerical mathematics than algebraic number theory. So I  
>don't quite see why you are focused on algebraic number theory.

I wonder if this is a case of "moving the goalposts."  Compare with the 
perennial question of whether machines can be intelligent, or can think, 
or can be conscious, or whatever.  Challenges are stated in the form of, 
"No machine could ever X, Y, or Z."  After machines do X, Y, and Z, the 
goalposts are moved: The fact that machines can do X, Y, and Z just proves 
that X, Y, and Z don't require intelligence, creativity, thought, 
consciousness, etc.

Similarly, we have the question of whether f.o.m. can be relevant to real 
mathematics.  Challenges are stated in the form of, "f.o.m. is totally 
irrelevant to X, Y, and Z."  After f.o.m. is demonstrated to be relevant 
to X, Y, and Z, the goalposts are moved: The conclusion is that X, Y, and 
Z are not real mathematics.

Algebraic number theory is currently considered to be real mathematics.  
Once f.o.m. is shown to intrude into algebraic number theory, that status 
will undoubtedly be questioned.


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