[FOM] 461: Reflections on Vienna Meeting

[Harvey Friedman]

I wasn't talking about why ZFC was created, or even all the things  
that ZFC does well, and/or better than anything else.

I am only talking about one important aspect of what ZFC does that is  
particularly easy to point to against those, like Angus, who claim  
that ZFC says nothing interesting about mathematical practice.

Namely, it is a superb model of mathematical practice in that it is  
sufficient to allow us to draw great dramatic conclusions with great  
confidence.

The comparison with physical theories is very focused on only one  
fundamental point: that we expect to gain from refining the model in  
order to handle more complex phenomena.

For example, the model has already been refined greatly in order to  
support Reverse Mathematics, and also refined greatly to study  
constructive mathematics.

On the other hand, the model will have to be refined considerably in  
order to deal with such notions from mathematical practice as "this  
proof is the same as that proof" or "this proof is simpler than that  
proof" or "this proof is trivial" or "this proof has four important  
ideas" or "this statement is mathematically natural" or "this  
definition is simple", or even "this proof is longer than that proof",  
etcetera.

Harvey Friedman

On May 12, 2011, at 1:13 PM, Walt Read wrote:

>
> 10. To begin with, ZFC forms an unexpectedly comprehensive and  
> definite
> model of mathematical practice, which is clearly sufficient to draw  
> a number
> of startling and deep conclusions about the nature of mathematical  
> practice.
> Our great physical theories also have this property - they are clearly
> sufficient to draw a number of startling and deep conclusions about  
> the
> nature of physical phenomena -  and none of them come close to  
> reflecting
> all physical phenomena that we seek to analyze.
>

   Is the defense of ZFC that it's a good "tool to inspire insight"?
Somehow it seems unlikely that that could have been the original
intent.

   There is a profound difference between mathematical "theories" and
physical theories and therefore between mathematical practice and
physical practice. The reality of physics is assumed to be outside of
us and only accessed indirectly. We need to be constantly testing our
hypotheses. The reality or "reality" of mathematics (depending on your
philosophical position) is assumed to be directly accessible. Unless
you're prepared to assert that PA is a set of working hypotheses which
are falsifiable as we get more experience with specific numbers, any
comparison of mathematical practice and physical practice is
superficial at best and misleading in the main. Surely Zermelo et al.
were looking for an actual foundation, something to reliably build
everything else on, not insights into mathematical practice,
falsifiable by sociologists of mathematics.

-Walt

_______________________________________________
FOM mailing list
FOM at cs.nyu.edu
http://www.cs.nyu.edu/mailman/listinfo/fom