[FOM] Replies to Putnam

[Paul Hollander]

On 6/1/09 1:13 AM, Harvey Friedman wrote:
> My point of view is more nuanced: I strongly deny that category theory
> forms a philosophically coherent autonomous foundation for mathematics
> - in the sense that set theory forms a philosophically coherent
> autonomous foundation for mathematics. MacLane disagreed with this
> position. I suspect that Hilary Putnam agrees with my position on this.
>    

Harvey,

I don't think I advocate category theory per se.

But I do question set-theoretic exclusivism.

Since category theory and set theory is each finitely axiomatizable, and 
each capable of representing '=' and '<', why should we prefer set 
theory as the official model-theoretic representation of '=' and '<', to 
the exclusion of category (or any other) theory?  It begs the question, 
with all due respect to Putnam, that there is a single "official" model 
theory in the first place.  Why not model-theoretic pluralism?

May I request a clarification and demonstrate of the above claim about 
category theory?  What is meant by "philosophically coherent autonomous 
foundation for mathematics," and how does it differ from Beth 
definability with respect to '=' and '<'?

Cheers,

-paul