[FOM] 827: Tangible Incompleteness Restarted/1

Harvey Friedman hmflogic at gmail.com
Wed Oct 2 21:44:50 EDT 2019

On Wed, Oct 2, 2019 at 4:21 PM Louis H Kauffman <kauffman at uic.edu> wrote:
> Yes, a category is a digraph with extra structure. And one of the more important collections of categories are the categories generated from digraphs via the structure of paths on the given digraph.
> My point is that the concept of a graph is involved through and through with just about all aspects of mathematics. People often feel that they can make a point by contrast, and it gets remembered for too long.

You "point" makes it all the more fascinating that my Fields Medalist
I referred to earlier said that they "didn't know what a graph is, and
never wanted to know what a graph is".

> J. H. C. Whitehead famously said “Graph theory is the slums of topology.” I’ll not comment on that. But I think it was an intuitive avoidance of the sort of complexity in topological invariants that can come from
> deep combinatorial definitions. And at the same time Whitehead made such definitions with the Whitehead Groups and the concept of CW complexes. The dam broke in the 1980’s with advent of the Jones polynomial, its relation with the Tutte polynomial and contraction/deletion relations, relations with statistical physics, topological quantum field theory and categorical formulations that span the gamut from graph theoretic structures to braided monodical categories, field theory and beyond into present day link homology and its relationships with graphs and with the homology of categories. Mathematics is a whole and
> it is silly to argue about why some people have prejudices.

It would be great if you could elaborate on your "Mathematics is a
whole". Do you know of any kind of coherent account of "mathematics is
a whole" along these or other lines, by you and/or other
mathematicians? I can do a coherent account of "f.o.m. is a whole"
FROM FIRST PRINCIPLES, which would be incomparably easier to pull off
than for "mathematics is a whole" - particularly FROM FIRST

Harvey Friedman

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