[FOM] 827: Tangible Incompleteness Restarted/1

[Louis H Kauffman]

No sense of humor? I did not say that graph theory was the slums of topology, nor do I assert that category theory is a branch of graph theory. Nevertheless, literally, a category is a digraph with the property that 
to every node (object) there is an edge (morphism) called an identity morphism, and given two edges so that the head of one is the tail of the other, there is a third edge from the tail of the first to the head of the second called the composition of these two morphisms. Composition of morphisms is associative and compositions with an identity morphism has no effect on the composee. A category is a special kind of directed graph. But the concept of categories and functors is quite different from the concept of graphs, as usually held by combinatorialists. This is not to say that a graph theorist might not construct or use a category. You could also say that combinatorial knot theorists are a species of graph theorists with special taste about the sort of graphs and the sort of questions that they consider. And indeed knot theorists, like most topologists, do not shy away from categories if they are useful.

Harvey Friedman asked "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
PRINCIPLES.”

I do see mathematics as a whole. Its division into fields and subjects is an organizational convenience. What is common among mathematicians is their
intent to create and explore formalisms, follow patterns and rely on reason in the articulation and explication of ideas and relationships. I was brought up to imagine that one 
could house all of mathematics in set theory and I was in fact brought up in the notion that it was good enough to use the level of set theory that Paul Halmos called 
naive (perhaps satirically, or perhaps with a nod to Naive Realism). Now I am not so naive. But I would not proselytize my views. I see 
mathematics as a pastiche of formal games, all played quite seriously by their practitioners, and that most of us feel that anyone else’s formal game is fair game for our 
use. I personally want the entire edifice of such structures to be consistent and I know that this desire should not be elevated to a belief. I’ve witnessed many events 
where different mathematical fields found significant connection, and I believe that the finding of such connections is the essence of mathematics.

The above paragraph is not a formal articulation for the assertion that mathematics is a whole. Please use your sense of proportion and do not write a long critique of its shortcomings.
Think about learning and doing basic mathematics and how all of those actions fit together, how there is no piece of mathematics that can be justifiably separated from the rest. When I use the word 
wholeness I am referring to that sort of coherence. I am not attempting to make a definition of what is mathematics that would be a formal definition. In my opinion that aim is seriously quixotic.
It would be like trying to define the concept of a distinction. Courant and Robbins wrote a beautiful book entitled “What Is Mathematics?”. Their answer was to take what they felt to be good and 
beautiful and understandable and put it together in a story to illustrate the range of ideas, connections, calculations, techniques and applications that is mathematics. 




> On Oct 3, 2019, at 9:10 AM, José Manuel Rodriguez Caballero <josephcmac at gmail.com> wrote:
> 
> Harvey Friedman wrote:
>>> Try giving such an account, which is totally understandable by
>>> "everyone", for path algebras and representation theory. This should
>>> definitely decide the relevant culture wars under a common
>>> understanding of the true nature of intellectual life. Such a common
>>> understanding is obviously missing, and an interesting question that I
>>> frequently think about is: why and what is to be done about it?
> 
> 
> My modest answer to this question is: it is important to write papers and to give lectures (available on YouTube) about the culture wars in the history of mathematics, including recent history.
> 
> As an example (concerning the infinitesimals) of such papers, I would like to cite:
> 
> https://arxiv.org/abs/1306.5973 <https://arxiv.org/abs/1306.5973>
> 
> In the particular case of the present discussion about graph theory, it is important to know whether or not the category-theoretical community was trying to change the names of already defined mathematical objects in a systematic way, e.g., digraph by quiver. Such a question may be the motivation of an interesting research in history of mathematics. I am not talking about conspiracy, but just of a spontaneous behaviour inside a community produced by some reason that a historian should discover.
> 
> Kind Regards,
> Jose M.
> 
> Sent from my iPhone
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