[FOM] 827: Tangible Incompleteness Restarted/1

[Louis H Kauffman]

It should be pointed out in this discussion that a category is a digraph with extra (compositional) structure.
Category theorists would not care to be categorized as studying a subcategory of graph theory.

> On Sep 30, 2019, at 9:20 AM, Harvey Friedman <hmflogic at gmail.com> wrote:
> 
> From Chow https://cs.nyu.edu/pipermail/fom/2019-September/021693.html
> 
>> It is true that everywhere in the literature, a quiver is defined as a
>> directed graph.  I find this unfortunate because it does raise the
>> question of why you would introduce a new word for something that already
>> has a name.
>> 
>> Associated to a quiver is something called its path algebra.  When people
>> use the word "quiver," they're signalling the fact that they're primarily
>> interested in the representation theory of the path algebra, and not in
>> the directed graph for its own sake.  Graph-theoretic facts about the
>> underlying directed graph are interesting only insofar as they lend
>> insight into the representation theory of the path algebra.  That's a very
>> narrow set of graph-theoretic facts, compared to the sort of things that
>> graph theorists might be interested in.
>> 
>> In my opinion, it would have been better to use the word "quiver" to refer
>> to the path algebra rather than to the directed graph itself, but it's too
>> late to change established terminology.
>> 
> These circumstances are highly suggestive of a math culture war. I get
> the feeling that the relevant core mathematicians are expressing their
> willful ignorance of digraphs - or maybe even marginalization of those
> interested in digraphs in and of themselves -  through inventing
> "quivers" for digraphs as Chow explains.
> 
> This whole embarrassing scenario (I'm taking for granted Chow's
> account) would have been avoided through an adherence of the
> fundamental driving criterion of "general intellectual interest". It
> is completely obvious that the notions of graph and digraph and
> related concepts like clique and independent set, are of a high level
> of gii. Just consider how many salient examples:
> 
> A likes B
> A has sent a message to B via the internet
> A,B are friends
> A,B are in internet correspondence
> {A_1,...,A_k} like each other
> {A_1,...,A_k} do not like each other
> A_1,...,A_k, forms a custody chain
> and so forth.
> 
> And labels on digraphs: A has sent a message n times to B via the
> internet, where the label is n for the edge going from A to B.
> 
> F.o.m. at its highest levels is motivated by matters of high gii. At
> least the basics of graph theory is also motivated by matters of high
> gii.
> 
> 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?
> 
> Harvey Friedman
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