[FOM] 827: Tangible Incompleteness Restarted/1

[Timothy Y. Chow]

On Wed, 25 Sep 2019, Joe Shipman wrote:
> But graph theory is quite strongly connected to computation theory, 
> which does not elicit contempt from Fields medalists.
>
> It’s my impression that every area of math has sub-areas which “are of 
> interest only to specialists”, and that graph theory is not particularly 
> bad in this respect.

Just to be clear, a significant proportion of my own research is, or can 
be thought of, as being in graph theory, so I'm not arguing that graph 
theory is contemptible.  But Friedman asked for an elucidation of the 
attitude exemplified by an anonymous Fields medalist (and for some reason 
cannot ask said Fields medalist to elaborate).

I don't have concrete statistics at my fingertips, but I seem to recall 
some statistics about the total number of graph theory papers, and the 
percentage of such papers that appear in specialized graph theory (or 
perhaps combinatorics or discrete mathematics) journals as opposed to 
general mathematics journals.  On the face of it, these statistics support 
the thesis that graph theory is an "outlier" of sorts---the literature is 
large compared to other subfields, and the fraction of that literature in 
specialized journals is high compared to other subfields.  Now, you could 
perhaps explain this by saying that when graph theory and discrete 
mathematics started to "explode," the generalist journals were overwhelmed 
by the volume and so the community was "forced" to develop specialist 
journals.  But even so, I think there is something to the argument that 
graph theory is not "just like any other subfield."

Regarding Shipman's comment about computation theory---it may not get as 
much contempt as graph theory, but I certainly have the impression that 
back when it was called "recursion theory," the purer areas of the subject 
were accused of being specialized and disconnected from the rest of math, 
and were similarly regarded disdainfully.

I didn't mention it before, but an additional contributing factor may be 
the perception that graph theory is "too easy."  To borrow a term from 
Gian-Carlo Rota, some might regard graph theory as a "Mickey Mouse 
subject."  Rota was talking about attitudes toward combinatorics, which 
has gained in respectability over the last century (in part because of 
Rota's efforts to systematize the subject and connect it with other 
subjects such as algebra and topology), but I think the situations are 
closely analogous.  The underlying assumption is that if someone is 
pumping out dozens of papers on graph theory then the papers can't be 
worth much.

As a final remark, I often disagree with Doron Zeilberger's opinions, but 
here's one of them that I like:

http://sites.math.rutgers.edu/~zeilberg/Opinion81.html

My paraphrase is, don't worry unduly about what famous mathematicians 
think about your research interests.  As long as you have the freedom to 
pursue what you regard as impotant and disseminate your results somehow, 
there's no point in wasting your energy worrying about what other people 
think of you.

Tim