Combinatorial briefs

I tend to write longish posts, in part for the sake of clarity, and in part because I can – it is easier to express yourself in a long form.  However, the brevity has its own benefits, as it forces the author to give succinct summaries of often complex and nuanced views.  Similarly, the lack of such summaries can provide plausible deniability of understanding the basic points you are making.

This is the second time I am “inspired” by the Owl blogger who has a Tl;Dr style response to my blog post and rather lengthy list of remarkable quotations that I compiled.  So I decided to make the following Readers Digest style summaries of this list and several blog posts.

1)  Combinatorics has been sneered at for decades and struggled to get established

In the absence of History of Modern Combinatorics monograph, this is hard to prove.  So here are selected quotes, from the above mentioned quotation page.  Of course, one should reade them in full to appreciate and understand the context, but for our purposes these will do.

Combinatorics is the slums of topology – Henry Whitehead

Scoffers regard combinatorics as a chaotic realm of binomial coefficients, graphs, and lattices, with a mixed bag of ad hoc tricks and techniques for investigating them. [..]  Another criticism of combinatorics is that it “lacks abstraction.” The implication is that combinatorics is lacking in depth and all its results follow from trivial, though possible elaborate, manipulations. This argument is extremely misleading and unfair. – Richard Stanlеy (1971)

The opinion of many first-class mathematicians about combinatorics is still in the pejorative. While accepting its interest and difficulty, they deny its depth. It is often forcefully stated that combinatorics is a collection of problems which may be interesting in themselves but are not linked and do not constitute a theory. – László Lovász (1979)

Combinatorics [is] a sort of glorified dicethrowing.  – Robert Kanigel (1991)

This prejudice, the view that combinatorics is quite different from ‘real mathematics’, was not uncommon in the twentieth century, among popular expositors as well as professionals.  –  Peter Cameron (2001)

Now that the readers can see where the “traditional sensitivities” come from, the following quote must come as a surprise.  Even more remarkable is that it’s become a conventional wisdom:

Like number theory before the 19th century, combinatorics before the 20th century was thought to be an elementary topic without much unity or depth. We now realize that, like number theory, combinatorics is infinitely deep and linked to all parts of mathematics.  – John Stillwell (2010)

Of course, the prejudice has never been limited to Combinatorics.  Imagine how an expert in Partition Theory and q-series must feel after reading this quote:

[In the context of Partition Theory]  Professor Littlewood, when he makes use of an algebraic identity, always saves himself the trouble of proving it; he maintains that an identity, if true, can be verified in a few lines by anybody obtuse enough to feel the need of verification.  – Freeman Dyson (1944), see here.

2)  Combinatorics papers have been often ostracized and ignored by many top math journals

This is a theme in this post about the Annals, this MO answer, and a smaller theme in this post (see Duke paragraph).  This bias against Combinatorics is still ongoing and hardly a secret.  I argue that on the one hand, the situation is (slowly) changing for the better.  On the other hand, if some journals keep the proud tradition of rejecting the field, that’s ok, really.  If only they were honest and clear about it!  To those harboring strong feelings on this, listening to some breakup music could be helpful.

3)  Despite inherent diversity, Combinatorics is one field

In this post, I discussed how I rewrote Combinatorics Wikipedia article, largely as a collection of links to its subfields.  In a more recent post mentioned earlier I argue why it is hard to define the field as a whole.  In many ways, Combinatorics resembles a modern nation, united by a language, culture and common history.  Although its borders are not easy to define, suggesting that it’s not a separate field of mathematics is an affront to its history and reality (see two sections above).  As any political scientist will argue, nation borders can be unhelpful, but are here for a reason.  Wishing borders away is a bit like French “race-ban”  – an imaginary approach to resolve real problems.

Gowers’s “two cultures” essay is an effort to describe and explain cultural differences between Combinatorics and other fields.  The author should be praised both for the remarkable essay, and for the bravery of raising the subject.  Finally, on the Owl’s attempt to divide Combinatorics into “conceptual” which “has no internal reasons to die in any foreseeable future” and the rest, which “will remain a collection of elementary tricks, [..] will die out and forgotten [sic].”  I am assuming the Owl meant here most of the “Hungarian combinatorics”, although to be fair, the blogger leaves some wiggle room there.  Either way, “First they came for Hungarian Combinatorics” is all that came to mind.