## The Unity of Combinatorics

I just finished my very first * book review* for the

*Notices of the AMS*. The authors are Ezra Brown and Richard Guy, and the book title is the same as the blog post. I had mixed feelings when I accepted the assignment to write this. I knew this would take a lot of work (I was wrong — it took a

*huge*amount of work). But the reason I accepted is because I strongly suspected that there is

*“unity of combinatorics”, so I wanted to be proved wrong. Here is how the book begins:*

**no**One reason why Combinatorics has been slow to become accepted as part of mainstream Mathematics is the common belief that it consists of a bag of isolated tricks, a number of areas: [very long list – IP] with little or no connection between them. We shall see that they have numerous threads weaving them together into a beautifully patterned tapestry.

Having read the book, I continue to maintain that there is no unity. The book review became a balancing act — how do you write a somewhat positive review if you don’t believe into the mission of the book? Here is the first paragraph of the portion of the review where I touch upon themes very familiar to readers of this blog:

As I see it, the whole idea of combinatorics as a “

slow to become accepted” field feels like a throwback to the long forgotten era. This attitude was unfair but reasonably common back in 1970, outright insulting and relatively uncommon in 1995, and was utterly preposterous in 2020.

After a lengthy explanation I conclude:

To finish this line of thought, it gives me no pleasure to conclude that the case for the unity of combinatorics is too weak to be taken seriously. Perhaps, the unity of mathematics as a whole is an easier claim to establish, as evident from [Stanley’s] quotes. On the other hand, this lack of unity is not necessarily a bad thing, as we would be amiss without the rich diversity of cultures, languages, open problems, tools and applications of different areas.

Enjoy the full review! And please comment on the post with your own views on this alleged “unity”.

P.S. A large part of the book is freely downloadable. I made this website for the curious reader.

**Remark** (ADDED April 17, 2021)

Ezra “Bud” Brown gave a talk on the book illustrating many of the connections I discuss in the review. This was at a memorial conference celebrating Richard Guy’s legacy. I was not aware of the video until now. Watch the whole talk.

## 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.