Saharon Shelah, logic juggernaut

There’s a great quote via Hunter Johnson on Quora arguing for the importance of mathematical logic to math:

Shelah is attending a mathematics talk. The presenter has offered, with great difficulty, a new example of some mathematical structure, let’s say a quasi-Hebrand reticular matrixoid. The existence of a new example of this object is significant in the field of quasi-Hebrand reticulation theory (I am making up these names). 

Shelah has come in late and missed most of the talk. When the time for questions comes, he raises his hand and says, “I can give you uncountably many of these objects. Now, tell me, what is a quasi-Hebrand reticular matrixoid?”

Mathematical logic is about the forest rather than the trees. When you look at the structure that different mathematical fields have in common, you see overarching themes that make the theory work.

The funny thing is that it’s entirely believable, because this is Saharon Shelah we’re talking about. I don’t know of anyone else alive more prolific in math than he is – 1,166 papers/preprints/books as of July 2019 with 260 coauthors.

I was pretty happy to see him being signal-boosted today on Quora, by Alon Amit. Alon wrote:

He is also recognized as one of the most powerful problem solvers around (I remember this actual phrase being used in a Scientific American article about Van der Waerden’s Theorem). As a result, some mathematicians are slightly relieved that Shelah focuses on infinite combinatorics and model theory and not on their field.

Shelah made tremendous contributions to model theory and set theory, solving a huge number of open problems and establishing major theories and directions for research, most notably PCF theory. In the well-known classification of mathematicians into “theory builders” and “problem solvers”, Shelah is a rare dual citizen.

A “dual citizen” of the highest order! I’ve only heard this explicitly said (in terms of praise) about one other modern-day mathematician, Akshay Venkatesh in his Fields laudatio, but of course that’s just memory failing me.

What is Shelah’s style? A Kanamori gave a beautiful description back in 1999. Unfortunately it’s just one solid wall of text, too much for my non-Shelahian working memory, so I’ve broken it up:

In set theory Shelah is initially stimulated by specific problems. He typically makes a direct, frontal attack, bringing to bear extraordinary powers of concentration, a remarkable ability for sustained effort, an enormous arsenal of accumulated techniques, and a fine, quick memory.

When he is successful on the larger problems, it is often as if a resilient, broad-based edifice has been erected, the traditional serial constraints loosened in favour of a wide, fluid flow of ideas and the final result almost incidental to the larger structure. What has been achieved is more than just a succinctly stated theorem but rather the establishment of a whole network of robust constructions and arguments. A telling point is that when some local flaw is pointed out to Shelah, he is usually able to come up quickly with another idea for crossing that bridge.

Shelah’s written accounts have acquired a certain notoriety that in large part has to do with his insistence that his edifices be regarded as autonomous mental constructions. Their life is to be captured in the most general forms, and this entails the introduction of many parameters. Often, the network of arguments is articulated by complicated combinatorial principles and transient hypotheses, and the forward directions of the flow are rendered as elaborate transfinite inductions carrying along many side conditions. The ostensible goal of the construction, the succinctly stated result that is to encapsulate it in memory, is often lost in a swirl of conclusions. This can make for difficult and frustrating reading, with the usual problem of presenting a mathematical argument in linear form exacerbated by the emphasis on the primacy of the construction itself and its overarching generality.

Further difficulties ensue from the nature of the enterprise. Shelah regards the written word as necessary and central for capturing and fixing a construction, and so for him getting everything down on paper is of crucial importance. The tensions among the robustness of the construction, the variability of its possible renditions, and the need to convey it all in print are inevitably complicated by the speed with which he is able to establish new results. The papers have to be written quickly, previous constructions are newly refreshed and modified, and so a labyrinthian network may result over a series of related papers.

In mathematics one often aspires to the most elegant or definitive treatment; in contrast, Shelah’s work features a continuing, dynamic self-dialogue, one that pushes to the limits of exposition. Many may consider Shelah’s work to be “technical”, but as T S Eliot has written “We cannot say at what point ‘technique’ begins or where it ends” [‘The Sacred Wood’]. While there is a particular drive to solve specific problems, Shelah with his generalizing approach is able to draw out larger, recurring patterns that lead to new techniques that soon get elevated to methods.

What’s an example of this approach?

One primary instance is the whole complex of approaches and results he developed under the general rubric of proper forcing. Shelah started out in model theory, developing an abstract classification theory for models which is a continuing research program for him and model theorists to this day. In the mid-1970’s, in his first major body of results in set theory, Shelah resolved a long-standing problem in abelian group theory, Whitehead’s problem, by establishing both the consistency and the independence of the corresponding proposition. It is through these beginnings, motivated by the set-theoretic problems that arose, that Shelah started to develop a general theory of iterated forcing for the continuum.

But what does Shelah “do all day at the office”? In his own words:

Since I have succeeded in demonstrating a substantial number of theorems, I have also a lot of work completing and correcting the demos. As I write, I have a secretary typing (I did have a lot of troubles concerning this) and I have to proof-read a lot. I write and make corrections, send to the typist, get it back and revise it again and again.

A great amount of time is used to verify what I wrote. If it is not accurate or utterly wrong, I ask myself what went wrong. I tell myself: there must be a hole somewhere, so I try to ll it. Or perhaps there is a wrong way of looking at things or a mistake of understanding. Therefore one must correct or change or even throw everything and start all over again, or leave the whole matter. Many times what I wrote first was right, but the following steps were not, therefore one should check everything cautiously. Sometimes, what seems to be a tiny inaccuracy leads to the conclusion that the method is inadequate.

I have a primeval picture of my goal. Let us assume that I have heard of a problem and it seems alike to problems that I know how to resolve, provided we change some elements. It often happens that, having thought of a problem without solving it, I get a new idea. But if you only think or even talk but not sit down and write, you do not see all the defaults in your original idea.

Writing does not provide a 100% assurance, but it forces you to be precise. I write something and then I get stuck and I ask myself perhaps it might work in another direction? As if you were pulling the blanket to one part and then another part is exposed. You should see that all parts are integrated into some kind of completeness.

Indeed, sometimes you are happy in a moment of discovery. But then you find out, while checking up, that you were wrong. There has been a joy of discovery, but that is not enough, for you should write and check all the details. My office is full of drafts, which turned to be nonsense.

Strong words, but it does explain A Kanamori’s remark that “Shelah regards the written word as necessary and central for capturing and fixing a construction, and so for him getting everything down on paper is of crucial importance“.

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