A Brilliant Idea In Publishing
Big Results Can Come In Small Packages
Susan Friedlander is currently the Editor in Chief of the Bulletin of the American Mathematical Society, and also the Director of the Center for Applied Mathematical Sciences. She works on partial differential equations such as the Navier-Stokes equations. These equations are famously hard to understand, partially because of their connection to turbulence. Of course another measure of their difficulty is that they are the core of one of the remaining Clay Millennium Prize Problems.
Today I would like to talk about one of Friedlander’s brilliant ideas that has nothing to do with either the Navier-Stokes equations or turbulence.
In the latest issue of the Bulletin, October 2011, Friedlander devoted the entire issue to an interesting experiment. She had her editors select six articles from past generations that were seminal and helped stimulate research for decades, and then republished them. The were printed exactly as they appeared before—even with one or two simple typos that were there the first time. On page 215 there is “ otherwise”: which seems like a typo to me.
Five dated from the period 1977-1984, one was from earlier. Each editor wrote a brief introduction to their selected paper, but the stars were the original papers:
- Poincaré Recurrence And Number Theory, by Harry Furstenberg.
- Internal Set Theory: A New Approach To Nonstandard Analaysis, by Edward Nelson.
- Invariants of Finite Groups And Their Applications To Combinatorics, by Richard Stanley.
- On The Parallelizability Of The Spheres by Raoul Bott and John Milnor.
- An Elementary Introduction To The Langlands Program, by Stephen Gelbart.
- Lectures On Morse Theory, Old And New, by Raoul Bott.
I think this idea of republishing important papers is terrific, and I personally greatly enjoyed the issue, and urge you to take a look at it. It was wonderful to see these famous papers—we should try this experiment in theory of computing. Congratulations to Friedlander.
The first paper is beautiful discussion of the modern consequences of one of the great results of Henri Poincaré, who made fundamental contributions to almost all aspects of math and physics, and was called a polymath. What I found so interesting about the paper is that Furstenberg shows how Poincaré’s theorem helped launch an entire area of research, yet it has a tiny and simple proof.
This made me think about the size of proofs and the impacts of proofs. There is, at best, a weak connection between the length of a proof and its impact and importance in mathematics. There are long proofs that have had small impact on math; there are long proofs that have had a large impact; there are short proofs that have had a small impact—no surprise with these statements. What may be surprising is that there are proofs of statements that have had huge impact, yet the proofs can be small, even tiny.
Long Proofs With Huge Impact
Finding examples of proofs that are large, even huge, that have had large impact on mathematics is pretty easy. A moment’s thought yields many examples—you may have your own, and your own may be even better than mine.
Fermat’s Last Theorem: Andrew Wiles proved the famous theorem that there is no solution to where and are non-zero integers. The proof is clearly long, and has had major impact on the understanding of number theory.
Simple Group Classification: This is the work of many mathematicians, and classifies all finite simple groups into certain infinite families with twenty-six special exceptions. The classification proof is beyond large; it is estimated to be tens of thousands of pages long. The theorem is used in many places in mathematics to solve problems invoking finite groups and even beyond.
Gödel’s Relative Consistency Of The Axiom Of Choice: Kurt Gödel proved that the Axiom of Choice (AC) can be added to set theory without causing it to become inconsistent. Set theory, as in ZF, is already inconsistent or remains so after the addition of AC. This was immensely important since AC while very useful is also quite powerful and leads to many surprising results. For instance, AC shows:
- There are sets that are not Lebesgue measurable.
- There is a way to cut up a solid sphere into five pieces and reassembly them into a sphere that is twice the volume. This is called the Banach-Tarski paradox after the great mathematicians Stefan Banach and Alfred Tarski.
Short Proofs With Huge Impact
Euclid’s Theorem: Euclid proved almost 2300 years ago that there are an infinite number of primes. Recall the proof is to assume that there are only a finite number of primes: let be all the primes. Then look at
The number and so is divisible by some prime, let it be denoted by . Then by assumption for some . But this implies that divides , which is a contradiction.
Halting Is Undecidable: Alan Turing of course proved that there is no Turing Machine that can tell if an arbitrary Turing Machine will halt on a given input. The proof of great importance is a clever diagonalization argument. Yet it can almost be stated as an afterthought.
Recurrence Theorem: This is due to Poincaré, and I will explain it and the proof in the next section.
Let’s state the theorem precisely and then give the almost shockingly short proof. Let be an abstract space, and let be a measure on it. We put and assume that the class of measurable sets, i.e. for which is defined, is sufficiently extensive. Also let map to so that the measure is preserved,
for all measurable sets. This is all we need to state the theorem:
Theorem: For any measurable set with , there exists a point so that for some .
I quote Furstenberg:
The proof is extremely simple. Assume that no point returned to . Then for all , and so whenever . But the sets all have the same measure , and they cannot be disjoint since .
The importance of this theorem is many-fold. It helps shed light on physical systems, which is why Poincare studied it in the first place. It also has helped yield insights into combinatorics. For example, generalizations of it are shown to prove the famous theorem of Bartel Leendert van der Waerden:
Theorem: Divide the integers into subsets, and let be a natural number. Then one of the subsets contains an arithmetic progresssion of length .
What are your favorite examples of short proofs with large impact? The fact that they exist may even suggest some ideas for automated discovery of proofs. It plausible that while getting computers to find proofs that are huge may be impossible, getting them to help find very short ones might be possible. What about an automatic search for short, tiny, proofs with huge impact?
Also do you like Friedlander’s republish idea?
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