Fitting Surprise in Mathematics

25 09 2012

In his faculty research colloquium this past Friday, my colleague, mathematician Andrew Shallue, gave a presentation titled “Constructing Large Numbers with Cheap Computers.”  As a part of this presentation, in which he discussed how he and his fellow researchers created the world’s largest Carmichael number, Andrew read a portion of the following, from chapter 18 of G.H. Hardy’s A Mathematician’s Apology:

”What ‘purely aesthetic’ qualities can we distinguish in such theorems as Euclid’s and Pythagoras’s?  I will not risk more than a few disjointed remarks. In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions.  There are no complications of detail – one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency.  We do not want many `variations’ in the proof of a mathematical theorem: `enumeration of cases,’ indeed, is one of the duller forms of mathematical argument.  A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.”

Needless to say, I was intrigued. The whole talk (at least as much as I could understand) was excellent, but Hardy’s idea that the famous theorems of Euclid and Pythagoras are noteworthy because they seem both inevitable and unexpected struck a chord: Hardy was impressed by fitting surprise in mathematics, and fitting surprise is something—an aspect of poetry and many other arts (fiction, drama, painting)—that interests me greatly. Now, here it is in mathematics. Seems fitting. And surprising.



4 responses

25 09 2012
Kristin Gecan

Seems math is a #poetry trend today – over at Scientific American there’s discussion of Zukofsky and “mathexpressive poetry,” and even surprise! – “I claim that nothing is more important for a poet than finding new ways to surprise people with the familiar.”

26 09 2012
Mike Theune

This is great, Kristin–thanks! Math does seem to be in the air… I encourage everyone to check out the Scientific American blog post–it’s good, and it’s by Bob Grumman, who’s been a welcome visitor to this blog. (Here is one of our exchanges.)

25 09 2012

The notion of inevitability has been isolated by Bertrand Russell on a number of different occasions. Perhaps the most notable case is his essay “The Study of Mathematics” published in his book Mysticism and Logic (which can be read here ). This essay is noted most frequently for containing one of the most vivid descriptions of mathematical beauty, one component of which is a strong and almost overwhelming sense of inevitability in argument regardless of where the conclusions might lead.

It is interesting to note that the method of analytic investigation employed religiously by Russell follows a sort of understated inevitability that is only surprising once you have followed his argument from beginning to end. For example, in his book The Scientific Outlook he makes a relatively simple, and entirely unsurprising, claim: a vast majority of the problems had by modern society would dissolve if one followed a doctrine of skepticism with dedication and vigor. The surprise occurs when one reads his descriptions and arguments that seem to inevitably show that such an understatement is in fact supremely surprising in its implementation. That is, if you follow through an investigation of the impact of a skeptical outlook on various facits of daily life there are subtle influences which when combined have a surprising impact on the way that humans might live their lives.

In mathematics we have a number of precise examples of what it means for something to be inevitably surprising. Paul Erdös is one of the greatest mathematicians to have lived, having published more papers in mathematics than any other human ever, and he had a rather interesting view about mathematical proofs. He imagined, in a purely playful way, that God had a book in which all of the most elegant solutions to mathematical problems had been transcribed. It is our goal, as puny earthlings, to uncover the proofs from THE book. So Paul spent most of his life searching for proofs from the book, and when he found them they were often just as beautiful as you might imagine (the type of thing that is so well orchestrated only Bach could have conceived of such a complexly interwoven structure). Often proofs from the book had the following features: they used elementary mathematics that was as close as possible to mathematical objects such as the natural numbers, graphs, counting arguments, and other such simple tools that you might find in the mathematical tool box of a high school student, and secondly, their conclusions were powerfully enlightening. Often such proofs have the feature of being over before it seems the conclusion has even been met: like that feeling of missing the last step of a staircase.

You should not be misled into thinking that the theorems he was referencing were noteworthy because they were unexpected: I have yet to meet a mathematician who is capable of appreciating a mathematical theorem simply because it is beautiful or unexpected. The theorems he’s referencing are in fact HIGHLY useful i.e. they are used frequently in the development of further mathematics, in fact they are the conceptual turning points in some cases. Often the notion of enjoying mathematics because of its beauty is relegated to those chosen few who have already proven themselves worthy of living such a mathematically enlightened life. As a graduate student in mathematics I’m continually confronted by the conflict between utility and beauty. Specifically, it is ritual suicide to declare that the reason you study mathematics is because you find it delightfully beautiful. The response you will get from your fellow professors is one of admiration for your naivety and “cuteness”: “isn’t he a darling, he thinks that you can be a mathematician and simply study it because it’s beautiful”. If you are not producing mathematics that is of public interest, meaning it is valued by the mathematical commons because of how it may be used, then you are shuffled off to a quite and dark corner of the university and left to think about what you have done. Sure you can live a life of mathematical beauty, and then cut your ear off…

I have met a number of mathematicians who claim to love mathematical beauty, but there are often two issues which arise. The first is that they do not actually mean they enjoy mathematical beauty, they actually enjoy being someone who enjoys “the finer things”. You might find them wining and dinning and talking about the beauty of their cigarets vapor trails. Secondly, they have done nothing to discover or describe what mathematical beauty might be. I don’t think it is possible to concretely understand mathematical beauty, but I have spent enough time trying to capture it in order to know, in at least a vague way, that it is not the type of thing that one wakes up and declares their dedication to. As far as I can tell the only way to come to some understanding of what it might mean for something to be mathematically beautiful is to try to construct descriptions of mathematical concepts or ideas which are in themselves beautiful. For example, if you wished to say something about the beauty of poetry, how better to say it than to construct a beautiful poem. If you are trying to demonstrate your understanding of beauty then one would hope that you could create it. I’ve tried to collect descriptions, which I find to be personally beautiful, of mathematical things here:

26 09 2012
Mike Theune

John, you are the man. The mathematish-man. And you are a poet: “like that feeling of missing the last step of a staircase”; “and then cut your ear off…” Thanks for your lively and thoughtful response.

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