This post is a commentary on Alexandre Borovik’s book Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice, published by the American Mathematical Society last year. The manuscript of the book was on the author’s web page for several years before publication; as a result, the book incorporates many comments from readers of the pre-publication version.
This is not a book review; I have put far more of my own opinions into it than I would do in a review. The book features childhood photos of many of the people quoted or referred to, since “I wish my book to bear a powerful reminder that we were all children once”. Sasha grew up in a village on the shore of Lake Baikal; I grew up in a village on the Darling Downs. Is it fanciful to think that some shared background helps the book to communicate to me?
Early on, a few pages into the introduction, Sasha endears himself to me by saying,
I take, as a working hypothesis, the assumption that mathematics is produced by our brains and therefore bears imprints of some of the intrinsic structural patterns of our minds. If this is true, then a close look at mathematics might reveal some of these imprints—not unlike the microscope revealing the cellular structure of living tissue.
I came to the same conclusion many years ago. Apart from the fact that it seemed right, it offered a passage between the points of view that mathematics is “discovered” or “invented”, the Scylla and Charybdis of the philosophy of mathematics. When we do mathematics we are neither discovering universal truths, nor inventing socially conditioned conventions, but are bringing forth what is best and most characteristic of ourselves.
This philosophical positioning is important to the book. Manin’s famous statement “A proof only becomes a proof after the social act of accepting it as a proof” is glossed as follows:
… as Yuri Manin stresses, … a proof becomes such only after it is accepted (as the result of a highly rigorous process) […] Manin describes the act of acceptance as a social act; however, the importance of its personal, psychological component can hardly be overestimated.
In the other direction, he says
When learning or doing mathematics, we quite frequently have to create mental images of mathematical objects with eidetic qualities as close to that of the images of real objects as possible. (If we are duped, as a result, into the belief that mathematical objects exist in some ideal or dream world, this happens only because we want to be duped.)
The book is full of gems of mathematics, the focus being on the mental “click” that occurs when we internalise the argument. Consider, for example, Coxeter’s proof of Euler’s theorem that a direct isometry with a fixed point in three dimensions is a rotation. First, observe that any isometry fixing a point is the product of at most three reflections. (We have to map an orthonormal basis to its image; each basis vector can be mapped to its image by a reflection fixing the vectors we have already put in their places.) Now a direct isometry is a product of an even number of reflections, necessarily two; the two reflecting mirrors meet in a line which is the rotation axis.)
Did you feel the click as you read that?
The book, then, is a challenge to neurophysiologists, psychologists, and similar people: mathematics exhibits important aspects of the structure and function of our brains; investigate these!
Sasha discusses some of the great divisions in mathematics: discrete and continuous (or “switch” and “flow”), finite and infinite, geometric and algebraic. In ech case, the mental activity is presumably associated with different activity in the brain. The examples used to explore the divisions are interesting. For example, what is the next simplest real function after linear functions? You might say x2, but Sasha’s answer is |x|. He explores the problem of iterating the function ||x|−1|, a problem which certainly invokes the “discrete” circuits (try it!)
(It has to be said that some of the research so far is counter-intuitive. For example, researchers claim that subitizing (recognising the number of items in a set directly, possible up to 4 or 5 items) and counting engage the same brain circuits. This is not what it feels like from inside!)
Some of the connections seem just a little far-fetched; possibly Sasha has tongue in cheek at some points. For example, in Chapter 2, watching a woman on a train solving a Sudoku puzzle leads him to suspect that the pleasure we get from working recursive algorithms such as long division or solving Sudoku comes from the same source as our pleasure in popping bubble-wrap, and indeed as our ancestors’ pleasure in grooming one another to remove lice. Moreover, he points out, grooming behaviour is controlled by the HOXB8 gene, one of the ancient and strongly conserved homeobox family, which encode transcription factors important in morphogenesis in all multicellular organisms. So should people with defective HOXB8 (if they don’t die in infancy) should have problems counting, maybe? Among several objections one could raise is the fact that it is misleading to describe Sudoku as solved by a recursive algorithm. Formally, of course, the algorithm is just “while there is an empty space, find a number that is forced, write it in, and repeat”. But any Sudoku harder than about moderate difficulty will involve branching.
Other points raised are more thought-provoking and testable; I hope that brain researchers will take up some of his challenges.
But, for a mathematician, the book has other attractions, not least the rich store of examples and connections. Chapter 3, for example, has the following topics: sentences which require a “stack” to parse, including a recursive construction of grammatical sentences with an arbitrarily large number of consecutive occurrences of the word “had”; the relation of number sense to grammar, including languages with several different number markers (Russian has separate forms for “one”, “two-three-four”, and “five or more”); a re-formulation of the notion of a Coxeter system in terms of palindromes; the fact that students trained with user-friendly logic software with automatic bracket matching do less well when tested on logic; a delightful proof that the numbers of dissections of a polygon and of bracketings of an expression are equal; and an explanation of a mysterious statement by Plutarch that Hipparchus counted 103,049 compound propositions made from ten simple propositions “on the affirmative side” and 310,952 “on the negative side”. (This chapter is by no means exceptional.)
A final comment. I was interested to see that Sasha has selected many of the same quotes that I have collected on my web page, which I discussed here in an earlier post. One of these he did take from my page: Myles Aston’s illustration of the “miserable wasteland of multidimensional space” with an experience in an MIT sandwich shop.
If you are a brain researcher, I hope you will read the book. But if you are a mathematician, I can confidently predict that you will enjoy it!