G. H. Hardy On The Supposedly ‘Second-Rate Mind’

In A Mathematician’s Apology G. H. Hardy wrote:

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

I make note of this famous excerpt today because I saw it, again, on a friend’s Facebook status. As I noted in response then, “Hardy says ‘second-rate mind’ like that’s a bad thing. I’d love to have a second-rate mind.” (A response stolen from George Mikes‘ ranking of writers where he says something like “I wish I was at least a fourth-rate writer.”)

But less facetiously, there are two confusions that afflict Hardy’s claim above.

First, Hardy is caught up in the mythology of the lone creator, artist, genius, whose thoughts and works spring ab initio from his or her mind alone, independent of history and context and antecedent work. Such a creature is entirely mythical; there is no fallow ground in the arts and sciences to be worked. All has been worked and tilled before; the creator, the genius, the artist builds on what came before. In one crucial sense, all supposedly ‘creative’ and ‘original’ work is exposition and explication and criticism and appreciation; departures depart from somewhere, they do not find an Archimedean point for themselves. (And can the genius’ work be understood without it being explicated?)

Secondly, a mathematician who writes about mathematical work may be doing philosophy of mathematics or perhaps noting connections between bodies of work that are not visible to those who might have worked on them individually. There is ‘insight’ here to be found, which might be as penetrating in getting to the ‘heart of the matter’ or in affording us a new ‘vision’ as the work of the ‘original creator’–perhaps achieved with flair and style that might lift the work out of the realm of the ordinary. Such might be the case with other kinds of explicatory work. Writing about writing is still writing, and still subject to the critical assessment we direct at the written word; we might find brilliance and innovation and style there too.

Ultimately, Hardy’s view speaks for too many, says too much, and yet manages to convey an impoverished and narrowed vision of the creative mind and its various endeavors. Moreover, it betrays its own trivial concerns in attempting to devise a hierarchy of values for such forays of the intellect: unsurprisingly, we find the work that Hardy saw himself as engaged in placed at the top of this hierarchy.   Philosophies are disguised autobiographies indeed. (My defense of the explainer now suddenly becomes comprehensible.)


The Greek Alphabet: Making The Strange Familiar

In his review of Patrick Leigh Fermor‘s The Broken Road: From The Iron Gates to Mount Athos (eds. Colin Thubron and Artemis Cooper, New York Review Books, 2014) Daniel Mendelsohn writes:

His deep affection and admiration for the Greeks are reflected in particularly colorful and suggestive writing. There is a passage in Mani in which the letters of the Greek alphabet become characters in a little drama meant to suggest the intensity of that people’s passion for disputation:

I often have the impression, listening to a Greek argument, that I can actually see the words spin from their mouths like the long balloons in comic strips…:the perverse triple loop of Xi, the twin concavity of Omega,…Phi like a circle transfixed by a spear…. At its climax it is as though these complex shapes were flying from the speaker’s mouth like flung furniture and household goods, from the upper window of a house on fire.

I first encountered Greek letters, like most schoolchildren, in my mathematics and physics and chemistry classes. There was π, the ratio of the circumference of a circle to its diameter; ω, the frequency of a harmonic oscillator and later, infinity in set theory; λ, the wavelength of light; θ, ubiquitous in trigonometry; Ψ, the wave function of quantum mechanics; Σ, the summation of arithmetic and geometric series; a whole zoo used to house the esoteric menagerie of subatomic particles; and many, many more. The Greek alphabet was the lens through which the worlds of science and mathematics became visible to me; it provided symbols for the abstract and the concrete, for the infinitely small and the infinitely large.

I never learned to read in Greek but the Greek alphabet feels intimately familiar to me. Perhaps the most familiar after English.

I first saw Greek texts in the best possible way: Greek versions of Aristotle and Plato in my graduate school library, intended for use by those who specialized in ancient philosophy. (These texts were in classical Greek.) I took down the small volumes from the shelf and opened their pages and looked at the text. It was incomprehensible and yet, recognizable. I could see all the letters, those old friends of mine: the α and the β used to denote the atoms of a language for propositional logic, the Γ of the generalized factorial function, the Δ of differences; they were all there. But now they were pressed into different duties.

Now, they spoke of ethics and metaphysics and politics, of generation and corruption; their forms spoke of the Forms. Now they were used to construct elaborate philosophical systems and arguments. But even as they did so, I could not help feeling, as I looked at the pages and pages of words constructed out of those particles, that I was looking at the most abstruse and elaborate mathematical text of all. It was all unknown quantities, an endless series of fantastically complex mathematical expressions, one following the other, carrying on without end. Yes, it was all Greek to me.  And yet, I still felt at home.

The Terror of the Formerly Utterly Incomprehensible

Yesterday’s post detailing my rough introduction to calculus in high school reminded me of another encounter with a forbiddingly formidable mathematical entity, one that in later times served as an acute reminder of how even the utterly incomprehensible can come to acquire an air of familiarity.

One reason for the rough ride I experienced in my physics class in the eleventh grade was that I had transitioned to studying a more advanced science and mathematics curriculum after my ninth and tenth grades. In those, I had been assigned to a ‘Arts and Humanities stream’; the science and mathematics I had studied were pitched at a more elementary level compared to those assigned to the ‘Science stream’. Not content with my placement, and overcome by the surrounding familial and social pressure to pursue a more ‘applied’ and ‘practical’ course of study, I switched schools and curricula. Unsurprisingly, this meant I had some catching up to do, which I intended to get started on in the break between my tenth and eleventh grades. My brother, who was about to graduate high school and move on to university, helpfully handed me his textbooks and study guides, throwing in the singularly unhelpful warning that I was about to be swamped.

Armed with this grim prognostication, I began what I hoped would be a rewarding period of autodidactic endeavor, one that would equip me with not just the requisite curricular background by the time regular classes began in the new academic year, but also some confidence.

Before I would begin systematic study, of course, I would take a peek at what lay ahead. And there, I was confronted by a terrifying, mysterious new entity, one that seemed so beyond my intellectual capacity that I almost resolved there and then to give up my dream of studying the sciences in high school. Perhaps I was meant, as I had originally intended, to study what seemed like the considerably friendlier humanities subjects. What had frightened me so?

Something called the ‘parallelogram law of vector addition.’ It appeared early in my physics textbook, in the second chapter, shortly after the one devoted to something called ‘dimensional analysis.’ While I knew what the geometrical figure termed a ‘parallelogram’ was, I did not know what a vector was, and I did not understand–could not begin to fathom!–what the former had to do with the ‘addition’ of the latter, especially as they seemed to be, from what I could make out, things with arrows, and were described by letters with, you guessed it, arrows above them. This was all black magic; perhaps there were mysterious potions and incantations handed out to initiates in order to enable their understanding of the dark arts of physics. So I retreated in panic; it took some mustering up of an elusive inner resolve to approach those books again.

Months later, deep into my eleventh grade, and having moved on well beyond the parallelogram law of vector addition, I was able to look back on my initial exposure to it with a complex mixture of mortification, relief and euphoria. I had not imagined that something so seemingly esoteric and inaccessible could ever be incorporated into my corpus of academic knowledge, into the grab bag of things known and grappled with. The knowledge of that transition from utter incomprehension to familiarity stood me in good stead on many occasions later; indeed, I would say I still rely on it when confronted with a seemingly insuperable intellectual task.

Manil Suri on the Beauty and Beguilement of Mathematics

Manil Suri has an interesting Op-Ed on math–How To Fall In Love with Math–in The New York Times today. As befitting someone who is both a mathematician and a novelist, there are passages of writing in it that are both elegant and mathematically sound. The examples he provides of mathematical beauty–the natural numbers, n-sided regular polygons that become circles as n approaches infinity, fractals–are commonly used, but for all that they have not lost any of their power to beguile and fascinate.

I fear though that Suri’s message–that math is beautiful, creative, elegant, not to be confused with routine number crunching, and worthy of wonder and exaltation and careful study–will not be heard through the haze of the very math anxiety he seeks to cure. Suri needn’t feel too bad about this though; many others have tried and failed in this very endeavor in the past.  If, as Suri suggests, we are wired for math, we also sometimes give the appearance of being chronically, congenitally, incurably anxious about it.

A personal note: I came to a realization of mathematics’ beauty late myself. Like many of the math-phobic that Suri refers to in his article, through my school years my attitude toward the study of math was fraught with fear and befuddlement. I was acceptably competent in the very junior grades but a harsh teacher in the seventh grade ensured that I would earn my worst grades then. My concerned father took it upon himself to drill me in algebra and I regained a little confidence. Not enough though, to want to study it at the higher levels that were made available to us in the ninth and tenth grades. But the pressure to study engineering at the university level meant I had to return to the study of more advanced mathematics for the eleventh and twelfth grades.

In those two years, I was exposed to calculus for the first time and came to love it; its connection with motion, the slopes of curves, the use of differential equations to model complex, dynamic systems; these all spoke to me of a logical system deeply implicated in the physical world around me. Once I had learned calculus, I saw it everywhere around me: in a stone’s dropping, a car’s acceleration, a rocket’s launch, an athlete’s push off the starting block.

I majored in mathematics and statistics at university, but if I saw any beauty in mathematics in those years, it came when I saw some good friends of mine working out complex problems with some style. I had lost motivation and interest and barely survived my college years. My graduate studies in computer science didn’t help; I was able to–unfortunately–skip the theory of computation and graduate.

Years later, when I encountered mathematical logic as part of my studies in philosophy, some of my appreciation for the world of abstract symbol manipulation came back. It helped that my dissertation advisor was an accomplished mathematician whose theorems and proofs sparkled with style and substance alike. From him, I regained an appreciation for the beauty of the symbolic world.

I was never a mathematician, and don’t work in logic–mathematical or philosophical–any more. But my brief forays into that world were enough to convince me of the truths that Suri refers to in his piece:

[Mathematics] is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall.