Mathematics for the Million?

The integer 1936 is not only a perfect square and the year of my birth, it was also the year in which a scientist called Lancelot Hogben published a popular book called Mathematics for the Million. (It is still in print. A 2017 reprint by Prelude Press is available on Amazon UK for £10.99 but the reviewers complain of the poor quality of the typesetting and layout.)  Hogben’s parents were Plymouth Brethren but he rejected his family’s religion during adolescence. After retiring from front-line ambulance service in the First World War and then being imprisoned as a conscientious objector, he went to Cambridge as a medical student. He became, in his own words, a ”scientific humanist” and subsequently followed successful careers in experimental zoology and medical statistics. The book’s didactic method was to develop mathematical ideas in the context of their history and their applications to everyday life. Readers found his approach refreshingly different from the dry textbooks used in schools at the time.  The book covers roughly the current secondary school mathematics curriculum for 18-year-old school-leavers in England. In its day, it was praised by the science fiction writer H.G. Wells (“a great book of first class importance”) and by Albert Einstein (”it brings alive the contents and elements of mathematics”).

Hogben’s writings stirred up a lively controversy about the relative values of pure and applied mathematics and the contrasting mindsets of their practitioners, and many took sides. Hogben made clear his dismissive view of pure mathematicians as a coterie of self-indulgent high priests, protected in their ivory towers from realities of everyday life. After condemning Greek mathematicians and philosophers for their fondness for abstraction and obsession with aesthetics, he wrote “The fact that mathematicians are often like this may be why they are so inclined to keep the high mysteries of their Pythagorean brotherhood to themselves.” His main sparring partner in this philosophical spat was the eminent Cambridge number theorist G.H. Hardy, perhaps best known for his collaboration with the Indian mathematical genius, Srinivasa Ramanujan (Their relationship is popularized in the film The Man Who Knew Infinity in which Hardy is played by Jeremy Irons).

Hogben viewed mathematics as the science of measurement, to be valued for its practical applications and its potential, through science, to make the world a better place. Hardy was passionate about the beauty of his discipline and dismissed applied mathematics as trivial:

“There are then two mathematics. There is the real mathematics of the real mathematicians, and there is what I will call the ‘trivial’ mathematics, for want of a better word. The trivial mathematics may be justified by arguments which would appeal to Hogben, or other writers of his school, but there is no such defence for the real mathematics, which must be justified as art if it can be justified at all. “ 

This quotation is taken from his personal credo A Mathematician’s Apology, published in 1940 towards the end of his career.  In further quotations Hardy continues:

“A mathematician, like a painter or a poet, is a maker of patterns”,  …  “Even Professor Hogben, who is out to minimize at all costs the aesthetic element in mathematics, does not venture to deny (the reality of its beauty).”  …  he proudly asserts “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating.” 

Posterity has shown him to have been dramatically wrong in claiming that pure mathematics could never serve the deadly purposes of warfare -- think of Alan Turing at Bletchley Park or Hiroshima and Nagasaki. But as the “two mathematics” have cross-fertilized and enriched each other over the years, this squabble about their relative worth has run its course. Nowadays there are two new kinds of mathematicians: those - a dying breed -- who still believe there are two kinds of mathematics and those who don’t.

This post was prompted by the recent death at 94 of John Tate, a distinguished American number theorist. Like Hardy, he relished the beauty of mathematics but realized it was not something that could be easily shared with those not in his field. 'Unfortunately it's only beautiful to the initiated, to the people who do it,' he said in an American Mathematical Society interview. 'It can't really be understood or appreciated much on a popular level the way music can. You don't have to be a composer to enjoy music, but in mathematics you do. That's a really big drawback of the profession. A non-mathematician has to make a big effort to appreciate our work; it’s almost impossible."

Perhaps my title for this post should have been “The Loneliness of the Long-Distance Mathematician”.