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The Glory Beyond: David Berlinski’s Elementary Mathematics


A video circulating on the Internet tries to show why atheism, compared to theism, is the smart opinion to adopt. The videographer presents his case by stringing together fifty distinguished scientists and philosophers in a series of interviews, all giving their assent to the view that the entirety of existence may be explained in reductive materialist terms. You’re supposed to be impressed not by the wisdom of anything that’s said — because no one says anything memorably wise — but by the prestige of the job descriptions, titles, awards and academic affiliations that appear in quick succession in a corner of the screen.

Typical is Oxford University mathematician Marcus du Sautoy, pictured walking briskly through a cemetery. He recalls how as a boy he sang in a church choir. “There was a lot of talk of souls, and spirits and things, and by the time I was 13 I realized the whole thing was pretty illogical. I mean” — he says this with a little “Well, duh!” sort of comical grimace — “I am just flesh and blood.”

In their grasp of what religious faiths claim, you don’t get the impression that any other of the fifty academicians in the video has got beyond what he thought he knew at age 13. They all seem united by a condition of callow spiritual adolescence to which even the briskness of du Sautoy’s stride bears subtle testimony. Being in a cemetery, surrounded by the mystery of death, is one of those experiences that prompts sensitive people to wonder if some things in our lives are really like fingers pointing us to hidden truths or meanings beyond the plane, the given, the literal and the material. Passing among the houses of the dead calls, if nothing else, for a slower, more contemplative pace.Probably a certain blindness or deafness to such things confers an advantage in specialized academic work. It keeps you focused on the tiny problems before you. A very different kind of mathematician or other scientific thinker would have an ear open to the musica universalis, the music, only intellectually audible, that philosophers once conceived as emanating through the proportions embodied in the movement of the celestial spheres.

Discovery Institute’s David Berlinski is such a mathematician. His new book, One, Two, Three: Absolutely Elementary Mathematics, is a beautiful, brief, and very funny introduction to the history and philosophy behind basic math. It returns again and again to the allusiveness of numbers and the operations we perform on them. They allude, they point to, they gesture to something beyond themselves. Just what that might be, of course — of course, if you know anything about David Berlinski — Berlinski won’t say.

Absolutely Elementary Mathematics, or AEM as he abbreviates it, begins with and in a sense is encompassed by the act of counting by one. How can we justify even so seemingly simple an act as adding two numbers together and relying on the result? Addition as well as subtraction, multiplication and division, numbers in their varieties and modes of representation, theorems and proofs, exponents and logarithms, structures and sets, the inventors and theoreticians of AEM themselves, from ancient, anonymous Sumerian merchants to the Persian al-Khwarizmi (c. 780-850) who introduced Hindu-Arabic numerals to the West, to the vaulting geniuses of the nineteenth century and a bit beyond: It’s all covered with remarkable grace and wit, a richness of authorial personality and soul that breathes through everywhere. And Berlinski does it in fewer than just two hundred pages.

There’s no mistaking David Berlinski’s writing for anyone else’s. What other mathematician would extend a recounting of the life of Sonya Kovalevsky (1850-1891) to a consideration of the anxiety a father in nineteenth-century Russia felt about sending “his darling daughter,” all “latent erotic power,” “easy indolence,” and “decorously shielded limbs,” to study abroad? The anxiety came from the fact that this meant Sonya, budding math prodigy, would be riding on trains unaccompanied.

A woman sitting alone and — of all things! — reading a treatise on mathematics was widely regarded among even educated men as an invitation to debauchery. Anna Karenina had spent a good deal of time traveling alone on the night sleeper from Saint Petersburg to Moscow, after all, and even though she was a married woman, no one could miss the alliterative clack of trains, traveling, time, and treachery.

Who else would note in a discussion of how definitional descent brings order to exponentiation that ten raised to the power of zero and expressed as a symbol, 100, calls to mind an unappealing fish, “one of those horrible flounders with eyes on the same side of its head”? Who else would observe in reflecting upon the operation of subtraction that taking a minus five away from ten, expressed as -10 –5, “suggests nothing so much as someone gagging in a steak house”? Look at it again. He’s right, it does!

We take not only the appearance of numerals but the existence of the numbers for granted, with their qualities and powers, as a given, something that’s just there. But then under scientism’s spell we take a lot of things for granted, like the development of life, its origin, the origin of the cosmos itself. Berlinski compares the unspooling of the ordinary counting numbers, from zero to one and onward, to the last of these. “The natural numbers,” he observes, “are like the Big Bang in marking the appearance of a complicated structure with no obvious antecedents.”

Sumerians and Babylonians figure prominently in Berlinski’s story, and the complicated, eccentric structure of the natural numbers may remind you of some renderings of the Tower of Babel as it was intended by its doomed builders, twisting upward indefinitely into heaven.

The result emerges in stages very much like a tower rising where none might ever have been expected.

The effect is eerie, because the numbers are wonderfully various, different in properties, riotous, even numbers against odd, perfect numbers against imperfect numbers, square numbers, abundant numbers, deficient numbers, Mersenne primes, prime numbers against all the rest, numbers that are small, and those that are large. No physical tower ever arises in this way.

Nor does anything else I can think of in nature.

The structures of algebra, elaborated as “groups, semi-groups, monoids, rings, ideals, modules, vector spaces, semi-latices, categories, fields — a long list, but one that is finite” — call to mind some of the mysteries that trail through the evolutionary history of living systems. Both are arrayed in archipelagos of coherence, surrounded by impossibly vast oceans where the constituent elements do not and cannot add up to anything.

“Why such conceptual stability,” asks Berlinski, “whether in biology or in modern algebra?” How nature discovered the islands in the archipelago remains a resounding mystery. To say with a careless wave of the hand that they are somehow inherent in the way of things, it just happened, is the characteristic Darwinian gesture.

Berlinski, as he himself notes, is far from the first mathematician to have sensed such things about math. Leopold Kronecker (1823-1891) regarded the natural numbers as nothing other than a gift from God. Gottfried Leibniz (1646-1716) speculated about the numbers zero and one that they “are the touchstones of creation itself, the universe — yes, the universe — forged from the conflict between two numbers, the first representing nothingness, the second being.”

The view, interestingly, echoes the numbers mysticism in kabbalistic thinking that attributes creative power to the Hebrew letters, serving double duty as numerals, which God continually combines and recombines, speaking the universe into existence.

Of his own beliefs on these matters, the enigmatic Berlinski will suggest only that AEM, like a rope connecting earth and heaven, is part human art and part something else. In the frontispiece he reproduces Jean-Baptiste van Mour’s Harem Scene with Sultan as a representation of the dichotomy, keeping you waiting till the end of the book for an explanation of how exactly the painter’s erotic and architectural symbolism works. Elementary mathematics is “at once near and divinely distant,” Berlinski writes, suggesting “as nothing else can the glory that is beyond.”