What is mathematics?
Review of The Mathematical Experience, by Philip J. Davis & Reuben Hersh (1981)

By Tony LongShanks LeTigre

Several years ago, during a period when my life had taken a strange turn & I began delving into physics books as an inquiry into the deepest ontological & philosophical questions, the question above began to interest me. One book I read was Murray Gell-Mann’s The Quark & the Jaguar, in the course of which he posed the question of whether mathematics was or was not a science in itself; if so, he states, it is “more fundamental than any other.” Gell-Mann deserves some credit for being open-minded enough to even pose the question, for some physicists—as well as logicians, & computer scientists, & engineers, & various others—hold a reductive or belittling view of math as merely a tool or language for other sciences. The more I read about physics, the more fascinated I became by the way that equations like those involved in Einstein’s theory of relativity could tell us things about the nature of the cosmos, could predict things & point to answers of mysteries that would otherwise lie far beyond our present grasp. I didn’t want to do mathematics—solving equations, measuring angles, formulating convergence proofs—but to read about the subject itself in a detached way; if math is the science of numerical abstraction, you could say that I had an abstract interest in an abstract science. I wanted to find a book that would explain the nature of mathematics & mathematical philosophy, if such a thing exists. It turns out it does—or did—& I found the perfect introduction in the form of The Mathematical Experience, by Philip J. Davis & Reuben Hersh, published in 1981.

Who knew the history of math could be so fascinating, so entertaining, even? Davis & Hersh trace that history from its beginnings with Thales (circa 600 BC), Pythagoras, Euclid, & Archimedes in the classical Greek era; into the relative darkness of the Middle Ages, followed by the enlightenment of the 1600s with Kepler, Galileo & Newton; & thence forward via Lagrange, Euler, Gauss, etc. to the modern era.

The original view of math was the Platonist or realist view, which holds that math is something with an objective reality that transcends humanity, something that humans only study, like any of the natural sciences. Davis & Hersh describe the way this view has evolved over time, tied in with the general evolution of philosophy, & it is here that the story comes alive. The Platonist view was challenged first by materialism & empiricism, then by Kant & logic, which led to the triumph of formalism, which then gave way to “the great foundationist controversies” of the 20th century (which are tons of fun to read about!) First, Descartes reduced geometry to algebra (& for that, I say we reduce Descartes to a pulp!) Then in the 1800s, it was found that Euclidean geometry was not the only conceivable kind; at least two forms of non-Euclidean geometry that explicitly spurned Euclid’s fifth postulate were developed. Then the formalists sought to make math as a whole synonymous with symbolic logic. Right around the turn of the 20th century a German mathematician named Frege had just completed a towering tome that would cement the conversion of math into logic. Meanwhile, Bertrand Russell—of Russell & Whitehead—discovered a logical paradox pertaining to definitions in set theory that turned his head white, in horror. He sent a note to Frege informing him of the ill news; Frege received it just as he was about to send his grand opus off to the printers & open a bottle of champagne to celebrate. Which is hilarious! But wasn’t, for Frege.

The cumulative effect of all these assaults on mathematical indubitability, on the age-old belief in math as the most infallible branch of human knowledge, was a sort of crisis of faith: the whole glittering edifice of mathematics that had stood since the ancient Greeks was suddenly seen to stand on a hill of loose sand, you might say. Raging controversies over the “foundations” of mathematics raged throughout the late 1800s & first half of the 1900s. A Dutch mathematician named Brouwer, driven mad by the uncertainties, locked himself in a room in Amsterdam, shouting something about “constructivism,” & vowed he would deal only with nice, natural, unambiguous numbers from that point on. (I’m not sure they ever talked him out of there; someone should probably check.)

The formalist/logical bubble was further popped by Karl Popper in the 20th Century, who helped usher in a sort of anti-authoritarian revolution in the natural sciences that spilled over to mathematics, physics, & all the rest. A book by Imre Lakatos called Proofs & Refutations—described as a masterpiece by Davis & Hersh, & the subject of one of the most interesting chapters of The Mathematical Experience—seems to point the way forward to a more “human,” fallible, heuristic & intuitive mathematics, & the revival of mathematical philosophy, which had been largely strangled by the formalists & their implacable quest to wring it dry of “meaning.”

After reading The Mathematical Experience, I understand many things that were only labels or vague concepts to me before—differential equations & their relation to mechanics; the difference between pure & applied mathematics; the way a scientific theory is formed from postulates & axioms; the difference between heuristics & rigor & (related thereto) between analog/intuitive & analytical/deductive modes of reasoning. It’s really a wonderful book, in the same vein as Gödel, Escher, Bach (which came out just a couple years before). And it has also answered the question that inspired my inquiry into the nature of math in the first place: mathematics—in my now somewhat informed opinion at least—is most definitely a science in itself, & not just a tool for use by the other sciences. I am left with a newfound respect for mathematicians & for the way that their branch of knowledge intersects with the one of my own deepest interest at present: philosophy, which I define as the study of the meaning of life & of the right way to live.

A few criticisms: Davis & Hersh don’t describe in any detail the beginnings of math stretching further back than the Greeks, to ancient Egypt, India, & China; in the section on the history of individual numbers, they missed the fascinating history of the number zero, which only appeared relatively late on the scene; & their fixation on the three divided schools of mathematical belief (or nonbelief, or indifference to belief) is a bit overextended & repetitive. But these are minor flaws in a major achievement.