WHAT IS MATHEMATICS, REALLY?<i> By Ruben Hersh</i> .<i> Oxford University Press: 344 pp., $35</i> : NUMBER SENSE: How the Mind Creates Mathematics.<i> By Stanislaw Dehaene</i> .<i> Oxford University Press: 288 pp., $25</i>
I
A physicist at M.I.T.
Constructed a new T.O.E. [Theory of Everything]
He was fit to be tied
When he found it implied
That seven plus four equals three.
Reviewing Reuben Hersh’s “What Is Mathematics, Really?” was an agonizing task because I have such high respect for him as a mathematician and such low respect for his philosophy of mathematics. Now retired, Hersh belongs to a very small group of modern mathematicians who strongly deny that mathematical objects and theorems have any reality apart from human minds. In his words: Mathematics is a “human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. I call this viewpoint ‘humanist.’ ”
Later he writes: “[M]athematics is like money, war, or religion--not physical, not mental, but social.” Again: “Social historic is all it [mathematics] needs to be. Forget foundations, forget immaterial, inhuman ‘reality.’ ”
No one denies that mathematics is part of human culture. Everything people do is what people do. The statement would be utterly vacuous except that Hersh means much more than that. He denies that mathematics has any kind of reality independent of human minds. Astronomy is part of human culture, but stars are not. The deeper question is whether there is a sense in which mathematical objects can be said, like stars, to be independent of human minds.
Hersh grants that there may be aliens on other planets who do mathematics, but their math could be entirely different from ours. The “universality” of mathematics is a “myth.” “If little green critters from Quasar X9 showed us their textbooks,” Hersh thinks it doubtful that those books would contain the theorem that a circle’s area is pi times the square of its radius. Mathematicians from Sirius might have no concept of infinity because this concept is entirely inside our skulls. It is as absurd, Hersh writes, to talk of extraterrestrial mathematics as it is to talk about extraterrestrial art or literature.
With few exceptions, mathematicians find these remarks incredible. If there are sentient beings in Andromeda who have eyes, how can they look up at the stars without thinking of infinity? How could they count stars, or pebbles, or themselves without realizing that two plus two equals four? How could they study a circle without discovering, if they had brains for it, that its area is pi times the radius squared.
Why does mathematics, obviously the work of human minds, have such astonishing applications to the physical world, even in theories as remote from human experience as relativity and quantum mechanics? The simplest answer is that the world out there, the world not made by us, is not an undifferentiated fog. It contains supremely intricate and beautiful mathematical patterns from the structure of fields and their particles to the spiral shapes of galaxies. It takes enormous hubris to insist that these patterns have no mathematical properties until humans invent mathematics and apply it to the outside world.
Consider 2 1398269minus one. Not until 1996 was this giant integer of 420,921 digits proved to be prime (an integer with no factors other than itself and one). A realist does not hesitate to say that this number was prime before humans were around to call it prime, and it will continue to be prime if human culture vanishes. It would be found prime by any extraterrestrial culture with sufficiently powerful computers.
Social constructivists prefer a different language. Primality has no meaning apart from minds. Not until humans invented counting numbers, based on how units in the external world behave, was it possible for them to assert that all integers are either prime or composite (not prime). In a sense, therefore, a computer did discover that 2 1398269minus one is prime, even though it is a number that wasn’t “real” until it was socially constructed. All this is true, of course, but how much simpler to say it in the language of realism!
No realist thinks that abstract mathematical objects and theorems are floating around somewhere in space. Theists such as physicist Paul Dirac and astronomer James Jeans liked to anchor mathematics in the mind of a transcendent Great Mathematician, but one doesn’t have to believe in God to assume, as almost all mathematicians do, that perfect circles and cubes have a strange kind of objective reality. They are more that just what Hersh calls part of the “shared consensus” of mathematicians.
To his credit, Hersh admits he is a maverick engaged in a “subversive attack” on mainstream math. He even provides an abundance of quotations from famous mathematicians--G.H. Hardy, Kurt Godel, Rene Thom, Roger Penrose and others--on how mathematical truths are discovered in much the same way that explorers discover rivers and mountains. He even quotes from my review, many years ago, of “The Mathematical Experience,” of which he was a co-author with Philip J. Davis and Elena A. Marchisotto. I insisted then that two dinosaurs meeting two other dinosaurs made four of the beasts even though they didn’t know it and no person was around to observe it.
A little girl makes a paper Moebius strip and tries to cut it in half. To her amazement, the result is one large band. What a bizarre use of language to say that she experimented on a structure existing only in the brains and writings of topologists! The paper model is clearly outside the girl’s mind, as Hersh would of course agree. Why insist that its topological properties cannot also be “out there,” inherent in what Aristotle would have called the “form” of the paper model? If a Hottentot made and cut a Moebius band, he would find the same timeless property. And so would an alien in a distant galaxy.
The fact that the cosmos is so exquisitely structured mathematically is strong evidence for a sense in which mathematical properties predate humanity. Our minds create mathematical objects and theorems because we evolved in such a world, and the ability to create and do mathematics had obvious survival value.
If mathematics is entirely a social construct, like traffic regulations and music, then Hersh argues that it is folly to speak of theorems as true in any timeless sense. For this reason, he places great importance on the uncertainty of mathematics, but not in the sense that mathematicians often make mistakes. The fact that you can blunder when you balance a checkbook doesn’t falsify the laws of arithmetic. Hersh means that no proof in mathematics can be absolutely certain. That two plus two equals four, he writes, is “doubtable” because “its negation is conceivable.” No proof, no matter how rigorous, or how true the premises of the system in which it is proved, “yields absolutely certain conclusions.” Such proofs, he adds, are “no more objective than aesthetic judgments in art and music.”
I find it astonishing that a good mathematician would so misunderstand the nature of proof. Benjamin Peirce, the father of philosopher Charles Peirce, defined mathematics as “the science which draws necessary conclusions,” a statement his son was fond of quoting. Only in mathematics (and formal logic) are proofs absolutely certain. To say that two plus two equals four is like saying there are 12 eggs in a dozen. Changing four to any other integer would introduce a contradiction that would collapse the formal system of arithmetic.
Of course, two drops of water added to two drops make one drop, but that’s only because the laws of arithmetic don’t apply to drops. Two plus two is always four precisely because it is empty of empirical content. It applies to cows only if you add a correspondence rule that each cow is to be identified with one. The Pythagorean theorem is timelessly true in all possible worlds because it follows with certainty from the symbols and rules of formal plane geometry.
In his worst attack on the absolute eternal validity of arithmetic, Hersh uses the analogy of a building with no 13th floor. If you go up eight floors in an elevator, then five more floors, you step out of the elevator on Floor 14. Hersh seems to think this makes eight plus five equals 14 an expression that casts doubt on the validity of arithmetic addition. I might just as well cast doubt on two plus two equals four by replacing the numeral four with the numeral five.
Hersh imagines that because the concept of “number” has been steadily generalized over the centuries, first to negative numbers, then to imaginary and complex numbers, quaternions, matrices, transfinite numbers and so on, this somehow makes two plus two equals four debatable. It is not debatable because it applies only to positive integers. “Dropping the insistence on certainty and indubitability,” Hersh tells us, “is like moving off the [number] line into the complex plane.” This is baloney. Complex numbers are different entities. Their rules have no effect on the addition of integers. Moreover, laws governing the manipulation of complex numbers are just as certain as the laws of arithmetic.
Within the formal system of Euclidean geometry, as made precise by the great German mathematician David Hilbert and others, the interior angles of a triangle add to 180 degrees. As Hersh reminds us, this was Spinoza’s favorite example of an indubitable assertion. I was dumbfounded to come upon pages on which Hersh brands this theorem uncertain because in non-Euclidean geometries the angles of a triangle add to more or less than a straight angle 180 degrees.
Non-Euclidean geometries have nothing to do with Euclidean geometry. They are entirely different formal systems. Euclidean geometry says nothing about whether space time is Euclidean or non-Euclidian. Hersh’s claim of triangular uncertainty is like saying that a circle’s radii are not necessarily equal because they are unequal on an ellipse.
Hersh devotes two chapters to great thinkers he believes were “humanists” (social constructivists) in their philosophy of mathematics. It is a curious list. Aristotle is there because he pulled numbers and geometrical objects down from Plato’s transcendent realm to make them properties of things, but to suppose he thought those forms existed only in human minds is to misread him completely. Euclid is also deemed a humanist without the slightest basis. (The person most deserving to be on Hersh’s list of maverick anti-realists is, of course, the mathematician Raymond Wilder. He and his anthropologist friend Leslie White were leading boosters of the notion that mathematical objects have no reality outside human culture. Hersh calls White’s essay “The Locus of Mathematical Reality,” a “beautiful statement” of social constructivism.
John Locke is on the list because he recognized the fact that mathematical objects are inside our brains. But Locke also believed--Hersh even quotes this!--that “the knowledge we have of mathematical truths is not only certain but real knowledge; and not the bare empty vision of vain, insignificant chimeras of the brain.” The angles, in other words, of a mental triangle add to 180 degrees. This is also true, Locke adds, “of a triangle wherever it really exists.” A devout theist, Locke would have been as mystified as Aristotle by the notion that mathematics has no reality outside human minds.
The inclusion of Peirce as a social constructivist is even harder to defend. “I am myself a Scholastic realist of a somewhat extreme stripe,” Peirce wrote in Vol. 5 of his “Collected Papers”. (All of Hersh’s arguments against realism, by the way, were thrashed out by medieval opponents of realism.) In Vol. 4, Peirce speaks of “the Platonic world of pure forms with which mathematics is always dealing.” In Vol. 1, we find this passage:
“If you enjoy the good fortune of talking with a number of mathematicians of a high order, you will find the typical pure mathematician is a sort of Platonist. . . . The eternal is for him a world, a cosmos, in which the universe of actual existence is nothing but an arbitrary locus. The end pure mathematics is pursuing is to discover the real potential world.”
Hersh devotes many excellent chapters to summarizing the history of mathematics, and he ends his book with crisp, expertly worded accounts of famous mathematical proofs. An odd thing about this final chapter, though, is that Hersh writes as if he were a realist. This is hardly surprising, because the language of realism is by far the simplest, least confusing way to talk about mathematics.
Over and over again Hersh speaks of “discovering” mathematical objects that “exist.” For example, the square root of two doesn’t exist as a rational fraction, but it does exist as an irrational number that measures the length of a diagonal of a square on one side. Mathematicians “find” complex numbers “already there” on the complex plane. After saying that Sir William Rowan Hamilton “found” quaternions while he was crossing a bridge, Hersh reminds us that quaternions did not exist until Hamilton “discovered them.” Of course, he means that until Hamilton “constructed them” on the basis of a social consensus of ideas, they didn’t exit, but his wording shows how easily he lapses into the language of realism.
We must constantly keep in mind that although Hersh talks like a realist, his words have different meaning than they have for a realist. Once humans have invented a formal system like plane geometry or topology, the system can imply theorems that had previously been difficult to “discover.” Their discovery is, therefore, of theorems that can be said to “exist” outside any individual mind, but have no reality beyond the collective minds of mathematicians.
Hersh closes this chapter with a beautiful new proof by George Boolos of Godel’s famous theorem that formal systems of sufficient complexity contain true statements that can’t be shown true within the system. Boolos’ proof is flawless, a splendid example of mathematical certainty, as are all the other proofs in this admirable chapter.
Let the great British mathematician G.H. Hardy have the final say: “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations,’ are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onward, and I shall use the language which is natural to a man who holds it. A reader who does not like the philosophy can alter the language: it will make very little difference to my conclusions.”
II
Stanislaw Dehaene, a neuroscientist at a Paris medical research institute, is, I regret to say, as down on mathematical realism as Reuben Hersh. “The Number Sense” is subtitled “How the Mind Creates Mathematics.” However, instead of seeing numbers as social constructs, he thinks they are present at birth inside each person’s head where the brain is genetically wired to “recognize” them. How the brain can recognize numbers unless in some sense they are “out there” is not made clear.
Dehaene is convinced that many animals--mice, raccoons, dolphins, pigeons, parrots and, of course, apes--also are capable of understanding very small integers and to perform simple arithmetic with them. His first chapter is a fascinating survey of recent experiments with animals that seem to support this view. Rats, for example, can be trained to take the fourth entrance to any maze. They can be trained to press a lever exactly n times if n is a small number.
In one experiment, a rat was conditioned to press a certain lever only after it heard two tones followed by two flashes of light. Dehaene takes this as evidence that a rat somehow knows that two plus two equals four. A chimp named Sheba, after finding the numeral two under a table, and the numeral four inside a box, was trained to select the numeral six from a set of cards. To Sarah Boysen, who designed this and similar sensational experiments, it shows that chimps can add two and four to get six. If true, it proves that mathematics is not restricted to human cultures.
“Never had an animal come any closer,” Dehaene writes, “to the symbolic calculation abilities exhibited by humankind.” I’m inclined to doubt this. Assuming Sheba was not responding to unconscious cues from her trainers, I think it is possible to believe that she was simply associating a combination of two meaningless symbols with a third symbol without the slightest inkling of their arithmetic meanings.
In his next chapter, “Babies Who Count,” Dehaene describes experiments with babies that he is convinced show that they can do simple arithmetic long before they can speak. Most of this research fails to impress me. For instance, 4 1/2-month-old infants are shown two Mickey Mouse toys. A screen is placed to conceal the toys. Behind the screen, two red balls are substituted for the toys. When the screen is removed and the babies see the two balls, they are not in the least surprised. But if they see only one red ball or three red balls, they appear shocked.
“Mickey Mouse turning into a ball. . . is an acceptable transformation as far as the baby’s number processing system is concerned,” Dehaene writes. “As long as no object vanishes or is created de novo, the operation is judged to be numerically correct and yields no surprise reaction in babies.” Twoness is preserved. But if two objects turn magically into one or three, the baby is startled. “The demonstration is irrefutable,” the author says. “Babies know that one plus one makes neither one nor three, but exactly two.”
How did psychologist Tony Simon, who supervised this test, decide when a baby is surprised? By measuring the average amount of time a baby takes when looking at objects. If it takes a few seconds longer when it sees that two objects have changed to one or three, this is taken to mean the baby is shocked. Measuring and averaging such times is not easy because babies look here and there and seldom keep a fixed gaze on anything. There is so much room here for an experimenter’s expectations to bias statistics that I find it hard to accept this measurement and similar tests as proof that very young babies, like chimps, have an inherited ability to do simple arithmetic.
Jean Piaget’s famous experiments with infants showed that children have to be several years old before they have any grasp of numbers. Dehaene believes Piaget was wrong. “Babies are much better mathematicians than we thought only 15 years ago,” he assures us. “When they blow the first candle on their birthday cake, parents have every reason to be proud of them, for they have already acquired, whether by learning or by mere cerebral maturation, the rudiments of arithmetic and a surprisingly articulate ‘number sense.’ ”
The rest of Dehaene’s book concerns the history of numbers, variations in the counting practices of different cultures, the mystery of idiot savants who perform rapid calculations with huge numbers, the intricate structure of the human brain, and how its workings differ from those of digital computers. Not until his final chapter, “What Is a Number?” does his anti-realism become explicit.
Platonic realism is branded a philosophy “no neurobiologist can believe.” Most mathematicians, he grants, feel as if they are exploring a jungle out there, but this is sheer illusion. Formalism is dismissed because it turns mathematics into a game played with strings of meaningless symbols. He forgets that those symbols can be interpreted as mathematical objects that in turn can be applied to the outside world. His sympathies are with the philosophical school called intuitionism (now called constructivism), which views mathematical objects as “nothing but constructions of the human mind.”
Nothing but? Dehaene now bumps into what he calls the “unfathomable mystery” of why mathematics tells us so much about the universe. “How is it possible,” Einstein is quoted, “that mathematics, a product of human thought that is independent of experience, fits so excellently to the objects of physical reality?” Dehaene does not add how Einstein answered. The Old One, as he liked to call the universe, is mathematically structured.
Dehaene admits that “reality is organized according to structures that predate the human mind,” but he cannot bring himself to say, as everybody else would say, that those structures are mathematical. If we restrict mathematical objects and theorems to what mathematicians say and write, it is trivially true that mathematics is entirely a human endeavor. We might just as well say that the laws of physics are limited to what physicists say and write, with no locus outside human culture. It’s hard to believe, but a few sociologists actually think that physical laws are entirely human inventions! The non-trivial question is whether it is meaningful to assert (I choose from millions of examples) that the shell of a dinosaur egg divided space into inside and outside long before a topologist evolved to prove it.
Like Hersh and other anti-realists, Dehaene loves to stress the uncertainty of mathematics. He closes his book by exhuming one of the oldest fallacies in geometry, a seeming proof that a right angle equals an obtuse angle. (Euclid wrote a book about such fallacies--alas, it did not survive the centuries--which may well have included this amusing chestnut. I first ran across it in high school in a book on mathematical recreations.) Does it suggest that plane geometry is uncertain? No, it shows exactly the opposite. Once the faulty construction of the diagram is recognized, it follows with iron certainty that the two angles are not equal.
Is the universe written in mathematical language, as Galileo said? Dehaene thinks not. All mathematical objects “are mental constructions whose roots are to be found in the adaptation of the human brain to the regularities of the universe.” True enough, but why refuse to say, as almost all mathematicians and everyone else says, that those regularities are mathematical?
It is not just that planets tend to move in elliptical orbits or that galaxies often spiral, but matter itself has now dissolved into pure mathematics. The entire universe, including you and me, is made of leptons and quarks, particles that are the quantized aspects of fields. Perhaps all particles are made up of still smaller particles called superstrings. And what are the fields and particles made of? Patterns. Nothing can be said about these patterns except to describe their mathematical properties. You and I, like the stars, are made of mathematics. If you think otherwise, then pray tell what these mathematical structures are made of.
“The universe seems to be made of nothing,” a friend recently remarked, “yet somehow it manages to exist.” This is what Ronald Graham of Bell Labs meant when he said: “Mathematics is the only reality.”
You would think that the testimonies of so many eminent mathematicians about the independence of mathematical objects from human culture might arouse a bit of humility, wonder, and mystery among the anti-realists, who make humanity the center of all being. Although it will have not the slightest effect, let me close with some statements made last year by the great John Conway, now at Princeton University, in a radio interview for the Canadian Broadcasting System:
“I don’t think there’s much disagreement about mathematics. We discover it. We might invent a particular method for solving a problem. So some mathematics is invented, but the real cornerstones of mathemat ics are discovered. They’re out there. . . . Take the whole numbers, one, two, three. People often call those ‘concepts’ . . . implying that they’re just in the mind. But they’re not! And so if we study those numbers we are studying an abstract part of the world, rather than some of its more concrete aspects. But anybody who studies numbers is studying something that is really there.”
Of course they are not “there” in the same way the moon is there. But the patterns of which the moon is made are eternal, more real than the poor old moon itself, destined like you and me to eventually vanish from the universe.