In precisely what sense do universals (such as blueness, goodness, cowness, squareness, and threeness) exist? For Plato they are transcendent things, independent of the universe. Aristotle agreed that they are outside human minds, but he pulled them down from Plato’s heaven to make them inseparable from the world. During the Middle Ages the nominalists and conceptualists shifted universals sideways from the outside world to the inside of human heads.
In the philosophy of mathematics, with which The Mathematical Experience is primarily concerned, this ancient controversy over universals takes the form of speculating on what it means to say that such abstractions as the number three, a triangle, or an infinite set “exist,” and the companion problem of what it means to say that a theorem about these ideal objects has been “proved.” Let us not get bogged down in the technical and ambiguous differences between such schools as the logicism of Bertrand Russell, the formalism of David Hilbert, and the constructivism (or intuitionism) of L.E.J. Brouwer. All of these are briskly discussed, along with many other central mathematical issues, by the book’s two distinguished authors, mathematicians Philip Davis and Reuben Hersh. Let us consider instead the more fundamental question which cuts across all the schools. Do mathematical structures have a reality independent of human minds?
It is easy to caricature what mathematicians mean when they call themselves realists. They certainly do not suppose (I doubt if Plato did) that were we transported to some far-off realm we would see luminous objects floating about which we would recognize as pi, the square root of minus one, transfinite sets, pure circles, and so on; not symbols or models, but the undefiled universals themselves. Realists mean something less exotic. They mean that if all intelligent minds in the universe disappeared, the universe would still have a mathematical structure, and that in some sense even the theorems of pure mathematics would continue to be “true.” On its ultimate microlevel (if it has one) the universe may be nothing but mathematical structure. “Matter” has a way of vanishing on the microlevel, leaving only patterns. To say that these patterns have no reality outside minds is to take a giant step toward solipsism, for if you refuse to put the patterns outside human experience, why must you put them outside your experience?
For a mathematical realist a tree not only exists when nobody looks at it, but its branches have a “tree” pattern even when no graph theorist looks at them. Not only that, but when two dinosaurs met two dinosaurs there were four dinosaurs. In this prehistoric tableau “2 + 2 = 4” was accurately modeled by the beasts, even though they were too stupid to know it and even though no humans were there to observe it. The symbols for this equality are, obviously, human creations, and our mental concepts of two, four, plus, and equals are by definition mind-dependent. If mathematical structure is taken to mean only what is inside the brains of those who do mathematics, it is as trivial to say all mathematics is mind-dependent as it is to define sound as a mental phenomenon, then proclaim that the falling tree makes no sound when nobody hears it.
Fortunately scientists, mathematicians, and ordinary people seldom talk this way. The existence of an external world, mathematically ordered, is taken for granted. I have yet to meet a mathematician willing to say that if the human race ceased to exist the moon would no longer be spherical. I suspect Davis and Hersh would not care to say this, yet the troubling thing about their book is that it does not make clear why.
Although there are hints of the authors’ philosophical perspective throughout the book, it is not explicitly stated until the last page but one:
Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus….
Mathematics does have a subject matter, and its statements are meaningful. The meaning, however, is to be found in the shared understanding of human beings, not in an external nonhuman reality. In this respect, mathematics is similar to an ideology, a religion, or an art form; it deals with human meanings, and is intelligible only within the context of culture. In other words, mathematics is a humanistic study. It is one of the humanities.
Davis and Hersh do not deny that mathematical concepts are objective in the sense that they are “outside the consciousness of any one person,” but they are not outside the collective consciousness of humanity. Mathematicians do not discover preexisting, timeless things like pi and dodecahedrons; they construct them. Once constructed, however, they can be studied in much the same way that astronomers study Saturn. They acquire from the culture’s consensus a permanence of structure that cannot be altered by the whims of individual mathematicians.
What is one to make of this extreme conceptualist view? All that mathematicians do is certainly part of culture for the simple reason that everything human beings do is part of culture. But to talk as if mathematical objects are no more than cultural artifacts is to adopt a language that quickly becomes awkward because it is so out of step with ordinary language. It is like insisting that all birds are pink, then distinguishing between the pinkness of cardinals and the pinkness of crows. Conceptualism in mathematics has its strongest appeal among anthropologists and sociologists who have a vested interest in making culture central. (See Leslie White’s naïve paper, “The Locus of Mathematical Reality: An Anthropological Footnote,” in Volume 4 of The World of Mathematics, edited by James Newman.) It is a language that also appeals to those historians, psychologists, and philosophers who cannot bring themselves to talk about anything that transcends human experience.
Mathematical realists avoid this language for a variety of reasons, one of which is its obvious clumsiness in explaining some things everybody knows are true. For example, why do mathematical theorems fit the universe so accurately that they have enormous explanatory and predictive power? The authors call attention to Eugene Wigner’s well-known paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” For a nonrealist this effectiveness is indeed an awesome mystery. And if mathematical concepts have no locus outside human culture, how has nature managed to produce such a boundless profusion of beautiful models of mathematical objects: orbits that are conic-section curves, snowflakes, coastlines that model fractal curves, carbon molecules that are tetrahedral, and on and on?
If mathematical entities are no more than cultural products, one would expect independent cultures to fabricate widely disparate laws of arithmetic and geometry. But they don’t. Number systems may differ in their notational base, but of course this is only a difference in how numbers are symbolized. If theorems of elementary geometry are created, not discovered, why has no culture found it expedient to suppose that the cube of the hypotenuse of a right triangle equals the sum of the cubes of the other two sides? Who can believe that on some distant planet intelligent beings have constructed a planar map of five regions, each pair sharing a common portion of a border? The mere existence of extraterrestrial mathematicians would at once place some mathematical objects outside human culture, but even here on earth are apes aware, albeit dimly, of the difference between a ball and a cube, and between one and two bananas? Of course if one believes in a God who knows all that can be known, then all mathematical objects are not only “out there,” beyond the folkways, they are way out there.
For the realist, mathematical progress, like scientific progress, mixes creativity with discovery. Never would Newton have entertained the fantastic notion that he had invented the law of gravity, or Einstein the wild belief that he had invented the law E = MC2. There is an obvious sense in which scientists create theories, but there is an equally obvious sense in which theories penetrate the secret chambers of what Einstein liked to call the Old One. Einstein did not impose his equations on the universe. The Old One imposed its equations on Einstein.
What does a conceptualist gain by talking as though the spirality of Andromeda is projected onto the galaxy by human experience? Of course if spirality is defined as entirely a mental concept, then the spirality cannot be “out there.” But what astronomer, seeing a photograph of a newly discovered galaxy, is likely to exclaim: “How astonishing! When I look at this photograph I perceive that lovely spirality stamped on my brain by the shared experience of my race”? Not that there is anything inconsistent about such a language. Rudolf Carnap was able to show, in his Logical Structure of the World, that a phenomenological language, never going beyond human experience, is capable of expressing the same empirical content as any realistic language, but he quickly opted for realism as the only workable language for science.
It is also the most efficient language for most mathematical discourse. Although I am an unabashed realist (for emotional reasons) I agree with Carnap’s application of his “principle of tolerance” to the various schools of mathematical philosophy. The choice of a language for talking about mathematics is not so much which language is “right” (in logic, said Carnap, there are no morals) as it is which language is most convenient in a given context. With reference to the book under review, the context is not a technical discussion about the flimsy foundations of set theory. As the authors make clear in their preface, the book is an attempt to convey to nonprofessionals what mathematics is all about.
No mathematician hesitates to speak of “existence proofs” about objects even when they are nowhere modeled, or known to be modeled, by the external world. And most mathematicians, including the very greatest, think of such objects as independent of the human mind, though not of course existing in the same way Mars exists. Last year Robert Griess, Jr., constructed a finite simple group called the “Monster.” It has 808,017,424,794,512,875,886,459,904, 961,710,757,005,754,368,000,000,000 elements, each a matrix of 196,883 by 196,883 numbers. Griess prefers to call it “The Friendly Giant from the 196,883rd Dimension” because it is a symmetry group of the packing of identical hyperspheres in a space of 196,883 dimensions. There is nothing “wrong” in thinking of the Friendly Giant as composed by Griess the way Mozart composed a symphony, but there also is nothing wrong in thinking of the Giant as having existed as timelessly as a large prime, waiting to be discovered.
Artists can paint anything they like, but if a Russian mathematician had constructed the Monster before Griess had, the group would have had exactly the same properties as Griess’s group. A conceptualist can explain this, but not without using a language both curious and cumbersome. Davis and Hersh, overwhelmed by the mysteries of infinite sets and modern proof theory, have chosen a language of considerable value in analyzing the obscure foundations of mathematics, but it serves only to confuse us ordinary folk when applied to all of mathematics.
Closely related to the antirealism of Davis and Hersh is their attack on the infallibility of mathematical reasoning. Most philosophers have found it useful to distinguish mathematics from science by saying that mathematicians can prove some things in ways scientists cannot. Awareness that all science is fallible (I think it was the mathematical realist Charles Peirce who first applied the term “fallibilism” to science) goes back to the ancient Greek skeptics and is taken for granted by all modern scientists and philosophers. (It is not a doctrine first stressed by Karl Popper, as the authors imply on page 345.)
This fallibilism follows at once from the absence of any logical reason why a natural law cannot alter tomorrow. Science has no way of establishing facts, laws, or theories except by assigning them what Carnap called degrees of confirmation and Popper prefers to call degrees of credibility. The borderline between this corrigible synthetic truth (based on observation of the world) and infallible analytic truth (based on consistency in the use of words) may not be as sharp as Hume thought, but the distinction is too useful to throw away. “There are three feet in a yard” clearly is not the same sort of statement as “Mars has two moons.”
Now a large portion of mathematics is analytic, and where it is, there is no harm in speaking of certainty. The truth of 2 + 2 = 4 does not depend (as John Stuart Mill contended) on the pleasant fact that two fingers plus two fingers make four fingers. It follows from the way terms are defined in a formal system that constructs integers. Davis and Hersh devote several pages to instances where arithmetic addition fails to apply—for example, a cup of milk added to a cup of popcorn does not produce two cups of the mixture. No realist would deny the authors’ assertion that “There is and there can be no comprehensive systemization of all the situations in which it is appropriate to add.” In relativity theory, to give another example, addition of relative velocities does not obey the usual arithmetical laws.
But it does not follow from the misapplication of mathematics to the world that there are no infallible proofs in pure mathematics. On this point the arguments of Davis and Hersh become careless. A good instance is on page 326 where they speak of Euclid’s theorem that the angles of a triangle add to a straight angle. This, they declare, has been “proved false” by non-Euclidean geometry. A better way to put it—it is what they really mean—is that in a formal non-Euclidean geometry the theorem is false. But in the Euclidean system it remains true for all possible (noncontradictory) worlds because it expresses a tautology that follows from the system’s axioms and rules. It says nothing at all about the structure of physical space.
To blur the distinction between analytic and synthetic truth (as Willard Van Orman Quine and others have done) is to blur the difference between science and mathematics. A colorful recent effort along these lines is a monograph called Proofs and Refutations1 by the Hungarian philosopher Imre Lakatos, who died suddenly in 1974, age fifty-one, of a brain tumor. Lakatos’s fiery broadsides against mathematical certainty have acquired something of a cult following, especially among social scientists. Davis and Hersh devote a chapter to this eccentric book, which they deem brilliant, overwhelming, and a masterpiece of complex reasoning and historical erudition.
Fascinating though this book by Lakatos is, in my opinion Davis and Hersh greatly overrate its merits. Lakatos had been a student of Popper. Impressed by Popper’s vision of science as an ever-growing body of constantly altering conjectures, Proofs and Refutations tries to show that mathematical progress follows a similar zigzag course. The book has been called more Popperian than Popper. Later Lakatos and Popper clashed over the problem of induction. (Lakatos’s acid tongue got him into brawls with almost everybody.) You will find Popper’s low opinion of Lakatos vigorously detailed in The Philosophy of Karl Popper, where he replies to Lakatos’s contribution to that anthology.
Now it is quite true, as Davis and Hersh emphasize, that mathematicians seldom use deductive reasoning to create theorems. First they have a hunch. Then, like scientists, they make experiments (in their heads or on paper) and search for proofs that the hunch is sound. (The fact that they can tinker with drawings and discover elegant theorems is not easily justified in a nonrealist language.) This fumbling process is unlikely to be reflected in their papers. As Davis and Hersh remind us, only after a published proof has met the approval of peers is it eventually accepted. Sometimes, as in the case of the celebrated four-color-map theorem, a proof is taken to be valid for years before someone punches a deductive hole in it.
Recently the four-color theorem was proved with the aid of a computer, but the proof is buried in such an ugly mass of printouts that it requires other computers to check it. Davis and Hersh are right, in my opinion, in denying that this reliance on computers adds a new empirical element to mathematics. Many proofs, especially in group theory, are so horrendously complex that the possibility of human error becomes large. To say such proofs may be invalid is not different in principle from saying that mortals can fumble when they do long division by hand or on an abacus. Because waitresses make mistakes when they add your check, however, it does not follow that the laws of arithmetic are corrigible, or that geometers should keep trying to trisect the angle.
Lakatos’s book takes the form of an entertaining dialogue between a teacher and his students. First the teacher gives Cauchy’s clever proof, using graph theory, of Euler’s famous conjecture that the number of vertices of any polyhedron, minus the number of edges and plus the number of faces, equal two. Thus for a cube: 8 – 12 + 6 = 2. This formula, with its apparently ironclad proof, is then shot down by the students, who describe a zoo of “monster” counterexamples. Consider a cube with a smaller cube glued to the center of one face. The number of vertices is 16, the edges 24, the faces 11. Plugging these values into Euler’s formula gives 16 – 24 + 11 = 3. Does this undermine Cauchy’s proof?
It does not. For Euler and Cauchy a polyhedron was assumed to be simply-connected (topologically like a ball), with nonintersecting faces that are simply-connected polygons (topologically like a circle). Lakatos writes as though Cauchy, had someone showed him the cube-on-cube monster, would have slapped his forehead and exclaimed: “What a fool I am! Euler’s formula is false!” But the formula is not false. The face around the base of the smaller cube is a polygon with a square hole, and therefore the solid is not what Cauchy meant by a “polyhedron.” And the same for the other monsters: polyhedrons with intersecting faces, polyhedrons joined at edges or at corners, polyhedrons with tunnels or interior hollow spaces, and so on. In a footnote Lakatos actually speaks of Cauchy’s “inability to imagine” a polyhedron not topologically equivalent to a ball, as though this eminent French mathematician could not conceive of a cube with a square hole through it!
What happened historically has little resemblance to the distorted history sketched in Lakatos’s seemingly learned notes. Mathematicians simply generalized Euler’s formula to other kinds of solids, and as this commonplace process continued, terms like polygon and polyhedron broadened in meaning. Steady generalization, with inevitable language modification, is more characteristic of mathematical growth than revisions forced by oversights and faulty proofs. The discovery of irrational numbers did not demolish proofs that all integers are either odd or even, nor did the discovery of quaternions invalidate the commutative law of arithmetic. Both discoveries simply pushed along the social process of enlarging the way mathematicians decided to use the word “number.”
Lakatos was aware of these obviosities. In fact, they are expressed by students in his dialogue. But he seemed to think that the final moral of his book—Euler’s formula holds only for “Eulerian polyhedrons”—is somehow an indictment of formalism. But this is just what formalism is all about. For a formalist, a theorem never holds except in a formal system in which it holds.
Although Lakatos’s historiography is, as Gerald Holton put it, “parody that makes one’s hair stand on end,” his book does suggest the shaggy, meandering way in which mathematics, like science, advances. As for providing any evidence that all proofs are suspect, as Davis and Hersh suggest, the book is irrelevant. Proofs naturally are fallible in the pragmatic sense, and they become ambiguous and controversial when applied to such queer objects as transfinite sets. Mathematicians do make errors, and proofs are often naïve, incomplete, and plain wrong. No complicated proofs are ever wholly formalized, because of printing costs and limits of time, space, and energy.
Moreover, thanks to the work of Kurt Gödel (whose Platonic realism was extreme) we know that in any formal system complicated enough to include arithmetic there are theorems that cannot be proved within the system. The structure of a brick may indeed have mathematical properties that can never be completely captured within a deductive system. None of this touches the realist view that the brick and its properties are independent of human minds, and that where proofs are simple enough to be formalized they can be considered “certain” in a way that does not apply to any scientific claim.
Many aspects of The Mathematical Experience deserve high praise. It contains discussions, often quite technical, of topics not usually found in books for general readers. The authors are skillful in describing the monumental task of classifying finite simple groups—a task completed after the book went to press. They do an excellent job on the notorious and still unproved Riemann conjecture. There are admirable chapters on non-Cantorian set theory and nonstandard analysis.
The book jumps around a lot from topic to topic, but this hopscotch effect was probably inevitable since many of the chapters are excerpts from previously published articles, some by Davis alone, some by Hersh alone, some by both, and some by one of them in collaboration with somebody else. An excellent chapter on Fourier analysis is by Reuben and Phyllis Hersh. Not least of the book’s merits are the many photographs of mathematicians whose faces are unfamiliar even to most professionals.
In my opinion The Mathematical Experience is a stimulating book that is marred by its preference for an ancient way of talking about mathematics that has recently become fashionable in some mathematical circles,2 but that seems to me so inappropriate in a book for general readers that it spreads more confusion than light. It is possible to scratch your left ear with your right hand, but why bother?
August 13, 1981
Imre Lakatos, Proofs and Refutations, edited by John Werroll and Elie Zahar (Cambridge University Press, 1976). ↩
See, for example, Morris Kline’s 1980 book, Mathematics: The Loss of Certainty, which takes the same extreme anthropocentric point of view as the book by Davis and Hersh. All mathematics, Kline tells us, is a “purely human creation,” all laws of logic are the products of human experience, and “today the belief in the mathematical design of nature seems far-fetched.” No mathematical design in nature? My mind reels at the infelicity of this phrasing. I am in complete agreement with Ernest Nagel’s criticisms, expressed in his restrained review (NYR, November 6, 1980) of this quirkish volume. ↩