Why Math Matters

This book is not an account of mathematical analysis, and is not intended to supply the reader with a survey of the various branches and achievements of the subject. In this respect it is notably different from books that attempt to do just this, such as the fascinating What Is Mathematics? by Richard Courant and Herbert Robbins (first published in 1941). As Professor Kline makes explicit, his object is to make clear the important part mathematics has played in the investigation of physical nature. The book is thus primarily a history of the development of physical knowledge, from the time of the ancient Greeks to the revolutionary discoveries of the present century.

Dr. Kline writes appreciatively and with enthusiasm about the advancement of knowledge gained with the help of mathematical analyses of natural phenomena; and except for the chapter on quantum theory his exposition is highly readable and informative, and generally illuminating. He makes evident that the Ptolemaic theory of planetary motion was a great triumph of mathematical analysis—a triumph that was repeated in the heliocentric theory of Copernicus and Kepler, in the mechanics and gravitational theory of Newton, in Maxwell’s fusion of electrical and magnetic forces into a comprehensive theory, in Einstein’s special and general theories of relativity, and in the revolutionary ideas employed in exploring atomic and subatomic processes.

Kline’s account of most of these developments in science before the early years of the twentieth century can be understood by readers with only moderate familiarity with elementary geometry and algebra. For example, he explains Newton’s unification of terrestrial and celestial mechanics with the help of some calculations that require no more mathematics than a high school student normally possesses. He gives an engrossing account of the development of non-Euclidean geometries (geometries with parallel postulates different from Euclid’s), and so prepares the reader to understand the geometrization of physics that characterizes relativity theory. To be sure, to appreciate Maxwell’s electromagnetic theory some familiarity with the differential calculus is necessary; and the structure of quantum theory involves more advanced mathematics, which Kline merely sketches in general terms without attempting to describe the actual content of the relevant theoretical equations. Kline has succeeded remarkably well in conveying to the layman the power, the penetration, and the indispensability of mathematical analysis in exploring physical nature.

Kline is far less successful in his explanation of the role of mathematics in quantum theory. His chapter on the subject describes the hierarchy of structures constituting the nature of matter and gives the reader a good idea of the small dimensions characterizing the basic components of physical reality. But the reader is also likely to be overwhelmed by the details presented in his account of subatomic structures, and may fail to obtain a satisfactory grasp of the pattern of analysis. Perhaps no one can do a better job in this respect than Kline has done—certainly not without a much more leisurely paced explanation of the details than can be given in a relatively brief chapter. Such explanations are found in the late Selig Hecht’s Explaining The Atom (1954) or in the more recent Physics in the Twentieth Century by Victor Weisskopf (1972).

However, Kline does not concern himself only with the substantive achievements of the mathematical investigation of nature. He also deals with some of the larger philosophical issues that have been raised in the history of physics, and these add greatly to the interest of his book. He surveys the diversity of views that have been advanced on the question whether there is an “external world” independent of human consciousness; on determinism and causality; on sensory illusions and their correction; on the nature of mathematics and mathematical method; and on why mathematics is such an effective instrument of scientific research. For the most part he does not commit himself fully to any of the ideas he thus presents, although he does indicate considerable sympathy (if not complete agreement) with some of them. The problem to which he repeatedly reverts is whether there is an objective world “outside” the human mind. He offers at the outset a thumbnail sketch of the history of philosophical thought on this question, and the issue is raised again in subsequent chapters, especially in the presentation of some widely accepted interpretations of quantum theory.

Although the subject has occupied the minds of a number of distinguished quantum theorists, it is somewhat of a puzzle why it is so fascinating to Kline. For his thesis that mathematical analysis is an essential tool in the investigation of nature presupposes that there is indeed a “nature” (or “external world”) to be investigated. As is well known, Einstein (though without convincing most of his professional colleagues) vigorously rejected the “subjective” construal of quantum theory, which holds that the attributes of subatomic phenomena depend on the measurements made by the experimenter. He could not subscribe to the eighteenth-century Bishop Berkeley’s contention that “to be is to be perceived,” and stoutly affirmed that there is a moon even when no one is looking at it. It is often said that physicists who subscribe to the Berkeleian view are currently the real pioneers in the philosophy of mind. But it should be added that few if any physicists who belong to this group have paid serious attention to the impressive criticisms that have been made of Berkeley’s thesis. Moreover, unlike Berkeley (who used his thesis to prove the existence of God, whose eternal thought sustains the continued existence of things even when human minds are not perceiving them), it rarely occurs to those who subscribe to Berkeley’s thesis to ask whether the physical world comes to an end when sentient life itself no longer exists.

Kline himself vacillates in this matter. For example in his discussion of Heisenberg’s Uncertainty Principle (which holds that if the position of an electron is determined to have a precise locus, its velocity becomes indeterminate or uncertain), he declares that it “refutes the classical idea of objectivity—the idea that the world has a definite state of existence independent of our observing it.” But earlier he asserted that, “Nature does not care much about our impressions. She continues on her course whether or not we are there.” He quotes Heisenberg with apparent approval (at any rate without demurrer): “What we observe is not nature itself, but nature exposed to our method of questioning.”

It is not clear, however, why it is not nature itself that is observed, even though it is we who question nature. We see with our eyes, but what we see is not determined exclusively by our eyes; and our eyes and what they see exist whether or not we use them. Kline further asserts,

Where classical mechanics appealed to either a particle picture or a wave picture [of fundamental processes], there was an objective reality independent of the observer. Today the laws of physics concern our knowledge rather than what may be true in the physical world.

Taken at face value, this would make physicists psychologists, since the subject matter of the latter is the human mind. But it is difficult to believe that Kline means what he says, though it is unclear what he does mean. He is evidently not entirely happy with Heisenberg’s subjectivism, for he expresses the hope that perhaps in fifty years a more satisfactory interpretation of quantum mechanics will be available. Moreover, there is what may be an unwitting conventionalism in his assessment of scientific theories. Thus, in comparing the relative advantages of the geocentric and heliocentric theories of planetary motion, he declares that in the modern view they are “equally valid.” According to him, despite the impressive observational and experimental evidence that confirms the heliocentric theory, there is no compelling logical reason for rejecting the geocentric one. But this is to fly in the face of present knowledge.

Kline discusses a number of so-called sensory illusions and seems to think that mathematical analysis has helped to correct them. But at least some of his examples are hardly instances of illusions. He characterizes as illusory the appearance as bent of a straight stick partly immersed in water, but this would usually not be classified as an illusion, for that appearance is the consequence of the optical law of diffraction and can be recorded by a camera independently of our eyes. Most readers are likely to regard as bizarre his designation as illusory of a realistic painting on a flat surface of a three-dimensional scene. Even more customary instances of sensory illusion (such as the Müller-Lyer effect of seeing as unequally long two equally long line segments having differently directed arrowheads at their extremities) are not purely sensory phenomena, but are mistakes in judgment. At the present stage of psychological inquiry it is at best dubious to suppose that there is such a thing as “immediate sensory knowledge,” since a nonsensory intellectual component enters into such supposed occurrences. Nor is it evident, as Kline holds, that mathematical analysis has had a central part in correcting these mistaken judgments. Thus, the apparent convergence of parallel railroad tracks can be shown to be spurious in the light of our familiarity with the behavior of solid bodies; and the equality of the lengths of the line segments in the Müller-Lyer figure can be established by juxtaposing the segments without performing any mathematical calculation.

Kline nowhere explicitly defines what he understands by “determinism,” but for the most part he has in mind the Laplacian conception, according to which an intelligence given the initial positions and velocities of the elementary particles of the universe, and given also the laws of their behavior, would be able to deduce all future and past occurrences in the world. So understood, determinism is an epistemic doctrine, of which the Laplacian version is a highly specialized instance. Kline rightly rejects it, in view of the essentially statistical character of quantum mechanics. But he also conflates determinism with causality, and cites with evident approval Heisenberg’s dictum that “quantum mechanics definitely shows the invalidity of the causal law.”

But the example Kline takes of an alleged acausal occurrence in nature is not persuasive. The example considers an unstable situation, such as a rock poised on a mountain peak which might start an avalanche if given a slight push. Kline declares this to reveal a “flaw in the deterministic world. Laws break down in these instances, and effects that are negligible in other circumstances can be dominating.” But this conflation of determinism and causality is not warranted; for as previously noted determinism is an epistemic doctrine, while causality denotes a relation of dependence between events. Certainly Kline gives no reason for believing that the course of the avalanche does not depend, among other things, on the initiating push, even if we do not know the precise form of the dependence.

One chapter of the book is devoted to the question “Why Does Mathematics Work?,” a question that was raised more than twenty years ago by the Nobel laureate physicist E.P. Wigner in his article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” But the question is a puzzling one, and it is not plain what could be a satisfactory answer to it. Mathematics is a highly specialized language, and the question can be raised about language in general: Why does language work? Why, for example, does the syllogism work: All men are mortal, Socrates is a man, hence Socrates is mortal? Two different issues must be distinguished. The first one is: Why is it the case that all men are mortal? Here the answer is in part that the available evidence relating to human mortality, including the evidence found in physics and physiology, strongly supports this proposition. The second question, on the other hand, is why, if the premises are taken for granted, the conclusion must also be accepted. The answer to the second question is not empirical, but is one of logic and meaning. For in virtue of the meaning embodied in the premises, the conclusion is implicitly “contained” in them, and the task of deductive logic is to make this explicit.

The same distinction must be made with respect to mathematically formulated reasoning. Thus the Newtonian laws of motion and gravitation are empirical assumptions which must be supported by empirical evidence that the laws are in reasonably good agreement with the findings of observation and experiment. But if the laws are accepted as sound, and if sufficient initial conditions about the relative positions and velocities of the sun and planets are supplied, further laws such as Kepler’s are the necessary consequences of these assumptions. There is surely not a mystery here, other than the “mystery” of why there are any regularities or uniformities in the universe and why the properties of things can be represented symbolically. It is, however, at least misleading to say, as Kline does, that “nature never ceases to accommodate itself to man’s mathematics.”

But Kline does raise an important question concerning the source of mathematical ideas. It would indeed be a mystery if the mathematics that is central in scientific inquiry had no roots in the physical world. John Stuart Mill believed that mathematical notions are “generalizations” of what observation reveals. But this is a much too simplistic thesis. For although mathematical conceptions may be, and often are, suggested by observable things and their properties, they are not just generalized transcriptions of the latter—any more than a melody suggested by a bird’s song is just a reproduction of what a composer experiences in listening to the twitter of birds. For example, the mathematical notions of continuity and infinity as developed by the great mathematician Georg Cantor—such as the idea that the aggregate of real numbers is both infinite and uncountable—are human inventions, and not just “abstractions” from the content of sensory observation; they are imaginative reconstructions and elaborations of what observation discloses. Cantorian infinite sets and continuous series are not found in sensory experience. Nor, as Kline makes it entirely clear, are they encountered in a Platonic heaven; they are created by mathematicians in order to perfect their analytic instruments. Cantor invented set theory and redefined the notion of continuity to resolve problems encountered in his work on the calculus.

However, in at least one passage of his book, Kline makes a larger claim. “The role of mathematics in modern science is now seen to be far more than that of a useful tool,” he declares. “Mathematics is the essence of scientific theories…. Although credit for the achievements of modern science…cannot be accorded only to mathematics, the role of mathematics is more fundamental and less dispensable than any contribution of experimental science.” This is an exaggeration that can be expected from a professional mathematician. As Einstein repeatedly pointed out, a mathematically formulated scientific theory is a “free creation” of the scientist; but mathematics cannot legislate which of the many free creations is factually sound, and the question can be decided only by experimental science. In any event, despite my doubts about some of Kline’s philosophical views, he has been eminently successful in achieving the main aim of his book.