Crises in Mathematics

Views concerning how reliable knowledge can be acquired usually reflect the achievements and methods in notably successful branches of inquiry. Newtonian mechanics was widely regarded for more than two hundred years as the paradigm for understanding the constitution of nature; in the nineteenth century, evolutionary biology often served as the model to be followed in the study of psychological and social phenomena; and in our own century, the extensive use of statistical notions in the natural sciences as well as in more recent theories for deciding which one of several possible courses of action is the best has inspired the construction of probabilistic conceptions of knowledge.

However, ever since the creation of demonstrative geometry in antiquity, perhaps the most profound and certainly the longest lasting influence on views concerning the character of genuine knowledge has been mathematics. The feature of demonstrative geometry, and of other branches of mathematics when they were axiomatized—i.e., when, as in the case of geometry, basic assumptions or axioms were found from which the theorems of the subject are deducible—that made them exemplars in the construction of theories of knowledge was the apparently “absolute” certainty of their propositions: of their axioms, which for centuries were generally regarded as self-evident truths about basic structural traits of space and of other things; and of their theorems, because in demonstrative reasoning the assumed truth of its premises is necessarily transmitted to whatever is deducible from them.

Nevertheless, although for more than two millennia Euclidean geometry was believed to embody this ideal of knowledge, it in fact did not do so, as was recognized in part in ancient times. In particular, Euclid’s parallel axiom was admitted to lack the “self-evidence” claimed for the remaining axioms; some of the so-called theorems do not really follow logically from the axioms; basic notions such as “point” and “straight line” are quite unclear and imprecise; and the axioms imply that there is a strange kind of “number”—“irrational” numbers such as the square root of two—that are neither integers nor the ratio of integers, for which no satisfactory account was available for more than two thousand years after their discovery.

The subsequent development of mathematics introduced even more puzzling “objects,” and led to innovations that eventually undermined traditional conceptions of the subject as well as of the nature of knowledge. The study of that development reveals the remarkable ability of human beings to construct increasingly abstract and intricate symbolic systems, many of which have been and continue to be indispensable tools in the effort to gain intellectual and practical mastery of causal dependencies between events. But such a study also shows that proposed interpretations of novel symbolic constructs are often seriously confused, and that the cogency of the logic employed in many of such constructions and proofs is frequently highly controversial. One conclusion can surely be drawn from that study: there has been a steady decline in the authority of self-evidence as a criterion of truth, as well as in the strength of the once widely held belief that unaided or “pure” reason is capable of grasping infallibly the nature of things.

The story of this decline, and of what he calls “the loss of certainty” in mathematics, is the theme of Professor Kline’s fascinating and readable book. The book is a selective history of mathematics, from its beginnings in ancient Greece down to the contemporary debates on what is sound mathematics, with primary emphasis on the major “crises” that have marked the expansion of the subject. Technical mathematical and logical ideas are not excluded from this story, but in general they are explained simply so that even non-mathematicians can acquire an understanding of what is at stake in the crises generated by the creative advance of mathematics.

Among the crises Kline discusses are some of the problems raised by the introduction of what were once called “impossible numbers” (i.e., the negative and imaginary numbers, such as minus 1 and the square root of minus 1). He also explains the dubious bases initially proposed for the differential and integral calculus as well as the difficulties encountered in attempts to validate the use of the calculus in the natural sciences, and the critical movement of the nineteenth century to provide a rigorous foundation for mathematical analysis. The nineteenth century also saw the momentous advent of non-Euclidean geometries and of set theory (i.e., the theory of aggregates or classes). The new geometries made untenable the long-entrenched belief that Euclidean geometry is the uniquely true theory of the properties of space; they created a profound uncertainty over which of the various geometries is the really true one; and they helped to undermine the widely held idea that geometry is about the structure of space rather than about spatial and various other properties.

On the other hand, set theory was built on the assumption that it makes sense to suppose there are “actual” infinite collections. For example, the number of integers in the set of all the integers 1, 2, 3,…is the smallest “transfinite” (i.e., nonfinite) number; on the other hand, the set of all sub-classes of the integers (i.e., the set consisting of sub-classes such as the class of the odd integers, the class of the even integers, the class of multiples of five, the class of primes or integers not divisible by any other integer except one, and so on) is also transfinite, but the number of classes in this set is greater than the smallest transfinite number. This construction can be repeated by forming the set of all the sub-classes of the set consisting of the sub-classes of the integers, and the process continued without limit, so that an endless array of increasingly large transfinite numbers is obtained. It was subsequently shown, however, that the construction as just stated involves serious contradictions.

Nevertheless, set theory came to be regarded as the “foundation” on which arithmetic rests. For as Bertrand Russell among others showed, the integers and the arithmetical operations on them (such as addition and multiplication) can be defined in terms of the ideas of set theory—for example, the cardinal number three was defined by Russell as the set of all triplets, while the addition of integers was defined by him in terms of certain operations on sets. But since set theory contained contradictions, the consistency of arithmetic, and indeed of all mathematical analysis in consequence, also came into question. Although a number of alternative ways of outflanking the set-theoretical and analogous contradictions have been proposed—Kline examines four of them—all have serious limitations and none has been universally adopted.

In a chapter with the ominous title “Disasters” he discusses some of the remarkable discoveries in the present century concerning the foundations of mathematics, and among other matters he explains two epoch-making theorems proved by the late mathematical logician Kurt Gödel. One of the theorems states that the consistency of arithmetic cannot be established unless rules of logical inference are employed that are more powerful (and hence no less open to doubt) than are the logical principles used in arithmetic. The other theorem is even more surprising and disconcerting, for it reveals an inherent limitation in the axiomatic method. The theorem states that no set of axioms can be specified from which all arithmetical truths are derivable; in other words, every set of axioms for arithmetic is essentially incomplete. Because of these and other comparable findings, Kline thinks that the present state of mathematics is “anomalous and deplorable.”

The last three chapters of the book are addressed to different though related issues. They present Kline’s reflections on what he regards as the growing isolation of mathematics from empirical science, and the grounds for his conviction that the price of the resulting specialization is sterility. They also explain his conception of what is “sound” mathematics, and of the place in it of intuition and proof; and they give his own views on how logic and mathematics are best validated, on the question why mathematics is so effective in scientific inquiry, and on the general nature of science.

It would be difficult to exaggerate the importance of the various “crises” that have occurred in the history of mathematics, for most if not all of them were a stimulus for putting the mathematical house in order, and in a number of cases they marked the beginning of fresh developments in the subject. On the other hand, it is easy to exaggerate the seriousness of the challenge that such crises have presented to the validity of mathematics as a whole. The discovery that the theory of transfinite numbers as initially presented by its creator Georg Cantor was inconsistent certainly required a revision of the theory. But that inconsistency did not by itself establish the inconsistency of other parts of mathematics, such as elementary arithmetic and geometry. To be sure, set theory is widely regarded today as the foundation for all of mathematics, because as has been already mentioned the concepts employed in different branches of the subject, in particular in arithmetic, can be defined in terms of set-theoretical concepts. However, it can be argued, and has in fact been argued, that arithmetic, for example, does not necessarily have a set-theoretical foundation, since arithmetic can be developed on the basis of its own distinctive axioms, and thus far at any rate no contradictory propositions have been deduced from them.

A similar comment can be made concerning other branches of mathematics, and other foundations for them that are not set-theoretical. The “foundation” that set theory provides for various branches of mathematics does not consist in making the latter better warranted or more certain, for it manifestly does not do this; it consists in bringing different parts of mathematics into systematic relation with one another, in being one way (and not necessarily the only way) of unifying them by defining their various concepts in terms of set-theoretical ones. But however this may be, it is not clear why discoveries such as the incompleteness of arithmetic or the limitations of the axiomatic method should be characterized as “disasters,” or as grounds for judging the present state of mathematics to be “deploration.” To my mind contrary evaluations should be more appropriate.

It seems to me also that Kline’s complaint about the “isolation” today of so much “pure” mathematics should be taken with many grains of salt. He may of course be correct in declaring that much pure mathematics is sterile—I am not qualified to assess this claim. But is applied mathematics, to which he has devoted his professional life, entirely free from this fault? If we grant that great creative talent is relatively scarce and that the number of professional mathematicians who publish fairly regularly is large, we should be surprised were this the case.

Moreover, is the distinction between pure and applied mathematics as clear and sharp as is often supposed? At what point does a problem suggested in the study of some natural phenomenon and explored by mathematicians cease to be one of applied mathematics and become one in pure mathematics? In particular, although Cantor developed his set theory in order to resolve some questions in the mathematical theory of beat, does the later preoccupation with the construction of a consistent set theory and with proving the consistency of arithmetic belong to pure or to applied mathematics? As for the alleged sterility of so much contemporary mathematics, the opinion Kline quotes from the mathematical physicist John L. Synge seems to be as just today as it was when Synge expressed it more than thirty years ago: “At present science [including mathematics] is humming as it never hummed before. There are no obvious signs of decay.”

Kline thinks that the still perplexing questions concerning what are the “proper” foundations for mathematics can be sidestepped, though not answered, by stressing the successful applications of mathematics. According to him, the soundness of mathematics should be judged by whether its use enables us to deal satisfactorily with problems encountered in the investigation of nature. Indeed, he subscribes to the idea that not only the principles of mathematics but even the laws of logic are “products of experience”—a view that was widely current in the nineteenth century, subsequently rejected as untenable, and recently revived by a number of philosophical mathematicians. However, he offers no explicit argument for this claim. Nor does he explain how it is to be reconciled with his earlier contention that “there are no truths in the axioms or theorems” of mathematics.

This contention is puzzling, since on the face of it there are innumerable true statements in mathematics, such as that there is no largest prime number; and it would be ridiculous to suppose that Kline is unaware of this. It seems plausible that what he means by his contention is that the symbolic and uninterpreted constructions of mathematics are not statements (and hence not true statements) about the natural world—and such a claim is undoubtedly correct.

On the other hand, when the symbolic constructs of pure mathematics are suitably interpreted—for example, when the terms of a geometric system are coordinated with certain physical configurations—the resulting statements may be true (within the limits of experimental error) of some sector of the world. Indeed, were this not often the case, of what use would mathematics be in the empirical sciences? Nevertheless, one should not be too confident that this suggestion about what Kline means by his contention is sound. For he also maintains that “science is rationalized fiction,” and that in particular the notions of mass and inertia in Newtonian mechanics are just such fictions. It is, however, not at all clear by what reasoning he reaches these and similar conclusions about the nature of science.

Under the apparent influence of Albert Einstein and Eugene Wigner, Kline also thinks it is a “mystery” why mathematics is so effective in the natural sciences—why, in an amplified version of the question, mathematics works so well “even where, although the physical phenomena are understood in physical terms, hundreds of deductions from the axioms prove to be as applicable as the axioms themselves.” However, it is far from evident just what is the mystery, or just what the question means. If the effectiveness of mathematics is a mystery, the successful use of syllogistic reasoning—that is, the application of an assumed law of nature to concrete instances—should also be so characterized.

For example, from the general assumption that copper expands when heated, coupled with the instantial statement that a given object is copper, it follows that the object will expand if it is heated. Accordingly, if the two premises in this miniature deduction are true, the conclusion must also be true. Why must the conclusion be true under the stated circumstances? The answer is that the rule used in obtaining the conclusion from the premises is demonstrably truth-preserving, as has already been mentioned. Should it turn out that when the given object is heated it does not expand, one or the other or both of the premises cannot be true.

Just where is the mystery in all this? To be sure, one may ask why, given the assumption about them, the premises are true; but this is a question different from, and not relevant to, the initial one. It is difficult to believe that “the problem of the unreasonable effectiveness of mathematics in the natural sciences” is not a pseudo problem.

A book covering so vast an array of different and difficult questions as the book under review does would be astonishing were it entirely free of mistaken, inaccurate, or misleading statements of fact. In this respect the book is not astonishing. However, it would be pointless to mention all the dubious assertions I have noted in it, especially since many of them (such as the statement that Freeman Dyson is a Nobel Laureate in physics) are minor ones. But it may be useful to call the reader’s attention to a few important ones.

It is an error to say that the completeness of predicate (or first order) logic cannot be established by finitistic means. It is inaccurate to declare that according to Gödel every axiomatic system is incomplete. It is misleading to suggest that nature’s laws are man’s creation, and that men are the lawgivers of the universe. And in the light of the book’s subtitle, it is perplexing to find in it Professor Kline’s belief that “insofar as certainty of knowledge is concerned, mathematics serves as an ideal, an ideal toward which we shall strive, even though it may be one that we shall never attain.” But despite such flaws, he has written an interesting, informative, and stimulating book.