Crises in Mathematics

Mathematics: The Loss of Certainty

by Morris Kline
Oxford University Press, 366 pp., $19.95

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 …

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