Although the great French mathematician Henri Poincaré wrote on topics in the philosophy of mathematics from as early as 1893, he did not come to consider the subject of modern logic until 1905. The attitude he then expressed toward the new logic was one of hostility. He emphatically denied that its development over the previous quarter century represented any advance whatsoever, and he dismissed as specious both the tools devised by the early logicians and the foundational programs they urged. His attack was broad: Cantor, Peano, Russell, Zermelo, and Hilbert all figure among its objects. Indeed, his first writing on...

Throughout his philosophical career, Carnap places the foundations of logic and mathematics at the center of his inquiries: he is concerned above all with the Kantian question “How is mathematics (both pure and applied) possible?”¹ Although he changes his mind about many particular issues, Carnap never gives up his belief in the importance and centrality of this question—nor does he ever waver in his conviction that he has the answer: the possibility of mathematics and logic is to be explained by a sharp distinction between formal and factual, analytic and synthetic truth. Thus, throughout his career Carnap calls for,...

To most mathematical logicians working in the 1980s, first-order logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician named Thoralf Skolem argued that set theory should be based on first-order logic, it was a radical and unprecedented proposal.

The radical nature of what Skolem proposed resulted, above all, from its effect on the notion of categoricity. During the 1870s, as part of what became known as the arithmetization of analysis, Cantor and Dedekind characterized the set R of real numbers (up to isomorphism) and thereby found a...

At a conference on the history and philosophy of mathematics, it seems especially appropriate to talk about Kronecker’s place in history. My objective is to show that the prevalence of one philosophical viewpoint in contemporary mathematics has had a distorting influence on the way that that place—and with it several major issues in the history and philosophy of mathematics—have been viewed in our time.

Consider the story of Georg Cantor (1845-1918) and Leopold Kronecker (1823-91) as it is told by present-day writers:

Cantor was the founder of set theory—he even gave it is name. As such, he...

Felix Klein’s “Erlanger Programm” (E.P.), listed in our references as (Klein 1872), is generally accepted as a major landmark in the mathematics of the nineteenth century. In his obituary biography Courant (1925) termed it “perhaps the most influential and widely read paper in the second half of the nineteenth century.” Coolidge (1940, 293) said that it “probably influenced geometrical thinking more than any other work since the time of Euclid, with the exception of Gauss and Riemann.”

In a thoughtful recent article, Thomas Hawkins (1984) has challenged these assessments, pointing out that from 1872 to 1890 the E.P. had a...

Historically, the dual concepts of infinitesimals and infinities have always been at the center of crises and foundations in mathematics, from the first “foundational crisis” that some, at least, have associated with discovery of irrational numbers (more properly speaking, incommensurable magnitudes) by the pre-socartic Pythagoreans¹, to the debates that are currently waged between intuitionists and formalist—between the descendants of Kronecker and Brouwer on the one hand, and of Cantor and Hilbert on the other. Recently, a new “crisis has as been identified by the constructivist Erret Bishop:

There is a crisis in contemporary mathematics, and anybody who has Bishop,...

This paper could have a slightly different title with the wordHowdropped, but I could argue both sides of that question. The present title presumes the optimistic answer, and while we all hope that this is the correct answer, the present time may not be the right one for this answer.

The history of mathematics is not an easy field, and it takes a rare person to be good at it. Keynes supposedly said it took a rare person to be a great economist: one must be a second-rate historian, mathematician, and philosopher. For him second-rate was very good,...

By almost all current philosophical accounts, the success of applied mathematics is a perpetual miracle. Neither formalist nor logicist nor Platonist (at least of the latter-day dilute variety) can provide a plausible explanation of why numbers should fit the world. (Here I usenumbersas a convenient shorthand for all elements of mathematics, since most—but not all—examples of applied mathematics involve numerical measurements.) Why should the relationships among undefined elements or logic or purely mental objects describe a multitude of phenomena with tolerable accuracy? Short of invoking preestablished harmony or a coincidence of mind-boggling improbability, we are philosophically at...

Mathematics underwent, in the nineteenth century, a transformation so profound that it is not too much to call it a second birth of the subject—its first birth having occurred among the ancient Greeks, say from the sixth through the fourth century B.C. In speaking so of the first birth, I am taking the wordmathematicsto refer, not merely to a body of knowledge, or lore, such as existed for example among the Babylonians many centuries earlier than the time I have mentioned, but rather to a systematic discipline with clearly defined concepts and with theorems rigorously demonstrated. It...

For over two decades, one of my major interests has been reading, teaching, and writing history of mathematics. During those decades, I have become convinced that ten claims I formerly accepted concerning mathematics and its development are both seriously wrong and a hindrance to the historical study of mathematics. In analyzing these claims, I shall attempt to establish their initial plausibility by showing that one or more eminent scholars have endorsed each of them; in fact, all seem to be held by many persons not fully informed about recent studies in history and philosophy of mathematics. This paper is in...

The principal thrust of this essay is to describe the current state of interaction between mathematics and the sciences and to relate the trends to the historical development of mathematics as an intellectual discipline and of the sciences as they have developed since the seventeenth century. This story is interesting in the context of the history of present-day mathematics because it represents a shift in the preconceptions and stereotypes of both mathematicians and scientists since World War II.

The notion that significant mathematical and scientific advances are closely interwoven is not particularly new. The opposing notion (associated, with whatever degree...

Virtually all the discussion of the “philosophy of mathematics” in our century has been concerned with the enterprise of providing a foundation for mathematics. There is no doubt that this enterprise has often been mathematically fruitful. Indeed, the growth of logic as an important field within mathematics owes much to the pioneering work of scholars who hoped to exhibit the foundations of mathematics. Yet it should be almost equally obvious that the major foundational programs have not achieved their main goals. The mathematical results that they have brought forth seem more of a piece with the rest of mathematics than...

Everywhere in our society—at the supermarket, in the stock market, in mathematics courses—we see the presence of the computer. We are told that we are entering a new age, that of the Computer Revolution, and that the field known as Artificial Intelligence is at the forefront of that revolution. Its practitioners have proclaimed that they will solve the ageold problems of the nature of human thought and of using technology to build a peaceful and prosperous world. These claims have provoked considerable controversy.

The field known as AI is, of course, vast, but the subject matters that have...

In 1896 the College of New Jersey changed its name to Princeton University, reflecting its ambitions for graduate education and research.

At the time, Princeton, like other American universities, was primarily a teaching institution that made few significant contributions to mathematics. Just four decades later, by the mid-1930s, Princeton had become a world center for mathematical research and advanced education.¹ This paper reviews some social and institutional factors significant in this rapid rise to excellence.²

The decade of the 1930s was a critical period for American research mathematics generally, and for Princeton in particular. The charter of the Institute for...