Since my title refers to “twentieth-century” foundations of mathematics, you may think I haven’t quite got the news that we live in the twenty-first century. The fact is, however, that what I could find to talk about are some tendencies in the thought about the foundations of mathematics of thetwentiethcentury, indeed more prominent in its first half than in its second. I will try to say at the end what relevance these tendencies might still have today.

There is a picture of the foundations of mathematics with which many English-speaking philosophers grew up in the period between roughly...

It is fair to say that the acceptability of impredicative definitions and reasoning in mathematics is not now, and hasn’t been in recent years, a matter of major controversy. Solomon Feferman may regret that state of affairs, even though his own work contributed a great deal to bringing it about. I see the work of Feferman and Kurt Schütte on the analysis of predicative provability in the 1960s as bringing to closure one aspect of the discussion of predicativity that began with Poincaré’s protests against “non-predicative definitions” in the first decade of the twentieth century and with Russell’s making a...

The name of Paul Bernays (1888–1977) is familiar probably first of all for his contributions to mathematical logic. Many of those were in the context of his position as David Hilbert’s junior collaborator in his proof-theoretic program inaugurated after the First World War. For those of us starting out in logic in the mid-twentieth century, the monumentalGrundlagen der Mathematikof Hilbert and Bernays was one of the basic works in mathematical logic that we were obliged to study. That was the more true for those like me who aspired to work in proof theory.

Bernays’ collaboration with Hilbert...

Kurt Gödel was born on April 28, 1906, in Brünn, Moravia, then part of Austria and now Brno, Czech Republic. His father held a managerial position in a textile firm, and Gödel grew up in prosperous circumstances. He was baptized in his mother’s Lutheran faith. He attended schools in his home city and in 1924 entered the University of Vienna, initially to study physics but eventually studying mathematics. He also attended lectures in philosophy and sessions of the Vienna Circle of logical positivists. He completed the Dr. Phil. in 1929 and qualified as aPrivatdozent(unsalaried lecturer) in 1933. He...

This paper was written forThe Philosophy of Bertrand Russell, a volume of Paul Arthur Schilpp’s series The Library of Living Philosophers. In his letter of invitation of November 18, 1942, Schilpp proposed the title of the paper and also wrote, “In talking the matter over last night with Lord Russell in person, I learned that he too would not only very greatly appreciate your participation in this project, but that he considers you the scholar par excellence in this field.” Gödel sent in the manuscript on May 17, 1943. There followed a lengthy correspondence about stylistic editing proposed by...

This essay is more about Gödel than about Quine. My excuse is that it concerns an aspect of Quine’s philosophy that is well known, while the related aspect of Gödel’s philosophy is comparatively little known; indeed, most of the texts where it is presented have remained unpublished until now.¹ I do hope, moreover, that seeing some of Quine’s arguments in comparison with Gödel’s will put some points in Quine’s philosophy into sharper relief.

Let me remind you of some points about Quine’s discussion of analyticity. It is useful to distinguish three strands.

1. All but the last section of “Two Dogmas...

The best known and most widely discussed aspect of Kurt Gödel’s philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell’s report in his autobiography of one or more encounters with Gödel is well known: “Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians might hope to meet it hereafter.”¹ On this Gödel commented:

Concerning my “unadulterated” Platonism, it is no more...

During the time since World War II, nominalism as a philosophical tendency or research program has been largely identified with what was inaugurated by Nelson Goodman in such works asThe Structure of Appearance. What was definitive of nominalism for Goodman was the rejection of the assumption of classes in philosophical and logical construction. Quine joined the research program in one well-known joint paper with Goodman, “Steps toward a Constructive Nominalism,”¹ which inaugurated post-war nominalism in the philosophy of mathematics. The opening sentence of that paper, “We do not believe in abstract entities,” could serve as the slogan for recent...

Quine’sRoots of Reference¹ is a puzzling book for a number of reasons. It naturally attracts interest because after his proposal that epistemology should be “naturalized,” this work represents an attempt actually todonaturalistic epistemology. But the focus of the work on reference means that Quine’s attention is drawn away from general epistemological issues.² This is not incompatible with its belonging to naturalistic epistemology, but it indicates that the book is concerned with a rather restricted part of what might in principle count as naturalistic epistemology.

It is helpful in understanding Quine’s enterprise to observe that the work is...

In this essay I attempt to convey an idea of Hao Wang’s style as a philosopher and to identify some of his contributions to philosophy. Wang was a prolific writer, and the body of text that should be considered in such a task is rather large, even if one separates off his work in mathematical logic, much of which had a philosophical motivation and some of which, such as his work on predicativity, contributed to the philosophy of mathematics. In a short essay one has to be selective. I will concentrate on his bookFrom Mathematics to Philosophy,¹ since he...

The origin of this essay lies in a puzzlement I have felt about how Putnam understands Quine’s views on existence and ontology. I have found it difficult to put my finger on what his quarrel with Quine about this has been. But one could also see it as a fragment of a commentary on the following recent remark of Putnam. Speaking of the lectures “Ethics without Ontology” published in the book of that name, Putnam writes that they

…provided me with an opportunity to formulate and present in public something that I realized I had long wanted to say, namely...

William Tait’s standing in the philosophy of mathematics hardly needs to be argued for; for this reason the appearance of this collection will be welcomed. As noted in his preface, the essays in this book “span the years 1981–2002.” The years given are evidently those of publication, although one essay (no. 6) was not previously published in its present form. It is, however, a reworking of papers published during that period. The introduction, one appendix, and some notes are new. Many of the essays will be familiar to the readers ofPhilosophia Mathematica;indeed two (nos. 4 and 12)...