Paul Bernays is arguably the greatest philosopher of mathematics in the twentieth century. His prominence in the field from the 1920's into the 1970's is widely acknowledged; but much of his work remains untranslated from the original German or (in one case) French. For this reason, neither the substance of his contribution to the development of twentieth century foundations of mathematics nor the positive insights he has brought to it are sufficiently well-known in the English speaking sphere.
Our project is to prepare and publish a volume of English translations of his papers on the philosophy of mathematics. The originals were written in either German or French, with Bernays himself supplying an English translation in one case. Some of these papers have been collected and published in German under the title Abhandlungen zur Philosophie der Mathematik.
Bernays played a pivotal role in the discussion of foundations of mathematics in the twentieth century. As David Hilbert's assistant from 1917 into the 1930's, he was active in the development of the finitist conception of mathematics as the basis upon which the consistency of all of mathematics was to be proved; he was responsible for the writing of the two-volume Grundlagen der Mathematik, which expounds the results of the Hilbert school (and much other work) in foundations. But also, after the discovery in 1931 by Gödel of his incompleteness theorems, which signaled the failure of Hilbert's project in its original form, Bernays was a leader in the search for a new understanding of contemporary mathematics.
In his lecture "Mathematical Problems", before the International Congress of Mathematicians in Paris in 1900, Hilbert, motivated in part by the discovery of the so-called paradoxes of set theory, listed as the second problem that of proving the `compatibility' or consistency of the axiom system for the real numbers. The consistency of this system can be reduced to that of the theory of sets of positive integers; but the consistency of that theory, or even of the elementary theory of the arithmetic of the positive integers, now called Peano arithmetic, cannot be further significantly reduced: it could only be reduced to a theory of some infinite domain, and the positive integers are the simplest example that we have of an infinite domain. So a proof of consistency for Peano arithmetic would have to involve a new idea. And it took until the early twenties before the methodological frame was set for Hilbert's proof theory.
It is here that Bernays played a pivotal role: among those who, in the first part of the century, were able to understand the technical mathematical issues, he was one of relatively few scholars in foundations of mathematics (as another, Hermann Weyl comes to mind) who had the humanistic background to understand the philosophical issues; and it is widely believed that Bernays was responsible for the idea of founding `finitist' or `contentual' mathematics on the Kantian conception of pure intuition.
A large number of the papers that we will include in the volume are concerned with this program of Hilbert's, to clarify its epistemological underpinnings, and to defend it against philosophical critics. However, as a result of Gödel's incompleteness theorem in 1931, the Hilbert program in its original form had to be abandoned. In the first place, as a result of the first incompleteness theorem, it is impossible to completely formalize, as Hilbert had envisioned, the theory of the positive integers or any extension of it. Moreover, by the second incompleteness theorem, even if we settle for incomplete formalizations, providing that they formalize reasonably strong fragments of mathematics (in particular, all of finitist mathematics), it is impossible that there be a finitist consistency proof for them.
Not all of Bernays' work in philosophy of mathematics, not even all of the work that we include in the proposed volume, is best understood in terms of Hilbert's program and reaction to Gödel's theorems. For example, his review of Wittgenstein's Remarks on the Foundations of Mathematics and his work on the foundations of geometry are not in this vein. But it is perhaps easiest to explain his significance in the field in these terms.
In consequence of Gödel's theorems, there were three possible roads to take. One was simply to restrict mathematics to what can be done finitistically or, since finitism represents too extreme a limitation on reason, what can be done constructively or intuitionistically. In finitist mathematics, one is restricted to discrete objects, viz. such as can be represented by positive integers, and to concepts for which one has an algorithm to decide for given objects whether or not the concept applies to it. Constructive mathematics also admits higher order objects, such as functions of numbers, functions of these, etc.; but the functions are restricted to ones which are given by an algorithm for computing their values. Moreover, concepts are admitted for which there is no such algorithm; one makes no assumption that they either hold or do not hold of given objects, so that the `law of excluded middle' is not a part of the logic of constructive mathematics. It is claimed that constructive mathematics, though more extensive than finitism, nevertheless possesses a kind of evidence that is lacking for ordinary unrestricted, sometimes called `Platonistic', mathematics.
A second direction to take is to retain Hilbert's formalistic conception of ordinary mathematics, but to extend the methods for proving consistency to include some constructive methods which go beyond finitism, though not necessarily all of them. From such consistency proofs, we can obtain an interpretation of the original unrestricted proof, telling us what it means constructively. In this direction are Gentzen's consistency proof for Peano arithmetic, obtained by adding to purely finitist methods the principle of transfinite induction on a certain simple well-ordering of the positive integers, and Gödel's consistency proof for the same system, obtained by adding to finitist methods the principle of defining functions of functions of … of functions of positive integers by means of simple induction. Bernays discusses both the first and second directions in many of his later papers.
The third direction is to abandon both constructivism and the formalistic attitude towards ordinary mathematics and to take the latter at face-value as a meaningful theory of mathematical objects and seek simply to find the right axioms for it. Of course, in view of Gödel's theorems–and, indeed, so long as we wish to retain Cantor's theory of transfinite numbers–we will never find `all' the axioms: no axiom system will ever be complete, and new axioms will always suggest themselves to us. Much of Bernays' later technical work, for example his papers and book on axiomatic set theory and his paper on reflection principles in set theory, are concerned with this direction, as is his paper on the so-called `Skolem paradox'.
All three of these approaches to foundations of mathematics are being actively developed still today. The actual development of constructive mathematics in recent times has been less in the intuitionistic style of L.E.J. Brouwer than in that of Errett Bishop; but the underlying philosophy is the same. A philosophical argument for constructive mathematics has been attempted in recent years by Michael Dummett and, following him, by Dag Prawitz, and, most recently, by Neil Tennant, based on a particular theory of meaning. As for the second approach, the general outlines of possible extensions of Hilbert's program, obtained by weakening the requirement of finitism, were laid out by Bernays and, especially, Gödel (1938a). On the technical side, after some success, it looked as though there was an insuperable barrier to obtaining constructive consistency proof for formal systems of any substantial logical complexity. In the last few years, there have been substantial advances which hold promise that this barrier can be circumvented; though that still remains to be seen. On the philosophical side, what remains at issue is the epistemological question of the gain obtained by constructive consistency proofs. But, aside from the general arguments for constructive mathematics already cited, the discussion of this question, which rather concerns restricted constructive methods (transfinite induction on certain well-orderings, functions of higher type, etc.). although lively, tends at the moment to be either at a sophisticated mathematical level or at the level of rather informal discussion. Surprisingly, there is also – at the mathematical level - a growing connection to the third approach (through the appeal to large cardinal axioms for the motivation of strong systems of constructive notation systems used in consistency proofs). The third approach has been, both technically and philosophically, the most lively. On the technical side there has been an intensive investigation of new axioms for set theory and, on the philosophical side, extensive debate about the kind of grounds upon which such axioms might be accepted.
From the point of view of on-going research in philosophy of mathematics, which we just briefly described, Bernays' philosophical papers are entirely contemporary, holding insights that are still relevant to today's issues and debates. For this reason alone, a volume of these essays, translated into the lingua franca of the debates would be a most important contribution. But, along with the purely philosophical importance of the essays, they are also of considerable historical interest. There is a waxing interest in the history of logic and foundations of mathematics from the late nineteenth century. The old labels logicism, formalism, intuitionism and Platonism are now seen to be entirely too crude to tell even a minimally adequate story of the various programs and trends, even just those arising from the work of Frege, Russell, Hilbert and Bernays, Brouwer, and Weyl. A finer grained story must not only begin earlier (no later than the first decade of the nineteenth century), but must recognize some variation and evolution in the positions so labeled, themselves. There have already appeared a number of works which help to serve this renewed historical interest with English translations of important documents–for example, the three volumes of Kurt Gödel's Collected Works, William Ewald's two-volume collection of papers From Kant to Hilbert: A Source Book in the Foundations of Mathematics (especially the second volume), and Paolo Mancosu's collection From Brouwer to Hilbert. The latter work contains four of the papers that we will include in the proposed volume; however, they are concerned only with the Hilbert program. A volume of Bernays' papers, all collected together, would be, in itself, a reflection on that history from 1917 into the 1970's through the eyes of one of the great philosophers of the century.