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.