Philosophy by numbers

Kurt Godel's place as one of the greatest logicians of the century, perhaps of all time, is secured by his celebrated Incompleteness Theorem of 1931, which put paid once and for all to the dream of creating a single formal system of logic within which the whole of mathematics could be proved.

The philosophical significance of this result is still hotly debated. Some, like Roger Penrose, believe it has important consequences for the philosophy of mind, demonstrating the impossibility of regarding the human mind as a formal system and thereby refuting the claims made on behalf of so-called Artificial Intelligence. Others believe it shows that mathematics cannot be regarded as a construction of the human mind, but must be regarded platonistically, as a body of objective truths. For the postmodernist philosopher, Jean-Francois Lyotard, it demonstrates the untenability of "meta-narratives", while for Wittgenstein, it had no philosophical importance whatever beyond confirming the "disastrous", "cancerous" nature of mathematical logic.

What did Godel himself think? Until now, Godel's own philosophical views have had to be inferred from suggestive, but fragmentary, titbits. That he took a platonist view of mathematical truth is evident from an article he contributed to The Philosophy of Bertrand Russell, edited by Paul Schilpp in 1944, called "Russell's mathematical logic", and also from his 1947 article, "What is Cantor's continuum problem?" And that he took seriously Kant's denial of the objectivity of time emerges from his contribution to another Schilpp volume, this time on Einstein, in 1949. But after that, he published little. From the work published by his friend Hao Wang in the 1970s and 1980s, however, it was known that Godel's interests in the last 20 years of his life (he died in 1978) were primarily philosophical, and that he had embarked on a detailed study of the work of the great German phenomenologist, Edmund Husserl.

Now, at last, with the publication of Volume III of Godel's Collected Works, we have a chance to read for ourselves the essays and lectures in which Godel attempted to present his philosophical perspective. It is a fascinating and enduringly important collection, superbly well-edited, each paper being prefaced with an excellent introduction providing background information, summaries of the arguments and, where necessary, criticisms of the views expressed.

The arrangement is chronological, beginning with the lecture Godel gave at Konigsberg in 1930 in which he announced his incompleteness result for the first time, and ending in 1970 with a series of decidedly eccentric papers, including some uncharacteristically faulty formal arguments intended to solve Cantor's continuum problem, and a page of symbols that purports to be a rigorous proof for the existence of God. This latter is a recasting of the notorious "Ontological Argument" for God's existence into the language of mathematical logic. He establishes first the "theorem" - [unavailable on database] - which says that, if God's existence is possible, then it is necessary, and then argues that God's existence is indeed possible. Therefore, necessarily, God exists.

Such curiosities aside, the interest of this collection rests chiefly on a series of philosophical papers that Godel wrote between 1951 and 1961. The first of these is the Josiah Willard Gibbs Lecture that Godel delivered at Brown University, "Some basic theorems on the foundations of mathematics and their implications", from which it emerges that Godel went some way towards endorsing the use made of his incompleteness result in the philosophy of mind. What the result implies, Godel claims, is that either there exist absolutely undecidable problems in mathematics, or the human mind "infinitely surpasses the powers of any finite machine".

As his rational optimism compelled him to think the first alternative unlikely, he was inclined to the second. The lecture also amplifies Godel's platonistic view of mathematics, which, like Frege's, sees mathematics as the study of concepts, which, in Godel's words, "form an objective reality of their own, which we cannot create or change, but only perceive and describe". Mathematical truths are analytic, but not tautological; that is, they are true, not because of the definitions we give to words, but because of the "nature of the concepts" they describe.

In 1953, Godel prepared the first draft of an article in which he sought to defend this notion of mathematics against the then influential view of the logical positivists that mathematics was tautological. More specifically, Godel's target was the view of Rudolf Carnap that mathematics was essentially linguistic, its propositions rules of the syntax of our language. The article, "Is mathematics syntax of language?", was intended for the Schilpp volume on Carnap, but, at the last moment, Godel withdrew it, despite having worked on several versions of it over a period of six years.

In this volume, two versions are given, the second much shorter. In both, the heart of Godel's case rests on an intriguing argument from his Second Incompleteness Theorem. This says that the consistency of a formal theory of arithmetic cannot be proved from within that theory. Godel's use of this result against Carnap's view runs as follows: if we are to say that mathematics is syntactical, then we must have a way of distinguishing syntactical from empirical propositions, one that does not allow us to derive one from the other. This means that our system of syntactical rules must be consistent, since, from a contradiction everything (including all empirical propositions) can be derived. But, by the Second Incompleteness Theorem, we know that this consistency cannot be established from within the system. Therefore, the belief that mathematics is solely contained in our rules of syntax is false.

The idea that we perceive the objective reality of mathematical concepts was one which Godel took quite literally, as emerges from one of the most interesting papers in this volume: a draft of a lecture that Godel had intended to deliver to the American Philosophical Society soon after his election in 1961. The draft was found in an envelope, which, characteristically, Godel never posted.

Called "The modern development of the foundations of mathematics in the light of philosophy", the lecture announces Godel's commitment to Husserlian phenomenology as a method of clarifying our knowledge of abstract concepts. If Godel's "conceptual realism" is correct, then our knowledge of mathematics cannot consist in a series of definitions; at some point we need to see the conceptual realities embodied in our mathematical axioms. This is where Husserl's notion of an "eidetic intuition" comes in: an act of perception, which, in Godel's words, "consists in focusing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the use of these concepts".

Phenomenology, as Godel understands it, is thus "a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us. I believe there is no reason at all to reject such a procedure at the outset as hopeless".

That the man hailed as the greatest logician of his day should subscribe to such a belief is so surprising that it is a pity this lecture draft is so short, and that Godel did not expand elsewhere upon his understanding of the phenomenological method. The editors of this volume tell us that Godel's copies of Husserl's works are heavily annotated and that "most of his comments are positive and expand upon Husserl's points". My only regret about the editing of this collection is that these marginal notes were not included as an appendix. They have, I think, at least as much right to be included as the "texts relating to the ontological proof", which appear as Appendix B.

On the whole, however, the editors are to be wholeheartedly congratulated on bringing to the public work which deserves careful study and which ought to do something to revitalise the philosophy of mathematics by presenting a point of view that, unusually, combines intellectual rigour with a willingness to make bold and sweeping metaphysical claims.

Ray Monk is lecturer in philosophy, University of Southampton, and the author of biographies of Wittgenstein and Russell.