Ingenious ingénue

A Logical Journey
April 3, 1998

By general acclaim, Kurt Godel (1906-78) was the greatest logician of the 20th century - and even stronger assessments of his stature are frequently made. In 1931 he virtually demolished formalism, that standpoint with regard to the foundations of mathematics whereby some precise and clear-cut system of axioms and rules of procedure are decreed to provide a complete description of mathematics and of correct mathematical reasoning. Formalism had been put forward as a way of avoiding certain paradoxes that arise when too free a use is made of the powerful methods of the theory of infinite sets - a theory that had been introduced in the 19th century by the great mathematician Georg Cantor. It was argued that these paradoxes can be avoided if mathematical reasoning is constrained to be in accordance with certain very strict rules, as provided by what is referred to as a formal system. What Godel showed was that no matter which formal system is chosen (so long as it is broad enough to include certain very basic operations), then either that system is itself inconsistent (and so contains its own internal "paradoxes") or else it is incomplete in the sense that there are true mathematical statements that lie beyond its scope.

This now-famous result, Godel's incompleteness theorem, has important philosophical implications. It shows, in a clear sense, that mathematical reasoning - that mental activity which represents thinking in its most precise and reliable form - cannot be reduced to any set of pre-assigned clear-cut rules. Some people, such as the philosopher John Lucas, have further argued that, accordingly, Godel's theorem shows that human thinking contains essential elements that lie beyond the scope of the action of any computing machine; and I have argued in a similar vein myself. Since some others have vehemently disputed such conclusions, it is of particular interest to know what Godel's own thoughts on this and other related matters might have been.

The distinguished Chinese/American mathematical logician Hao Wang had numerous conversations and extended correspondence with Godel on matters of mathematics and philosophy, and he kept careful records. But Godel was a recluse, a very strange and private individual who did not readily reveal his inner thoughts. Thus, Wang appears to have been in a virtually unique position to address the question of Godel's thinking on such matters, and we are fortunate that he has been able to relate this thinking to a wider audience in three books in which his conversations with Godel are described. Unfortunately, Wang died before he could finalise all the details in the manuscript of this third and last volume, and he was unable to assemble the material for a proposed further volume to be entitled The Formal and the Intuitive: From Computation to Wisdom in which he intended to relate his own extended reflections on many of these central issues. The final checking of A Logical Journey had to be carried out by Palle Yourgrau and Leigh Cauman. They have done this in a very professional way, but they did not deem it appropriate to remove some of the repetition which remains and which would presumably have been dealt with by Wang himself had he lived and been in good health for a longer period.

It should be made clear that this volume is not a biography of Godel, although it does contain a somewhat brief and austere account of his life. Wang's concern is primarily with Godel's ideas and opinions on philosophy and mathematics. Yet, a little of Godel's strange personality does come through, such as his unworldliness, his hypochondria, his fastidiousness and fussiness over details, and his simplistic views of other people's motivations. Yet Wang's account makes no mention of some of his more extreme eccentricities, such as Godel's almost scuttling his application for US citizenship because of an insistence on pointing out anomalies in the US Constitution, or his refusing to eat during his final illness in the belief that his doctors were trying to poison him. It is important, if we are to assess the validity of Godel's opinions on various matters, for us to gain some understanding of the nature of these eccentricities.

What seems clear is that the value of Godel's views is enormously the greater when there is minimal input from the specifics of the world's activities, and particularly of human behaviour. When Godel is concerned with pure matters of abstract reason he is supreme, and his views must be treated with exceptional regard. When he addresses matters of physics, then his opinions are of definite interest and importance - and occasionally of deep originality - but not, in my opinion, always revealing of a full appreciation of important developments that were taking place in the 20th century. His understanding of politics and of individual personalities seem sometimes to be very odd and wide of the mark; naive in a childlike way.

An illustration of this final point was his comment concerning the suicide of Alan Turing - the brilliant logician, seminal computer scientist, and codebreaker, whose difficulties with the British authorities concerning his homosexuality had caused him continued anguish, until he took his own life in 1954. Godel asked "Was Turing married?" and, upon being told that he was not, remarked "Maybe Turing wanted but failed to get married". Godel also took the view that it was no coincidence that Robert Taft and Joseph Stalin died not long after Eisenhower became president, because "Eisenhower's policies might have brought distress to Taft and Stalin".

With regard to physics, Godel made at least one highly significant and tantalising contribution to Einstein's theory of general relativity, and to the interpretation of time within that theory. No doubt stimulated by his close friendship with Einstein, he produced some very unusual and ingenious models for cosmology, incorporating an omnipresent rotation and an inconsistent global notion of time, in which it is theoretically possible to travel into one's past. Apart from this, his viewpoint concerning physical reality seems to have been a very "Newtonian" one ("The backbone of physics remains in Newtonism"), and my impression is that he never really came to terms with quantum mechanics - which is particularly difficult, in any case, for someone with a thoroughly logical turn of mind!

His views concerning the foundations of mathematics, on the other hand, must carry an enormous weight. Godel was a strong mathematical Platonist, in the sense that he firmly believed that mathematical notions have an existence of their own, quite independent of whether or not these notions are actually perceived by any particular individuals. In fact, most mathematicians are Platonists to some degree, but they may differ significantly with regard to how far their Platonism may be carried. Godel frequently pointed out that his own strongly Platonistic, or realistic, viewpoint was crucial to his own discoveries. Indeed, his famous incompleteness theorem shows that we do not access mathematical truth merely by following humanly devised (but trustworthy) mechanical rules, since we always have the capability to step outside those rules in order to ascertain the truth of certain statements that lie beyond their scope. Moreover, he argued that even the great mathematician David Hilbert was hampered by taking a more "positivistic" than "realistic" point of view in his bid to establish the acceptability of Cantor's "continuum hypothesis": an important still-unresolved mathematical problem to which Godel made a fundamental contribution by adopting his "realistic" philosophy.

The crucial issue confronting mathematical philosophy can be stated in deceptively simple terms: any multitude (within reason) is also a unity! This is the essence of set theory, and provides its mathematical power - collections of mathematical entities are themselves individual entities. The basic issues of dispute between different mathematical standpoints lie in the interpretation of my inserted phrase "within reason". The stronger one's mathematical Platonism, the less restrictive is one's interpretation of that phrase. Deleting it altogether would lead to contradiction, but Godel's inclination would be to fall barely short of that ideal. His "big jump" is to pass from individual natural numbers 0, 1, 2, 3, 4I to the totality of all of them. This is the first step into the infinite and, according to Godel, the acceptance of this first step leads on to an unending acceptance of this notion to ever-increasing degrees.

Godel's opinion was that such an unrestricted attitude to set theory is free of difficulties, yet more is needed in order that a satisfactory overall philosophical standpoint can be obtained. He strove for an all-embracing concept theory, but was unable to achieve this goal to his satisfaction. Nevertheless, he thought that it should be possible eventually to place philosophy on a sound basis, so that its questions can be answered with the same kind of certainty as they are in mathematics.

Godel makes clear that he regards the human mind to be superior to any computing machine, and that his incompleteness theorem lends strong support to this view. His rather "Newtonian" picture of physical action, however, leads him to support the common opinion that a physical brain must act computationally. Accordingly, he views the mind as something separate from the physical brain, which is "a computing machine connected with a spirit". His philosophical position follows that of Gottfried Leibniz, according to which the basic entities are "monads" which "have an inner life or consciousness", and where there is a "central monad" - namely God. "Undoubtedly it was for several decades a major wish of Godel's to develop such a Leibnizian monadology and to demonstrate convincingly that it is a true picture of the world (but) most of us today are inclined to think that neither Leibniz nor Godel has offered convincing reasons to believe that a 'monadology' can be developed far enough even to be considered plausible."

Wang relates Godel's opinions of the views of many philosophers and mathematicians: Bernays, Cohen, Hegel, Hilbert, Husserl, Kant, Rawls, Russell, Skolem, Spinoza, Tarski, Wittgenstein - as well as his views on many topics in philosophy and mathematics. Much of this material was never published, as Godel was loath to set forth what he could not rigorously prove. Much survives only in Godel's form of an obsolete German Gabelsberger shorthand. Wang has provided a momentous service in bringing all this material together and giving us his own authoritative assessments. I did not find the book an easy read, but it will provide an invaluable source for others who wish to study the views of one of mankind's deepest and strangest thinkers.

Sir Roger Penrose is professor of mathematics, University of Oxford.

A Logical Journey: From Gödel to Philosophy

Author - Hao Wang
ISBN - 0 262 23189 1
Publisher - MIT Press
Price - £33.95
Pages - 391