# The Stymied Philosopher

*Since most human knowledge is acquired, rather than being instinctive, most of our knowledge of the world is likely to be culturally conditioned. This means that error (in the form of preconceptions and preconditioning) is as likely to come from this source as any other. One of the most basic ways of extending understanding, then, is by making mistakes; if we don't make mistakes (or if we don't come to realise that we have) then we do not learn. But facing up to error, or even trying to see where we have gone wrong, can be a very painful process.*

*In science the process of experimentation and testing of hypotheses are essential because they open up new areas in which mistakes can be made, and subsequently exposed. This procedure is bound to generate further 'problems', which will, in turn, lead to greater insights. Submerged beneath every conventional 'certainty' there are a multitude of discarded errors; behind every scientific 'fact' there are any number of abandoned hypotheses, half-developed ideas and false leads. Errors are a routine part of life, but the realisation of error is the very life-blood of science, and the surest path to greater knowledge. This can be very problematic however, and nowhere more so than in the abstract realms of philosophical speculation.*

In 1902 the German mathematician Gottlob Frege was overseeing the publication of the second volume of his Grundgesetze der Arithmetik (Fundamentals of Arithmetic), when he received an abstruse, but thoroughly disquieting communication from England, from the young philosopher Bertrand Russell.

Frege had been developing the theories that he presented in the Grundgesetze for some three decades. It was to be the culmination of his attempt to place the whole of mathematics on an unassailable foundation of formal logic. Russell had been working along similar lines in his own masterwork, the Principia Mathematica. He had followed Frege's explanations of the numbers as sets. Like so many familiar concepts, numbers seem obvious, self-evident things until some philosopher questions what they really are; how are the notions of 'one', 'two', 'three' etc. to be objectively defined? Frege's rather ponderous solution was that 'one is the set of all sets that have only one member, two is the set of all sets that have only two members' and so on.

Russell extended this principle into a notion of classes of things, and he was well advanced with his Principia when, turning the matter over in his mind, it occurred to him that a class sometimes is, and sometimes isn't, a member of itself - 'The class of teaspoons, for instance, is obviously not another teaspoon - but the class of things that are 'not teaspoons' is itself clearly one of the things that are not teaspoons'. An irritating conundrum then presented itself - was the class of things that were 'not members of themselves', a member of itself or not? This problem (meaningless to any but a finely tuned logical mind) turned out to be a version of a truly ancient paradox, that of the Liar. In this, Epimenides the Cretan, declares that 'All Cretans are liars' - so is he a liar or is he telling the truth? It seems that if he is, he isn't, and if he isn't he is. Annoyingly, Russell found that the same problem applied to his theory of classes - each alternative solution appeared to lead to its contradictory opposite. If it was a member it wasn't, if it wasn't it was. When Russell referred the problem to his friend, the mathematician and philosopher A.N. Whitehead, he failed to console him, quoting Browning's line 'Never glad confident morning again!'.

It was this absurd, almost puerile, logical glitch that Russell had conveyed to Frege, who replied 'Your discovery of the contradiction has surprised me beyond words'. For the German mathematician the news was indeed devastating. He had immediately grasped its implications - that his laborious scheme to establish a basis for mathematics that was entirely free of paradox and contradiction was fundamentally flawed. Although the second volume of his life's work was already at the printers he felt compelled to add an appendix. It began with the melancholy observation that ... 'A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell...'

Russell's own scheme to establish a realm of absolutely certain and demonstrable truths (based on his assumption that mathematics was 'the only thing we know of that is capable of perfection') now seemed to have a Damoclean sword hanging over it. But he was determined to resolve the problem. He had already spent the latter months of 1901 working on it - 'At first I supposed that I should be able to overcome the contradiction quite easily, and that there was probably some trivial error in my reasoning. Gradually however, it became clear that this was not the case ... by the end of the year I had concluded that it was a big job.' Other demands on his time then obliged him to leave the matter in abeyance for a while.

But the problem continued to haunt him. Like Frege, he found it impossible to reconcile these paradoxical inconsistencies with his scheme for mathematical purity. He was finally able to return to the problem in the summer of 1903, fresh and determined to sort it out. But it proved as intractable as ever. 'Every morning I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty.' This was an extraordinary situation. One of the greatest minds of his generation was utterly baffled by a problem that might have fallen out of a Christmas cracker! After some months of working on the problem, and having made no headway at all, he was once again obliged to break off.

Russell took up the problem the following summer, by now it seemed to be impinging on his entire career and reputation. 'It was clear to me that I could not get on without solving the contradictions, and I was determined that no difficulty should turn me aside from the completion of Principia Mathematica, but it seemed quite likely that the whole of the rest of my life might be consumed in looking at that blank sheet of paper. What made it all the more annoying was that the contradictions were trivial, and that my time was being spent in considering matters that seemed unworthy of serious attention.' There was a dangerous precedent for Russell in this dilemma - the poet Philetus of Kos (3rd century BC) was reputed to have grown thin, and eventually to have died, as a result of brooding over the Liar paradox.

In fact Bertrand Russell never was able to resolve his paradox of classes. It turned out that self-referential posers of this sort are simply not answerable within the framework of classical logic. But it took an extraordinary leap of mathematical imagination to establish this - and in the process mathematics was brought from the classical into the modern era. Russell's struggles were not entirely in vain. In 1930 a brilliant young Austrian logician, Kurt Gödel, motivated by the Frege's and Russell's failure, took up their quest to establish the 'completeness' of mathematics. However, in a wonderfully short time, he came instead to a quite opposite view, and was soon providing a definite proof that there could, in fact, be no general method for verifying mathematical propositions in the way that had been imagined. Gödel's Incompleteness theorem dealt a deathblow to Russell's Principia, but it proved to be the most important proof in mathematical logic of the 20th century. In essence it demonstrated that it was impossible to determine in advance whether a given mathematical problem has a solution or not. But Gödel's theorem is not easy - it has been described as one of the most impenetrable works of genius ever written - and it is believed that even Russell never fully grasped it.

Ironically the theorem has had little effect on the actual practice of mathematics, since it does not affect procedure - it simply lays bare the possibility that a solution to a problem might never actually be achieved. The whole episode perfectly illustrates Herman Kahn's notion of scientific paradigms, which observes that sooner or later those in any branch of science who try to extend the existing paradigm find that there are puzzles that they cannot solve; there is then a crisis and a new paradigm is required, allegiances are transferred, and one world view is exchanged for another. When the new theory is accepted many 'insoluble problems' simply evaporate.

There is one area in which Gödel's abstruse theorem does have a bearing on ordinary day-to-day existence, namely in computer science. Computers, it would appear, are prone to be flustered by variations of the 'Liar' paradox and, in order to function properly, have to be proofed against precisely the same sort of is/isn't oscillations that so plagued Russell in the early years of the 20th century.