Part 4 of Turing: a natural philosopher (1997)
Thinking the UncomputableTuring then studied at Princeton for two academic years, with a break back at Cambridge in summer 1937. It was a period of intense activity at a world centre of mathematics. Turing was overoptimistic in thinking he could rewrite the foundations of analysis, and added nothing to the remarks about limits and convergence given in On Computable Numbers. (One reason for this might be the following: if x and y are computable numbers, as specified as Turing machines, the truth of the statements x=y, or x=0 cannot tested by a computable process.) But besides wide-ranging research in analysis, topology and algebra, and the 'laborious' work of showing the equivalence of his definition of computability with those of Church and Gödel, he extended the exploration of the logic of mental activity with a paper Systems of Logic based on Ordinals .
This, his most difficult paper, is much less well known than his definition of computability. It is generally regarded as a diversion from his line of thought on computability, computers and the philosophy of mind, and I fell into this assumption in Alan Turing: the Enigma, essentially because I followed Turing's own later standpoint. But I now consider that at the time, Turing saw himself steaming straight ahead with the analysis of the mind, by studying a question complementary to On Computable Numbers. Turing asked in this paper whether it is possible to formalise those actions of the mind which are not those of following a definite method — mental actions one might call creative or original in nature. In particular, Turing focussed on the action of seeing the truth of one of Gödel's unprovable assertions.
Gödel had shown that when we see the truth of an unprovable proposition, we cannot be doing so by following given rules. The rules may be augmented so as to bring this particular proposition into their ambit, but then there will be yet another true proposition that is not captured by the new rules of proof, and so on ad infinitum. The question arises as to to whether there is some higher type of rule which can organise this process of 'Gödelisation.' An ordinal logic is such a rule, based on the theory of ordinal numbers, the very rich and subtle theory of different ways in which an infinite number of entities may be placed in sequence. An ordinal logic turns the idea of 'and so on ad infinitum' into a precise formulation. Turing wrote that: 'The purpose of introducing ordinal logics is to avoid as far as possible the effects of Gödel's theorem.' The uncomputable could not be made computable, but ordinal logics would bring it into as much order as was possible.
Turing's work, in which he proved important (though somewhat negative) results about such logical schemes, founded a new area of mathematical logic. But the motivation, as he himself stated it, was in mental philosophy. As in On Computable Numbers, he was unafraid of using psychological terms, this time the word 'intuition' appearing for the act of recognising the truth of an unprovable Gödel sentence:
Turing then explains how the axiomatization of mathematics was originally intended to eliminate all intuition, but Gödel had shown that to be impossible. The Turing machine construction had shown how to make all formal proofs 'mechanical'; and in the present paper such mechanical operations were to be taken as trivial, instead putting under the microscope the non-mechanical steps which remained.
In consequence of the impossibility of finding a formal logic which wholly eliminates the necessity of using intuition, we naturally turn to 'non-constructive' systems of logic with which not all the steps in a proof are mechanical, some being intuitive. An example of a non-constructive logic is afforded by any ordinal logic... What properties do we desire a non-constructive logic to have if we are to make use of it for the expression of mathematical proofs? We want it to show quite clearly when a step makes use of intuition, and when it is purely formal. The strain put on the intuition should be a minimum. Most important of all, it must be beyond doubt that the logic shall be adequate for the expression of number-theoretic theorems...It is not clear how literally Turing meant the identification with 'intuition' to be taken. Probably his ideas were fluid, and he added a cautionary footnote: 'We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions.' But the evidence is that at this time he was open to the idea that in moments of 'intuition' the mind appears to do something outside the scope of the Turing machine. If so, he was not alone: Gödel and Post held this view.
Turing and WittgensteinAs it happened, Turing's views were probed by the leading philosopher of the time at just this point. Unfortunately their recorded conversations shed no light upon Turing's view of mind and machine. Turing was introduced to Wittgenstein in summer 1937, and when Turing returned to Cambridge for the autumn term of 1938, he attended Wittgenstein's lectures — more a Socratic discussion group — on the Foundations of Mathematics. These were noted by the participants and have been reconstructed and published.  There is a curious similarity of the style of speech — plain speaking and argument by question and answer — but they were on different wavelengths. In a dialogue at the heart of the sequence they debated the significance of axiomatizing mathematics and the problems that had arisen in doing so:
Wittgenstein:... Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone — much more extraordinary than you might think... Because the thing works like this: if a man says 'I am lying' we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn't matter. ...it is just a useless language-game, and why should anyone be excited?
Turing's responses reflect mainstream mathematical thought and practice, rather than showing his distinctive characteristics and original ideas. In 1938, it should be noted, he was an untenured research fellow whose first application for a lectureship had failed, and whose chance of a conventional career lay in the mathematics studied and taught at Cambridge. His work in logic was but a part of his output, by no means well known. His Fellowship was for work in probability theory; his papers were in analysis and algebra. That year, he made a significant step in the analysis of the Riemann zeta-function, a topic in complex analysis and number theory at the heart of classical pure mathematics.
Getting statements free from contradictions is the very essence of mathematics. Turing perhaps thought Wittgenstein did not take seriously enough the unobvious and difficult questions that had arisen in the attempt to formalize mathematics; Wittgenstein thought Turing did not take seriously the question of why one should want to formalize mathematics at all.
There are no letters or notes which indicate subsequent contact between Turing and Wittgenstein, and no evidence that Wittgenstein influenced Turing's concept of machines or mind. If influence in the next ten years is sought, it should be found in the Second World War and Turing's amazing part in it.
 Systems of logic based on ordinals, Proc. Lond. Math. Soc
(2) 45 pp 161-228 (1939).
© 1997, Andrew Hodges.