43740 Turing’s mentor, Max Newman

he interaction between mathematicians and mathematical logicians has always been much slighter than one might imagine. This chapter examines the case of Turing’s mentor, Maxwell Hermann Alexander Newman (1897–1984). The young Turing attended a course of lectures on logical matters that Newman gave at Cambridge University in 1935. After briefly discussing examples of the very limited contact between mathematicians and logicians in the period 1850–1930, I describe the rather surprising origins and development of Newman’s own interest in logic.

One might expect that the importance to many mathematicians of means of proving theorems, and their desire in many contexts to improve the level of rigour of proofs, would motivate them to examine and refine the logic that they were using. However, inattention to logic has long been common among mathematicians.

A very important source of the cleft between mathematics and logic during the 19th century was the founding, from the late 1810s onwards, of the ‘mathematical analysis’ of real variables, grounded on a theory of limits, by the French mathematician Augustin-Louis Cauchy. He and his followers extolled rigour—most especially, careful definitions of major concepts and detailed proofs of theorems. From the 1850s onwards, this project was enriched by the German mathematician Karl Weierstrass and his many followers, who introduced (for example) multiple limit theory, definitions of irrational numbers, and an increasing use of symbols, and then from the early 1870s by Georg Cantor with his set theory. However, absent from all these developments was explicit attention to any kind of logic.

This silence continued among the many set theorists who participated in the inauguration of measure theory, functional analysis, and integral equations.1Close The mathematicians Artur Schoenflies and Felix Hausdorff were particularly hostile to logic, targeting the famous 20th-century logician Bertrand Russell. (Even the extensive dispute over the axiom of choice focused mostly on its legitimacy as an assumption in set theory and its use of higher-order quantification:2Close its ability to state an infinitude of independent choices within finitary logic constituted a special difficulty for ‘logicists’ such as Russell.) Russell, George Boole, and other creators of symbolic logics were exceptional among mathematicians in attending to logic, but they made little impact on their colleagues. The algebraic logical tradition, pursued by Boole, Charles Sanders Peirce, Ernst Schröder, and others from the mid-19th century, was nothing more than a curiosity to most of their contemporaries. When mathematical logic developed from the late 1870s, especially with Giuseppe Peano’s ‘logistic’ programme at Turin University from around 1890, Peano gained many followers there but few elsewhere.3Close

Followers of Peano in the 1900s included Russell and Alfred North Whitehead, who adopted Peano’s logistic (including Cantor’s set theory) and converted it into their logicist thesis that all the ‘objects’ of mathematics could be obtained from logic. However, apart from the eminent Cambridge mathematician G. H. Hardy, few mathematicians took any interest in Russell–Whitehead logicism.4Close From 1903 onwards, Russell publicized the form of logicism put forward from the late 1870s by Gottlob Frege—which had gained little attention hitherto, even from students of the foundations of mathematics, and did not gain much more in the following decades.

In the late 1910s David Hilbert started the definitive phase of his programme of ‘metamathematics’, which studied general properties of axiom systems such as consistency and completeness, and ‘decision problems’ concerning provability. Hilbert’s programme attracted several followers at Göttingen University and a few elsewhere; however, its impact among mathematicians was limited, even in Germany.5Close

The next generations of mathematicians included a few distinguished students of the foundations of mathematics. From around 1900, in the United States, E. H. Moore studied Peano and Hilbert, and passed on an interest in logic and metamathematical ‘model’ theory to his student Oswald Veblen, and so to Veblen’s student Alonzo Church, and from Church in turn to his students Stephen Kleene and Barkley Rosser.6Close At Harvard, Peirce showed his ‘multiset’ theory to the philosopher Josiah Royce, who was led on to study logic and (around 1910) to supervise budding logicians Clarence Irving Lewis, Henry Sheffer, Norbert Wiener, Morris Cohen, and also Curt John Ducasse—the main founder of the Association of Symbolic Logic in the mid-1930s.7Close In central Europe, John von Neumann included metamathematics and axiomatic set theory among his concerns,8Close but in Poland a distinguished group of logicians did not mesh with a distinguished group of mathematicians, even though both groups made much use of set theory (even their joint journal Fundamenta Mathematicae, published from 1920, rarely carried articles on logic).

The normal attitude of mathematicians to logic was indifference. For example, around 1930 the great logician Alfred Tarski and others proved the fundamental ‘deduction theorem’;9Close this work met with apathy from the mathematical community, although it came to be noted by the French Bourbaki group, who were normally hostile to logic. (Maybe the reason was that French logician Jacques Herbrand had proved versions of the theorem—if so, this was his sole impact on French mathematics.) Also, while logicians fairly quickly appreciated Kurt Gödel’s theorems of 1931 on the incompletability of (first-order) arithmetic, the mathematical community did not become widely aware of Gödel’s results until the mid-1950s.10Close

Turing’s own career provides a good example of the cleft between logic and mathematics. When he submitted his paper ‘On computable numbers’ to the London Mathematical Society in 1936 the journal could not referee it properly because Max Newman was the only other expert in Britain and he had been involved in its preparation (Fig. 40.1).11Close Nevertheless, they seem to have accepted it on Newman’s word.12Close But this detail prompts historical questions that have not so far been explored. Why was this logical subject matter so little known in Britain? Why was Newman, a mathematician, a specialist in it? Answers to these questions are suggested below. 13Close

A crucial event, Newman’s lectures to Turing, occurred in the mid-1930s, when he taught a new final-year course on ‘Foundations of mathematics’ for the Mathematical Tripos. The tripos examination questions he set show that he had handled all the topics in logic and metamathematics mentioned in the opening section of this chapter, and also the intuitionist mathematics and logic of L. E. J. Brouwer. Turing, newly graduated from the tripos, sat in on the course in 1935 and learnt about decision problems and Gödel numbering from one of the few Britons who was familiar with them.14Close There seems to be no documentation about contact between Turing and Newman following the lecture course, but presumably the two met on a regular basis during 1935 and 1936 as Turing prepared his paper on computability.

A sense of isolation hangs over Newman’s lecture course. Launched in the academic year 1933–34, Newman ran the course for only two more years before it was closed down, possibly because of disaffection among dons as well as students. In 1937 Hardy opined to Newman:15Close

though ‘Foundations’ is now a highly respectable subject, and everybody ought to know something about it, it is (like dancing or ‘groups’) slightly dangerous for a bright young mathematician!

Somehow Newman continued to set examination questions on foundations for five of the six years that he remained at Cambridge before moving to Bletchley Park.16Close The questions for 1939 may have been set by Turing, who was invited—presumably in a spirit of resistance against Hardy’s coolness—to give a lecture course on foundations in the Lent Term of 1939. He was asked to repeat it in 1940, but by then he was at Bletchley Park.17Close Newman arrived there in the summer of 1942, to join the fight against Tunny. During his wartime period he published three technical papers in logic, one written jointly with Turing. Two, including the one written with Turing, were on aspects of type theory, which was Russell’s solution of the paradoxes of logic and set theory,18Close and one on the so-called ‘confluence’ problem (concerning the reduction of terms).19Close

How did Newman become so involved with logic in the first place? Born Max Neumann in London in 1897, to a German father and an English mother, he gained a scholarship in 1915 to St John’s College, Cambridge, taking Part I of the Mathematical Tripos in the following year.20Close^,21Close During the First World War, Max’s father Herman was interned by the British; when released he went back to Germany, never to return. In 1916, Max and his mother Sarah changed their surname to ‘Newman’. A pacifist, Max served in the Army Pay Corps, returning to his college in 1919. He completed Part II of the tripos with distinction in 1921.

Against the odds, Newman spent much of the academic year 1922–23 at Vienna University. He went with two other members of his college. One was the geneticist and psychiatrist Lionel Penrose (father of the distinguished mathematician Roger Penrose), who seems to have initiated the trip to Vienna; his family was wealthy enough to sustain it, especially as at that inflationary time British money went a long way in Vienna. Penrose had been interested in Russell’s mathematical logic as a schoolboy, and had studied traditional Aristotelian logic as an undergraduate at Cambridge. He had also explored modern mathematical logic and it might even have been Penrose who alerted his friend Newman to the subject. Penrose wanted to meet Sigmund Freud, Karl Bühler, and other psychologists in Vienna. The third member of the Cambridge party was Rolf Gardiner, an enthusiast for the Nazis, later active in organic farming and folk dancing, and father of the conductor Sir John Eliot Gardiner. Gardiner’s younger sister Margaret came along too. She would become an artist and companion to the biologist Desmond Bernal. Margaret recalled ‘the still deeply impoverished town’ of Vienna, where Penrose and Newman would walk side-by-side down the street playing a chess game in their heads.22Close

Of Newman’s contacts with the Vienna mathematicians all that tangibly remains is a welcoming letter of July 1922 from Wilhelm Wirtinger.23Close^,24Close Yet it seems clear that Newman’s experience of Viennese mathematics was decisive in changing the direction of his researches. His principal mathematical interest was to become topology, which was not a speciality of British mathematics. Some of Wirtinger’s own work in Vienna related to the topology of surfaces, and in 1922 the University of Vienna recruited Kurt Reidemeister, who, like Newman himself, went on to become a specialist in combinatorial topology. Most notably, Hans Hahn, later a leading member of the famous ‘Vienna Circle’, ran a preparatory seminar on ‘Algebra and logic’ while Newman was in Vienna.25Close^,26Close Hahn was not only a specialist in the topology of curves and in real-variable mathematical analysis but also regarded formal logic as an important topic, both for research and teaching. In later years, Hahn held two full seminar courses on Whitehead and Russell’s Principia Mathematica, one of the earliest major publications in modern mathematical logic. Russell’s approach to philosophy and logic strongly influenced the Vienna Circle. Hahn also supervised the young Kurt Gödel, a doctoral student in Vienna.

Newman became a pioneer of topology in Britain, with a serious interest in logic and logic education, and also in Russell’s philosophy. One surely sees strong Viennese influence here, especially from Hans Hahn.

Newman applied for a College Fellowship at St John’s in 1923, a year that saw him publishing in the traditional area of mathematical analysis (in particular, on avoiding the axiom of choice in the theory of functions of real variables), and also writing a long unpublished essay in the philosophy of science, entitled ‘The foundations of mathematics from the standpoint of physics’. This could well have originated in a Viennese conversation. Maybe he even wrote some of it in Vienna.27Close In the essay, Newman contrasted the world of idealized objects customarily adopted in applied mathematics (smooth bodies, light strings, and so on) with the world ‘of real physical objects’. He distinguished the two worlds by the different logics that they used. The idealizers would draw on classical two-valued logic, for which he cited a recent metamathematical paper by Hilbert as a source,28Close but those interested in real life, he said, would go to constructive logic, on which he cited recent papers by Brouwer and Hermann Weyl.29Close This readiness to embrace logical pluralism and to put logics at the centre of his analysis of a problem was most unusual for a mathematician, and far more a product of Vienna than Cambridge.30Close^,31Close

An occasion for Newman to exercise his logical and philosophical talents arose when he attended a series of philosophical lectures that Russell gave in Trinity College Cambridge in 1926. These lectures were the basis for Russell’s book The Analysis of Matter.32Close Newman helped Russell to write two chapters, and, when the book appeared in 1928, he criticized its philosophical basis most acutely.33Close^,34Close Russell accepted the criticisms, which stimulated Newman to write Russell two long letters on logic and on topology, featuring some ideas from his 1923 philosophical essay.35Close Newman continued to pioneer both topology and logic at Cambridge and, doubtless with topology in mind, Hardy successfully proposed Newman as a Fellow of the Royal Society, with J. E. Littlewood as seconder—even though Newman was no analyst in the Hardy–Littlewood tradition.

Newman used the Royal Society to support the cause of logic. In 1950, he proposed Turing as a Fellow, seconded by Russell, and Turing was duly elected in 1951. Just five years later, in 1955, Newman wrote Turing’s Royal Society obituary.36Close In 1966 Newman proposed Gödel as a Foreign Member, and again Russell seconded; Gödel was elected to the society two years later.37Close When Russell died in 1970, Newman agreed to write the society’s obituary of Russell, together with the philosopher Freddie Ayer, but failing health prevented him from fulfilling his obligation. He died in 1984.

I conclude with some reflections about this very unusual history. As a result of Penrose’s early interest in mathematical logic, and the unusual mixture of mathematics, logic, and philosophy in Vienna, Newman changed direction: had he stayed in Cambridge in 1922–23, he would surely have continued on the path indicated by his 1923 paper about avoiding axioms of choice—namely, Hardy–Littlewood mathematical analysis. Even then his interest in logic could have waned and he might not have set up a foundations course to be taken by Turing. The existence of this course in 1935 was a highly improbable event, in that Newman might easily not have accompanied Penrose to Vienna, and might never have had an inclination to teach logic. After Turing’s graduation in 1934 he was himself another budding Hardy–Littlewood mathematical analyst, working on ‘almost periodic’ functions and with interests in mathematical statistics and group theory. One can imagine Turing remaining happily engaged in these branches of mathematics—particularly in analysis, with the influential Hardy back at Cambridge from 1931 after twelve years at Oxford. Turing would not have come across recursive functions or undecidability, and would not have invented his universal machine.

Could Newman and Turing have come to these topics by another route—via some of the Cambridge philosophers, perhaps? Frank Ramsey, who visited the philosopher Ludwig Wittgenstein in Austria in 1923 and 1924, and died in 1930, was along with other Cambridge philosophers largely concerned with revising the logicism of Russell and the early Wittgenstein.38Close Wittgenstein himself, back in Cambridge from 1929, was philosophically a monist, and so distinguished between what can be said and what can only be shown. By contrast, the Hilbertians depended centrally on the concept of a ‘hierarchy’, mathematics and metamathematics, upon which Turing seized. (The logician Rudolph Carnap, another member of the Vienna Circle, coined the term ‘metalogic’ in 1931, on the basis of reflecting upon Gödel’s procedures in his proof of the incompleteness of arithmetic, and Tarski was starting to speak of ‘metalanguage’ at about the same time.) Russell was in a strange position on this issue. Back in 1921, in his introduction to Wittgenstein’s Tractatus, he had advocated a hierarchy of languages to replace the Wittgensteinian showing–saying distinction—a move that Wittgenstein rejected—but he never envisaged a companion hierarchy of logics, and so neither understood (nor even stated) Gödel’s theorems properly.39Close It thus seems highly unlikely that either Turing’s contacts with Russell or the contacts over logic and logicism that he had with Wittgenstein would have led him to metamathematics and decision problems.

If neither Turing nor Newman had found their way to those crucial topics in logic and metamathematics, then—while they might have been recognized as clever analysts who were also good at chess—they would not have been such obvious choices for Bletchley Park. If they had got there, they would probably not have been as effective as the actual Newman and Turing were, because they would not have known much, if any, of the key mathematics. So a crucial part of the British wartime decoding effort, especially Turing’s Enigma-breaking bombe, came about as a consequence of Turing’s change of direction years earlier, which was inspired by the happenstance of Newman’s course on the foundations of mathematics, offered because of his own earlier change of direction towards this unusual topic, a change in turn brought about as an unintended consequence of some decisions that Penrose took to develop his own career. In short, what a fluke!40Close