The Life, Death and Miracles of Alan Mathison Turing Settimo Termini The life of Alan Turing is described in many biographies. The best and most encyclopaedic of these is that of Andrew Hodges; quite pleasant is the agile volume by Gianni Rigamonti, Turing, il genio e lo scandalo (Flaccovio editore, Palermo, 1991). Both of these also make mention of his tragic end, which certainly casts a shadow on the mores English society at the time; but of course, who knows how other societies might have behaved? And the miracles? Well, yes, he worked those too, or at least if we want to give credit to Kurt Gödel he worked hand in glove with the other logicians of the first half of the twentieth century, and as far as we know even a single miracle is enough to merit beautification. In the introduction to one of his books on the theory of computer science, Martin Davis wrote that: It is truly remarkable (Gödel...speaks of a kind of miracle) that it has proved possible to give a precise mathematical characterization of the class of processes that can be carried out by purely mechanical means. It is in fact the possibility of such a characterization that underlies the ubiquitous applicability of digital computers. Davis a great American logician and computer scientist to whom are owing many important results, making it possible for Yuri Matiyasevich to take the final step towards solving Hilbert s tenth problem was a student of Emil Post, another great mathematician and logician who (between one depression and the next) contributed to the development of the admirable symphony that is the theory of computability. Post, like Turing, had some very unhappy moments, but perhaps we just can t understand how the creation of such high constructions can make any kind of difficult moment fade into the background. But what part does Turing play in any of this? He plays a part because he was one of the creators and founders of this theory, along with Alonzo Church, Stephen Cole Kleene and Gödel. Of all of these, Turing was the one who, from the very beginning, most firmly believed in the generality of this theory and its revolutionary C. Bartocci et al. (eds.), Mathematical Lives, DOI 10.1007/978-3-642-13606-1_13, # Springer-Verlag Berlin Heidelberg 2011 91
92 S. Termini importance. By analysing the way in which a human being proceeds when he has to perform an arbitrary computation, he extracted some basic, essential elements and, idealising them, created an abstract model of a machine, called the Turing machine. Not content with this, he also stated a thesis, known as the Church-Turing thesis, which states that any function that is intuitively computable that is, a function such that we have the impression or the conviction of its being solvable in one way or another, using whatever ideas or techniques that spring to mind at the moment is also computable with a Turin machine. In that very same period, Alonzo Church had proposed a different model of calculus (the so-called lambda calculus), which was more formal and less intuitive. Turing, in an appendix to his article, proved that the two models were equivalent. Gödel, besides being timid and introverted, was also more cautious than the other members of this bunch. When they began to talk to him about these things and of the possibility of constructing a theory that would grasp the intuitive notion of computable in a completely general way, he was sceptical. In those years, logic had taken many steps forward, some quite disconcerting. Of some of those steps crucial ones he himself had been the greatest and sole artificer. But the results were always tied to a particular formalism, to a specific formal system. This had been the case with the notion of definable and with that of provable. Why was it necessarily different for that of computable? But reflective and honest as he always was, Gödel thought and rethought and in the end, he became convinced that the opposite was true. Once convinced, he was the one who most forcefully underlined the importance of these results every time he returned to the subject.
The Life, Death and Miracles of Alan Mathison Turing 93 In 1946 Gödel wrote: This importance [of Turing s computability] is largely due to the fact that with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e. one not depending on the formalism chosen....for the concept of computability, however, although it is merely a special kind of demonstrability or decidability, the situation is different. By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion. Here is the miracle that we can present to the petitioner in the case for beatification. It is worthwhile noting that Gödel, in presenting his results, had never before spoken of a miracle. Again, in 1965 he wrote: The concept of computable is in a certain sense absolute, while practically all other familiar meta-mathematical concepts (e.g. provable, definable, etc.) depend quite essentially on the system with respect to which they are defined. The miracle, then, consists in having formulated a theory that catches hold in an integral way of an intuitive notion that of computable as well as in the fact that in this theory we are also able to demonstrate several interesting things. Let s mention only two. One positive result and one negative. The positive result tells us that there exists a UNIVERSAL Turing machine, that is, a unique model capable of doing the work of any other particular or specific Turing machine. This is what we are accustomed to today: any computer whatsoever, even our own laptop that weighs less than 2 kg, can do everything (everything, that is, that can be done on a computer; let s not exaggerate!). It s not that my computer can do, in principle, different things than what my friend s computer can do, not taking into consideration concrete limitations such as memory and the like. That is, our computers are in certain sense universal Turing machines. Now we can better understand what Martin Davis meant in the passage quoted earlier. The negative result tells us that there are problems that are undecidable, that is roughly speaking there exist well posed questions to which it is not possible to give algorithmic answers. One example is given by the theory known as the halting problem : there is no algorithm that can tell us whether a generic program which we have given input certain initial values for the variables will sooner or later finish running, providing us with the result of the finished computation, or if it will continue to run forever (as might happen if we ask the computer to give us a value for a function at a point where it is not defined). An example that is mathematically filled with this kind of problem is Hilbert s tenth problem. The theory of computability came out of Turing s head (and other great heads like his). One strange coincidence is that everyone came together in order to find these results independently in the mid-1930s (the works appeared in 1936). With respect to his companions, Turing did something more. We have already mentioned that his model his machine could be visualised, in contrast to other proposals, though these were mathematically equivalent. We have also mentioned the determination with which he sustained his ideas. During the war, he was successful in deciphering the secret codes of the German navy and, after the war, he concerned
94 S. Termini himself with constructing computers while contemporaneously delineating the fundamental mathematical elements of a theory of morphogenesis. In 1950 he then wrote an article provocatively entitled Computing Machinery and Intelligence for the English philosophy journal Mind. He jokingly described what computers would be able to do, introducing a game (the imitation game) as an empirical test to establish a machine s intelligence. The day when we can no longer distinguish from the answers given by a human being and a machine which is the human and which the machine, will be the day that machines have achieved an acceptable level of intelligence (Fig. 1). In a word, in his free time, he also invented artificial intelligence, 5 years before its name was invented. His student, Robin Gandy, who passed away some years ago, recalled that Turing had a lot of fun writing this article, and roared with laughter as he read bits of it to him. More signs of his greatness the ability to laugh even about his own work, and to have fun while doing important things absolute greatness. Gandy, in reconstructing the birth of the theory of computability, noted that the existence of a profound theory is helpful for the development of the technologies related to it. This was the case with electricity, which was based on Maxwell s theory. This has also been the case for computer science, which was based on the theory of computability. But this was not the case for internal-combustion engines, which contributed to the development of thermodynamics instead of finding it already ready and waiting. It is no coincidence that they developed much more slowly. Up to now we have been lucky with computer science, but the new developments, Internet and distributed systems, have not had a true theory to base themselves on. For the essential problems in these sectors, the theory of computability as a point of reference is too remote or generic to play a significant role. If we want the continued development of our technologies to be carried out quickly, as it has been so far (and not totally alien, as technological development not based on general and profound theory threatens to be), we would do well to invest in fundamental research, inviting everyone to reflect on the important fundamental problems, hoping that sooner or later one of Turing s heirs will lend a hand in providing us with a theoretical point of reference for what is happening. Who Is A. M. Turing Alan Mathison Turing was born near London on 23 June 1912. Son of an officer in the Indian Civil Service who spent long periods abroad with his wife, Alan was fostered by family friends and attended English Public School, showing talent and a specific interest in following his own ideas, independent of the teaching he received. In spite of this (or perhaps thanks to it) Turing won every single scholastic competition in mathematics. His first interests and extra-curricular readings concerned Einstein s articles on relativity, then recently published, and the newborn field of quantum mechanics. In 1931 he won a scholarship and entered King s College in Cambridge, where he
The Life, Death and Miracles of Alan Mathison Turing 95 Fig. 1 The Enigma machine
96 S. Termini turned his attention to logic and the philosophy of mathematics, under the influence of Bertrand Russell. He was a sympathiser with the pacifist movement, but he never joined an organisation. In 1934 he completed his studies and the next year attended an advanced course in the foundations of mathematics taught by Max Newman, with whom he was to remain in contact. During the course he came to know Gödels s theory of incompleteness and Hilbert s problems of decidability, and began to work on his own original approach to them. He became a fellow of King s College in 1935 with a thesis on the calculus of probability, but he also continued to work on decidability. In 1936 he published the fundamental article On Computable Numbers with an Application to the Entscheidungsproblem, where he introduced an ideal machine (today called the Turing machine) which formalised the intuitive idea of algorithm beginning with the elementary operations that are characteristic of all calculations. His work in decidability brought him into contact with Alonzo Church, who at that time was working on the same problems. From 1936 to 1938 Turing studied under Church at the Institute for Advanced Study in Princeton. Back in England, he was invited to Bletchley Park by the Government Code and Cipher School (GCCS) to participate in the project of deciphering the German Enigma code. Here he was able to put to use all of his skills in logic and statistics joined with his talent for constructing computer machinery. The result was a remarkable contribution to the construction of some bombes, electromechanical devices for calculating, named for their characteristic ticking. As early as 1941 they were able to decipher the secret messages sent by the German navy. In 1945 Turing received the OBE for his wartime service. After the war, he took part in a project to construct a computer for the National Physical Laboratory, and returned to academic life in Cambridge and to mathematics. In 1948 he was invited by his former teacher Newman to transfer to the University of Manchester. In 1950 he published another memorable article, Computing Machinery and Intelligence in the journal Mind, introducing the topic of artificial intelligence. He was also something of an athlete. He participated in marathons and decathlons, achieving world-class standards. He was elected a fellow of London s Royal Society in 1951, mainly for his work on decidability of 1936, but his curiosity also drove him to investigate the mathematical structures in biology. In 1952 he published a study on the evolution of living organisms. In the meantime, this being the Cold War period, he had secretly begun to work again for the GCCS. In 1952 he was convicted of gross indecency for homosexual acts, and as an alternative to prison was sentenced to undergo oestrogen treatment. Because of the conviction, he lost his security clearance and could no longer work on deciphering codes. He and his colleagues and scientific correspondents, both British and foreign, were kept under constant surveillance. Turing died on 7 June 1954, apparently from eating an apple containing cyanide. The conclusion of the official inquest was that he had committed suicide, but his mother always claimed his death an accident due to carelessness while conducting chemical experiments.