#### Commentary by M. Atiyah

On numerous occasions I have given “general” talks and some of these have been published. The degree of generality has varied, depending on the nature of the audience. In some cases I was talking to professional mathematicians but in others I was almost the only mathematician in the room. Giving such general talks (and, even more, writing them up!) is hard work. It requires much greater thought on the material and presentation than for a normal seminar, but it is a worthwhile and important activity.

Paper [1] was a lecture given at the centenary meeting of the London Mathematical Society, and it gave me an opportunity of explaining my views on algebraic topology. Not having had an orthodox training as a topologist I tended to see topology as a powerful tool that should be more widely appreciated and used. My lecture was a plea for this point of view.

In 1968 I received an honorary degree from the University of Bonn on the occasion of its 150th anniversary. While I was of course delighted by the honour, particularly because of my close relation with
Hirzebruch
and his *Arbeitstagung*, I was less than pleased when I discovered that I had to deliver a lecture to the entire Science Faculty (plus spouses). I did my best to explain in very broad terms why general ideas like symmetry, continuity, and probability were now so prominent in mathematics. The lecture (delivered in English) was in sure course translated into German, suitably embellished with illustrations and, at Hirzebruch’s suggestion, published in a glossy popular science magazine
[2].

My lecture [4] at the IMA meeting provided an occasion for an expression of my philosophical views on the styles of mathematical research. The text is essentially the transcript of a tape recording, which explains its rather loose and verbose form. It contains nothing particularly profound or provocative.

In 1975 I was asked to deliver the Bakerian lecture [8] of the Royal Society. Looking back through the records I failed to find any pure mathematician among my predecessors, and this only confirmed my view of the difficult task a mathematician faces when addressing a really general scientific audience. Clearly one could not talk down patronizingly to such distinguished scientists, and the talk should also have some intellectual weight. On the other hand most scientists, particularly on the biological side, would know only rather basic classical mathematics. How was I to cope with this dilemma? After much thought I decided to focus on algebraic equations, as being familiar, and to talk about general ideas of structure, culminating in the Weil conjectures. It was such a difficult task that I felt compelled to write the talk verbatim in advance and then to read it — the only occasion when I have gone to this length of preparation.

Another Royal Society publication, but of quite a different kind, was my obituary [6] for Hodge. As his student and disciple, this was a task I readily undertook. I had no problem dealing with the mathematical side and fortunately Hodge had left extensive notes on the personal side which I relied on heavily. A few years before, on the occasion of Hodge’s seventieth birthday conference, I had given a lecture trying to summarize Hodge’s mathematical contribution, put into its historical context. Hodge was in the audience and was kind enough to say afterwards that I had given a very accurate description of how his mathematical ideas had evolved. I hope the same is true of the obituary.

In 1975 the IMA organized a meeting in Cambridge to celebrate Littlewood’s 90th birthday, and [7] is again the transcript of a tape recording. The lack of polish will be immediately apparent to the reader. I was supposed to talk about something “exciting”, so I decided to discuss the Bernstein polynomial and related questions. I was by no means an expert on this subject, though it was related to my earlier paper [e1], but I found it a very fascinating area and had tried to follow developments. I had first met the Bernstein polynomial in Gelfand’s house in Moscow. Gelfand introduced Bernstein to me and told him to explain his results. However, Bernstein was young and bashful, so his slow presentation exasperated Gelfand, who then interrupted and carried on the explanation himself with great verve. No wonder I associated the topic with “excitement”!

At the ICME Congress in Karlsruhe in 1976 I was invited to give a plenary address [9]. The audience consisted largely of school teachers and others interested in mathematical education. This was clearly an occasion for a very broad-brush presentation, and I tried to put the development of modern mathematics into some sort of historical perspective. There had been a great deal of nonsense talked (and implemented!) about modern mathematics in schools, and I thought I should attempt to clear the air.

On my return from Princeton in 1973 I joined the Council of London Mathematical Society and from 1974 to 1976 I was President. The final duty of the President is the presidential address, and this is clearly a rather special and challenging occasion. In my case the challenge was enhanced by the presence in the audience of my wife and two sons, all mathematically trained! I decided to illustrate some important and deep mathematical developments by using very simple elementary examples. The published form [11] of my presidential address is essentially the same as the spoken version. This is a rule I generally follow. I resist the temptation to enlarge or improve the published version of a talk, in the belief that the informal style of a lecture has much to commend it. Too often mathematics in print is heavy, formal, and pedantic, making impossible demands on the reader.

The Mathematical Association has a long tradition of choosing its presidents alternately from the universities and the schools. In 1981–82 I became president, and for some time afterwards my educational involvement increased, culminating in membership of the Cockcroft Committee. This was a good example of the snowball route to becoming an “expert” — each committee adding a layer to one’s credentials and hence leading to the next committee. Presidents, in their presidential addresses, frequently speak on educational policy and usually castigate the Government of the time. Not feeling an authority on these matters I chose instead, in my address [13], to make a plea for geometry. All my research has had a pronounced geometrical flavor, and I tried to explain my view of geometry in terms that would be intelligible to school teachers, and might be relevant in the classroom.

The record [16] of my interview with Minio is hardly one of my “works”, but it expressed my views on many mathematical topics in a spontaneous and uninhibited way. To have omitted it would have looked like censorship! Of course, some of my off-the-cuff remarks are a bit outrageous, and outrage has been duly registered particularly for my remarks about the classification of finite simple groups. While I do not wish to retract anything I said, I could have been more diplomatic and a written article would have been more balanced. Perhaps I should add here my belief that the most important outcome of the search for classification was the discovery of the Monster group. This is clearly a mysterious object having deep connections with important topics such as modular forms, and remains to be properly understood.

In my interview with Minio I had commented with approval on the absence of Nobel Prizes in mathematics. Ironically a short while later I heard that I had been awarded the Feltrinelli Prize by the Italian Academy (the “Lincei”). While lacking the fame or notoriety of the Nobel Prizes it was comparable financially and my mathematical predecessors were Hadamard, Lefschetz, and Leray: I was in excellent company. The presentations were to be made in Rome by the President of the Republic, and I was asked to prepare a speech of about 25 minutes on my work. This was clearly a serious task and I gave the matter much thought. Presumably I was supposed to survey my mathematical work in a way that would not be entirely incomprehensible to the distinguished audience at the Italian Academy. Eventually, with some misgiving, I produced [26], and set off with my wife for the ceremony in Rome. Fortunately, Rome is not Stockholm and the Italians have a different way of doing things. There was no lack of ceremony with imposing guardsmen in uniform and a beautifully furnished salon. Moreover, President Pertini duly turned up. However, as the proceedings progressed, time was running short, the President had other obligations to attend to and I was asked to compress my carefully prepared speech into 10 minutes! I hastily decided which passages to select. I cannot now remember what I omitted but no doubt I left out all the more mathematical bits. In a way it was a great relief. So, although my speech was essentially not given, the written version now serves a different purpose by providing a brief overview of the papers collected here.

In October 1983 the Institut des Hautes Études Scientifiques celebrated its 25th anniversary. As someone who had been closely involved with its development I attended the celebrations and gave one of three scientific lectures (I narrowly escaped having to give one of the formal ministerial-type addresses!). I chose to survey the current interaction between geometry and analysis [25]. I justified my choice partly by the fact that the three Fields Medallists at the Warsaw Congress (Connes, Thurston, Yau) all worked in this area. I also referred to the work of Donaldson and Freedman: it did not take much foresight to guess that they would figure amongst Fields Medallists at the next Congress.

George Orwell’s 1984 made such an impact on our time that the eventual arrival of that year was greeted with some foreboding. I was persuaded (against my better judgement) to take part in a public meeting in Locarno, where a number of speakers from different fields were asked to describe the challenge of 1984. I found myself in unusual and distinguished company. It was indeed a pleasure to make the acquaintance of Sir Karl Popper, the well-known philosopher, and to meet Sir John Eccles, who had shared the Nobel Prize for Physiology with Hodgkin and Huxley. This was however severe competition, from the point of view of the general public. Although I had prepared a careful address [22], [24] on the computer revolution, as seen by a mathematician, I doubt if this is what the audience wanted. The proceedings were sponsored by, and eventually published in, an Italian magazine. They reached a perhaps more receptive and appropriate audience when Kahane, President of ICMI, suggested that the article be presented to a conference on computers and mathematical equation.

Another but more serious occasion when I had to address a non-specialist audience took place at Colmar in 1983, at a small meeting organized by the European Science Foundation. This consisted of representatives of different disciplines, from biology to art history, trying to describe the nature of progress in their field: what are the criteria used to assess “progress”? It was a serious and interesting meeting with high-level participation. Papers were written and circulated in advance, with subsequent commentary and discussion at the meeting, so I did not actually deliver my own contribution [21]. My analysis of the situation in mathematics repeats many of the points I had made in previous articles, but it represents perhaps the most carefully thought-out version of my views on mathematics and its place in society.