## Extracts from Michael Atiyah's popular papers

Below we list eight papers written by Michael Atiyah and give a brief extract from each of them. We encourage the reader to consult these very interesting papers and read the complete texts. We list the papers in chronological order:

**1.**What is geometry? The 1982 Presidential address,

*The Mathematical Gazette*

**66**(437) (1982), 179-184.

Of all the changes that have taken place in the mathematical curriculum, both in schools and universities, nothing is more striking than the decline in the central role of geometry. Euclidean geometry, together with the allied subject of projective geometry, has been dethroned and in some places almost banished from the scene. While educational reform was certainly needed there is always the danger that the pendulum may swing too far the other way and that insufficient attention may be paid to geometry in its various forms. Much of the difficulty here centres round the elusive nature of the subject: What is geometry? ... Broadly speaking I want to suggest that geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words "insight" versus "rigour" and both play an essential role in real mathematical problems. The educational implications of this are clear. We should aim to cultivate and develop both modes of thought. It is a mistake to overemphasise one at the expense of the other and I suspect that geometry has been suffering in recent years. The exact balance is naturally a subject for detailed debate and must depend on the level and ability of the students involved. The main point I have tried to get across is that geometry is not so much a branch of mathematics as a way of thinking that permeates all branches

**2.**Address of the president, Sir Michael Atiyah, given at the anniversary meeting on 29 November 1991,

*Notes and Records Roy. Soc. London*

**46**(1) (1992), 155-169.

As a mathematician, I am conscious of my somewhat singular position in being President of the Royal Society. A number of my illustrious predecessors were mathematicians, but the practice seemed to die out about 100 years ago with Sir George Stokes. I take my presence here today to be a reaffirmation of the basic role mathematics plays in the sciences, although this role is a constantly changing one. In addition to its traditional and intimate relation to all the physical sciences - the A-side in our jargon - it is increasingly finding application on the B-side. The recent award of the prestigious Balzan Prize to John Maynard Smith reminds us that mathematical models have taught us something about the process of natural selection. The spread of epidemics lends itself to sophisticated mathematical analysis. Problems of vision and visual interpretation involve several areas of science, but there is certainly an important mathematical component. In all these fields old mathematical ideas are finding new application and new branches of mathematics are being created to attack new scientific problems. We all know that the traditional barriers between scientific disciplines have been rapidly breaking down, and this trend tends to emphasise the unifying role of mathematics. Perhaps one of these days the Royal Society will even abolish its division into an A-side and a B-side! But mathematics is not restricted to the natural sciences. It plays an increasingly important part in the social sciences, particularly in economics. The younger generation of economists tend to be skilled in mathematics, and a young mathematician who coverts a Nobel Prize could do worse than move into the field of economics. In a different direction mathematics has a traditional link with logic and philosophy, a link which has acquired greater importance through the growth of computer science.

**3.**Mathematics: Queen and servant of the Sciences,

*Proc. Amer. Philos. Soc.*

**137**(4) (1993), 527-531.

Just as primitive thought (e.g., in animals) is non-verbal, so primitive science (e.g., associating heat and light) is non-mathematical. Moreover, the mind has produced several types of language in which to express itself, music being one example. At present mathematics is by its depth and scope the pre-eminent language of science, but it remains to be seen whether other types of language will be needed. If we accept this analogy then we can begin to understand how mathematics has a creative/aesthetic side, like poetry, where the imagination is being stretched, and a more utilitarian side, as used by engineers in routine calculations. Also mathematics has its own internal analysis, like linguistics and philology. Finally, mathematics is constantly being enlarged by the addition of new concepts in response to the advancing needs of science. This may be compared with the growth and development of language, having to deal with the needs of sophisticated modern society. As with language, where thought and word interact with one another, so science and mathematics interact with each other. It is difficult to separate contents and framework: each influences the other in a complex symbiosis. It is for this reason that I have no difficulty in describing mathematics as the language of science. Some of my colleagues might feel that this gives mathematics too humble a status, that of the "servant," and they would prefer the loftier position of the "queen" from whom all authority and beauty emanates. But if we reflect on the power of words, and the role they play in organizing, refining, and transmitting ideas, then we see that the role is an honourable one. Ideas without words remain vague and ineffective, and science without mathematics remains similarly handicapped.

**4.**Address of the President, Sir Michael Atiyah, O. M., given at the anniversary meeting on 30 November 1994,

*Notes and Records Roy. Soc. London*

**49**(1) (1995), 141-151.

In thinking of mathematics as the language of science, it is perhaps helpful to reflect on its main characteristics. How and why does it work? In the first place mathematics develops by a process of abstraction. In every scientific model one simplifies, ignoring what one hopes are irrelevant or minor factors, to concentrate on the main features. Mathematics takes the process to its ultimate conclusion: the identity of the players is ignored, only their mutual relations are studied. It is this abstraction that makes mathematics such a universal language: it is not tied to any particular interpretation. A wave (something which oscillates or vibrates) is a good example of such an abstract and universal concept. The second characteristic of mathematics is that it is entirely open-ended. It is very hard to define or confine mathematics. At one time one might have defined it as the study of quantities and their mutual relation, but modern mathematics abounds in non-quantitative branches such as topology or group theory (the abstract study of symmetry). A wider definition is that mathematics concerns itself with patterns or order, and this makes it clear why it is relevant to science. But the study of disorder and chaos, which grapples with random behaviour, is also an important branch of mathematics. The truth is that mathematics keeps developing in different directions as the result of the needs of potential applications. Each scientific development is likely to require a new theoretical framework and, if classical mathematics does not already provide the right language, a new one has to be worked out. This is likely to have sufficient links with parts of existing mathematics for it to form a natural extension and to be absorbed in due course into the total mathematical corpus. In fact mathematics progresses in two ways: either by broadening itself in response to external needs or by deepening itself through internal analysis. Finally and perhaps most remarkably there is the longevity or permanence of mathematics. Mathematical theories in principle persist indefinitely, although they may frequently be absorbed in larger ones. ... Frequently mathematical results find unexpected applications decades or centuries later, demonstrating vividly that the distinction between basic and applied research in mathematics can be ephemeral.

**5.**Geometry and Physics,

*The Mathematical Gazette*

**80**(487) (1996), 78-82.

[I will] discuss the interrelation of Geometry and Physics. There are two very good reasons for doing this. One is historical and arises from the close ties between the two subjects in their early evolution. A second and more topical reason is that, over the past two decades, there has been a remarkable burst of interaction of a quite unexpected kind between Geometry and Physics. ... Over the past two decades ideas from quantum physics have led directly to remarkable new mathematical discoveries across a very wide range of problems in Geometry. Usually the physical theories are 'formal' in the sense that they are not yet in rigorous mathematical form. Mathematicians have therefore to produce proofs based on alternative ideas and techniques. However, without the physical intuition and background the results in question would probably not have been discovered. The physics also provides an overall unifying conceptual framework, whereas the mathematics frequently degenerates into uninformative and varied techniques. The reverse benefits in which the physicists benefit from the mathematics are also present though more difficult to assess. In physics the ultimate test is whether the theory explains all the experimental data and that stage has not yet been reached. But what is certainly true is that a new dialogue has been set up between mathematicians and physicists and ideas are constantly flowing both ways. The future looks exciting.

**6.**Science for Evil: The Scientist's Dilemma,

*British Medical Journal*

**319**(7207) (1999), 448-449.

You are a scientist You see your science being put to evil purposes, with disaster looming ahead. What do you do? This was the dilemma that confronted nuclear physicists in 1945 after the atomic bomb had been exploded over Japan. The search to understand the ultimate nature of matter, driven by intellectual curiosity and carried out in an abstruse mathematical frame work, had produced the ultimate weapon of war. Whether the development and use of these first atomic bombs was morally justified is today deeply disputed, but the problem that confronted the scientists at the end of the war was how to control the genie that had escaped from the bottle. They had produced a weapon that threatened the future of humankind. Their collective responsibility (or guilt) was unambiguous, but what should they do? This was the genesis of the Pugwash organisation, an informal group of scientists who saw it as their responsibility to prevent the catastrophe of nuclear war in the future. From the beginning they were international, including, crucially, scientists from both sides of the Iron Curtain; they were experts, in both scientific and military spheres; and they conducted their discussions out of the limelight The aim was to bring calm academic thinking to the complex scientific, military, and political issues involved. ... As we face the next century, in which science will inevitably move to new discoveries and potential applications, it is essential that the scientific community regains the trust of the public. This can come only from a policy of openness and humility. We must tell the public that science is full of uncertainties, that the uncertainties are greatest on the frontiers of scientific advance, and that decisions have to be made on the best evidence available, not on a mythical certainty. But the public will want to involve itself in these decisions, a process in which both the media and the political establishment must play their part.

**7.**Mathematics in the 20th century,

*Amer. Math. Monthly*

**108**(7) (2001), 654-666.

If you talk about the end of one century and the beginning of the next you have two choices, both of them difficult. One is to survey the mathematics over the past hundred years; the other is to predict the mathematics of the next hundred years. I have chosen the more difficult task. Everybody can predict and we will not be around to find out whether we were wrong. But giving an impression of the past is something that everybody can disagree with. All I can do is give you a personal view. It is impossible to cover everything, and in particular I will leave out significant parts of the story, partly because I am not an expert, and partly because they are covered elsewhere. I will say nothing, for example, about the great events in the area between logic and computing associated with the names of people like Hilbert, Gödel, and Turing. Nor will I say much about the applications of mathematics, except in fundamental physics, because they are so numerous and they need such special treatment. Each would require a lecture to itself. Moreover, there is no point in trying to give just a list of theorems or even a list of famous mathematicians over the last hundred years. That would be rather a dull exercise. So instead I am going to try and pick out some themes that I think run across the board in many ways and underline what has happened. Let me first make a general remark. Centuries are crude numbers. We do not really believe that after a hundred years something suddenly stops and starts again. So when I describe the mathematics of the 20th century, I am going to be rather cavalier about dates. If something started in the 1890s and moved into the 1900s, I shall ignore such detail. I will behave like an astronomer and work in rather approximate numbers. In fact, many things started in the 19th century and only came to fruition in the 20th century. One of the difficulties of this exercise is that it is very hard to put oneself back in the position of what it was like in 1900 to be a mathematician, because so much of the mathematics of the last century has been absorbed by our culture, by us. It is very hard to imagine a time when people did not think in our terms. In fact, if you make a really important discovery in mathematics you will get omitted altogether! You simply get absorbed into the background. So going back, you have to try to imagine what it was like in a different era when people did not think in our way. ... Let me look at the history in a nutshell: what has happened to mathematics? I will rather glibly just put the 18th and 19th centuries together, as the era of what you might call classical mathematics, the era we associate with Euler and Gauss, where all the great classical mathematics was worked out and developed. You might have thought that would almost be the end of mathematics, but the 20th century has, on the contrary, been very productive indeed and this is what I have been talking about. The 20th century can be divided roughly into two halves. I would think the first half has been dominated by what I call the "era of specialization", the era in which Hilbert's approach, of trying to formalize things and define them carefully and then follow through on what you can do in each field, was very influential. As I said, Bourbaki's name is associated with this trend, where people focused attention on what you could get within particular algebraic or other systems at a given time. The second half of the 20th century has been much more what I would call the "era of unification", where borders are crossed, techniques have been moved from one field into the other, and things have become hybridized to an enormous extent. I think this is an oversimplification, but I think it does briefly summarize some of the aspects that you can see in 20th-century mathematics.

**8.**Advice to a Young Mathematician, in

*The Princeton Companion to Mathematics*(Princeton, 2008), 1000-1004.

A research mathematician, like a creative artist, has to be passionately interested in the subject and fully dedicated to it. Without strong internal motivation you cannot succeed, but if you enjoy mathematics the satisfaction you can get from solving hard problems is immense. The first year or two of research is the most difficult. There is so much to learn. One struggles unsuccessfully with small problems and one has serious doubts about one's ability to prove anything interesting. I went through such a period in my second year of research, and Jean-Pierre Serre, perhaps the outstanding mathematician of my generation, told me that he too had contemplated giving up at one stage. Only the mediocre are supremely confident of their ability. The better you are, the higher the standards you set yourself - you can see beyond your immediate reach. Many would-be mathematicians also have talents and interests in other directions and they may have a difficult choice to make between embarking on a mathematical career and pursuing something else. The great Gauss is reputed to have wavered between mathematics and philology, Pascal deserted mathematics at an early age for theology, while Descartes and Leibniz are also famous as philosophers. Some mathematicians move into physics (e.g., Freeman Dyson) while others (e.g., Harish Chandra, Raoul Bott) have moved the other way. You should not regard mathematics as a closed world, and the interaction between mathematics and other disciplines is healthy both for the individual and for society.