The invitation to the International Congress of Mathematicians to meet in Edinburgh in 1958 was sponsored by the City of Edinburgh, the University of Edinburgh, the Royal Society and the Royal Society of Edinburgh; it was conveyed to the Amsterdam Congress in September 1954 by Professor W V D Hodge, and was unanimously accepted. After informal discussions on the procedure to be adopted, the first (and only) meeting of the Congress Committee was held in Edinburgh on 29 April 1955; this Committee contained representatives of the four sponsoring bodies and of the London Mathematical Society, the Edinburgh Mathematical Society and the British National Committee for Mathematics. It was announced at this meeting that H.R.H. the Duke of Edinburgh had graciously consented to become Patron of the Congress.

The main Congress excursions were held on Sunday, 17 August. The most popular was the steamer cruise from Glasgow down the Clyde and round the island of Bute, ending at Gourock; this was a full-day excursion. The alternative was an afternoon excursion by coach to Loch Lubnaig and Loch Earn.

On the afternoon of Tuesday, 19 August, a wide variety of excursions was available to members of the Congress; scenic, historic, artistic and technological interests were all catered for. Sight-seeing bus tours of the City of Edinburgh were provided on two evenings during the Congress.

Special excursions were provided for associate members on three mornings; most of these were to various firms, factories and workshops, but there were also trips down the Royal Mile and to the Royal Botanic Garden, and visits to Hopetoun House and Lennoxlove. In addition, all associate members were invited by the Royal Zoological Society of Scotland and the Royal Society of Edinburgh to visit the Royal Scottish Zoological Gardens on the morning of Friday, 15 August.

The inaugural session of the Edinburgh Congress took place in the McEwan Hall on the morning of Thursday, 14 August 1958. The Right Honourable Ian Johnson-Gilbert, Lord Provost of the City of Edinburgh, and Chairman of the Congress Committee, presided over the meeting.

The Lord Provost opened the proceedings by welcoming the Congress on behalf of the City of Edinburgh. His address was followed by other speeches of welcome: these were given by Sir Edward V Appleton, Vice Chancellor and Principal of the University of Edinburgh, and Vice Chairman of the Congress Committee, on behalf of the University; by Sir David Brunt on behalf of the Royal Society; and by Professor N Feather on behalf of the Royal Society of Edinburgh.

My Lord Provost, Ladies and Gentlemen: Professor Schouten, President of the Amsterdam Congress of 1954, being prevented by reasons of health from coming to Edinburgh, to his and our deep regret, has asked me to transmit a message to you, a message which contains a proposal. I think the best thing I can do is to read you the letter he wrote to me on the 6th of August. I should only like to add the remark that Professor Schouten does not mention any motive for his proposal, presumably for the trivial reason that in the eyes of all of us such a mention would be superfluous. After having heard his letter you will all, I am sure, agree with Professor Schouten's views.

Epe, August 5, 1958

Dear Colleague,

It is the custom that the President of the last Congress proposes the name of the person to be elected by the Congress as its President.
As I cannot attend the Congress, I beg you to bring my best wishes for the Edinburgh Congress, and to propose Professor W V D Hodge as President.

Yours sincerely,
J A Schouten.

I am most grateful to Professor Koksma for the kind words he has used about me, and I am deeply honoured by the manner in which you have accepted me as your President.

When the idea of holding this Congress in Edinburgh was first mooted, it was the hope of all British mathematicians that we should have as our President Sir Edmund Whittaker, that great figure in the life of the City and University of Edinburgh, so much respected and loved in the mathematical world. But in March 1956 he passed away. I count it not the least of my claims to be President of this Congress that I was one of those fortunate enough to receive their first introduction to higher mathematics in Sir Edmund's classes.

The preparations for this Congress have been onerous, and I would first like to pay tribute to all who have helped to make it possible for us to bring our plans to maturity. It has been a most moving and pleasant experience to find that so many were willing to contribute both their time and their money to the enterprise. Our four hosts, the City of Edinburgh, the University of Edinburgh, the Royal Society, and the Royal Society of Edinburgh, have proved far from merely formal sponsors. Each in its own way has contributed practical help to an extent which cannot easily be measured, and the goodwill which they have shown to us throughout has been quite indispensable. Next, I should like to pay my tribute to all those individuals, mathematicians and others, throughout the country, who have laboured long for the success of the Congress. If I do not name them individually it is simply because the list is too long. I should also like to thank the various institutions to which these people are attached, who have so generously allowed them to use their facilities for the work of the Congress. And, finally, we are most grateful to the International Mathematical Union and the many learned societies, industrial organisations, and individuals who have contributed most generously to the cost of this enterprise.

At the Harvard Congress of 1950 Professor Veblen referred to the difficulties encountered by the organisers of International Congresses, caused by the ever increasing number of people professionally engaged in the study of mathematics, and at Amsterdam in 1954 Professor Schouten spoke of the same problem. As you can well imagine, the organisers of the present Congress have had to face this problem once again. I should like to take up a little of your time by giving my own personal reflexions on this matter.

The International Congresses of Mathematicians, which are held every four years serve a number of purposes. The most important is to get together the leaders in all branches of mathematics so that they may discuss their common problems and exchange ideas on them. In saying this, I wish to emphasise the phrase "all branches of mathematics". In recent years there has been a steady growth in the number of symposia held, many with the support of the International Mathematical Union. These symposia have done excellent work in advancing research in special fields. But this is not enough. It is essential for the well-being of mathematics that there should be periodic gatherings attended by representatives of all branches of the subject, and this for several reasons: in my personal opinion, the most important reason is that gatherings such as this serve as an invaluable safeguard against the dangers of excessive specialisation.

The problem of specialisation is a difficult one. Mathematics is now so vast that few can hope to cover the whole range, and much of our progress has been due to the efforts of men and women who have devoted their lives to work in a narrow field of research. Most of us must continue to work in specialised fields, and with good fortune we can make our contribution to mathematics as a whole in this way. But there are dangers in this. There is always the risk that we may come to regard our own special problems as all-important; and to regard mathematics simply as a system of conclusions drawn from definitions and postulates that must be consistent, but otherwise may be created at the free will of the mathematician. As Professor Courant has justly remarked: "If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms without motive or goal. ... Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can the free mind achieve results of scientific value."

I believe that mathematicians are now much less likely to fall into this danger than they were some time ago. But over-specialisation also produces a practical difficulty. As we all know from our own experience, in order to make progress in our own field we must know what is going on in other fields, and what new techniques are being developed elsewhere in mathematics. The problem we are faced with is simply that of maintaining contact with all the main developments going on in mathematics while working intensively in our own specialised field. Some solution of this problem is essential, and International Congresses can go a long way towards giving the required answer. These Congresses provide an opportunity for periodic stocktaking, and the opportunities they provide for surveying the whole field of mathematics are a way of counteracting the evils of excessive specialisation, and of determining the 'intrinsic necessity' to which Professor Courant refers: they may thus vitally influence the whole course of mathematics in the succeeding years.

The organisers of this Congress have planned our meetings so as to pass under review all the main developments in mathematics and to try to get things into perspective. In the main, we have followed the traditional divisions into sections, but we have, surely not before time, given topology a section to itself, and we have somewhat changed the emphasis in the sections dealing with applied mathematics. But the one-hour speakers are not assigned to sections. They have been picked as a team so that a continuous spectrum will be presented and they have been asked to make their lectures broad surveys of recent developments. In this way it is our hope that their contributions will present a general survey of all that is important in modern mathematics, and that when our Proceeding are published, they will form a focus from which many of the developments of mathematics in the next few years may begin.

Over one week, it is not possible to cover the whole range of mathematics, and at the same time to deal adequately with the wide applications of mathematics to other fields of intellectual endeavour. No mathematician can be indifferent to the ever growing number of applications of mathematics to the various sciences - physical, biological, and social - and in industry the present generation has witnessed with pride the revolution brought about by the introduction of statistical methods, and by the spectacular development since the war of the science of computing. We should like to include in the business of this Congress a thorough study of all the applications of mathematics to Science and Technology. But factors of time and space make this a practical impossibility, and our business is primarily concerned with that abstract science of mathematics whose laws govern so much of our knowledge. Hence most of our work will be concerned with pure mathematics. But not all. In our sections dealing with applied mathematics we have endeavoured to overcome, to some degree, the limitations of time-table, by inviting some distinguished exponents of other sciences to talk to us about their mathematical problems: we are at least establishing contact with them on ground which gives hope of fruitful cooperation.

In one instance, we have gone further. The youngest child of mathematics, the science of computing, perhaps because of its youth, has presented the mother science with many fascinating new problems, and we have considerably enlarged the amount of space given to this subject. Practical considerations have, however, forced us to confine ourselves to the mathematical side of this science, leaving the engineering side for other Congresses.

You will see that we have again included a section dealing with history and education in mathematics. Mathematics has a great history, and mathematicians should know something of it. The problems of mathematical education are many, and the International Commission on Mathematical Instruction has devoted much time to some of these problems. The meetings of the Commission form part of the work of the section on Education. The work of the Commission will be concerned with three problems of importance in the field of mathematical education which have been selected for special study during the last few years; but, at the same time, there are other problems in mathematical education, particularly on the higher levels, which concern us all. It is part of our duty to see that our pupils who go on to walks of life outside the academic field understand that mathematics is an integral part of world culture; not only a pillar of the technological civilization of today, but an essential item in the intellectual equipment of the good citizen. To achieve this state, it is first necessary that the training we give our young men and women should be aimed at developing this understanding of principles and encouraging their interest, instead of crushing it beneath a mass of technicalities ; and secondly, that we should be prepared to take the trouble to give accounts of our work to the mathematically educated layman.

Another respect in which our Congresses differ from symposia lies in the fact that membership of a Congress is open to all mathematicians, while that of a symposium is by invitation, and is therefore confined to those who have been or are making a name for themselves in their particular field. Hence Congresses offer almost the only opportunity for many young mathematicians to meet and listen to the leaders in their subject. We welcome the large number of young people who are attending a Congress for the first time. Many will be here just to listen, but they will be able to meet and discuss problems with more mature mathematicians. Others, and the number of them is very large, are presenting papers and it is to be expected that, amongst the 650 papers offered, a number will attract attention from the more senior of us, and may prove a foretaste of great things to come. On this occasion we do not propose to publish these short papers; it is better that they should follow the normal channels of publication, but, in the expectation that many of the papers read at this Congress will excite considerable interest, we have made arrangements for a number of small discussion rooms to be available where groups can get together informally and discuss their ideas more fully.

Believing, as I do, that we have provided for those essential needs of mathematicians which only an International Congress can satisfy, I wish you all a profitable and enjoyable week in Edinburgh.

My Lord Provost! Ladies and Gentlemen! As on the occasion of the last three International Congresses of Mathematicians, so at this Congress two Fields Medals are to be awarded. It is already a tradition that the recipients of the medals are young mathematicians. This is not expressly prescribed in the memorandum of the donor, the late Professor Fields. It is only said that the awards should be made "in recognition of work already done, and as an encouragement for further achievement on the part of the recipients" - and this has been interpreted to imply that the recipients should be young. However, the other day, a friend of mine made the remark that when one looks at the present situation in mathematics and the developments in recent years, one feels that it is the old rather than the young who need encouragement. But even the point of this bon mot persuades us again to applaud and reward youth. Thus the Committee on the Fields Medals 1958 agreed, from the beginning, to keep to the tradition of awarding the medals to mathematicians of the younger generation.

This Committee on the Fields Medals, which was set up by the Organising Committee of the International Congress of Mathematicians, Edinburgh, consists of eight members, namely: Chandrasekharan, Bombay; Friedrichs, New York; Hall, Cambridge (England) ; Hopf, Zürich; Kolmogorov, Moscow; Schwartz, Paris; Siegel, Göttingen; and Zariski, Cambridge (U.S.A.). Each of us first wrote down his own list of nominees. The combined list contained thirty-eight names. Let me address these words to the thirty-six who will not be named here: "The Committee on the Fields Medals wishes to express its sincere appreciation and admiration for the work you have done. The high quality and the great variety of your achievements augur well for the future of mathematics. These very attributes have created considerable trouble for our Committee; again and again have we regretted that more than two medals could not be presented."

The great variety within mathematics is due not only to the multiplicity of the branches of mathematics, but also to the diversity of the general tasks that face a mathematician in any branch. A task which is particularly fundamental, is: to solve old problems; and another, no less fundamental, is: to open the way to new developments. Our Committee is glad to have found two young mathematicians who have done unusually good work, one in each of these directions. As Chairman of the Committee on the Fields Medals 1958, I have the honour and pleasure to announce that the Committee has decided to award the Medals to

Klaus Friedrich Roth, of the University of London, for solving a famous problem of number theory, namely, the determination of the exact exponent in the Thue-Siegel inequality;

and to

René Thom, of the University of Strasbourg, for creating the theory of "Cobordisme" which has, within the few years of its existence, led to the most penetrating insight into the topology of differentiable manifolds.

Detailed reports on the work of the laureates will be given in a special session; Professor Davenport will speak on Dr Roth's work, and I on Professor Thom's.

May I now ask Dr Roth and Professor Thom to come forward to receive the Medals from the hands of the Lord Provost of Edinburgh?

The closing session of the Edinburgh Congress took place in the McEwan Hall on the afternoon of Thursday, 21 August 1958. Professor W V D Hodge, President of the Congress, was in the chair. Professor Hodge made the following statement about the 1962 Congress and the procedure to be followed in deciding where the 1966 Congress should be held:

Before passing to the business of this meeting, may I, on behalf of all members of the Congress, express our sincere sympathy with the delegation from the Netherlands in the tragic accident which occurred to one of its members on Tuesday. It is the fervent hope of all of us that Professor van Wijngaarden will make a good recovery from his injuries.

As those who were present at the International Congress in Amsterdam will remember, a committee consisting of representatives of the International Mathematical Union and of the organisers of the 1958 Congress was appointed to consider the location of the Congress of 1962. This committee consisted of Professors Hopf, Chandrasekharan and Mac Lane representing the Union, and Dr Smithies and myself representing this Congress.

The committee has discussed with the representatives of a number of countries the possibilities of holding the next Congress in a number of places. I am authorised by the committee to say that while for reasons of a technical nature it is not possible to make any announcement today of the name of the host country for 1962, the prospects of holding a Congress in that year amount to a certainty. In order to remove any element of mystery from this statement, I will explain that one country represented here is very anxious to be our host but is unable to issue a formal invitation until certain consultations are completed at home; while another country has generously expressed its willingness to await the conclusion of these consultations and has promised to issue an invitation if, but only if, the first country finds itself unable to do so. The necessary consultations will be completed in a few months. The joint committee, therefore, undertakes to make an announcement by 1 January 1959. This announcement will be sent to the Adhering Organisations of the International Mathematical Union and subsequently published in International Mathematical News. If any country which is not a member of the International Mathematical Union wishes to be informed directly, will its representatives please send to Professor Eckmann, Secretary of the Union, the name and address of the organisation which should be informed.

Another matter concerning future Congresses must now be decided. I should like to propose that a committee of five, three to be appointed by the Executive Committee of the Union and two by the organisers of the 1962 Congress, be appointed to consider the location of the Congress of 1966.

The opportunity has been given to me of saying some words at this closing session of our congress. I shall try to express the feelings of gratitude to our hosts that I am sure are shared by all members of the congress. We must all be happy that our British colleagues, when inviting the congress to meet in Britain, chose Edinburgh as the meeting place, thus making it possible for us to enjoy for a while the special atmosphere of this ancient and beautiful city and to become acquainted also with other parts of Scotland. We have reason to be most grateful to the city of Edinburgh, to its famous university, and to its people for the hearty hospitality with which they have received us. I believe that only those who have tried it quite know what it means to organise a congress of the size of our international mathematical congresses. It must be an enormous amount of work that the organising committee and its helpers over a long period of time have put into the planning of the congress. I must express the admiration that I am sure we all feel for the way in which everything has been arranged. You certainly have done an excellent job. I wish that we could thank you all individually, including all the young people who have been around to help us with the many practical problems that invariably arise. But you have been very shy about it, and have not even printed the names of the organising committee in the membership list.

It is certainly not possible at the present stage to sum up the results of this congress. Through the choice of the invited speakers and through the large number of communications of other members the congress has presented a picture of mathematics today with all its trends. But the international congresses have another purpose, which I believe is just as important, that of promoting the fellowship between mathematicians of all countries. This fellowship has its roots in our common love for our science, to whose growth we all try to contribute. It is the responsibility of each generation to take care that this fellowship is maintained and strengthened, and extended to the new generation. It increases our joy in our work and, like the similar fellowship among scientists of other fields, sets an example for international collaboration. I am sure that, through the way in which this congress has been prepared, it has also admirably served its purpose in this respect.

The song of Auld Lang Syne has as its theme friendship and kindness. Our British colleagues and friends have at this congress given us both in full measure. Thank you.