The Organising Committee invited the following mathematicians to address the Congress:

**Wednesday 30 August:** A Beurling, H Hopf, Henri Cartan, R L Wilder.

**Thursday 31 August:** S Bochner, K Gödel, O Zariski, A Weil.

**Friday 1 September:** M Morse, A Rome.

**Saturday 2 September:** Abraham Albert.

**Monday 4 September:** A Tarski, J von Neumann.

**Tuesday 5 September:** A Wald, H Whitney, W V D Hodge, J F Ritt, H Davenport, L Schwartz, S Kakutani, S S Chern,

**Wednesday 6 September:** Norbert Wiener.

The Opening Address and welcome to the participants was given by the President of the Congress **Oswald Veblen** on 30 August 1950.

**OPENING ADDRESS OF PROFESSOR OSWALD VEBLEN**

In taking the chair today I feel that I am just acting as deputy for my friend, George Birkhoff, whose untimely, death has kept him from performing this duty. It was he who could have best welcomed the mathematicians of the world both on behalf of his University and on behalf of the American Mathematical Society.

If this Congress could have been held, as originally planned, in 1940 it would have marked in rather a definite sense the coming of age of mathematics in the United States. At the time of the International Congress in Chicago, in 1893, there was no indigenous mathematical tradition in this country, but there were a few active mathematicians, some of whom were beginning to diverge a bit from the lines laid down by their European teachers. By the time of the Oslo Congress, which was so admirably conducted by our Norwegian colleagues, a notable growth and transformation had taken place. Important discoveries had been made by American mathematicians. New branches of mathematics were being cultivated and new tendencies in research were showing themselves. Some American universities were receiving students and research workers from overseas, and interchanges of all sorts tended to be more and more on terms of equality. The colonial period was ending. At the same time mathematics had attained a small but growing amount of recognition from the rest of the American community - enough, at least, to encourage us to invite the mathematicians of the world to a congress in this country in 1940.

Now, fourteen years have elapsed since the invitation was issued, and we are approaching the end of another epoch. I mean the period during which North America has absorbed so many powerful mathematicians from all over the world that the indigenous traditions and tendencies of mathematical thought have been radically changed as well as enriched. These American gains have seemed to be at the cost of great losses to European mathematics. But there are so many signs of vitality in Europe that it is now possible to hope that the losses will be only temporary while the American gains will be permanent.

We are holding the Congress in the shadow of another crisis, perhaps even more menacing than that of 1940, but one which at least does allow the attendance of representatives from a large part of the mathematical world. It is true that many of our most valued colleagues have been kept away by political obstacles and that it has taken valiant efforts by the Organizing Committee to make it possible for others to come. Nevertheless, we who are gathered here do represent a very large part of the mathematical world. I will also venture the much more hazardous statement that we represent most of the currents of mathematical thought that are discernible in the world today. I hope that this remark will be dissected and, if possible, pulverized in the private conversations that are so valuable a part of any scientific meeting.

I have referred to the political difficulties which have harassed this Congress, but think that if there are to be future international congresses, an even more serious difficulty will be the vast number of people who have a formal, and even an actual, reason for attending. This makes all meetings, even for very specialized purposes, altogether too large and unwieldy to accomplish their purposes.

Mathematics is terribly individual. Any mathematical act, whether of creation or apprehension, takes place in the deepest recesses of the individual mind. Mathematical thoughts must nevertheless be communicated to other individuals and assimilated into the body of general knowledge. Otherwise they can hardly be said to exist. But the ideal communication is to a very few other individuals. By the time it becomes necessary to raise one's voice in a large hall some of the best mathematicians I know are simply horrified and remain silent.

The Organizing Committee of the present Congress has tried to meet this problem by means of a series of conferences, more informal than the regular program, but even in the conferences the problem of numbers will remain. It is to be hoped that our colleagues who have been meeting in New York to consider organizing an International Mathematical Union will have something to say to us on this and other problems before this Congress adjourns.

The solution will not be to give up international mathematical meetings and organizations altogether, for there is a deep human instinct that brings them about. Every human being feels the need of belonging to some sort of a group of people with whom he has common interests. Otherwise he becomes lonely, irresolute, and ineffective. The more one is a mathematician the more one tends to be unfit or unwilling to play a part in normal social groups. In most cases that I have observed, this is a necessary, though definitely not a sufficient, condition for doing mathematics. But it has made it necessary for mathematicians to group themselves together as mathematicians. The resultant organizations of various kinds have accomplished many important things known to us all. Of these accomplishments I am sure that the most important is the maintenance of a set of standards and traditions which enable us to preserve that coherent and growing something which we call Mathematics.

To our non-mathematical friends we can say that this sort of a meeting, which cuts across all sorts of political, racial, and social differences and focuses on a universal human interest will be an influence for conciliation and peace. But the Congress is, after all, just a meeting of mathematicians. Let us get about our business.

HARVARD UNIVERSITY

CAMBRIDGE,

MASSACHUSETTS

AUGUST 30, 1950