I. Algebra and number theory.

II. Analysis (3 subsections: IIa, IIb, IIc).

III. Geometry (2 subsections: IIIa, IIIb).

IV. Probability calculation, actuarial mathematics and statistics.

V. Mathematical-technical sciences and astronomy.

VI. Mechanics and mathematical physics (2 subsections: VIa, VIb).

VII. Philosophy and history.

VIII. Pedagogy.

The International Mathematical Teaching Commission met in Zurich during the Congress. Their negotiations took place in section VIII.

Monday, 5 September 16.00: Tea in the Lyceum Club, Rämistrasse 26.

Tuesday, 6 September 10.00-12.00: Visit to the Swiss National Museum, with a guided tour.

Wednesday, 7 September 10.00-12.00: Various tours in the city of Zurich. 15.00: ride on the Forchbahn (from Stadelhoferbahnhof) to Zumikon, tea in the golf clubhouse.

Friday, 9 September 14.00: Excursion in the Autocar to Wildegg Castle.

Saturday, 10 September 10.00-12.00: Visit to the Kunsthaus, Heimplatz, with
Guide.

The Lyceum Club was open to the Congress ladies throughout the day for the duration of the Congress.

During the entire duration of the Congress, an exhibition of mathematical books was organised in the Swiss Federal Institute of Technology, organised by Dr J J Burckhardt. It gave an overview of all of today's mathematical world literature. Books were also sold there or orders for books were accepted.

An exhibition of Swiss mathematical instruments was also held at the same time.

Ladies and gentlemen,

I have the great honour to bid you a hearty welcome in the name of the Swiss mathematicians. Almost 700 mathematicians of both sexes from 41 different countries have accepted our invitation and we hope that they will feel at home in our city and that the Congress will be a success in every way. In spite of the economic crises we have succeeded in organising the Congress in a simple but dignified manner owing to the support of the Swiss government, of the cantonal and municipal authorities of Zurich, and of the different financial and industrials concerns in our town. In this way we trust to have rendered a service to our highly beloved science who will be during the congress-days the uniting element of all our thoughts and interests. We hope that the mathematical science will be greatly developed in these days by the mutual exchange of knowledge and the general researches.

Ladies and gentlemen!

The International Mathematicians' Congress has already met once in Zurich. In 1897 the first international meeting of mathematicians took place under the presidency of Professor Geiser, whom we are pleased to greet among us. We are fortunate that Professor Geiser, despite his almost ninety years, did not miss being with us today. We also have the honour of seeing leading mathematicians from abroad who already worked here in 1897. From the large number, I would just like to mention the well-deserved President of the unforgettable congress in Bologna, Professor Salvatore Pincherle, the illustrious mathematician of Belgium Charles de la Vallée-Poussin, the great French mathematician Emile Borel, the old master Alfred Pringsheim from Munich and the representative of Poland, Samuel Dickstein. All these colleagues will be able to state that the development of the organisation of our congresses has not stopped. Nowhere else other than in scientific endeavours does standing still mean ossification and death. Almost all the numbers have more than doubled compared to 1897. Instead of 4 days, the congress takes place over 9; instead of 2000 we sent more than 6000 invitations; instead of 240 we have 800 participants today, instead of 5 sections there are 8 and instead of in two languages we operate in 4. Only the price of the congress card has hardly changed: Fr. 30.- instead of Fr. 25.-, while the women's card was Fr. 15 for both congresses.

In 1897, neither the Swiss Federal Institute of Technology was in its current form, nor was the university building. At that time, the E.T.H. hosted the Congress in its old buildings. However, this beautiful hall had not yet been built. We thank the Federal School Council, and in particular its honoured President, Professor Rohn, for providing the premises in the new building that has now been completed. We also thank the rector of the university, the eminent constitutional law teacher Fritz Fleiner, for making the halls of the University of Zurich available to us for the afternoons.

World history events have taken place in the 35 years since the first congress. But the meaning and purpose of the congress has remained the same. The unforgettable Hurwitz characterised it in 1897 in the most exquisite way, and in Toronto de la Vallée-Poussin emphasised how Hurwitz's words still apply. The main purpose of the Congress is to facilitate personal discussion, personal learning and understanding. This personal contact is based on the love of truth and scientific knowledge common to all mathematicians. No difference in race or social stratification can prevent the mathematician from cheering passionately every new fruitful thought and fully recognising it. In this regard, our science is eminently humanitarian, and thus connecting the masses and the peoples. The mathematician indulges in the joy of speaking with colleagues about his scientific ideas; he gives and receives the indispensable stimulation.

May our Zurich Congress today serve this purpose and thus promote mathematics, the science to which we all dedicate our life's work.

Dear Assembly!

Zurich is very honoured to receive and welcome the International Congress of Mathematicians in these days.

I take great pleasure in this, its opening session, because I have a particularly appealing task to perform: on behalf of the state authority, the government council of the canton of Zurich, I offer a warm welcome to your Congress and to all of you personally, and express our joyful satisfaction that you have chosen this place for your event, because we are receptive to everything friendly and appreciative that is intended for our Zurich and strive to reward it gratefully. - I especially greet the ladies, the delegations and the guests.

As you can see from your Congress papers, you have a large work programme to deal with these days. If that means a lot of work for you, it is also a serious justification for your conference and your cooperation. And in these times, which so often have the characteristics of contradictory diverging, planned work for international cooperation in the service of culture is particularly valued and deserves public funding and grateful recognition.

For your orientation, I would like to point out that the city of Zurich is home to two universities, the Swiss Federal Institute of Technology, to which this Auditorium maximum belongs, and the cantonal University.

It is easy to see the age of these two institutes: the E.T.H. celebrated its 75th anniversary two years ago and the cantonal University will complete its first century next year. The two institutions have developed from simple organisations. In the buildings of the E.T.H. the cantonal university was also housed until 1914. Today, this building has been greatly enlarged and expanded and is used exclusively by the E.T.H. It is surrounded by a number of additional institutes and laboratories, as well as the work schedule of the E.T.H., which until 1909 was called the "Federal Polytechnic", for scientific teaching investigation and research on a large scale, demonstrating and adding laboratory activities.

In the last quarter of a century, more than 5,000 diplomas were awarded based on the courses completed under regulations (as architects, civil engineers, mechanical engineers, electrical engineers, chemical engineers, cultural, forestry and agricultural engineers, and to specialist teachers in mathematics, physics and natural sciences). The number of doctoral degrees awarded by the E.T.H. - introduced in 1909 - exceeds 700.

The courses are taught by teachers: 75 professors, 42 assistant teachers and holders of special lectureships, and also 51 private lecturers, and the number of students is around 2000.

The Confederation, to whose immediate sovereignty the E.T.H. is subordinate and to which the special Swiss school board serves as a link, spends around 4 million francs annually for this federal institute.

The cantonal university, which has had its own new monumental buildings in the neighbourhood since 1914, comprises six faculties (theological, legal, medical, veterinary and two philosophical). It is also surrounded by a number of special institutions, laboratories, clinics, libraries and museums. The number of teachers is: professors of the various categories 103, private lecturers and lecturers also 103.

The number of enrolled students is around 2000, of which around four fifths are Swiss and one fifth are foreigners.

The operation of the University and its institutes requires around 3 million francs a year from the cantonal treasury.

Individual mathematical subjects also feature in the lecture schedule of the cantonal university.

What I wanted to tell you is that the subject you represent, which I would like to call not only a science, but also an art, which requires special talent, in view of the peculiarities of the requirements it places on its disciples who undertake teaching, research and practice, are regularly kept and promoted.

The speaker is not one of these scientific artists, although as administrator of the cantonal finances he has a lot to do with numbers. There he can do simple operations without algebra and logarithms, and the difficult task for him is to bring in money from the state then appropriately and justifiably reissue it.

But the way a finance director deals with numbers and quantities establishes a closer and more appealing relationship with the mathematician, who traces his family tree back to the oldest cultures, where, of course, there were no state accounts yet, and who classified the greatest in their history as ancestors. The mathematician - I imagine - has, as already indicated, a talent of his own kind. And this also gives him the meaning and content of an artist with a special vocation.

The mathematician has the advantage of having graphical means of expression in his numbers and symbols that are universally understood and do not need to be translated into other languages, and then mathematics is approaching the art whose symbols, musical notation, are also a universally accepted means of communication and no translation is required of the music.

In this way, a conceptual connection is involuntarily established, in which one hears something like pleasant, facilitating music, even under the most difficult mathematical problems.

I would like to wish you such a mood that you are carried by a high quality of tones and melodies for your Zurich days!

You are warmly welcome; and a lot of good things for your work and your whole Zurich programme!

Dear Festive Assembly!

As President of the International Congress of Mathematicians, I have the great honour of warmly welcoming you this evening. Our festive gathering would like to publicly express the fact that mathematics, our science, is one of the foundations of today's culture.

First of all, I welcome Federal Councillor Dr Meyer, our esteemed head of the Federal Department of Home Affairs. We are proud that he wants to spend the evening with us and will speak to us.

I greet the authorities of the Canton of Zurich, Dr Streuli and our director of education Dr Wettstein. We appreciate the interest with which you have supported our efforts.

I greet the Mayor Dr Klöti. The city council also provided powerful assistance to our Congress, which we would like to emphasise.

I also greet the members of the Honorary Committee. Among them are the leading personalities of banking and industry, who with foresight recognised the importance of the Congress and helped us. We would like to express our sincere thanks to everyone, but especially to the Swiss Credit Corporation, its well-deserved President Dr Stoll and his general manager Dr Jöhr and the Swiss Bank Association, who have been with us from the start.

I greet representatives from 41 countries, as well as those from countless academies, universities and societies. It would be going too far to name them all individually. We feel deeply honoured by their presence. It is a sign of the importance that is attached to the Congress. With our sincere thanks, we combine the hope that Zurich will be remembered well.

Today's event replaces the usual banquet. We hope that the freedom of movement that these spaces allow will make it easier for everyone to get to know and understand each other; yes, quite a lot of friendships will to be made tonight. Everyone can still make discoveries among the large number of scholars present and find those to whom he has not yet spoken.

I hope that you will feel comfortable and that you will have fond memories of the Swiss entertainment programme which follows. Perhaps we can also say a word today for historical reference: "The Congress is dancing."

A pleasant task brings me to you: to deliver a greeting on behalf of the Federal Council, from our highest authority and our country, to the International Congress of Mathematicians. We were pleased to see what a large number of highly regarded scholars from around 40 foreign countries came to Zurich to hold this professional conference here. We are convinced that this gathering is an important event for science. The Federal Government does not have a lot of direct relationships with science in our country. For us, cultural affairs are the privilege of the cantons. The federal government owns and maintains only one school, the Swiss Federal Institute of Technology. It cherishes it with all the greatest love and strives to expand it to meet modern needs. This institution (in addition to the universities) endeavours to support the field of mathematics. We are happy to have a special relationship with your high level of assembly thanks to the resources of this university. The decentralisation of our education system up to the universities is in the style of our entire state structure. Our people are made up of groups of different languages and cultures. These groups have to cultivate the cultural interests in their own spirit. This brings with it a fragmentation of means; our seven universities place heavy burdens on the cantons that have to maintain them. But this means that the schools are closely related to the cultural groups, they are closer to the people and are more deeply rooted in them. This is important for democracy, because in it you can never do enough for the education of the people, and every strength that stimulates this is beneficial. So we accept the extensive decentralisation of the education system and believe in its beneficial influence.

Mathematics is not one of those sciences that can be praised for its immense popularity. Some don't think much of it, and even in the year of the Goethe anniversary, the immortal German poet's judgment did not shed any light on this science. But we want to be aware that the exact sciences are the source of scientific technology has grown out of it, and that research into the laws of nature would be unthinkable without it. However, cultural history also shows mathematics has an intimate relation to the problems with which the great minds of all time have struggled. Mathematics is at the beginning of our intellectual culture. Today we are amazed at the achievements that emerged centuries ago from the research of those in whom the spirit of numbers lived. Our Zurich poet Konrad Ferdinand Meyer lets the chorus of the dead speak to the living in one of his poems:

And what we found in valid sentences,
All earthly change remains bound to it.

This is obviously tailored to mathematics. Its truths have lasted through the centuries, unaffected by the great catastrophes that at times threaten to revaluate all values. But mathematics is a shining example for all of us to undertake scientific work. Its research is done for its own sake. Mathematical thinking finds lasting truths. In this respect, mathematics, if you will put it that way, is an "impractical" science. But it is precisely for this reason that in so many cases it produces results that go deep into the life of mankind and that have also influenced the material development of the human race with the spiritual. If mathematics had become the servant of practical branches of the national economy, it would only be because it enabled the insurance that is used today in all areas and thus allowed an important step to be taken to eliminate chance as much as possible and to ensure the economic continuity and security of the individual. It is similar with science in general: the more that research is done in an ideal sense, the deeper it is cultivated for its own sake, the more magnificent are not only its theoretical but also produces the best practical results.

There is another reason that makes my obligation to offer our greetings particularly pleasant. Switzerland is the seat of the League of Nations. We see its efforts and experience its difficulties. We are deeply serious about living in peace with all peoples. Our location in the heart of Europe and the composition of our four-language population urgently lead us to this endeavour. We are wholeheartedly attached to the League of Nations because we are witness to the mutual beneficial fertilisation of different languages and cultures in our own people every day. We in our country have no language issue, we have no minority problems. We want to grant this reconciliation of races and cultures to the whole world. It is a special satisfaction for us to look at the international composition of your Congress. The peoples are most likely to get to know and respect each other by working together. And if this work takes you to such high heights as yours, then the congenial force of such a Congress is a double and triple blessing. I thank you for coming to our country and congratulate you on your happy meetings.

The Federal Polytechnic, which already opened its doors to the first International Congress of Mathematicians, was happy and proud to open them again, to receive, at 35 years apart, mathematicians from all parts of the world taking part in the Zurich congress.

The Polytechnic intended above all to train engineers, practitioners for whom mathematics, although essential, is only a working tool, the Federal Polytechnic however has two sections of pure sciences: the school of mathematical and physical sciences, and the school of natural sciences. I must give back to the men who have succeeded each other at the presidency of the Council of the School the testimony that they did not measure the solicitude they devoted to these two sections to the number of their students; the Polytechnic did not shrink from any sacrifice to maintain their high scientific level and to make them centres of research. The names of Raabe, Dedekind, Christoffel, H Weber, Schwarz, Frobenius, Hurwitz, to name a few of the masters who taught our Polytechnic for many years during the 19th century, are proof.

By chance, the person who speaks to you and brings you the greetings of the faculty of the Federal Polytechnic here is one of your own. He will therefore be allowed to sketch broadly, before you, the picture of some of the progress of our science in the interval between the two congresses in Zurich.

The first congress in Zurich was barely over when, following the work of G Cantor, we see the theory of sets and that of functions of real variables take a new boom in the hands of Baire, Borel, Lebesgue, Young, and de la Vallée-Poussin. The concepts which they introduced and which they made familiar soon proved to be indispensable in all areas of analysis.

The theory of integral equations to which the names of Fredholm, of Hilbert, of Volterra will remain forever attached, was born and made it possible to solve problems which Riemann, Neumann, Schwarz and Poincaré had attacked. The lecture which Torsten Carleman gave showed us that it is far from having exhausted the field of its applications.

Geometry, on the one hand, scrutinises its foundations thanks to the axiomatic method; it received, on the other hand, from theoretical physics, a new impulse which in the works of Weyl and Cartan makes it break out of its framework, yet already enlarged by Riemann.

In the theory of analytical functions, the problem of uniformisation was solved by Poincaré and Koebe; it bridges the points of view of Riemann and Weierstrass. Picard's theorem is the starting point for a series of fine researches on entire and meromorphic functions. The theory of conformal representation is coming to an end and recent works throw some light on the still obscure domain of analytical functions of two variables. After having heard on these subjects the lectures of Julia, Valiron, Nevanlinna, Bieberbach, Carathéodory and Severi, let us read again the beautiful lecture that Hurwitz gave at the congress of 1897; we will better be able to measure the distance travelled.

I will say nothing about algebra and topology, the revival of which we have been witnessing for a few years and whose role is becoming more and more preponderant.

If we take a look at theoretical physics, we see that, by pre-established harmony, we would be tempted to say, the theories it develops, however revolutionary they may be or appear to be, find their mathematical expression in group theory, that of functional transformations and, by extension, that they offer mathematicians new problems and cause new progress.

To these advances contributed geometers of different races, from different nations, a comforting testimony to the unity of mathematical thought. "Truth on this side of the Pyrenees, error on the other side," these words from Pascal does not apply to our science. Why does it have to be verified in so many areas?

Between the two congresses in Zurich the war passed, sowing hatred, accumulating ruins, enslaving science to its work of destruction. In a few decades Zurich may receive for a third time the elite of mathematicians from around the whole world. I hope that my successor, in this post, no longer has to evoke this spectre, but that he can give our generation of mathematicians the testimony that it too has brought its stone to the construction of science and its good will to the work of loyal collaboration and intelligent understanding of the people.

The University of Zurich is very grateful to you for holding your Congress with us. Mathematics had a decisive influence on the development of the intellectual life of Zurich in the 18th century. Under the sign of the immortal name of Newton, mathematical and scientific thinking succeeded in breaking the frozen forms of spiritual and state orthodoxy and preparing the ground for the enlightenment. From this nourishment came the critical meaning that led public institutions through the French Revolution into the new era. At the entrance to the new era in Zurich stands the lonely figure of the great reformer of education, Heinrich Pestalozzi. He taught that the education of every child must begin with the simple view of things. All of our perception must be expressed in number, form and language, and one of the elements of human upbringing is to connect number concepts with spatial perceptions. Intended as a means for elementary school, one of Pestalozzi's most ardent pupils, Jakob Steiner, took the idea of the simple poor teacher and built the system of his geometry on his ideas.

No wonder that the old Zurich school of scholars, the Carolinum, gave mathematics a proper place. When in 1833 the canton of Zurich built our own university from the Carolinum, the anniversary of which we will celebrate in the coming year, mathematics was also given a permanent place as part of the Faculty of Arts. Twenty years later, our cantonal university was joined by the Federal Polytechnic, now the Swiss Federal Institute of Technology, one of the first major foundations of the Swiss state. At first, for economic reasons, a connection between the mathematics professorships at both universities appeared to be the solution. But the mathematical impetus of the university made the teaching of mathematics independent at the university by establishing its own university professorships, and this dualism of the mathematical professorships at both of our universities is thanks to the rich development of mathematical teaching and mathematical research in Zurich, which has registered an honour for us today in hosting the International Congress of Mathematicians. In this area, as in other areas of our state life, the best is based on tradition. However, the other universities in Switzerland have consistently held on to their own mathematical professorships. Since we are meeting this evening in the spirit of mathematics, it is necessary for me to send a special greeting to our sister university in Basel and to the city of Basel, the home of Leonhard Euler.

Just as mathematical thinking gave the impetus to transform public spirit in Zurich two centuries ago, respect for number is a hallmark of our democratic state in our political framework today. The people have the highest decision in all factual questions of state life. Constitutional laws, like simple laws, require the approval of a simple majority of the voting citizens. In the legal exploration of the sentence that the majority is king, the representative of public law feels a particular pride in the awareness that he is at least in the forecourt of the mathematical temple. Yes, our federal law on the proportional election procedure for the National Council uses a mathematical phrase when it speaks of the term "proportional representation" (according to the Hagenbach-Bischoff system) and shows you that even democracy can not manage without the addition of mathematical theology.

Switzerland only pushed mathematical laws aside on one point. The share of our three national languages and cultures in the development and further education of our public life is not based on the principle of the numerical strength of German, French and Italian. We have only one formula for this: full equality of the three national languages. This understanding of the peculiarity and cooperation of different ethnographic elements is also fully endorsed by you. Because in the field of mathematical research, barriers go up from country to country. We admire a science that has been freed from all the gravity of the earth and yet can approach the last riddles of life and nature. We rightly give praise that the personal element which so often poisons life must stay away from the threshold of research. However, we are unable to get rid of it entirely, and we hope that there will be a memory of Zurich here and there during your future studies, and a soft Zurich tone. Rest assured that you will get a loud echo from the slopes of the Zürichberg.

Professor Veblen spoke informally without notes. He expressed the gratitude which all mathematicians must feel to Professors Fueter and Andreas Speiser for the generous way in which they came forward at Bologna, under very difficult circumstances, and invited the Congress to Zürich. He felicitated them on their brilliant success in carrying out the arduous task which they had assumed at that time. He also thanked the Swiss Government, the city of Zürich, the University of Zürich, and the Federal Technical School for their hospitable cooperation. Finally, he referred to the friendliness and pleasant manners of the citizens of Zürich of all classes which he illustrated by a pleasant little incident.

Ladies and gentlemen,

My first words will be to thank, on behalf of French-speaking mathematicians, the federal authorities, the cantonal authorities and the municipality of Zurich for the interest they showed at our Congress; it is their collaboration which made it possible to achieve this perfect organisation which we all admired and thanks to which our stay in Zurich will leave us with only charming memories.

The city of Zurich is one of the intellectual capitals of Europe. It is one of those places where art and science, thought and action, theory and practice are most harmoniously combined. This union finds its symbol in this magnificent Federal Polytechnic School which, along with the University, gave refuge to our sessions. Without having yet managed to be a hundred years old, it has long since acquired a worldwide reputation for the beauty of its laboratories, the excellence of its teaching and the liberalism with which it welcomes students and teachers from all countries. For us mathematicians, it is above all the School where Dedekind, Christoffel, Schwarz, Frobenius, Minkowski, and Hurwitz taught, not to mention those who still live here some of whom are among the masters of contemporary mathematical thinking. Finally we were reminded at all times of the day, in case we could have forgotten, that Switzerland is the homeland of Leonard Euler, one of the greatest mathematical geniuses of all time, to whom the Swiss Society of Natural Sciences is building a grandiose monument by the publication of his Complete Works. May I add that to the admiration that we all have for Euler is added a shade of affection, since his effigy freely opened for us the doors of all the trams of Zurich!

Fortunately, not all of our time was spent on the trams, or even listening to lectures or talking about mathematics. The Organising Committee has worked to introduce us to the most beautiful sites in Switzerland, a country which has so many. I will not make a fool of myself by singing, after so many others, of the forests, lakes and mountains of Switzerland. One impression that struck us all is that nowhere, it seems to me, is nature no longer without human influence. On our walk last Tuesday on Lake Zurich, an essential element would have been missing from the charm that emerged from the harmonious landscape which unfolded before our eyes if man had not manifested his presence through all these villas tiered on the mountainside and by the lights which, in the evening, began to shine like so many stars, thus forming a whole from which it would have been impossible to detach the smallest detail without detracting from the beauty of the whole. An ancient philosopher, I don't know which one [it was Francis Bacon!], gave art the following definition: 'homo additus naturae' [art is man added to nature]. As far as this definition is correct, Switzerland is a huge work of art.

But Switzerland is much more than that. Those who, like me, took part in the excursion to Lake Lucerne could not prevent feeling a deep emotion as they passed the Tellsplatte and the Rutli plain. These are sacred places; the feelings which animated a few hundred peasants there are now part of the moral heritage of all humanity. From these very simple feelings, but fraught with far-reaching repercussions, came the present Switzerland, a land of stability in the storm, an asylum of harmony for all men of good will. Let's not doubt it! This is the deep reason why the first International Congress of Mathematicians met in Zurich in 1897; this is also the deep reason why we met in Zurich in 1932.

Those of us who attended the Bologna Congress four years ago, chaired by our dear and revered colleague Mr Pincherle, cannot recall without emotion the closing session held in the magnificent hall, steeped in history, Palazzo Vecchio, the session in which our colleague Mr Fueter, responding to the eloquent appeal of Mr Pincherle, officially invited us to hold our next congress in Zurich. We were too well aware of the heavy work that we imposed on our Swiss colleagues and we were grateful to them in advance for their dedication. If we seek by extrapolation to determine the number of congresses which are held annually in Zurich, we arrive at astronomical figures since, according to what we could see for ourselves, there is a new one every three days. It is extraordinary that our colleagues, and in particular the eminent Rector of the Federal Polytechnic, can survive it. It is true that, at this rate, a marvellous technique had to be created for the organisation of congresses, and this is what our experience verifies. But there is one thing that the most perfected technique will never give, it is the cordiality of the reception and, much more, this 'je ne sais quoi', difficult to express in words, which makes us feel that we are received by friends, happy to welcome us, happy to feel we are at home, happy to find in our joy the reward for their efforts. If our dear president M Fueter, if the Rector Plancherel, if all their collaborators had experienced, before the opening of the Congress, any apprehension, then may they be reassured. My dear colleagues, we are very happy with our stay with you, and I am very happy to have been chosen to tell you this.

Dear assembly!

We are witnessing an extremely unlikely event. For the number n of International Congresses of Mathematicians opened before the present moment, the inequality is 7 ≤ n ≤ 9; unfortunately, our axiomatic foundations are not sufficient to give a more precise statement. If you compare the number of existing cities, it is highly surprising that the second of these ≤ 9 congresses is now taking place here in Zurich. However, I believe that for the quite vaguely formulated probability statement, teaching about the basics of probability calculation is right, which says that the probability is relative to our knowledge or ignorance. You only need to know a few facts to understand the amazing character of the event we attend.

The round of International Congresses of Mathematicians was opened in Zurich in 1897. At that time, the President of the Congress, Mr Geiser, formulated the one and most fundamental of the facts that must be relied on to understand the present astonishing event in his opening address as follows: "After the project had begun to take on a more solid shape due to numerous verbal and written correspondences and the question of location had been repeatedly considered, it was generally described as expedient that the first attempt should start from a country which, due to its location, its circumstances and because of its tradition, is particularly suitable for establishing international relations. So soon all eyes turned to Switzerland and especially Zurich." After the nameless and shameful confusion of feelings in the wake of the great war, Switzerland became more than ever before the haven of European cultural unity, which persists despite all the gaps in language, history and national struggle.The threads which were torn off were tied back on. As representatives of a science that, in terms of universality, in terms of content and means of expression, can only be compared with music, we mathematicians are natural allies of the ideal, that embodies Switzerland in the life of the state of Europe. So in a sense we are starting again in Zurich for the second time, I hope, ready to make the pledge: we never want to betray each other again as people and as mathematicians for the sake of political struggle.

But Switzerland is at the same time dear to us mathematicians as the birthplace of some of our greatest heroes. The Bernoulli family, among whom Jakob Bernoulli, Johann Bernoulli and Daniel Bernoulli shine, belongs to that wonderful dynasty of mathematicians. Switzerland gave us the outstanding genius Leonhard Euler in the 18th century, and the primitive geometric spirit of Jakob Steiner in the 19th C. Even in times when such tremendous talents did not leave their mark - they always remain a rare, precious gift from nature - mathematics found constant care in teaching and research; and we mathematicians of the German tongue are delighted with the blossoming freshness that Swiss mathematical life is showing right now.

As a third reason why we mathematicians met again and in such large numbers in Zurich, I consider to be the location and incomparable beauty of this country and especially this city. Now I'm tempted, if I don't keep hold of my heart, to turn to my German colleagues and to point and praise without end. Because this is the city where, for many years, as the successor to Mr Geiser, I had the good fortune to work as a teacher at the Technical University. Memories rush back to me like a swarm of seagulls: the city on a walk across the Gemüsebrücke on winter days, afternoons on the Dolder, short walks along the edge of the forest along Susenbergstrasse on windy days, when the entire alpine chain from Säntis to Jungfrau lies in clear splendour, quiet summer evenings on our balcony, from where we gazed at the waving sea of lights on the shores of the darkening lake beyond Thalwil, a morning on the Uetliberg, hiking along the Albiskette; but above all the lake with boating, rowing, sailing and swimming; the landscape of Lake Zurich, which Goethe sang in a few lines so beautiful that any other description must be silenced beforehand. I hope that all those Congress members who have been blessed with the weather god have seen and enjoyed a little of all that has been stored in me as a memory of many years; in some, the hymn of the poet Klabund, who found refuge here, will make a feeling like a string vibrating in me:

I can sleep peacefully in the arm of freedom,
And Zurich means more to me than: City by the lake.
It says: flower eye, Alpine port,
And sun gold-shining Danae.

But none of this would have brought us to Zurich if there had not been a fourth reason: the courage of Mr Fueter, who dared to go to Bologna four years ago, in an extremely difficult situation, to send out the invitation to Zurich, together with the initiative of his Zurich and Swiss colleagues. He knew that apart from this he could rely on the extensive support of the authorities and other circles in the city, canton and the Confederation. Still, it was no small venture. The terrible economic depression that has occurred in the meantime has made the task much more difficult. Many an international congress has been cancelled in the last few years, but the mathematicians congress is taking place. We owe you a big thank you for having achieved this, dear Swiss colleagues, and for preparing this wonderful conference for us.

Despite the thick line that emergency and foreign exchange regulations currently draw between the Reichsmark, Schilling and the Swiss franc, we Germans and Austrians cannot help but feel warmly connected to you Swiss. We speak the same language with the greater part of your people, even if you have held on to your particular dialect more faithfully than most other German tribes. The rediscovery of the essence of German poetry emanated from Zurich in the 18th century, and in the 19th century one of the greatest masters of German prose was the state secretary of Zurich. The relationships over and between Hesse are many.

From Göttingen, where I am now located, I brought two things with me, which I can cite as a testimony to our close ties. The first is a pretty, old volume that I found in a Göttingen antiquarian bookshop, containing Albrecht von Haller's "Attempts at Swiss Poems", printed in a tenth edition in the Göttingen university bookstore in 1768. Haller, a down-to-earth Bernese and a scholar of almost universal scope in the area of research, founded in the first decades the fame of the University of Göttingen, where he was later to find not unworthy successors in men like Gauss and Riemann. Among the poetry combined in that booklet we find the great poem "The Alps", verses to the Zurich friend Bodmer and the wedding of the Bernese Schultheissen von Steiger in addition to cantatas for the inauguration of the "Göttingen High School" and to greet its founder, the English king George II. Also a poem to the Zurich Canon and Mathematician Gessner, from which I want to quote a stanza here:

Soon you ascend through Newton's path
To Nature's private council,
Where your art of measurement leads you.
O art of measurement, bridle of imagination!
Whoever wants to follow you is never wrong;
Whoever shuns you will be lost in dark.

His biographer tells us that Haller himself always had a classic writer in front of him when at tables, on the street, on horseback and while walking, and worked on differential calculus on his wedding day. "We peoples on a gentle leash" - as he once called us - always kept something of this sober scholarly spirit of our first great Bernese teacher, even if we can hardly reach such a high standard. From his book of poems, the cosmopolitan outlook of the 18th Century, the gorgeous expression in Lessing's "Nathan the Wise" I experienced again the day before yesterday in the opening performance of the local theatre. We, children of an era that national zealotism invented, look back on it with envy.

The second witness of the close connection between German and Swiss, which I brought along with this book of poems, is myself. Don't interpret this as an exaggeration! The German Mathematical Society has chosen me as chairman this year with an express view of the Zurich Congress. Take this gesture, dear Swiss friends, instead of all my words, as an expression of the affection and thanks of the German mathematicians! People knew how close I am to Zurich and you, among whom I spent the happiest and best years of my work. For 17 years I was a servant of the Swiss Confederation, whose Federal Council greeted us so warmly here today. If you want to allow me this compliment, Mr Federal Councillor Meyer, I am still letting you teach me about world events, including those of my own country, through the "Züri-Zitig" [the Neuen Zürcher Zeitung]. At the first session, which served to prepare this congress, I still took part as a Zurich resident, so that it is about $\large\frac{1}{10^{10}\normalsize}$ self-praise if I congratulate our Swiss colleagues on the success of this conference.

I ask the guests of this country, by acclamation, to approve the following resolution of our assembly: Switzerland and our current Swiss friends have made a contribution to mathematical science.

Federal Councillor; Ladies and Gentlemen!

I have the honour of bringing to the noble country that hosts us, to its government and to the Swiss mathematicians, the cordial greeting of Italy and its government, as well as that of all Italian-speaking mathematicians.

We spent in these happy districts, widely smiled on by nature, embellished by the grace and organisational capacity of the people, made even more attractive by the noble sense of Swiss hospitality, delicious days, which we will not forget, and which have contributed to the making of our works.

As an Italian and as a lover of algebraic geometry, I feel I must here reverently remember the name of Jacob Steiner, the great man of Basel, who was one of the founders of modern synthetic geometry, and the names of Wilhelm Fiedler and Adolf Hurwitz, who illustrated the Polytechnic of Zurich, and made valuable contributions to the mathematical direction, which is dearest to us.

I must also remember with particular satisfaction that one of ours, Professor Eugenio Togliatti, taught here in fraternal intimacy with his Swiss colleagues, and finally that, precisely in the noble and peaceful environment of the Zurich Polytechnic, it was in 1915 accomplished by the intervention of your learned mathematician, Professor Grossmann, to whom we send our best wishes, the fusion between the absolute differential calculus of Ricci and Levi-Civita and Albert Einstein's theory of relativity.

We Italians therefore have more than a spiritual share, strengthened also by the common language with a part of your people, to the intimate friendship with Switzerland. I would like to thank the Italian mathematicians the organisers of the very successful Congress, which was held here, and especially our illustrious and tireless President Professor Fueter and the diligent Secretaries.

These plants, which surround the speaker's table with a delicate act, are the symbol of the kindness of our guests; the green of these palms is the symbol of the ever-living hope in the continuity of the progress of science to which we have devoted our lives; the red of these flowers is the symbol of the warmth of our gratitude and our thanks for your welcome.

Dear Sir or Madam!

It may seem somewhat 'post festum' when the speaker addresses you today to welcome you to our city, where you have been meeting for a week and are about to leave. It might almost seem that we still had to think about it.

Of course there is no question of that. The mathematicians are such respected scientists everywhere and such harmless people that we cannot imagine that they would not be welcome everywhere. So it was not we started today, but long before the opening of the Congress, that we were pleased to be able to host such an illustrious international scholarly meeting in which our most prominent representatives from almost every state and continent attended.

The reason why we are only able to express this joy today is because the careful organisers of the Congress cleverly ensured that you, dear attendees, did not receive too many official welcoming speeches, which are known to be so entertaining, which you suddenly had to endure.

So I ask you to accept the late but no less sincere welcome from the authorities and the population of our city, and I hope that in the past week you were granted not only useful but also nice days in Zurich. Our city has endeavoured to show itself to the honoured guests in beautiful September sunshine, so we can hope that it has managed to instil respect not only for its high rewards but also for its scenic beauty.

There are also rainy days here and there; but I want to keep their annual number discreet, because I don't want to run the risk of distracting you into the mathematical field by giving a number.

This danger is supposed to be quite big; the reporter of a local newspaper wrote that on the trip to Rapperswil last Tuesday you only dealt with the most serious mathematical problems and didn't appreciate the beautiful surroundings, and you drank tea in Rapperswil "always solving formulas and changing theories".

Of course, this is one of the well-known journalistic exaggerations. Had the man observed more closely, he would certainly have noticed that you had not missed the beauty of Lake Zurich, which had already inspired Goethe, but that you understood it with refinement, two pleasures, the enjoyment of nature and the enjoyment of witty entertainment in added skilful combination.

Most of the honoured guests were hardly allowed to get to know the people of Zurich in the few days of the Congress. When you look at the city and its surroundings, you may have thought that a happy people must live here.

Upon closer contact with the residents, this ideal idea would have been shaken. Zurich is certainly not an ugly and not a poor city. It is even considered one of the richest cities in the world, where every inhabitant has a respectable average fortune. But as a mathematician you know what an average means. As you know, it can be made up of very extreme sizes, after all it can be said that the living standards of the lower classes are much higher for us than in some other places. It is questionable whether it can be maintained. The crisis, which has spared our country for a long time, continues to spread from day to day. Exports, a necessity for the existence of our small, inland resource-free country, are shrinking. The number of unemployed, far from being as terrifying as in some other countries, grows from month to month in the middle of summer, and we see with concern the coming winter and the next year with their need and their struggles against the lowering of living standards and to counter the distribution of the burden of the crisis on individual social classes. So you will find more restlessness and dissatisfaction in our population than contemplative happiness.

Unfortunately, the seclusion of the individual states is not only limited to the economic area, it also extends to the intellectual area. In this difficult time, it is particularly valuable if the individual sciences maintain and strengthen their international character. The speaker is unable to appreciate the scientific output of your Congress. But the value of international congresses, especially in the exact sciences, lies less in the lectures, but above all in the creation of personal contact and in the cultivation of collegiality among the representatives of the individual sciences, regardless of the diversity of race and nationality.

I hope and have no doubt that the Congress which is approaching its end has fulfilled this task and thus helps to hold onto and defend the idea of internationality against the currents that want natural love for home to degenerate into unrestrained nationalism and chauvinism.

To be sure, the international cohesion of mathematicians is only a modest counterweight. However, even if this spirit of the universality of science is cultivated in other fields of knowledge, the representatives of science as a whole can exert a significant influence and help us soon to get out of this unpleasant period of national closure. Every group working in this direction deserves recognition and thanks. The City Council is allowed to express this feeling of gratitude towards your Congress with a symbolic expression by presented the art portfolio to the City of Zurich to the eminent representative of the United States of America, Professor Veblen, Princeton, the outstanding representative of the German mathematicians, Professor Hilbert, Göttingen, and the very deserving President of the Organising Committee, Professor Fueter, Zurich. And now, ladies and gentlemen, I wish your Congress a harmonious conclusion and a happy journey home to all of you.

May the days of Zurich remain in your memory and encourage you to honour us with your visits from time to time.

Dear Mr Mayor, Ladies and Gentlemen!

The congress management has given me an honourable mandate to thank you, Mayor and the City of Zurich, both for the friendly words of welcome that you have dedicated to us and for the other benefits that the city has given us. I have not been able to ask each of the 800 members of the Congress for their consent to speak for him, but I do hope that you will all agree if I try to give our thanks as warmly as possible. A beautiful, suitable frame is part of a valuable picture, and, in a corresponding way, a city is this to a Congress that has enriched us scientifically and personally. And who wanted to deny that Zurich is an ideal congress city? It is a big city with all the amenities of such a city, but it is a city that is closely related to nature everywhere: everywhere it offers the eye the green of leaves, the shimmering green-blue of the lake, plus the shining light of a wonderful Central European September. This gave us the necessary relaxation. Zurich is also a city where, as a child said, you can see the stars both above and below you in the evening. The view of Zurich from its inhabited heights is as enchanting at night as it is during the day. Zurich is a big city, an extraordinarily busy city with high-rise buildings and everything modern that belongs to a big city. But it is also a city that has a Grossmünster and memories of Charlemagne and knows how to preserve and value them. Zurich is a big city with the noise and bustle that goes with it, but it is also a city of universities, the heroes Zwingli and Pestalozzi, and a city of poets, Konrad Ferdinand Meyer and Gottfried Keller. And Richard Wagner also lived in it for a long time.

We have grasped all of this and absorbed it. You have complimented us, Mr President, that we are harmless people and, further, that we are clever people who do mathematics on a sea journey and yet see the beauty of the landscape. With the sure eye of the politician, you have correctly recognised what appears to be the opposite as a unit: we mathematicians are indeed clever as thinkers and connoisseurs of the spirit who see the beautiful outside of our heads; we unite the outer harmlessness that is our protective cover. I can return half of your compliments to you. Perhaps it was the city's special sophistication to give us this beautiful afternoon at the end of the conference. Because of the beauty that made an impression on us, the last of course has the greatest chance of remaining within us. But be assured, Mr President, that such a refinement would not have been necessary: the beautiful city of Zurich, what it wanted to do for us and did unwanted, will always remain in vivid association with all the other valuable impressions that we have had here. Receive our warmest and sincere thanks, Mr President of the City of Zurich and your city!