## 1954 International Congress of Mathematicians - Amsterdam

**1. Secretary's Report on Preparations.**

Immediately after its return, The Netherlands delegation reported this decision to the Netherlands Mathematical Society (Het Wiskundig Genootschap). It was obvious that the International Congress of Mathematicians 1954 should be held under the auspices of the "Wiskundig Genootschap", the Society which unites all Netherlands mathematicians and whose 175th anniversary was to be celebrated in 1953.

In October 1950 the Board of the "Wiskundig Genootschap" appointed a small committee with the task to draw up a report on structure and regulations of the Congress. This Committee consisted of the professors:

H D Kloosterman (chairman),

J Haantjes (secretary),

O Bottema,

J F Koksma,

J Popken.

Professor D van Dantzig, at that time President of the Wiskundig Genootschap, took part in the discussions.

The Committee submitted its report at the beginning of 1951. Its main points may be summarised as follows:

A description of the aims of the Congress:

The most important aim of the Congress is to stimulate scientific research in the various branches of mathematics and to further good understanding and more intensive cooperation among the mathematicians of all countries. In order to attain this aim, mathematicians of all nations should be invited to meet for a short period for exchange of ideas, which should be realised by organising lectures and by creating an informal contact between the participants in the Congress.

His Royal Highness, Prince Bernhard of the Netherlands, consented to give his Patronage to the Congress.

**2. Report of the Inaugural Session.**

**2.1. Message from Prince Bernhard of the Netherlands.**

H. R. H. Prince Bernhard of the Netherlands, who was unable to attend the session, sent a message which was reproduced in the Congress Guide and which also is also given here:

Soestdijk Palace

July29,1954

Not having the opportunity to be present in person at the opening of the "International Mathematical Congress1954", I would like to offer this greeting, a warm welcome to the many hundreds of mathematicians from all parts of the world who are now meeting in the Netherlands.

The purpose of this Congress is joint; to practice and promote a profession which, like few others, recognises the unity of the human race and which - also in the light of recent applications - is still gaining importance.

I wish all participants good and fruitful days.

Bernhard

Prince of the Netherlands.

**2.2. Welcome from the President of the Organising Committee.**

The President of the "Wiskundig Genootschap" and of the Organising Committee, Prof Dr Ir J A Schouten, spoke the following words of welcome:

As President of the Mathematical Society of the Netherlands and of the Organising Committee of this Congress, my first duty is to welcome all participants of this Congress and to express the sincere hope that their stay in Amsterdam will be a most successful and most pleasant one. It would be impossible to mention here the names of all official bodies, all institutions and all private persons, who by financial help or in other ways have made the organisation of this Congress possible and I can only express here our warmest thanks to all.

[This was then repeated in Dutch, French, German, Italian, Swedish and Russian.]

Of course I can not go on in this way because I have already gone beyond the limit of those languages that I can really speak to the limit of the languages that I can only pronounce a little. So you will excuse me if I use from now on the English language.

But first I have to use once more my own language to address Mr d'Ailly Burgomaster of the city of Amsterdam, who will do us the honour to open this Congress.

**2.3. The Burgomaster of Amsterdam opens Congress.**

Professor Schouten then invited Mr Arn J d'Ailly to open the Congress.

After the Congress was opened Professor Oswald Veblen asked for permission to speak and addressed the Congress in the following speech:

**2.4. Speech by Professor O Veblen.**

The series of International Congresses of which this is to be one are very loosely held together. They are not congresses of mathematics, that highly organised body of knowledge, but of mathematicians, those rather chaotic individuals who create and conserve it. At the end of each congress they somehow agree on the country where the next one is to be held and then leave it to their colleagues in this country to work out a programme.

To symbolise the tenuous continuity which is thus achieved, the President of the old congress emerges for a moment from the obscurity in which he belongs, to propose the name of the person selected by the hosts of the new congress to preside over it.

In this case, it is a pleasure for me to utter the name of an old friend and colleague who has distinguished himself by long and successful devotion to one of the most important branches of our sciences. He has also displayed what is not too common a trait, a willingness to do more than his share of the drudgery which is necessary to our common effort. I present the name of Professor J A Schouten for President of the Congress of Amsterdam. Will you manifest your approval?

**2.5. The Presidential Address.**

The Congress applauded Professor Veblen's proposal and Professor Schouten, accepting his election, delivered the following presidential address:

**Presidential Address by Prof J A Schouten.**

My first task as President is a sad one. I have to ask your attention for the memory of two of our congress members: Professor Fabio Conforto from the University of Rome, who died on the 24th of February and Professor Rodolphe Henri Joseph Germay from the University of Liège, who died on the 16th of May. Their work will live in our minds and our thoughts are with their families.

Then I have to inform you that our Patron, His Royal Highness Prince Bernhard of the Netherlands, is abroad and cannot attend this opening-session. His Royal Highness gave us a message for the Congress, the text of which, with some translations, you will find in the program. I propose to send His Royal Highness the following telegram:

Finally I wish to draw your attention to a fact which was perhaps not so clear four years ago, but which is absolutely clear now: the place of mathematics in the world has changed entirely after the second world war. Before, mathematics had an honourable place among the sciences because of its central position, its history and its traditions, but there were in those times not many mathematicians and most people had only some bad memories from their school years and the comforting idea that in real life they would meet mathematics never more. Even some older engineers propagated the idea that the mathematical training of technical students was only a kind of quasi-scientific ornament that could be dropped before long for the greater part because technical methods themselves had by now developed into real sciences!To His Royal Highness Prince Bernhard of the Netherlands.

The International Congress of Mathematicians1954at Amsterdam, at the opening session, expresses its warmest thanks for the kind words of welcome and encouragement contained in the message of Your Royal Highness as Patron of this Congress.

J A Schouten, President.

It was not really necessary to discuss such ideas because real life gave the answer to questions of this kind in a very short time and with the utmost clearness.

During and after the war it became obvious to every one that nearly all branches of modern society in war and in peace need a lot of mathematics of all kinds, from the simplest school arithmetics up to the highest developed theoretical parts. In fact, there is nowadays no big factory without its computing machines and no investigation involving series of experiments or observations is possible without an elaborate application of modern statistics. But computing machines do not work without a staff of very good mathematicians for the programming and modern statistics need also mathematicians of a very high standard. Also the so-called "applied mathematics" came to new life and asked for more men well trained in mathematics and physics, because modern computing machines had made it possible to make use of solutions that formerly only had theoretical value on account of the impossibility of doing the computing work in a reasonable time. It is very remarkable that in connection with all these activities a development of many parts of pure mathematics was necessary, thus making true the word of Felix Klein that all which is mathematically pure will find sooner or later some practical application.

Thus mathematicians of all kinds are needed in numbers our ancestors could not have dreamt of and universities all over the world are constantly busy producing more of them. Even technical universities, instead of dropping a great deal of their mathematics, are now training mathematical engineers, who have to fill the gap between mathematics and practical engineering sciences.

Now this is all very satisfying and we could be content that our science got so prominent a place in the structure of modern society. But some difficulties arise. Sixty one years ago the first mathematical congress at Chicago was attended by 25 mathematicians. In 1936 we had the congress in Oslo with 500 and after the war Cambridge (Massachusetts) with 2316 and this congress at Amsterdam with 1550 attendants, notwithstanding the fact that in Cambridge there were 1410 Americans and among us only 240.

On the one hand we may be happy with this progress, but on the other hand it is wise not to shut the eyes for the fact already pointed out by Professor Veblen in his opening address at Cambridge, that there is a limit to congresses of this kind. This limit will perhaps be reached very soon if the number of mathematicians goes on increasing as rapidly as it does now and if in the future, as I fervently hope, big countries with a great number of good mathematicians will break with the system of sending a very small delegation, the extent of which is in no way proportional to the mathematical importance of the country involved. This system of sending a small delegation only is entirely wrong, the chief aims of a mathematical congress being, as Professor Störmer pointed out in his presidential address at Oslo, to enable the direct exchange of ideas from man to man and to give a great number of younger people the opportunity to get the personal contacts they need for orientation and stimulation. The average age of participants at our congress is 40^{1}/_{2} and that is too old.

As Professor Veblen put it, mathematics is so "terribly individual" that a man practically can only speak for himself and this means that instead of a small delegation we need an adequate number of scientists and among them many younger men, to get all the personal contacts we are so urgently looking for.

But if the number of participants increases the question arises: shall we have in future one big congress or instead several smaller meetings on definite topics. In the years after the war we have already had several very small meetings called colloquia and as far as I can see they were a great success. But what I mean here is a splitting up of the big congress into a small number of parts to be held separately but with one central organisation. Personally I think that the mutual induction of the several branches of mathematics is so very important that we should try as long as possible to save the idea of one big congress. But, if from purely technical considerations such a congress would become impossible, the splitting up should be done very carefully and with an open eye for the structure of the science of mathematics as a whole.

It is good to remember here a word of Poincaré's, who stated explicitly that in mathematics there are two kinds of mental acting, one above all occupied with logical deduction and the other guided by a more intuitive faculty for arranging or rearranging known facts in a new way satisfying some principle of aesthetics or of unification. Poincaré laid particular stress on the point that the choice of the method is by no means fixed by the matter treated and that it has nothing to do with the difference between analysis and geometry. There are famous analysts using largely the more intuitive faculty and famous geometers working as a rule with deductional methods.

Exaggerating one aspect of mathematics and neglecting the other part invariably leads in the end to undesirable results. It is my personal opinion that the lack of interest in mathematics among young people is for the greater part due to the fact that the more intuitive aspect of mathematics is sometimes neglected, for instance where geometry is reduced to a system of axioms and deductions only, thus over-stressing just that aspect of geometry which is most uninteresting for young people of an age between 12 and 18. I am glad that in section VII of this congress this point will be discussed.

In 1905 "L'Enseignement Mathématique" started an inquiry into the methods of working of mathematicians. The results of this inquiry augmented and developed later by several authors, for instance Carmichael and Hadamard, can be expressed shortly as follows. The faculty of deduction belongs more to the conscious mind, the subconscious being in general only able to perform very simple and trivial deductions. On the contrary the faculty of rearranging is typical of the work of the subconscious and is described by Carmichael as consisting of an extremely rapid passing over of innumerable useless combinations till a vital one or some vital ones rise to consciousness, to bring, after a severe control of the conscious mind, new truth to light.

It is remarkable that our modern computing machines can imitate some of the lower parts of both faculties of our mind. In fact, there are machines, effecting a few simple logical deductions, and other machines, especially constructed for the investigation of big molecules, which are able to pass in a short time over say a million possible combinations of phases in order to single out some twenty five most suitable ones for a more detailed examination.

The development of any part of mathematics always involved the action of both faculties of the mind in the same or in different investigations. This we should take into serious consideration if we wish to organise congresses in the future, be it a big one or several smaller ones. It is certainly very important to create the possibility of mutual induction of several sections during a congress, but it is far more important that both faculties of the mind come into their right in every section. So a section for analysis should try to stimulate the influence of the more intuitive faculty and a section for geometry should make sure that sufficient place is reserved for deductive investigation. In order to come to a practical result I should like to ask all of you to give your attention to this point during this congress and especially to observe how the two faculties of mind are really working in every section and to give your opinion as to the sufficiency of their interaction. In this way we might get a scientific inquiry, the results of which could be very valuable for an efficient organisation of future congresses.

**2.6. Musical Interlude.**

After the presidential address a musical interlude took place, viz. a piano solo by Mrs Fania Chapiro:

F Chopin

Impromptu in a flat major op. 29

Nocturne in c minor op. 48

Scherzo no. 2 in b flat minor op. 31.

**2.7. The Fields Medal Committee Report.**

Then Professor Herman Weyl, President of the Fields Medal Committee 1954 addressed the Congress on behalf of the Fields Medal Committee, expounding the grounds on which the Fields Medals 1954 were awarded to Mr K Kodaira and Mr J P Serre. [The Fields Medal Committee 1954 consisted of: Prof H Weyl (President), Prof E Bompiani, Prof F Bureau, Prof H Cartan, Prof A Ostrowski, Prof A Pleijel, Prof G Szegö, Prof E C Titchmarsh.]

**Address of the President of the Fields Medal Committee 1954.**

Professor Hermann Weyl, President of the Fields Medal Committee 1954, delivered the following address:

That at each International Mathematical Congress two gold medals be presented to two young mathematicians who have won distinction in recent years by outstanding work in our science: this was the intention of the late Professor J C Fields, the donor of the trust fund for these medals, and such the resolution adopted at the International Congresses in Toronto 1924 and Zurich 1932. In his Toronto memorandum Professor Fields expressed the wish that scientific merit be the only guide for the award of the medals, and he expressed the hope that the prizes would be considered by the recipients not only as an acknowledgment of past, but also as an encouragement for future, work.

Owing to the turbulent conditions of the world, award of the Fields Medals has up to now taken place but twice, at the Oslo Congress in 1936 and at Harvard, in 1950. As Chairman of a Committee consisting of the following members: E Bompiani, F Bureau, H Cartan, A Ostrowski, A Pleijel, G Szegö, E C Titchmarsh and the speaker, I have the great honour and pleasure to perform this ceremony here-now for the third time. Our Committee owes its origin to the action of Prof J G van der Corput as nominee of Het Wiskundig Genootschap for the presidency of this Congress, Prof J A Schouten as Chairman, Prof H D Kloosterman as Vice-Chairman and Prof J F Koksma as Secretary of its Organising Committee. Not for the composition, but for the judgment of our Committee the responsibility is ours, and we assume it gladly and with a good conscience indeed. After much deliberation during which many names were discussed and which made us fully aware of the arbitrariness involved in the selection of just two from the array of mathematicians of outstanding merit, we finally agreed, and this decision was reached as unanimously as anyone could reasonably expect, to present the two gold medals, together with an honorarium of $ 1,600 each, to:

**Professor Kunihiko Kodaira**,

and

I hope the Congress as a whole will approve our choice. In justification of it let me say this: by study and information we became convinced that Serre and Kodaira had not only made highly original and important contributions to mathematics in recent years, but that these hold out great promises for future fruitful non-analytic (will say: non-foreseeable) continuation. Carrying out the Committee's resolution, I now call upon Professor Kodaira and Dr Serre to receive from my hands the prizes awarded them - awarded them, to repeat the donor's words, as recognition of past, and encouragement for future, research work. In the name of the Committee I extend to you, Professor Kodaira, and to you, Dr Serre, my heartiest congratulations.

Two precedents have established the custom to combine with the award of the Fields Medals a brief survey of the recipients' main mathematical achievements, in particular of those which most attracted the Committee's attention. In Oslo it was not its Chairman, Elie Cartan, but another of the Committee's members, Carathéodory, who discharged this duty. In 1950, at Harvard, Harald Bohr was Chairman of the Fields Medals Committee, and he did both: handed over the medals to Atle Selberg and Laurent Schwartz and delivered the laudatio. I am going to follow his example, though with considerable hesitation; for I realise how difficult it is for a man of my age to keep abreast of the rapid development in methods, problems and results which the young generation forces upon our old science; and without the help of friends inside and outside the Committee I could not have shouldered this burden at all. It rests more heavily on my than on my predecessors' shoulders; for while they reported on things within the circle of classical analysis, where every mathematician is at home, I must speak on achievements that have a less familiar conceptual basis. A report like this cannot help reflecting personal impressions. Hence I now speak for myself and no longer as Chairman of the Committee. Thus freed from irksome bonds, I start by confessing that I am deeply satisfied by the Committee's choice - though I hope they will admit that it was reached without undue pressure from my side.

**2.8. Second Musical Interlude.**

A second musical interlude by Mrs Fania Chapiro then followed:

C Debussy

Suite pour le Piano

(prélude - sarabande - toccata)

**2.9. Secretary of the Executive Committee of I.M.U. addresses Congress.**

Professor Bompiani, Secretary of the Executive Committee of I.M.U. then addressed the Congress with the following speech on behalf of I.M.U., which had held its General Assembly at the Hague on August 31 and September 1.

**Speech by Prof E Bompiani.**

The International Mathematical Union has the honour to announce the election of officers and other holders of office for the period January 1, 1955 - December 31, 1958, as follows:

President of the International Mathematical Union: H Hopf (Switzerland)

1st Vice-President: A Denjoy (France)

2nd Vice-President: W V D Hodge (U.K.)

Secretary: E Bompiani (Italy)

Elected Members: K Chandrasekharan (India), J F Koksma (The Netherlands), S Mac Lane (U.S.A.)

**2.10. Address by representative of Dutch Government.**

The final address in the opening session was delivered by the representative of the Dutch Government, Mr H R Woltjer, Head of the Department of Advanced Education and Sciences of the Ministry of Education, Arts and Sciences:

**Address by Mr H R Woltjer.**

Mr President, Mr Burgomaster of the City of Amsterdam, Ladies and Gentlemen,

The Minister of Education, Arts and Sciences, His Excellency Mr Cals deeply regrets that official duties prevent his being personally among you this morning. His Excellency has asked me to state that the Netherlands Government feels greatly honoured about this highly important Congress being held in the Netherlands, and to convey to you his best wishes for its success.

The Government may express her gratitude to the Board of the Wiskundig Genootschap and the Organising Committee who gave all their energy and time for the organisation.

Besides, I want to offer my heartfelt congratulations to the young winners of the Fields Medals on obtaining this high distinction granted for outstanding scientific merits. I could imagine that a real scientist attaches but little importance to congratulations offered by a Government's official, not so much so, because activity of the public authorities with science so often implies a threat to its autonomy - the attention and respect of politics, business and finance is directed often wholly to its results - as because science finds its reward in itself. In the spirit of science resides its chief value. This can be asserted without abating anything of the claim for the value of its results. Knowledge for the sake of knowledge, as the history of science proves, is an aim with an irresistible fascination for mankind, which needs no defence. The mere fact that science does, to a great extent, gratify our intellectual curiosity, is a sufficient reason for its existence. Science gives therefore a great satisfaction to those who apply themselves to it.

Johannes Kepler, telling about a discovery he had made, says the intense pleasure he had received from his discovery never could be told in words. He regretted no more the time wasted; he tired of no labour; he shunned no toil of reckoning days and night spent in calculating until he could see whether his hypothesis would agree with the orbits of Copernicus or whether his joy was to vanish into air. Every scientist, exerting himself to advance science, will experience something like that. It is the aesthetic value of science which gives this inner satisfaction in all scientific work. If scientific knowledge consisted of a mere inventory of facts, it might still be interesting and even useful, but it would not be one of the major activities of the mind. It would not be pursued with passion. For science to have inspired such ardour and devotion in men it is obvious that it must meet one of the deepest needs of human nature. This need manifests itself as the desire for beauty. It is in its aesthetic aspect that the chief charm of science resides. This is true for scientific men themselves. To the majority of laymen, science is valuable chiefly for its practical application. But to all the greatest men of science practical applications have emerged incidentally, as a sort of by-product. This is, perhaps, most obviously true of the men who created the mathematical sciences. In the work of mathematicians, in particular, the aesthetic motive is very apparent. Many mathematicians have written about their work in a sort of prose poetry, and the satisfactions they get from it seem indistinguishable from those of an artist. The language of aesthetics is never far to seek in the writing of mathematicians. In their frequent references to the "elegance", "beauty", and so on of mathematical theorems they evidently imagine themselves to be appealing to sensibilities that all mathematicians share. Nearly all mathematicians show themselves uneasy in presence of a proof which is inelegant, however convincing, and, sooner or later, endeavour to replace it by one which approaches closer to their aesthetic ideal. Some of them have gone so far as to remark that the actual solution of problems interests them much less than the beauty of the methods by which they found that solution. If mathematics is to be ranked as a science, then it is, of all the sciences, the one most akin to the arts.

I hope that in the strain of these thoughts, derived from Sullivan, the congratulations with the Fields Medals will be understood and accepted.

This involves also some of the reasons why the Dutch Government has such a vivid interest in this International Congress of Mathematics and highly appreciates that so many mathematicians from all over the world assemble here. In this circle I may deem myself discharged of going into the value and the significance of mathematics in science and society. It is fascinating to see how history is showing again and again the mutual influence which always existed between mathematics and the other sciences and how mathematics have changed the world outlook. Nothing is more impressive than the fact that as mathematics withdrew increasingly into the upper regions of ever greater extremes of abstract thought, it returned back to earth with a corresponding growth of importance for the analysis of concrete fact. The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact. For this very reason it makes mathematics a living science. It was Whitehead who warned for dead knowledge. He says, What is wanted is "activity in the presence of knowledge". In all abstract thinking there must be a final connection with the living reality. This activity in the presence of knowledge, is an activity of the mind, in which the human factor is more important than an outsider would expect. Poincaré has told us that even in pure mathematics, where reason, one would think, is most pure and undefiled, a proof which is quite satisfactory to one mathematician is often not at all satisfactory to another. Indeed, Poincaré was led to divide mathematicians into psychological types, and to point out that a kind of reasoning which would convince one type would never convince another. In view of these facts it is obviously misleading to present science as differing fundamentally from the arts by its "impersonal" character. Therefore I can't see this Congress as an efficient organisation of exchange of scientific information only, but also of human understanding. Seen in this way the social and artistic part of the program of this Congress has also a more profound sense. Also scientifically human contacts can still strengthen considerably the success of the Congress. Lucien Price in his recent book *Dialogues of Alfred North Whitehead* reported that Whitehead once said: "By myself I am only one more professor, but with my wife, I am first-rate".

May human understanding contribute also to the success of this Congress and to the strengthening of the bonds between the countries you have come from and the Netherlands.

**3. The first lecture.**

**4. Closing Ceremonies.**

**4.1. Professor Schouten's Address.**

After Professor Kolmogorov's lecture, Professor Schouten addressed the Congress and read first the following telegram, with a translation in English, from the Royal Palace in Soestdijk:

Then Professor Schouten spoke as follows:At the request of His Royal Highness, Prince of the Netherlands, I thank you very much for your telegram and offer you the best wishes from His Royal Highness for the success of the Congress.

As Professor Bompiani told in the opening session there is a joint committee consisting of the President and Secretary of the I.M.U., the President and Secretary of the Organising Committee of the Congress and Prof Iyanaga as a fifth member for the preparation of the discussion on the place of the next Congress. This joint committee has received one letter only from Professor W. V. D. Hodge, authorised by the Royal Society, the Lord Provost and Town Council of the City of Edinburgh, and the Principal of the University of Edinburgh, inviting the International Congress of Mathematicians to meet in Edinburgh in 1958. There was also a letter from the Department of Mathematics of the Hebrew University in Jerusalem concerning the Congress 1962. But as this matter belongs to the competence of the Congress 1958 it will be forwarded to the Organising Committee of this latter Congress. The joint committee recommended that Professor Hodge from Cambridge be invited to speak.

**4.2. Invitation for 1958 from W V D Hodge.**

Professor Hodge spoke as follows:

**Speech By Prof W V D Hodge.**

I am speaking as the representative of the delegates from Great Britain and Northern Ireland. I have the honour to convey an invitation from the mathematicians of Great Britain and Northern Ireland, sponsored by the City of Edinburgh, the University of Edinburgh, the Royal Society of London and the Royal Society of Edinburgh, to the mathematicians of the world to hold their next Congress in Edinburgh in 1958. If this invitation is accepted, I can assure mathematicians of every country of a warm welcome, and we in Britain will do our best to organise a Congress worthy of the high standard set by our present hosts.

In issuing this invitation, I want to mention two particular points. As is well known, a Festival of Music and Drama is held each year in Edinburgh, over a period which includes the first week of September, which seems to have become the accepted time for our congresses, and this will necessitate some departure from recent practice. While we cannot be expected to fix the dates of the Congress precisely at this moment, it is our intention to try to arrange to hold it during the earlier part of August. From enquiries I have made, I have formed the impression that this change of date will be welcomed by many mathematicians, and we hope that it will be possible to time the Congress so that those mathematicians who are also interested in the Musical Festival will be able with as little inconvenience as possible to attend both gatherings.

The second point which I have to make refers to something which our President, Professor Schouten, said in his address, and which was also mentioned by Professor Veblen. The steadily increasing size of our Congresses has caused some people to wonder whether they are not in fact becoming too big, and in danger of getting out of control. It would be most improper for me to enter into any argument here, and I shall only express my personal conviction that the purpose of international congresses and of specialised colloquia are quite different, and that there is a real danger that if the complexities and cost of organisation continue to increase it will become more and more difficult to find countries able and willing to undertake the burden of arranging a congress, and eventually there might only be one or two of the few remaining rich countries able to do so. There can be no doubt that this would be very bad for mathematics.

Therefore, if the invitation to meet in Edinburgh is accepted, we propose to give serious thought to the question whether we can in some way give a lead towards achieving some simplification in congress arrangements, without sacrificing anything essential in the scientific value of the Congress, and at the same time providing, in some measure, for the social side of our activities. But I can at the same time promise that we shall make every endeavour to ensure that the Edinburgh Congress will be both successful and enjoyable.

Ladies and gentlemen, I have the honour to invite you to meet in Edinburgh in 1958.

The invitation of Professor Hodge was received with great applause. The Chairman stated that the Congress had unanimously decided that the Congress 1958 will take place in Edinburgh.

After this Professor Hopf, Zürich, addressed the Congress.

[We omit the speech by Hopf]

The last words of the Congress, spoken by the Chairman, were the following:

**4.3. Closing Speech By Prof J A Schouten.**

First speaking as President of the Organising Committee, I wish to distribute the general thanks given by Prof Hopf. There are a great number of persons and organisations that have to be thanked for their kind interest and help. I remind you first of the interest that was taken in the Congress by Her Majesty the Queen who was kind enough to receive a delegation of the Congress and of the honour His Royal Highness Prince Bernhard of the Netherlands did us by consenting to be our Patron. The Ministry of Education helped us not only financially but in every way possible. The City of Amsterdam gave us its financial help and the never failing assistance of all local authorities during the whole period of preparation. Without this assistance a Congress like this with so many participants and with all its excursions and entertainments could not have been organised.

A great number of industries in the Netherlands helped financially. From the great number of other organisations that have to be thanked for their kind cooperation only allow me to mention the I.M.U., the C.I.E.M., I.C.M.I., or I.M.U.K., the Committee for the Fields Medals and the Organising Committees of the three symposia that took place in connection with the Congress.

The Organising Committee formed sub-committees and sub-sub-committees, till at last nearly every mathematician in the Netherlands belonged in some way to the big committee resulting in this way. To all these cooperators thanks must be given. Last but not least all mathematicians who have come from abroad to Amsterdam must be thanked, because they made this Congress. The Organising Committee only provided the opportunity of doing it. I express the hope that they have all found here something interesting and something that gives them much stimulus for their future work.

In the opening session I drew attention to the fact that mathematics nowadays has a much more important place in the world than before. Now something about the consequences. It is an experimental fact that mutual understanding is very difficult as long as problems can not be formulated in an exact way. For instance it is much easier to avoid misunderstanding in the field of physics than in the field of economics or political sciences. Now mathematics is in one of its aspects the science of exact formulation and that means that better understanding may arise where problems can be formulated mathematically.

In our days several branches of mathematics as for instance mathematical statistics, can be applied not only to physical problems but also to problems from many other fields, for instance, economics. Of course, problems in real life can never be reduced to calculation alone, but exact formulation and some calculations, where they are possible, can do much good. So we may hope that mathematics in its modern development will become more and more useful for the mutual understanding of mankind as a whole. With these words full of hope for the future I close the Congress 1954.