### PRESIDENTIAL ADDRESS

**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 Liége, 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: "To His Royal Highness Prince Bernhard of the Netherlands. The International Congress of Mathematicians 1954 at 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".

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 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!

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 (Mass.) 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 40Y2 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 organization. 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 overstressing just that aspect of geometry which is most uninteresting for young people of an age between 12 and 18. 1 am glad that in section 1711 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 organize 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 organization of future congresses.