# Editorial

The Twelfth General Assembly of the International Mathematical Union (IMU) was held in Lucerne (Switzerland) on July 31st and August 1st, 1994, just before the International Congress of Mathematicians, ICM'94, in Zurich.

It was on the occasion of the previous General Assembly in Kobe (Japan), August 18-19, 1990, that the following Resolution was voted:

"Since the IMU wishes to mark the turn of the century in a manner appropriate to the standards set up by David Hilbert in 1900, the General Assembly directs the Executive Committee (EC) to set up a committee to report to the adhering bodies by September 1991 on how to accomplish this, so that in 1994 the Assembly can discuss it and decide how to proceed".

Following this Resolution, the EC of IMU created a Turn of the Century Committee", chaired by Professor Jacob Palis Jr, Secretary of IMU, and having as members Professors V. Arnold, F. Hirzebruch, L. Lovasz, B. Mazur, S. Mizohata, W. Thurston, J. Tits, and S. Varadhan.

The IMU launched in May 1992 the World Mathematical Year (WMY 2000) with the sponsorship of UNESCO (Professor Federico Mayor), of the Third World Academy of Sciences (Professors Abdus Salam and Carlos Chagas), of the French Minister of Research and Space (Professor Hubert Curien), of the Brazilian Secretary of State for Science and Technology (Professor Helio Jaguaribe), of the Brazilian Academy of Sciences (Professor Israel Vargas) and, ICM'94 being organized in Switzerland, of the Swiss Federal Counsellor (Dr. Flavio Cotti). The document issued after this meeting was named "Declaration of Rio de Janeiro for Mathematics".

At the Lucerne General Assembly, a report was presented on the work of the Turn of the Century Committee and on the plans for WMY 2000.

The corresponding proposals were unanimously endorsed by the General Assembly:

## Resolution 2

"The General Assembly thanks the Turn of the Century Committee for its report. It asks the new Executive Committee to proceed with the planning of World Mathematical Year 2000, and to organize and coordinate activities such as:

1. inviting a select group of outstanding mathematicians to present their views on topics they expect to be central to mathematical activity in the next century.
2. selecting a number of symposia, some possibly organized together with other scientific bodies, dedicated to mathematics, its applications and its role in society.
3. events to be held under the auspices of the International Commission for Mathematical Instruction (ICMI), the Commission for Development and Exchange (CDE) and the International Commission for History of Mathematics (ICHM).

The Executive Committee is asked to explore the possibilities provided by communications technology for uniting activities the world over".

It is to be noted that, in her address at the Opening Ceremony of ICM'94, the Federal Counsellor, Mrs Ruth Dreifuss, endorsed WMY 2000.

Following these important decisions, there was considerable exchange of views on the subject during ICM'94, where many ideas and proposals were discussed.

A world-wide coverage for the Newsletter should therefore be systematically organized.

The Editors of the Newsletter, starting 1995, are :
Professor Mireille CHALEYAT-MAUREL (FRANCE)
Professor Gérard TRONEL (FRANCE)

World Correspondents for the Newsletter are :
Professor A. A. ASHOUR (ARAB REPUBLIC OF EGYPT)
Professor Mohammed H. A. HASSAN, Third World Academy of Sciences (ITALY)
Professor Angelo MARZOLLO , UNESCO (ITALY)
Professor Mitsuo MORIMOTO (JAPAN)
Professor Mogens NISS, Secretary of ICMI (DENMARK)
Professor Rolando REBOLLEDO, Chairperson of CDE (CHILE)
Professor Anna SIERPINSKA, Vice President of ICMI (CANADA)

All ideas and proposals connected with WMY 2000 should be sent to the Editors and to the World Correspondents.

It may be useful here to comment on part c) of the Resolution voted at the General Assembly of Lucerne. We recall, among the goals of WMY 2000:

• help for less developed countries. It is essential that the countries that are members of UNESCO be gradually able to reach a level justifying their admission to IMU by the turn of the century. This implies considerable additional efforts in the field of education, training and access to scientific information (Aim No. 2 of the Declaration of Rio de Janeiro).
• promotion of the image of Mathematics. Mathematics should be systematically present in the world of Information Sciences, thanks to examples and applications which will be scientifically exact and open to the largest number of people (Aim No. 3 of the Declaration of Rio de Janeiro).

Here it is important to note the complete agreement of these goals with those of the American Mathematical Society (AMS). They are all related, in one way or another, to the task of teaching mathematics, hence to the key role played in this respect by ICMI, CDE and ICHM.

We also want to stress the part to be played by the various electronic communication technologies. Witness the growing importance of e-mail and the ever-rising number of electronic journals.

Note that WMY 2000 Newsletter is now available on the IMU's WWW server and on the server of European Mathematical Society.

Electronic libraries and tele-conferences should be a set of fundamental tools for the end of this century and for the century to come. These tools should be made available to all mathematical centres in the world as soon as possible.

After our international discussions in Zurich, let us now all work together to achieve these goals.

Jacques Louis LIONS

# ICMI's INVOLVEMENT IN WMY 2000

## GENERAL PERSPECTIVES

The International Commission on Mathematical Instruction (ICMI) is very enthusiastic about WMY 2000 and is eager to contribute to activities up to and in that year as best it can. This article offers a brief outline of ICMI's ideas and measures so far. ICMI believes that it is important to establish rather clear overall purposes and general goals for the WMY 2000. The very idea of a World Mathematical Year is an extrovert one, attentive to the relations between mathematics, in all its manifestations, and the world in which it evolves. It seems that the key "problematics" is that in spite of the social and cultural significance of mathematics, its nature, roles, and functions are, to a considerable extent, invisible to the world outside the mathematical community. We therefore propose that the main task of the WMY 2000 be to "make mathematics and its role in the world visible to society in general, and to the general public in particular". The primary aim should not be to advertise and propagate the marvels of mathematics - lobbying and propaganda will hardly convince the kind of audience we would want to reach; on the contrary, they are more likely to be counter-productive. Instead, the task should be to reveal the five-fold nature of mathematics, as a pure science, an applied science, a system of instruments for decisions and actions, a field of aesthetics and, last but not the least, one of the major teaching/learning subjects in modern times. We should strive to present and demonstrate, and not just claim, these properties of mathematics. In so doing it should be emphasised that the creation and use of mathematics as we know it rely on human activity; that mathematics have a social and cultural history, and that mathematics has intimate relations with philosophical, scientific and practical issues.

## A GLOBAL CONGRESS OF MATHEMATICAL EDUCATION

It is a matter of course for ICMI, as IMU's commission on mathematical education, to place particular emphasis on educational activities in the WMY 2000. A key event of the year will be the Ninth International Congress on Mathematical Education, ICME-9. The congress will take place in a city, yet to be chosen by the ICMI Executive, by and large in accordance with the "classical format" of such congresses. However, in addition to the congress proper, ICMI intends to explore ways to expand and enhance ICME-9 into an event of even greater global significance by means of international communication networks. Thus, it might be possible to organize a number of simultaneous satellite conferences in strategically located places around the world, and provide interactive communication links between them and the main Congress. Such an arrangement would make it possible for huge numbers of people to participate actively in one global Congress and so place mathematics education on the agenda in all parts of the world. Should technological or financial constraints turn out to make this project unrealistic, a more modest scheme could be adopted in its place. A sequence of regional ICMI conferences could be organized, each devoted to a crucial issue or theme of mathematical education in that region, preparing to culminate in ICME-9.

## THE ROLE OF MATHEMATICS IN CULTURE AND SOCIETY

A great variety of public media should be activated to present, as concretely as possible, the ways in which mathematics continues to play an important part in modern society and culture, as it has done in the past. One idea could be to involve in this task the greatest communicators among mathematicians, mathematics educators, scientific writers, producers of scientific films and TV programmes, designers of exhibitions, etc.. If measures are taken in due course, it might be possible for IMU, ICMI and its affiliated study groups, to commission books and articles, public lectures, films, videos, TV programmes, computer-based information technology materials (including CD-roms), museum design, travelling exhibitions, and so forth, all touching on various aspects of the role of mathematics in the world of today and yesterday. As a very concrete example of what could be done, WMY 2000 organizers could invite a number of internationally well-known and influential politicians, composers, industrialists, journalists, film or theater directors, writers, scientists, and many more, all outside what could be labelled as the mathematical community, each to write a chapter of a book telling about his personal encounters and relationships (whether positive or negative) with mathematics. Again, the intention should not be just to produce rosy stories and propaganda lauding the marvels of mathematics but to give honest and authentic accounts of serious people's mathematical experiences. Another concrete idea is to establish a prize for the best talk given in public, and for the best book appealing to the general public, on the role of mathematics in the world.

## ICMI's PARTICIPATION IN WMY 2000

ICMI looks forward to contributing to the setting up of WMY 2000 in various ways, some of which are sketched above, while others are yet to be identified. The Executive Committee has decided to establish an Ad Hoc Committee to make specific plans for ICMI's involvement in the WMY 2000. The EC is in the process of appointing the members of this Ad Hoc Committee. Readers of this Newsletter will be informed of the composition of the Committee when it is complete.

Mogens Niss, Professor, Secretary of ICMI

# MATHEMATICAL INSTITUTIONS' PARTICIPATION IN WMY 2000

## EUROPEAN MATHEMATICAL SOCIETY (EMS)

The European Mathematical Society wishes to take a very active part in the preparation of WMY 2000. The third EMS Congress will be held in 2000 and should be included in the activities of WMY 2000. A special Committee with correspondents in each European country will shortly be appointed by the Executive Committee.

## BERNOULLI SOCIETY

The Bernoulli Society for Mathematical Statistics and Probability is in the process of establishing a committee to enable the Society to participate in the activities of WMY 2000. The committee will be chaired by past president O.E. Barndorff-Nielsen and its scientific secretary will be Professor Rolando Rebolledo. The scientific secretary of the Bernoulli Society, presently Professor Richard Gill, will be an ex officio member of the committee. Once the committee is in place, the names of all its members will be given in the next issue of the Newsletter. The 5th World Congress of the Bernoulli Society will be held in 2000, and it should be included as part of the WMY 2000 activities.

# INTERVIEW OF DAVID MUMFORD, PRESIDENT OF IMU

This interview took place in May, 1995, during the IMU Executive Committee meeting in Paris.

Q : How does IMU feel, and more precisely, what do you think about the WMY 2000 initiative?

D.M. : I think it is an excellent initiative and I think it offers a great opportunity to try and improve the image of mathematics and to try and give some inspiration to the mathematical community. This should be done in terms of several factors like Hilbert looking ahead to where we stand in mathematics and what the major challenges are and, at the same time, looking at our relations with the applications of mathematics, looking at the issues in mathematical instruction and looking at the hopes of involving the Third World more intensely in mathematical activities.

Q : Specifically on these three topics which have been defined for the WMY, have you a personal view on what the challenging problems in mathematics will be in the 21st century ?

D.M. : Ah! That is really a great question ; one would want to sit down and think about it much longer! I personally feel that a very important issue is to restore the free interchange of ideas between pure mathematicians and applied mathematicians. During the 19th century, you see that most mathematicians were both. Fourier series, for instance, were inspired by applications. But there has been a divergence, especially in the US, although I think to some extent in other countries too. I hope that the present emphasis on the usefulness of science in general (if the public is going to put money into scientific research, then it will be useful), will not lead simply to pushing applied subjects at the expense of work on pure subjects, but rather to a sense of common purpose where pure researchers can find inspiration in applications and use their theoretical ideas to grapple more effectively with applied problems. I think there has always been an interchange, but that will really be a major challenge.

Q. : What is your involvement in mathematical education ?

D.M. : It is a vast question. I'm involved in a particular project in the US, called "The Calculus Reform movement", where the goals have been to make instruction in Calculus more relevant to people who are specialized in all kinds of science and engineering and financial professions. Specifically, there is a group lead by Deborah Hughes-Hallet and Andrew Gleason which has written a series of texts now. They have what they call rule three : instead of simply teaching formulas and algebraic manipulations which are often the hardest things for people to apply, to understand their relevance, they are putting equal stress on numerical work and stating every general concept in terms of "How actually do you compute it on a calculator, how can you approximate when you have the data in front of you ?". That's one thing. Secondly, there is the visualisation approach that seeks to make effective use of graphical techniques to see the geometrical side of concepts. Finally, one must not forget algebraic manipulation. I should add that with these three technical methods of tackling the subjects, they also combine the idea of always including meaning for applications at every stage, so drawing applications from physics and chemistry, from economics. For instance, teaching multivariable calculus, when they start with integration. They start off with a graph representing the fox population in South West England which comes from this bible book of mathematical biology by Murray. And the question is : "How do we estimate from this graph giving the density of fox presence, the entire fox population?". So, this, of course, leads to and motivates integral calculus. I strongly believe that mathematical instruction should make a very serious effort to try to have real applications in mind. If you go back to the 19th century, you will find that almost all elementary school teachers often used examples with money : lending money and borrowing money, and so on, and every child understands money. This is concrete. I think statistics is a very important subject to bring into the school curriculum, because the kind of errors that schoolchildren are apt to make are very often errors in numerical judgement ; if you introduce the subject in schools, they will be better prepared.

Q. : What would you like to see in the Newsletter WMY 2000?

D.M. : What I hope will happen is that more concrete activities will gradually be formulated, with concrete plans for involving different groups, and so I think the Newsletter will evolve around these initiatives. We talked in the Executive Committee of IMU about the issue of the image of mathematics and tried to identify methods for explaining what mathematics is to the general public. I think there is no simple answer to this question. I think it is a very difficult challenge, I believe the most important thing is to find some individuals who are highly motivated and have the ability to put themselves in the place of the scientific public. Some people have this skill, and some don't. I think we have to resist a strong tendency to give lectures that are always more and more specialized.

Q. : Have you projects to change a bit the way IMU is organized?

D.M. : I have had two projects that I felt to be very important since people asked me to be President, that we are moving ahead on. One of these concerns the perception from the point of view of the US that the Union is some mysterious rather secret organization that is never clearly understood. Maybe one goal is to publicize what the Union is and what it does: we hope to have next April in New York City a meeting in connection with the meeting of the Executive Committee, to have a public forum, a round table discussion on Union activities, so the general public will see the valuable things it does. The other issue that I have been involved with is trying to increase the participation of applied mathematicians in the congresses and the method for doing this that we want to explore is to have joint activities with applied mathematical organizations. So we are not just talking about a subcommittee of the Union itself, but about involving groups like ICIAM, computer scientists, statisticians, mathematical physicists.

# LETTERS TO THE EDITORS

## In spite of the scientific value and the historical interest of Hilbert's famous Twenty-Three Problems enounced in 1900, one must agree that mathematics have developed in unforeseen directions throughout the XXth century.

In 1992, a symposium held in Luxembourg addressed the comprehension of that evolution during the first half of the present century. One issue was the publication of the volume "Development of Mathematics 1900-1950" (Birkhäuser, Basel, 1994, 3-7643-2821-5, 0-8176- 2821-5)" with articles by J. Dieudonné, J.L. Doob, G. Fichera, M. Guillaume, W. Hayman, C. Houzel, J.-P. Kahane, A. Lichnerowicz, J. Mawhin, L. Nirenberg, J.-P. Pier, W. Schwarz. The book presents a chronological list of major results obtained during the period under study (P. Dugac, B. Eckmann, J. Mawhin, J.-P. Pier); it also contains a list, drawn up by P. Dugac, of original and reference sources.

Whereas it is already hazardous to undertake this investigation of evolution up to 1950, a still more ambitious task would consist in tracing these movements during the later period, as former classifications and structures do not seem to apply any more. A group of persons has started on this historical and thematic study in view of World Mathematical Year 2000. They should appeal to specialists for contributions covering a wide spectrum of themes which would ultimately be linked by logical connections.

Liaison address :
Jean-Paul Pier, Mathématiques
Centre universitaire de Luxembourg
162 A, Avenue de la Faïencerie, L-1511 Luxembourg (Luxembourg)
Fax : (352) 46.66.44.237.
e-mail : Den här e-postadressen skyddas mot spambots. Du måste tillåta JavaScript för att se den.

## MUSIC AND MATHEMATICS : DREAM OR REALITY ?

Some years ago, I used to be irritated by the commonplace saying that mathematicians had a privileged relationship with music. However, for some time now, a number of contacts with people whose intellectual life was devoted to these two poles, and my own experience as musician and mathematician, have led me to rethink this view. In the Middle Ages, the Quadrivium included the four basic sciences which were arithmetic, geometry, astronomy and music. And Leibniz still spoke about the "secret calculation". Could this close tie between the two subjects over many years account for the symbiosis which we seem to note today? This question leads to another : although the medieval learning mentioned earlier derived from the Greek system, it was widely dominated by the Western European schools. Did this give rise to a qualitative or a quantitative difference in the relatedness of the two subjects in Western Europe and in countries with a completely different culture like China or India for instance? These are questions we would like to attempt to answer in the context of a study for the World Mathematical Year 2000. A questionnaire was proposed to the members of the Société Mathématique de France (SMF) with a view to collecting statistical data that will allow us to confirm or to infirm this music-mathematics relationship. We are planning to carry out a counter-test in another scientific area (possibly chemistry) to see if differences emerge.

This operation has already been successful among members of the SMF. We would now very much like to submit it to a wider sample. The questionnaire is now available in English on the web : Questionnaire Mathematics and Music . In advance we wish to thank all those who agree to spare a little of their time (not more than fifteen minutes) to fill in the questionnaire. Also, as I am new in the field, I am slowly discovering that many people have already delved into this issue from a historical, psychological or philosophical point of view. So I do hope that the people I have not yet had a chance to get in touch with will kindly let me have their opinion (and a bibliography, if they wish) in order to give more weight to the project.

Liaison address:
Laurent Mazliak
Laboratoire de Probabilités
Université Pierre et Marie Curie
Tour 56 3ème étage
4, Place Jussieu
F-75252 Paris Cedex 05
France

# ANNOUNCEMENTS, NEWS FROM SCIENCE MUSEUMS

The Editorial board would like to open a column for information on the two last aims of the "Rio de Janeiro Declaration". It would be very interesting to collect more for the next issues.

ICME8
The Eighth Congress on Mathematical Education (ICME8) will be held in the city of Seville (Spain), from July 14 to 21, 1996. Electronic information is available on Mosaic. The URL is :http://icme8.us.es/ICME8.html

## FIRST SCIENCE CENTRE WORLD CONGRESS

Vantaa (Finland), June 14-18, 1996.
The Association of Science-Technology (ASTC) and the European Collaborative for Science, Industry and Technology Exhibitions (ECSITE) invite Science Centre professionals to the First Science Centre World Congress on the general theme Learning for tomorrow ".
Informations can be obtained from :

Ms. H. Von Troil,
HEUREKA po Box166,
FIN-013001 Vantaa (Finland).

## CITÉ DES SCIENCES ET DE L'INDUSTRIE, PARIS (France)

In the Science Museum of Cité des Sciences et de l'Industrie (Paris, La Villette) was opened a new presentation of the permanent exhibition in the mathematical department. The originality and the aims of this presentation are to offer a more attractive and better image of mathematics to the general public visiting the museum. Starting with models from daily life or Pythagoras' Theorem, the visitor can go on to the theory of complexity used in the technical world (computers, planes,...). He can also enter domains using probability theory, statistics, data analysis in economical and social sciences. There are short movies on "Proofs" and "Modelling in Applied Sciences". The staff in charge of the mathematical department hopes to improve the transparency of mathematics in daily life.
Address :

Cité des Sciences et de l'Industrie
30, Avenue Corentin Cariou
F-75019 PARIS (France).

## NATMATH

The ECSITE organization plans a travelling exhibition (called NATMATH) on the theme : "Mathematical Modelling of Nature ". The first goal of the exhibition is to cultivate interest in mathematics among the visitors to European museums and science centres, and to instil confidence in their understanding of the value of mathematics and its role in their comprehension of the world around them. NATMATH is planned for stays of three to six months at ECSITE museums.
For information, contact the project leader :

J. Wagensberg,
Director of the Science Museum
Teodor Rovilralta
SP-08022 BARCELONA (Spain).

## A LOGO FOR WMY 2000 ACKNOWLEDGEMENTS

All ideas and propositions are welcome. We suggest the following basic principles :

• all symbols of the abreviation "WMY 2000 " must be used,
• the pictures must represent the notions of  "WORLD " and " MATHEMATICS ".

## ACKNOWLEDGEMENTS

The Editorial board wishes to thank the following institutions for help and sponsorship : UNESCO, IMU, Comité National Français des Mathématiciens, Collège de France, École Polytechnique, Institut des Hautes Études Scientifiques, Institut de mathématiques (Jussieu, Paris).