Mischa Cotlar's Master Class

After accepting the Senator Domingo Faustino Sarmiento Award - conferred by the Senate of the Nation on the proposal of Senator Rodolfo Terragno - the eminent mathematician Mischa Cotlar delivered a master class on Tuesday 11 April 2006 in the Blue Congress Hall. We note that he was 92 years old at this time.

Although Professor Cotlar improvised during the class, expanding the concepts enunciated in the script he used as an aid, we find it interesting to offer that script. What follows is the text that served as the basis for the master class, which the audience applauded with a standing ovation lasting several minutes. We give an English translation:

The Master Class Script

I am moved by such a high honour that is conferred on me through this award whose name is associated with nothing less than Sarmiento, to whom we owe the educational system and secular education.

I express my sincere gratitude to the President of the Honourable Senate Nation Lic Daniel Scioli and Senator Dr Rodolfo Terragno. As well as all of you who have come to gently accompany me.

Given the characteristics of this award, you would expect me to talk about general culture, and particularly about what the University was during the period 1955-1966, since I witnessed the beginnings of a National University project in a golden age and I was also the witness of its lamentable destruction, begun on the 'Night of the Long Batons'.

In this era, in which the market seems to be the goal of life, it is important not to forget that every aggressive attack that attempts to destroy the University will produce damage that will take decades to return to the path traced by those who were the true university, in love with science and culture, with different ideologies and in spite of that with long lasting ties of friendship. Where the student groups were not coloured by the interests of political parties.

Do not forget that the training of scientists and intellectuals in that university project required not only remarkable people but also an exceptional moment that we should reflect on.

However, to talk about those times there are few people prepared both scientifically and socially that can do it with the appropriate depth. While my knowledge is limited to some particular areas of science.

But there is another problem of great importance that relates to the area of culture that I could develop in a little more detail and I want to share it with you.

I am going to refer to an aspect of mathematics that attracted me to it since my youth, which is often called the Pythagorean-Platonic aspect of mathematics. And that later I was surprised by the similarity of these ideas with the principles of the Upanishads and Buddhism.

The Pythagoreans of the sixth century B.C. observed that musical harmony, geometric entities and astronomical phenomena obeyed laws that were expressed through numbers; that numbers and their proportions ruled in such different and remote areas as geometry, music and astronomy.

And even more, that abstract statements of a very different and remote nature can be logically equivalent and this equivalence is established through logical reasoning which is also abstract.

They thought like this, and they affirmed that all the areas of the universe were connected to each other by a universal Unity that manifested and realized itself through numbers and abstraction, and that abstraction was the manifestation of Unity in the mental world.

As I progressed in my mathematical learning it was a joy to see that the development of mathematics fully confirmed those Pythagorean-Platonic ideas.

And that also the great founders of current science, such as Heisenberg, Pauli, Schrödinger, André Weil, Alexander Grothendieck, had a deep knowledge of the Upanishads and Buddhism.

That Whitehead, the collaborator of Bertrand Russell and greatest expert on Platonic Pythagoreanism, gathered his best ideas in a theory known as "Process Philosophy" and later it became clear that this theory essentially coincides with the basic principles of Buddhism.

Einstein's knowledge of Buddhism, driven by the philosophies of Spinoza and Schopenhauer, was perhaps deficient and sometimes wrong, but he pointed out a fact of fundamental importance: the goal of mathematics or of mathematical physics is the same as that of Ethics, but with applications to different fields.

That Ethics without Science is blind and Science without Ethics is lame.

The objective of both is to realize the Unity, but this realization will be complete only if they do not contradict when both fields are applied in a common area.

That is, Einstein seeks to unify the Platonic Pythagoreanism, that realizes the unity by abstraction in the world of mental objects, with Buddhism and the Judeo-Christian tradition that seek unity in the world of living beings, the basic problem of Ethics.

The world in which we live contains these other two worlds and here it is essential that the applications of Ethics and science do not contradict each other.

Although Einstein did not write a treatise on this, he made his ideas very clear in brief informal observations, sometimes metaphorical or mystical.

It is interesting that his ideas are very close to those of Buddhism in the interpretation given to them by Krishnamurti and the physicist David Bohm.

Einstein did not leave any formal work in which he formulated some system of axioms that allows one to deduce theorems about the Unity of Science and Ethics.

And it is clear that Plato still believed that it was possible to develop an ethics and a science that is not based on any axiom.

These problems are of utmost importance at the present time.

To think that even today, at the beginning of the twenty-first century, many people of great intelligence who are well-intentioned believe that ethics requires that every citizen must blindly obey his government and that he can glorify his country by killing living beings from the enemy country, and today there are not many people who worry about the fact that science is used in projects that endanger the life of the Earth.

On the other hand Einstein and Russell warned that the alliance of science with militarism destroys the essence of it, and that the priority problem of humanity is to seek an agreement among scientists on the correct use of science, prohibiting its application to destruction or exploitation.

Without an ethical culture based on love and cooperation, and not on rivalry, competition and the desire for power, there will hardly be a solution to the great problems of humanity.

Without an ethical culture based on love and cooperation, the development of technology will be like a dangerous weapon in the hands of a child.

The great riches are in the interior of men.

If one needs luxuries and excessive pleasures it is because their interior is still empty.

It is necessary to have an intense and urgent diffusion of the basic notions of Ethics and a clear understanding that human problems are not solved by violence but by the use of the higher faculties latent in all human beings.

Before finishing I would like to express my gratitude to the University of Buenos Aires and in particular to the Institute of Mathematics, as well as the Central University of Venezuela, and also of Dr Saúl Drajer and my friend César Orda for their constant help in the difficult moments of recent years. And to my lifelong partner Yanny, who is the greatest wealth I have.

And finally I must highlight the fact, particularly important in my general education, that my teachers, great mathematicians, as well as my Argentine and Uruguayan friends and collaborators, were people of exceptional human qualities with deep and noble concerns that transcend cold professionalism.

My gratitude to those for all that they taught me in the mathematical field and in the understanding that human quality comes before science.

Thank you so much.

Mischa Cotlar - April 2006

Last Updated May 2018