Round Table
Shaping the 21st Century
3ECM, 11-14 July 2000, Barcelona
Some ethical aspects in the mathematical activity
Miguel de Guzmán
Universidad Complutense de Madrid
There are many aspects of our mathematical activity that strongly involve our sense of duty and of responsibility.
There are also many ethical values that could be very necessary for the adequate development of society now and ever that can and should be fostered through the dedication to mathematics.
I consider that we, in our present time, have disregarded many of these aspects of mathematics that in some other periods of the history have been at the forefront. I think it would be necessary to restore many of the attitudes that were commonplace among the mathematicians of other periods.
This is the reason I think it worthwhile to talk about ethical aspects and ethical values in our mathematical activity in a place like this, in which we try to look towards the future. Our mathematics will be more healthy internally and more influential in society if we try to reinstate the ethical values which are deeply rooted in it.
Of course, we, mathematicians have particular duties which derive from our belonging to the community of researchers, educators, or to a particular institution or business. These duties, in which I am not going to enter in detail, are very well outlined in some interesting articles and authoritative documents, for example in the Ethical Guidelines of the American Mathematical Society.
As belonging to a global society we have also particular duties of fostering social justice not only in our own country but also in the community of countries. To this task we can efficiently contribute from our particular position as mathematicians under the basic assumption that social justice implies much more than sharing material goods.
Since mathematics will play, and is already playing, such an influential role in the progress of our civilization we have ahead of us the task of trying to make the knowledge and application of mathematics adequately shared by all different communities in our world.
It is our responsibility not to permit that any country lacks the necessary means to give its citizens the correct mathematical education through which it might advance towards a just situation in its cultural and eonomical development. We have a duty to apply our personal and social solidarity in order to try to achieve a better situation of equity and justice in this respect.
But let us come to what is more internal. What are the specific values that mathematics can foster?
Mathematics, from long ago, has been strongly linked to a peculiar view of the universe, which makes one suspect that at the same root of mathematics one can find certain aspects that stimulate a deeper vision of our science.
Mathematics, as we conceive it today, was born within the Pythagorean religious-scientific community at the end of the sixth century BC Then it was mainly considered as a means through which man could pierce some of the many enigmas which the contemplation of the physical world presents to the human mind. Mathematics was for them a way by which man could perceive «the roots and sources of nature» as they expressed it.
At that time mathematics was rather far from being the mere routine technique in order to master some aspects of our physical world that it is today for most of us. What Pythagoras and his followers started to perceive in their mathematical reflections were the deep harmonies present in this world where we live. And in these considerations they based their ethic and religious way of life.
If the universe is so harmoniously constructed as we see it through our mathematical exploration, it seemed a clear consequence to them that our lives should try to conform to such harmony, first of all by contemplating it with veneration and then by respecting and fostering it. And this not only in the more external physical aspects but also in what is more specifically human, through the special respect toward life in all its forms, and much more through the mutual relations with the persons around them, both human and divine.
The mathematical life was in a certain way among Pythagoreans a guide to contemplation and even to behaviour. Their attitude might constitute for us a good lesson in ecological conduct, but we have missed it by converting our mathematical task in a god proportion into a rather empty routine. And this is even more regrettable in the mathematical education of our youth, where the capability of mathematics and of science in general in order to offer a deeply integrated human education is most needed.
It is obvious, of course, that mathematics has also been, and should continue to be, a science in search of truth, a tool helping all other sciences, technologies and the most varied activities in our civilization, and also a creative art which tries to produce a beauty only perceived by the eyes of the soul, as Plato said. And it is quite clear that to be efficient in all these respects one has to acquire a basic mastery of the most important tools. But it would be good not to forget the richness of the original conception of the Pythagoreans.
The same nature of the mathematical task makes it capable of stimulating important ethical aspects that we should foster in us and that we should try to instill in any healthy educational system of our days.
Mathematics can be conceived as a peculiar exploration of certain structures, physical as well as mental, that appear in the world around us and in our mind. These are the objects that admit the specific rational approach through symbols that we call mathematization. Through it we obtain a certain mastery of the realities concerned.
For example, the mathematical mind approaches the multiplicity of things and creates the number and the arithmetic to master it. It approaches the form and figures in space and creates the geometry. Mathematics explores more deeply the symbols already present in the numbers it has created and constructs the algebra. It analyzes the changes and transformations in space and time and there arises the mathematical analysis, and so on.
In this task the role of the human mind consists in interpreting, as well it can, the structures of reality, facts that appear to it as previously given. This constitutes one of the profound experiences each mathematician lives every day: the feeling of being following some tracks that were there before and that have been illuminating his or her work. This submission to the truth and to reality, that normally so deeply rooted is in the scientist, is without any doubt one of the important features we should appreciate and foster in us and in others.
This passionate search for the truth, for the reality of the situation, should be one of the typical traits of the scientist, and much more so of the mathematician, for whom it is usually easier to distinguish a working hypothesis from an incontrovertible truth.
The joyful acceptance of this truth, whoever might be the first one in finding it, and no matter whether the true situation contradicts or not our expectations, is one of the generosity features that may ornate the mathematician’s attitude. The joy in the contemplation of the mathematical truth and in the sharing of its beauty is the bonus he receives from this open and generous behavior.
The sense of profound humility one feels before the multitude of truths still to be discovered is another one of the positive ethical attitudes mathematics can foster. Newton expressed it in beautiful words: I seem to have been only like a boy playing on the sea-shore and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lays all undiscovered before me.
The mathematical quest makes us feel, more so than in any other science, closer to all those who have been working with enthusiasm today and also with our more distant antecessors. The theorems that were found by the Mesopotamian or by the old Greek mathematicians are today as valid as ever. Mathematical work is a sharing task. Newton himself said: If I have reached anything it is due to the fact that I have been on the shoulders of giants. And for this reason through our work in mathematics we can learn to clearly perceive this common responsibility of making our culture advance.
Mathematics is based from its own beginning on consensus. Their first sentences are called postulates and the definitions of their new objects we slowly introduce are also conventions that require agreement. On this basis we raise the mathematical building. The acceptance of the agreement and consensus is another one of the ethical attitudes so necessary in our future society. And mathematics is a very adequate field to foster them.
Mathematics is consensus, is submission to reality, but also, and in a very important part, it is creative freedom. As George Cantor solemnly said at the beginning of the 20th century, «the essence of mathematics is freedom». And it truly happens that in the same way that the artist who intends to express for others’ profit a personal experience he has achieved, the mathematician also has at his disposal plenty of different tools and ways to do it. Mathematics is, without doubt, discovery, but also an adventure, a free creation.
And now let us say something about some of the risks to which our mathematical activity and the general trend towards mathematization is exposed.
The successes that mathematics has achieved along its history, and specially in our times, are so important that they can make us forget the limitations of the mathematical thought that come, together with its power, from the innermost of its nature.
The success of mathematics is very much due to the facta that mathematics necessarily must mutilate reality in some way, has to abstract from it. By such mutilation we are able to control some interesting aspects of reality, but not the reality as a whole. We sometimes feel tempted to believe that we control reality completely by our mental constructions forgetting that we have omitted some aspects from our considerations which may be very important to man as such, not merely to man as producer of artifacts.
In what follows we shall try to schematically describe some of the major risks to which both general and scientific thought and even philosophy are exposed if they allow themselves to be deceived by the brilliance of mathematic’s success and try to adopt its methods of exploring reality indiscriminately. Unfortunately, some important aspects of human culture seem to have started to move along this dangerous road.
First of all, there are some risks in the mathematization of the sciences. In an interesting paper entitled The Pernicious influence of Mathematics on Science J.T. Schwartz presents several reasons why the style of thinking to which the mathematization is prone may not be imitable by other sciences. Mathematics tends to be single-minded in its exactness and rigor. For most of the other sciences it is normally necessary to work with more or less ill-understood approximations towards which the scientist must maintain an appropriate skepticism. Mathematization tends to be literal-minded in the persistence to maintain the departure axioms and rules no matter what happens, while other scientists have to work with the constant perception that perhaps what he has taken for an intial set of axioms is a mere first approximation to the reality behind the phenomena he is contemplating.
There are also some verious risks in the mathematization of philosophy. The success of mathematics may present to philosophers the temptation to think that the mathematics’ method should be closely followed by them. But mathematics is succesful by its abstraction, by its precision, by its rigor, which are are obtained thanks to the mutilation of reality. In a certain sense mathematics can at times forget the reality. The philosophers’task task is to deal with reality itself as it is. Precision is of course important in philosophy, but it should not be for it and idol. G.-C. Rota expressed it in his peculiar style: «How much longer will the presnet folie for precision in philosophy last? Need a concept be precise in order to be meningful and effective? Or do philosophers wish to commit hara-kiri on the altar of mathematics?»
Finally let us mention some risks in the mathematization of our everyday life.
«To believe naively that everything can be mathematized without leftovers» If mathematics itself, as Gödel theorem shows, necessarily leaves open important questions, how many things will have to remain unanswered in the attempt at mathematizing something like physics or biology?
«To allow our lifes to drown in symbols and mathematical formulation» The computer environment is filled with recipes, precise languages, formalisms, whose poit of interest is the mere operative aspect rather than the meaning of such operations. The danger in this is not that the computer becomes pseudo-human, as in certain science-fiction movies, but that man, adapting himself to his machine, turns into a robot.
«To allow the mathematicians to play the sorcerer’s apprentice» Very often in our present civilization, people think that mathematics has or will have an adequate model for each real situation, weithout taking into account that mathematization implies , as has been said, a certain amputation of reality. Many of the elements left aside in the mathematization process can be quite important for the human being, and their ommission can become disastrous. There are many aspects in human life which are too important to be left in the hands of the mathematician with impunity.
«To confuse manipulation with wisdom» Our computers make us capable of successfully manipulating important fragments of reality even if do not understand them very deeply. We can be very proud of our success. Also we can operate our own brain without understanding very well how it works. But we had better become aware that our manipulative success is rather far from the true comprehension to which we aspire. Let us not loos the meaning and attraction of the mystery.
