One of the goals for the WMY 2000, as stated in the Rio de Janeiro declaration, is that efforts should be made to improve the image and presence of mathematics in todays " information society ".
This goal is sometimes understood as follows : Mathematics has a poor image in the society because most people see it only as a boring school subject which they either failed or barely passed. So, at the occasion of the WMY 2000, mathematicians should undertake actions to show how exciting mathematics can be, and how widely applicable and actually applied it is in the modern "information society". Mathematical competitions, exhibitions, popular lectures should be organized to serve this purpose.
This line of thinking seems to be based on the assumptions that (a) peoples apprehensions vis à vis mathematics are ill-founded, and (b) that mathematicians know what mathematics is and what has been or should be its role in society and the only thing that remains to do is to convey this clear understanding to " the general public ". I think that, instead of taking these assumptions for granted, the WMY 2000 could be an occasion to discuss them within the mathematical community itself. It may well be that our own " image of mathematics " does not always agree with reality. I am thinking here, in particular, about the forever repeated clichés that " the language and values of mathematics are universal ", and that the learning of mathematics plays " a key role for the development of rational thought ", which found their way to the " draft resolution " about the WMY 2000 presented to the UNESCO Conference in Paris in November 1997 (see WMY 2000 Newsletter, no. 5). In the following, I shall look at each of these statements in turn.
Cultural roots and universality of the language and values of mathematics.
The above mentioned " draft resolution " contained also a claim about the " cultural roots of mathematics ". I certainly agree with this, but I think that, if one takes a cultural view of mathematics, then one has to be cautious in the interpretation of the claim about the " universality of the language and values of mathematics ".
If we see mathematics as a specifically human way dealing with change and complexity of the natural and social world then we can make some claims about its universality : Mathematics is a universal cultural phenomenon, in the sense that every culture creates a mathematics. For imagine a world, in which change and complexity have not been mentally organized into relationships, dependencies, patterns, order and tools for the comparisons of magnitude, all of which transcend the immediate historicity of the human life. Such a world would be unbearable and terrifying : it would be a Chaos, an antithesis of Cosmos. Because the mental " organizing systems " have to do with abstractions such as pattern, order, relationship, measure, they can be seen as prototypes of " roots " of mathematical ideas. As the mental "organizing systems " are a lifes necessity for humans living in social groups, they do appear in all cultures. But this is where universality ends, because the ways in which these " organizing systems " developed differ widely from culture to culture. The most differences are between oral cultures and cultures in which the written language plays an important role. But there are differences within cultures with written language as well. This is true in the historical sense, as when we compare ancient Western and Eastern mathematics (see, e.g. Joseph, G., 1991, Crest of the Peacock : Non-European Roots of Mathematics; Tauris, London). But it is also true when we look at the different cultural and institutional " niches " in which mathematics is practiced and developed in todays world. Research mathematicians, architects, construction engineers, brokers, financial advisors, stock market employees, actuaries, are speaking different languages and what they value in the mathematics they are practicing is not the same.
Mathematics, rationality and power
It is said that the learning of mathematics is useful because it promotes the development of " rational thinking ". Two assumptions could be underlying this belief. One is that rationality should be modeled on mathematical thinking ; another is that a widespread rationality creates a society of intellectuals or a " government of reason ", free from criminal violence and wars. These assumptions were quite strong some thirty years ago, in the international New Math curricular movement. Teaching mathematics to all children, not as a set of computational and algebraic rules but as a theory-in-the-making, was supposed to turn these children into " little mathematicians ", focused on beautiful ideas and not on bodily matters. This was assumed to be and easy task, because, according to Piagets psychology (a major reference at that time), the childs " natural rationality " is mathematical (roughly speaking, in the sense that human thinking is based in structures of mental operations analogous to groups of transformations in mathematics).
With all due respect for the ideals and hopes of these people, we see today that they were wrong on several counts. It is very difficult to speak about something like " human natural rationality ". People normally develop a range of " rationalities ", situated in a variety of contexts of practice with which they engage. There is the " pragmatic logic " of everyday conversations at home, which is different from the logic of professional argumentation and instruments of persuasion in the workplace, which is yet different from the logic of mathematical proofs. One obvious difference is that the ordinary language does not satisfy the law of tertium non datur, and, for example, two negations do not cancel each other out. There are many negations, not just one negation in the ordinary language (e.g. " unhappy " and " not happy " do not mean the same thing). But even " the logic of mathematical proofs " is not ahistorical or acultural, nor is it independent from the socially constructed standards and styles. In the XIXth century, it was still all rigtht to say, for Abel, that Cauchys theorem about the sum of a series of continuous functions " souffre des exceptions ". So even " mathematical rationality " is historically and culturally situated, and is not, in some sense, " absolute ".
It can be a dangerous thing to postulate a hierarchy of these different " rationalities ", and tell the child who comes to school, that the way he or she thinks in everyday situations is all wrong or inferior and that the school, and especially learning of mathematics, is going to straighten it up. This, however, happens at school all the time, and is the cause of many school failures. It is also one of the reasons why so many people grow with a deep dislike of mathematics.
The hope that a " democratic government of reason " is going to prevent wars is also bound to failure. Wars are caused not by the paucity of rationality but by an excess of greed for money and power.
While academic mathematics may have little to do with the rationality situated in peoples family of professional lives, it seems to have a lot to do with power. According to some researchers on situated cognition (e.g. V. Walkerdine, 1988, The Mastery of Reason, Cognitive Development and the Production of Rationality, London and New York : Routledge), the very " joy of doing mathematics ", so often evoked by those trying to motivate students to learn mathematics consists in the pleasure of control : " the learner of mathematics is not caught in the play of desire in the Imaginary, but believes himself to have control of it. ... It is extremely powerful and it involves the manipulation of a universally applicable symbolic system - a fantasy of playing God, " the Divine mathematician ", the fantasy inscribed in the Cogito, the Ratio " (ibid.). - The US based National Council of Teachers of mathematics overtly state in their " Standards " that the aim of mathematics education should be the achievement of " mathematical power " by all children. For them, the " mathematical power " is the key liberating factor in todays technological world, opening the gate to a better life and a better society.
But " mathematical power " is seen by some people in a negative way. If success in mathematics is perceived as " empowering " the individual, the failure is associated with an overwhelming feeling of powerlessness, lack of control (strangely absent when one fails, for example, in geography or philosophy!). Mathematics is seen as the " gatekeeper " (rather than as a " gateway ") to many professions and well paid jobs. It is hard to dispel this apprehension because the entry into many academic programs is based on a selection through mathematics examinations or courses with material and approach to its teaching that have little to do with the target profession. This apprehension is expressed in no uncertain terms by words such as " mathematical imperialism " or " colonialism ".
On a more global plane, " mathematical power " is seen by some as destructive, when the involvement of mathematicians in war industry is evoked : mathematicians are sometimes thought responsible for much of what is called the " destructive power " of the Western culture. And this responsibility will have the tendency to grow in the future " information societies " with their menace of " wars in the cyberspace ".
All in all, mathematics seems to be a mixed blessing, implicated in many cultural and political issues which it would be useful to discuss at the occasion of the WMY 2000.
" The WMY 2000 Newsletter could become a forum for this discussion, but interested Readers are invited to propose other forms and modes of exchange of thoughts on the themes raised in this editorial (for example, a special conference in the year 2000, informal meetings of mathematicians working in industry, business, inner city schools, mathematics departments at universities, etc.) ".
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