Aporism, and the problem of democracy in mathematics education.
Ole Skovsmose
The following notes are based upon two papers ‘Aporism: Uncertainty about Mathematics’ and ‘Linking Mathematics Education and Democracy’ which both will appear in Zentralblatt für Didaktik der Mathematik. The notes may serve as an introduction to my lecture.

A Paradox

"In the last 100 years, we have seen enormous advances in our knowledge of nature and in the development of new technologies. ... And yet, this same century has shown us a despicable human behaviour. Unprecedented means of mass destruction, of insecurity, new terrible diseases, unjustified famine, drug abuse, and moral decay are matched only by an irreversible destruction of the environment. Much of this paradox has to do with the absence of reflections and considerations of values in academics, particularly in the scientific disciplines, both in research and in education. Most of the means to achieve these wonders and also these horrors of science and technology have to do with advances in mathematics." This is how Ubiratan D’Ambrosio, in ‘Cultural Framing of Mathematics Teaching and Learning’, introduces a section about mathematics and society.

According to the Enlightenment, scientific development and human progress are closely related. Therefore it seems a paradox that science can be related to human destruction. This paradox questions the optimistic assumption, that science also sustains progress in an economic and political sense. Has science come to play a dual role? Is science related not only to human progress but also to human disaster? Does mathematics play a double-role, representing both reason and unreason in social development?



The Greek word aporeo means ‘being in a loss’ or ‘being without resources’. Aporism represents an uncertainty about how to understand and criticise the ‘social agency’ of mathematics. Aporism is an expression of a concern for decoding also the horrors that might be associated with applications of mathematics.

Aporism acknowledges the possibility that pure reason may turn into perverted forms, meaning that the ideal harmony between reason, scientific development and human and social progress is broken. As part of the rationalistic perspective, reason ensures the progressive qualities of knowledge, but aporism accepts the possibility that pure reason develops pathological cases, and that some of these are connected to the development of mathematics.

Aporism elaborates on the paradox mentioned by D’Ambrosio. On the one hand, mathematics is a condition for technological wonders, on the other hand, mathematics appears to be part of a destructive force also associated with technology. Reason, in the shape of ‘instrumental reason’, becomes problematic.


The Formatting Power of Mathematics

If mathematics is interpreted as language, the speech act theory of language will raise the question: What can be done by means of mathematics? Mathematics can be interpreted not only as a descriptive tool, but also as a source for action. This brings into focus the notion of ‘symbolic power’ and the theme of ‘knowledge and power’. Mathematics as a possible source for technological action and we may consider the thesis of the formatting power of mathematics: Social phenomena are structured and eventually constituted by mathematics.

In Descartes’ Dream: The World According to Mathematics, Davis and Hersh provide a long list of examples of prescriptive use of mathematics which leads to some sort of human or technological action: "We are born into a world with so many instances of prescriptive mathematics in place that we are hardly aware of them, and, once they are pointed out, we can hardly imagine the world working without them. Our measurements of space and mass, our clocks and calendars, our plans for buildings and machines, our monetary system, are prescriptive mathematisations of great antiquity. To focus on more recent instances ... think of the income tax. This is an enormous mathematical structure superposed on an enormous pre-existing mathematical financial structure. ... In American society, there are plentiful examples of recent and recently reinstated prescriptive mathematisation: exam grades, IQ’s, life insurance, taking a number in a bake shop, lotteries, traffic lights ... telephone switching systems, credit cards, zip codes, proportional representation voting ... We have prescribed these systems, often for reasons known only to a few; they regulate and alter our lives and characterise our civilisation. They create a description before the pattern itself exists." This illustrates the scope of the thesis of mathematical formatting.

However, my claim is not that the thesis of the formatting power of mathematics is true. The only claim is that the thesis expresses a possible truth, and that this possibility is important to consider when mathematics and mathematics education are investigated from a social and political point of view. Nor is the claim that the thesis is simple. Naturally, it does not make sense to claim that mathematics per se has a formatting power. The thesis concerns mathematics in context. Social, political and economic interests can be pursued by means of the powerful language of mathematics. In this way the thesis of the formatting power of mathematics becomes a thesis of the existence of an interplay between mathematics as a source for technological actions and other sources for social development.


The Vico-Paradox

According to Giambatista Vico, the rationalist idea that it is possible to come to understand nature and the whole universe, expresses a blasphemy: How can humankind imagine that, by its limited resources, it could come to understand the creations made by an almighty and omniscient God? Each individual human being has only a limited knowledge and a limited power. God, as the creator of the universe, can understand how it works, but only the creator will be able to understand his work. What human beings can hope to understand is what they themselves have been able to create.

The Greek techne refers to human creation. Following Vico’s line of ideas, we should expect it possible for the human mind to grasp technology which is the paradigm of human creations. But when we consider the functions of technology we are lost. Humankind is not in control of technology, not even from a conceptual point of view. We are unable to express effects of technology, whether intended or unintended. This, I want to call the Vico paradox: Not even what we ourselves have constructed are we able to grasp and to understand.

We no longer live in ‘nature’. Our environment is structured and organised into a ‘techno-nature’. Science has provided us with means for describing and predicting natural phenomena, which can be used for technological inventions. But when we face techno-nature, which includes our own constructions, then natural sciences fail. Scientific knowledge of nature is not sufficient for interpreting the totality of nature and human construction. Neither sciences nor ‘critique of culture’ do provide us with means for clarifying the effects of science.


Mathematics Education

Does mathematics education produce critical readers of mathematical formatting? Or does mathematics education prepare a general acceptance of the formatting, independent of the critical nature of the actual formatting?

Essential functions in the technological society depend on how a competence in mathematics is distributed by means of the educational system. Mathematics education can serve as a ‘blind’ instrument for providing the mathematical competence in a form that is ‘adjusted’ to the present technological development. The structure of the educational system can make sure that the mathematical competence is distributed in such a way that, for instance, the ‘adequate’ number of people, needed in developing the information technology, in fact receive a sufficient mathematical competence.

Mathematics education can also make sure that the ‘inverse competence’ is in place and distributed in a functional way, meaning that a sufficient number of people come to understand that mathematics is not their business. Excluding a certain number of people from a competence can also be ‘functional’. (A potential group of critics are eliminated.) In ‘Mathematics by All’, John Volmink writes: "Mathematics is not only an impenetrable mystery to many, but has also, more than any other subject, been cast in the role as an ‘objective’ judge, in order to decide who in the society ‘can’ and ‘cannot’. It therefore served as the gate keeper to participation in the decision making processes of society. To deny some access to participate in mathematics is then also to determine, a priori, who will move ahead and who will stay behind."

Mathematics education is facing a problem of democracy. In my lecture I want to discuss aspects of this problem.