MATHEMATICS: A CRITICAL RATIONALITY?
Ole Skovsmose
Osk(at)learning.aau.dk
Aalborg University, Denmark
Abstract: One can view mathematics as a sublime rationality, as a malignant outlook, or as an insignificant way of thinking. In order to open up a different interpretation of mathematics, I investigate mathematics in action in terms of the following issues: (1) Technological imagination which refers to the possibility of constructing technical possibilities. (2) Hypothetical reasoning which addresses consequences of not-yet-realised technological initiatives. (3) Legitimation or justification which refers to possible validations of technological actions. (4) Realisation which signifies that mathematics itself comes to constitute part of reality,and (5) elimination of responsibility which might occur when ethical issues are eliminated from the general discourse about technological initiatives and their implications.These investigations of mathematics in action highlight that mathematics rationality is critical as it both significant and undetermined. Furthermore I point out that through investigations of mathematics in action the notionsof structuration, liquid modernity, knowledge-power dialectics, deconstruction and contingency are amplified.
Key-words: mathematics in action, technological imagination, hypothetical reasoning, legitimation, justification, realisation, responsibility, critical rationality.
“Mathematics”is an open conceptwith many possible meanings. In Philosophical Investigations, Ludwig Wittgenstein talks about the variety of language games, and “mathematics”mayoperate in a huge number of such games. While mathematics as a research field includes a vast domain of unsolved issues and conceptions in development, mathematics as a school subject refers to a well-defined body of knowledge parcelled out in bits and pieces to be taught and learned according to pre-formed criteria. Mathematics could, however, also refer to domains of knowledge and understanding that are not institutionalised through research priorities or curricular structures. Thus, we can locate mathematics in many work practices.[1] It is part of technology and design. It is part of procedures for decision making. It is present in tables, diagrams, graphs, and we can experience a lot of mathematics just leafing through the daily newspaper. According to the language-game metaphor, such occurrences of mathematics need not be different expressions of the same underlying “genuine mathematics”; instead very different formats of mathematics might be in use with only the name in common. As a consequence, perhaps we had better give up the assumption that it is possible to provide a defining clarification of mathematics. Well-intended definitions, as suggested by classic positions within the philosophy of mathematics – where logicism describes mathematics as a further development of logic[2]; formalism describesit as a formal game governed by explicitly stated rules[3]; and intuitionism describes mathematics as a particular mental process[4]– might simply be concealingthe fact that there are no unifying characteristics of mathematics to be indentified. I shall try to keep this observation in mind when,in what follows,I continue to use the word“mathematics”.[5]
Differentperspectives on mathematics have been presented. One can see mathematics: (1) As a sublime form of rationality, which represents the pinnacle of human intellectual enterprise. Such a perspective has been elaborated, although in different set-ups, by logicism, formalism, and intuitionism. It is deeply rooted in both Platonism and in the conception of mathematics as propagated by the scientific revolution. (2) As a malignant rationality through which instrumental forms of thinking spread to different forms of life. This conception has, for instance, been elaborated with itsoutset in the critique of positivism as formulated by the Frankfurt School. And (3) as an insignificant way of thinking, a perspective which, although indirectly, has been propagated by much recent social theorising.
In the following three sections, we are going to consider these three perspectives more carefully before we discuss mathematics in action in preparation for the formulation of a fourth possibility, namely seeing mathematics as a critical rationality.
1. A divinerationality?
One basisfor considering mathematics a divine rationality is found in Platonism. This represents a broadly accepted philosophy of mathematics assuming a reality of ideas with which mathematics is concerned.[6]We do not have access to this reality through our senses, yet we can grasp itscharacteristicsthrough our rationality. Thus a triangle, asbelonging to the world of ideas, has many properties. While these properties might appearonly in an approximate formatas properties of triangles of our sense perceptions, they apply exactly totriangles of the world of ideas.Only through our thinking we can come to grasp that these properties apply with certainty to the ideal triangles.
Certainty has been associated withthe Euclidian paradigm, according to which a body of knowledge should be formulated in anaxiomatic system.The axioms should be few and simple, and from these axioms deductions will take us to theorems. The simplicity of the axioms would ensure that human intuitioncould be reliable forassigning truthto the axioms of the system, while the properties of deduction would ensurethat truth propagatesto all theorems of the systems. In this wayaxiomatics ensures a body of knowledge to be true with certainty.[7]
The scientific revolution brings a further dimension to this paradigm. It appeared that the properties of nature could be expressed in mathematical terms, meaning that God had createdthe world within a mathematical format. It has to beremembered that all the representatives ofthe scientific revolution, Copernicus, Kepler, Galileo, Descartes, Newton expressed a firm belief in the existence of God;atheism as an intellectual possibilitydid not come about until later. As God had followedmathematical patterns, the secrets to God’s creation, i.e. the secrets of Nature, could be grasped mathematically. The essentialpoint, then, was toformulate the laws according to which Nature was operating.
In The World, Descartes tried to formulate such laws, and he talked about Laws of Nature as being imposed on Nature by God.[8]According to Descartes, God was the creator of the universe, while after the creation God left things to themselves, meaning that the universe was running like a clockwork according tothe imposed laws.
Descartes found that the Laws of Naturewere both simple andfew: The first law states that “each particular part of matter always continues in the same state unless collision with others forces it to change its state” (1998: 25). In other words, there are no tendencies in nature, as formulated within the Aristotelianphysics: a stone is not searching for it natural place, etc. Nature operates as a mechanism, and not as an organism.The second law states that when a body pushes another “it cannot give the other any motion except by losing as much of its own motion at the same time; nor can it take away any of the other’s motion unless its own is increased by the same amount” (1998: 27). This is a formulation of a principle of action and reaction: there can be neither more nor less in the reaction, than was in the action itself. This law ensures that the material unities of which nature is assumed to consistoperate like a system of billiard balls. The whole universeis comparable to a game of billiards, where Godmade the initialstroke.The third law states that “when a body is moving, even if its motion most often takes place along a curved line … each of its parts individually tends always to continue moving along a straight line” (1998: 29). This law includes the formulation of the principle of inertia. This lawnegates the Aristotelian idea that heavenly bodiesmove in circlesaccording to some particular laws applicable only to heavenly bodies. According to the third law there is nothing called a “natural circular movement”. Instead there must be some force which causes the circular movement, in particular there must be a force that ensures the rotation of the earth around the sun.
After enumerating these three laws, all having to do with mechanical movements, Descartesstates: “…I shall be content to tell you that, apart from the three laws that I have explained, I wish to suppose no others but those that most certainly follow from the eternal truths on which mathematics have generally supported their most certain and most evident demonstrations...” (1998: 31) This is really a profound insight Descartes claimed to have reached. He had identified the laws of nature, three in total, and this means that, according to theEuclidian paradigm, one would be able to deduce all true statements about nature from these three laws, taken as axioms (and simultaneously observing all mathematical truths).
Descartes provided averbalformulation of the Laws of Nature, but they could be restated in a mathematical format. This means that one can achieve tremendousinsight intonature by means of mathematics. God created Nature as he imposed the three Laws of Nature (and only these three including their consequences) onit, and as soon as these laws are grasped and formulated, there are no secrets of Nature which humankind could not come to grasp.Mathematical rationality had really taken a divine form.
Through mathematics a perfect harmony between knowledge and what is to be known can be established, and knowledge in the sense of true-with-certainty can be obtained.[9] This conception brings about a certain set of preoccupationswithin the philosophy of mathematics which can be condensed into the following two questions: What is the domain of mathematics? What is the nature of certainty in mathematics? These two questions – concerning ontology and epistemology – establish a broad paradigm within the philosophy of mathematics, where logicism, formalism, and intuitionism have taken up their positions.This paradigm prolongs the celebration of mathematical rationality.[10] It does not leave much space for a social-political critique of this rationality.
2. A demonic rationality?
While thescientific revolution has symbolised what scientific progress could mean, the industrial revolution, which was soon to follow, symbolised progress in a broader technological and economic form. A defining idea of modernity was that scientific progressis a “motor” of progress on a grand scale.[11]However, it became obvious that the industrialrevolutioncould hardly serve as an enduring example of progress.One only had to consider the working- and life conditions of the workers and their families. Sharp observations have been presented, for instance, by Friedrich Engels inThe Condition of the Working Class in England, Émile Zola in Germinal from 1885, and George Orwell in The Road to Wigan Pier, first published in 1937. There is much more included in science-based social changes than progress.
The approach of the Frankfurt School includes a critique of positivist science and the development of the modern state, and inOne Dimensional Man, first published in 1964, Herbert Marcuse points out problematic aspects of scientific rationality.[12]He finds that the very rationality of science, shaped as it is according to positivist standards, is problematic. According to logicalpositivism all sciences belong to the same family. The basic science is physics, while other sciences, like chemistry, biology, psychology can bereduced to dealing with the physical reality. This reductionism is basic to the positivist claim that a universal science – observing the same standards, the samecriteria for quality, the same methods – is possible.And as mathematics can be considered the language of physics (an idea that is immanent in Descartes’ formulation of the mechanical world view) it can be seenas the language of science in general. Itrepresents the rationality of science. According to Marcuse, however, this rationality brings about adevastatingformation of the social sciences and, in turn, of society in general.This rationality, which Marcuse refers to as instrumental reason,turns sciences into schemes of suppression and manipulation.
Recently this observation was formulated by Ubiratan D’Ambrosio with directreference to mathematics.In “Cultural Framing of Mathematics Teaching and Learning”, he makes the following comment: “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 an 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.” (D’Ambrosio, 1994: 443).
With this formulation we leave behind any assumptionof mathematics representing a universal logic of progress. Instead D’Ambrosio points out that mathematics is part of not only the achievement of wonders, but the production of horrors as well. In fact his formulation expresses that mathematical rationality is critical, a point that we will return to later.
3. An insignificant rationality?
When I refer to “social theorising”, I do not have particular sociologicalstudies in mind, but rather the formulation ofbroader conceptual perspectives through which one tries to grasp basic features of our “social condition”. Let me refer to a few representatives of such social theorising.
Anthony Giddens presents the notion ofstructuration through whish he tries to capture how actions and structures are related in complex social processes.[13] The notion of structuration highlights a general quality of social phenomena, namely as both given and constructed. They are both facts and fabrications.In Giddens’ sociological writings there is no elaborated reference to mathematics. It appears that the very concept ofstructuration can be developed without reference to any form of operation of mathematical rationality. In this sense I see Giddens as representing the position that mathematicsisinsignificantfor social theorising.
Zygmunt Bauman elaborates on the notion of post-modernity and makes profound observations about the social conditions of our time.[14] Through the notion of liquid modernity he tries to grasp a characteristic feature of these conditions.[15] While social structures and priorities of a more permanent character might have been characteristic of what can be referred to at classic modernity,life-conditions of today have lost solidity. Not only social institutions, but also social priorities and conceptions dissolve as foundations with respect to human priories. Values are taken by the stream of changes. While we, during modernity, might at least have had the illusion of being on firm ground – for instance with respect to notions of progress, improvement, and knowledge –liquid modernity has thrown us into the open sea. Bauman provide his interpretation of liquid modernity with many references: to philosophy, to sociology, to literature. However, it is not easy in any of his writings to find references to mathematics. This rationality seems to have nothing to do with the liquid turn of modernity.
Michael Foucaulthasexplored the knowledge-power dialectics, and heprovides a new opportunityfor investigating the role of science in society.[16]With reference to Giddens, Foucault’s overall point could be formulated as:a knowledge-power dialecticsis part of a structuration. And with reference to Bauman, one could claim that this dialectics turns modernity fluid.Foucault investigates the knowledge-powerinteraction without any particular reference to mathematics and natural sciences. He does not explore the possibility that a mainsitefor such interaction could be the exact sciences and in particular mathematics. In this way, I see Foucault as representing the position that mathematics is of little significance for social theorising. Foucault located the sites forhis knowledge-power archaeology within the humanities and in vast distance from the so-called exact and formal sciences. In this way the paradigmaticformat of Foucault’s work has de facto fortified the assumption that mathematics is insignificant for excavatingrelationshipsbetween science and power and for social theorising in general.
Jacque Derridareferred to mathematics when he commented on Husserl’sOrigin of Geometry.[17]However, these comments donot help toidentify mathematics as part of significant social processes. Derrida does not provide anopeningfor seeing mathematics as being relevant for social theorising. Derridahas inspired the development of different notions, and let me just refer to deconstruction.[18] This notion refers to the investigation ofa social phenomenonand the notions of which this phenomenonis constituted. A deconstructionmay reveal the profound depths of assumptions, ideas, and discursive fragments that form the phenomenon in question. The post-structuralist position, as associated withDerrida’s work, has provided a broad inspiration for the deconstruction of a variety of social phenomena. However, within post-structuralism it is difficult to find examples of the deconstruction of formal techniques or of any form of applied mathematics rationality.[19] In this way this position has not assigned any significance to mathematicsfor understanding the social condition.
Richard Rorty establishes a complexintegration of different philosophic positions in order to grasp features of our social condition. A particularly important notion in this respect iscontingency, which serves to emphasise that socialdevelopment does not run alongalready constructed railsmaking social forecasting possible.[20] Contrary to any form of social determinism, Rorty finds that contingency represents a social condition. The future is not anticipated in the past, nor in the present. In his development of the notion of contingency, Rorty does notrefer to mathematics, which appears insignificant for understanding contingencies.
Certainly there are many differences between Giddens, Bauman, Foucault, Derrida, and Rorty. However, my shortpresentation served to point out two similarities. First,they all developperspectives for reading the most general features of oursocialcondition; and, second, they do not pay any particular attention to the role of mathematics in developing these perspectives. In this (indirect) way they have helped position mathematicsas being insignificantfor social theorising.
4. Mathematics in action
There are at least two points I want to emphasise when talking about a mathematical rationality as being critical. First, I see thisrationality as being significant in the sense that it has an impact on all spheres of social life. Something can be done through this rationality, not least through technology.
Second, I see the impact of mathematical rationality as undetermined in the sense that it could go in all possible directions: it mayprovide “wonders”as well as “horrors”. Thus, I do not associate any essence withmathematical rationality which ensures that it will operate in particular ways. Mathematics has no nature that ensures that applications of mathematics will be for the sake of everybody.[21]It might be that mathematics may provide wonders, sometimes, and that it might provide horrors, sometimes. However, we should not be trapped by any kind of dualism, including a horror-wonder dualism. It might be better to give up all dualisticframeworks, andassociate being undetermined with a much more complex set of possibilities. One could think of mathematicalrationality asopening up an indeterminism surpassing any form of dualism. Mathematical rationality might make availableunexpected possibilities; bring about devastating risks, serve particular business interests;be a part of schemes of surveillance and domination;etc.