Mathematics in Action

Ole Skovsmose

<osk@learning.aau.dk>

Aalborg University, Denmark

Mathematics: Insignificant?

Is it true that mathematics has no social significance? Or does also mathematics provide a crucial resource for social change? In other word: How may mathematics and power be interrelated? Here I think of academic mathematics more than on school mathematics, acknowledging, however, that the notion of mathematics is open and includes a broad range of possible meanings.

In A Mathematician’s Apology, Hardy discusses the usefulness of mathematics, and his general conclusion is: “If useful knowledge is […] knowledge which is likely, now or in the comparatively near future, to contribute to the material comfort of mankind, so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless.” (Hardy, 1967: 135) Could mathematics, nevertheless, do any harm? Hardy concludes: “[…] a real mathematician has his conscience clear; there is nothing to be set against any value his work may have; mathematics is […] a ‘harmless and innocent’ occupation” (1967: 140-141). In the final pages of his Apology, Hardy draws conclusions about his own work in mathematics: “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or for bad, the least difference to the amenity of the world.” (1967: 150) Hardy provides a picture of ‘pure’ or ‘higher’ mathematics as an intellectual enterprise that cannot be judged by its effects on society, for the simple reason that there are no such effects. According to Hardy’s interpretation, mathematics is insignificant in the sense that mathematics does not have any structuring impact on social development.[1]

Let me contrast this perspective on mathematics with the following claim made by D’Ambrosio in ‘Cultural Framing of Mathematics Teaching and Learning’: “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.” (D’Ambrosio, 1994: 443) D’Ambrosio strongly indicates that mathematics is positioned in the nucleus of social development. The role of mathematics is not significant but crucial and must be considered when interpreting a wide range of social phenomena. However, we also find that the role is undetermined, as the achievements of technology and science, including mathematics, may turn into wonders as well as horrors. What might be done through mathematics is not predetermined by the nature or essence of mathematics.[2] To bring mathematics in action is a risky affair.

However, what is the response in the most overall social theories to the question of whether mathematics is indeed insignificant or crucial for social development? Naturally no simple answer is found, but if we study works such as The Constitution of Society and Social Theory and Modern Sociology by Giddens, we do not find any reference to mathematics.[3] Surprisingly, mathematics is not referred to in Castells’ comprehensive study The Information Age I-III, where we find a broad description of the network society; nor in Gibbons, Limoges, Nowotny, Schwartzman, Scott, P. and Trow, M. (1994) or in Nowotny, Scott and Gibbons (2001) where the Mode-2 society, characterised by a new form of production of knowledge, is discussed. However, Lyotard (1984) refers to mathematics in his discussion of the post-modern condition, although not for its social relevance. So, judged by the silence about mathematics, the conception in much social theorising appears to be effectively that of Hardy’s: The social impact of this science is insignificant. There is no reason to consider mathematics in particular in order to interpret social affairs. Knowledge in the form of mathematics and power is not interrelated.

In what follows, I shall discuss how mathematics in action can be interpreted as an integrated part of technological planning and decision making, and how mathematics therefore operates as technology and therefore plays a powerful role in social development, which cannot be ignored by social theorising.[4] By developing the notion of mathematics in action, I try to point out that mathematic is significant, being both crucial to socio-technological development as well as being undetermined. I include a variety of aspects within the notion of technology: the artefacts of technology (be it a car, a computer or any other device) as well as strategies for action (a plan of production or any other product of ‘systems development’ – tailorising being one classic example). Thus, technology refers to knowledge, techniques, artefacts, organisational structures, and economic resources and priorities – all linked together into systems of fabrication and design.[5] In order to emphasise this broad concept of technology, I sometimes choose to talk about socio-technology.

I find that mathematics in action is significant for social development, as technology and socio-political action s can be structured by mathematics; and this observation turns into a challenge for social theorising. Furthermore, I find that mathematics in action provides a challenge for the philosophy of mathematics, which has to address the uncertainty linked to this form of action. Finally, I find that this uncertainty reveals a need for reflection on and critique of any form of mathematical activity, and this becomes a challenge for mathematics education. In the concluding remarks of this chapter I shall return to these challenges. However, before we get so far, we have to develop the notion of mathematics in action. I will do so in three steps, first, by considering ‘reflexivity’; secondly I will provide some examples of mathematics in action; and, thirdly, I will summarise what this action might include by pointing out three aspects of this form of action.

Reflexivity

In Reflexive Modernization, Beck, Giddens and Lash present (in individual written chapters) a discussion of modernisation. According to Beck, we now face “the possibility of creative (self-)destruction for an entire epoch: that of industrial society. The acting ‘subject’ of this creative destruction is not the revolution, not the crisis, but the victory of Western modernization” (Beck et al., 1994: 2). In fact, it does not seem possible to identify more specifically any acting subject for this creativity. And Beck continues: “This new stage, in which progress can turn into self-destruction, in which one kind of modernization undercuts and changes another, is what I call the stage of reflexive modernization.” (1994: 2) So, reflexive modernisation is not about radical changes taking place as a result of certain critical dysfunction of modernity. Beck does not follow a variant of Marx’s analysis, that “capitalism is its own gravedigger”; instead he finds that it is “the victories of capitalism which produce a new social form” (1994: 2ff.). So this new social form is born within the existing social structures. Reflexive modernisation includes an unplanned change of industrial society which harmonises with existing political and economic orders. Nevertheless, reflexive modernisation breaks up the contours of industrial society and opens ‘paths to another modernity’. Although there will be no revolution, there will be a new society.

If we want to understand the dynamics of social development, then we should not seek for that understanding from within the institutions which represent this development. The mechanisms of reflexivity bypass the democratic institutions and operate as part of the social subconsciousness. This problem is significant to sociology: “The idea that the transition from one social epoch to another could take place unintended and unpolitical, bypassing all the forums for political decisions, the lines of conflict and the partisan controversies, contradicts the democratic self-understanding of this society just as much as it does the fundamental convictions of its sociology.” (1994: 3) Beck indicates that sociology has not been able to grasp the basic principles of reflexivity, which characterises reflexive modernization.

Beck introduces the notion of risk society which “designates a developmental phase of modern society in which the social, political, economic and individual risks increasingly tend to escape the institutions for monitoring and protection in industrial society” (1994: 5).[6] Risk society is symbolised by many events such as the Chernobyl disaster, financial crises, pollution of food, etc. According to Beck: “Society has become a laboratory where there is absolutely nobody in charge.” (Beck, 1998: 9) In this return of uncertainty a new frame of social life is established. Risk society is however formed by basic elements of industrialised society: “One can virtually say that the constellations of risk society are produced because the certitudes of industrial society […] dominate the thought and action of people and institutions in industrial society. Risk society is not an option that one can choose or reject in the course of political disputes. It arises in the continuity of autonomized modernization processes which are blind and deaf to their own effects and threats.” (1994: 5-6) Industrial society accumulates its own products, including their effects and side-effects, and eventually this turns society into a new form. In particular, due to a false ‘certitude’, industrial society produces risks, which transform the industrial society into a risk society. But how to understand the nature and the reflexive processes  leading to the emergence of new risk structures?

What about mathematics? Let us take a look at the index of Reflexive Modernization: No reference to mathematics. However, we find the following sentence in Beck’s chapter: “Risks flaunt and boast with mathematics.” (1994: 9)[7] In Reflexive Modernization this sentence is left as a passing remark. If reflexive modernisation can be discussed and analysed in depth, without any reference to mathematics, then mathematics might be insignificant to social development and, therefore, irrelevant to social theorising. But I want to illustrate that this is not the case. I will try to indicate that the recent development of the industrialised society – establishing a reflexive modernisation, a risk society, a Mode-2 society, a network society, a postmodern society – is linked to a mathematical-resourced development. To simplify my argument, I shall, however, first of all relate to the vocabulary presented by Beck. Mathematics makes part of that mischievous ‘certitude’, which transforms industrial society into a risk society. Thus, I find that mathematics in action makes part of socio-technological processes which result in effects and side-effects, and which turns society into a new form characterised as reflexive modernisation. In other words, I find that reflexive modernisation can only be grasped if we become of aware of what forms mathematics in action might take.

Examples of Mathematics in Action

Foucault has explored the relationship between knowledge and power, although without addressing issues involving mathematics. To me this gives an awkward distortion to the discussion of the knowledge-power complex. In fact I find that mathematics in action provides a principal site for investigating this complex.

By means of a couple of examples, I try to illustrate how mathematics may operate as part of technological planning and decisions processes, and how mathematics becomes part of technology itself. I want to illustrate how mathematics can be brought in action and in this way set the scene for our daily-life activities. Mathematics may operate without the persons, who are working with or who are affected by mathematics, being aware of this – and the sociologists studying these phenomena might not be aware either.

Booking models. My first example of mathematics in action refers to a model presented by Clements in ‘Why Airlines Sometimes Overbook Flights’. Clements does not claim that his model is identical to any actually used model (such models are commercial in confidence), but certainly it is similar to such models. Booking strategies may have developed considerably since Clements constructed his model; nevertheless this model illustrates several basic aspects of mathematics in action.[8]

Airlines deliberately overbook?! Why? Naturally, in order to maximise profit or, to put it more gently, to make sure that the prices of tickets are kept to a minimum. It is essential to try to prevent flying with empty seats. The costs associated with flying a full airplane or one with empty seats are approximately the same. The salary or working hours for pilots and cabin staff does not depend on whether the airplane is full or not. The cost for service onboard may differ, but this is a marginal cost. A full airplane may consume a bit more fuel compared to a half empty one, but this is as well a small difference. This simply implies that travelling with empty seats is a costly affair for any airline company. A good sale-strategy must try to prevent from this.

For every departure, it is most likely that some of the passengers who have already booked will fail to turn up, the so-called ‘no shows’. A passenger, paying a full fare ticket, is normally allows to change the ticket without any extra costs, also in case he of she was late for the departure. As a consequence, it appears possible to overbook flights. Certainly, there must be an upper limit to this, as the company is going to compensate those passengers who might be refused, or ‘bumped’, if more than the expected number of passengers turn up. Furthermore, it must be considered that the probability of a passenger being a ‘no show’ depends on, for instance, the destination, the time of the day, the day of the week, and, not to forget, the type of his or her ticket.

All this can be incorporated into a mathematical model containing parameters such as the cost of providing a flight, the fare paid by each passenger, the capacity of the airline, the number of passengers booked on a flight, the costs of refusing a passenger who has booked, the probability of a booked passenger being a ‘no show’, the surplus generated by a flight, etc.[9] With reference to the model, it becomes possible to plan the overbooking in such a way that revenue is maximised. Essential information, of course, is the probability, p, that a booked passenger will in fact be a ‘no show’. If this probability is equal to 0, then there is no point in overbooking, but if p is greater than 0, then we can devise an overbooking strategy. The actual value of p for particular departures can be estimated by means of statistical records concerning previous departures, and in this way the degree of overbooking can be graduated according to a set of relevant parameters. For instance, the degree of overbooking the last evening flight from Copenhagen to London should be kept lower than that of an afternoon flight, as the compensation for bumping a passenger in the first case would include hotel costs.

This example illustrates that mathematics may serve as a basis for planning and decision-making. The traditional principle: ‘Do not sell any more tickets than there are seats’ becomes substituted with the much more complex principle: ‘Overbook, but do it in such a way that revenue is maximised, considering the amount of money to be paid as compensation, the destination, the time of departure, the day of the week, as well as the long term effects of having sometimes to bump passengers who in fact have made valid bookings.’ This new principle cannot be created or come to operate without mathematics. Its complexity presupposes that applications of mathematical techniques are ‘condensed’ into a booking programme. The principle illustrates what, in general, can be called mathematics-based action design.

A mathematical booking-model does not only describe a certain situation, in this case, patterns of reservation, cancellations and ‘no shows’. Mathematics does not only provide a ‘picture’ of reality, as suggested in several philosophies of mathematical modelling. In fact, many descriptions of mathematics as language assume a picture-like theory of how mathematics might relate to a non-mathematical reality. In this way the descriptions embark on the metaphysics from Wittgenstein’s Tractatus Logico-Philosophicus. However, should mathematics be compared with language, then the speech act theory, as suggested by Austin and Searle, invites the following question: What is in fact done by means of mathematics?[10] This question introduces also the idea of linguistic relativism: What world view is provided by a specific language? Applied to the language of mathematics, the question becomes: What world views are made available by means of mathematics? Or: How is the world constructed, according to mathematics? A booking model does not just describe some principles of queuing. It actually establishes new types of queues. And it might create a situation in which some people suddenly have to make new travel plans. Mathematics becomes part of a technique, here represented by the management of booking of flights.

An adequate understanding of the actions carried out in the process of selling tickets is not possible unless we pay attention to the existence of the booking-model. What interpretation to make of the airline assistant’s exclamation, when he or she has to address the newly ‘bumped’ passenger: “Oh, I’m so sorry, but unfortunately we have some problems with the computer system…” How would a sociological interpretation of this particular situation look like? Without awareness of the existence of a booking model, the assistant’s explanation may appear plausible: certainly there could be problems with the computer system. But this explanation does not capture the fundamental rationality of the situation. In many cases, ‘bumping’ a passenger is not a computer mistake. Instead it is a well-calculated consequence, occurring when the passenger in question comes to represent a statistical ‘deviation’ from the expected norm. If we want to interpret the episode, we need to understand how mathematics operates behind the desk. This is the case as well with many other situations where mathematical models facilitates decision making. The example of overbooking is not a unique example of mathematics-based action design. Instead it can be seen as paradigmatic of any (complex) business management. Mathematics becomes part of the language of management as well as of the execution. Without being aware of mathematics being in place, sociological explanations of such enterprises will become superfluous, if not misleading. To me sociology must be aware of mathematics-based action design in order to interpret a wide range of social phenomena.

And to conclude in even more general terms: Mathematics may operate as part of the processes of reflexivity. It can make part of the construction of unintended implications. It can operate along the route bringing about new risk formations.

ADAM. Mathematics is certainly involved in grand scale economic management. This can be illustrated by the Danish macro-economic model ADAM (Annual Danish Aggregated Model), which is used by the Danish Government as well as by other institutions, private as well as public.[11] One of the principal aim of ADAM is to promote ‘experimental reasoning’ in political economy. In this way, ADAM provides a basis for political decision-making. One way of doing so is to provide economic prognoses. Another, maybe even more important application of the model, is to provide different scenarios. Experimental reasoning tries to address the question: If a certain set of decisions is made and the economic circumstances develop in a particular way, what would be the consequence? Implications of a scenario can be investigated by a comparison between applications of the model to different sets of values of the parameters in question. In this way it becomes possible to observe the implications of a political action without having first to carry out the action (at least this is the assumption). Naturally, such reasoning is basic in politics. However, by relying on the model, the political discourse changes because the experimental reasoning which refers to the model acquires a new authority. Experimental reasoning can help to discover which economic initiatives are ‘necessary’ in order to achieve some economic aims, say, within a definite time limit. Certainly, ‘necessary’ has to be put in inverted commas, as necessity refers to the space of possibilities produced by the model.

As emphasised by the builders of ADAM, the quality of the scenarios provided by the model depends on the accuracy of the estimations of the variables providing the foundation for the calculations. It naturally has to be added that the quality of the presented scenarios also depends on the quality of the model itself. What, then, does a model like ADAM consist of? An awful lot of equations! These equations can be summarised in different ways, one possibility is to group them into seven clusters having to do with commodity demands, commodity supply, labour market, prices, transfers and taxes, balance of payments, and income. In fact, ADAM can be considered as a set of sub-models addressing certain aspects of the Danish economy. The system of equations in ADAM is constructed around different types of variables, exogenous and endogenous. The value of an exogenous variable is determined from outside the model; the population of Denmark is as an example of such a variable. To estimate the employment-unemployment ratio, this number is essential. Endogenous variables are those which are determined by the model itself, and many variables, which appear exogenous in some part of the ADAM-complex, are determined by other parts of the model, so when ADAM is considered in its totality, they become endogenous.

When such a system of equations is constructed and accepted, experimental political reasoning can be carried out. The problem is, of course, how to present such reasoning. Obviously, the detailed structure of the model cannot be presented, nor grasped, in actual political discussions. A possibility is to let experimental reasoning take the particular form of a multiplier analysis. Let us assume that the equation y = f(x1,…, xn) belongs to the model. If the variable x1 is multiplied by a certain factor c, the result would be yc = f(cx1,…, xn). By calculating d = yc /y, it can be claimed that when the input x1 is multiplied by c, the output y will be multiplied by the factor d. Questions inviting multiplier analysis are raised everywhere in political discussions. For instance, if the government tries to carry out an expansive finance policy, and expand public demand, what effect would such a policy have on the degree of unemployment? In particular, if the government increases its public demand by 5%, how much would the unemployment then decrease? A multiplier analysis would provide an estimation.

ADAM is certainly not merely providing a description of some part of socio-economic reality. It also imposes certain theoretical assumptions about this reality. Taken together, ADAM “displays features which are characteristically Keynesian” (Dam, 1986: 31). Thus, the choice of the basic equations, which supply the model with a ‘soul’, does not simply reflect certain economic reality; it also prescribes a particular perception of economic affairs. ADAM provides a new example of mathematics-based action design. By being a resource for actions, the model becomes part of economic reality. It even comes to dominate this reality, to the extent that its assumed economic linkages establish real linkages. ADAM was created by mathematics, but ADAM got life. And, as we all know, ADAM did not stay alone, and a huge variety of mathematical models have been developed.[12] Human beings become part of a reality structured by economic principles formulated in mathematical terms. We observe the same phenomenon associated with the booking-model: the mathematical model becomes part of a social reality.

Many other models. Mathematics does not only influence the economic part of our reality. In 1995, the Danish Council of Technology (Teknologirådet) published the report, Magt og Modeller (Power and Models), discussing the increasing use of computer-based models in political decision making. The report refers to 60 models, which cover the following areas: economics, environment, traffic, fishing, defence, population. The models are developed and used by public as well as private institutions in Denmark.[13]

The report Magt og Modeller emphasises that political decision-making concerning a wide range of social affairs is closely linked to applications of such models, and that this development may erode conditions for democratic life: Who construct the models? What aspects of reality are included in the models? Who have access to the models? Are the models ‘reliable’? Who is able to control the models? In what sense is it possible to falsify a model? If such questions are not clarified in an adequate way, traditional democratic values may be hampered. As an illustration of this problem, I shall summarise the comments of the report related to traffic and environmental issues. In this case models are often used in support of decisions which cannot be changed, like the construction of a bridge, say, between two major Danish islands. Decisions concerning traffic are almost exclusively based on models developed in private companies. It is not usual to develop more than one model to illuminate a certain issue. Finally, it happens that models are used in order to legitimate de facto decisions, in the sense that a model-construction provides numbers and figures which justify a decision already made.

Beck claims that the process of reflexivity, which leads to a risk society, occurs outside of democratic control, and that it eludes contemporary sociology. The extensive use of mathematical modelling, as discussed in Magt og Modeller, exemplify this claim. How to obtain a democratic access to decision-making, which is integrated with mathematical modelling processes? The conditions for democratic life may be eroded by the spread of mathematical based action design.[14] Thus, it becomes difficult to ignore the role of mathematics, if we want to establish a sociological discussion of conditions for democracy regarding the nature of technological development.

Three Aspects of Mathematics in Action

Many different interpretations of mathematics have been particularly interested in processes of abstraction, i.e. on processes though which mathematics idea are extracted from a richness of empirical phenomena and observations. By talking about mathematics in action, however, I concentrate on the inverse process: seeing how mathematical abstractions can be projected into reality. When we use mathematics as a basis of technological design, we bring into reality a technological device that has been partly conceptualised by means of mathematics. First, it exists in the world of mathematics, later it is brought into reality by an actual construction. A mathematical ‘speech act’ has been carried out. I see such acts as being part of (almost) any socio-technological action. Mathematics in action becomes part of our environment.

            In order to clarify further processes of mathematics in action, I shall concentrate three  aspects of t his process: (1) How mathematics plays a role in originating a technological imagination; (2) how mathematics may facilitate hypothetical reasoning; and (3) how mathematics might become realised for instance as algorithms or principles of a functioning technology. In other words, I shall try to address mathematics based power through the notions for imagination, reasoning and realisation.

Technological imagination. In order to specify aspects of this particular imagination, let us consider the notion of ‘sociological imagination’, which expresses a capacity to separate what is necessary from what is contingent and, therefore, possible to change. A fact is not only a fact but also a (social) necessity, when it is impossible to imagine that the fact is not the case. If we consider a particular culture where a certain work process is carried out in a particular way (maybe obeying some ceremonial traditions), and no alternative to that approach is identified (within the culture), then this process would appear to be a (cultural) necessity. The existence of an imagination that describes alternatives to an actual situation makes a difference. In this case, the fact is ‘reduced to’ a contingent fact. The experienced necessity is revealed as an illusion when an alternative is conceived. This is the power of sociological imagination: A social given has been identified as available to change.[15]

A process of design includes the identification and the analysis of hypothetical situations, and mathematics helps by providing material for constructing such situations. By means of mathematics, we can represent something not yet realised and therefore identify technological alternatives to a given situation. Mathematics provides a form of technological freedom by opening a space of hypothetical situations. For instance, by means of mathematics it is possible to imagine new forms of encryption, even before such forms have been constructed.[16] In this sense, mathematics becomes a resource for technological imagination and, therefore, for technological planning processes including mathematical based action-design. However, as we shall come to see, the stipulated attractive qualities associated to sociological imagination is not simply transposed to technological imagination. This is important to keep in mind.

          The space opened by a technological imagination might very well contain hypothetical situations which are not accessible via common sense. A mathematical framework provides us with new alternatives. For instance, when a booking model is established, it becomes possible to specify special fare schemes, and at present there is no lack of special offers, linked to a variety of conditions for cancellation. Thus, the model makes clear the relevance of establishing a grouping of passengers, which makes a prediction ‘no shows’ within each group reliable. In order to do a more detailed planning, it becomes essential to have a booking model, as well as many other models, available. In fact, any elaborated price policy – having to do with airlines, tele-companies, travel agencies, etc. – is based on experimentation with mathematical models. In a similar way the set of equations in ADAM constitutes hypothetical situations, which makes it possible to establish political thought experiments; this means conceptualising details of situation, which is not possible to identify by common sense.

          In other respects, the space of hypothetical situations might be very limited. A technological imagination can be a restricted imagination. Thus, ADAM does not support political thought experiments which contradict the political priorities, installed in ADAM in terms of its basic equations. When a technological imagination relays on mathematics, it may provide a very particular space of hypothetical situations. Political and economic interests can express themselves in the set of technological alternatives that are established as mathematical well-defined. Therefore, mathematics as part of a technological imagination can interact with other power structures. As mentioned previously with reference to models for traffic planning, the set of alternatives established by mathematics can be so limited that the modelling in fact serves as a legitimisation of a de facto decision. By providing one and only one alternative, this alternative appears to be a necessity within the space of hypothetical situations provided by the model. This situation helps to establish credibility in the political claim that a certain political decision is a ‘necessary’ decision.

Thus, the first aspect of mathematics in action concerns technological imagination: By means of mathematics, it is possible to establish a space of hypothetical situations in the form of (technological) alternatives to a present situation. However, this space may contain serious limitations. Technological imagination not only draws on mathematical resources; it also provides a combination of analytical capacities, structures of power, interests and priorities. Technological imagination sets out possible directions for technological initiatives and decision making. Imagination signifies an aspect of power (remember still that sociological imagination represents a form of empowerment).

Hypothetical reasoning. Mathematics provides the possibility for hypothetical reasoning, by which I refer to analysing the consequences of an imaginary scenario. By means of mathematics we seem to be able to investigate particular details of a not-yet-realised design. Thus, mathematics constitutes an important instrument for carrying out detailed thought experiments. Because of ADAM, it is possible to carry out hypothetical reasoning related to economic policy. This reasoning is counterfactual, as it addresses implications of the form: ‘p implies q, although p is not the case’. A representation of p is provided by ADAM in terms of equations including the values of the relevant parameters. The hypothetical reasoning can then address a particular situation ‘realised’ by ADAM. Some conclusions of the hypothetical reasoning can then be simplified and expressed in multipliers that are easily included in the common political discussion. Without mathematically based hypothetical reasoning, the political discussion would take a completely different form. It would lose a great deal of so-called ‘precision’. Hypothetical reasoning represents an essential element in the mathematics-bases analysis of particular implications of particular actions.

The strength of the hypothetical reasoning is demonstrated by the level of details to which the hypothetical situation is specified. However, hypothetical reasoning, supported by mathematics, also lays a trap, because we are investigating details represented only within a specific mathematical construction of a given alternative. Furthermore, the actual hypothetical reasoning is limited by the fact that the reasoning itself is supported by mathematics. As clearly illustrated by ADAM, the weakness of the hypothetical reasoning is that the decisions made on the basis of hypothetical reasoning will operate in a real life situation, not grasped by ADAM. So, when q is found attractive, and p is realised, we will see that the ADAM-supported hypothetical reasoning, does not operate straightforward in a real life context. The hypothetical situation, p, is an imaginary situation created only by the model, and it need not have much in common with any actual situation. The problem of hypothetical reasoning is caused by the ‘gap’ between the model-constructed virtual reality and the ‘complexity of life’.

The second specification of mathematics in action concerns hypothetical reasoning: By means of mathematics, it is possible to investigate particular details of a hypothetical situation, but mathematics also causes a severe limitation of the hypothetical reasoning. This means that the quality of mathematically-based thought experiments might be highly problematic. The similarity gap between mathematically identifies virtual reality, including its virtual effects, might be very different form any real events. The similarity gaps might provide openings for new risk formations to turn real.

Realisation. When an alternative is chosen and realised, our environments changes. What is the nature of this new situation? As already emphasised, a booking model that structures airline bookings is certainly not simply a description of what takes place when tickets are booked and sold. When introduced, the model becomes part of the passengers’ reality. And this story can be continued: Insurance companies also offer insurance for cancellation of tickets. They, therefore, need a model telling about the likelihood that a ‘sure passenger’ will in fact become a ‘no show’. In this sense, models create models, and one layer after another of mathematics sinks into our social reality.

Tymoczko has summarised this point in the following way: “Business does not just apply various already existing mathematical theories to facilitate an activity that is, in principle, independent from such mathematical application (although it can do that). Business could not exist in anything like its historical form without some mathematics. Certainly we cannot imagine a modern economy struggling along without mathematics then suddenly becoming more efficient because of the introduction of mathematics!” (Tymoczko, 1994: 330) That mathematics becomes part of reality is a general phenomenon. At his lecture at the 7th International Congress on Mathematical Education in Québec, Tymoczko mentioned the relationship between mathematics and war. His point was that war and mathematics are interrelated in an intimate way. We may talk about modern warfare as constituted by mathematics. Not in the sense that mathematics is the cause of war; but we cannot imagine a modern warfare to take place without mathematics as an integrated part. The same statement can be made if we, instead of ‘war’ or ‘business’, talk about ‘travel’, ‘management’, ‘communication’, ‘architecture’, ‘insurance’, ‘marketing’, etc. In their present form such types of social phenomena are modulated if not constituted by mathematics.[17]

Whenever we talk about mathematics-based design, we have to remember that the realised situation need not have much in common with the hypothetical situation presented and investigated in mathematical terms. Any technological design has implications not identifies by the hypothetical reasoning. This is a basic problem related to any kind of mathematical based investigation of counterfactuals. When p is represented by a mathematical based vision, and the implications of p is identifies by a hypothetical reasoning as q, and found attractive, then the realisation of p may nevertheless contain heavy surprises. By means of mathematics, we might grasp only the consequences, which can be formulated mathematically. Not surprisingly mathematics-base hypothetical reasoning has blind spots, behind which might hide dramatic side-effects of a certain technological invention. When implemented, these effects might emerge in dramatic form, making sense of Beck’s observation that society have turned into a laboratory where there are absolutely nobody in charge. As already indicated, risks emerge in the gap between the mathematical based reasoning related to the hypothetical situations and the really functions of the contextualised realisation. Certitude turns into risk.

Still, the realisation maintains mathematics as an operating element. In this sense we come to live in a environment, produced by integrating a model-supported virtual reality with an already constructed reality. For instance, much information technology materialise in ‘packages’. Such packages can be installed and come to operate together with other packages, and they contain mathematics as a defining ingredient. In particular, Hardy’s research has made a significant contribution to the area of cryptography, which addresses the question of ‘trust’ and security of electronic communication. Knowledge about the distribution of prime numbers and about the efficiency of mathematical algorithms, is essential for estimating the likelihood of maintaining privacy. Also in this case mathematics has become inseparable from other aspects of society.[18]

This bring us to the third aspects of mathematics in action which concerns realisation: Mathematics modulates and constitutes a wide range of social phenomena, and in this it becomes part of reality. As mentioned, mathematical modelling has been discussed as if a model provides a more of less accurate ‘picture’ of a piece of reality.[19] However, by means of mathematics we become involved in a fabrication of our lived-through reality. Mathematics is not simply a language of description; it serves also as a discourse for execution. It turns into the very execution. Following the discourse theory, as for instance presented by Torfing (1999), it becomes impossible to separate mathematics from a reality structured by a mathematical discourse. Mathematics in action operates as a discourse by means of which we constitute and elaborate our ‘reality’.[20] This brings a new perspective to the discussion of mathematical modelling.

Put together, the three aspects of mathematics in action tell: (1) By means of mathematics supporting a technological imagination it is possible to establish a space of hypothetical situations in the form of possible (technological) alternatives to a present situation. However, this space may have serious limitations, at the constructed set of hypothetical situation might represent economic interest or other priorities. (2) By means of mathematics, in the form of hypothetical reasoning, it is possible to investigate particular details of a hypothetical situation, but this reasoning may also include limitations, and therefore also uncertainties for justifying technological choices. (3) As part of the realisation of technologies, mathematics itself becomes part of reality and inseparable from other aspects of society. Being part of this process, mathematics is positioned in the centre of social development, and in the production of wonders as well as of horrors. The three aspects condense the idea that mathematics and power operate in unity.

Concluding remarks

As conclusion I want to point out how an awareness of mathematics in action as being significant (crucial an undetermined) brings a challenge social theorising, the philosophy of mathematics and mathematics education: (1) I find it problematic when, within the conceptual framework of social theorising, the possible different roles of different types of knowledge are not addressed in particular. To me this appears a prerequisite to understand ‘knowledge society’, ‘learning society’ and other such overall sociological constructs; in particular to address such processes as Beck has referred to in terms of reflexivity. I find that mathematics in action provides a challenge to social theorising, as this represents a particular form of knowledge-power in action. (2) I find that mathematics in action makes part of techniques, design, and fabrications of very different qualities and with very different socio-political functions. There is no essence in mathematics (say, representing pure rationality), which brings certain qualities to mathematics-based actions. I find that this uncertainty represents a basic challenge to the philosophy of mathematics, which cannot escape addressing the ethical dimension of such actions. (3) I find that a duty to reflect on and to criticise mathematics-based actions provides a challenge to mathematics education, which in this was come to deal with both empowerment and disempowerment. Let me consider the three challenges one by one.

Challenging social theorising. In his study ‘The Information Society’, Bell emphasises that “information and theoretical knowledge are the strategic resources of the postindustrial society, just as the combination of energy, resources and machine technology were the transforming agencies of industrial society” (Bell, 1980: 545). In his impressive work, The Information Age: Economy, Society and Culture I-II-III, Castells both develops and modifies this idea. He describes knowledge and information as “critical elements in all modes of development, since the process of production is always based on some level of knowledge and in the processing of information” (Castells, 1996: 17). Such statements are certainly crucial to the whole understanding the information age. However, the significance of these statements rests upon a specification of what can be understood as information and as knowledge. Castells adds a footnote to this parts of his text: “For the sake of clarity of this book, I find it necessary to provide a definition of knowledge and information, even if such an intellectual satisfying gesture introduces a dose of the arbitrary in the discourse, as social scientists who have struggled with the issue know well.” Following these preliminaries he characterises knowledge as set of organised statement, which includes some kind of justification, and which is transmitted to others. ‘Information’ he described as a concept even broader than knowledge. It is clear that Castells does not take this intellectual gesture seriously, and he does not apply this definition in any profound way later in his work. Instead he lets ‘knowledge’ and ‘information’ stay as cloudy concepts throughout his whole study of the information age. (I am sure that Castells has realised this.) But I find that it is essential to make a much stronger specification of the notion of knowledge in order to get a deeper understanding of some of the basic social process of the information age (and I am afraid that Castells has not realised this).

       By being kept on a general level, the discussion of knowledge and information makes it difficult to raise questions about the particular roles played by different types of knowledge in the construction of new technologies. In this way, the idea of mathematics being insignificant regarding social affairs becomes incorporated in the sociological discussion of the information age. However, I simply do not think that any kind of knowledge and information operates as ‘strategic resources’. Quit contrary, I find that particular types of knowledge operates in particular ways as resources for developing and realising technologies. Thus, the use of ‘knowledge’ and ‘information’ as dummies obstruct the possibility of identifying how particular forms of knowledge and information procession plays particular roles in technological and socio-political development. Beck did emphasise that the risk society is produced because the certitudes of industrial society dominates thought and action. As I have tried to argue, this phenomenon is related to mathematics-based action design and, in particular, to the application of mathematics in investigating counterfactuals. To me, a basic challenge to social theorising is to grasp the scope and particularities of mathematics in action, which I find adds important insight in the knowledge-power complex. I conceive this as a condition for any adequate interpretation of the basic processes which brings about reflexive modernisation, and for interpreting how ‘certainty’ turns into free growing risk structures, which are going to accompany us into the future.

       A main point with respect to knowledge-based economy is that it does not operate as classic forms of economy.[21] For instance, it is not determined by consumption and investment in any straightforward way. The amount of knowledge cannot be measured as can the amount of investment. There is a big vagueness linked to this economy. My point is that it does not make sense to try to conceptualise implications of knowledge and knowledge development, without more specific references to the types of knowledge involved. I find that mathematics in action represents a significant economic factor, which becomes open to demand-supply considerations. In particular mathematics based technological imagination can be crucial for conceptualising new possibilities for design and fabrication. I see this imagination as an important economic factor. Hypothetical reasoning becomes essential for the judgement of what to realises, and accompanying the blind spots of this reasoning, new risk structures may emerge. Finally, the realisation includes mathematical techniques, for instance in forms of algorithms put in operation in those ‘packages’ by means of which we construct the and reconstructs the network society. In short, mathematic in action fits into any scheme of demand-supply dynamics. Mathematics in action, unifying both power and knowledge, must be considered in any social theorising.

Challenging the philosophy of mathematics. According to classic philosophy of mathematics, mathematical thinking is a model for human thought. And following the ‘modern condition’ and the spirit of the Enlightenment, reason can be interpreted as a powerful resource for progress.[22] Reason, in the shape of science and of mathematics, represents an ‘ultimate good’. The classic philosophy of mathematic is based on such assumptions, and to a great extend it has dealt with the possibilities for identifying what certainty could mean, and in what sense it could be obtained, or at least approached, within the domain of mathematics. To concentrate on certainty has meant to concentrate on assumed pure aspects of mathematics, in particular as expressed through logical structures.

However, a glorification of the queen of science can no longer be the object of philosophies of mathematics. The ambiguity in the functioning of reasoning is well revealed by mathematics in action. Reason in the form of mathematics does not represent an epistemological ideal, which could serve as model for other science and modes of thought. Nor does it represent an unquestionable resource for socio-technological actions. Mathematics in action does not demonstrate universal attractive qualities. Mathematics, and reason in general, cannot any longer be considered a reliable carrier of progress, although, certainly, they represent powerful resources for technological and socio-political dynamics. As any for of action, so also any form of mathematics in action has to be accompanied by reflections and considerations: addressing the content and scope of technological imagination, the possible blind spots of any hypothetical reasoning, and the social impacts of realising mathematics based technologies. However, there is no well-defined platform to be found for such reflections and considerations. Instead a philosophy of mathematics has to operate above an abyss. This apori signifies a challenge to the philosophy of mathematics.

Aporism, as a philosophy of mathematics, acknowledges that ‘pure reason’, in terms of mathematics, can turn into ‘disastrous reason’.[23] Aporism see mathematics as an essential element in social and technological development; at the same time it acknowledges that the presence of mathematics does not provide any guarantee for a particular ‘quality’ of this development. Wonders mix with horrors, and this turns into uncertainty regarding the construction of our future. I see this uncertainty as a principal challenge to the philosophy of mathematics. Instead of trying to grasp certainty, a philosophy of mathematics could try to grasp what uncertainty with respect to reason, rationality and in particular to mathematics in action could mean.

Challenges to mathematics education. Uncertainty with respect to mathematics in action calls for reflection and critique, and this represents a challenge to mathematics education. This means that mathematics education cannot simple serve as an ‘ambassador’ of mathematics, aiming at bringing mathematics to students, or to facilitate students construction of mathematics. Mathematics education must also deal with a form of knowledge which, as part of an technological enterprise, provide wonders and horrors.

Mathematics education provides an introduction to forms of knowledge, which makes parts of many different technologies and techniques. Not only in form of advanced patterns of design and fabrication, but to a variety of daily-life technologies and techniques. It is a challenge to mathematics education to provide not only access to this powerful technological resource, but also to address any mathematics in action by reflection and critique: What kind of possibilities and alternatives are opened by a technological imagination? Does a certain mathematics-based technological imagination establish limitations and blind spots? How are analyses of implications of technological proposals carried out? What might have been overlooked? How is mathematics in fact put in operation in technological affairs and in out daily-life environment? Reflection and critique can address aspects of mathematics in action, and, certainly, there are many more aspects than the three pointed out in this paper. There is no simple way out of an aporia, but in case this challenge is faced by mathematics education, I will talk about critical mathematics education.[24]

I find that the duty of mathematics education is not only to help students to learn certain forms of knowledge and techniques, but also to invite students to reflect on how these forms of knowledge and techniques might be brought in action. Such reflections can address reliability and responsibility.[25] Thus, it becomes important to make it possible for students to consider the reliability of mathematics put in operation: Are the calculations reasonable? Has something been overlooked when relevant numbers and figures were identified? are there something mathematics might not be able to grasp? It is important to consider the limits of mathematics put in action. And next it becomes important to consider that mathematics in action it put in action by somebody and it operating in a certain context. This raised the question of responsibility. What does it mean for somebody to act responsible with reference to figures and numbers (that might be more or less reliable)? There is not simple answer to be found to such questions. But a mathematics education cannot ignore such question, in case it wants to face the challenge provoked by mathematics in action.

Acknowledgements

I want to thank Morten Blomhøj, Leone Burton, Dick Clements, Anna Chronaki, Tony Cotton, Arne Juul, Miriam Godoy Penteado, Sal Restivo, Susan Robertson, Paola Valero and Keiko Yasukawa for critical comments and suggestions for the improvement of preliminary draft for this chapter. The present paper is a revised version of ‘Mathematics in Action: A Challenge for Social Theorising’ which has been presented at the Annual Meeting of the Canadian Mathematics Education Study Group, University of Alberta, Canada, May 2001. See E. Simmt and B. Davis (Eds.): Proceedings: 2001 Annual Meeting, Canadian Mathematics Education Study Group, Groupe Canadien d’Étude en Didactique des Mathématique, University of Alberta. A Portuguese version of the paper is published in M. A. V Bicudo and M. C. Borba (Eds.) (2004), Educação Matemática: Pesquisa em Movimento (30-57). São Paulo: Cortez Editoria; and a Greek version in Themata stin Ekpedefsi, Volume 4, 2004.

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Ole Skovsmose

Department of Education and Learning

Aalborg University

Langagervej 8

DK-9220 Aalborg East

Denmark

E-mail: osk@learning.aau.dk