SCRIPTING THE WORLD IN MATHEMATICS AND ITS ETHICAL IMPLICATIONS

Ole Skovsmose Keiko Yasukawa Ole RavnAalborg University University of Technology

Sydney Aalborg University

osk@learning.aau.dk keiko.yasukawa@uts.edu.au orc@learning.aau.dk

Abstract: We examine the inseparability of mathematical reasoning and ethics by showing how mathematics is a powerful script that influences how people experience the world and act on their world. We examine four aspects of mathematics as a script: the way mathematics is used to describe social and natural phenomena; to inscribe particular values and beliefs into the way we view and understand the world; to prescribe certain actions to be taken; and how mathematics establishes a discourse that people come to subscribe to. However, by the time the numbers and measures behind mathematical descriptions are translated into actions in the real world, those values and beliefs that have been inscribed into them come obscured. The lack of transparency of the values and beliefs underpinning mathematical models raises profound ethical issues.

Keywords: Mathematics, Values, Ethics, Mathematical modelling, Mathematics in action

Mathematics is a language, a formalism, a school discipline, a research discipline, an engineering tool, a logic for reasoning, a social practice. Mathematics can be applied, but it can also be claimed to be pure; it can be dynamic, but it can also be frozen; it can inform and illuminate, but also hide and obscure. So what is this “it” that can be so many different things?

We can try to describe the nature of mathematics from many perspectives. One might think that if we observed mathematics from different, for example, sociological, philosophical, and educational angles, we would come to understand what “it” is, and how “it” may act. However, this idea is not quite adequate because we cannot assume that the different perspectives focus on the same thing. So, instead we could argue that a perspective determines, at least in part, what it is we are observing. We could even imagine that mathematics itself represents a certain kind of perspective rather than being an object of description.

In this paper, we investigate mathematics as an active constituent of society. Like language, mathematics is intimately linked to ideology and power: it affords us with a perspective, and in this way creates part of our life-world. We refer to this constructive aspect of mathematics as “mathematics in action”.1 It follows that if mathematics is “active” in society, there are also

ethical implications concerning its actions, particularly where they interact with relations of power.

In order to understand the nature of the relationship between mathematics and power, we need to examine not just the technical features of mathematical problems and solutions, but also the context in which they arise, how and why people interact with, produce and interpret mathematics, and how the influence of mathematics extends beyond the immediate context in which it emerges and shapes people’s values and beliefs.

We will argue that mathematics can be understood as “language”: as a discourse, where we use the term discourse to mean a frame for seeing, interpreting, designing and acting. We also see discourses as dynamic because it is itself shaped by the processes arising from within its own frame. Mathematics carries an image as being a universal language free of any ideological and ethical issues. We argue, however, quite the contrary: when we consider mathematics in action, we come to address actions that are informed and shaped by mathematical reasoning and mathematically derived information, and like any action they also require ethical considerations. Indeed we cannot think of any action operating in an ethical vacuum. In the next sections, we will outline what we mean by mathematics in action and four of its dimensions.

1. Dimensions of mathematics in action

We will illustrate that mathematics is involved in the production, enactment and legitimisation of design and decision and that power relations are established based on scripts written in the language of mathematics. We shall identify and examine four different dimensions of these scripts.

First, through modelling processes mathematics is thought to be a powerful language for description. Mathematical models can facilitate everyday communication as well as a specialised discourse about otherwise elusive ideas, phenomena or systems. However, it is questionable, as we will argue below, whether mathematics can ever fully and objectively describe a given reality.2 Secondly, we suggest that inscription is part of scripting the world mathematically. An inscription provides a means for building in a particular lens to see what exists and what could exist in a certain way and according to certain priorities. Thus, mathematics has the power of inscribing into visions and imaginations particular ideologies. Thirdly, mathematics provides a means for prescribing3 certain actions or authority to particular human and non-human actors; their actions are authorised and legitimised by what is written in the mathematical scripts. Mathematics in this prescriptive role becomes a tool for calculating what to do. Finally, we talk about subscription in relation to mathematics in action. To subscribe to a

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mathematical script means that an institution, a group or an individual applies or accepts a certain mathematical model in a given setting as a tool for decision-making. In many cases one is forced to subscribe, and in many cases one does so without knowing it. When one become ready to assume mathematical scripting as being the principal and most effective way of dealing with any type of problem, we witness what we could call “pre-subscription”. A whole set of potential practices may, through a pre-subscription, be preconfigured mathematically, even before they in fact start operating. This dimension of mathematics in action deals with the choice to use a mathematical script in the first place, as a means to act in the world.

These aspects of scripting represent ways of seeing, believing, deciding and doing; they represent ways of acting in and on the world, and we see them as four dimensions of mathematics in action. As in other forms of actions, these ways of scripting reality can have different qualities. They can be doubtful, unselfish, risky, dominating or dangerous. They can have any quality, and they cannot maintain any sublime neutrality or objectivity by virtue of their mathematical basis. We find that an analysis of a mathematical script brings us directly to considering the ethical dimension of mathematics. In the following sections we will as well discuss ethical issues in each of the four dimensions of mathematics in action.

2. Mathematics in action: description

Mathematics is a powerful resource for describing what is, what might be, and what could be. Mathematics affords a language for modelling aspects of the world. These models might describe an aspect of the material world, for example the average rainfall at different times of the year in different parts of the world; an aspect of a piece of machinery, for example of a motor in an automobile; an aspect of the social world, for example the growth or decline in levels of unemployment; the performance of a socio-technical system, for example an economic model of water recycling; or aspects of the virtual world, for example the traffic flow over the internet.

An example of a mathematical model is the budget for an organisational unit such as a household, a university department, or a state. The budget provides a picture of what has been and what could be. At the university, a departmental budget is developed based on the current financial income and expenditures and shows, among many things, how much could be spent on wages for different categories of staff; on travel and conferences; on office equipment and printing; and how much income could be expected from different sources, for example, student fees; research grants; and commercialisation of innovation. On the basis of the budget, members of the department can see what they can and cannot do during the following financial

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year. The mathematical model also provides a picture of what the department must do in order to retain a particular balance of income and expenditure for the following year; for example, it might have to increase the number of research grants by at least 10% to keep up wage increases and increased travel and equipment costs. The budget enables the people in the department to imagine what their working conditions are going to be like in the year covered by the budget. Furthermore, comparing the year’s financial accounts, based on the actual income and expenditure, with the budget provides a picture of new possibilities or restrictions. Thus the budget is a tool for both reflection and imagination.

But budgets and income-expenditure statements are not the same as the working lives or conditions of the members of the department. They do not provide a complete picture of the phenomena they model. A departmental budget does not show, for example, that 60% of the journal papers that were written by department members were done on the weekends, or that academics have to buy their own stationary because the budget no longer allows for such expenses. It does not show the real income from the research grant because 15% of it was taken off by the central university offices for administrative costs, and so on. Mathematical models provide part of the picture of reality, and on that basis allow us to imagine a partial picture of what could be in the future. However, what is not shown in the picture is often forgotten even if they are essential (in some way) features of reality.

Mathematical models of other kinds also provide descriptions, but always a “partial description”. This is revealed for example in a map of a city or a floor plan of a house. A map is a scaled version of the physical reality, and as we scale up the map, you see more of the details of what is there, and as you scale it down, you see less. However, even if we map in a 1:1 relation to reality, the map is far from identical with reality. Maps are partial in terms of spatial dimensions compared to the realities they are describing; they are two-dimensional representations of three dimensional spaces. A map is a snapshot of a physical configuration, of a place taken at a particular point in time. But physical spaces are dynamic; an example is the way national boundaries shift as a result of geo-political conflicts and negotiations.

But mathematical models can be partial in other ways as well. For example, a risk assessment based on the probability of a risky event occurring – say a car accident, and the cost of the damages, including hospital bills if there is an injury and the repairs or replacement of the car – is a mathematical description of a risk linked to driving a car. But such a risk assessment is limited, not only in what it regards as possible consequences but also in time. It does not, for example, consider the longer term psychological trauma that the accident victim might face, the cost to the environment of getting rid of the damaged car. Thus mathematical models have limitations including both spatial and temporal limitations.

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One could in fact argue that what we refer to as describing a given phenomenon through mathematics is a deceitful use of words. The normal “description” metaphor with respect to mathematical modelling is too simple to capture the process taking place when mathematics is put into action. A description is only partial, but the previous examples also illustrate that we should not assume that part of reality is re-presented in a mathematical model; how can a part (re)present the whole? It might be more adequate to think of a mathematical description as a reformulation, a re-scripting of elements of reality.

Mathematical descriptions are powerful but also inherently deceptive: they present aspects of reality which can be confused with reality itself. They include some features and not others of what is being modelled; and they are used – sometimes consciously and sometimes unconsciously – to imagine, evaluate and create new realities. This means that the notion of mathematics having the ability to produce descriptions might be a problematic if it is underpinned with the assumption that there exist some principal similarities between the real phenomenon and the mathematically described phenomenon.

It might appear commonsense that “picturing” is a neutral activity. A photo reveals things as they are, and photos can be used in documenting an episode. However, this is only part of the story. A photo is taken from a certain perspective; it includes something and excludes other things. A photo is framed, and framing provides illusions. A photo is taken at a particular moment and not another. It can be black and white; it can be in colour. It can be subjected to any kind of photographic techniques.

Such considerations also apply to pictures provided through mathematical modelling. Any such modelling means including some aspects and excluding others. Something becomes celebrated as important while other things become relegated as insignificant. In this sense mathematical modelling includes a framing. Furthermore, like a black-white photo a mathematical model leaves out something due to the very nature of the description. Both forms of picturing rely on a reduced colour spectrum. A mathematical model excludes everything that cannot be measured and put in numbers. Measures of size, position, speed, acceleration, as well as abstract measures such as means and standard deviations, probabilities, and risks can be formulated; many things that once might have defied mathematisation, for example obesity, intelligence, happiness, progress now have one or more measures associated with them so they also can be included in mathematical descriptions. Any mathematical description imposes a strong formatting on what is described.

An important ethical dimension of any description concerns its inclusion and exclusion. The very process of description, not least when we think of mathematical descriptions, includes powerful ways of prioritising. They provide a grid or a format in which reality is configured. Any process of inclusion and exclusion includes an ethical challenge: whose reality counts, and whose reality does not.

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Moreover, we must raise the question: Who has access to the description? Who has the power to impose the inclusion-exclusion processes that accompany any mathematical description? One can argue that a photo is a most democratic form of description: putting aside likes and dislikes, anybody can make some sense of a photo (if they have access to the photo). Who can seriously engage with a mathematical description; it is far from everybody that can make sense of it. It is meaningful to only a few.

In most budget statements, there would be some kind of formula behind each number that is shown. The faculty budget for wages might, for example, include the various percentage pay rises that are due during the year, allowance for leave that staff might need to take, and increases in wages for some staff who are expected to be promoted during the year. The travel expenses might have been set because a prior agreement had been reached between one professor and the Dean about their travel to an international conference and the budget needs to at least cover that amount. These calculations may be revealed in some cases, or they may not be revealed at all. However, even if they are revealed, the underlying calculations are tied up with the politics of the department and academia generally. The mathematical discourse that characterises a particular description is not independent of other types of discourses of the field; they interact to form a meta-discourse of power.

By describing material and social phenomena within a mathematical discourse, decisions are made, implicitly or explicitly, to include those who are within this particular discourse community and to exclude those who are not. An ethical position is taken by virtue of this choice of language. A script assigns authority to those who master, and indeed author the script and to those who have easy or exclusive access to it. In this sense a mathematical script can be discussed in terms of a holy script: Who has the power to write, read, interpret and draw implications from the script?

3. Mathematics in action: inscription

When a person is describing, they are not only representing aspects of what they are looking at; they are also describing aspects of themselves. Their perspective constitutes part of the description. As a consequence some of our formulations in the previous section need to be reconsidered. We have talked about reality as an entity that could be separated from a description of reality. But reality is also constituted through the process of description and the choice of perspectives.

A perspective is afforded according to one’s position, but the notion of position can be understood in many different ways. It need not only refer to one’s physical position, but also to one’s historical, social, cultural and ideological position. We could also talk about a conceptual positioning, and, as

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a consequence, the grammar (or discourse) one uses in a description might come to shape part of the world. Thus, the grammar of mathematics may imply that a certain “order of things” becomes imposed on or included in a description. In The Order of Things,4 Foucault emphasises that the order of what we experience is not simply an order which is inherent in material or natural objects. Order is also imposed on things. It is also “that which has no existence except in the grid created by a glance, an examination, a language.”5 We can see mathematics as providing a grid, which entails a certain order of what we are addressing. In The Order of Things, Foucault also makes the following observation:

The fundamental codes of a culture – those governing its language, its schemas of perception, its exchanges, its techniques, its values, the hierarchy of its practices – establish for every man, from the very first, the empirical orders with which he will be dealing and within which he will be at home.6

Mathematics can be seen as being part of a fundamental code of culture. Therefore we should not be surprised that mathematics could govern schemas of perception as well as techniques, values, and hierarchies of practices, and in this way establish an order for everybody. This brings us to the notion of inscription.

By inscription we refer to all the elements of a positioning, which are encoded in a description and in this way “gridding” reality. A positioning can refer to the choice of tools or media used for the description. As a consequence a description refers, on the one hand, to elements of an object which is described; and, on the other hand, to the subjective elements of the person creating the description, i.e. to the positioning of the describer. Thus, an inscription makes part of a description. Certainly an inscription need not be seen as an individual act. Social, economic, cultural or ideological priorities can be inscribed. In particular, a broad range of fundamental codes of culture or particular world views can be inscribed through mathematics.

In his study of the emergence of technological artefacts,7 Latour identifies “inscription” of goals and values of human actors involved in the technological development as a critical stage in the design process. In the production of mathematical models, there is also a stage of inscription where values held by model makers about what is important and what is not becomes embedded. Depending on the values held by the script writers, different scripts will emerge; and different inscriptions will be completed.

The budget, as a primary representation of what an academic department does and could do, is increasingly based on a business model of academic work. The finance of the department dictates and defines what is the legitimate, efficient and effective way of doing things, not the other way around. Of course it is possible that if wide and critical scholarship, democratic decision

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making, equitable distribution of resources were to be foregrounded, a different sort of a script might emerge. In this latter scenario, the financial “bottom line” might not be set in advance. Decisions for conference attendance might be made on the basis of the scholarly value of the conference rather than simply on the basis of what is affordable. Inscribing these values into a mathematical model may be more difficult, but some sketch of the department that works on the basis of these values is conceivable, and would be different to that produced by an economic business model. Another obvious example from the academic world would be the focus on the number of publications achieved by a scholar. Even the most productive scholars would probably admit that quality is not the same as quantity. However, when one models the amount of research time per year to each scholar as a function of the number of publications from this person one inscribes certain values into the model. Speedily written articles rather than deeply considered ones that hold new contributions of course constitute an inscribed value in such popular models.

Not only social but also physical and natural phenomena can be seen in different ways depending on the descriptive tool one chooses. Also in such cases “reality” will be constructed according to a particular format that reflects the principal tool of description. In our paper, “The Mathematical State of the World: Explorations into the Characteristics of Mathematical Descriptions”,8 we analyse how Lomborg9 measures the “real state of the world” in the global environmental debate. We see the measurement provided by Lomborg as a paradigmatic case of inscription. In particular, the measurements that Lomborg utilises explicitly differ from what a large number of other researchers in the field of study would welcome as satisfactory measurements of the “state of the world”. A mathematics-based descriptions make it possible to highlight certain elements, and to ignore others. The case of Lomborg displays how different statistical approaches – different inscriptions of values in the mathematical reasoning – can lead not only to different conclusions but rather to opposing views. The Lomborg-environmental case was accompanied by a widespread political discussion where politicians sided with one or the other in the scientific debate. In that way the results of the scientific process come to reflect inscribed political priorities.

Mathematics as a language allows many kinds of positioning to be inscribed in mathematical models. The “grammar” of mathematics influences what can be inscribed and what not. A construction standard for buildings is based on scientific knowledge and experiments. However, there are other factors such as environmental, aesthetic and economic factors that influence what is accepted as the minimal standard. Built into the standard, say in an area with frequent earthquakes, are decisions made on some cost-benefit analysis of the collapse of the building – for example, how many lives would be lost, and what is the cost of the insurance payout; or what is the cost of rebuilding. There are layers of scripts involved in the production of such a standard. The insurance costs are based on complex calculations including parameters like

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age, life expectancy, the economic contributions that are likely to be made by a person of a certain age. In life insurance policies and standards is inscribed the assumption that human lives can be valued in quantitative terms. This is in turn an inscription of an ethical position. In general, mathematical scripts can have written and inscribed within them, ethical positions on fundamental questions about human dignity and the value of a human life.

According to utilitarian principles, ethical standards for actions can be expressed in terms of “pleasure” and “pain”. The formulation of this utilitarian interpretation of ethics in Western philosophy has helped people to make ethical considerations free from the heavy burden of religious ideas and assumptions. Instead of thinking about the “right action” in term of religious obligations, utilitarianism located the ethical discourse within a human domain. We as human beings can determine which actions are right or wrong in terms of how they affect us: do they cause pleasure or pain, and in what amounts? In order for a utilitarian principle to be brought into operation, rather than remaining as a philosophical idea, pleasure and pain have to be measured. And why not interpret pain and pleasure as costs and benefits that can be expressed mathematically? And as soon as this step is taken there is no end to the models that can be brought into operation.

Cost-benefit analysis can be used in the analyses of impact, first of all the financial impact, but other units of measure could be applied as well, with respect to actions of any kind in any kind of situations. We can think of economic decision making in small companies, big companies, international companies; we can think of actions concerning a national economic policy. Should the cost-benefit analysis take into account the burden of the policy on the poor and unemployed, or on the rich shareholders? We can think of local impact, or of global impact. Should the costs of the financial crisis be contained within the countries most responsible or shared globally? We can think of decision making in medicine. Medical or health economics is a rapidly growing discipline and one of its foci is evaluating the cost of different medical treatments and care. Should the cost be borne by the state or the individual?

A mathematical inscription provides a way of seeing, believing and doing. The way one frames a discussion through a particular cost-benefit analysis defines what is taken into consideration and what is left out; it defines the ways decision-making is carried out. Mathematical inscriptions are taking place in the most ethically-sensitive contexts. There is nothing neutral about mathematical inscriptions.

4. Mathematics in action: prescription

Mathematical models are used to make decisions, to prescribe certain actions. For example, on the basis of a variety of economic data, models can be

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developed that calculate the risks of investment for foreign companies working in particular Third World countries. Decision about possible investments – how much, in what, and for how long – can then be based on these numbers. Statistics on the results of international tests of children’s mathematical performance might show that a certain country has performed poorly, so the state might introduce policies to make schools “work harder” to ensure that the children perform better in the next round of tests. Mathematical models help to write a script for what and how people should perform. It is also the case that mathematical models provide a script of how machines and other non-human actors should perform, for instance in terms of reliability of computer controlled devices and energy efficiency of household appliances. Mathematics writes a powerful script for prescribing actions.

What makes the scripts powerful is not just the severity of the prescription itself, but the legitimacy with which the prescription is issued. Although neither the numbers nor the values inscribed within the model are visible, mathematical scripts give the illusion of objectivity and indifference to ideological orientations. The models or scripts are presented as a given, a prescriptive “black box” that is not intended to be opened to uncover the human judgements, values, errors, fetishes that informed what is written into it. The idea of mathematical model as a “black box” is akin to Latour’s characterisation10 of technological artefacts. Latour describes the process of technological development as a series of complex negotiations between actors who bring different interest and visions, but leading to a “black-boxing” once the negotiations are completed.

Let us consider a different example from construction engineering. What standards would be appropriate for ensuring the structural stability of a certain building? As an example, consider the case of constructing earth-quake-proof buildings. How should the engineers achieve a balance between, on the one hand, something that is “reasonably safe” and, on the other hand, something that is “reasonably cost efficient”? What does it in fact mean to balance such different forms of reasonableness? Such a balancing act seems to presuppose an ability to make a suitable kind of measurement, but how can one measure “reasonable safety” and how can one find a unit for measuring “reasonable cost efficiency”?

These considerations bring us to consider the notion of risk. How can we conceptualise a risk? Here mathematics brings a powerful perspective to the situation. A risk, R(A), associated to an event, A, can be expressed as R(A) = P(A)xC(A), where P(A) refers to the probability that the event A will take place, and C(A) refers to the consequences of the event A taking place. If the event A indicates the collapse of a building, then we can imagine very many different calculations in order to estimate both P(A) and C(A). But the first “magic” is that the idea of risk is assumed to be mathematisable, and in the form of estimating two numbers, P(A) and C(A).

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An estimation of P(A) might be based on statistics of say, previous occurrences of earth quake of different magnitudes, the number of similar constructions being submitted to and collapsing under an earth quake of particular magnitudes. How then to estimate C(A), the costs of the event A in fact taking place? There are in fact many different factors to consider: the price of making a new construction being one of them. However, people could be killed or injured because of the collapse. How does one estimate the severity or the costs of this? What is the price of a person? One can look at the estimate as a question of insurance: how much money must be paid in case a person gets killed? Or one could consider: what is the average value of the productive output of an average person during their estimated remaining working life? Using such lines of analyses, one can get an estimation of R(A).

The general point is that there is no longer an obvious and accessible entity which can be considered the risk that is detached from the modelling of the risk: the very modelling shapes the constitution of the risk. Thus, the risk symbolised with R(A) is characterised as a product of a probability, P(A), a number between 0 and 1, and the cost, C(A). In other words, a risk is a price less than C(A), or at most equal to C(A). When such an inscription has taken place, we are able to formulate a statement about what to do and not to do, and we can produce detailed analyses of questions like: Should the building be made safer? How should we, for example, “measure” the lives of people in different countries and regions? Could it be that the value of a person in the USA should be estimated differently than the value of a person in Mexico, considering, say, the insurance payouts? The modelling of risks opens a space for analysis of such questions, and therefore a space for decision making. And if we were able to retrace the origins of some of the decisions that have been made, we may find that highly disputable assumptions are hidden behind the carefully constructed calculations.

Decisions and policies about how things “need to be” are derived from mathematical scripts. This is how the prescriptive function of mathematics operates. Such prescriptions can be based on cost-benefit analysis or on risk estimation; in many cases the prescriptive use of mathematics combines both. One kind of prescription takes the form of norm-setting, including establishing the assumption that in case a phenomenon falls outside the stipulated norm, actions need to be taken. We can think of many examples from medicine where decisions are based on a diagnosis that draws on norms set by precedence. Any norm-setting using mathematics is based on what is mathematisable – that is, describable in numerical forms and relationships. However, in enacting this norm-setting one is not only dealing with the variables that have been mathematised, but implicitly at least, the values and beliefs that are determining what is worth mathematising, and what is not.

Prescription has to do with decision making, and when mathematics is brought into operation certain forms of justification is formulated. Mathematics provides a particular structure to the nature of justification for instance by

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enveloping the discussion in a discourse of objectivity and neutrality. Our principal point is that any form of prescription is in need of ethical consideration. No prescription is objective or neutral, including any mathematics-based prescription.

Boltanski and Thevenot11 argue that in many situations where there is a dispute, people resort to what they call a “regime of justification” to argue and resolve it.12 This is a different way of formulating the assumed “objectivity” that is produced by relying on mathematical measures. The regime of justification is based on the principle of equivalence: what is being experienced is or is not “equivalent” to what it should be (payment for labour, rights to inheritance, speaking order in a meeting, etc.). Mathematical scripts are very effective resources for resolving arguments about equivalence. Indeed, Boltanski and Thevenot argue that disputes can be resolved with reference to a regime of justification (not always mathematical regimes) because in many instances there are initial agreements13 before the dispute arises that are based on devices and instruments (budgets, contracts etc) that allow equivalences to be argued when a dispute emerges later on.

We see mathematics as a powerful “regime of justification”, that often eludes ethical questioning. We find this situation to be a most problematic feature of the prescriptive use of mathematics.

5. Mathematics in action: subscription

An “efficient” mathematical model is one that is generalisable and can be applied to a number of different related problems. As an analogy, one could think of the automated production line. When a mathematically configured conveyor belt has been put in operation, it will continue to function, even after one has forgotten about the purpose and goal of its original instalment. Automated mathematically-based decision making turns into part of a daily-life routine. Mathematical algorithms can establish routines and industrial regimes; they can make certain decision procedures automatic. Many construction or design processes are mathematically-guided. In such cases mathematics-in-action becomes part of the wider shared social reality.14

We refer to the non-reflective stance to a mathematical script as a subscription. Naturally, the very notion of “subscription” seems to include an element of free choice, but in most cases one is “doomed” to subscribe. Mathematics takes part in the structuralisation of our social sphere, and in this way it will be lived out in reality. Mathematical prescriptions materialise as technological artefacts or systems of all forms of industrial fabrication, as well as of hierarchies of practices as referred to by Foucault (in the quotation we brought in section 3).

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Returning to our example on the departmental budget one could imagine that rather than thinking about the scholarly value of a research initiative, academics learn to think about research in terms of prospective research income. Instead of thinking about education as both a source of individual and social empowerment, academics might begin to think about – and subscribe to the idea that –courses, or knowledge and skills are commodities that they can “sell” at the highest price (and deliver at the lowest expense) to their “customers”.

The propensity of many Western societies to use mathematical reasoning in an increasingly widening sphere of their lives is not entirely a new phenomenon.15 The quantification of the social world, and in particular, the increasing commodification of social life – learning, caring, healing – have been critiqued as part of the broader critique of neo-liberalism. Why have we been experiencing what appears to be this increasing mathematisation of society in recent years? This is of course a very hard and complex question but also an important one that takes seriously the concern about subscription to mathematical models in a broader sense. We are using mathematical modelling more and more to support political decision-making in all spheres of our life world. Once produced, mathematical models are easy to implement because one only needs to pay attention to numbers and codes. Mathematical models can be implemented as algorithms on computers. It is easy to routinise the scripts written in mathematics. And once the script turns into routines it becomes even more difficult to question it.

We subscribe to mathematics in this broader cultural sense on a large scale. What we have defined as subscriptions can in this sense become infectious. When this occurs we shall talk about a ‘pre-subscription’ to mathematics. This is the situation where a person or a social entity subscribes to the idea of using mathematical modelling for future actions in relation to areas of decision-making that have not hitherto been subject to mathematical reasoning.

We could reconsider the example from the department budget above. Today, in many Western universities, everything that an academic does is expected to be accounted for and justified in terms of some workload formulae that might be expressed in terms of research outputs and/or student load. An academic’s work can be measured in terms of what number of courses and how many sessions he or she has to do with regard to teaching obligations, and how many publications he or she has to produce to meet some standard. As a consequence, one could try to standardise the number of articles he or she has to write in order to be a top-researcher instead of a mediocre researcher. Such a system of ranking is a good example of pre-subscription to mathematical rationality.

When social practices increasingly subscribe to mathematical modelling we at the same time subscribe to the general process of making our life world measurable and calculable. It appears ideal for politicians at all levels of the

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social sphere to apply the “regime of justification through numbers” to silence critique or hide the real decision making process that takes place when the mathematical model is put into action. Let us take a look at an example from education.

Denmark has, with some variation, done not so well in international scores on mathematical skills among children in primary school. Many efforts have been made to counter the bad exam results in the international competitions in mathematics. Among these efforts is the publication of the average performance of students at each school in Denmark. The government’s public and mathematical description of the state of Danish schools has inscribed in it a number of values: that all schools should in principal be able to produce the same results irrespective of the diverse social contexts of the schools, including, for example, the levels of wealth and educational attainment of the parent community of the schools, the degree of cultural and linguistic diversity of the student community. Prescriptive actions that can be taken on the schools according to this model are easily imagined.

But let us focus on how subscription to the mathematical model is in play here. First, it is a major step to subscribe to the international competition in mathematics as if this very limited and highly specific test could ever give us any clear picture as to the state of Danish schools. It is much more likely that Danish pupils are not thoroughly or exclusively taught the competencies favoured by the international test system, as may be the case in other nations. Secondly it is clear from the Danish case that having subscribed to the quantitative international model for determining how well the school system is doing, the responses to change the situation are influenced by these measures. Seemingly, the consequence is locating the problem in the Danish school system and introducing more refined and pointed measurements that will eventually force teachers to conduct mathematics teaching in accordance with the international test system. Subscribing to the demands of measurability and “teaching to the test” may risk losing less quantifiable kinds of learning like creativity, deep understanding of concepts, social awareness of the role of mathematical modelling in society and other kinds of learning that are not addressed in the international examination regime.

Subscribing has to do with one’s worldview or one’s perspective. Kuhn16 refers to the notion of paradigm, when he describes how a scientific community subscribes to certain standards and assumptions when addressing scientific issues. Naturally, one can subscribe to standards and assumptions which reach far beyond scientific investigations and one can talk about, not paradigms, but discourses. For instance, one can subscribe to certain ways of looking at particular groups of people, particular groups of problems, particular ways of treating people and problems, etc. There are many ways of subscribing to discourses.

When mathematics is imbued in descriptions, inscriptions and prescriptions, particular priorities and justifications are brought into operation.

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The world becomes viewed from a certain perspective and then subscribes to it. When one addresses environmental problems in terms of cost-benefit analyses, one subscribes to a certain discourse about environmental issues. When one addresses how to evaluate academic productivity in terms of standards that can be expressed mathematically, one subscribes to a whole range of particular priorities. When one provides certain measures for decision-making about whether on not to recommend an abortion, one subscribes on a certain standard with respect to the nature of justification of the most fundamental ethical decision – life or death. In all such situations one subscribes to the idea that a discourse referring to measures provides an adequate frame for addressing the issues in question. 17 However, this is far from a neutral assumption. The very act of subscribing to such a regime of justification is another ethical position.

6. Conclusion

Instrumentalists who view that technologies are simply tools used by people to serve their purposes, and as such are in themselves value-neutral may argue a similar case for mathematics. Technological determinists who believe that technologies acquire a momentum of its own once they are “released” may argue the same about mathematics. For both the instrumentalists and the determinists, discussing the ethics of mathematics in action would be pointless – for the instrumentalists, it would not have meaning, and for the determinists, it would be a futile exercise. Understanding mathematics-in-action in terms of scripts enables us to uncover the ethical dimensions of the role of mathematics in generating human imaginations and socio-political practices.

We find that the discussion of mathematics in action in term of description, inscription, prescription and subscription might bring the discussion of both mathematics and mathematics-based technology into a new theoretical framework. We find that mathematics can be interpreted as a powerful technology.18 (Naturally we do not claim that mathematics is nothing but a powerful technology.) As such mathematics can be part of very many different actions. Many discussions in social theorising concern the role on technology in social development. Thus, it has been assumed that technology provides an open “frame” within which social priorities and economic interests can be acted out. According to such an interpretation technological development reflects other forms of social development. It has, however, also been argued that technological development demonstrates its own powerful determinism, which conditions all other forms of development. In particular, the discussion of the development of Information-Communication Technologies (ICT) has often been formulated as if ICT provides a basic format of other developmental processes.

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Our observation is that the investigation between social and technological development can be reformulated when mathematics-in-action becomes interpreted as a principal technological feature. Furthermore, as soon as we see mathematics as connected to action we enter an ethical domain. We have tried to illustrate this by emphasising how the four aspects of mathematics in action – description, inscription, prescription and subscription are all entangled with ethical issues. We do not, however, make a point of trying to maintain a hard distinction between these different forms of scripts, nor of maintaining principal distinctions between different sets of ethical implications. Our aim is first and foremost to provide a terminology which might help to clarify the nature and scope of mathematics-in-action. In this way we try to rework a social analysis of mathematics from being of interest only to the philosophy of mathematics into being of principal relevance to the investigation of technology and to social theorising in general.

Notes

References

Bijker, W. B., T. P. Hughes, and T. Pinch, ed. The Social Construction of Technological Systems: New directions in the sociology and history of technology, Cambridge, MA: MIT Press, 1997.

Boltanski, Luc and Laurant Thevenot. “The Sociology of Critical Capacity.” European Journal of Social Theory 2, no. 3 (1999): 379-96.

Christensen, Ole R. and Ole Skovsmose. “Mathematics as Measure.” Revista Brasileira História de Matemática. Especial No. 1 Festschrift Ubiratan D’Ambrosio, (2007): 143-156.

Christensen, Ole R., Ole Skovsmose, and Keiko Yasukawa. “The Mathematical State of the World: Explorations into the Characteristics of Mathematical Descriptions” Alexandria: Journal of Science and Technology Education 1, no. 1 (2008): 77-90.

Davis, P. and R. Hersh. Descartes’ Dream: The World According to Mathematics. London: Penguin Books, 1988.

Foucault, M. The Order of Things: An Archaeology of the Human Sciences. New York: Vintage Books, 1973. (First French edition 1966.)

Kuhn, Thomas S. The Structure of Scientific Revolutions. Chicago: University of Chicago Press, 1970. (First edition 1962.)

Latour, B. Science in Action: How to follow scientists and engineers through society. Cambrdige, MA: Harvard University Press, 1987.

Latour, B. Pandora’s Hope: Essays on the reality of science studies. Cambridge, MA: Harvard University Press, 1990.

Lomborg, Bjørn. The Skeptical Environmentalist: Measuring the Real State of the World. Cambridge: Cambridge University Press, 2001.

MacKenzie, D. Mechanizing Proof: Computing, Risk and Trust. Cambridge, MA: The MIT Press, 2001.

MacKenzie, D. and J. Wajcman, eds. The Social Shaping of Technology, 2nd edition, Buckingham: Open University Press, 1999.

Porter, T. Trust in Numbers: The pursuit of objectivity in science and public life. Princeton: Princeton University Press, 1995.

Skovsmose, O. Travelling Through Education: Uncertainty, Mathematics, Responsibility. Rotterdam: Sense Publishers, 2005.

Skovsmose, O. In Doubt: About Language, Mathematics, Knowledge and Life-Worlds. Rotterdam: Sense Publishers, 2009.

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Skovsmose, O and Keiko Yasukawa. “Formatting Power of ‘Mathematics in a Package’: A Challenge for Social Theorising?” In Critical Issues in Mathematics Education, edited by P. Ernest, B. Greer and B. Sriraman, 255-281. Charlotte, NC, USA: Information Age Publishing, 2009.

Winner, L. “Do artifacts have politics?” Daedalus, 109 (1980): 121-36.

Yasukawa, Keiko. “Looking at Mathematics as Technology: Implications for numeracy” In Mathematics Education and Society: Proceedings of the First International Mathematics Education and Society Conference, 6-11 September, Nottingham, UK, edited by P. Gates, 351-359, Centre for the Study of Mathematics Education, Nottingham University, 1998.

Yasukawa, Keiko, Ole Skovsmose, and Ole Ravn Christensen (draft). Is Mathematics Important for Social Theorizing? Aalborg: Department of Education Learning and Philosophy, Aalborg University.

19

Ole SkovsmoseDepartment of Education, Learning and Philosophy

Aalborg UniversityFibigerstræde 10

DK-9220 Aalborg Eastosk@learning.aau.dk

Keiko YasukawaFaculty of Arts and Social SciencesUniversity of Technology, Sydney

PO Box 123 Broadway, NSW 2007, Australiakeiko.yasukawa@uts.edu.au

Ole Ravn Department of Education, Learning and Philosophy

Aalborg UniversityFibigerstræde 10

DK-9220 Aalborg Eastorc@learning.aau.dk

1. The notion of mathematics in action has been developed in Ole Skovsmose, Travelling Through Education: Uncertainty, Mathematics, Responsibility (Rotterdam: Sense Publishers, 2005); in Ole Skovsmose and Keiko Yasukawa. “Formatting Power of ‘Mathematics in a Package’: A Challenge for Social Theorising?” In Critical Issues in Mathematics Education, edited by P. Ernest, B. Greer and B. Sriraman, 255-281(Charlotte, NC, USA: Information Age Publishing, 2009); and in Ole Christensen, Ole Skovsmose and Keiko Yasukawa, “The Mathematical State of the World: Explorations into the Characteristics of Mathematical Descriptions” Alexandria: Journal of Science and Technology Education 1, no. 1 (2008).

2. It is also uncertain in what sense we use “reality” in this formulation. Any “reality” may be constructed through a description and a perspective, so in what sense can we talk about reality as being an external object to a description? We will return to this question several times in the following.

3. In P. Davis and R. Hersh, Descartes’ Dream: The World According to Mathematics (London: Penguin Books, 1988), the authors have also used the terms description and prescription in their taxonomy of functions of applied mathematics. They identified prediction as a third function.

4. Michel Foucault, The Order of Things: An Archaeology of the Human Sciences. (New York: Vintage Books, 1973).

5. Foucault, The Order of Things, xx.6. Ibid, xx.7. B. Latour, Science in Action: How to follow scientists and engineers through society

(Cambridge, MA: Harvard University Press, 1987).8. Ole Ravn Christensen, Ole Skovsmose and Keiko Yasukawa, “The Mathematical State of

the World: Explorations into the Characteristics of Mathematical Descriptions” Alexandria: Journal of Science and Technology Education 1, no. 1 (2008).

9. Bjørn Lomborg, The Skeptical Environmentalist: Measuring the Real State of the World (Cambridge: Cambridge University Press, 2001).

10. Latour, Science in Action.11. Luc Boltanski and Thevenot Thevenot, “The Sociology of Critical Capacity.” European

Journal of Social Theory 2, no. 3 (1999).12. In Keiko Yasukawa, Ole Skovsmose, and Ole Ravn Christensen (draft), Is Mathematics

Important for Social Theorizing? (Aalborg: Department of Education Learning and Philosophy, Aalborg University), it is argued in detail how in the conceptual framework of math-in-action, the concept of subscription can be related to Boltanski and Thevenot’s conception of the “regime of justification”.

13. This “initial agreement” is a form of subscription, which we are going to address in the next section.

14. D. MacKenzie, Mechanizing Proof: Computing, Risk and Trust (Cambridge, MA: The MIT Press, 2001) is a study of how the mechanisation of mathematical proofs (through computing technologies) has changed the notion of trust in human judgement.

15. Narrow scientific or technological rationalism in the field of Science and Technology Studies has been the subject of critiques that have led to new ways of understanding science and technologies as human endeavours. See for example B. Latour, Science in Action: How to follow scientists and engineers through society (Cambridge, MA: Harvard University Press, 1987); B. Latour, Pandora’s Hope: Essays on the reality of science studies (Cambridge, MA: Harvard University Press, 1990); W. B. Bijker, T. P. Hughes, and T. Pinch, ed. The Social Construction of Technological Systems: New directions in the sociology and history of technology (Cambridge, MA: MIT Press, 1997); L. Winner, “Do artifacts have politics?” Daedalus, 109 (1980); and D. MacKenzie and J. Wajcman, eds. The Social Shaping of Technology (2nd edition, Buckingham: Open University Press, 1999).

16 Thomas S. Kuhn, The Structure of Scientific Revolutions (Chicago: University of Chicago Press, 1970).

17. See also Ole R. Christensen and Ole Skovsmose, “Mathematics as Measure.” Revista Brasileira História de Matemática. Especial No. 1 Festschrift Ubiratan D’Ambrosio, (2007).

18. Yasukawa has examined the consideration of mathematics as technology in Keiko Yasukawa, “Looking at Mathematics as Technology: Implications for numeracy” In Mathematics Education and Society: Proceedings of the First International Mathematics Education and Society Conference, 6-11 September, Nottingham, UK, ed. P. Gates (Centre for the Study of Mathematics Education, Nottingham University, 1998). T. Porter, Trust in Numbers: The pursuit of objectivity in science and public life (Princeton: Princeton University Press, 1995) provides case studies of how mathematics operates as a technology of distance and objectivity.