CRISIS, CRITIQUE AND MATHEMATICS Ole Skovsmose
Aalborg University, Denmark and State University of São Paulo, Brazil osk@hum.aau.dk
ABSTRACT
The notion of crisis can be applied in different contexts and with reference to a variety of situations. I find it important to relate the notion of crisis to the notion of critique, which provides some further insight in the perspective of critical mathematics education.
Mathematics can be a part of the very formation of a crisis, thus mathematics is not just a descriptive tool, but it also has a performative power. Mathematics can make part of the readings and treatment of a crisis, which however could turn out to be misreadings and mishandlings. Furthermore, mathematics can operate as a political pacifier by making controversial readings and handlings appear neutral and objective. The perspective of critical mathematics education highlights the importance of addressing such features of mathematics not only with respect to school education, but also, for instance, with respect to technical education, university mathematics education, and journalist education. In conclusion, I provide a summary in terms of some recommendations.
Key words: crisis, critique, critical situation, mathematics, mathematical modelling, critical mathematics education.
In 2017, the Ninth Mathematics Education and Society Conference (MES 9) took place in Volos in Greece. It had the title Mathematics Education and Life at Times of Crisis.1 I joined the conference and got much inspiration for rethinking critical mathematics education.
Paying attention to the news, one learns about economic crises that affect families, companies, countries (such as Greece, for instance); about security crises which may turn everyday life terrifying for many people; about energy crises that could turn into military conflicts; and about environmental crises caused by an ever-increasing pollution. Much evidence indicates that we live at times of crisis.
In the article “The Anthropocene,” Paul J. Crutzen and Eugene F. Stoermer (2000) add to this evidence. The notion of anthropocene refers to the recent historical period where the human influence on the atmosphere of the earth is significant. So significant, that we might be witnessing a new geological epoch that needs a name, the anthropocene. A characteristic of this epoch is that we are going through a multitude of crises, and further that it is an illusion to assume that the occurrence of crises are temporary; rather we have to deal with crises as ongoing phenomena. I became aware of the conception
1 See Chronaki (2017).
of anthropocene, through the article “Mathematics Education in the Anthropocene” by Alf Coles (2016). Here Coles opens a profound discussion of the notion of crises with respect to mathematics education.
In the following, I will proceed in four steps. First, I find it important to address the notion of crisis and relate it to the notion of critique.2 Second, I show how mathematics can contribute, making a part of the very formation of a crisis; that mathematics can itself make part of the readings and handlings of a crisis, which in fact can turn out to be misreadings and mishandlings; and that mathematics can operate as a political pacifier by making controversial readings and handlings appear neutral and objective.
Third, I will address critical mathematics education, which highlights the importance of addressing such features of mathematics, not only within school education, but also in, for instance, technical education, university mathematics education, and journalist education.3 Fourth, I will provide a summary in terms of some recommendations.
Crisis and critique
At MES 9, various contributions added to the conception of crisis. In particular, I want to refer to the symposium “Crisis” and the Interface with Mathematics Education Research and Practice: An Everyday Issue organised by Aldo Parra, Arindam Bose, Jehad Alshwaikh, Magda Gonzales, Renato Marcone, and Rossi D’Souza (2017).4 At the symposium, it was pointed out that discourses of crises might represent particular world-views. When we hear about security crises, we might hear about actions of terror taking place in London, or Boston, or Paris. When we hear about economic crises, we might hear about difficulties in the European Union. And when we hear about refugee crises, we hear about the many people from Syria and neighbouring countries crossing the border into Europe. All such references tell us about severe difficulties, but principally about difficulties as experienced by people from the wealthiest countries in the world.
The organisers of the symposium pointed out that crises constitute everyday life for many people around the world, not least from the so-called Third World countries. It was also pointed out that particular strategic priorities can be associated to a crisis terminology. As crises call for urgent actions, references to security crises can be used for justifying implementations of new patterns of control and surveillance; references to economic crises can be used for justifying austerity measures; and references to refugee crises can be used for justifying constructions of new walls and legal barriers.
Furthermore, we cannot ignore the possibility that some crises might be discursive fabrications. All this brings me to acknowledge both the relevance and the complexity of addressing crises.
2. Previously, I have tried to relate mathematics education and crisis (Skovsmose, 1994), but now I feel inspired to rethink the whole issue.
3 I do not make a distinction between critical mathematics education and mathematics education for social justice. Both educational approaches are concerned with addressing forms of social injustices. In the following, however, I choose to talk about critical mathematics education. See Skovsmose (2011).
4 See Greer, Gutiérrez, Gutstein, Mukhopadhyay and Rampal (2017), who also organised a symposium at the MES 9 that addressed the conception of crisis.
The notion of critique is used in many contexts. Here, I will pay attention to the notion as marked by the works of Immanuel Kant and Karl Marx. While Kant provided critique with a profound analytical depth, Marx related critique with real-life crises.
Critical investigations were conducted by Kant in three monumental works: Critique of Pure Reason (Kritik der reinen Vernunft) of 1781, Critique of Practical Reason (Kritik der praktischen Vernunft) of 1788, and Critique of Judgment (Kritik der Urteilskraft) of 1790.5 Kant was deeply rooted in the Enlightenment movement, according to which knowledge has a principal role to play in human development. As a consequence, it seems obvious to ask:
What is knowledge? Kant tried to clarify this question through profound philosophical investigations. He tried not to assume anything about the nature of knowledge based on empirical evidence, but to establish a critique of knowledge as a pure analytical enterprise. To Kant, a critique is a philosophical activity addressing our basic epistemological conditions, and he wanted to provide a once-and-for-all critique of knowledge with permanent validity.
Marx also wrote a book that contained the word critique in the title, namely A Contribution to the Critique of Political Economy (Zur Kritik der politischen Ökonomie) that was published in 1857. Marx’s principal work, The Capital (Das Kapital), which first volume appeared in 1867, has the subtitle Critique of Political Economy (Kritik der politischen Ökonomie).6 So, just like Kant, Marx made heavy use of the notion of critique. However, Marx’s critique was of a different nature. His critique of the political economy addressed a range of economic and political theories, but simultaneously it addressed the very economic and political structures themselves.
Marx’s conception of critique can be related directly to the notion of crisis. He characterised the principle socio-economic crisis as a tension between the dominant class and the working class. This tension would get deeper and deeper due to the ever-growing degree of exploitation. Marx tried to capture this exploitation in terms of the “tendency of the rate of profit to fall,” which he incorporated as a crucial explanatory feature of the collapse of capitalism. When the exploitation cannot be brought any further, the result will be an open crisis, a revolution, which opens for a new economic and political order.
I am inspired by the way Marx relates crisis and critique, but several features of his outlook I do not embrace. I do not assume that the outcome of certain crises might be predictable.7 Marx also assumed the existence of a certain hierarchical order among crises, defining the tension between the dominant class and the working class as being the principal one, and interpreting other crises as being “derivatives” from this “top crisis.” Contrary to this, I do not make assumptions about a pre-given structural relationship among crises. I see the development of crises as being interrelated in unpredictable ways. Finally, Marx wanted to provide his critique with a solid theoretical foundation. In this sense, he wanted to present his version of a once-and-for-all critique, not an analytical version, but a political version.
However, the very idea of providing a critique with a “solid foundation” is to me an illusion.
I see crises as contingent phenomena, and related critical activities I see as tentative and preliminary. I see critique as an expression of uncertainties.
5 Kant’s works have been published in many editions, see, for instance, Kant (1933, 2007, 2010).
6 Marx’ works have been published in many editions, see, for instance, Marx (1970, 1992, 1993, 1993).
7 Marx was inspired by Newton’s formulation of exact laws grasping the movements of physical bodies. In a similar way, his aspiration was to formulate laws, which grasped economic development. In this way, Marx enveloped his conception of crises and critique in a deterministic outlook.
The relationship between crisis and critique can be highlighted, when we take a look at the origins of the words. The Greek words kritikós, kritikê and kritikón refer to the capacity of judging and deciding. These adjectives are derived from the verb kríno, which means to separate, to distinguish, or to decide. Krísis is a noun with meanings like “act of distinguishing,”
“act of choosing,” and “act of deciding.” Like kritikós, it also refers to a decisive phase in an illness. The words kríno and krísis have the same root, namely kri, which means to choose or to separate.8
That crisis and critique can be related, brings us to the notion of critical situation. A patient at a hospital might be in critical situation. Things could go “both ways:” the patient might become cured, or the situation might end with fatal consequences. In the most direct way, a critical situation represents a crisis. A critical situation is in need of being critically addressed. This applies to the person at the hospital; it applies to any critical situation. In the following, I am going to use the notion of critical situations as referring to a crisis that is calling for a critique. I will allow myself to be rather free in sometimes talking about a crisis and sometimes about critical situation.
Mathematics
Mathematics has often been described as a pure science by means of which we become able to provide neutral and objective descriptions. Contrary to this, I want to provide a performative interpretation of mathematics by showing how mathematics can make part of the formation of crises; how mathematics can become part of readings and handlings of crises; and how mathematic can serve as a political pacifier. 9
Mathematics can make part of the formation of crises
In the book Weapons of Math Destruction, Cathy O’Neil (2016) discusses the role of mathematics in handling huge data material.10 O’Neil is a research mathematician who quit her job at the university in order to work in a private hedge fund company. She became involved in the administration of huge investments. In 2008, after one year in the company, came the economic crash, and O’Neil observes:
That crash made it all too clear that mathematics, once my refuge, was not only deeply entangled in the world’s problems but also fuelling many of them. The housing crisis, the collapse of major financial institutions, the rise of unemployment – all had been aided and abetted by mathematicians wielding magic formulas. What’s more, thanks to the extraordinary powers that I loved so much, math was able to combine with technology to multiply the chaos and misfortune, adding efficiency and scale to the systems that I now recognized as flawed. (p. 2)
After having experienced the economic crash as a mathematical expert, O’Neil points out that mathematics not only makes up part of the world’s problems, but actually fuels them. This 8 In this clarification of the Greek words, I got important help from Irineu Bicuco, State University of São Paulo at Rio Claro.
9 Previously, I have provided a performative interpretation for mathematics by exploring mathematics in action, see, for instance, Part 4 “Mathematics and Power” in Skovsmose (2014). See also how Ole Ravn and I develop the ethical dimension of a philosophy of mathematics, Ravn and Skovsmose (2019).
10 In a discussion at MES 9, Peter Gøtze draw my attention to this book.
observation bring O’Neil to coin the notion of weapons of math destruction, which refers to huge scale mathematical models, which are opaque and damaging.
As an example, O’Neil refers to IMPACT, which is a programme implemented in 2009 by the school administration of Washington DC with the purpose of evaluating teachers. By the end of the 2009-2010 school year, when this evaluation was completed, the 2 percent of the teachers with the lowest IMPACT-scores were identified, and they became fired. For the teachers, IMPACT certainly was damaging. It was opaque as well, as it remained unknown how the scores became calculated, except that they first of all were based on numbers, as for instance the scores in tests of the teachers’ students.
With respect to models of this type, O’Neil points out that “many of these models encode human prejudice, misunderstandings, and bias into the software systems that increasingly managed or lives” (p. 3). Throughout the book, she presents several examples of such weapon of math destruction operating in, for instance, financing, advertising, and management. She describes how they are causing a range of critical situation for individuals and groups as well as for whole sections in society.
Let me now try to be more specific and ask: What features of a mathematical modelling could bring about a critical situation? I have in mind not only modellings applied to huge data material, as considered by O’Neil, but any kind of mathematical modelling.
As an example, let us imagine that we are going to construct a sky scraper in an area close to a swamp. We have to consider the size of the building as well as the solidity of the ground.
Certainly, a trial and error approach is not applicable. Before any construction does begin, we have to consider the parameters that are relevant for making decisions with respect to the construction. The only way of doing so is to operate with mathematical models. We have to provide models of the ground as well as of the building. Within this model-world, we are going to estimate the conditions for completing the construction.
This applies to the construction of a skyscraper. It applies to any form of technological construction being a bridge, ferryboat, airplane, airport, drone, coffee machine, etc. However, the scope of mathematical modelling reaches much further that to the construction of physical objects. Mathematical modelling is crucial when we consider, for instance, the implementation of new economic initiatives, production schemes, or medical programmes.
Also in such cases, mathematical modelling is used for clarify consequences of what one might be doing, before doing it.
This observation brings us to the notion of similarity gap, which refers to the difference between the mathematical model and the completed technical construction (being physical or not). Such a gap might be minimal in case the model and the real construction were similar.
But this is never the case. The existence of similarity gaps is a direct implication of the fact that mathematical models always include simplifications and stipulations, if not prejudice and misunderstandings.
In order to characterise our present social condition, Ulrich Beck (1992, 1999) talked about the risk society. By this notion, he wanted to highlight that – while previously risks were caused by nature as floods, drought, epidemics – risks today are also caused by human beings, in particular through the application of advanced technologies. I am in line with Beck’s observations, but want to add that mathematics plays a particular role in the formation
of risks. Due to similarity gaps that accompany any form of mathematical modelling, we are not able properly to predict the implications of what we are doing. While a laboratory is a closed environment where technological research experiments can be performed, the whole society today becomes subjected to technical experiments, which implications can only be address through mathematical models. As similarity gaps cannot be eliminated, mathematics comes to make part of the very formation of the risk society.
A risk represents a potential crisis, which may turn into an actual crisis. Acknowledging the idea of the anthropocene, instead of risk society, one could talk about a risk/crisis society, and observe that the extensive use of mathematical modelling drives us into this society.
For how long a time has this drive taken place? Beck made his observations with particular reference to the implementation of atomic energy. One can as well refer to the science-based development of war technology, as taking place during the 20th century. Mathematics made part of this development, but new chapters become added. With references to the mathematics-based handling of huge data material, O’Neil (2016) observes: “By 2010 or so, mathematics was asserting itself as never before in human affairs, and the public largely welcomed it.” (p. 3). While I consider crises a general part of human history, the role played by mathematics in the formation of crises might be a relatively new phenomenon. However, this issue needs careful consideration.11
Mathematics can make part of the readings and handlings of crises
Mathematics may not only make up part of the formation of crises, but also of our readings and handlings of crises. I will illustrate this point by referring to the discussion of climate change.
In order to provide any weather forecast, advanced mathematical modelling is brought into action. This also applies when the forecast does not just have to do with the weather tomorrow, but with environmental forecasting addressing the future climate of the planet.
Such forecasting cannot be based on any well-controlled laboratory experiment. The only possibility is to provide some mathematical modelling, and on this basis to try to look into the future. Climate models provide the departure for such predictions. With such models at hand, one can change the values of some of the parameters, for instance with respect to the level of pollution by certain materials, and calculate the implications with respect to the level of global heating. Evidently no such model-based predictions are reliable, as there is no way of closing down similarity gaps.
In the article “The Mathematical Formatting of Climate Change: Critical Mathematics Education and Post-Normal Science”, Richard Barwell (2013) makes observations with respect to the use of mathematics in the discussion of climate change. I will highlight two of his observations.
11 One can, for instance consider the Quebra-Quilos Revolt (Breaking the Kilo Uprising) that took place in the in middle of the 19th century in the North East part of Brazil. The region faced serious economic difficulties, and at the same time the government changed the traditional metric with the French system. This caused an uprising which the government violently repressed. Could that be an example of a crises formed through mathematics?
(See Knijnik, 1998, 2012)
First, Barwell points out that the identification of possible climate changes depends on mathematics. With reference to the book Mathematics of Climate Change: A New Discipline for an Uncertain Century by Dana McKenzie (2007), Barwell states: “The prediction of likely future course of climate change is based on more advanced mathematics. Developing predictions about future global, regional or local effects of climate change draws on a range of advanced mathematical methods, including mathematical modelling, differential equations, non-linear systems and stochastic process.” (p. 2).
Second, Barwell observes that any mathematics-based identification of climate change, reflects not only some possible physical phenomena, but also the very nature of the mathematics being used. Mathematics provides a particular reading of climate change, making space for a selective set of possible actions. In Barwell’s formulation: “Mathematics also formats how we interact with the climate. Through the mathematised, model-based perspective prevalent in climate research, the climate is constructed in particular ways: as, for example, measurable, predictable, technical and controllable by humans, rather like the temperature in a high-tech sensor controlled greenhouse.” (p. 2).
Barwell’s two observations lead me to highlight the fact that mathematics makes up part of any reading of climate change. There is no way of doing this without mathematical modelling. However, we cannot expect mathematics to provide any neutral reading; rather it establishes a particular perspective that may serve particular political, economic or industrial interest, by for instance portraying climate change as being manageable.
Any reading opens a space for action, but maybe it also does the opposite. As an example, I can refer to a study of climate change, The Skeptical Environmentalist: Measuring the Real State of the World by Bjørn Lomborg (2001). He based his study on mathematics and statistics, and he reached the conclusion that climate changes is not caused by human inventions. His readings explicitly contradict the readings that lead to the conception of the anthropocene, and Lomborg’s reading was welcome by industry.
In many situations, the notion of similarity gap might be inadequate for capturing how mathematical modelling is operating, as it assumes that the model can be compared to reality itself. However, we cannot make any such assumption in general. A mathematical model might create its own reality in accordance with assumptions and presumptions that it incorporates. A mathematical modelling might provide a reality fabrication, which might be taken as reality itself. Lomborg’s investigations might be an example of this.
Mathematics can function as a political pacifier
In his study “Truth” as the Pacifier of Mathematical Education: How Middleclass Students’
Argumentation can Reveal their Position about Social Injustices, João Luiz Muzzinati (2018) formulates the idea that mathematics can serve as a political pacifier. By this he means that numbers, including diagrams and graphs, may provide illusions of objectivity and neutrality.
A political action, whose justification becomes accompanied by numbers, becomes more smoothly put in action. Due to the numbers, the suggested action might appear the most reasonable, if not the only thing to do.
Let me illustrate this point by referring to Dimitris Chassapis (2017) plenary lecture at the MES 9 conference. The lecture had the title: “Numbers have the Power” Or the Key Role of Numerical Discourse in Establishing a Regime of Truth about Crisis in Greece. In the
lecture, Chassapis showed how the Greek crisis was portrayed in the media, and the extent to which numbers made part of this portrayal.
As the title signalises, Chassapis has found inspiration in the work of Michel Foucault. The expression “regime of truth,” as coined by Foucault, highlights that “truth” is not anything epistemologically sublime, rather it can be related to the mundane notion of “regime.”12 As any regime is an expression of power, so is the regime of truth. Such a regime can have certain extensions, both in time and space, but it does not include ahistorical features; its power is temporary.
The regime of truth that Chassapis is referring to is constituted through numerical discourses.
It establishes a way of seeing the Greek crisis, and it opens a way of interpreting the crisis, where one possible way out becomes identified as the only thing to do.
In fact, techno-mathematical discourse premises an action it covertly wants to recommend as policy, and then cites “evidence” and “reasoned arguments” which show that this action as the only feasible option. (p. 47)
A mathematical discourse can create false necessities and stipulate “evidence” and “reasoned arguments” that turn the action in question into the only option.
This observation brings Chassapis in the direction of the notion of political pacifier. Although he does not use this notion, he makes observations that clearly resonate with Muzzinati’s conception:
In situations of economic and social crises, in which a plurality of political forces, interest groups and societal views are contesting, numerical discourse used by the media may produce a public rhetoric of interest or disinterest. Such rhetoric may play a crucial role in the creation of a public sphere where technical expertise dominates political debate excluding people not only from political debates but also from acting towards or reacting against political decisions and policies. (pp. 53-54).
By excluding people from political debates and preventing them from acting against political decisions, number-based discourses come to function as pacifiers.
That mathematics functions this way is due to the image of mathematics as a pure science by means of which one is able to provide neutral and objective descriptions. Considering that mathematics may form crises as well as providing (mis)readings and (mis)handlings of crises, this image is wrong. However, the image of mathematics as being a neutral arbiter as still broadly assumed.
Critical mathematics education
I have tried to characterise critical mathematics education in different ways (see, for instance, Skovsmose 2011), but here I only want to highlight that I see this critical mathematics education as an expression of several concerns: What is the role of mathematics in social affairs? How is mathematics brought in action in technology, in economy, in daily life? What functions might mathematics education serve in adjusting new generations to the given social
12 See Foucault (2000).
order? What potential does mathematics education have for addressing cases of social injustice?
I find that critical mathematics education is important when one wants of address how mathematics makes up part of the formations of crises and brings about mis(readings) and (mis)handlings of crises. Critical mathematics education as engaging in the crisis-critique dialectics, and I find that its concerns are relevant, not only with respect to school education but to many other forms of education as well (see Skovsmose, 2016).
School Education
Taking a look at the vast literature of critical mathematics education, one finds descriptions of project work addressing how unemployment is affecting different groups of people; how poverty causes damage to people’s health; how racism effects all sectors in social life; how salaries are gender biased; how displacement is taken place around the world today; and many more topics.13
Inspired by Paulo Freire, Eric Gutstein (2006) published the book: Reading and Writing the World with Mathematics: Toward a Pedagogy for Social Justice. “Reading the world” means interpreting the world from the perspective of the oppressed, making it possible for them to identify forms of injustices wherever they might occur; and “writing the world” means engaging people in political actions and making changes. Gutstein shows how this can be done through mathematics.
We can make a small modification of the formulation and talk about reading and writing critical situations with mathematics. This formulation will apply nicely to most of the examples that have been developed with reference to critical mathematics education, if not all of them. I find that critical mathematics education to a large extent has been developed with a high sensitivity, both with respect to critical situations that concerns society in general as well as to critical situations that concerns communities, families and individuals. I see critical mathematics education as elaborating on the close connections between crises and critique.
The observations concerning mathematics-based readings and handlings of crises makes us aware of a necessary self-critique of critical mathematics education. As pointed out, mathematics can contribute profoundly to the readings of crises. Such a reading can take any format and include preconceptions and misunderstandings. We cannot be sure that when we apply mathematics, we will make any adequate reading of a crisis. We can have to do with a misreading, and the case of Lomborg might illustrate this possibility. In fact, we might not be able to distinguish clearly between readings and misreadings, nor between writings and miswritings.
The very notions of reading and writing with mathematics was coined within the outlook of critical mathematics education. However, the metaphor can be applied in many other cases.
Right-wing policy, for instance in the form on neo-liberal initiatives, is also based mathematics-based readings and writings of the world. We have to deal with powerful readings-writings that might have disastrous implications.
13 See, for instance, Alrø, Ravn, and Valero (Eds.) (2010); Avci (2019); Ernest, Sriraman and Ernest (Eds.) (2015); Frankenstein (2012); Greer, Mukhopadhyay, Powel and Nelson-Barber (Eds.) (2009); and Wager and Stinson, (Eds.) (2012).
Technical education
Critical mathematics education has first of all been explored with respect to school education, but its relevance reaches much further. Here I will address any form of technical education, referring to studies such as engineering, business, medicine and economy.
When crises are addressed through mathematics, mathematics itself needs to be addressed critically. This point has been carefully elaborated in the article “Categories of Critical Mathematics Based Reflection on Climate Change” by Kjellrun Hiis Hauge, Peter Gøtze, Ranghild Hansen and Lisa Steffensen (2017). The authors outline a range of reflections needed when climate changes become explored through mathematical modelling. Some reflections concern the technical aspects of the model, others more general aspects.
“Categories of Critical Mathematics Based Reflection on Climate Change” is formulated with school education in mind, but to establish critical reflections on mathematics concerns any technical discipline where mathematics-based forecasting becomes applied. In technical disciplines it is routine to engage in mathematical modelling, and there are many issues that become addressed as part of this practice, for instance the sensitivity of the involved parameters. It is important to clarify how the general result of a modelling might change when the value of some of the individual parameters becomes changed. Such a clarification makes it possible to identify the parameters the whose value need to be estimated with the highest degree of accuracy. Such clarifications are part of any technical evaluation of a model.
However, it is not sufficient only to conduct such technical evaluations. The use of mathematics brings about similarity gaps and possible fabrications of “reality.” Any technical discipline might contribute to the formation of the risk/crisis society. Like technical evaluations, broader socio-political evaluations of mathematical modelling should also be a part of technical education. Addressing critically how we read and write the world is not only relevant with respect to school mathematics, it is relevant with respect to any technical education. Also in this case, reading could imply misreading and writing could imply miswriting.
University mathematics education
We now turn our attention to university studies in mathematics. Most often, they are not conducted from the perspective of critical mathematics education, but recognising that mathematics may operate as a political pacifier we need to reconsider this possibility. 14 The pacification caused by mathematics emerges from the glorification of mathematics, which has deep historic roots back to the so-called scientific revolution. The people engaged in this revolution – Copernicus, Kepler, Galilei, Newton, etc. – were all deeply religious. The universe was considered God’s creation. They believed that we can come to understand this creation by means of mathematics, for it appeared that mathematics captures the rationality of God. Mathematics represents a sublime form of thinking.
Later the belief in God was removed from the scientific outlook, but the glorification of mathematics continued. This glorification can be captured in terms of an ideology of purity.
14 One finds exceptions. See, for instance, Vithal, Christiansen and Skovsmose (1995). See also Skovsmose, O.
(2016).
An explicit formulation of this ideology is found in A Mathematicians Apology, written by G.
H. Hardy (1967) and first published in 1940. Hardy was aware that the First World War demonstrated a broad application of scientific insight in physics and chemistry in the very construction of the war machinery. He certainly has acknowledged that this could become repeated. However, he claimed:
[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. (pp. 140-141)
At the final pages of his Apology, Hardy writes the following about his own particular 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 ill, the least difference to the amenity of the world. (p. 150)
Hardy considers the work in pure mathematics as being a form of art. But he is explicitly wrong in this claim with respect to “usefulness.” His work in number theory has huge applications in cryptography, among other applications it is crucial for the functioning of communication within a military organisation.
Normally, the ideology of purity is not stated explicitly as done by Hardy, but it can be acted out in different ways, in particular through the very education of pure mathematicians. This ideology becomes formed, not through what is included in the curriculum, but through what becomes excluded. By strictly focussing on mathematical issues, one relegates a range of issues as being without significance. Applications of mathematics become addressed only at a different place: the department for applied mathematics. It becomes taken for granted that issues related to mathematical modelling is not significant for pure mathematicians. And broader issues are not addressed as, for instance, how mathematics makes part of the overall cultural structures in society, or how mathematic notions and ideas have been formulated and reformulated during time. Certainly, nothing is said about the possible formation of crises through mathematics. The ideology of purity becomes cultivated by all such omissions.
The ideology of purity is composed by different assumptions, of which I will refer to three.
Mathematics ensures neutrality. Mathematics is not associated to any particular interest, but represents pure rationality. Mathematics ensures objectivity, as it presents things as they really are, and not as one might think they are. Mathematics removes subjectivity from any scientific enterprise. Finally, mathematics ensures certainty, as any calculation is definite.
The ideology of purity has devastating implications. It is basic for mathematics functioning as a political pacifier. As illustrated by Chassapis, due to this ideology the mathematical reading of a crisis appears neutral and objective, and the actions taken become identified as necessary course of action.
For me, it is crucial to challenge the political pacification provoked by the ideology of purity.
It is important to challenge how departments of pure mathematics function as factories producing an ideology of purity. It becomes crucial, also in this education, explicitly to address the possible socio-political functions for mathematics. It becomes important to challenge any attempt to maintain a distinction between pure and applied mathematics. The
perspective of critical mathematics education thus also becomes relevant for university mathematics education.
Journalist education
Chassapis’ discussion of the Greek crisis turned our attention to how the media presents a crisis. Here we also find a reading of crises that reflects assumptions and priorities as well as a solid dose of the ideology of purity. This turns our attention to the education of journalists.
The relevance of addressing journalist education has been pointed out to me by Miriam Godoy Penteado, and together we are preparing the project Critical Mathematics Education for Journalists.
In the daily media, one finds a broad range of information put in numbers. One finds the weather forecast; one finds forecasts of any kind, for instance with respect to economy, employment, elections. One finds all kind of critical situations presented in terms of numbers, as was the case with respect to the Greek crisis. However, if we look at the actual education of journalists, we do not find much attention paid to the role of mathematics – as, for instance, how it might operate as a political pacifier in the time of crises.
Certainly, we are not suggesting that mathematics as a discipline should enter the curriculum of journalist education, but the socio-political role of mathematics needs to be addressed. It is important that the ideology of purity becomes challenged and that the assumptions of neutrality, objectivity and certainty become questioned. Numbers are just one way of expressing also misconceptions and prejudices, and they need to be critically addressed like any other expression of possible misconceptions and prejudices.
Let us look at an example. Working conditions are a recurrent topic in the media. Back in time, Japanese companies in the car industry launched the just-in-time approach. The idea was to eliminate the need for huge storage capacity. Instead of storing the different components from which a car becomes assembled, the idea was to organise the overall production scheme in such a way that these components arrived next to the assembly line just when needed. The just-in-time approach, implemented through advanced mathematical modelling of the production process, added enormously to the productivity in the car industry.
The idea can be applied with respect to many other types of production; and it can be applied not only with respect to mechanical components, but also with respect to human beings.
Thus, one can imagine a new efficiency established when the people needed in a production process are present just when needed. Clearly, the companies’ expenses on salaries could be kept to a minimum.
In order to implement a just-in-time approach handling human beings as components in a production process, mathematical modelling becomes put in operation in terms of scheduling software. O’Neil makes the following observation:
Scheduling software can be seen as an extension of just-in-time economy. But instead of lawn mower blades or cell phone screens showing up right on clue, it’s people, usually people who badly need money. And because they need money so desperately, the companies can bend their lives to the dictate of a mathematical model (O’Neil, 2016, p 128).
The mathematical model defining the scheduling software forms the daily life of the employees. This is an example of a mathematics-based fabrication of reality – a reality experienced by many people, not least those who “badly need money.”
There are many cases of such mathematics-based reality fabrications. Insurance companies provide different offers, based on a careful modelling of different segments of people.
Advertising takes new forms; targeting advertising being a recent approach which presupposes that huge data material becomes processed and continuously revised and expanded. Banks offers loans on different conditions to different groups of people, being the result of extensive mathematics-based risk calculations. This is just to mention some of the examples to which O’Neil refers to in terms for weapons of math destruction.
Journalists are addressing the daily life-reality of very many different groups for people. This reality might be formed by mathematical fabrications, which cannot be ignored by journalism.
Recommendations
The perspective of critical mathematics education is relevant for school education, technical education, university education, and journalist education. However, it is relevant for other contexts as well. Let me summarise why I find this to be the case through the following seven recommendations.
First: As mathematics educators, we must be ready to address crises of any kind: it could be emergent global crises, as referred to through the anthropocene; it could be local crisis as experienced by different groups of people; and it could be crises experienced by specific families. Crises can be interconnected in many ways, and we must be ready to look at crises from a variety of perspectives. We must be ready to address critical situations, also in cases where they are not articulated as such: crises can be ignored, or even hidden away. We must also be be aware that a crisis-terminology can be invented, serving the advancement of certain political initiatives.
Second: In all its formats from the elementary to the most advanced, mathematics can contribute to the very formation of crises. Mathematics is not just a simple and transparent descriptive device; it is performative as well. When brought into action, mathematics might deepen and accelerate a crisis. It can also bring about new critical situations.
Third. Mathematics might provide a reading of a critical situation that is not possible otherwise. Mathematics provides a unique tool that cannot be substituted by any other approach. A mathematics-based reading of a crisis might be unique and insightful. However, it might also provide a limited understanding of the situation. Mathematics might convey problematic assumptions and misguiding simplifications.
Fourth: A reading of a crisis might lead to a writing of it, referring to ways of handling the crisis. As any mathematics-based readings might be insightful, so mathematics-based handlings of crises might demonstrate unique qualities. However, as just as misreading of a crisis is a possibility, so too is a mishandling. Mathematics-based interventions in a critical situation might initiate a dramatic slide into deeper risk structures.
Fifth: Mathematics may operate as a political pacifier. It might tend to substitute a broader discussion of a critical situation by technical specifications of what to do. It might present a particular way of responding to a critical situation as being the only possible thing to do, and this way turn further discussions superfluous. Mathematics may provide a discourse that serves decorative purposes by bringing about a glitter of neutrality, objectivity and certainty.
Sixth: Critique is always preliminary and uncertain. I do not assume the possibility of identifying a solid theoretical foundation for critique (as assumed by Marx), nor to identifying a proper analytical approach to critique (as assumed by Kant). Mathematics might bring new approaches to critical investigations, but it does not bring about new degrees of certainties. Any form of critique, also mathematics-based critique, is in need of self-critique.15 Seventh: Crises might include an overwhelming dynamics that we seem unable to cope with, and we have to recognise the brutality of crises. Simultaneously, we have to recognise the fragility of critique, as we cannot expect that through a critical activity we will be able to adequately interpret and manage a crisis. To me the brutality of crises combined with the fragility of critique set the parameters for our human condition. This is also the condition for critical mathematics education.
Acknowledgement
I want to thank Daniela Alves, Denner Barros, Ana Carolina Faustino, Peter Gates, Renato Marcone, Amanda Queiroz Moura, João Luiz Muzinatti, Aldo Parra, Miriam Godoy Penteado, Celia Roncato, and Débora Vieira de Souza for their helpful comments and suggestions.
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