Proceedings on Privacy Enhancing Technologies

Skovsmose O.

In this final chapter I present a dialogic theory of learning mathematics in terms of the following learning interacts: getting in contact, exploring, positioning, foregrounding, externalising, and doubting. This theory captures both the learning of mathematics and learning about mathematics. When having the latter kind of learning in mind, I also talk about processes of reflective inquiries. They might bring to the forefront the ways in which mathematics is involved in all kinds of social affairs, and question the glorifications of mathematics. Reflective inquiries are important for bringing mathematics out of an ethical vacuum. I present examples of how students can be engaged in reflective inquiries. Such inquiries might concern the nature of mathematical truth and oppose absolutist positions. They might concern socio-political issues and reveal how mathematics can be a means for identifying cases of social injustice. Reflective inquiries never result in definite conclusions. They are of an aporetic nature.

Skovsmose O.

The close connection between mathematics and power is expressed in many ways. One way is the formation of the digital panopticon that is making use of big data. We are all potentially open to being observed as the data about our behaviour is collected, stored, and made available for analysis. Big-data analysis demands new mathematical and statistical techniques; it is of huge interest to financial markets, for social control, and for the military. Economic decision-making is an expression of the mathematics-power amalgamation. Algorithmic procedures can ascribe a credit score to a person, and based on this score a bank can judge whether the customer should be offered a loan or not. The FICO system was an original way to do so, but recently e-score systems have been created that make use of big data. Social power concerns the governing of and conditions for democratic life. The crucial question is whether further mathematisation of society tends to support or obstruct democratic life. Mathematics contributes to research on enhancement of military efficiency, and much of this research is directly financed by the military.

Skovsmose O.

The modern project embraces a belief in progress, a belief nourished by a trust in human rationality. It encompasses the conviction that after the scientific revolution, science found its proper form and has made ongoing advances ever since. The belief in progress also includes a technological optimism, seeing technology as a solid engine of progress. However, the modern project includes profound ambiguities. As part of Modernity, one sees brutal forms of colonisations, the development of slave trade, and the formulation of explicit racist worldviews. The industrial revolution established new forms of exploitation and oppression of workers and their families. Also during Modernity, several acts of genocide have taken place. The Holocaust is just one gruesome example. Mathematics is an integral part of all features of Modernity, and the ambiguities inscribed in Modernity might be inscribed in mathematics as well. A critical philosophy of mathematics must bring to the forefront the diversity of roles, including the horrific ones, that might be played by mathematics.

Skovsmose O.

The so-called “big discoveries” started a race to establish colonies worldwide. Colonisation harboured imperialism, which facilitated brutal exploitation through the extraction of natural and human resources. Attempts to “justify” colonisation gave rise to further racist explanations. With the phrase “the irony of globalisation”, I refer to some ambiguity encapsulated in this notion. On the one hand, globalisation seems to indicate solidarity, to embrace the perception that we are all in it together, and to signal a universal concern for each other. On the other hand, processes of globalisation demonstrate brutal forms of exclusion and exploitation. The globalised world is a world apart. Processes of globalisation include the automatisation of production processes, financial transactions, and the free movements of ownership. Such processes are rooted in mathematics-based technologies. Mathematics is deeply embedded in global networking and in all its accompanying ambiguities.

Skovsmose O.

Copernicus proposed a heliocentric cosmology which initiated the scientific revolution. Further important contributions came from Galilei, Kepler, and Descartes, before a coherent mathematics-mechanical worldview was presented by Newton. According to this view, nature is operating according to laws that can be formulated mathematically. This inspired the idea, strongly advocated by logical positivism, that mathematics is the language of science. This brought together the Modern glorification of mathematics, claiming mathematics ensures scientific unity, objectivity, and neutrality. This view coagulated as a paradigmatic dogma portraying science and mathematics as isolated phenomena not involved in any social or political complexities. This idea engenders an inadequate conception of science and mathematics.

Skovsmose O.

A critical philosophy of mathematics considers mathematics as being indefinite. Mathematical constructions are not heading towards definite formats; they are tentative and always open to change. In this chapter I concentrate on showing mathematics as being indefinite with respect to concepts and proofs. I illustrate this with reference to the changing roles of the notion of infinitesimal. Before the nineteenth century the term infinitesimal was used liberally, but in the nineteenth century it became a concern to ensure rigour in mathematical reasoning, and to get the infinitesimals under control. The notion of function is indefinite; in fact, it has not always been part of the mathematical vocabulary. Neither Newton nor Leibniz operated with the notion of function. The very word function was first used by Bernoulli, and Euler made some further clarifications of the notion. The notion of infinity has, in its own way, brought about huge controversies in mathematics. The notion is indefinite, and this indefiniteness has a huge impact on what to consider as a valid mathematical proof.

Skovsmose O.

Mathematical performatives are found in all spheres of life. Mathematics does not simply depict reality, but rather formats the natural as well as the social reality that it is assumed to describe. Mathematics fabricates the mechanical worker as a gear to be incorporated in the clockwork-like production machinery. Mathematics contributes to the sanctification of norms, for instance with respect to health, beauty, and productivity. Mathematics is used for splitting up practices into a surface practice requiring a few competences, and a deep practice operating with complex sets of mathematical algorithms. Mathematics is formatting our future—for instance, by providing means for constructing climate models and for completing experimental forecasting. Mathematics and power are interacting phenomena, but mathematics tends to disguise this interaction. By means of mathematics it is possible to identify cases of injustice and, in this way, articulation of social justice also becomes one of the possible performatives of mathematics.

Skovsmose O.

Much philosophy of action has focused on what could be called individual or collective actions. One can, however, also consider structural actions, which need not be deliberately performed by any individual, team, group, or institution. Structural actions can be extremely powerful and have a profound and also devastating social impact. In this chapter I am going to elaborate on the performative interpretation of mathematics by relating mathematics to individual, collective, as well as structural actions. A variety of different consequences can be brought about by actions, and so these actions need to be critically addressed. I see critique of action as being an ethical endeavour. Logical positivism has insisted on the existence of a sharp distinction between, on the one hand, mathematics and science, and, on the other hand, ethics. Logical positivism inserts mathematics and science into an ethical vacuum. A critical philosophy of mathematics opposes the elimination of ethics from mathematics, and highlights that mathematics is involved in all spheres of life, including the formation of wonders as well as of horrors. This brings mathematics face to face with an ethical challenge.

Skovsmose O.

Logicism tried to show that the foundation of mathematics is in logic. The logicist programme was launched by Frege, and elaborated in technical details by Whitehead and Russell. Logicism confronts the idea that mathematical notions and theorems are grounded in empirical observations and personal experiences; logicism confronts any such form of psychologism. A key point in the logicist programme is Frege’s definition of number in terms of set theoretical notions. According to logicism all mathematical concepts can be defined by logical concepts, and all mathematical theorems can be derived from logical theorems. Consequently, logicism sees mathematics as logic. Since logical theorems can be shown to be tautologies, all mathematical theorems consequently become tautologies. Logicism inspires the idea that the proper language of science is a formal mathematical language, where the meanings of concepts are defined in terms of sets, and the meanings of propositions are defined in terms of their truth values. A similar theory of meaning has found its way into mathematics education dressed up like the Modern Mathematics Movements.

Skovsmose O.

According to modern self-understanding, the industrial revolution was important because of the way it increased human productivity and, consequently, secured human welfare. Accordingly, much philosophy of technology has established the optimistic understanding that technology, together with science, is a reliable engine of progress. However, technology does not operate as a transparent tool that we human beings can dominate and freely use for our selected purposes. Technology enters into human life; it enters the human being; it transforms society. Technological rationality formats production processes, management approaches, and economic strategies. To a critical philosophy of mathematics, it is important to consider how the implementation of a mathematics-mechanical worldview might serve particular political and economic priorities, and how this view shapes our living conditions.

Skovsmose O.

The colonisation of mathematics is engraved in the version of the history of mathematics that presents it as a Western achievement. As part of the process of colonisation, Eurocentrism developed into an all-dominating perspective on the world, and the white-supremacy ideology became part of the modern outlook. The European colonisation of mathematics was accomplished by cutting away the non-Greek roots of mathematics, and by describing Greek culture as European. The Eurocentric presentation of the history of mathematics is, however, a myth that has transformed into an unquestionable given, repeated in many mathematics textbooks, including brief sketches of the history of mathematics. Ethnomathematical studies have contributed profoundly to the recognition of the multiplicities of the enculturation of mathematics. For a critical philosophy of mathematics, it is crucial to acknowledge this multiplicity and to recognise mathematics as being indefinite with respect to culture.

Skovsmose O.

Mathematics is indefinite with respect to topics. This is illustrated by the Modern Mathematics Movement, which brought about revolutionary changes to the mathematics curriculum. These changes were facilitated by an incoherent combination of political preoccupations, military interests, economic priorities, and intrinsic mathematical ideas. That mathematics is indefinite with respect to applications is illustrated by the radical change in research related to algorithms. The Entscheidungsproblem is a specific metamathematical problem, whose solution presupposed that the notion of algorithm was clearly specified. This specification was done by means of the Turing machine, a conception tremendously important in the development of computing. That mathematics is indefinite with respect to applications is also illustrated by the development of modern cryptography, which refers to cryptography based on computing. In the development of modern cryptography, number theory plays a huge role. Results from apparently harmless pure number theory have acquired massive economic and also military interests.

Skovsmose O.

Intuitionism accuses formalism of confusing mathematics with formal systems. Mathematics is a metal act, and not a string of symbols. According to intuitionism, many of the proofs used in mathematics are not valid. The paradoxes that brought mathematics into a foundational crisis would all evaporate if the process of proof had been “constructive”. This brings intuitionism to articulate a logic that is different from classical logic; for instance, the principle of the excluded middle is not valid in intuitionistic logic. In classical logic the meaning of the logical connectives can be clarified in terms of truth tables, but this is not so in intuitionistic logic. Through a discussion of language, mathematics, and meaning, Brouwer inspired Wittgenstein to return to philosophy, and he inspired Freudenthal to see mathematics as a human activity.

Skovsmose O.

The formalist concept of mathematics was built in steps. The first step was taken by Hilbert when he investigated the foundation of geometry. It had long been recognised that Euclid’s Elements had flaws, as some proofs were not only reached through logical deduction, but also based on intuitive readings of figures and diagrams. While Euclid presented five axioms, Hilbert presented 20 axioms as the foundations of geometry. A second step was taken by the metamathematical programme, which turned mathematical theories into objects for systematic study. They were concerned with the independence of mathematical axioms, the consistency and completeness of mathematical theories, and the possibility of solving the decision problem. A third step consisted of specifying what a formal system is in terms of: the alphabet the system; the sequences of symbols that count as formulas; the set of formulas that serve as axioms; the rules of inference to apply when making deductions; the notion of proof; and the definition of a theorem. With this clarification of a formal system, formalism declares that mathematics is made up of formalisms.

Skovsmose O.

The philosophy of mathematical practice attempts to give the philosophy of mathematics a new beginning. Essential for this beginning is to acknowledge that mathematics is constructed by humans. Hence, as with any other kind of human knowledge construction, mathematics is fallible, corrigible, tentative, and evolving. According to the philosophy of mathematical practice, mathematics can be defined simply as the mathematicians’ practice—it is what mathematicians do. For a philosophy of mathematical practice, it is important to address features of this practice instead of being preoccupied with problems defined through philosophic traditions. It is important to explore the concept of proof, not by embracing any specific ideals, but by considering the variety of ways proof is accomplished in practice. As with logicism, formalism, and intuitionism, so also the philosophy of mathematic practice operates in an ethical vacuum by not considering what might be the social impact of bringing mathematics into action.