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Policies, Technology and Markets: Legal Implications of Their Mathematical Infrastructures

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Abstract

The paper discusses legal implications of the expansion of practical uses of mathematics in social life. Taking as a starting point the omnipresence of mathematical infrastructures underlying policies, technology and markets, the paper proceeds by attending to relevant materials offered by general philosophy, legal philosophy, and the history and philosophy of mathematics. The paper suggests that the modern transformation of mathematics and its practical applications have spurred the emergence of multiple useful technologies and forms of social interaction but have impoverished access to meanings originating in the lifeworld. The paper also argues that, as part of devices of interest aggregation and expert networks, mathematical infrastructures can be scrutinized by a revised form of legal practice that subjects them to legal critique and reconstruction in order to overcome conditions that have eroded the moral self-awareness of individuals and communities and their existential meanings.

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Notes

  1. Kant opposes ‘intuition’ (that which is given to the mind through sensibility) to ‘concept’. In Kant’s own words: ‘In whatever way and through whatever means a cognition may relate to objects, that through which it relates immediately to them, and at which all thought as a means is directed as an end, is intuition. (…) Objects are (…) given to us by means of sensibility, and it alone affords us intuitions; but they are thought through the understanding, and from it arise concepts’. (Kant 2009, p. 355).

  2. According to Haakonssen (2006), in the Anglican Church there was ‘an unbroken realist tradition in moral thought’ that sought inspiration from Aristotelian, neo-Platonist and Stoic sources. When confrontation of this tradition with voluntarist ideas coming from Hobbes and Pufendorf arose, ‘a set of eclectic compromises was struck, beginning with the Cambridge Platonists but developed mainly by Scottish thinkers, most notably Francis Hutcheson’ (Haakonssen 2006, p. 253).

  3. At that time, as remarked by Blackstone, ‘the gradual influence of foreign trade and domestic tranquility’ led to the decline of military tenures. As Blackstone points out, ‘the judges quickly perceived that the forms and delays of the old feudal actions, (guarded with their several outworks of essoins, vouchers, aid-prayers, and a hundred other formidable entrenchments) were ill suited to that more simple and commercial mode of property which succeeded the former, and required a more speedy decision of right, to facilitate exchange and alienation’ (Blackstone 1893, pp. 267–268).

  4. Kennedy (2006) characterizes these alliances as links formed between law and the scientific study of ‘the social’, which did not focus on the will of individuals, but on social ‘interdependence’. In Kennedy’s words: ‘The social and its studies were scientific in the way characteristic of the social science of that period, which was a mish mash of evolutionism, pragmatism in the Dewey tradition, and diverse forms of positivism, such as statistics-based empirical surveying’ (Kennedy 2006, p. 39).

  5. Among Greek intellectuals, a mathematical ‘ratio’ expressed a philosophical meaning. As explained by Boyer (1968, p. 5): ‘For the modern word “ratio” the Greeks had two expressions: diastema, which meant literally “interval,” and logos, which meant “word,” especially in the sense of conveying meaning or insight. The latter term generally was used in mathematics, pointing to the Pythagorean idea that ratios express the intrinsic nature of things’.

  6. Vamvacas has described this aspect of the evolution of mathematics as follows: ‘For the Pythagoreans the discovery of incommensurability would constitute an astonishing, shattering experience. The Pythagorean mind, which “likened all things to numbers” and which discerned in simple arithmetic ratios (logos) the deepest meaning (Logos) of the harmony in all things, suddenly found itself confronted with magnitudes that were a-logos, absurd in their relations, and thus, “unutterable”, that is, “inexpressible”’ (Vamvacas 2009, p. 46).

  7. Newton was well versed in Viète’s and Descartes’ algebra. It may come as a surprise the fact that, in formulating his ‘method of fluxions’, Newton self-consciously avoided the use algebra, employing classical geometry instead (Guicciardini 2006, pp. 1736–1737). Actually Newton overtly criticized the algebraic method of Descartes and his followers. Guicciardini (2006) suggests that Newton’s motives were partly religious, since he believed that God intervened in nature and he therefore could not accept the deistic mechanical philosophy of Descartes.

  8. In Klein's words, ‘[a] new kind of generalization, which may be termed “symbol-generating abstraction,” leads directly to the establishment of a new universal discipline, namely “general analytic,” which holds a central place in the architectonic of the “new” science’ (Klein 1992, p. 125).

  9. Just to take one crucial example, Viète’s life circumstances were marked by Huguenot disputes with Catholics. See Klein (1992, p. 150).

  10. The most famous of such mathematicians was Muḥammad ibn Mūsā al-Khwārizmī (circa 780-850 CE), who wrote in the preface to his Compendious Book on Calculation by Completion and Balancing the often quoted lines: ‘That fondness for science, by which God has distinguished the Imam al-Ma’mun, the Commander of the Faithful (…), that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and almuq abala, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.’ Quoted in Katz (2009, p. 271).

  11. For an interpretation of the procedures involved in the establishment of modern algebraic structures and relations, see Serfati (2010).

  12. Hence the remark by the famous mathematician Hermann Weyl: ‘The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematicians interested in the constructs of his imagination.’ Weyl, quoted in Ferreirós and Gray (2006, p. 15).

  13. Such connections between mathematics and society have been shown by the so-called ‘externalist’ (in contrast to ‘internalist’) historians of mathematics. As pointed out by Stedall, ‘Historians of mathematics have increasingly moved away from a purely “internalist” view (…) [It has been shown that] mathematical activity has for centuries manifested itself in a variety of ways, all of them socially and culturally determined’ (Stedall 2012, p. 110).

  14. For discussions see MacKenzie (1993); Frans and Kosolosky (2014).

  15. Interest in bringing mathematics into the study of economics was clear since the so-called forerunners of marginalism, such as Cournot and Dupuit. See Sandmo (2011, pp. 146–159).

  16. Nicolas Bourbaki was the name adopted by a group of French mathematicians that became known for their extremely formalist approach to the discipline. See Corry (1997).

  17. This strategy, into which converged the research agendas of Gottlob Frege, Bertrand Russell and David Hilbert, sought to ‘[turn] philosophical questions about mathematics into logico-mathematical questions’ (Ferreirós and Gray 2006, pp. 4–6). The Hilbertist program itself aimed at proving mathematically that mathematics was free of contradictions (Snapper 1979, pp. 212–214).

  18. Mirowski refers to many ‘cyborg sciences’ that were impacted by von Neumann’s development of applied mathematics partly unfurling from military efforts associated with the Second World War. He describes these as ‘a set of regularities observed in a number of sciences that had their genesis in the immediate postwar period, sciences such as information theory, molecular biology, cognitive science, neuropsychology, computer science, artificial intelligence, operations research, systems ecology, immunology, automata theory, chaotic dynamics and fractal geometry, computational mechanics, sociobiology, artificial life, and, last but not least, game theory.’ He also stresses that ‘[m]ost of these sciences shared an incubation period in close proximity to the transient phenomenon called ‘cybernetics’ (Mirowski 2002, p. 12).

  19. Some exceptions are Quack (2010), Cutler (2010), Cohen and Cutler (2013) and Kennedy (2016).

  20. Formula (1) was taken from Malinovsky et al. (2017), which is a technical study on how to improve safety conditions of road traffic in the metropolitan region of Moscow. In this formula, the inputs of the function are: the number of vehicles present in Moscow from each considered metropolitan or further outlying region (A); the population of each region covered by the research (P); the geographical area of a given region (measured in square kilometres) from which vehicles travel into Moscow (S); the number of vehicles per population (AP); the number of vehicles per area (measured in square kilometres) (AS); the number of vehicles per distance from origin (i.e., place of registration as indicated in licence plate) to Moscow (AD); number of vehicles per ‘sequence number’ of suburban or further outlying region (AN); distance from Moscow to the administrative centre of given suburban or further outlying regions (measured in kilometres), shown as denominator (D); ‘sequence number’ of suburban or further outlying regions counting from Moscow, shown as denominator (N); population density, also shown as denominator (PS).

  21. Formula (2) shows the structure of the so-called ‘Water Poverty Index’ (WPI) that appears in Sullivan and Meigh (2003, pp. 45–46). The authors explain that the WPI was developed from pilot projects in South Africa, Tanzania and Sri Lanka as ‘an integrated and holistic tool, capturing the wide range of issues influencing effective water management and policy’. The authors also explain that the index was elaborated to represent five key aspects of water provision: resource (R), access (A), capacity (C), use (U) and environment (E), whereas “wrwe” denote weights (W) applied to each of these components.

  22. A famous example is the so-called ‘Black–Scholes option pricing equation.’ See (MacKenzie 2006).

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de Castro, M.F. Policies, Technology and Markets: Legal Implications of Their Mathematical Infrastructures. Law Critique 30, 91–114 (2019). https://doi.org/10.1007/s10978-018-9236-9

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