The Establishment of 'Mixed Mathematics' and Its Decline 1600–1800

The Establishment of ‘Mixed Mathematics’ and Its Decline 1600–1800 Sayaka O KI∗ Abstract The term ‘mixed mathematics’ originally derived from the Aristotelian framework of sciences in which mathematics treated abstract entities and could be ‘mixed’ with sensible properties in varying proportions. Its history is deeply concerned with major events in history of science: the mathematicians’ manifesto on mathesis universalis at the end of the sixteenth century, the impact of Newtonian sciences at the end of the seventeenth century and the development of algebraic analysis in continental Europe in the latter half of the eighteenth century. The first two contributed to extending the scope of mathematics to the cognitive territory of the natural philosophers, and the third encouraged the further enlargement of its scope to the fields of engineering and even those of social human activities such as economics and demographics. It was at the beginning of the nineteenth century that the notion of ‘mixed mathematics’ gradually disappeared and was replaced by a set of modern terminologies. Key words: dorcet. mixed mathematics, physico-mathematics, Bacon, Encyclopédie, Con- 1. The Establishment of ‘Mixed Mathematics’ in Europe (Seventeenth Century) The term ‘mixed mathematics’ can be traced back at least as far as the beginning of the seventeenth century.1 The most well-known and influential references are Francis Bacon’s two works, Of the Proficience and Advancement of Learnings (1605) and De Dignitae et Augmentis Scientiarum (1623). Bacon represents the term ‘mixed mathematics’ in his tree of knowledge (see Fig. 1 for a portion of the tree), which classifies all the fields ∗ Hiroshima University, Graduate School of Integrated Arts and Sciences, 1–7–1 Kagamiyama 739–8521 Hiroshima, Japan. This paper is part of the work supported by the Grant-in-Aid for Scientific Research (C) No. 23501200 of the Ministry of Education, Science, Sports and Culture (MEXT) from 2011 to 2013. I am grateful for their support. I would also like to thank Eric Brian, Shinnichi Nagao, Satoshi Nakazawa, and Sathishi Nozawa for their useful comments on drafts of this paper. 1 The root of this classification itself goes back to Aristotle’s discussion of ‘the more physical of the branches of mathematics’, which are optics, harmonics, and astronomy, and its appropriation in the Arabian view of learning. See, Gary I. Brown, “The Evolution of the Term ‘Mixed Mathematics”’, Journal of the History of Ideas, Vol. 52, No. 1 (Jan.–Mar., 1991): 81–83; Nobuo Kawajiri, “Francis Bacon’s view of mathematics—Bacon’s concept of mixed mathematics”, Proceedings of the Faculty of Science of Tokai University, XV (1980): 7–21. HISTORIA SCIENTIARUM Vol. 23–2 (2013) The Establishment of ‘Mixed Mathematics’ and Its Decline 1600–1800 83 Figure 1. Francis Bacon’s Tree of Knowledge (Human Leaning), cited in Brown, “Evolution”, 81. of human knowledge by the faculties of the mind.2 In Bacon’s system, mathematics is divided into two categories: ‘pure mathematics’ and ‘mixed mathematics’. This distinction is based on the traditional Aristotelian argument that mathematical entities cannot be truly separated from sensible things, but only abstracted from them. In this framework, the category ‘pure mathematics’ corresponds to the fields with the highest levels of abstraction, such as geometry and arithmetic. The category ‘mixed mathematics’ literally means ‘mathematics mixed with sensible properties in varying proportions’, and is considered to have the following six branches: perspective, music, astronomy, cosmography, architecture, and engineering. According to Bacon, both categories of mathematics belong to the more general category, ‘metaphysics’, which concerns the inquiry of ‘quantity determined or proportionable’.3 It is also noteworthy that the expression ‘mixed mathematics’ itself had attracted Dutch and French mathematicians in a series of discussions concerning mathesis universalis at the turn of the seventeenth century. For example, Dutch mathematician Adriaan van Roomen mentions mathematica mixta in his Universae mathesis idea (1602) and some mathematicians in the sixteenth century, such as Rudolf Snellius and Petrus Ramus, also share this notion, coupled with their vast knowledge on ancient philosophical and mathematical literature.4 The mathematicians’ concerns were directly linked to the intellectual manifesto that elevated the status of those mathematical sciences, an attitude typically discerned in the Jesuit mathematician professor Christopher Clavius. They had challenged the 2 Brown, “Evolution”, 82. Bacon divides all human learning into history, poesy, and philosophy, which correspond respectively to the three intellectual faculties, ‘memory’, ‘imagination’, ‘reason’. 3 Brown, “Evolution”, 82. See also Lorraine J. Daston, “Fitting Numbers to the World: The Case of Probability Theory”, History and Philosophy of Modern Mathematics, William Aspray and Philip Kitcher ed. (Minneapolis: Univ. Minnesota Press, 1988), 222–228. 4 Chikara Sasaki, Descartes’s Mathematical Thought (Dordrecht: Kluwer Academic Publishers, 2003), ch. 7; Jean-Marc Mandosio, ‘Entre mathématiques et physique: note sur les ‘sciences intermédiaires’ à la Renaissance’ in Comprendre et maîtriser la nature au moyen âge, Mélanges d’histoire des sciences offerts à Guy Beaujouan (Genève, Paris: Droz, 1994), 131. 84 Sayaka O KI traditional Aristotelian definition of sciences, in which mathematics, with all its branches, was considered subordinate to physics (also called ‘natural philosophy’) defined as qualitative science on the natural world.5 The term ‘physico-mathematics’ was also used mostly by mathematicians such as Isaac Beeckman, Marin Mersenne, John Wilkins, and Isaac Barrow to designate fields almost identical to ‘mixed mathematics’, with the image of the mathematical sciences at an epistemically equal level to physics in the traditional framework.6 It seems that Bacon himself simply adopted the terminology circulating among mathematicians, ‘pure’ and ‘mixed’ mathematics without taking much interest in their arguments on the necessity of quantitative investigations, partly because of his limited capacity in those fields.7 However his systematic classification of sciences was to be the most influential in light of the awakening interest in the encyclopaedic vision of human knowledge, and indeed it came to be well recognized around the middle of the seventeenth century.8 2. ‘Mixed Mathematics’ and Newtonian Natural Philosophy (Mid-Eighteenth Century) In the course of the eighteenth century, much had changed in the classification of mathematics, especially in the category ‘mixed mathematics’ itself, as is represented in the famous “Figurative system of human knowledge” in Diderot and d’Alembert’s Encyclopédie (see Fig. 2).9 The category ‘mixed mathematics’ then contained mechanics, geometric astronomy, optics, acoustics, pneumatics, and the art of conjecturing (analysis of games of chance). These changes reflected the developments in mechanics, optics, and astronomy and the rise of the probability theory in the latter half of the seventeenth century. ‘Pure mathematics’ kept the same two traditional branches, arithmetic and geometry, and also acquired new elements at the level of subcategories under those two, such as the ‘theory of curves’ under the subcategory ‘geometry’.10 What’s more, in the category ‘mathematics’, a third subcategory ‘physico-mathematics’ was introduced. Diderot, the effective author of the “Figurative system of human knowledge” tree, did not give any precise definition of the disciplinary contents of this ‘physico-mathematics’.11 It was d’Alembert who further explained it in his Discours 5 Peter Dear, Discipline and Experience: The Mathematical Way in the Scientific Revolution (Chicago: University of Chicago Press, 1996), 2–3, 32–46. 6 Dear, Discipline and Experience, 8, 168–170. 7 Bacon conceived the usefulness of pure mathematics exclusively in terms of helping the concentration and did not have much to say on practical ‘mixed’ mathematics. See Stephen Gaukroger, Francis Bacon and the Transformation of Early-Modern Philosophy (Cambridge: Cambridge Univ. Press, 2001), 23–25. 8 Brown, “Evolution”, 83. 9 There are different versions of this tree of knowledge, but little changed in the category of mathematics, whereas many modifications had been made in the categories related to chemistry and the experimental sciences. 10 Brown, “Evolution”, 87–91. 11 Michel Malherbe, “Mathématiques et Sciences physiques dans le Discours préliminaire de l’Encyclopédie”, Recherches sur Diderot et sur l’Encyclopédie (RDE), 9, octobre (1990): 109–146; Malherbe, “Introduction”, in d’Alembert, Discours préliminaire de l’Encyclopédie, introduit et annoté par Michel Malherbe (Paris: Vrin, 2000). The Establishment of ‘Mixed Mathematics’ and Its Decline 1600–1800 85 Figure 2. “Figurative system of human knowledge” in Diderot & D’Alembert, Encyclopédie, ou dictionnaire raisonné des sciences, des arts et des métiers, etc. (Paris, 1751), t.1. préliminaire in the first volume of the Encyclopédie. According to him, all the sciences called ‘physico-mathematics’ were born from the use of geometry and mechanics for “acquiring the most varied and profound knowledge about the properties of bodies”. He argued also that “the physico-mathematical sciences, by applying mathematical calculations to experiments, sometimes deduce from a single and unique observation a large number of inferences that remain close to geometrical truths by virtue of their certitude.”12 However, as recent studies suggest, his definition is not helpful in distinguishing ‘mixed mathematics’ from ‘physico-mathematics’. What’s more, it seems that Diderot and d’Alembert do not share the same point of view on the two categories. In d’Alembert’s argument ‘physico-mathematics’ is treated almost as a branch of ‘mixed mathematics’. For example, his Discours préliminaire mentions astronomy as the representative example of the physico-mathematical sciences, for the reason that it could precisely predict the movements of the heavenly bodies, successfully joining observation and calculation. Then, in his article “Mathematique” in the tenth volume of the same Encyclopédie, he defines ‘mixed mathematics’ as a science concerning properties that had concrete measurable or calcu12 D’Alembert, Preliminary Discourse to the Encyclopedia of Diderot, Translated and with an Introduction by Richard N. Schwab (1st ed., 1965; rev. ed. Chicago: Univ. Chicago Press, 1995), 24 [Discours préliminaire, Encyclopédie, t.1:vii]. 86 Sayaka O KI lable quantity, and mentions as its components, again astronomy, along with ‘mechanics, optics, . . . geography, chronology, military architecture, hydrostatic, hydraulic, hydrography and navigation, etc’.13 By contrast, Diderot makes a stronger distinction between the two categories. Although his view is not clearly indicated in the figurative system of human knowledge, a careful reading of his other articles in the Encyclopédie and some of his writings on mathematics reveal this distinction. In his opinion ‘mixed mathematics’ concerns only the objects in which the abstract geometrical methods and the observation of the physical bodies can be firmly united, such as mechanics and optics, while the category ‘physico-mathematics’ concerns the objects that need special precaution for the use of mathematics. In a document written ten years later, he refers to the application of calculus of probabilities as the “physico-mathematical science of life”.14 This position contrasted with d’Alembert’s, who, in fact, does not support the extension of the probability theory to the fields concerning life, such as medicine and demography. D’Alembert prefers to put them under the category of ‘general and experimental physics’, which was not mathematical and consisted of “only a systematic collection of experiments and observations”.15 It seems that this unclear relationship between ‘mixed mathematics’ and ‘physicomathematics’ in the Encyclopédie derives partly from the difficulty to properly locate the dominant trend culminated in the work of Isaac Newton in the accepted classification of sciences in France. Newton’s endeavour, which forged the nexus between mathematics and natural philosophy,16 effectively instigated the reform of natural philosophy in that his efforts upset the traditional Aristotelian subordination of mathematics to natural philosophy.17 Then, as I. Passeron and T. Houquet suggest, one of the problems identifying the nature of the relationship between the two fields was the difficulty of translating this reformed ‘natural philosophy’ in a way adapted to the French conceptual framework of sciences. Direct translation was not possible, because the term philosophie naturelle was not common in French. One of the options was to use the word physique (physics) which was in principle synonymous to ‘natural philosophy’ in English. However, the word physique, both as noun and as adjective, firmly maintained a traditional Aristotelian connotation: “the 13 Preliminary Discourse, 2 [Discours préliminaire, Encyclopédie, t.1:vi]; Mathematique, Encyclopédie, t. 10: 188–189. See Irène Passeron, “D’Alembert refait le MONDE (Phys.). Parcours dans les mathématiques mixtes”, RDE, octobre (2006), 40–41; Alain Firode, Les catégories de la mécanique dans l’Encyclopédie, RDE [En ligne], 40–41 |octobre 2006, mis en ligne le 01 octobre 2008, consulté le 11 octobre 2013. URL: http:// rde.revues.org/339; DOI: 10.4000/rde.339. 14 Denis Diderot, Œuvres complètes, édition critique et annotée par J. Fabre, H. Dieckmann, Jacques Proust, Jean Varloot (Paris, 1975), vol. II, 341; Malherbe, “Mathématiques et Sciences physiques”, 124–125. 15 Preliminary Discourse, 24 [Discours préliminaire, Encyclopédie, t.1:xvii]. See Jean Mayer, “Diderot et le calcul des probabilités dans l’Encyclopédie”, in Revue d’histoire des science (1991), t. 44, no 3–4: 374– 391; Sayaka Oki, “Sugaku to shakai-kaikaku no yutopia: Buffon no dotoku-sanjutsu kara Condorcet no shakaisugaku made” (Mathematics and Utopia for social reform: From Buffon’s moral arithmetic to Condorcet’s social mathematics), in Kagaku-shisoshi, O. Kanamori ed. (Tokyo: Keiso-shobo, 2010), 135–157. On d’Alembert’s view on the probability theory, see Michel Paty, “D’Alembert et les probabilités” in Sciences à l’époque de la Révolution française. Recherches Historiques, R. Rashed éd. (Paris: Albert Blanchard, 1988), 203–265; Eric Brian, La mesure de l’Etat. Administrateurs et géomètres (Paris: Albin Michel, 1994), 94–111. 16 Dear, Discipline and Experience, 35–8, 161–8; John Henry, The Scientific Revolution and the Origins of Modern Science (London: Macmillan, 1997), 18–21. 17 Richard Yeo, “10 Classifying the Science”, in The Cambridge History of Science, Roy Porter ed., Vol. 4: Eighteenth-Century Science (Cambridge: Cambridge Univ. Press, 2003), 244. The Establishment of ‘Mixed Mathematics’ and Its Decline 1600–1800 87 science of all natural beings” without mathematical reasoning.18 For example, the Royal Academy of Science of Paris, where d’Alembert was a member, divided their members into the six following categories: three in sciences physiques which included chemistry, botany, and anatomy, and three in sciences mathématiques including astronomy, geometry, and mechanics.19 Faced with the inaptitude of the word physique in relation to Newtonian (mathematical) natural philosophy, the term ‘physico-mathematics’ should have been a good alternative. Another, more serious problem was that the reception of Newtonian natural philosophy had resulted in different positions among French scholars on the epistemic status of mathematics and its limit of application to the physique. On one hand, there were efforts to set the foundation of physique as a solid science without abstract mathematical reasoning, such as Buffon’s attempt at systematic observation of natural beings in Histiore Naturelle,20 and Diderot’s several writings on chemistry. They believed in the importance of ‘physical’ certainty that was based on the recurrent observations of sensible objects. In their opinion, this kind of certainty was different from ‘mathematical’ certainty, which was considered to be dependent on the level of mathematical abstraction. On the other hand, d’Alembert and his fellow mathematicians tended to perceive the fields not adapted to mathematical reasoning as sciences with limited certainty.21 The definition of ‘mixed mathematics’ and its relationship with ‘physico-mathematics’, as mentioned above, touch the very heart of the debate between those two parties. Diderot took many questions concerning probability theory, such as inoculation, human life expectancy, as a part of physico-mathematics, not of mixed mathematics, because those 18 Passeron, “D’Alembert refait le MONDE”, 158–159; Thierry Houquet, “History without Time: Buffon’s Natural History as a Nonmathematical Physique”, Isis, Vol. 1 (March, 2010), note 4, 23. We see Ephraïm Chamber’s Cyclopaedia (1728) put “otherwise call’d Physics” in the item “Natural Philosophy”. We can understand why the French authors of the Encyclopédie, which originally stemmed from the project of translating Cyclopaedia, did not adopt Chamber’s tree of knowledge, in spite of the fact that it had been directly inspired by Bacon. In Chamber’s classification, ‘pure mathematics’ and ‘natural philosophy’ both belonged to the category of ‘natural and scientific’ knowledge and they found themselves among branches such as ‘religion’ and ‘metaphysics’ at the same level of hierarchy. ‘Mixed mathematics’ was separated from them, under the category of ‘Artificial and Technical’ knowledge. See Maurizio Mamiani, “The Map of Knowledge in the Age of Volta Jacques”, in Nuova Voltiana: studies on Volta and his times, Fabio Bevilacqua, Lucio Fregonese ed., 4 (Pavia and Milan, Hoepli, 2002), 1–10; Bernet, “La Cyclopaedia d’Ephraïm Chambers (1728), Ancêtre direct de l’Encyclopédie de Diderot et d’Alembert”, in Les sources anglaises de l’Encyclopédie, Sylviane Albertan-Coppola and Madeleine Descargues-Grant ed. (Paris: Presse Universitaires de Vancennes, 2005), 25–38. As to Newton’s special position in the Encyclopédie, see Sandra Lasne, “L’Encyclopédie et Newton (1642–1727)”, in Les sources anglaises de l’Encyclopédie, 71–85. 19 The origin of this division goes back to Christiaan Huygens’s plan for the Academy. It was institutionalized in the structure of the Academy at the end of the seventeenth century and lasted until the reform in 1785. Historians of the Academy have emphasized that a boundary remained between the mathematical and physical sections, one that very few scientists crossed. During its history, only three scientists were promoted from the classe de mathématiques to the classe de physique or vice versa: Buffon, Jean-Nicolas de La Hire, and La Condamine. See Christian Gilain, “La classification des mathématiques à l’Académie royale des sciences (1699– 1785)” in Règlement, usages et science dans la France de l’Absolutisme, éd. C. Demeulenaere-Douyère et E. Brian ed. (Paris: Tec & Doc, 2002), 509–518. 20 See Houquet, “History without Time”. 21 George-Louis Leclerc de Buffon, Histoire naturelle, générale et particulière, avec la description du cabinet du Roy (Paris, 1749), t.I, 1er discours, 53–55; Diderot, De l’interprétation de la Nature, in Œuvres complètes, 1754 (Paris, 1875), t.II. 88 Sayaka O KI questions dealt with the observations of real and sensible phenomena in which he thought a rather intuitive use of probability calculus should be tolerated and even considered useful. As recent studies show us, he was in fact an eager advocate of the probability theory because he thought it was an exceptional branch of mathematics that could serve to consolidate the ‘physical’ certainty of observations, as opposed to other highly theoretical (or ‘abstract’, in his expression) mathematical branches, which he regarded as useless to most of the complex natural phenomena treated in the physique. However d’Alembert thought the extensive use of the probability theory risked the abuse of mathematics. Diderotian ‘physico-mathematics’ was absent in d’Alembert’s mind. As a mathematician faithful to the Cartesian ‘geometrical spirit’, he thought that the use of probability theory should be restricted to cases with a simple structure, such as gambling games using a coin toss or a die, in which all the causes constituting an event are clearly recognizable and countable, supported by explicit evidence.22 Finally, it was two different attitudes based on the same premise that, in introducing a greater ‘mixture’ of sensible ideas, mathematicians ran the risk of error.23 On one hand, Diderot regarded insight from the observation of sensible properties as important, and wished the development of the physical ‘physicomathematics’ useful for the fields where traditional ‘mixed’ mathematics could not give a proper answer. On the other hand, d’Alembert who heavily privileged results obtained from rigorous mathematical reasoning, wanted to limit the level of ‘mixture’ between mathematics and sensible ideas to that of traditional ‘mixed mathematics’. We can easily imagine that, in this context, putting the branch ‘physico-mathematics’ without any details, into the tree of knowledge should have been a good compromise between them.24 3. The Decline of Baconian Classification (Late Eighteenth Century–Nineteenth Century) D’Alembert and Diderot managed to preserve Baconian classification of sciences, despite their seemingly different positions as regards the interpretation of Newtonian natural philosophy. However, another element gradually undermined the Baconian structure, and that was the development of ‘analysis’ in mathematics, to which d’Alembert had also contributed, but of course without intending this effect. In the latter half of the eighteenth century, analysis had become an almost autonomous branch from geometry and algebra, and had demonstrated its potentiality as the veritable language of science by its application to many objects related to mechanics and the engineering sciences.25 Although the volume 22 See Jean Mayer, “Diderot et le calcul des probabilités dans l’Encyclopédie”, in Revue d’histoire des science (1991), t. 44, no 3–4: 374–391; Oki, “Sugaku to shakai-kaikaku no yutopia”, 135–157. On d’Alembert’s view on the probability theory, see Michel Paty, “D’Alembert et les probabilités”, in Sciences à l’époque de la Révolution française. Recherches Historiques, R. Rashed éd. (Paris: Albert Blanchard, 1988), 203–265. 23 Daston, “Fitting Numbers to the World”, 222–228. 24 See also Malherbe, “Mathématiques et sciences physiques”, 124–125. 25 Regarding the evolution of the category ‘analysis’ during the period from the Cyclopaedia to the Encyclopédie Méthodique, see Christian Gilain, “La place de l’analyse dans la classification des mathématiques: de l’Encyclopédie à la Méthodique”, Recherches sur Diderot et sur l’Encyclopédie 1/2010 (n 45), p. 109–1028. URL: www.cairn.info/revue-recherches-sur-diderot-et-sur-l-encyclopedie-2010–1-page-109.htm. The Establishment of ‘Mixed Mathematics’ and Its Decline 1600–1800 89 on mathematics in the Encyclopédie méthodique published in 1785 retained the category ‘mixed mathematics’ almost as it was in the Encyclopédie in the 1750s,26 we clearly sense hesitation concerning the use of this term, at least in the writings of one influential author of that age: Jean Antoine Nicolas de Condorcet, d’Alembert’s disciple mathematician and the perpetual secretary of the Royal Academy since the 1770s. In effect, since the beginning of his career, Condorcet tirelessly endeavoured to establish a new classification of sciences, mainly for the two following motives. First, he was seeking a way to reconcile the tradition of probability theory with d’Alembert’s concern with mathematical rigor, the same issue that had divided his master and Diderot. Second, the classification of science itself continued to interest him, especially as the permanent secretary of the Royal Academy of Sciences in Paris, in the 1770s and 1780s, and then as a revolutionary reformer of education in the 1790s. We can see one of his first attempts in an unaccomplished manuscript written probably in the early 1770s. He tried to classify the fields related to mathematics, dividing them into two branches without recourse to the traditional terminology: Mathématiques proprement dites (mathematics themselves) and Mathématiques phisique [sic] (physical mathematics) (see Fig. 3).27 The category Mathématiques proprement dites included the theoretical investigation of the entities abstracted from the mind, independently from outside phenomena, such as geometry, arithmetic, and the theories of analysis used to calculate or approximate quantities. That of Mathématiques phisiques corresponded to the fields created by the application of mathematics to “phenomena in front of us”, that is to Figure 3. Condorcet, Table in the manuscript, reproduced in Condorect, Tableau historique des progrès de l’esprit humain, 237. 26 D’Alembert, Bossut, Lalande, Condorcet, Charles et al., Encyclopédie méthodique. Mathématiques, t.II (Paris, 1785), “Mathematique ou Mathematiques”, 366. 27 Condorcet, Arithmétique politique: textes rares ou inédits (1767–1789) (Paris: INED, 1994), “Ébauche de division des mathématiques”, 242. 90 Sayaka O KI say, natural sciences and some fields related to human life and society, such as Science des rapports moraux ou Politiques (science of moral or political relationships) and Science de l’usage des Corps (science of the usage of bodies).28 Although this schema still echoed the traditional definition of mathematics as an ‘abstract’ science, it is interesting to see that he introduced another criteria of division—whether or not the objects of investigation concerned phenomena outside the mind, or not—and this enabled him to put social phenomena and natural phenomena together under the same category ‘Mathématiques phisiques’. As historians argue, it seems that Condorcet noticed at the beginning of the 1780s that the level of abstraction and the degree of certainty of a science were in fact different questions, a point which neither d’Alembert nor Diderot had clearly recognized. With his colleague Pierre-Simon Laplace’s extensive application of analysis to probability theory, Condorcet realized that one could use probability theory inductively for any kind of object, that is to say, to reason from observed events to their unknown causes without any violation of mathematical rigor, even with respect to the most complex phenomena such as medical problems or human social behaviour (for example, the application of the Bayes-Laplace theorem to the statistical estimation of population by the number of annual births was methodologically rigorous, even if the certainty of its results could vary according to the quality of data). This insight, for Condorcet, eliminated the traditional boundary between the ‘mathematical’ sciences and all the other sciences, including even so-called ‘moral and political’ ones. In this vision, mathematics could not be ‘mixed’ with sensible properties, but theoretically ‘applied’ to some particular objects, without losing methodological certainty.29 In the 1790s, in the midst of the French Revolution, Condorcet clearly abandoned Baconian classification, taking it as an arbitrary system. As a reformer of the education system, he tried inventing a new classification according to the objects of scientific investigation, not by the faculties of mind, or by the use of a particular method. The very definitions of ‘arts’ and ‘sciences’ and their unclear relationship came at the centre of his concern.30 He writes: It is simpler and more exact to attach the theory of an art to the art itself, than to make of the truths of one same science applicable to different arts a part of this science.31 We can recognize that his argument is applicable to the relationship between the mixed mathematics and technical objects, even if his definition of the term ‘art’ as “the 28 Condorcet, “Ébauche de division des mathématiques” [1770–71], 237, 240–246. It seems that by “usage of bodies” he means a sort of anthropologic observation on customs and culture concerning bodies. 29 As to the details of this argument, see especially Brian, La mesure de l’Etat, 213–229 and the related bibliography presented in ibid., 404–407. 30 Condorcet, Tableau historique des progrès de l’esprit humain (Paris: INED, 2004), Note 4 “Sur les sens des mots sciences et art, sur les classifications des sciences et des arts” [1793–94], 768. On Condorcet’s classification of sciences, see Brian, La mesure de l’Etat, 55–58, 218. 31 The citation is a literal translation of the original French text: “Il est plus simple, plus exact d’attacher la Théorie d’un art à l’art même, que de faire des vérités d’une même science applicables à differrens [sic] arts une portion de cette science” [Condorcet, Tableau, 766]. I kept it this way to avoid overinterpretation of the original draft, which remains unaccomplished. The Establishment of ‘Mixed Mathematics’ and Its Decline 1600–1800 91 collection of the means to achieve a purpose”32 connotes not only all sorts of technical instruments or machines but also several mental skills, such as the “art of presenting ideas” or the “art of education”. Mixed mathematics was indeed a part of mathematics, consisting of the mathematical theories that can be applied respectively to different technical or material entities. Then Condorcet thought it better to put each of those theories together with their object of application, rather than to treat each as a part of mathematics. Though most of his writings on this issue remain unpublished, Condorcet’s argument was not the voice of a lone philosopher, but that of an engaging politician at the time of the French Revolution, a period when a series of radical social reforms created a new system of scientific education. In effect, the committee of public instruction, which Condorcet headed, discussed several topics related to the new classifications of the sciences, such as the design of course materials in engineering education for the École Polytechnique, and the structure of new academies of sciences.33 It was after this turbulent period that the term ‘mixed mathematics’ gradually began to disappear. The term appeared as late as the third volume of the Encyclopédie méthodique, on physics, published in 1819, but the author explained it as synonymous to ‘physicomathematics’ then, saying, “It is the application of pure mathematics to results of observations or experiments. . . .”34 We see that the nuances of the terms ‘pure’ and ‘mixed’ had shifted slightly from the contrast between abstract ‘pure mathematics’ and ‘mixed mathematics’ treating sensible entities to that between ‘pure mathematics’ as a body of theories and ‘mixed mathematics’ as their application. The same tendency was observed in European countries in general. As Brown and L. Daston argue, ‘mixed mathematics’ went on to be replaced by a new set of terminologies during the first half of the nineteenth century: that is to say, the familiar twin notions of ‘pure’ and ‘applied’ in mathematics, the modern definition of ‘physics’, the emergence of the category ‘engineering sciences’, and so on.35 (Received on 8 December 2013; Accepted on 15 December 2013) 32 Condorcet, Tableau, 765. For example, in a report to the Revolutionary government in 1792, he proposed to create a new academy of sciences that had the class of ‘application of mathematical and physical sciences to the arts’, which covered the medicine, the mechanical arts, the agriculture and the navigation [Condorcet, Rapport et projet de décret sur l’organisation générale de l’instruction publique (Paris, 1792), 36–37]. 34 Monge, Cassini, Bertholon, and Hassezfratz, Encyclopédie méthodique. Physique, t. III (Paris, 1819), “Mathematiques”, 811. 35 Brown, “The Evolution”, 99–102; Daston, “Fitting Numbers to the World”, 231. On mathematics, further details should have been discussed also in this workshop: “From ‘mixed’ to ‘Applied’ Mathematics: Tracing an important dimension of mathematics and its history”, organized by: Moritz Epple, Tinne Hoff Kjeldsen and Reinhard Siegmund-Schultze, Oberwolfach Reports, vol. 10, Issue 1 (2013): 657–733, consulted on 9 December, 2013; URL: www.mfo.de/document/1310/OWR_2013_12.pdf; DOI: 10.4171/OWR/2013/12. As to the notion of ‘applied science’, see Ann Johnson, “Everything New Is Old Again: What Place Should Applied Science Have in the History of Science?”, in Science in the Context of Application, M. Carrier, A. Nordmann, eds., Boston Studies in the Philosophy of Science 274 (Dordrecht: Springer, 2011), 455–466. 33