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Infinitesimal Knowledges

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Abstract

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues.

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Notes

  1. Foucault (2010, pp. 51-52). An earlier version in French is available online as La fonction politique de l’intellectuel, at http://1libertaire.free.fr/MFoucault133.html.

  2. Aristotle (2004, p. 5).

  3. Alexander (2014).

  4. See Jesseph (1999) for the clash between Hobbes and Wallis, but also a later section in this article.

  5. On the confusions over divisibility arising from a blurring of the distinction between the actual and the potential, see Waterfield’s further comments in Aristotle (2008, pp. lii-liii). The distinction mentioned by Waterfield was recognised by Aristotle (1995, 316b lines 19-21, p. 517). Also, note that historically there are notions of the infinitesimal that include entities other than points. However, whereas a point has a comfortable geometric interpretation as a ‘nothing’ indicating (perhaps) position, an infinitesimal, as an entity that is not a point but has infinitesimal length, resists geometric intuition.

  6. Barnes (1987, p. 251). See also Aristotle (2008, pp. lii-liii and 138-139).

  7. Aristotle, (1975, pp. x-xi).

  8. Aristotle, (1975, p. 17).

  9. Alexander, (2014, p. 64).

  10. Tamvakis, (2014, p. 703).

  11. On Clavius see Alexander (2014, Chapter 2). Concerning Clavius and the Ratio Studiorum see also Feldhay (1995, Chapter 11). The status of mathematical knowledge was in wider dispute over this period. See Schöttler (2012), where he regards the issue as ‘...the incompatibility of Euclidean geometry with the Aristotelian understanding of science’. A major issue was the apparent lack of ‘cause’ in mathematics.

  12. Alexander (2014, pp. 67–68 and 119–120). On the value placed on Euclid by the Jesuits see also H. Bosmans (1927, p. 77), where we read ‘... le grand nombre des mathématicians de la Compagnie de Jésus resta jusqu’à la fin du XVIII siècle profondément attaché aux méthodes euclidiennes. Par un respect un peu outré, un peu suranné, pour la belle géométrie des Grecs, la plupart dentre eux ne surent, ou ne vouleront pas évoluer.’

  13. Alexander (2014. pp. 112, 317), where Alexander mentions the originator of the paradox as Evangelista Torricelli (1608–1647).

  14. See Alexander (2014, pp. 90, 124), and the Stanford Encyclopedia of Philosophy entry Continuity and Infinitesimals, available at https://plato.stanford.edu/entries/continuity.

  15. A discussion of Cavalieri’s method is in Jesseph (1999, pp. 40–42). On contemporaneous criticisms of Cavalieri’s method and their possible motivations see Alexander (2014, pp. 152–157). On Torricelli, see Alexander (2014, pp. 111–116) and Jesseph (1999, pp. 185-187).

  16. Alexander (2014, pp. 17–21, 147–149), and Palmerino (2003, Introduction and pp.187–188).

  17. Palmerino (2003, p. 188.)

  18. Alexander (2014, p. 148) and Galileo (1914, pp. 33–38).

  19. Alexander (2014, pp. 154–157).

  20. Sherry (2018, p. 368).

  21. Redondi (1987, pp. 163–165), Heilbron, (2010, pp. 264–265), Hellyer (2005, chapter 5).

  22. Alexander (2014) does not pursue the possible place of eucharistic doctrine in the Jesuit opposition to indivisibles, concerning which see the subsequent discussion in Sherry (2018), Alexander (2018) and Radelet-de Grave (2018).

  23. Redondi (1987, pp. 157–165, 333–335).

  24. Sherry (2018, p. 368), regards as hypothetical the two main views of the reasons behind Jesuit opposition to indivisibles. One is stated by Alexander as the struggle of the Jesuits to impose ‘..... a true, eternal, and unchallengeable order upon a seemingly chaotic reality’, Alexander (2014, p. 67). The other is ‘A different hypothesis ..... a conflict between the method of indivisibles and the Catholic doctrine of the Eucharist’, Sherry (2018, p. 368). Whereas there is direct evidence that the Jesuits opposed indivisibles because of their perceived challenge to Eucharistic doctrine, the extent to which the challenge posed by indivisibles to Euclidean-style reasoning in mathematics was significant in this opposition is less clear. One possible reason for the importance of Euclid for the Jesuits might be the influence of Clavius, given his high regard for Euclid. But in the absence of direct evidence, one is inclined to agree with Radelet-de Grave (2018, p. 602) who says: ‘Did Clavius impose Euclidean mathematics on the Jesuits and prevent them from learning the the new mathematics of indivisibles in order to maintain the hierarchy within the Company? I don’t think so.’ One of the points at issue in this discussion is the importance at the time of any distinction between infinitesimals and indivisibles. However, noting the subtlety of any such distinction, noting the resistance of infinitesimals to geometric intuition, and noting that the term ‘infinitesimal’ was not really current until later, it seems unlikely that the Jesuits were interested in the finer distinctions that might be made between indivisibles that were a threat to Eucharistic doctrine and those that might not be. However, note Sherry (2018, p. 16): ‘...it is not out of the question that a handful of Jesuit mathematicians understood that there was a middle ground between Euclid and Cavalieri, one that did not invoke the horror of indivisibles’.

  25. See Alexander (2014, pp.178–180) and Sherry (2018, p. 367).

  26. For the differing interpretations of the Galileo affair see Feldhay (1995), especially Chapter 1.

  27. Palmerino (2003), Introduction. Also, on Jesuits and science see Feingold (2003), and Feldhay (1995) especially Chapters 11 and 12.

  28. Drake (2001), Introduction.

  29. Galileo (1957, pp. 206–210).

  30. Gaskin (1996, p. xliii) , Grant (1966, pp. 110–111) and Macpherson (1985, pp. 17–30).

  31. Alexander (2014, pp. 218, 255–256, 277–278).

  32. Quoted in Jesseph (1999, p. 135).

  33. On Wallis see Alexander (2014, pp. 231–242). See Jesseph (1999), especially Chapter 4), for the overall dispute between Hobbes and Wallis. On Hobbes’ criticism of Wallis and indivisibles, see Jesseph (1999, pp. 179–182, pp. 187–188). See also Jesseph (2018).

  34. Berkeley, (1734, p. 8).

  35. Cajori (1919, p. 57). See also the discussions in Boyer (1949, Chapter 6), and Jesseph (1993, Chapter 6).

  36. See Pourciau (2001) on Newton’s implied but definite notion of limits, and see also Grattan-Guiness (1997, pp. 242–298). Concerning Leibniz’s arguments, see Katz and Sherry (2012, pp. 1550–1558).

  37. Jesseph (1999, pp. 199–200 and 226–230).

  38. Hacking (2004).

  39. Galileo (1914, p. 194) reveals an awareness by Galileo of the limitations of the new science.

  40. Newton (1846, p. 384) (from Rule III in the ‘Rules of Reasoning in Philosophy’). Newton’s principle of the ‘consonance of Nature with itself’ is fundamental to scientific knowledge, as it underlies the derivation of general behaviour from only particular observations of that behaviour.

  41. Collingwood (1992, pp. 81–85).

  42. Hume (1854, chapter LXXI, p. 476), also available at http://oll.libertyfund.org/titles/793 p. 542. See also Hume’s Newtonianism and Anti-Newtonianism in The Stanford Encyclopaedia of Philosophy, available at https://plato.stanford.edu/entries/hume-newton, section 4.2: ‘Hume ..... does not view Newton’s achievement as a decisive advance in knowledge of nature but, instead, as decisive evidence for the claim that nature will remain unknowable in principle’. See also Chomsky, (2002, pp. 52–53) where, in referring obliquely to the body-mind dualism of René Descartes he says: ‘Newton exorcised the machine, he left the ghost intact’. Chomsky comments on the criticism of Newton on the grounds that gravitation was ‘occult’, and notes that Newton’s defence involved the acceptance of a ‘weaker model of intelligibility’.

  43. Nikolai Lobachevsky and János Bolyai independently discovered non-Euclidean geometries in the 1820s. They are important in Einstein’s theory of general relativity, according to which space is ‘curved’ in the presence of matter, so enabling elliptical motion to be perceived as ‘normal’.

  44. Cantor (1955).

  45. There is a discussion of changing notions of mathematical proof and certainty in mathematics in Vavilov (2019). Euclid provides an example where there is an incomplete axiom system—in 1882 M. Pasch identified an assumption used in some of Euclid’s proofs that could not be derived from the axioms. On this, see Davis and Hersch (1981, pp. 159–161).

  46. Alexander (2014, p. 276).

  47. Alexander (2014, pp. 289–294).

  48. Butterfield (1931).

  49. Cronin (2012).

  50. Lewis, (2004, pp. 92–94). Lewis quotes the Middle English from Chaucer’s Hous of Fame, II, lines 730– 736. The translation here is by A. S. Kline. Lewis also says ‘ ..... this is the medieval synthesis itself, the whole organisation of their theology, science, and history into a single, complex, harmonious mental Model of the Universe ..... They are bookish. They are indeed very incredulous of books. They find it hard to believe that anything an old auctour has said is simply untrue.’ Lewis’ comments are pertinent for considering the response of the Jesuits and the Aristotelian philosophers to the new ideas that could not be accommodated by the old synthesis. Also pertinent is Foucault (1981 p. 58): ‘It was indispensable, in the Middle Ages, that a text should be attributed to an author, since this was an index of truthfulness. A proposition was considered as drawing even its scientific value from its author. Since the seventeenth century this function has steadily been eroded in scientific discourse.’

  51. Galileo (1914, p. 194.)

  52. Sprat, (1667, pp. 111–113). It is perhaps worth noting that Sprat’s own prose remains intermediate and not infrequently rhetorical, and bears marks of the more florid style he is rejecting as inimical to knowledge.

  53. Arnold (1964, pp. 235-260.)

  54. Eliot (1975, pp. 64-65).

  55. Robinson (1982, pp. 260–272).

  56. On the problem of non-objectifying language see Ott (1967). Although set in a Christian and religious context, it has secular relevance concerning language and its objects and also indicates possibilities for thinking about metaphysical (non-objective) entities. On language and its effect upon our conceptual range see also Robinson (2018), especially in the chapter on The Divine.

  57. Alexander (2014, p. 277). On Hobbes and the Royal Society see Skinner (1969).

  58. Note the view ‘ ....never ....expect more precision in a subject than the nature of that subject permits’, Aristotle (2004, p. 5). Note also the recognition of the varying epistemological profiles for different areas of intellectual enquiry in Foucault (2010, pp. 51-52).

  59. Barnes, Bloor and Henry (1996, p.193).

  60. Foucault (1981). This is from an inaugural lecture given at the Collège de France in 1970. The lecture contain sharp historical and analytic insights, but in this brief section the analytic and descriptive tone changes abruptly and becomes more strident and rhetorical. Whereas the word constrainte had been used to refer to the constraints discourses impose, the word now used is the stronger assujettisements, translated in English as ’subjection’ or sometimes ‘subjugation’. It’s as though Foucault has suddenly suddenly decided to reveal his ‘true’ agenda, or is it simply a lapse that has that effect? In any case, it gives a strong indication of the place from which Foucault speaks. The lecture in French is at https://litterature924853235.files.wordpress.com/2018/06/ebook-michel-foucault-l-ordre-du-discours.pdf.

  61. Varying epistemological profiles are recognised in Foucault (2010, p. 51).

  62. Hanna and Jahnke (1996, p. 890).

  63. Hanna and Jahnke (1996, pp. 890–892). The point about proof not demanding any particular view of mathematics is important, for proof is compatible with an ‘absolutist’ or Platonic concept of mathematics but also with one that regards mathematical knowledge as socially constructed. In any case, ‘socially constructed’ knowledge can have a high degree of objectivity, depending on the type of knowledge and the constructive process.

  64. This point is also made in Hacking (1999, p. 89): ‘..... As real as anything we know. People who have never experienced a mathematical proof (the feeling of, as Wittgenstein puts it, “the hardess of the logical must”) seldom grasp what Platonistic mathematicians are on about.’

  65. Berlin (1958, p. 26). It is not clear that this is Berlin’s own view, as he is describing a certain philosophical attitude towards liberty. But that it was his view seems likely from the context.

  66. The issue is examined in Plato (1956), in which Socrates argues, in effect, that mathematical knowledge is latent in every individual.

  67. Weil (2002, p. 69).

  68. Burke, (1999, pp. 288–289).

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Acknowledgements

I thank Amir Alexander for his account in Infinitesimal of a fascinating period of seminal mathematical and intellectual history. His work has provided motivation for this article. I am also indebted to the other scholars whose work I have consulted. I am indebted to Philip Broadbridge, Michael Cwikel and Neil Powers for encouragement and helpful suggestions. Parts of this paper were presented as work in progress to members of the Independent Scholars Association of Australia in Sydney and Canberra, and their interest is appreciated.

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    Rodney Nillsen

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Nillsen, R. Infinitesimal Knowledges. Axiomathes 32, 557–583 (2022). https://doi.org/10.1007/s10516-021-09540-z

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