The Eighteenth and Early Nineteenth Centuries: The Age of Continuity

Bell, John L.

doi:10.1007/978-3-030-18707-1_3

John L. Bell¹⁹

Part of the book series: The Western Ontario Series in Philosophy of Science ((WONS,volume 82))

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Abstract

The leading practitioner of the calculus, indeed the leading mathematician of the eighteenth century, was Leonhard Euler (1707–83). While Euler’s genius has been described as being of “equal strength in both of the main currents of mathematics, the continuous and the discrete”, philosophically he was a thoroughgoing synechist. Rejecting Leibnizian monadism, he favoured the Cartesian doctrine that the universe is filled with a continuous ethereal fluid and upheld the wave theory of light over the corpuscular theory propounded by Newton.

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Notes

1.
Bell (1965), Vol I, p. 152
2.
Euler (1843), Vol 1, pp. 83–84.
3.
Ibid., pp. 254–5.
4.
Euler (1843), Vol. II, p. 39.
5.
Ibid., p. 39–40.
6.
Ibid., p. 41.
7.
Ibid., pp. 50–51.
8.
But here Euler seems to have missed the point that, if from the assumption that bodies are extended it follows logically that they are unextended, then it can be concluded that bodies are unextended. This is an instance of the logical tautology (p → _¬ p) → _¬ p.
9.
Quoted from Euler’s Institutiones of 1755 in Kline (1972), p 429.
10.
This is a reigning principle of smooth infinitesimal analysis, see Chap. 10 below. See also Lawvere (2011).
11.
Weyl (1950), p. 92.
12.
Euler , Principia motus fluidorum and Principes généraux du mouvement des fluides, 1755. Summarized in Dugas (1988), pp. 301–304.
13.
Kline (1972), p. 340.
14.
Art. Function, Encyclopedia Britannica, Eleventh Edition, 1910–11.
15.
Ibid.
16.
Boyer(1959), p. 243.
17.
Ibid., p. 247.
18.
Quoted in Boyer (1959), pp. 248.
19.
Quoted ibid., pp. 247–8.
20.
Ibid.,p. 250.
21.
Maclaurin , Treatise of Fluxions , Preface, in Ewald (1999) From Kant to Hilbert, p. 93.
22.
Ibid, pp. 107–8
23.
Ibid., p. 108.
24.
Ibid.
25.
Boyer (1959), p. 251.
26.
First introduced in 1715 by the English mathematician Brook Taylor (1685–1731).
27.
Hence the term derivative. The notation f ‘(x) was also introduced by Lagrange.
28.
Carnot (1832), p. 14.
29.
Ibid., p. 33
30.
Ibid., p. 34.
31.
See below.
32.
Carnot (1832), pp. 101–2.
33.
Jesseph (1993), p. 37.
34.
Ibid., p. 57. The doctrine of perceptible minima is, of course, subject to the same objections that had been raised against previous attempts at analyzing extension in terms of atoms.
35.
Berkeley (1960), §124.
36.
According to Jesseph (1993, p. 67), Berkeley was largely unaware of the tradition of “mathematical atomism ” in ancient and medieval philosophy.
37.
Berkeley (1960), §130. It is of interest here to note that the final sentence of this quotation is an explicit rejection of the concept of nilpotent infinitesimal which had been defended by Nieuwentijdt against Leibniz . 250 years later, that concept was to be revived in smooth infinitesimal analysis. See Chap. 10 below.
38.
Ibid., §132.
39.
Berkeley , De Motu, §61. In Ewald (1999).
40.
Likely the astronomer Edmund Halley (1656–1742).
41.
By “difference” Berkeley means “differential”.
42.
Berkeley , Analyst, §7. In Ewald (1999).
43.
Ibid., §35.
44.
Ibid., §§5, 6.
45.
Ibid., §22.
46.
Cajori (1919), p. 89.
47.
Furley (1967), Ch. 10.
48.
Ibid., p. 137.
49.
Hume (1962), II, 1.
50.
Ibid. But presumably the idea of a grain of sand is separable into the ideas “grain” and “sand”.
51.
Ibid.
52.
Ibid.
53.
Ibid.
54.
Ibid., II, 2.
55.
Ibid., II, 3.
56.
Ibid., II, 4.
57.
Furley (1967) p. 142.
58.
Körner (1955), p. 94.
59.
Kant (1964), p. 204.
60.
Ibid., p. 203.
61.
Ibid., p. 203.
62.
Ibid., p. 208.
63.
Ibid., p. 270.
64.
Kant (1970), p. 53. Cf. Critique of Pure Reason, Observation on the Second Antinomy:

Though it may be true that when a whole, made up of substances, is thought by the pure understanding alone, we must, prior to all composition, of it, have the simple…
65.
Ibid., p. 53.
66.
Kant’s thus echoes Leibniz – with the exception that Leibniz’s monads were not physical.
67.
Physical Monadology , Proposition II, in Kant (1992).
68.
Physical Monadology , Scholium to Prop. IV.
69.
Ibid., Scholium to Prop. V.
70.
I am grateful to my colleague Lorne Falkenstein for pointing this out to me.
71.
Kant (1970), p. 55.
72.
Ibid., p.54.
73.
Kant (1977), p. 83.
74.
Ibid.
75.
As already observed, Kant would probably maintain the truth of the Thesis in that event.
76.
Kant (1964), p. 448.
77.
Ibid., p. 459.
78.
Hegel (1961), p. 204.
79.
Ibid., p. 200.
80.
Ibid.
81.
Ibid.
82.
Ibid., pp. 202–3.
83.
Ibid., p. 204.
84.
Ibid., p. 211.
85.
Ibid., p. 213.
86.
Ibid., p. 214.
87.
Ibid., p. 214.
88.
Russell (1964), p. 346.
89.
Quoted in Dauben (1979), p. 170.
90.
Quoted ibid., p. 221.
91.
By Quantum Hegel means determinate Quantity , that is, Quantity of a definite size.
92.
Hegel (1961), p. 269.
93.
Ibid., pp. 269–70.
94.
Ibid., p. 271.
95.
Ibid., p. 274.
96.
Ibid.
97.
Ibid., p. 275–6.
98.
Ibid., p. 280.
99.
Ibid., p. 280–1. Hegel regards as highly dubious the procedure of omitting terms in a sum because of their “relative smallness”.
100.
Ibid., p. 281–2.
101.
Ibid., p. 287.
102.
Ibid.
103.
Ibid.
104.
I.e., the differential triangle .
105.
Ibid., p. 287–8.
106.
Hegel distinguishes between extensive and intensive magnitude . When a magnitude is regarded as a multiplicity, it is extensive; regarded as a unity, it is intensive .
107.
Ibid., p. 288.
108.
Hegel ’s disciple Karl Marx was also preoccupied by the infinitesimal calculus. See Marx (1983).

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Department of Philosophy, University of Western Ontario, London, ON, Canada
John L. Bell

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Bell, J.L. (2019). The Eighteenth and Early Nineteenth Centuries: The Age of Continuity. In: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics. The Western Ontario Series in Philosophy of Science, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-030-18707-1_3

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DOI: https://doi.org/10.1007/978-3-030-18707-1_3
Published: 10 September 2019
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