Abstract
Cosgrove shows that the conceptual obstacles to the assimilation of algebra into mathematical physics were successfully overcome from the late seventeenth through the eighteenth century, yielding a new mathematical language for the science of physics and a new conception of nature inseparable from symbolic mathematics. He traces some salient features of this development, first in the received Greek mathematical tradition of Euclid and Diophantus and then in the writings of Vieta and Descartes, the principal architects of modern symbolic algebra in the seventeenth-century.
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31 August 2018
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Notes
- 1.
Newton 1769 [1707], 470.
- 2.
Newton 1769 [1707], 11.
- 3.
That modern mathematical physics is essentially determined by its symbolic mode of cognition is a thesis put forth by Jacob Klein in his classic Greek Mathematical Thought and the Origin of Algebra: “The intimate connection of the formal mathematical language with the content of mathematical physics stems from the special kind of conceptualization which is a concomitant of modern science and which was of fundamental importance in its formation” Klein 1992 [ 1934–1936], 4. Klein elaborates this thesis in a number of essays, particularly “The World of Physics and the ‘Natural’ World” (Klein 1985, 1–34).
- 4.
- 5.
Euclid 1956, 2:277.
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See Klein 1992 [1934–1936] , Chapter 6, on the Greek understanding of number. With respect to the pure units of scientific arithmetic, Klein argues that the Platonic notion of separately existing pure units impeded the development of theoretical logistic or the science of arithmetical calculation, since the indivisibility of the pure units precluded calculation with fractions . According to Klein , Aristotle’s critique of the Platonic separation thesis thus opens the possibility for theoretical logistic, in that Aristotle’s pure units are abstracted measures that can be changed at liberty. For our purposes nothing hinges on the particular ontological conception of the pure units . Klein stresses that the concept of the pure unit itself is metaphysically neutral in Greek thought, in the sense that it “precede[s] all the possible differences of opinion regarding the mode of being of the ‘pure’ number units themselves …” (54).
- 7.
Euclid 1956, 2:278.
- 8.
Klein 1992 [1934–1936], 4–5 and 147–149.
- 9.
Diophantus 2009, 129.
- 10.
Heath 2009, 52.
- 11.
Bashmakova 1977 [1972], 6.
- 12.
Diophantus 2009, 131.
- 13.
Diophantus 1893, 1:6.
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Diophantus 1893, 1:276.
- 15.
Klein 1992 [1934–1936], 133–135.
- 16.
Klein 1992 [1934–1936], 144–145.
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Vieta 1992 [1646], 319.
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Diophantus 1893, 9.
- 19.
Vieta 1992 [1646], 328.
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Vieta 1992 [1646], 345.
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Vieta 1992 [1646], 321.
- 22.
Vieta’s emphasis on the “tediousness” of the ancient procedure suggests, however, that he is interpreting the arithmos of Diophantus in terms of his own symbolic conception of number such that the difference is a simply a matter of degree of generality rather than change in the conception of number itself.
- 23.
Vieta 1992 [1646], 324.
- 24.
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Vieta 1992 [1646], 324.
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Third precept of Vieta’s rules for reckoning by species (Introduction, Chapter 4): “The denominations of products made by magnitudes ascending proportionally from genus to genus are related to one another in precisely the following way: A side multiplied by itself produces a square,” and so one through the various powers (Vieta 1992 [1646], 334). Therefore magnitudes of heterogeneous dimension do have a ratio to one another.
- 27.
Vieta 2006 [1646], 8.
- 28.
Kline 1972, 279.
- 29.
“When the equation of the magnitude which is being sought has been set in order, the rhetic or exegetic art, which is to be considered as the remaining part of the analytical art and as one which pertains principally to the application of the art (since the two others [zetetics and ‘poristic’] are concerned more with general patterns than with precepts …), performs its function both in regard to numbers if the problem concerns a magnitude that is to be expressed as a number, and in regard to lengths, surfaces, and solids if it is necessary to show the magnitude itself ” (Vieta 1992 [1646], 346).
- 30.
“And so, a proportion can be called the composition of an equation, an equation the resolution of a proportion” Vieta 1992 [1646], 324. That is, by cross multiplication, if y : x 2 ∷ x 2 a, for instance, then ay = x 4.
- 31.
Kline 1972, 264–265.
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Vieta 1992 [1646], 349.
- 33.
Klein 1992 [1934–1936], 173–174.
- 34.
Klein 1992 [1934–1936], 174.
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Stevin 1585. Although Stevin’s algebra has chronological precedence over Vieta’s, Klein gives Vieta priority as the inventor of the first completely general symbolic calculus.
- 36.
Stevin 1585, 495.
- 37.
Stevin 1585, 501–503.
- 38.
Stevin 1585, 495.
- 39.
Klein 1992 [1934–1936], 191–192. Stevin’s line of argumentation is reminiscent of the amusing observation, by the eponymous Parmenides in Plato’s dialogue, that “mastery itself ” must contain only mastery and so is related solely to “slavery itself,” not to “a man” (who presumably possesses other attributes than being a slave). Similarly, since “number” for Stevin contains solely the numerical, any part of number is also number. The difference is that Plato’s Forms are beings, while Stevin’s “number” is a reified concept.
- 40.
Vieta 1992 [1646], 324–325.
- 41.
Descartes 1985–1991 [1637], 1:120–121.
- 42.
Descartes 2001 [1637], 177.
- 43.
Grosholz 2007, 167.
- 44.
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The “locus problem” is to find a locus of points from any one of which lines can be drawn intersecting given lines at a given angle, such that figures constructed from those lines have a fixed ratio.
- 46.
There was precedent of a sort for nominalizing of ratios as numbers in the medieval practice of regarding the numerical quotient of the numbers comprising a ratio as the “denomination” of the ratio (Roche 1998, 46–47).
- 47.
Klein 1992 [1934–1936], 220.
- 48.
Weyl 1922, 8.
- 49.
Descartes 1985–1991 [1637]: 1:121.
- 50.
Descartes 1985–1991 [ca. 1628]:1:58.
- 51.
Descartes 2001 [1637], 178–179.
- 52.
Descartes 2001 [1637], 178.
- 53.
The medieval algebraist Omar Khayyam too used multiplication by the unit to achieve homogeneity of dimension, but specifically for geometrical figures used to obtain algebraic solutions. Unlike Vieta and Descartes, however, Khayyam imposed no homogeneity requirement on algebraic equations themselves. See Jeffrey Oaks, “Al-Khayyām’s Scientific Revision of Algebra.” 2011. Accessed July 11, 2017. https://www.researchgate.net/publication/265780088_Al-Khayyam’s_Scientific_Revision_of_Algebra.
- 54.
- 55.
Descartes to Mersenne, 27 July 1638. Descartes 1985–1991, III:119.
- 56.
Klein 1992 [1934–1936], 211.
- 57.
Einstein, “The Problem of Space, Ether, and the Field in Physics,” Einstein 1982, 276–285.
- 58.
Einstein, “Physics and Reality,” Einstein 1982, 290–323.
- 59.
Einstein 1982, 297.
- 60.
Einstein 1982, 279.
- 61.
Einstein 1982, 279–280.
- 62.
- 63.
Einstein 1961 [1916], 176.
- 64.
Jacob Klein, “The World of Physics and the ‘Natural World’,” in Klein 1985, 21.
- 65.
Newton is a noteworthy exception, for he actively resisted numerical methods and employed a non-numerical geometrical algebra in physics. However, historically the main current of algebraic development in modern physics is Vietan-Cartesian and numerical. Moreover, the fact that Newton resisted numerical methods in physics does not preclude his having operated, as Klein maintains, with a Cartesian concept of space.
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Cosgrove, J.K. (2018). The Historical Sense-Structure of Symbolic Algebra. In: Relativity without Spacetime. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-72631-1_4
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