Abstract
In the early modern period a crucial transformation occurred in the classical conception of number and magnitude.1 For the Greeks, the unit, or one, was not a number; it was the beginning of number and it was used to measure a multitude. Numbers were merely collections of discrete units that measured some multiple. Magnitude, on the other hand, was usually described as being continuous, or being divisible into parts that are infinitely divisible. A distinction was made between arithmetic and geometry; arithmetic dealt with discrete or unextended quantity while geometry dealt with continuous or extended quantity.2 In the early modern period a transformation occurred in this classical conception of number and magnitude. For instance, Simon Stevin (1548 – 1 620) insisted in his 1585 Arithmétique that the traditional Greek notion of numbers was wrong: he believed that numbers were continuous rather than discrete.3 Stevin also developed a system for indefinite decimal expansions of number that implicitly contained the idea of a numerical continuum.4 Moreover, François Viète (1540–1603) introduced an improved form of symbolism: unknowns were distinguished from given magnitudes by being labelled with capital vowels while the given magnitudes were denoted by capital consonants. This powerful new symbolism was used to denote both unknown magnitudes and numbers; it indicated that numbers and magnitudes were, in a sense, interchangeable.5 This association of numbers with magnitudes encouraged the notion that numbers could also be treated as though they were continuous in the Aristotelian sense.
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The term “number” is elusive and difficult to work with because its meaning changed often through time (this of course is one of the aspects that makes the subject historically interesting). Different meanings might be used by the protagonists of this book, secondary authorities cited such as Jacob Klein, and differently trained readers. For instance, mathematically trained readers may interpret ‘number’ as an element of the set of real numbers, defined either constructively (by the usual constructions including an equivalent of Dedekind cuts) from the natural numbers assumed given, or axiomatically by a set of axioms including some equivalent of a completeness property. These readers will therefore have a well-defined continuum in mind, in which the usual types of numbers as natural, integer, rational etc. are embedded. Readers without this mathematical background will probably have a different number concept in mind. Most likely this would be the common number concept of routine practitioners of letter-algebra: numbers are what letters stand for in algebra, they can be pictured along a scaled straight line (the so-called number line). Often the precise definition of these numbers is left to the experts, as is the precise definition of the global properties (such as continuity) of the number set. Less mathematically oriented readers usually locate the abstractness of numbers in the fact that it is not necessary to say what they are, and yet one can do algebra with them. In this book, terms such as numbers and continuity will be mainly used as defined by the actors in question.
Olaf Pedersen, “Logistics and the Theory of Functions: An Essay in the History of Greek Mathematics,” Archives Internationales d’histoire des Sciences 24 (1974): 29–50, has pointed out that logistics, or practical reckoning, although it was denied the status of being a theoretical science and thus not written about for its own sake, was carried to a greater state of perfection than has previously been suspected. For example, Ptolemy’s Almagest depended upon numerical calculations whose methods and procedures were never explained in the text. However, no Greek practical mathematics texts were available in the early modern period.
Continuous in an Aristotelian sense as will be discussed in chapter 2.
Meaning in this case that Stevin noted that numbers can be pictured along a number line.
Later in the chapter there will be a discussion about the distinction between geometrical and abstract magnitude.
For a wonderful discussion of the impact of humanism on mathematics in the 15th and 16th centuries, in particular the collection and translation of Greek mathematical authors such as Apollonius, Archimedes, Diophantus, Euclid, Hero, Pappus and Proclus, see Paul Lawerence Rose “Humanist Culture and Renaissance Mathematics: The Italian Libraries of the Quattrocento” Studies in the Renaissance 20 (1973)
Simon Stevin, The Principle Works of Simon Stevin, ed. D.J. Struik, vol. 2 (Amsterdam: C.V. Swets and Zeitlinger, 1958). The Principal Works of Simon Stevin includes his Tables of Interest. In the early modern period banking houses pursued varied activities involving questions of insurance, of annuities and other payments at set intervals, of discounting of sums due at a later date and related transactions. Many of the examples in Stevin’s book utilise fractions and complicated (for the period) interest computations.
See also N. Z. Davis, “Sixteenth Century French Arithmetics and the Business Life,” Journal of History of Ideas 21(1960): 18–48, and the discussion in Dijksterhuis’ Simon Stevin for details.
The dedication reads, “To Astronomers, Land-meters, Measureres of Tapestry, Gaugers, Stereometers in general, Money-Masters, and to all Merchants, SIMON STEVIN wishes health,” The Mathematical Works of Simon Stevin, p. 389.
Cossic algebra and Viète will be discussed in detail in chapter three.
Viète believed that algebra, as it was practiced in his period, had been “spoiled and defiled by the barbarians.” Introduction to the Analytic Art, trans. Reverend J. W. Smith, in J. Klein’s Greek Mathematical Thought and the Origin of Algebra (New York: Dover Publications, 1992), pp. 318–319.
I would like to thank an anonymous referee for introducing me to Henk J. M. Bos’s magisterial Redefining Geometrical Exactness: Descates’ Transformation of the Early Modern Concept of Construction (New York: Springer-Verlag, 2001). Much of what follows is taken from Bos’s chapter six “Arithmetic, geometry, algebra, and analysis” pp. 119–132.
Bos, Redefining Geometrical Exactness, p. 120.
Bos, Redefining Geometrical Exactness, p. 121.
See John E. Murdoch, “The Medieval Language of proportions: Elements of the Interaction with Greek Foundations and the Development of New Mathematical Techniques,” in Scientific Change (ed) A. C. Crombie (London: Heinemann, 1963), pp. 237–271;
E. D. Sylla “Componding Ratios: Bradwardine, Oresme and the First Edition of Newton’sPrincipia,” in (ed) E. Mendelsohn (Cambridge: Cambridge University Press, 1984), pp.11–43.
Bos, Redefining Geometrical Exactness, pp. 127–128,
These definitions have been taken from Redefining Geometrical Exactness pp. 128–130.
Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (Cambridge: M.D.T. Press, 1968), P. 7.
Klein, p. 124., p. 148. Klein claims that the algebraic tradition, from Leonardo of Pisa via the “cossic” school, and up to such figures as Michael Stifel (1544), Cardano (1545), Tartaglia (1556–1560), was separate from the traditional disciplines of the schools, and it struggled for a place in the system of western science. This school, he further claims, only became aware of its own “scientific” character and the novelty of its “number” concept when it came into contact with the Arithmetic of Diophantus. He cites as an example Bombelli’s modifying the “technical” character of his manuscript to a generalised form of the first five books of Diophantus after reading the Arithmetic. Moreover, practitioners such as Viète felt their work was an extension of a received ancient tradition because they believed that Diophantus had used a form of analysis similar to their own, but that Diophantus had hidden his method so that his “subtlety and skill might be more admired.” It is unclear, however, in what sense researchers needed to be “self-conscious” in order to broaden their number concepts or how their self-awareness actually influenced the course of research.
Klein, p. 125.
Klein, p. 213.
In some ways, of course, numerical solutions had always been available. As Pedersen pointed out in the context of Ptolemy, logistics was carried to a greater state of perfection than has previously been suspected. As we shall see, even scholars who maintained classical definitions of numbers often used fractions.
See, for example, Jacob Klein’s Greek Mathematical Thought and the Origin of Algebra; Charles Jones’ “The Concept of One as a Number,” diss. University of Toronto, 1978; Jones extends Klein’s work by taking a more detailed look at how the classical concept of number and the Greek distinction between number and magnitude, was overthrown by Stevin. See also, Joann Morse’s “The Reception of Diophantus’ “Arithmetic” in the Renaissance,” diss. Princeton University, 1981.
For another interesting discussion, see: Michael Mahoney, “The Beginnings of Algebraic Thought in the Seventeenth Century,” Descartes Philosophy, Mathematics and Physics, ed. Stephen Gaukroger (New Jersey: Barnes and Noble, 1980), p. 147. Michael Mahoney associates, like Klein, the broadening number concept in Viète and Descartes with the burgeoning algebraic mode of thought. Mahoney parts from Klein in attributing Viète’s and Descartes’ willingness to move beyond Greek mathematical concepts to the pedagogy of Peter Ramus and the search for a universal symbolism
Helena Pycior, “Mathematics and Philosophy: Wallis, Hobbes, Barrow and Berkeley”, Journal for the History of Ideas 42 (1987): 263–286, does briefly mention the unusualness of Isaac Barrow’s number concept, and she attributes Barrow’s point of view to his recognising only geometry as a mathematical science. The main focus of her article, however, is the seventeenth-century roots of Berkeley’s mathematical views. Thus she does not explore the details of how the dispute between ancients and moderns influenced the transition to an algebraic mode of thought.
C. Sasaki, “The Acceptance of the Theory of Proportion in the Sixteenth and Seventeenth Centuries,” Historia Scientiarum 29 (1985): 83–116, notes the differences between Wallis’ and Barrow’s views on algebra and number, and he relates their differences to a propensity towards ‘modern’ versus ‘traditional’ points of view.
This is an observation made by an anonymous referee.
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Neal, K. (2002). Transformation of the Number Concept. In: From Discrete to Continuous. Australasian Studies in History and Philosophy of Science, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0077-1_1
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