Part III Mathesis Universalis and Charateristica Universalis

Part III Mathesis universalis and charateristica universalis Chapter 8 The “merely mechanical” vs. the “scab of symbols”: seventeenth century disputes over the criteria of mathematical rigor Douglas M. Jesseph Abstract This paper deals with seventeenth-century understandings of rigorous demonstration. Although there was a widely-shared concept of rigor that has its origins in classical Greek sources, philosophers in the early modern period were divided over how to characterize the ultimate foundation for mathematics. One group (whom I term the “geometric foundationalists”) held that seeming physical concepts such as space, body and motion, were properly foundational. The other group (whom I call “algebraic foundationalists”) claimed that the true foundations of mathematics must be abstract notions of quantity which were the subject of algebra, or even a more general mathesis universalis that encompassed all reasoning about number and measure. The geometric foundationalists faced the objection that they had introduced “merely mechanical” or insufficiently abstract principles into the foundations of mathematics. In contrast, the algebraic foundationalists needed to rebut the accusation that they based mathematics on a “scab of symbols”, or empty notation divorced from anything real or substantial. I argue that this episode offers some useful insights into general questions about foundations, and can help us understand what is at stake in disputes over foundational issues, as well as how such disputes rise to prominence and then fade away. 8.1 Introduction One of the things that philosophers find most intriguing about mathematics is the rigor of its demonstrations. A properly developed mathematical theory is one which starts with clearly conceived and well-understood first principles, then unfolds via deductively valid inferences to reach conclusions that, however outré or surprising they may seem to untutored common sense, are 273 274 Douglas M. Jesseph every bit as secure as the first principles on which they depend. This picture of mathematics has convinced many philosophers that it is the pinnacle of hu- man intellectual achievement, something that can serve as a bulwark against the challenges of skepticism, and whose deductive structure might serve as a model for philosophical theorizing. We can see this dynamic at work in the famous anecdote concerning Thomas Hobbes’s discovery of mathematics as related by his friend John Aubrey: He was forty years old before he looked on geometry; which happened accidentally. Being in a gentleman’s library Euclid’s Elements lay open, and ’twas the forty-seventh proposition in the first book. He read the proposition. “By God,” said he, “this is impossible!” So he reads the demonstration of it, which referred him back to such a proof; which referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry. (Aubrey 1898, p. 322) As my concern here is with the interaction between philosophy and mathematics in the seventeenth-century, it is fitting to begin with an anecdote involving Hobbes, even if his mathematical misadventures featured numerous outrageously poor attempts to solve such problems as the squaring of the circle or the duplication of the cube.1 The issue I wish to discuss concerns seventeenth-century understandings of what it is that makes mathematical (and perhaps more specifically geometric) demonstrations rigorous. Although there was a widely-shared general characterization of rigor that has its origins in classical Greek approaches to the philosophy of mathematics, philosophers and mathematicians in the early modern period were divided over whether the ultimate foundation for mathematics should be sought in seemingly physical concepts such as space, body and motion, or in more abstract notions of quantity which were the subject of algebra, or perhaps a more general mathesis universalis that encompassed all reasoning about number and measure. Those who sought to ground the whole of mathematics in concepts such as space and motion had to confront the objection that they had introduced “merely mechanical” or insufficiently abstract principles into the foundations of mathematics. On the other hand, proponents of a mathesis universalis needed to rebut the accusation that, in doing so, they based the whole science on a “scab of symbols,” or a jumble of empty notation divorced from anything real or substantial. I begin with a brief overview of the classical Greek point of view on the subject, as exemplified in the works of Euclid and his neo-Platonist commenta- tor Proclus Diadochus. I then consider the “geometric foundationalism” pro- pounded by Isaac Newton and Isaac Barrow (among others), which seeks to ground all of mathematic in fundamental geometric notions. I will then inves- 1 For an account of Hobbes’s failed campaign for mathematical glory, see Jesseph (1999) 8 Criteria of mathematical rigor 275 tigate the “algebraic foundationalism” put forward by John Wallis, Descartes, and others. In the end, I think this episode offers some useful insights into general questions about foundations, and it can help us understand what is at stake in disputes over foundational issues, as well as how such disputes rise to prominence and then fade away. 8.2 Euclid, Proclus, and the Classical Point of View Classical Greek authors took pure mathematics to consist of two essentially different subjects: arithmetic on the one hand and geometry on the other.2 The applied mathematical sciences, which include such things as astronomy, music, optics, mechanics, were taken to differ from the pure sciences because the pure part is – to quote Proclus – “concerned with intelligibles only,” while the applied part of mathematics “work[s] with perceptibles and [is] in contact with them” (Proclus 1970, p. 31). Classically conceived, then, the pure sciences of geometry and arithmetic are distinguished from one another by having different objects. The object of arithmetic is discrete quantity (“number” or ἀριθμός in the Greek), while the object of geometry is continuous quantity (“magnitude” or μέγεθος in Greek). Thus understood, the two branches of pure mathematics can be developed in separate ways and there is no question of attempting to “reduce” one to the other or to show that one is in some sense the genuine foundation of all mathematics. This pluralistic approach to the philosophy of mathematics does not presume or require that the different mathematical sciences be reduced to some underlying foundational science. This does not, however, imply that classical authors always conceived of the two branches of mathematics as of equal importance. Proclus took the view that, although pure geometry was entirely independent of arithmetic, geometry nevertheless “occupies a place second to arithmetic,” because (in keeping with his neo-Platonist epistemology) everything knowable must ultimately be expressed in terms of commensurable ratios (Proclus 1970, p. 39). The princi- pal point here is that classical authors saw no relation of dependence between the truths of arithmetic and the truths of geometry. This becomes clearer in the presentation of the theory of ratios in Euclid’s Elements. Book V develops the theory of proportions in the context of continuous magnitudes, and Book VII offers an account of ratio and proportion as applied to discrete quantities or “numbers” (ἀριθμόι). It was, of course, well understood in the ancient world that the theory of proportions in Book 2 On the Greek classification of the geometrical sciences, see Heath (1931), Klein (1968, Part I) and Knorr (1981) 276 Douglas M. Jesseph VII could be taken as just a special case of the theory developed in Book V. Aristotle remarked that it might seem that the proportion alternates for things as numbers and as lines and as solids and as times – as once it used to be proved separately, though it is possible for it to be proved of all cases by a single demonstration. But because all these things – numbers, lengths, times, solids – do not constitute a single named item and differ in sort from one another, it used to be taken separately (Posterior Analytics 74a 18-22) Thus, rather than assuming that there is one underlying notion of proportion that applies equally to continuous magnitudes and discrete multitudes, the tradition in Greek geometry was to develop the two theories separately. This simply reflects, and indeed reinforces, the notion that geometry and arithmetic are autonomous sciences. Given that they are separate sciences, arithmetic and geometry will have to have separate first principles. The first principles of a demonstrative science were traditionally distinguished into three classes: axioms, definitions, and postulates. Axioms or “common notions” apply to any science whatever and include such general principles as “things which are equal to the same thing are equal to one another” or “if equals be subtracted from equals, the remainders are equal” (Elements 1, Ax. 1, 3). These apply universally to any subject matter, but they are insufficiently specific to develop a particular science. Definitions are principles specific to a given science and set out the essential properties of the objects of that science. So, in the case of arithmetic we have the definition of a unit as “that by virtue of which each of the things that exist is called one,” and a number as “a collection of units” (Elements 7, Def. 1, 2). In the case of geometry, we require

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