A Companion to Hobbes. Группа авторов
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Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of the latter equimultiples respectively taken in corresponding order.
(Elements, Book V, Def. 5)
This definition attracted a great deal of commentary over the centuries,6 and Hobbes was not alone in thinking that it was too intricate and obscure to be numbered among the first principles of geometry.
In Hobbes’s view, sameness of ratio can be properly defined only in terms of the motion of bodies. Thus, rather than following Euclid and explicating this concept in terms of order preservation under arbitrary equimultiples, he seeks to find a more general principle from which the Euclidean definition can be derived. As he puts the matter in PRG chapter 12: “I define “the same ratios” to be “those which uniform motion exposes in two straight lines in equal times”; or more universally, “those things are in the same ratio which are determined by some cause producing equal effects in equal times” (OL IV.420). Hobbes’s “more universal” definition cited here comes from De corpore (2.13.6), and he bragged that Euclid’s definition of ‘same ratio’ could be demonstrated from it. Speaking of himself in the third person in the second dialogue of Examinatio, he insists that the Euclidean definition “is true, I say, but not a principle, because it is demonstrable and demonstrated by Hobbes in article 12 of chapter 13 of the book De corpore, but demonstrated from a definition of the same ratio by generation that is different from this of Euclid” (OL IV.76–7).7
Hobbes also critiques the Euclidean definition of the circle, which defines it as “a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another” (Elements, Book I, Def. 15). In Hobbes’s estimation, this has the virtue of stating something true about the circle, but it fails to be a proper definition because it does not identify its causal generation, as he claims in SL Lesson 1:
But if a man had never seen the generation of a Circle by the motion of a Compass or other æquivalent means, it would have been hard to perswade him, that there was any such Figure possible. It had been therefore not amiss first to have let him see that such a Figure might be described. Therefore so much of Geometry is no part of Philosophy, which seeketh the proper passions of all things in the generation of the things themselves.
(EW VII.205)
In the Hobbesian scheme of things, the correct definition of the circle – found in De corpore XIV.4 – is in terms of the rotational motion of a straight line about one of its termini (EW I.180–1). Such a definition not only identifies the cause of the circle, but also enables the geometer to investigate its properties.
Hobbes’s methodology holds that demonstrative knowledge must be based on definitions that identify the causes of things. This is summarized in the Dedicatory Epistle to the SL, where he insisted that “where there is place for Demonstration, if the first Principles, that is to say, the Definitions contain not the Generation of the Subject; there can be nothing demonstrated as it ought to be” (SL Epistle). As he explains a greater length in PRG 12:
”But”, you will ask, “what need is there for demonstrations of purely geometric theorems to appeal to motion?” I respond: first, all demonstrations are flawed, unless they are scientific, and unless they proceed from causes they are not scientific. Second, demonstrations are flawed unless their conclusions are demonstrated by construction, that is, by the description of figures, that is, by the drawing of lines. For every drawing of a line is motion: and so every demonstration is flawed, whose first principles are not contained in the definitions of motions by which figures are described.
(OL IV.421)
In Hobbes’s view, once proper causal definitions are in place, the development of a science is a matter of relative routine. The guiding thought here is that knowledge of causes should provide easy access to knowledge of effects. In particular, since geometric objects are brought into being by motions through which the geometer literally constructs the object of investigation, so that “in his demonstration [the geometer] does no more but deduce the Consequences of his own operation” (EW VII.183–4). This approach to demonstration led Hobbes to believe that, once proper geometric definitions were in place, the solution of any geometric problem should follow with ease. Thus, the failure of previous generations of geometers to solve such problems as the quadrature of the circle did not arise from intractability of the problems, but from flawed first principles. Once the Euclidean definitions have been replaced by Hobbes’s causal definitions, the royal road to the solution of geometric problems was open. Or so Hobbes believed. As we will see, this confidence in the efficacy of his geometric first principles ultimately led Hobbes to serious error.
3.1.3 The Status of Algebraic and Infinitesimal Methods
Hobbes’s approach to geometry clearly set him against the traditional view of the subject, but he also opposed some of the innovations that were the most important advances in seventeenth-century mathematics. In particular, he was quite hostile to the algebraic methods characteristic of Descartes’s analytic geometry, and he found fault with some presentations of the “method of indivisibles” that is an important precursor to the calculus. A brief summary of these issues can bring our investigation into the Hobbesian philosophy of mathematics to a close.
The 1637 publication of Descartes’s Géométrie as one of the three Essais that accompanied the Discours de la méthode is generally taken to be the advent of analytic geometry. The details of this approach are widely enough understood that they need not detain us.8 The fundamental idea is that geometric curves can be represented algebraically, first by taking algebraic operations (addition, subtraction, multiplication, division, and the extraction of roots) as geometric constructions, and then identifying curves with indeterminate equations in two unknowns that correspond to the familiar co-ordinate axes in the Cartesian plane. Curves that are truly geometrical and susceptible to analysis can be given a “precise and exact” formulation expressed as a single equation showing the relation between the curve and the points on one axis (AT 6:392).
The ‘analytic’ aspect of Cartesian geometry stems from its use of a so-called analytic method that begins with the assumption that a problem has been solved, and then proceeds to investigate the conditions under which such a solution can be achieved. As he put the matter
If, then, we wish to solve any problem, we first suppose the solution already to have been effected, and give names to all the lines that seem needful for its construction – to those that are unknown as well as to those that are known. Then, making no distinction between known and unknown lines, we must unravel the difficulty in any way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these equations are together equal to the terms of the other. And we must find as many such equations as we have supposed lines which are unknown.
(AT 6: 372–3)
The use of algebraic techniques to investigate the solution conditions for geometric problems was not an entirely Cartesian innovation. The procedure often went under the name of the “analytic art” and was widely regarded as a universal method for the solution of geometric problems. François Viète’s 1591 In Artem analyticem Isagoge