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appendIx who is a geometer? wallis, algebra, and a humanist defense of mathematical practice I have argued against the view that Hobbes’s turn to mathematics and geometry was a turn away from humanism. I have also argued that Hobbes’s affinity for mathematics bears the marks of the humanist culture. Hobbes and the humanist educators that preceded him admired the same things in the mathematically trained mind. Finally, I have shown how Hobbes’s promotion of his mathematically inspired philosophy also bears the marks of what I have called the mathematical “high culture” of the Stuart Court. But how deeply enmeshed is Hobbes’s humanism within his mathematics? Here I will offer evidence that his humanism—the practices, habits, and framework in which Hobbes understood his claim to intellectual mastery—is thoroughly woven into his concept of mathematics. Although I will not offer a comprehensive treatment of Hobbes’s mathematics , I can make this point in light of a key part of the controversy in which Hobbes defended his conception of mathematics and attacked that of his opponent, John Wallis. Inevitably, this will also raise the larger question of the relationship of Hobbes’s mathematics to his politics generally. More might be said in linking Hobbes’s promotion of what he claimed was an infallible philosophical method, and his preference for absolutist government, but in this appendix I will limit my treatment to the more immediate political considerations that attended the Hobbes-Wallis conflict. Until recent efforts by historians of mathematics, there has been little to supersede George Croom Robertson’s account of it. That has now changed.1 Among most mathematically centered commentary, however, reflections on the politics of the conflict have been relatively slim. Douglas Jesseph has recently pursued these matters in greater detail, but he has also insisted upon a relatively bright line between the political and mathematical disputes between Hobbes and Wallis.2 I have already commented on Jesseph’s approach in Chapter 5, and here again it seems to me that the evidence Jesseph discusses undermines his claim to a firm distinction. Consideration of the question of 222 p appendix Hobbes’s conflict with Wallis from the point of view of humanism will further reinforce the case. (Unlike Jesseph I will not attempt to settle the question of Hobbes’s atheism, an issue raised by Wallis’s attack.) I shall first describe the crux of the key element of their conflict, the dispute over the merits of algebra. Then, after a discussion of the political contexts (some of which I have already described in Chapter 7), I will illustrate that this debate between Hobbes and Wallis was also, and at once, a debate over who was entitled to call himself a proper geometer. Moreover, in taking on this characteristic of a debate over an office or title, Hobbes pursued his side of the argument in a framework provided to him by his humanist contemporaries and predecessors. The Problem with Algebra Wallis was a strong advocate of algebraic methods in geometry, and has been understood as a pioneer. The enthusiasm for algebraic technique was something he shared with Descartes—although Wallis was loath to give Descartes credit for the development of algebraic techniques.3 Algebra, as developed by Descartes and also practiced by Wallis, entailed the reduction of all geometrical entities to a single unit, the line. Algebraic technique, moreover, violated the guiding principles of Hobbes’s practice (principles I described in Chapters 3 and 5). It did not start with elemental parts and work its way up to the construction of figures. It did not proceed one carefully placed step at a time. Indeed, from the point of view of Hobbes and those who favored more traditional geometrical practices (such as Newton’s teacher, Isaac Barrow), the audacity of early modern algebra was signaled at the very start of Descartes’s La Geometrie.4 His work did not begin with postulates, axioms, or definitions. Rather, he began with a reductive assertion that he claimed would facilitate the creation of a much more useful, and universalizable, mode of reasoning. He was not interested in studying and defining given figures. Rather, he proposed that mathematicians reduce and redescribe all figures in terms of a given (arbitrarily chosen) basic unit: Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction. Just as arithmetic consists of only four or five operations, namely...

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