By Paolo Mancosu
The 17th century observed dramatic advances in mathematical conception and perform. With the restoration of a few of the classical Greek mathematical texts, new ideas have been brought, and inside a hundred years, the principles of analytic geometry, geometry of indivisibles, mathematics of infinites, and calculus have been built. even though many technical reports were dedicated to those concepts, Mancosu offers the 1st complete account of the connection among mathematical advances of the 17th century and the philosophy of arithmetic of the interval. beginning with the Renaissance debates at the simple task of arithmetic, Mancosu leads the reader in the course of the foundational matters raised via the emergence of those new mathematical concepts, together with the impact of the Aristotelian notion of technology in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation among geometrical and epistemological theories of the limitless, and the Leibnizian calculus and the competition to infinitesimalist methods. within the method Mancosu attracts a worldly photo of the sophisticated dependencies among technical improvement and philosophical mirrored image in 17th century arithmetic
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Extra resources for Philosophy of mathematics and mathematical practice in the seventeenth century
The conclusion from both cases is that O can only be equal to 3V. We will see how Cavalieri's indivisibles and Guldin's centers of gravity allow us, among other things, a more direct approach to the proof of theorems that the Greeks had proved by exhaustion. 2 Cavalieri's Two Methods of Indivisibles The exhaustion method of the Greeks was logically very rigorous but it had the defect of not giving a sense to the working mathematicians of the way in which the result had been obtained. The mathematicians of the sixteenth and seventeenth centuries often voiced their dissatisfaction with proofs by exhaustion, claiming that the ancients had hidden their methods of discovery.
Both works are very long and not easy to read. Today the task of studying Cavalieri is made easier by two excellent works, Andersen (1985) and Giusti (1980), which give a thorough analysis of Cavalieri's work. I will limit myself to the essential technical points that will allow me to introduce the philosophical discussion. Cavalieri had two methods of indivisibles. To one of them he devoted the initial six books of the Geometria; the other he covered in the seventh book. He also gave an exposition of both methods in the first and second investigation of the Exercitationes, written as a defence of his indivisibilist approach against the attacks it had received.
But this cannot be, since clearly BDC < BAC. Thus we reject the assumption AB AC, and we conclude that AB = AC. Proofs by contradiction are often needed to prove geometrical converses of theorems. 5, which asserts that in an isosceles triangle the equal sides subtend equal angles. It is exactly the issue of converse propositions that focused attention within the Quaestio on proofs by contradiction. Piccolomini in chapter 11 of his Commentarium had given several arguments aimed at showing that mathematical propositions could not be proven by causal proofs.
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