Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

by Paolo Mancosu

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

Genres: Philosophy

288 pages, Paperback

Published July 8, 1999

About the author

Paolo Mancosu, Ph.D., Stanford University, is Professor of Philosophy. His interests lie in the philosophy of mathematics and its history, in philosophy of logic, and in mathematical logic. His written work is currently focused upon neologicism and the philosophy of mathematical practice.

Reviews

[Kristopher · Rating 4 out of 5]

This is a well-written, interesting read on the history and development of mathematics during a particularly volatile time in the development of the field. The book is not particularly accessible, however. Not being a mathematician myself, I found the first 10-15 pages of each chapter particularly difficult to follow, as Mancuso demonstrated technical nuances of various proofs that were at issue. I do not intend this as a criticism, as I'm certain that if I understood what was happening in the proofs, then the rest of the chapters would have been more rewarding. However, I found that I could glean what was at issue philosophically from the rest of the chapter, which I found fascinating. Again, not being a mathematician, I am unsure how accessible the second half of each chapter would be to a mathematician, as it is pretty difficult philosophical material.

This makes the text something of an odd piece. Unless one is both a mathematician and a philosopher, this may not be the text for you. Again, that's not to say that it's not worth reading unless you're both, but just prepare yourself for some dense material.

Here are a few nit-picky points. Mancuso does quotes from primary sources often, which is great, but he often quotes in the original language without providing a translation. I understand that this is a trend in some academic writing, and while I have passable French reading skills, when large passages are quoted it still takes me time to work through them. My Latin is significantly worse than my French, and while the Latin quotes tend to be shorter, it does unnecessarily break up the reading of what is already dense material.
Also, throughout the text, Mancuso provides an interesting narrative about the events of the seventeenth century, but there is literally no conclusion to the book. It just sort of ends. While there are themes that run throughout the text, and arguments made, they are all left open at the end of the book. It would be nice to have some brief note at the end that tied all those themes together.

[Sam · Rating 5 out of 5]

First of all, a lot of this book is composed of articles published in other places, but given its current price, it's still worth owning.

It's a very dense read, but I found it to be an incredibly rewarding piece of scholarship.