The Mathematical Career of Pierre de Fermat, 1601-1665

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Summary: Mathematics in transition: Pierre de Fermat
Review: Thanks to the widely publicized proof of Fermat's Last Theorem by Andrew Wiles, there is a growing public interest in the 17th- century mathematician and his famous theorem. However, readers interested in the theorem and its history should look elsewhere as this book is directed towards the history and the transition of mathematics as a science in the 17th century. Studying the evolution of concepts and methods in mathematics, Michael Mahony sets a standard with this excellent work. A student of Thomas Kuhn, Mahony is meticulous in his treatment and interpretation of historical data about Fermat within its historical context. He is very careful in the use of notational systems and at appropriate times he uses the same symbolism as Fermat used in his correspondence with Mersenne and others. These subtleties are important as the evolution in algebraic symbolism has precisely been functional in the foundation of analytic geometry. Further in line with Kuhn's Structure of Scientific Revolutions Mahony first situates Pierre de Fermat within Viète's analytic program and continuously refers to this program of scientific research. He clearly shows where Fermat departs from this program and lays the foundation of something new as with number theory as a discipline on its own.

The career of Pierre de Fermat coincides with an important revolution within mathematics during the 17th century: together with Descartes he fundamentally altered the balance between the visual and the abstract in mathematics. The Greek tradition of visually stating problems about curves, loci and triangles gave way to an abstract characterization in algebraic equations which allowed a more general treatment of these original problems. Where the Greek tradition provided the starting point for Fermat, he moved that far beyond his original sources that by the end of his carreer the original Greek texts became history for mathematicians.

However, the book is even more than an illustration of science in transition: it reveals the mathematician in transition. The author is very convincing in recreating step by step the way Fermat arrived at new ideas and theorems and how, over time, he became the father of new disciplines in mathematics. Mahony does so by reconstructing all the conceptual steps needed in arriving at new ideas. For the method of maxima and minima, the doctrine of tangents and the quadrature, he relies on available historical data, such as Fermat's letters to Mersenne, Descartes and Frenicle. For Fermat's contributions on number theory however, only a few historical documents are available as Fermat was very secretive about his findings and reluctant to publish anything. With some help from Euler and Weil, Mahony shows very convincingly and fits together how theorems and proofs on number theory emerged over time for the man who was Pierre de Fermat.

Everybody who is keen on mathematics and its history should read this book. Being an excellent case study of science in transition, the book will highly appeal to students in the philosophy of science.

Rating:
Summary: A fine account of Fermat's work
Review: There is a tremendous mythology surrounding Fermat, and interest in Fermat is high because of Wiles' proof of "Fermat's last theorem." Fermat was undoubtedly a genius, but exactly what he did and didn't do will never be clear, because he wrote so little about his methods and his proofs. Mahoney's book comes as close as I think it's possible to come in figuring out what Fermat's mathematical interests and methods were, what he proved and what he didn't. Only high school math is required to read this book easily; the writing style is clear, and the structure of the book is well organized.

Some readers may be disappointed to learn that Fermat's abilities were only human, and that he made a number of mistakes. But even a genius is entitled to be less than perfect, and the real Fermat is in many ways more interesting than the myth. I recommend this book to anyone curious about Fermat's mathematical abilities and achievements.

Rating:
Summary: A fine account of Fermat's work
Review: There is a tremendous mythology surrounding Fermat, and interest in Fermat is high because of Wiles' proof of "Fermat's last theorem." Fermat was undoubtedly a genius, but exactly what he did and didn't do will never be clear, because he wrote so little about his methods and his proofs. Mahoney's book comes as close as I think it's possible to come in figuring out what Fermat's mathematical interests and methods were, what he proved and what he didn't. Only high school math is required to read this book easily; the writing style is clear, and the structure of the book is well organized.

Some readers may be disappointed to learn that Fermat's abilities were only human, and that he made a number of mistakes. But even a genius is entitled to be less than perfect, and the real Fermat is in many ways more interesting than the myth. I recommend this book to anyone curious about Fermat's mathematical abilities and achievements.