The Mathematical Career of Pierre de Fermat, 1601-1665 [Paperback]

Michael Sean Mahoney (Author)

Book Description

Publication Date: October 17, 1994 | ISBN-10: 0691036667 | ISBN-13: 978-0691036663 | Edition: 2 Revised

Hailed as one of the greatest mathematical results of the twentieth century, the recent proof of Fermat's Last Theorem by Andrew Wiles brought to public attention the enigmatic problem-solver Pierre de Fermat, who centuries ago stated his famous conjecture in a margin of a book, writing that he did not have enough room to show his "truly marvelous demonstration." Along with formulating this proposition--xn+yn=zn has no rational solution for n > 2--Fermat, an inventor of analytic geometry, also laid the foundations of differential and integral calculus, established, together with Pascal, the conceptual guidelines of the theory of probability, and created modern number theory. In one of the first full-length investigations of Fermat's life and work, Michael Sean Mahoney provides rare insight into the mathematical genius of a hobbyist who never sought to publish his work, yet who ranked with his contemporaries Pascal and Descartes in shaping the course of modern mathematics.

Editorial Reviews

Review

Mahoney's sensitive handling of the material, his sharp appreciation of conceptual and notational subtleties, and his willingness to detail or reconstruct proofs and procedures, now make possible an appreciation of the real power and variety of Fermat's invention. -- Alan Gabbey, British Journal for the History of Science



A remarkably satisfying and cogent analysis. -- Carl B. Boyer, Science

Product Details

• Paperback: 438 pages
• Publisher: Princeton University Press; 2 Revised edition (October 17, 1994)
• Language: English
• ISBN-10: 0691036667
• ISBN-13: 978-0691036663
• Product Dimensions: 9.2 x 6.1 x 0.9 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #1,822,851 in Books (See Top 100 in Books)

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8 of 8 people found the following review helpful:

5.0 out of 5 stars Mathematics in transition: Pierre de Fermat, February 6, 2001

By 

Albrecht Heeffer (Belgium) - See all my reviews
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This review is from: The Mathematical Career of Pierre de Fermat, 1601-1665 (Paperback)

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.

7 of 7 people found the following review helpful:

4.0 out of 5 stars A fine account of Fermat's work, February 19, 2001

By 

Victor A. Vyssotsky "mandvav" (Orleans, MA USA) - See all my reviews
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This review is from: The Mathematical Career of Pierre de Fermat, 1601-1665 (Paperback)

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.

2 of 3 people found the following review helpful:

4.0 out of 5 stars "To have been a 'second Viète' was at once Fermat's glory and his tragedy" (p. 71), February 23, 2011

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Viktor Blasjo - See all my reviews
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This review is from: The Mathematical Career of Pierre de Fermat, 1601-1665 (Paperback)

This useful book illustrates how illuminating it can be to take mathematicians rather than mathematical theories as units of historical analysis, for it gives a remarkably coherent view of Fermat's diverse contributions as resulting from his close adherence to the research program of Viète, which he imbibed during his mathematical apprenticeship under the devoted followers of Viète in Bordeaux during his formative years.

The Viètean program was first and foremost a program defined by the use of algebra; that is, it emphasised method over content. Secondarily, it was driven by the belief that this method unlocked the hidden method of analysis of the ancients, and that therefore the reconstruction in algebraic terms of the most opaque and secretive aspects of ancient works would lead to a watershed of mathematical progress (Fermat spoke of "those long buried monuments of geometry in which so many great findings of the Ancients lie with the roaches and worms"; p. 119 [my page references are to the hardcover edition]).

Perhaps Fermat's proudest accomplishment in completing the Viètean program was his creation of coordinate systems. Like Descartes, he was led to his discovery by studying locus problems, in particular the reconstruction of Apollonius' Plane Loci. "One of the prettiest propositions in geometry" (p. 95), in particular, led his to his discovery: namely to find the locus (which will be a circle) of all points from which the sum of the squares of the distances to a given set of fixed points is constant.

Fermat's theory of maxima and minima is based on the idea of extrema being double roots. Let's say for example that we want to maximise f(x)=x-x^2 (cf. esp. pp. 153-155). Fermat's method goes like this. Pick some Y smaller than the maximum. Then Y=f(x) will have two solutions (one for each branch of the parabola), call them X and X+D. Thus f(X)=Y=f(X+D), i.e. X-X^2=(X+D)-(X+D)^2, which simplifies to 2X=1-D. But for the maximal Y the two roots coincide (at the vertex of the parabola), i.e. the maximum corresponds to the condition D=0, which gives the solution X=1/2.

Fermat's theory of tangents was a byproduct of this method. Let's say for example that we want to find the tangent to y=x^2 at the point (2,4) (cf. esp. pp. 166-167). The tangent line is below the curve everywhere except at the point of tangency. In other words, among all points on the tangent, the point of tangency minimises the quantity x^2-y. If we think of the tangent line as determined by its y-intercept -Y, its slope is (4+Y)/2. Using the theory of optimisation above, we suppose, counterfactually, that there is another x-value, say x=2+D, for which the quantity x^2-y is the same as for x=2. In other words, 2^2-4=(2+D)^2-(4+D((4+Y)/2)), which when we simplify and divide by D gives 0=2-D-Y/2. But since the minimum is actually unique, D=0 after all, so we get the answer Y=4.

Fermat never broke free from this algebraic approach in favour of a truly infinitesimal theory, although he later managed to supplement his approach with some infinitesimal reasoning to extend it to problems such a the determination of tangents to cycloids (pp. 212-213).

Fermat's theory of integration was closer to modern ways, as it basically consisted in approximating areas and arc lengths by sums that could be computed algebraically. But of course this idea had ample ancient precursors, so it is not Fermat's most original work. Whereas the ancients has focused on specific problems of great intrinsic interest, such as the volume of a sphere or the area of a segment of a parabola, Fermat seemed more willing to go wherever the algebra took him. In particular he studied the "higher parabolas" y^p=kx^q, seemingly for no other reason than that he could do the algebra for them.

The notion that Fermat's work marks the beginnings of the calculus is rejected in no uncertain terms by Mahoney: "One cannot say with any degree of fairness or objectivity that Fermat's work in analysis of curves was even heading in the direction of the calculus. ... For all the personal animosity that lay behind the observation, Descartes was probably quite right when he said to van Schooten that Fermat was a brilliant problem-solver but basically inept at conceiving systematic questions." (p. 279)

Mahoney has no distinctive point to make about Fermat's number theory, writing that "Like Winston Churchill's Russia, Fermat's number theory is 'a riddle wrapped in a mystery inside an enigma', and will probably largely remain so" (p. 282). Nevertheless the Viètean background may explain something, namely the puzzling fact that Fermat was fascinated by number theory whereas all of his contemporaries were utterly indifferent to it. This could be because the Viètean algebra that Fermat so adored was a method in search of field, so to speak. Thus the very fact that number theory allowed substantial applications of these new methods was enough to entice Fermat, whereas his contemporaries, who judged number theory by the intrinsic interest of its problems, remained unimpressed. Thus we can think of Fermat's number theory as the natural extrapolation of the tenets of Vièteanism outlined above: it is eminently algebraical and it involves the reconstruction of a secretive ancient source (in this case Diophantus).