The Mathematics of Plato's Academy: A New Reconstruction (Oxford Science Publications series) [Paperback]

D. H. Fowler (Author)

Book Description

Publication Date: 1987 | ISBN-10: 0198539479 | ISBN-13: 978-0198539476 | Edition: 1st

This book presents a reinterpretation of early Greek mathematics, one of the most tantalizing intellectual subjects of the last 2,000 years. The first part offers several new interpretations of the idea of ratio in early Greek mathematics and illustrates them in detailed discussion of several texts. Part Two discusses the historical context of the subject--what we know of Plato's academy during his lifetime, the origin of our text of Euclid's Elements, and what we know of early Greek numerical practice. The book finishes with an account of the theory of continued fractions and its history since the 17th century.

Editorial Reviews

Review

From reviews of the first edition: "...a real treat." --Greece and Rome

"...cites an impressive array of evidence...The result should be widely read by classicists and mathematicians as well as historians of mathematics." --ISIS

"...this fascinating book...will arouse the interest and command the admiration of any historically minded lover of mathematics with a taste for the unorthodox." --Institute of Mathematics and its Applications

--This text refers to the Hardcover edition.

About the Author

David H. Fowler is at Mathematics Institute, University of Warwick. --This text refers to the Hardcover edition.

Product Details

• Paperback: 401 pages
• Publisher: Clarendon Press; 1st edition (1987)
• Language: English
• ISBN-10: 0198539479
• ISBN-13: 978-0198539476
• Product Dimensions: 9 x 6 x 1 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #2,620,933 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

15 of 15 people found the following review helpful:

5.0 out of 5 stars A brilliant, sprawling book, August 16, 2001

By 

Colin McLarty (Chardon, OH USA) - See all my reviews

This review is from: The Mathematics of Plato's Academy: A New Reconstruction (Hardcover)

Two things are certain if you really want to know what mathematics was done in Plato's Academy, and before Euclid: Your heart will break at the lack of evidence, and you will have to read this book.

Fowler details how thin the surviving evidence is, even for such basics as when Euclid's ELEMENTS were written. Drawing on other careful classicists he demolishes now traditional stories about the Pythagoreans and the irrational, Plato's Academy, even Euclid's own style in the Elements. He shows them coming from heavy interpretations of extremely vague (and often late) sources. Plates in the book show how desperately scanty are the physical remains of any mathematical writing within centuries of Plato's death. Even the first and second century AD leave us only a few scraps of Euclid.

On the positive side, Fowler gives a persuasive account of a method of reciprocal subtraction which he calls "anthyphairesis". It lay within the grasp of Athenian geometers, and suits some remarks Plato makes on mathematics, and suits traditions on geometers Plato knew, and goes far to unify and explain much of Euclid. It was apparently cited by Aristotle (under the name "antanairesis"). Probably, it really was used in the period. It also makes some very pretty geometry. Regular pentagons make a lot of sense anthyphairetically. Anyone trying to read the later books of Euclid, especially books X and XIII, will get tremendous help from this book. Conversely, you can hardly read much of this book without reading Euclid.

The book is not well organized. It spends many pages at a time on mathematical reconstructions that could not possibly have been used by the Greeks, so as to show beyond question that they could not have been. And it probably pushes its point too far. That is what classicists do. They push a point for all it is worth and perhaps more. These flaws are inevitable when you work on such important questions on so little evidence. Fowler assembles enormous amounts of classical textual evidence and later scholarship. He gives some nice mathematics including an appendix on the later arithmetized incarnation of anthyphairetic methods as continued fractions.

If you are determined to ask what math Plato knew and promoted, and what existed before Euclid--and so you are determined to break your heart--then you must read this book.

10 of 10 people found the following review helpful:

5.0 out of 5 stars A new landmark, January 24, 2002

By 

G. J. Argyros "gja1238" (Houston, Texas) - See all my reviews
(REAL NAME)   

This review is from: The Mathematics of Plato's Academy: A New Reconstruction (Hardcover)

"THE MATHEMATICS OF PLATO'S ACADEMY"
Second Edition
Fowler.

The first impression on receiving this book in your hands is the heavy weight. But this is not only true physically, due to the high quality of the cartridge paper, it is also true intellectually. Thus the second impression reinforces the first. The caliber of the scholarship exhibited in this tome is of the highest order, doing full justice to an investment in so expensive a paper.

Nothing less than the most complete exposition possible of ancient Greek mathematics as taught at the Platonic Academy in Athens, is presented, based on all currently available sources.

The author labors to guide the reader with diagrams, definitions, explanations, cross-references, commentaries and modern mathematical symbols to provide a clear, detailed and thorough account. He even starts from the photographic plates of Greek papyri. This is a major work of scholarship that itself deserves to become a classic; a model of its kind.

Just in case amazon readers accuse me of obsequious flattery, abject servility and distasteful onesidedness, allow me one criticism. The influence of the Ionian philosopher-mathematicians, Thales, Anaxagoras, Anaximander and Anaximenes on Plato's Academy is not covered.

A magnificent twenty-one page bibliography testifies to the author's detailed background research, and whets the reader's appetite for further reading.

Finally, three separate indexes show that the author is making every effort to help his reader as much as he can. Could one ask for more ?

3 of 4 people found the following review helpful:

4.0 out of 5 stars Why Books II and X of the Elements look like they do, July 10, 2009

By 

Viktor Blasjo - See all my reviews
(REAL NAME)   

This review is from: The Mathematics of Plato's Academy: A New Reconstruction (Hardcover)

This book claims that a major concern in ancient days was "anthyparesis," which is basically the Euclidean algorithm (though with subtraction in place of division and some other minor corrections for anachronisms). Given two quantities the anthyparesis algorithm gives as an output of series of numbers, namely the numbers corresponding to "how many times the smaller goes into the larger" reiterated with the remainder as the new "smaller." "On the one hand, there is the spectacular success of the anthypaireic exploration of the ratios of sides of squares sqrt(n):sqrt(m) in which a wide range of arithmetical and geometrical techniques are bound together into some apparently coherent whole. [See esp. ch. 3. Such ratios will always produce sequences that are periodic according to a general pattern; p. 75.] On the other hand, there is an equally spectacular lack of success in the search for any similar regular, predictable anthypaireic behaviour of any other geometrical ratios, apart from the extreme and mean ratio [which is the simples case anthypaireically: the case which produces periodic 1s; see section 3.5(b)]. Such a contrast might lead to the growth of an attitude in which these sides of squares come to be regarded as the basic underlying understandable geometric objects in terms of which everything else should be described." (pp. 166-167). It is proposed that the otherwise obscure Books II and X of Euclid's Elements be understood as partial attempts at these problems. There is no "smoking gun" evidence; the case is based primarily on showing that some of Euclid's proposition are reasonably well-suited to a research programme along these lines. Incidental evidence includes Euclid's focus on two dimensions ("he nowhere shows any interest in giving any similar results for cubes or other three-dimensional figures" (p. 190), even though this seems very obvious too us, e.g. a formula for (x+y)^3) and the lack of any evidence for the alleged "crisis" as a result of the discovery of incommensurable magnitudes (cf. 304).