Greek Mathematical Thought and the Origin of Algebra

by Jacob Klein

Overview

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

Product Details

ISBN-13:	9780486272894
Publisher:	Dover Publications
Publication date:	09/11/1992
Series:	Dover Books on Mathematics
Edition description:	REPRINT
Pages:	384
Product dimensions:	5.50(w) x 8.50(h) x (d)
Age Range:	18 Years

Table of Contents

Author's note
Translator's note
Short titles frequently used
Part I
1. Introduction: Purpose and plan of the inquiry.
2. The opposition of logistic and arithmetric in the Neoplatonists.
3. Logistic and arithmetic in Plato.
4. "The role of the theory of proportions in Nicomachus, Theon, and Domninus."
5. Theoretical logistic and the problem of fractions.
6. The concept of arithmos.
7. The ontological conception of the arithmoi in Plato.
A. The science of the Pythagoreans.
B. Mathematics in Plato-logistike and dianoia.
C. The arithmos eidetikos.
8. The Aristotelian critique and the possibility of a theoretical logistic.
Part II
9. On the difference between ancient and modern conceptualization.
10. The Arithmetic of Diophantus as theoretical logistic. The concept of eidos in Diophantus.
11. The formalism of Vieta and the transformation of the arithmos concept.
A. The life of Vieta and the general characteristics of his work.
B. Vieta's point of departure: the concept of synthetic apodeixis in Pappus and in Diophantus.
C. The reinterpretation of the Diophantine procedure by Vieta:
I. "The procedure for solutions "in the indeterminate form" as an analogue to geometric analysis"
2. "The generalization of the eidos concept and its transformation into the "symbolic" concept of the species."
3. The reinterpretation of the katholou pragmateia as Mathesis Universalis in the sense of ars analytice.
12. "The concept of "number."
A. In Stevin.
B. In Descartes.
C. In Wallis.
Notes
"Part I, Notes 1-125"
"Part II, Notes 126-348"
"Introduction to the Analytical Art, by Francois Viète (Vieta)."
Letter to Princess Mélusine.
I. On the definition and division of analysis and those things which are of use to zetetics.
II. On the stipulations governing equations and proportions.
III. Concerning the law of homogeneity and the degrees and genera of the magnitudes that are compared.
IV. On the precepts of the reckoning by species.
V. Concerning the laws of zetetics.
VI. Concerning the investigation of theorems by means of the poristic art.
VII. Concerning the function of the rehetic art.
VIII. The symbolism in equations and the epilogue to the art.
Index of names
Index of topics

What People are Saying About This

From the Publisher

"Historians of ideas will be reminded of numerous struggles in other fields paralleling these attempts to overcome or accommodate vestiges of the traditional but no longer useful concept of the number. The thoroughness of this work will ensure that it will remain a standard reference for a long time to come." British Journal for the Philosophy of Science

The MIT Press

"It is to be hoped that someone before long will make searching a study of Arabic algoristic impulsion toward modern algebra as Klein has made of Greek philosophical stimuli.

Historians... will be grateful... for the very welcome inclusion of a long-desired English translation of Vieta's Introduction to the Analytical Art." Science

The MIT Press

Editorial Reviews

It is to be hoped that someone before long will make searching a study of Arabic algoristic impulsion toward modern algebra as Klein has made of Greek philosophical stimuli. Historians... will be grateful... for the very welcome inclusion of a long-desired English translation of Vieta's Introduction to the Analytical Art.

—Science

From the Publisher

It is to be hoped that someone before long will make searching a study of Arabic algoristic impulsion toward modern algebra as Klein has made of Greek philosophical stimuli. Historians... will be grateful... for the very welcome inclusion of a long-desired English translation of Vieta's Introduction to the Analytical Art.

Science

Historians of ideas will be reminded of numerous struggles in other fields paralleling these attempts to overcome or accommodate vestiges of the traditional but no longer useful concept of the number. The thoroughness of this work will ensure that it will remain a standard reference for a long time to come.

British Journal for the Philosophy of Science