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Jacob Klein: Greek Mathematical Thought and the Origin of Algebra

Lecturer: David Sweeney

Originally Taught: Summer School 2011

Jacob Klein's great work Greek Mathematical Thought and the Origin of Algebra (1934-1936) is one of the major works in the history of mathematics in particular and the history of ideas in general. This text puts modern science in its place historically and conceptually.

We must start, Klein argues, by understanding what number and mathematics were for the Ancient Greeks. Thus the first half of the text, and the course, covers the mathematical concepts and questions of Plato and Aristotle, the neo Platonists, and the Pythagoreans.  Questions such as what are the numbers? what is the role of the unit? how are fractions to be understood? Through this exposition Klein gradually opens a window on Greek mathematical thought, its ideas, goals and limitations that reveal a conceptual world completely different from our own. A primary difference being that for the Greeks a number, or arithmos, was, or rather intents, a definite number of definite things. Thus "three sheep" or "five bowls". While for us a number now is, or intents, a symbol. Thus "3" and "5". This attention to the ancient perspective seems to be almost unique to Klein within the history of
mathematics.

How did this change come about? The second half of the text, and the course, studies the development of the concept of arithmos (number), through the Renaissance and up to the present day. The development is traced though the works of Diophantus, Vieta, Stevin, Descartes and Wallis. Here Klein shows how during the Renaissance the history of Greek mathematics was 'rewritten' at the same time as a new (symbolic) concept of number was developed.

The resulting understanding of the development of mathematics allows for a more mature critique of modern science. 

This text and course should be of interest to anyone who has interests in: the Greek conception of mathematics and how mathematical ideas were used by the Greek philosophers; the historical development of mathematics; the phenomenology of mathematics; the nature and meaning of modern science.

No prior knowledge of mathematics or the history involved is required.
The course will be focused on the text, with the time divided between explanations and discussion of the readings.

Recommended Reading
The 1961 translation of Klein's Greek Mathematical Thought and the Origin of Algebra is inexpensively available in various editions.

Difficulty

Intermediate