Greek Mathematical Thought and the Origin of Algebra

by Jacob Klein, Eva Brann (Translator)

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.

Genres: Mathematics, Philosophy, History, Stem, Nonfiction, Science, Physics

384 pages, Paperback

First published June 15, 1968

Reviews

[Pater Edmund · Rating 5 out of 5]

Klein's book on the origin of algebra is about the great change in conceptualization that was achieved by Descartes's transformation of mathematics from a contemplative looking at form, to a method for manipulating symbols. Cartesian symbols are something unknown in Greek mathematics--they do not "intend" any concrete object, but are "indeterminate" quantities that are treated as determinate objects. Klein expresses this by saying that they are second intentions treated as first intentions, but Sean Collins has argued that they can better be understood as the kind of intentionality that the mind has when it is not apprehending the order of being, nor of its own act, nor of the moral acts of the will, but rather the order that it produces in external things. "Symbolic representation... signifies that which has existence through the very act of symbolizing."

[Eliot · Rating 5 out of 5]

This is one of the most carefully argued books I have read, honestly, ever. It ranks with the greats of philosophy and is an important book for all who wish to fully understand the power of ideas to shape our whole society and its trajectory.

[Arturo · Rating 5 out of 5]

Don't be misguided by the title, this is the Essential book for anyone who writes and researches about Modernity, the history of the West and the genesis of the scientific method, just like for anyone (who perhaps need it the most) criticizing the views that go along with those. A towering scientific, intellectual and philological effort of a rather neglected author, who instead should be introduced in higher education programs. It is also a difficult read, but who cares?!

[Alfia]

In this somewhat overwrought but certainly thought-provoking tome, philosopher Jacob Klein attempts to map the influence of Greek philosophers and mathematicians onto the development of modern algebra in 16th and 17th century Europe. The work examines when and how the concept of number went from being merely a determinate "assemblage of monads" (kind of like conceptual tally marks) to a symbolic abstraction that can be indeterminate and extended into infinity. Klein presents abundant evidence of varying approaches to number and notation from the Greek tradition. He then attempts to demonstrate how these coalesced in the algebraic notation of Francois Vieta, as well as the conceptual leaps of Simon Stevin, Rene Descartes, and John Wallis.

A key part of the book is the influence of 3rd c. mathematician Diophantus on Francois Vieta (1540-1603), who used letters for variables, and who articulated rules for performing operations on different geometric degrees, what I think we'd now call exponents. I'm not gonna lie that I got lost in all the handwaving about how D's use of M for myriad and some ways he presented problems "algebraically" was so revolutionary. I had nothing to compare it with in the work of the people who brought the term "algebra" to the west, as this was out of scope of this book. And yet..

There was repeated mention of the "barbaric" algebra from the "Arabs" that was abused throughout the work, especially in contexts where anyone was making a case for their advances hailing actually from Greek sources. I suppose it would make sense not to focus too much on "Arab" philosophy or notation in a book focusing on the Greek contribution. But wouldn't it make sense to at least present it in some form other than simply mentioning the significant shift away from word or tally-mark notation, or at the very least to address the socio-political influences that would have motivated such apparent (and possibly lip-service-level) antipathy towards non-European methods?

I was told that this is a notoriously difficult read, but now that I have read it (in the company of several others more-philosophy-adept than myself), I have to say that there are some modifications that could be made to it that would render it considerably less daunting.

1) Relegate original quoted text to footnotes. Sometimes half a page is taken up with quotes in the original Greek, Latin, French, German, or Italian. I'm a polyglot, but this is annoying when trying to grasp the train of a thus strangulated argument.

2) Remove unnecessary bombast from an updated translation. Readers should decide for themselves whether a point being made is credible without the incessant larding of phrases like "most certainly", "without a doubt", and "it should be obvious". Likewise abstruse sentence construction stating what something is *not* in order to claim what it is, reeks not at all of trying to make the author's arguments accessible.

3) Include more concrete examples of the Diophantine problems. Referencing "the cattle problem" to make a point and then not even including it in the footnotes isn't helpful.

4) Shockingly, this book has no bibliography or list of references at the end.

Perhaps proving I am indeed a base member of the hoi polloi, it irked me that Klein chose to uncritically foreground the belief that the marvels of algebra, and consequently calculus, should be kept as arcane sciences, far from the masses who would ostensibly abuse them. There are passages devoted to latter-day scientists claiming that the Greek ancients actually *did* have algebra; they just didn't want to reveal its secrets lest the wretched masses get a hold of them. This was given as the reason behind there not being real evidence of a developed algebra being present among the ancients, and behind Newton (and others) allegedly choosing to present their findings in the most obscure way possible. Wtf. This did not pair admirably with Klein's style of presentation, and certainly does nothing to advance the cause of greater access to advanced mathematics.

All this said, there are plenty of fascinating factoids about the history of ancient Greek and 16th-17th century European math, as well as thought-provoking propositions about the nature of how we conceptualize number, geometric progression (and thus dimension/exponents), mathematical operations, and more. I just wish reading it didn't feel like being hazed. ;^}

[Will Spohn · Rating 5 out of 5]

Way over my head, though I do think I have gained from reading it. The central issue is the reinterpretation of arithmos and how greek mathematics differs decisively, ontologically, one may say, from modern mathematics. The first part of the book is dedicated to a recovery of greek mathematics as it understood itself, and in this regard it is extremely useful toward understanding Plato; it will prove beneficial to return in that regard. I would consider the book of undeniable importance for the first part a alone. The second part focuses on the modern shift, and this part was much more difficult in that I simply didn’t have the mathematical knowledge to form much of a grasp at all. Still, there were brief moments of some insight. All in all, extremely challenging and, in the same measure, masterful.

[Steve Wilkerson · Rating 5 out of 5]

Difficult, as deep scholarship often is, this book is worth the effort. Klein or his editors, have inserted lengthy original language into the text itself which disrupts the reader's attention, assuring it will never be a bestseller, but it is fascinating if one gives it a go. I read to the end, well beyond what spurred my initial point of interest.

[Gabe · Rating 4 out of 5]

Can't say I've grapsed all of it, it'll take some time to digest. Very thorough if much drier than Klein's writing normally is.

[Kajia · Rating 4 out of 5]

Klein takes us on a (very complex) journey from the Pythagorean and Platonic concept of "number" as having separate Being beyond sense perception, through Aristotle's "number" as the product of a reduction performed in thought, to Descartes' "number" as a wholly symbolic concept of the "pure intellect" that has no relation at all to the world or things in it. In the words of Husserl, modern mathematical science "takes for true being what is actually a method."

The only reason I couldn't give this a 5 is because Klein's run-on sentences (he was German, after all) and insistence on using terms in the various authors' original languages were very distracting and made this book an absolute struggle. It took me exactly one year to read, go figure. It was worth it.

[Eric · Rating 5 out of 5]

Abstruse but immensely rewarding, this dealt with questions about number and representing them clearly and definitively. Algebra is NOT merely shorthand for geometry as I had long thought.
This is a great book on the history of science that should be required reading for math teachers.

[Andrea Valente · Rating 2 out of 5]

Very heavy book. I was looking for a book on the history of math, which is a subject I usually enjoy, but this one is waaaay too philosophical... not my cup of tea :D

[Aaron]

I have to say, I'm scared to start this book...