» Unknown Quantity: A Real and Imaginary History of Algebra

Unknown Quantity: A Real and Imaginary History of Algebra Reviews

Customer Rating:
Summary: a better kind of popularization
Comment: I would consider this a popularization but probably not an overpopularization. By this I mean, somebody with little mathematical background will be lost pretty quickly. For a person with a fair amount of mathematics or even a degree in mathematics (in a different field per se) this is a good introduction into (modern) algebra from a historical approach.

So yes, there is a lot of math in this book and the primers might help a little but by the time I got to the end, I kept on forgetting what terms meant and found myself looking back to see what a variety is for instance. To get the most out of this book, it probably wouldn't hurt to have a pen and pad nearby to work out some of the material yourself.

I enjoy these higher level popularizations which are starting to become more common.
I was always annoyed when authors rigidly shunned having more than one equation in their book (see physics popularizations).

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Customer Rating:
Summary: fascinating
Comment: The histoey of algebra,as told in this book, is full of very interestig biographies of famous matthematicians.But what is even more interesting is the history of development of human pure thougt."Cogito ergo sum".

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Customer Rating:
Summary: Historical Context
Comment: So many reviewers have written wonderful reviews -- I'll just add that I really appreciate the way that the author has woven a general historical context into the history of algebra. The author also doesn't hesitate, at times, from expressing opinions that may not reflect the main-stream thinking on a subject. Though the main focus of the book is the development of algebra, it is interesting to read how social, political, economic, and religious forces might have impacted the philosophers and mathematicians in their quest for algebra. For example, until reading this book, I didn't appreciate that modern algebraic notation wasn't even fully developed until roughly the time of Descartes.

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Customer Rating:
Summary: Fascinating History of Algebra
Comment: Fascinating History of Algebra

"Unknown Quantity: A Real and Imaginary History of Algebra" by John Derbyshire

Readers who enjoyed "Prime Obsession" will find "Unknown Quantity" irresistible. In this very readable text John Derbyshire covers the broad history of modern algebra. The history starts four thousand years ago in Egypt and Mesopotamia.

The author tells the lives of the men and women who created modern algebra. Their stories are fascinating.

The people who make up the history of algebra include (from the photographic plates after page 184):
01 - Otto Neugebauer - found algebra in old Babylonian tablets
02 - Hypatia
03 - Omar Khayyam - wrote poetry and tackled the cubic equation
04 - Girolamo Cardano - found a general solution for the cubic
05 - Francois Viete - separated things sought from things given
06 - Rene Descartes - algebrized geometry
07 - Sir Isaac Newton - saw symmetry in solutions
08 - Gottfried von Leibniz - found relief for his imagination
09 - Joseph-Louis Lagrange - carried symmetry forward
10 - Paulo Ruffini - believed the quintic was unsolvable
11 - Augustin-Louis Cauchy - made an "arithmetic" of permutations
12 - Niels Abel - proved Ruffini right
13 - Evariste Galois - found permutation groups in equations
14 - Arthur Cayley - abstracted the group idea
15 - Ludwig Sylow - delved into the structure of finite groups
16 - Camille Jordan - wrote the first book on groups
17 - Sir William R. Hamilton - found a new algebra
18 - Herman Grassman - explored vector spaces
19 - Bernard Riemann - launched two geometric revolutions
20 - Edwin A. Abbot - took us to Flatland
21 - Julius Plucker - based his geometry on lines not points
22 - Sophus Lie - mastered continuous groups
23 - Felix Klein - mastered the group-ification of geometry
24 - Henri Poincare - algebraized topology
25 - Eduard Kummer - used algebra on Fermat's Last Theorem
26 - Richard Dedikind - discovered ideals
27 - David Hilbert - a geometry of tables, chairs and beer mugs
28 - Emmy Noether - pulled it all together
29 - Solomon Lefschetz - harpooned a whale
30 - Oscar Zariski - refounded algebraic geometry
31 - Saunders Mac Lane - attained a higher level of abstraction
32 - Alexander Grothendieck: - as if summoned from the void

Just as before, the author takes a field of mathematics interesting for expert and layman alike. This is a very fresh perspective on the history of algebra.

See Also:
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

I thoroughly enjoyed and highly recommend "Unknown Quantity."

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Customer Rating:
Summary: Algebra - the deluge
Comment:
It is a brave author (to say nothing of the publisher) who could summon the audacity to write a popular book on the history of algebra, with real mathematics thrown in. This is especially true if describing algebra's modern developments. But the author is John Derbyshire, and he has shown excellent form in this area, evidenced in his book, "Prime Obsession".

Once more, Derbyshire negotiates the perilous middle-way of conveying the beauty of mathematical ideas by using the mathenmatics but always engaging the reader's interest. Exactly how much interest depends of course on who we take the reader to be. Those with, or about to acquire, an undergraduate background in applied or engineering mathematics will enjoy this book the most. They will be best placed to follow the theoretical explanations and learn quite a lot of new things about pure mathematics along the way. I found the historical account of permutations in the solvability of polynomial equations, of great help in understanding their appearance in Galois' theory.

Derbyshire's historical accounts of developments in algebra also convey a "tip of the iceberg" impression. He sets the developments against the social and political realities of the era in which they arose, and appears to have acquired a great deal of history scholarship in order to do so accurately and credibly. To his credit, he does not burden the accounts with excessive background detail, although one suspects he had much more information at his disposal if he had wanted to.

I wish I could believe that mathematical virgins will completely appreciate this book, as Derbyshire does lighten the topic quite well with gentle humour and the human stories behind the ideas. However, I believe the middle of the book would, sadly, be lost on them. The book tends to move across theoretical territory at a pace that presupposes a fair mathematical education and a certain quickness of wit. (The daunting phrase, "just by noticing the following simple algebraic fact....." appears on page 58. See what follows and judge for youself, dear reader!) Even so, a complete explanation of topics, such as Galois' theory of the solvability of polynomials, realistically, cannot be given in a book such as this and in that instance, it is not. Readers who enjoyed "Prime Obsession" might be disappointed in this respect, as that book dealt more completely with its subject using a smaller and more accessible mathematical knowledge base. However, Derbyshire has a way of summarising matters so clearly, one could probably understand more detailed explanations given elsewhere.

Inevitably, the humour, the history and the human face of mathematics predominate over the mathematics itself as the story swings into the 20th century. Nevertheless, Derbyshire provides a very useful overview of modern topics and some understanding of their inter-relationship. This is no mean feat when describing such exotica as algebraic geometry, homotopy, homology, cohomology, varieties, category theory,.... and this list, we are informed, is by no means exhaustive. Derbyshire also confirms their importance to modern theoretical physics. I found these two aspects of the book the most stimulating, as they provided a good guide to pursuing the topics further (My copy of "Algebra" by Saunders Mac Lane is already on order).

Overall, this is a book I will be glad to keep in my library. If I have any (small) criticism to make, I wish the theoretical primer sections had not been scattered amongst the chapters of the main story, but instead placed in an Appendix, as their placement tended to interrupt the enjoyable flow of the narrative.

It was startling to realise that most undergraduate mathematics, particularly as taught to applied physicists and engineers like me, dates back, at latest, to the 19th century. It is going to take a while to catch up.

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