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Title: Unknown Quantity: A Real and Imaginary History of Algebra
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Manufacturer: Plume
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| Unknown Quantity: A Real and Imaginary History of Algebra by Plume Algebra - the deluge |
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. | | Unknown Quantity: A Real and Imaginary History of Algebra by Plume Huguenots | | Not a review, just a remark on endnote 51 of this fascinating book: Collins English Dictionary offers a plausible etymology of "Huguenot", a mixture of "Eidgenoss" (oath companion, i.e., Swiss) and "Hugues", 16th-century mayor of Geneva. | | Unknown Quantity: A Real and Imaginary History of Algebra by Plume Worthy attempt at a difficult task | There's an inherent difficulty in writing a book of this kind; a significant portion of the material that the author is expected to cover is simply out of the range of readers that lack an extensive background in mathematics. It is, in fact, worse than physics, in which metaphors can be used to give the reader some inkling of what's going on, even if they don't completely understand the reasons behind it. That being said, Derbyshire does a worthy job at a devilishly difficult task.
The first half is a sparklingly written account of the early history of algebra going back to ancient times. In the second half the author starts to get into territories that many readers will have trouble following, and finally in the chapter on Alexander Grothendieck, gives up entirely on explaining the math, and sticks to the personal story of its creator. Some of these slower parts might have been enlivened by the stories of the mathematicians themselves, but with a few exceptions, mathematicians tend not to live scintillating lives outside of their work. Still, aside from some more abstruse portions of the latter half, Unknown Quantity should provide fascinating reading for most educated readers. | | Unknown Quantity: A Real and Imaginary History of Algebra by Plume Great book | | Great reading. This guy does know how to write good math books! As good as "Prime Obsession"? Well, that one is a classic. This one is just a teeny shade below. I look forward to more books written by this author, I love them! | | Unknown Quantity: A Real and Imaginary History of Algebra by Plume Where is algebra going? | "Unknown Quantity: A Real and Imaginary History of Algebra" is about the history of algebra. Generally speaking, the book is divided into three parts. They are "Part I: ... the adoption of ... symbolism--letters representing numbers--around the year 1600 ... Part II: the ... victories of ... symbolism and the ... detachment of symbols from ... arithmetic and geometry ... the discovery of new mathematical objects ... Part III: modern algebra--... placing ... the new mathematical objects on a firm logical foundation and the ascent to ever higher levels of abstraction."
One of the most distinctive features of the book is that the book has six math primers. The math primers are (1) numbers and polynomials, (2) cubic and quartic equation, (3) roots of unity, (4) vector spaces and algebras, (5) field theory, and (6) algebraic geometry. The first five primers are related to Galois Theory, which is a landmark of the history of algebra. Many new mathematical objects such as group, ring, and field are invented through the search of a general formula for the roots of a polynomial with order greater than or equal to five.
The author introduces algebra many times. " ... the complex numbers ... 'an algebra' ... quaternions can only be made to work as an algebra ... if you abandon this rule [commutative rule] ... eight-dimensional algebra, a system of numbers called octonious ... To make these work [as an algebra] ... abandon ... the associative rule ... for multiplication ... A vector space with this additional feature--that two vectors can not only be added but also [be] multiplied, giving another vector as the result--is called an algebra."
In addition, the author introduces many new concepts, theories, and conjectures: Poincare Conjecture, the Birch and Swinnerton-Dyer conjecture, homotopy group, homology groups, cohomology, manifold, category theory, and Lie group. The new branches of mathematics such as algebraic geometry, algebraic topology, and algebraic number are introduced, which are the products of utilizing the new mathematical objects on geometry, topology, and number theory respectively. One of the most interesting algebraic topology theorem is introduced: " ... Brouwer's Fixed-Point Theorem (Any continuous mapping of an n-ball into itself has a fixed point.) ... For instance: Stir the coffee in your cup smoothly and carefully ... one ... point of the coffee ... will end up exactly where it started."
Other than algebra, the book provides a brief account for the major revolutions in the 19th century geometry. The five major revolutions are (1) Projective Geometry, (2) Non-Euclidean Geometry, (3) Riemann Surface ("replacement for the ordinary complex plane when investigating certain kinds of functions"), (4) Differential Geometry, and (5) "Group-ification" of Geometry. Several "new" concepts are introduced. They are (a) variety, (b) Erlangen Program, and (c) transformation group.
On the other hand, John Derbyshire covers the dilemmas of a mathematician. L.E.J. Brouwer (1881-1966) is considered as Poincare's most important successor in algebraic topology. He developed "... a doctrine called intuitionism, which sought to root all of math in the human activity of thinking sequential thoughts ... Leopold Kronecker objected bitterly on grounds ... call intuitionist ... that uncountable sets like R do not belong in math--that math can be developed without them ... and that mathematics should be rooted in counting, algorithms, and computation. ... Brouwer's version was called "intuitionism," Bishop's "constructivism." It is ... "constructivism" that these ideas are known nowadays ... leading exponent in the United States is Professor Harold Edwards ... Professor Edwards argues that, with easy access to powerful computers, constructivism is now coming into its own and that ... much of the mathematics done from 1880 onward will come to seem misconceived. ... his [Brouwer's] most important research contributions were in topology, Brouwer never gave courses on topology, but always on ... intuitionism. It seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism ... " Interested readers on what mathematics should be rooted shall refer to professor Edward's 2004 book, Essays in Constructive Mathematics, which "illustrates the approach very well."
| | Unknown Quantity: A Real and Imaginary History of Algebra by Plume Product Description | | For curious nonmathematicians and armchair algebra buffs, John Derbyshire discovers the story behind the formulae, roots, and radicals. As he did so masterfully in Prime Obsession, Derbyshire brings the evolution of mathematical thinking to dramatic life by focusing on the key historical players. Unknown Quantity begins in the time of Abraham and Isaac and moves from Abel’s proof to the higher levels of abstraction developed by Galois through modern-day advances. Derbyshire explains how a simple turn of thought from “this plus this equals this” to “this plus what equals this?” gave birth to a whole new way of perceiving the world. With a historian’s narrative authority and a beloved teacher’s clarity and passion, Derbyshire leads readers on an intellectually satisfying and pleasantly challenging journey through the development of abstract mathematical thought. |
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