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Unknown Quantity: A Real and Imaginary History of Algebra
Perfect for history buffs and armchair algebra experts, Unknown Quantity tells the story of the development of abstract mathematical thought.
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 historical and mathematical journey.
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 historical and mathematical journey.
374 pages, Paperback
First published May 2, 2006
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1885 people want to read
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Displaying 1 - 30 of 61 reviews
March 18, 2010
Though Derbyshire is a dimwitted douchenozzle on many, many subjects, he managed to write a decent book on algebra.
My original review—before I realised this John Derbyshire was also John Derbyshire, the racist/homophobe/theotard/hypocrite/all-round dipshit who writes for the National Review—was going to mention how the book takes a naïve attitude towards history that's refreshing in this age of nuance and relative rigor (something that's only remotely acceptable because the book isn't about, and doesn't try to make points based on, history), but that feels a bit wry when you realise it's not the delightful innocence of professional deformation that's at the base of it, but plain old right-wing ignorance. The handful of offhand references to race, religion, and antisemitism also take on a more bitter tone; as does the general eurocentrism (though of course he can't get away entirely without mentioning the person algebra was named after, he does manage to avoid talking about any of his work).
On the other hand, at least it explains his accusation of "sour anti-Americanism" directed at the French.
In the end, the only thing I can say is that while as popular mathematics books go, Unknown Quantity is by no means a bad or uninteresting book, I also can't recommend that anyone spend money on it, because part of it will go to a bigoted blowhard. If you can find it in a library, by all means go for it, but otherwise, stick to people like Paulos and Stewart. Their books are easily as good, and they managed to write them without being retrograde scumbags.
My original review—before I realised this John Derbyshire was also John Derbyshire, the racist/homophobe/theotard/hypocrite/all-round dipshit who writes for the National Review—was going to mention how the book takes a naïve attitude towards history that's refreshing in this age of nuance and relative rigor (something that's only remotely acceptable because the book isn't about, and doesn't try to make points based on, history), but that feels a bit wry when you realise it's not the delightful innocence of professional deformation that's at the base of it, but plain old right-wing ignorance. The handful of offhand references to race, religion, and antisemitism also take on a more bitter tone; as does the general eurocentrism (though of course he can't get away entirely without mentioning the person algebra was named after, he does manage to avoid talking about any of his work).
On the other hand, at least it explains his accusation of "sour anti-Americanism" directed at the French.
In the end, the only thing I can say is that while as popular mathematics books go, Unknown Quantity is by no means a bad or uninteresting book, I also can't recommend that anyone spend money on it, because part of it will go to a bigoted blowhard. If you can find it in a library, by all means go for it, but otherwise, stick to people like Paulos and Stewart. Their books are easily as good, and they managed to write them without being retrograde scumbags.
December 28, 2007
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.
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.
August 5, 2017
According to Wikipedia, there are 13 branches of math, 6 of which fall into the realm of pure mathematics. Algebra is just one of these. So is there any merit in reading a book about history of this small corner of the mathematical universe? Of course, this is a large understatement on part of algebra – it definitely is not a “small corner” of math. On the contrary, it perhaps the largest and the most interconnected field of it. But it is still not whole of it, right? Why not go for some general math history book? Surely, there are better historians than John Derbyshire outside there?
These were the thoughts on my head when I picked up Unknown Quantity. Having finished the book, I have answer to these questions, all in favor of the infamous author, alongside with countless questions unanswered despite hours of enterprise. Let me begin by settling down one thing:
This book is by no means an alternative for math history. If you read it with this expectation in your mind, then you might not get much out of this book. It is a history of algebra. Not mathematics. Period.
Now about the aforementioned question. Is this book worth to read? It is, if you are interested in history of algebra or algebra or mathematics in general. This book introduced me to completely new fields of mathematics and taught me a great deal about their connections.
I am not quite sure who fascinated me more – Derbyshire or the topic itself. There is not much to say about algebra – it is truly…fascinating. Derbyshire is an excellent writer, too: witty and as simple as it gets, as if trying not to confuse readers who are already baffled by algebra.
These were the thoughts on my head when I picked up Unknown Quantity. Having finished the book, I have answer to these questions, all in favor of the infamous author, alongside with countless questions unanswered despite hours of enterprise. Let me begin by settling down one thing:
This book is by no means an alternative for math history. If you read it with this expectation in your mind, then you might not get much out of this book. It is a history of algebra. Not mathematics. Period.
Now about the aforementioned question. Is this book worth to read? It is, if you are interested in history of algebra or algebra or mathematics in general. This book introduced me to completely new fields of mathematics and taught me a great deal about their connections.
I am not quite sure who fascinated me more – Derbyshire or the topic itself. There is not much to say about algebra – it is truly…fascinating. Derbyshire is an excellent writer, too: witty and as simple as it gets, as if trying not to confuse readers who are already baffled by algebra.
June 21, 2014
As someone who has already been exposed to many, if not most, of the ideas in this book, I was hoping that it would be more interesting to me than the usual popular math book. "Unknown Quantity" definitely exceeded my expectations on this. Though there were a couple of parts that annoyed me (e.g., he defines a prime number in a ring as being one with no factors besides units and itself, which was especially bad because he was discussing a non-UFD, where "prime" and "irreducible" are not the same), there was still a lot of interesting math and interesting history to redeem it.
Other reviewers mentioned being unable to follow the latter third of the book. I definitely agree that this part of the book is the most abstract, although I was able to follow what he was saying, and it only inspired me to learn Algebriac Geometry!
Finally, I was slightly put off by some of his political comments, but they do not appear often enough to detract from the book.
Other reviewers mentioned being unable to follow the latter third of the book. I definitely agree that this part of the book is the most abstract, although I was able to follow what he was saying, and it only inspired me to learn Algebriac Geometry!
Finally, I was slightly put off by some of his political comments, but they do not appear often enough to detract from the book.
Currently reading
January 28, 2011"The story of algebra, is the story of civilization itself..."
I stumbled on this book somewhere at Amazon while searching for books to help me become a better TA in undergraduate Discrete Math class. The class is entirely in Japanese, so imagine studying sets and groups and lattices using symbols (read: kanji) you've never seen and had no clue on the reading and meaning.
I need a good English textbook to keep me sane, and being a fiction-lover, I certainly hope this book could lift my mood in the attempts to befriend Abstract Algebra - both in English, and in Japanese.
I stumbled on this book somewhere at Amazon while searching for books to help me become a better TA in undergraduate Discrete Math class. The class is entirely in Japanese, so imagine studying sets and groups and lattices using symbols (read: kanji) you've never seen and had no clue on the reading and meaning.
I need a good English textbook to keep me sane, and being a fiction-lover, I certainly hope this book could lift my mood in the attempts to befriend Abstract Algebra - both in English, and in Japanese.
December 9, 2015
This book is another good work of John Derbyshire;the history of algebra from the babilonians to our days making things understable for those with a background of high school,it makes understable concepts as the complex numbers ,vector spaces ,quaternions ,algebraic structures as rings and gives a very elemental introduction to galois theory and algebraic topology
June 7, 2024
a great book for lovers of math and gossip (me). this has been my plane read for the past 6 months and i finally finished otw to cali today.
November 15, 2010
Nella matematica che si fa a scuola la geometria quanto quanto è comprensibile: le figure almeno le si vede. L'analisi matematica, con derivate e integrali, è appannaggio di pochi (s)fortunati. Ma quello che probabilmente fa odiare a tutti la matematica sono le equazioni e i polinomi; quello che viene chiamato algebra. Un libro come questo, che racconta la storia dell'algebra partendo dai babilonesi per arrivare al ventunesimo secolo, potrebbe essere visto come il fumo negli occhi. Non è così, per fortuna. Il punto di vista di Derbyshire, che fa lo scrittore ma in fin dei conti è laureato in matematica, si può sintetizzare dicendo che l'algebra è il modo che la matematica ha per rendere astratte le cose concrete. Così le formule numeriche babilonesi ed egizie sono i primi esempi "algebrici", ancora legati a esempi assolutamente concreti: col passare dei secoli si è inizialmente riusciti a immaginare che ci possano essere delle incognite, cioè dei valori che non conosciamo ancora ma che possiamo trattare come numeri; dei coefficienti, degli enti che sono sì dei numeri ma non ci interessa quali siano; dei nuovi tipi di numeri, negativi e immaginari; fino ad arrivare alle strutture come matrici, gruppi, anelli che nascono da esempi concreti e poi si iniziano a studiare come enti per conto proprio da cui si può addirittura proseguire nell'astrazione.
Gli sviluppi della seconda metà del '900 sono almeno a mio parere incomprensibili e si possono tranquillamente saltare, ma il resto del libro è piacevole, e tra l'altro Derbyshire sembra farsi un punto d'onore a fare conoscere tutti i matematici che hanno fatto scoperte che poi sono state chiamate coi nomi di altri matematici. Questo significa che finalmente non sarete costretti a sorbirvi solo i soliti Abel e Galois: vi pare poco?
Gli sviluppi della seconda metà del '900 sono almeno a mio parere incomprensibili e si possono tranquillamente saltare, ma il resto del libro è piacevole, e tra l'altro Derbyshire sembra farsi un punto d'onore a fare conoscere tutti i matematici che hanno fatto scoperte che poi sono state chiamate coi nomi di altri matematici. Questo significa che finalmente non sarete costretti a sorbirvi solo i soliti Abel e Galois: vi pare poco?
May 10, 2012
I have written before about my propensity in high school to avoid being challenged in math: once I became intimidated by the work, by 8th or 9th grade, I took the easy way out, never challenged myself, and did altogether poorly in the subject. However, in college I had brilliant professors in the math classes I was required to take and they inspired me to take more than was required and instilled in me a love of the beauty of numbers, formulas, equations, etc. Algebra has always been my favorite branch of mathematics, I love the polynomials, the algebraic structures and the equations. All that being said, I still found this book difficult. A lot of it was like a refresher course in algebra, and I loved the narratives that explained the people, events, ambitions, and cultures that gave rise to algebra, indeed the history of it was the best part of the book, but there was a lot that was difficult, that I remember only vaguely from college, and while Derbyshire can on many levels write for someone who does not work with algebra every day, there was still a lot that was over my head. I have not read his other book, Prime Obsession, but I will try it. Also, this is just an aside, but it did not take me as long to get through Unknown Quantity as what is suggested by the dates here (although goodness knows it took long enough); I was behind in magazines by a month, so in between chapters of Derbyshire's book, I was catching up on The Nation, Martha Stewart Living and Alfred Hitchcock Mystery Magazine. :)
November 22, 2008
Unknown Quantity is an interesting book about the history of algebra, but I think its major failing is that it concentrates sufficiently heavily on the mathematics that it's hard to read sections if you aren't already knowledgeable about them. It claims to be aimed at the non-mathematician, but even as someone who has good knowledge of algebra, there were portions of the book (such as the topology sections) that I got very little out of because I wasn't already familiar with the particular branch of algebra being discussed. The Math Primer sections are helpful for reviewing subjects you've already been exposed to, but I doubt they would be sufficient for a non-mathematician to achieve sufficient understanding to understand the rest of the book (which makes sense, since something as intricate as Field Theory just isn't going to be explained in 10 pages).
January 1, 2022
As the title says the book is an account of development of Algebra starting with Babylonians and all the way to Grothendieck. As other reviewers have mentioned that the work suffers from eurocentrism (Regarding Indians, the author have to say that we discovered 0 and Sanskrit have words for every big number, just this) therefore I had to deduct one star. Except for this, the book is perfect for laymen and students of mathematics alike.
A History of Abstract Algebra also deals with the same topic but in a more concise and academic way, for those who want that type of presentation can choose to read that instead.
A History of Abstract Algebra also deals with the same topic but in a more concise and academic way, for those who want that type of presentation can choose to read that instead.
January 21, 2012
5/10
The book falls between the stools of "popular math" and math treatments but is not rigorous enough to satisfy those interested in the latter and loses those drawn to the former in splurges of (incomplete) equations and hard to follow
On the plus side the potted histories of the various mathematicians encountered are entertaining and the book is reasonably well written.
It did serve the purpose of illustrating the arcane geography of modern algebra but didn't make me interested in it. Somewhat disappointing, I felt that the author was writing for himself rather than the reader.
The book falls between the stools of "popular math" and math treatments but is not rigorous enough to satisfy those interested in the latter and loses those drawn to the former in splurges of (incomplete) equations and hard to follow
On the plus side the potted histories of the various mathematicians encountered are entertaining and the book is reasonably well written.
It did serve the purpose of illustrating the arcane geography of modern algebra but didn't make me interested in it. Somewhat disappointing, I felt that the author was writing for himself rather than the reader.
May 5, 2019
This book was excellent. It was recommended to me by a math professor who found out that I enjoy the history if mathematics as much as I enjoy the math itself. This book strikes a nice balance between explaining the material and maintaining a compelling narrative. Other math history books have learned a bit too far in one direction or another, but this one felt right. It does focus specifically on algebra rather than analysis or geometry, but those topics do appear from time to time.
January 24, 2010
I like the history of mathematics... [turns head away in shame, moist eyes brimming with hot tears of disgrace]
December 16, 2019
Amazing book! SO tight.
November 20, 2021
Another very good book on mathematics history by John Derbyshire that does a great job of explaining what algebra is, how it developed, and, in particular, opening a crack into the modern development of the subject as abstract algebra (group theory, fields, rings, etc.)
Despite having a vaguely mathematical degree myself, I quickly get lost when the interests of a modern mathematician are outlined - it's all so abstract and far from things that someone with just school mathematics can relate to. The author manages to introduce the subjects in such a way as to give the reader some insight to connect the abstract algebraic concepts with the more familiar.
The book is consistently entertainingly written with plenty of historical and biographical anecdote. He also allows his own personality to peek through more than most writers of popular science do, which adds to the enjoyment of reading the book. Concepts are illustrated with examples but generally the author does not make the mistake of trying to lead readers through actual mathematical proofs. This is probably a book for people with some pretensions to mathematical understanding rather than the truly mathematically ignorant, but I'd say that it doesn't require the reader to have proper mathematical education or training.
Despite having a vaguely mathematical degree myself, I quickly get lost when the interests of a modern mathematician are outlined - it's all so abstract and far from things that someone with just school mathematics can relate to. The author manages to introduce the subjects in such a way as to give the reader some insight to connect the abstract algebraic concepts with the more familiar.
The book is consistently entertainingly written with plenty of historical and biographical anecdote. He also allows his own personality to peek through more than most writers of popular science do, which adds to the enjoyment of reading the book. Concepts are illustrated with examples but generally the author does not make the mistake of trying to lead readers through actual mathematical proofs. This is probably a book for people with some pretensions to mathematical understanding rather than the truly mathematically ignorant, but I'd say that it doesn't require the reader to have proper mathematical education or training.
August 12, 2024
super interesting read! i thought the beginning was a little slow but that’s probably because the topics were basic (with respect to modern day math that we know about now) and there wasn’t much information on the mathematicians themselves, on account of it happening thousands of years ago and stuff was lost to time. things speed up quickly tho and there’s lots of interesting stuff happening (especially in the 19th century and on, altho i’m biased bc i love that area already). math is collaborative and builds upon itself and it was very cool to hear about the context of the mathematical world when these famous theorems were worked out!
some people might be upset at how he simplified some of the more “advanced” topics, but i think that simplification becomes necessary at a certain point if you want to appeal to people who haven’t studied math. besides, i think his reductions were good at explaining only what was necessary to understand the points he was making and what was relevant to the story. as someone who knows nothing about algebraic geometry, i’m sure this book wouldn’t be a good reference point for it, but it doesn’t claim to be! so long as you can leave pedantry aside when necessary, i think this is a cool book that portrays mathematicians as people (though very smart people who lived in worlds outside most of our experiences) and im always down to read about that :)
some people might be upset at how he simplified some of the more “advanced” topics, but i think that simplification becomes necessary at a certain point if you want to appeal to people who haven’t studied math. besides, i think his reductions were good at explaining only what was necessary to understand the points he was making and what was relevant to the story. as someone who knows nothing about algebraic geometry, i’m sure this book wouldn’t be a good reference point for it, but it doesn’t claim to be! so long as you can leave pedantry aside when necessary, i think this is a cool book that portrays mathematicians as people (though very smart people who lived in worlds outside most of our experiences) and im always down to read about that :)
December 29, 2018
I bought this book 10 years ago to the date with a Chapters gift card I received for my birthday. I've had enough math training in the interim to appreciate what this book tries to offer some perspective into, and thought it would be time to finally open up this dusty novel.
Ultimately this book does a good job for a popular mathematics book, a bit of history, a few interesting anecdotes, good overall approachable descriptions of some tricky concepts in algebra. But as good as the descriptions of some of the introductory material may be, it can all feel like a buildup to nothing when the climax and genius of proofs (like the Abel-Ruffini theorem) are only broadly referred to. If there's one thing this book does really well, it's in outlining the nonlinear and dynamic progress that a once-neglected branch of mathematics had through thousands of years of mathematical history.
Ultimately this book does a good job for a popular mathematics book, a bit of history, a few interesting anecdotes, good overall approachable descriptions of some tricky concepts in algebra. But as good as the descriptions of some of the introductory material may be, it can all feel like a buildup to nothing when the climax and genius of proofs (like the Abel-Ruffini theorem) are only broadly referred to. If there's one thing this book does really well, it's in outlining the nonlinear and dynamic progress that a once-neglected branch of mathematics had through thousands of years of mathematical history.
September 20, 2020
Unknow quantity gives an overview of algebra history and the main thinkers along the evolution of math. John Derbyshire as a mathematician showed his enthusiasm and knowledge at writing. It is very interesting to see how he tries to describe category group and other theories in a simpler way. But, the book is not for a non-math reader and it requires a math background like analysis and set theory knowledge. If you are akin to math, mainly analysis, set theory and other things, I think its a good book for a history of algebra and its envolvement.
April 11, 2021
Interesting and comprehensible, despite the complex algebraic concepts of the book. My one issue with this book is how frequently the author uses phrases like "about whom I shall say more later" and "which I am going to describe in just a moment." The book is full of these promises, and it often feels as if the author goes on tangents and sometimes fails to circle back to the original topic. Some sections feel disorganized and incomplete. However, overall it was a pleasant read, and it does a good enough job of capturing in only 300+ pages a very large field of math.
August 15, 2022
Interesting but hard, given how little math I know. I started reading this ten years ago when I was teaching Algebra 2 to some homeschoolers. Earlier this year I started over, and this time made it to the end. Derbyshire is funny and writes well, which livens the book up. I recommend his novel Seeing Calvin Coolidge in a Dream as well.
March 24, 2021
An interesting little history of Algebra, showing how the main streams emerged and connect in this huge branch of mathematics. It would have been helpful if I could have read this during my university days, to bolster my understanding of the "overall thrust of algebra". The text itself is quite math-heavy (yet never becomes a textbook) and the historical background of even key figures are not very much fleshed out, which makes this book a bit less interesting to the historian.
March 18, 2023
I have made a video review of this book in case you might be interested:
https://youtu.be/KRGwFbEwoJ8
https://youtu.be/KRGwFbEwoJ8
January 25, 2025
Excellent! Every bit as good as his prior book, Prime Obsession. Both are required reading.
October 10, 2015
Subtitle: A Real and Imaginary History of Algebra. This is, more or less, the story of how math got away from us. How it went from a way of counting clay casks of grain given as tribute in Mesopotamia, to a system for analyzing entities which have no physical existence, the nature of which cannot easily be explained, and the usefulness of which (while often, it is eventually discovered, quite substantial) is not apparent even to those who are working on it. It's basically the history of how math stopped being about the numbers.
This is a problem for an author such as Derbyshire who wants to tell the story of algebra. Contrast it with, say, a writer who wants to tell us the story of physics. If you explain to someone just how weird the consequences of the Theory of Relativity are, for example, you don't normally have to explain first what you're talking about. In the famous story of twins, one of whom gets in a spaceship that travels at near the speed of light, he won't first have to explain what twins are, or even what space flight is.
Derbyshire has a harder task here. In the first few chapters, certainly, he is able to refer to things which an educated reader is probably at least passably familiar with, for example quadratic equations. When we move past that to vectors and matrices, there are still some non-mathematicians who have at least a hazy idea of what that is. Then we are into things like groups, rings, spaces, even algebras, that have very different and specific meanings to mathematicians.
We also hear about people like Galois, who died in his early 20's in a duel (the reason for which is murky at best), Emmy Noether, who was the first woman mathematician to achieve prominence in Germany (perhaps the world), and Alexander Grothendieck, whose mathematical brilliance was rivaled only by his eccentricities, like filling a math lecture with pacifist rants or deciding to live entirely on dandelion soup. But the heart of the tale is how we, the species, piled one abstract idea on top of another, until we left the world of concrete objects not only far behind, but more or less out of sight entirely.
Derbyshire does a creditable job of walking us through all this, and when in the midst of (say) the chapter on rings, I can just convince myself that I know what we're talking about. Once I close the book and sleep on it, however, it becomes clear to me that I haven't any idea. I can still remember that, when some mathematicians were scandalized at the thought of Emmy Noether becoming a full math professor and teaching to male students, Hilbert (one of the biggest names in math at that time, and another teacher there), tried to silence the objections by observing, "after all we're a university, not a bathing establishment". When that didn't work, he just decided to teach the class himself, then paid Emmy Noether to teach in his place, and more or less dared the university establishment to object.
The problem with remembering the math, then, is not Derbyshire's writing (which is sparkling and easy to read when discussing the personalities and history involved), nor his enthusiasm for the topic (which is palpable), but just the basic reality that Math Is Hard. To really understand what rings are, I'd have to be doing some work on (and with) them myself. Someone would probably have to double-check my work, and point out to me when I had erred. Periodically through the book we would need to have a status check, to see if I had gotten everything up to this point.
And you know, if Derbyshire did write textbooks, I'd wager they'd be better than the ones I used in school. This book, however, does something else. It tells us a bit about how, even in the field of mathematics, the prejudices and predispositions of the human beings involved can take generations to overcome. Imaginary numbers, irrational numbers, even negative numbers were first used as a bit of a stopgap, just a convenient mental prop until we figure out how things actually work. Fast forward a generation or two, and it is accepted that that IS how the math works, irrational or imaginary as it may be. Math is still a human field, and overcoming a bias towards integral numbers took nearly as long as overcoming a bias towards men in the field. Derbyshire shows both aspects of this mental struggle, the mathematical and the social, and as a result we understand each one just a bit better.
This is a problem for an author such as Derbyshire who wants to tell the story of algebra. Contrast it with, say, a writer who wants to tell us the story of physics. If you explain to someone just how weird the consequences of the Theory of Relativity are, for example, you don't normally have to explain first what you're talking about. In the famous story of twins, one of whom gets in a spaceship that travels at near the speed of light, he won't first have to explain what twins are, or even what space flight is.
Derbyshire has a harder task here. In the first few chapters, certainly, he is able to refer to things which an educated reader is probably at least passably familiar with, for example quadratic equations. When we move past that to vectors and matrices, there are still some non-mathematicians who have at least a hazy idea of what that is. Then we are into things like groups, rings, spaces, even algebras, that have very different and specific meanings to mathematicians.
We also hear about people like Galois, who died in his early 20's in a duel (the reason for which is murky at best), Emmy Noether, who was the first woman mathematician to achieve prominence in Germany (perhaps the world), and Alexander Grothendieck, whose mathematical brilliance was rivaled only by his eccentricities, like filling a math lecture with pacifist rants or deciding to live entirely on dandelion soup. But the heart of the tale is how we, the species, piled one abstract idea on top of another, until we left the world of concrete objects not only far behind, but more or less out of sight entirely.
Derbyshire does a creditable job of walking us through all this, and when in the midst of (say) the chapter on rings, I can just convince myself that I know what we're talking about. Once I close the book and sleep on it, however, it becomes clear to me that I haven't any idea. I can still remember that, when some mathematicians were scandalized at the thought of Emmy Noether becoming a full math professor and teaching to male students, Hilbert (one of the biggest names in math at that time, and another teacher there), tried to silence the objections by observing, "after all we're a university, not a bathing establishment". When that didn't work, he just decided to teach the class himself, then paid Emmy Noether to teach in his place, and more or less dared the university establishment to object.
The problem with remembering the math, then, is not Derbyshire's writing (which is sparkling and easy to read when discussing the personalities and history involved), nor his enthusiasm for the topic (which is palpable), but just the basic reality that Math Is Hard. To really understand what rings are, I'd have to be doing some work on (and with) them myself. Someone would probably have to double-check my work, and point out to me when I had erred. Periodically through the book we would need to have a status check, to see if I had gotten everything up to this point.
And you know, if Derbyshire did write textbooks, I'd wager they'd be better than the ones I used in school. This book, however, does something else. It tells us a bit about how, even in the field of mathematics, the prejudices and predispositions of the human beings involved can take generations to overcome. Imaginary numbers, irrational numbers, even negative numbers were first used as a bit of a stopgap, just a convenient mental prop until we figure out how things actually work. Fast forward a generation or two, and it is accepted that that IS how the math works, irrational or imaginary as it may be. Math is still a human field, and overcoming a bias towards integral numbers took nearly as long as overcoming a bias towards men in the field. Derbyshire shows both aspects of this mental struggle, the mathematical and the social, and as a result we understand each one just a bit better.
June 30, 2023
From the point of view of a Non-Mathematician fellow who is interested in maths, I would pretty much not recommend this book to ‘learn’ math. At the start, this book clarifies that this book won’t contain a lot of equations, and will be focused on the history. Even though this is in part true, I really had a hard time trying to understand a lot of equations in this book, the explanation about them is pretty poor, and the history part is explained decently, although, not amazing. I really can not give this book any rate higher than 3.
November 17, 2008
I am reading a book on Maths, I am about to finish it, and in this book, I found a superb passage:
"I remain completely confident that the labor I have expended on the science presented here and which
hasd emanded a significant part of my life as well as the most strenuous application of my powers, will
not be lost. It is true that I am aware that the form which I have given the science is imperfect and must
be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if
this work should again remain unused for another seventeen years or even longer, without entering into
the actual development of science, still that time will come when it will be brought forth from the dust of
oblivion and when ideas now dormant will bring forth fruit. I know that if I also fail to gather around me
(as I have until now desired in vain) a circle of scholars, whom I could fructify with these ideas, and
whom I could stimulate to develop and enrich them further, yet there will come a time when these ideas,
perhaps in a new form, will rise anew and will enter into a living communication with contemporary
developments. For truth is eternal and divine."
Stirring words.
Hermann Gunther Grassman was a high-school teacher, who lived between the years 1810 and 1877. He
had studied theology and philology, and then studied maths on his own.
He published his breakthrough in a book, "The Theory of Linear Extensions...". That work laid the ground
of what eighty years later became known as the theory of vector spaces. He defined much of its basics,
and was instrumenta in the invention of the modern concept of "algebra".
But his work did not find recognition. The only review of it was written by Grassman himself, no one else
noticed him. He tried his best to promote the book, but Mobius, who read the book, described it as
unreadable, though he helped and praised Grassman. Cauchy, to whom he had sent his work for it to be
forwarded to Jean Claude Saint-Venant, who had developed similar ideas, failed to do it. Instead, six
years later Cauchy published a paper which could have been derived from Grassman's work. Grassman
complained, and a three man committee of enquiry was set up. Since one of them was Cauchy himself,
we can well know what the finding was. Hamilton praised Grassman's book, but promoted his own
method over its.
Eight years later, Grassman reprinted the book again, changing its language to make it more readable.
The quoted passage is taken from the preface he wrote to that edition. But still he went unnoticed.
Disillusioned, Grassman turned to philology, and translated Rig Veda into German: this work was
supported by a lengthy commentary, and was a massive 3000 page volume. He found recognition for this
achievement: The University of Tubingen awarded him a honorary doctorate.
Seventeen years later, in the year after Grassman's death, William Kingdon Clifford published a paper,
"Application of Grassman's Extensive Algebra". These Clifford algebras were applied in 20th century
theoretical physics. The modern theory of spinors is derived from them.
All this, I read in this book by John Derbyshire.
His words, "There will come a time when these ideas, perhaps in a new form, will rise anew and will
enter into a living communication with contemporary developments. For truth is eternal and divine.",
proved prophetic. His conviction in the value of his ideas, and the faith that truth, which is eternal and
divine, will come alive and illuminate the contemporary life of a later age, are moving in face of the
discouragement and defeat that he encountered.
For every one of us, these words should ring a message of inspirational tone: we might be neglected and
consigned to a dusty corner, but that should not deter us from pursuing our passion- truth, eternal and
divine, is certain to rise anew when the times calls for it.
October 19, 2024
Thoroughly enjoyed the early chapters, which covered the first developments of mathematics and algebraic applications, societies obsessed with large numbers, and the fascinating challenge of zero and negative numbers being accepted into the mathematical community. The latter 2/3 of the book descended into an unenjoyable experience that read like Wikipedia entries of obscure mathematicians interspersed with mathematical proofs.
March 4, 2011
Unless you already know what Nine Zulu Queens Ruled China has to do with anything and solve the occasional recreational quadratic equation (as I confess I have been known to do from time to time), this book may not be for you.
However, there are two approaches to this history of algebra. One is for those who are tickled to death with Edwin Abbott’s Flatland and know the significance of 1,1,2,3,5,8,13,21,34,55,89,144,233,377, . . . The other is for folks who would like to know which mathematician died at 27 in a duel and the Hibbert solution to the Noether problem (more on Noether later.)
John Derbyshire does a good job of dumbing down algebra to the level of the “curious non-mathematician” for whom the book is written. He includes little tutorials in the text, explaining numbers (NZQRC is a mnemonic to remember the types of numbers) and polynomials, differentiating between cubic and quartic equations and so on.
If he lost you with the natural numbers and integers do not give up hope. It is possible to read this book while ignoring (a+bi)+(c+di)=(a+c)+(b+d)i and any page with a cube root on it. You can read the book for the history and biography with which it is crammed, ranging from the amusing to the truly awesome.
Early mathematicians (and I mean Ur, Babylon, 38 centuries ago) were handicapped. Seriously, with no plus or minus sign, no zero or equal sign, math was a lengthy and complex undertaking. All math problems were word problems. But Derbyshire shows that they were doing what could legitimately be called algebra that long ago and recording it on cuneiform tablets.
Slowly, over the centuries, colorful characters added to our knowledge of algebraic theory, imported Arabic (actually Indian) numerals, devised the decimal point, and began practicing what every high school freshman knows as algebra. Fibonacci strung out those numbers, Fermat scribbled his theorem in the margin and promised to prove it later, and Newton’s “mighty claw scratched one great mark across the history of algebra.”
Derbyshire’s algebra lost me somewhere around the leap into the fourth dimension, but the history continued to fascinate. Near the end where I had no idea whatever what he was talking about I appreciated the beautiful illustrations of a stellated polyhedral, the ampersand curve, and a Calabi-Yau manifold.
The Noether problem? Emmy Noether was a first-class German mathematician with a doctorate from Erlangen. She was supervising graduate students at Gottingen, but how could a self-respecting university put a woman on the faculty in the second decade of the 20th century? “What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?”
To which David Hilbert, who judged mathematicians solely by their talent, responded: “I do not see that the sex of a candidate is an argument against her admission. . . . After all, we are a university, not a bathing establishment.“
“Hilbert’s solution to the Noether problem was characteristic: He announced lecture courses in his own name and then allowed Noether to give them.”
When the Nazis came to power Emmy no longer had a problem because she was a woman but rather because she was a Jew. She immigrated to the United States and taught briefly at Bryn Mawr.
2011 No 40 Coming soon: Sense and Sensibility, By a Lady
However, there are two approaches to this history of algebra. One is for those who are tickled to death with Edwin Abbott’s Flatland and know the significance of 1,1,2,3,5,8,13,21,34,55,89,144,233,377, . . . The other is for folks who would like to know which mathematician died at 27 in a duel and the Hibbert solution to the Noether problem (more on Noether later.)
John Derbyshire does a good job of dumbing down algebra to the level of the “curious non-mathematician” for whom the book is written. He includes little tutorials in the text, explaining numbers (NZQRC is a mnemonic to remember the types of numbers) and polynomials, differentiating between cubic and quartic equations and so on.
If he lost you with the natural numbers and integers do not give up hope. It is possible to read this book while ignoring (a+bi)+(c+di)=(a+c)+(b+d)i and any page with a cube root on it. You can read the book for the history and biography with which it is crammed, ranging from the amusing to the truly awesome.
Early mathematicians (and I mean Ur, Babylon, 38 centuries ago) were handicapped. Seriously, with no plus or minus sign, no zero or equal sign, math was a lengthy and complex undertaking. All math problems were word problems. But Derbyshire shows that they were doing what could legitimately be called algebra that long ago and recording it on cuneiform tablets.
Slowly, over the centuries, colorful characters added to our knowledge of algebraic theory, imported Arabic (actually Indian) numerals, devised the decimal point, and began practicing what every high school freshman knows as algebra. Fibonacci strung out those numbers, Fermat scribbled his theorem in the margin and promised to prove it later, and Newton’s “mighty claw scratched one great mark across the history of algebra.”
Derbyshire’s algebra lost me somewhere around the leap into the fourth dimension, but the history continued to fascinate. Near the end where I had no idea whatever what he was talking about I appreciated the beautiful illustrations of a stellated polyhedral, the ampersand curve, and a Calabi-Yau manifold.
The Noether problem? Emmy Noether was a first-class German mathematician with a doctorate from Erlangen. She was supervising graduate students at Gottingen, but how could a self-respecting university put a woman on the faculty in the second decade of the 20th century? “What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?”
To which David Hilbert, who judged mathematicians solely by their talent, responded: “I do not see that the sex of a candidate is an argument against her admission. . . . After all, we are a university, not a bathing establishment.“
“Hilbert’s solution to the Noether problem was characteristic: He announced lecture courses in his own name and then allowed Noether to give them.”
When the Nazis came to power Emmy no longer had a problem because she was a woman but rather because she was a Jew. She immigrated to the United States and taught briefly at Bryn Mawr.
2011 No 40 Coming soon: Sense and Sensibility, By a Lady
March 28, 2013
I've read a number of "Math for the layman" books in recent years (including this author's Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, which I reviewed here a while back: http://www.goodreads.com/review/show/...). This one covers a number of topics and history that I've seen covered in many of those other books.
Surprisingly, I have found the history sections of these books often to be more interesting than the math sections -- I say surprisingly because I disliked history in high school. This book is no exception to that. In fact, Derbyshire seems to delve more deeply into some of the conventional stories (particularly about Galois, for example) and does not automatically follow the conventional, sometimes romanticized, stories.
The mathematical sections were reasonably well written, although towards the end of the book, the density of math became a little much for my taste. Others seem to have had similar experiences, based on some of the reviews here.
Surprisingly, I have found the history sections of these books often to be more interesting than the math sections -- I say surprisingly because I disliked history in high school. This book is no exception to that. In fact, Derbyshire seems to delve more deeply into some of the conventional stories (particularly about Galois, for example) and does not automatically follow the conventional, sometimes romanticized, stories.
The mathematical sections were reasonably well written, although towards the end of the book, the density of math became a little much for my taste. Others seem to have had similar experiences, based on some of the reviews here.
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