Customer Reviews

Mathematics and its History

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This is a brilliant book that conveys a beautiful, unified picture of mathematics. It is not an encyclopedic history, it is history for the sake of understanding mathematics. There is an idea behind every topic, every section makes a mathematical point, showing how the mathematical theories of today has grown inevitably from the natural problems studied by the masters of the past.

Math history textbooks of today are often enslaved by the modern curriculum, which means that they spend lots of time on the question of rigor in analysis and they feel obliged to deal with boring technicalities of the history of matrix theory and so on. This is of course the wrong way to study history. Instead, one of the great virtues of a history such as Stillwell's is that it studies mathematics the way mathematics wants to be studied, which gives a very healthy perspective on the modern customs. Again and again topics which are treated unnaturally in the usual courses are seen here in their proper setting. This makes this book a very valuable companion over the years.

Another flaw of many standard history textbooks is that they spend too much time on trivial things like elementary arithmetic, because they think it is good for aspiring teachers and, I think, because it is fashionable to deal with non-western civilisations. It gives an unsound picture of mathematics if Gauss receives as much attention as abacuses, and it makes these books useless for understanding any of the really interesting mathematics, say after 1800. Here Stillwell saves us again. The chapter on calculus is done by page 170, which is about a third of the book. A comparable point in the more mainstream book of Katz, for instance, is page 596 of my edition, which is more than two thirds into that book.

Petty details aside, the main point is the following: This is the single best book I have ever seen for truly understanding mathematics as a whole.

Stillwell covers a lot of ground in a short undergraduate text intended to unify various mathematical disciplines. Naturally, _Mathematics_and_its_History_ begins with the early Greeks and in particular geometry (which is how mathematics was typically expressed then). The development of algebra and polynomial forms is described followed by perspective geometry. The invention of calculus and the closely related discovery of infinite series provide the backdrop for short biographies of prominent mathematicians (mostly dead white males to multicultural deconstructionists). The development of elliptic integrals (used in solving functions with specified boundary conditions such as a Neumann problem found in fluid mechanics). The treatment then diverges to physical problems including the vibrating string and hydrodynamics, together with a note on the renown Bernoulli family. Then Stillwell returns to the esoteric in complex numbers, topology, group theory and logic with some comments on computation at the end. Some mathematicians may find the overview to lack comprehensiveness, but the book's brevity for each topic and biographical notes present a balanced approach to the more casual reader about this important field of study and how it developed.

It is a very good book. It has presented very clearly some difficult-to-understand relationship especially the link between algebra and geometry. It is a very good balance - history, Mathmatics, biography all mixed very well together. Highly recommended.

Every page is filled with fresh insights, genuine scholarships, clarity, connections, and understandings. Leaves all other textbooks on history of math in the dust. Never blindly follows the crowd of other authors to repeat after each other the muddled, and often untrue, interpretations and stories. Makes me want to have a photographic memory to take in everything in the book and use them to motivate and inspire my own teaching. Also makes me want to read many of the original sources Professor Stillwell's vast scholarship has traveled through.

It's a great page-turner and at the same time a fine wine to be sipped and appreciated sentence by sentence.

The book is based on one good idea - that by making mathematics the subject of mathemtics and not say geometry exclusively, or algebra exclusively, we can make connections and explain simple and complex ideas at the same time. This is probably it's biggest and best idea.

From there, I felt the books desire to be a technical history was contradictory. He'd say on the one hand that he doesn't want it to really be a history of mathematics book; but, then he makes short biographies at the end of each chapter. He sometimes introduces some more interesting mathematics in them; often, he uses them to make corrections on the history of certain mathematicians like Galois. I like the biography of Gregory myself. Mr Stillwell might want to see what E.T. Bell says about Isaac Newton; Isaac wasn't reclusive just to be a jerk; he was tired of the many intellectuals around him that couldn't handle new ideas or think rigorously. That's a biography more intellectuals today need to read, absorb, and kick their you know what themselves! Seems to me that Mr Stillwell takes this biographies at the end of chapters format from Coxeter's Regular Polytopes.

Coxeter's "Regular Polytopes" is a little bit of a good comparison. Like John Stillwell's "Mathematics and its History", Coxeter relates many fields of mathematics. Only, Coxeter packs in much more mathematics in less pages(even if John Stillwell covers mathematics from the ancient times up to some of the most recent developments . . . although not all major new developments). I mean John Stillwell makes 'some' good observations of ancient mathemtics, but if you've read Van Der Waerden's "Science Awakening"(and I'm sure John Stillwell did), you can see a real difference in 1) the quality of writing(people are talking this book "Mathematics and its History" as if it's a literary masterpiece) is I almost hate to say the word cheap; but, untill you read E.T. Bell's "The Development of Mathematics"(a book Stillwell curiously does not reference), or even Van Der Waerden anything, and come and say how John Stillwell is such a great writer; well, maybe those people havn't seen Coxeter's "Regular Polytopes" or Van Der Waerden's "Science Awakening." I stress this because reading through this book, that's what it felt like. I just know that anybody who's read anything on any given subject covered by Mr Stillwell must have some misgivings at the presentation. John Stillwell will argue he's trying to make some stuff accessable. But, Coxeter's "Regular Polytopes" shows you can make a book accessable to both advanced and beginning people to the subject at the same time! And John Stillwell's effort to make connections throughout all mathematics reminds me of E.T. Bell's "Development of Mathematics", but Mr Stillwell does not reference it. How many mathematics history books are there out there, and Mr Stillwell does not know of it? Sounds like somebody has some personal problems with E.T. Bell's effort! And if you want to argue that well E.T. Bell's Development is not nearly as technical as Stillwells; well, I beg to differ a little bit.

When reading through the book, there were times when I wondered about some of John Stillwell's logic. For instance, he starts out with Pythagorases theorem and says its the first real piece of mathematics; i thought "what about the quadratic equation". I'd do this over and over again(for instance Dedekind cuts), and he always pretty much made up for it as things developed. E.T. Bell in his development of mathematics book shows how the calculus brought the nature of number into question. E.T. Bell doesn't consider the tally bones of tens of thousands of years ago; but, then again, at the time of his writing, maybe the facts weren't known to him. Considering that one could introduce groups, rings, and fields, and vector spaces from deriving each higher number system(integers from natural numbers, rational from integers, and reals from rationals), why would John Stillwell first, either not know E.T. Bell's "Development of Mathematics", two, not reference it, and three not take what's positive from it and use the fact that one can relate higher mathematics like abstract algebras from the derivation of higher number systems from one another? Something is fishy for me here.

Another mystery to me in John Stillwell's effort to show mathematics is he never has anything to say about the fundamental nature of mathematics. I'm talking about abstraction, generalization, idealization. E.T. Bell in his "Development" book makes strong points about the remarkable nature of mathematics - that of abstraction and how it works in mathematics. How, in mathematics, previous mathematics gets absorbed into vaster abstractions and he shows beautiful of a phenomenon this is. Ignorabilises of today, then, and tomorrow hate mathematics because they don't understand how this works. They say well Newton was wrong. Newton wasn't wrong wrong; when Einstein created his theories of relativity, he showed how his theory generalizes the Newtonian theory and how Newton's theory can then subsequently can be derived from it. This is one of the remarkable things about modern mathematics. The Calculus of Newton and Liebniz absorbed, generalized and made more precise, trigonometry, logarithms, geometry, algebra. But, John Stillwells macho technical history was mute about the nature and origins of mathematics. He missed a great opportunity probably because of some personal problem with E.T. Bell. I don't know what it is, but I detect it. John Stillwell does touch on something; an idea that seems to me to be 'in the air', but I won't go into it right here right now.

But, like I hinted at, while I was going through his book for the first time, I'd have some questions, but usually, he would make up for them down the road like in quadratic equations not being the first mathematics. But, he has some more odd contradictions. He'd relate some of the new major developments of matheamtics but leave much else as a shadowy hint of other stuff out there. He covers the recent Poincare conjecture(although poorly compared to other online reviews I've read) and the classification of finite groups. Despite his great intruduction to the number theory throughout mathematics history and Fermat and algebraic geometry, he strangely hardly touches of Emmy Noether's (non) commutative algebra. He points out how William Thurston's three dimensional hyperbolic geometry gets proved by the Poincare conjecture effort, but not anywhere close to relating Emmy Noether's non-commutative algebra for hypercomplex systems? He stresses once again the relations between number theory and algebraic geometry but not the relations between Emmy Noether's commutative algebra? And then, because he doesn't even get into Emmy's commutative algebra much less her non-commutative algebra, he has no chance to bring up the Alain Connes non-commutative geometry; from there he could bring up fractal geometry which he doesn't mention at all(and he shows the hints of fractals in Poincare and Felix Klien's geometric investigations). E.T. Bell on the other hands integrates Emmy Noether's mathematics beautifully throughout his "Development of Mathematics."

E.T. Bell also goes into much more of real analyses and Functional analyses than John Stillwell; in fact, John Stillwell steers well clear of real analyses and Lebegue integration. From my perspective, I would have hoped for the latest of real analyses and functional analyses done since Lebesgue's work. Mr Stillwell says those things are way beyond presentation in his book; and yet, when he gets to Poincare conjecture and the Classification of finite groups, he's often referencing off things that cannot be covered in his book. Maybe he could have learned a thing or two of how to explain and note higher mathematics from E.T. Bell effort - the Development of Mathematics.

The ultimate technical history of Mathematics has yet to be written. I don't mean that such a book cannot be written so long as there is mathematical intelligence to keep making more mathematical discoveries(and Godel's theorem ensures that matheamtical development and adventure is infinit). I mean the right approach to doing so at a given time has not been accomplished.

P.S.

I forgot to mention thins like Knot theory of the twentieth century, Langlands conjecture work, the Hilbert problems.

This book covers a lot of territory in the history of mathematics, which does a lot to show how various topics developed over time. My biggest beef is that throughout he keeps saying to read other books for details. Since one has to eat and sleep it's all but impossible to follow up on these suggestions.

This is an overall good text. It offers a very in depth history of many many mathematical ideas. It gets quite technical at times, which can be a good or bad thing, depending on what you are looking for.