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This volume contains 27 refereed research articles and survey papers written by experts in the field of stochastic analysis and related topics. Most contributors are well known leading mathematicians worldwide and prominent young scientists. The volume reflects a review of the recent developments in stochastic analysis and related topics. It puts in evidence the strong interconnection of stochastic analysis with other areas of mathematics, as well as with applications of mathematics in natural and social economic sciences. The volume also provides some possible future directions for the field. ... Read more

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Chapters: Carl Friedrich Gauss, David Hilbert, Gottfried Leibniz, Johannes Kepler, Georg Cantor, Bernhard Riemann, Christian Goldbach, Friedrich Bessel, Felix Klein, Gottlob Frege, Felix Hausdorff, August Ferdinand Möbius, Max August Zorn, Johann Heinrich Lambert, Karl Weierstrass, Wilhelm Ackermann, Gerhard Gentzen, Erasmus Reinhold, Georg Joachim Rheticus, Abraham Robinson, Hermann Grassmann, Franz Mertens, Johann Friedrich Endersch, Emmy Noether, Regiomontanus, Oswald Teichmüller, Julius Plücker, Karl Wilhelm Feuerbach, Georg Von Peuerbach, Ferdinand Von Lindemann, Heinz Hopf, Emanuel Lasker, Nicholas of Kues, Wilhelm Cauer, Constantin Carathéodory, Hermann Weyl, Bruno Augenstein, Richard Dedekind, Caspar Isenkrahe, Paul Cohn, Alfred Pringsheim, Manfred Wagner, Adam Ries, Ernst Schröder, Carl Gustav Jacob Jacobi, Hermann Minkowski, Friedrich Kambartel, Konrad Knopp, Valentin Naboth, Simon Von Stampfer, Johann Peter Gustav Lejeune Dirichlet, Lorenz Christoph Mizler, Ferdinand Eisenstein, Erich Kretschmann, Ernst Zermelo, Eduard Study, Ferdinand Georg Frobenius, Richard Courant, Adam Olearius, Hans Georg Bock, Michael M. Richter, Andreas Floer, Petrus Apianus, Ehrenfried Walther Von Tschirnhaus, Franz Ernst Neumann, Oskar Bolza, Leopold Kronecker, Erhard Weigel, Adolf Hurwitz, Joachim Lambek, Friedrich Heinrich Albert Wangerin, Volker Oppitz, Edmund Landau, Reinhard Selten, Kurt O. Friedrichs, Heinz-Otto Peitgen, Carsten Niebuhr, Ernst Kummer, Matthias Bernegger, Heiko Harborth, Hans-Egon Richert, Theodor Vahlen, Johannes Werner, Werner Fenchel, Augustin Hirschvogel, Carl Gottlieb Ehler, Michael Maestlin, Abraham Gotthelf Kästner, Christian Wursteisen, Carl David Tolmé Runge, Gerhard Frey, Herbert Busemann, Karl Christian Von Langsdorf, Karl-Rudolf Koch, Johann Christian Martin Bartels, Volker Strassen, Christopher Clavius, Johann Jacob Zimmermann, Johann Karl Burckhardt, Max Dehn, Gunter Sachs, Friedrich L. Bauer, Ludwig B...More: http://booksllc.net/?id=244649 ... Read more

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Chapters: Christian Goldbach, Max August Zorn, Karl Wilhelm Feuerbach, Werner Fenchel, Carl Gottlieb Ehler, Wilhelm Killing, Johann Faulhaber, Carl Neumann, Herwart Von Hohenburg, Carl Wilhelm Borchardt, Otto Hesse, Robert Fricke, Ernst Steinitz, Gaspar Schott, Paul Albert Gordan, Hans Grauert, Martin Wilhelm Kutta, Lazarus Fuchs, Arthur Wieferich, Lothar Collatz, Johannes Praetorius, Johann Friedrich Pfaff, Fritz Noether, Albrecht Beutelspacher, Emanuel Sperner, Rudolf Lipschitz, Karl Seebach, Friedrich Engel, Christian Zeller, Arthur Moritz Schönflies, Oskar Anderson, Alfred Clebsch, Bernhard Neumann, Karin Erdmann, Hans Rademacher, Eduard Heine, Elwin Bruno Christoffel, Heinrich Scherk, Kurt Reidemeister, Wolfgang Heinrich Johannes Fuchs, Philipp Ludwig Von Seidel, Johann Tobias Mayer, Georg Simon Klügel, Michael Stifel, Leo Königsberger, Ernst Witt, Max Deuring, Richard Rado, Jakob Rosanes, Felix Otto, Carl Theodor Anger, Johann Georg Tralles, Heiner Zieschang, Heinrich Martin Weber, Nicholas Kratzer, Martin Ohm, Hermann Hankel, Valentine Bargmann, Carl Gustav Axel Harnack, Friedrich Schottky, Heinrich Behmann, Hans Julius Zassenhaus, Paul Epstein, Ulrich Kohlenbach, Heinrich Brandt, Leopold Löwenheim, Georg Hamel, William Prager, Francesco Cetti, Magnus Pegel, August Beer, Gerhard Hessenberg, Erhard Schmidt, Hans Carl Friedrich Von Mangoldt, Gustav Herglotz, Arnold Schönhage, Felix Bernstein, Heinrich Emil Timerding, Eberhard Melchior, Johann Benedict Listing, Valentinus Otho, Rodolphe Radau, Ernst Jacobsthal, Johannes Widmann, August Leopold Crelle, Johann Gottlieb Friedrich Von Bohnenberger, Walther Von Dyck, Bernd Fischer, Hans Fitting, Nicolai Reymers Baer, Franz Breisig, Reinhold Remmert, Christoph Rudolff, Oskar Perron, Otto Blumenthal, Philipp Furtwängler, Karl Georg Christian Von Staudt, Martin Eichler, Alfred Enneper, Heinrich Behnke, Johann Bauschinger, Kurt Schütte, Paul Schatz, Hans Werner Ballmann, Christi...More: http://booksllc.net/?id=4784468 ... Read more

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Correction to the text about C.S. Roero printed on the inside front cover page:

Clara Silvia Roero began her research with Tullio Viola, full Professor of Analysis at Turin University. From 1987 to 2000 she was associate professor of Matematiche Complementari and of History of Mathematics at the University of Cagliari (1987-1990) and at the University of Turin (1990-2000). From 2000 she is full professor of History of Mathematics at Turin University, Faculty of Mathematical and Physical Sciences. She is currently President of the Italian Society of History of Mathematics (Società Italiana di Storia delle Matematiche). She is author of several articles and books on the history of mathematics from antiquity to 20th century, in particular on the history of the Leibnizian Calculus; and she is a member of the editorial board of the Bollettino di Storia delle Scienze Matematiche.

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Text in German. Illustrated with black-and-white prints and color maps. ... Read more

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G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician ... the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940 as his mathematical powers were declining, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. C. P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his aphorisms and idiosyncrasies, and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times.Amazon.com Review
A Mathematician's Apology is a profoundly sad book, thememoir of a man who has reached the end of his ambition, who can nolonger effectively practice the art that has consumed him since he wasa boy. But at the same time, it is a joyful celebration of thesubject--and a stern lecture to those who would sully it bydilettantism or attempts to make it merely useful. "Themathematician's patterns," G.H. Hardy declares, "like the painter's orthe poet's, must be beautiful; the ideas, like the colours orthe words, must fit together in a harmonious way. Beauty is the firsttest: there is no permanent place in the world for uglymathematics."

Hardy was, in his own words, "for a short time the fifth best pure mathematician in the world" and knew full well that "no mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game." In a long biographical foreword to Apology, C.P. Snow (now best known for The Two Cultures) offers invaluable background and a context for his friend's occasionally brusque tone: "His life remained the life of a brilliant young man until he was old; so did his spirit: his games, his interests, kept the lightness of a young don's. And, like many men who keep a young man's interests into their sixties, his last years were the darker for it." Reading Snow's recollections of Hardy's Cambridge University years only makes Apology more poignant. Hardy was popular, a terrific conversationalist, and a notoriously good cricket player.

When summer came, it was taken for granted that we should meet at the cricket ground.... He used to walk round the cinderpath with a long, loping, clumping-footed stride (he was a slight spare man, physically active even in his late fifties, still playing real tennis), head down, hair, tie, sweaters, papers all flowing, a figure that caught everyone's eyes. "There goes a Greek poet, I'll be bound," once said some cheerful farmer as Hardy passed the score-board.

G.H. Hardy's elegant 1940 memoir has provided generations of mathematicians with pithy quotes and examples for their office walls, and plenty of inspiration to either be great or find something else to do. He is a worthy mentor, a man who understood deeply and profoundly the rewards and losses of true devotion. --Therese Littleton ... Read more

Customer Reviews (31)

A Non-Mathematician's Apology
It is a melancholy experience for one to find himself reading about mathematicians and not mathematics. It is a sad experience to travel down memory lane and note that any interest one had in mathematics during the school and college days were for grades and grades only. It is, perhaps, even sadder to realize that the aesthetic aspect of it was never really and truly detected; still more to realize that that abstractness, though acceptable in mild doses, was considered as some sort of weirdness. The case for my mathematical life, then, or for that of any one else who has been a non-mathematician in the same sense which I have been one, is this: that, though the chance to make a contribution is lost, there, perhaps, exists still the opportunity to stroll through it and enjoy the walk.

A Mathematician's Apology

G.H. Hardy (1877-1947) is a famous mathematician and this book is famous, perhaps because he is its author, but the question of whether the book is especially remarkable and worthy of praise is separate from the fact of its being famous. Hardy loved mathematics, and he loved it for itself, not for what it offered to the physical world of a man's life. Useful mathematics, in his view, lacks the aesthetic and moral purity of mathematics that has no worldly use. Spin out the implications of this and add in that the quality of person's life (the ongoing value of his life as he lives it) is determined by the aesthetic and moral value of what he creates and you get Hardy's view of himself and others. The value of his life lay, he believed, in his mathematically creative past. Once Hardy lost his mathematical creativity, he felt himself to be of no ancillary value. His life, as a thing of ongoing value, was finished. From that point on, his life was "trivial".

Add one star to make four if C.P. Snow's introduction is taken into account.

INTERESTINGBUT DATED
Thetitle" A mathematiciansapolgy" occurs so often in bibliographysthat when I saw a copy of this famous bookI decided that I should read it as I have always had a particular love for mathematics.Not being a mathematicecian myself but having read many similar expositions of mathemematics and being a user of much mathematics I was keen to see what all the fuss was about.
I must say I was rather disapointed.This is a short book .The introduction by the Writer /Phyicist CP Snow being almost as long as the book.
Hardy comes accross as a the sort of Englishman that has long since been extinct.A prewarboffin type ,living in his own little world.A genius , true, but a rather sad one.
The book is permeated with Hardy's sadness that he has lost his mathematical prowess.True mathematics is a young mans game .But isn't this true of most of life? At least be thankfuul that once you had a gift.Be happy that now other young men will carry on the tradition.
Hardy's predictionthat his mathematics was of no practiccal use shows how wrong he was.Even as he wrote in the 40'snumber theory was being used in code breaking, Relativity theory has implications in nuclear energy and Quantum theory has helped to change the world in semiconductor physics.
Of couse maths is not donefor purly practical purposes .But neirther is any study of Science or nature.Man yearns to understand.That is a very human quality.
There is no in deep discussion of why mathematics works.
In short we get a picture of a rather shallow man.The passing of 70 years have done little to enhance his reputation.
Compare this book to that of the MathematicianJacobBronofski.A man a multiple intersstswho loved his fellow mankind.His writings are still as fresh as there were when they were writen 60 years ago

A Well- Titled Classic
This is a sad thoughtful book, slightly off-base due to recent developments, but still full ofsad and wonderful truths about mathematics and mathematicians.

What is mathematical beauty?
In the most famous phrase of the book, Hardy proclaims that "Beauty is the first test: there is no permanent place in this world for ugly mathematics." Quite so, but wherein does mathematical beauty consist? Here is Hardy's answer:

"What 'purely aesthetic' qualities can we distinguish in such theorems as Euclid's [on the infinity of primes] or Pythagoras's [on the irrationality of the square root of two]? ... In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of [1] unexpectedness, combined with [2] inevitability and [3] economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail; one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many 'variations' in the proof of a mathematical theorem: 'enumeration of cases', indeed, is one of the duller forms of mathematical argument." "The beauty of a mathematical theorem depends a great deal on its [4] seriousness ... The 'seriousness' of a mathematical theorem lies ... in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

I say: these four conditions are neither individually necessary nor jointly sufficient. In support of which I offer the following considerations.

Unexpectedness. Counterexample to the necessity of this criterion: Bernoulli's theorems on the logarithmic spiral. Bernoulli discovered numerous remarkable properties of this spiral: that it is "equal to its caustics by reflection and refraction, to its evolute, and to numerous other derived or conjugate curves" (Le Lionnais). Surely the unexpectedness would have worn off as his study went on. But there is no indication that Bernoulli's appreciation for the theorems diminished with it. On the contrary he wrote that "This marvelous spiral gives me such overwhelming pleasure that I can scarcely satisfy my desire to contemplate it" (ibid.).

Inevitability. There are many senses in which there can be ``no escape from the conclusion'' in a proof. A specification is needed for meaningful discussion. It seems to me that the only way to narrow down Hardy's intention is to take his phrase regarding "one line of attack" as his explication of inevitability. This is still very vague, but I think it is precise enough to admit counterexamples. A counterexample, then, would have to be a proof which we admire not for charging at the Achilles heel of the enemy, but rather for winning the battle by masterful deployment of forces and exquisite interplay between cavalry and infantry. I believe there is mathematical beauty of this type, for example Galois theory, and in particular its proof of the insolubility of the quintic. Another example is the theory of Riemann surfaces, which any general enjoys deploying with a deft touch, but which is nevertheless unlikely to win a battle all by itself. These are examples of a more general phenomena: the application of ideas from one field to a seemingly unrelated one (in our cases: group theory in algebra, and topology in complex analysis). Such interconnections are a considerable source of aesthetic pleasure. But this can hardly be said to be due to their enabling a single line of attack to replace a plurality of such lines. On the contrary, such interconnections often, as in our examples, function as simpler and more elegant alternatives to some aspect of a previously perceived "brute force" line of attack. (In fact, such connections are often discovered in precisely this manner; again this is the case in our examples.)

Economy. This, I take it, is what Hardy defines when he writes that "the weapons used seem so childishly simple when compared with the far-reaching results." A counterexample, then, would consist in beautiful use of advanced weapons to prove simple results. But surely there can be no doubt that advanced theories can yield beautiful spin-off proofs of rather basic results. For example, there are a number of beautiful solutions of the ancient isoperimetric problem based on modern theories such as complex analysis, vector analysis, etc. (for which, see my article in the Am. Math. Monthly).

Seriousness. As we see above, seriousness amounts to connectivity. Now, a theorem with no connectivity with the rest of mathematics is not likely to count as mathematics at all. However, it seems to me that there are enough beautiful theorems whose seriousness is so unexceptional (e.g., Bernoulli's theorems on the logarithmic spiral, the theorem that there are five regular polyhedra) that a criterion weak enough to include them would be too weak to have any teeth. Another source of examples of beautiful theorems with very limited connectivity is classical number theory, of which Euler wrote that "I must confess that I derive nearly as much pleasure from investigations of this kind as from the deepest speculations of higher mathematics".

All criteria at once. The theorem on integration by parts (which is surely not beautiful) scores highly on all four counts, thus proving that they are not jointly sufficient. ... Read more

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This revised and greatly expanded second edition of the Russian text Tales of Mathematicians and Physicists contains a wealth of new information about the lives and accomplishments of more than a dozen scientists throughout five centuries of history: from the first steps in algebra up to new achievements in geometry in connection with physics. The heroes of the book are renowned figures from early eras, such as Cardano, Galileo, Huygens, Leibniz, Pascal, Euler, Lagrange, and Laplace, as well some scientists of last century: Klein, Poincaré, and Ramanujan.

A unique mixture of mathematics, physics, and history, this volume provides biographical glimpses of scientists and their contributions in the context of the social and political background of their times. The author examines many original sources, from the scientists’ research papers to their personal documents and letters to friends and family; furthermore, detailed mathematical arguments and diagrams are supplied to help explain some of the most significant discoveries in calculus, celestial mechanics, number theory, and mathematical physics. What emerges are intriguing, multifaceted studies of a number of remarkable intellectuals and their scientific legacy.

Written by a distinguished mathematician and accessible to readers at all levels, this book is a wonderful resource for both students and teachers and a welcome introduction to the history of science.

Customer Reviews (1)

A glimpse into the biographies of some great minds!
Simon Gindikin did an excellent job in this new edition of his book. I got acquainted with his research not too long ago when reading an essay on Penrose's Twistor theory in the "Mathematical Intelligencer" by Springer dating from early 80's.
Eventually I found out the first edition of his book, which was already quite delightful.
The book, besides being filled with witty historical facts, contains also a few interesting problems to improve one's mathematical culture.
I'm particulary kind of the chapter on Huygens and his mechanical works,Poincare's ideas on non-euclidean geometry and finally and the above mentioned article on twistor theory(which could be included on a course on analytical/projective geometry, example).
As my usual cliche, "Two thumbs up!".
... Read more

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A GREAT BOOK! BUT NOT FOR ENGLISH READERS!!!!!
I am a fanatic mathematical fan of Euler,to me his the greatest mathematician of all time. Of course, you would challange me. How about Newton, Gauss, and Archimedes, the universally claimed 3 of the greatest mathematician of all time. But read more, I give Euler this honor is including all other aspects of his life, and not just on maths and his achievement. Euler enjoyed his life, his maths, he adventured maths, even with one eye blind followed by the other,he never gave up. Even under the bad relationship with Fredrich the Great, he still did what he should do and never involved in politics. He enjoyed his family, he was approachable, he helped mathematicians of new generation. ( He held up his paper on calculus of variation in order to give full credit to Lagrange was just one aspect ). He was humble, he almost had good relationship with all his contempory mathmaticians. ( except a tiny quarrel with D"alembert on the problem of vibrating string ).
Can you find another scientist or mathematician in history like him ? But why I do not recommend this for english readers?? THIS IS BECAUSE THE BOOK IS NOT WRITTEN IN ENGLISH. HOW DISAPPOINTED WHEN I WAITED FOR 6 MONTHS AND FOUND THAT I COULD NOT READ THIS BOOK OF MY IDOL!!THE REASON I DID NOT RETURN IS THAT I PUT THIS BOOK ON MY SHELF AS A RESPECT TO EULER.I SINCERELY HOPE THAT SOMEONE ( WILLIAM DUNHAM IS A GOOD CANDIDATE ) WOULD TRANSLATE THE WHOLE BOOK IN ENGLISH...A SALUTE TO THE GREAT ANALYSIS INCARNATE, LEONHARD EULER !!!!!!!! ... Read more

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This volume contains the work of the great Swiss mathematician on differential geometry, a field marked by some of his greatest achievements. Between 1690 and 1700, Jacob Bernoulli published twelve treatises in the scientific journal Acta Eruditorum on the use of infinitesimal methods to answer geometrical questions. Preparatory notes for most of these papers and on many other themes are found in Bernoulli's scientific diary Meditationes, from which twentynine texts are published here for the first time. Among the curves considered are the isochrones (lines of constant descent), the parabolic spiral, the loxodrome, the cycloid, the tractrix, and the logarithmic spiral (Bernoulli's spira mirabilis, which also adorns his tombstone). The description of these curves by differential equations and by geometrical constructions, their rectification and quadrature, and the determination of their evolutes and caustics offered Bernoulli and his colleagues a range of challenging problems, many of them relevant for mechanical or optical applications. The French mathematician André Weil, who lived in the United States until his recent death, has greatly influenced 20th century mathematics, among other things, as a founding member of the Bourbaki group. For many years he has pursued intensive studies of the history of mathematics, especially number theory and algebraic geometry. Weil's introduction to this volume places Jacob Bernoulli's contribution to differential geometry in a line of development from Descartes, Huygens and Barrow through Newton's und Leibniz's epochal innovations right up to the codification of the subject by Euler. Martin Mattmüller, secretary of the Bernoulli Edition at Basel, edited the source text. His commentaries consider particular topics in differential geometry with reference to their historical context at the end of the 17th century. ... Read more

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Emil J. Gumbel (1891-1966) began his career simply as a professor of mathematical statistics in Heidelberg, but he is most remembered as a political activist militantly advocating for pacifism during the complicated and volatile times of the Weimar Republic in Germany. As a Jew with left-wing socialist and democratic sensibilities, he was exiled to France and later America. Ironically, the same writings on political terror and politicized justice in Nazi Germany that caused his ostracization saved his life. A courageous man, Gumbel spoke out passionately against the Nazis and came to symbolize a "one-man party" at the centre of controversy in German academia. His intellectual and moral vigour never waned, and despite his significant scientific contributions, it is his legacy of political ideology that endures for later generations to learn from. This biography chronicles the public life of a man not entirely part of the political or the academic world, but who has earned his place in history nonetheless. ... Read more

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Customer Reviews (10)

The most interesting book I've ever read!
I CANNOT believe the intellectual insight described in this book.

I am a Mathemetician Master's school graduate from CSULB and USC and have been tested to have exceptional mathematic ability.
You would think with my background, I could breese through this book. (albeit, as a mathemetician, I might be bored reading the book!)
Quite the opposite. This book is NOT about Mathematics. It is about why SCIENTISTS (mathemeticians as an example) hate ambiguity so much that they "create" never-before-invented solutions to resolve that ambiguity. This book also explains the process of solving those problems.

This book explains why and how our human scientific itellect has evolved. AMBIGUITY is the impetus. SOLVING the ambiguity is the goal and "engine" for evolution.

Without the existence of ambiguity, contradiction, and paradox, we humans would never have raised our combined intellect beyond "nature" or "God".

To me, the "ah ha" teachings in this book MUST become a classroom experience for young scientific minds that leads them to look for ambiguities in their life and motivates them to solve them.

Not an easy read, but an interesting one nonetheless
Mathematics is a fascinating subject. I am not a mathematician, but deal enough with it in my chosen profession to be constantly amazed by how logical the application of mathematics to proving a theorem or analyzing an algorithm turns out to be. But wait ... is it really logical? Or does it merely seems so and what is actually happening is that the author of the said proof is using creative tricks and techniques from the mathematical tool box to somehow tie everything up with a nice red bow-tie? In this book, the author argues that mathematics is creative more than algorithmic, and that mathematicians use a good dose of ambiguity mixed with equal parts of contradiction and paradox to create mathematics. Now, I shall point out that the term "ambiguity" here does not mean vagueness, rather it refers to a central truth that is perceived in two self-consistent but mutually incompatible contexts. The author takes the reader on this journey of ambiguity, paradox and contradiction on the way to discovering a lot of interesting mathematics. There is a section on counting numbers and cardinality, complete with Hilbert's Infinity Hotel; there is an interesting section on how to approach geometry through Euclid's Elements, and so on. I don't suppose that this is the sort of book you would pick up for a plane ride -- contemplating the philosophy of mathematics at 35,000 feet is enough to induce stupor. But if you are interested in the field and still remember the Central Limit Theorem from Calculus-I or Series and Sequences from Calculus-II, then you will definitely enjoy this book. I know I did.

Mathematical Philosophy
I would classify this book as a Mathematical Philosophy Book. The author definitely places Philosophy more than the hard-core Mathematics, so don't be disappointed if the reader's main goal is on Math.
Overall this bookis a great book, but definitely not for the weaker math students. It brings you to the higher platform to look down on Math issues in a pensive way - ie "Switch on the light" à la Andrew Wiles.

This book should be best read not in sequential manner, because of the writing style of the author which is quite verbose.

Some chapters are very well written:
Chap 8: (Pg 363) Obstacles to Learning Mathematics : Many great ideas and truths are hidden behind the math theoretical structures, unfortunately in the university math profs emphasize more on structures and leave the poor students to find out the 'beauty' of truth themselves - because 'Beauty' is not tested in Exams :(
Chapter 4: Paradoxes of Infinity. The "Cantor Set" Construction example is very refreshing.
Chapter 5: on "Quotient Space" (X/R) is excellent.
I also like the Isomorphism ideas (Pg 216-217): as Isometry (in Geometry), homeomorphism (in Topology), besides various isomorphism in Groups, Rings, Fields...

In summary, this book will elevate your math philosophical thinking like a Mathematician.

Extremely simpleminded
This is yet another naive rehash of the same old pop-math clichés. Since Byers knows nothing about mathematics beyond the meat-and-potatoes undergraduate curriculum, he has to use dishonest tricks to make these tired topics seem interesting. One time-tested trick is to claim that every single theorem is "surprising". For example, "we usually forget how surprising" it is that logic can be applied to geometry (p. 219). Only eight pages earlier we were surprised that the natural numbers have the same cardinality as the even numbers. And only four pages later we are surprised again, this time that there are infinitely many primes (p. 223). And so it goes. What a thrilling ride! Another underhand trick is to make completely unsubstantiated claims as to some mysterious metaphysical importance of every mathematical concept, e.g., "the notion of countable infinity captures something quite fundamental about the human nature and limits of what human beings can know" (p. 165).

Stylewise, Byers' amateur prose is made all the more unbearable by his obnoxious habit of putting at least four or five words per page in quotation marks for no apparent reason. We learn, for example, of a conjecture which "seems" true (sic, p. 281); elsewhere we study "infinity", "zero", and the notion of continuity, which means that f(x) gets "close" to f(a) when x gets "close" to a (sic, p. 239).

Byers' pathetic use of footnotes is a parody of scholarship. For example, the claim that "Poincaré call[ed] Cantorism a disease" (p. 286) is backed up by a footnote saying "Gardner (2001)" with no page reference. This is Gardner's "Colossal Book of Mathematics", obviously a deeply unscholarly source, and a colossal one at that. In this case, of course, Byers could not have provided the original reference because there is none. Poincaré never made this statement in print; and even if he did make it informally he was most likely referring to the paradoxes of set theory and not Cantor's transfinite numbers, as Byers would have known if he had not done his research at a high school library.

Bits of interesting mathematics mixed with unremarkable philosophy
I suspect that Prof. Byers is an excellent mathematics teacher and I very much enjoyed the snippets of mathematics in this book.However, most of the book was devoted to philosophy, which I found to be at least one of the following: (1) repetitive, (2) unoriginal, or (3) wrong.Repetitive for sure.Unoriginal in that he repeats many of the points made more eloquently and clearly by folks like Lakatos (though Byers does do a good job of giving credit where it is due).Wrong in the philosophy of mind, as in the section toward the end of the book where he tries to argue (a la Searle?) that machines can't think, and that computers might be able to write proofs but they can't do the inherently creative aspects of mathematics.It's very strange to me to run into a mathematician who holds these kinds of mystical views about minds, that they are not machines!

I feel like this would make a truly excellent 50 page book, with just a few of the key philosophical points clearly explained and illustrated with some of the excellent mathematical examples in this book.It could even be expanded -- but with more of Byers' mathematical illustrations, not his philosophical ramblings.

If I focused just on my favorite 50 pages of this book, it would get at least four stars; but the other 300 pages average it down to two. ... Read more

Product Description
This book traces the history of the MIT Department of Mathematics one of the most important mathematics departments in the world through candid, in-depth, lively conversations with a select and diverse group of its senior members. The process reveals much about the motivation, path, and impact of research mathematicians in a society that owes so much to this little understood and often mystified section of its intellectual fabric, exemplified by names like Euclid, Newton, Euler, and Goedel.

At a time when the mathematical experience touches and attracts more laypeople than ever, such a book contributes to our understanding and entertains through its personal approach.

From the book:
The usual thing is there were these Thursday colloquiums at MIT or Harvard, and even the mathematicians from Brown would come. They would invite some speaker to give a mathematical talk. And afterward there was a dinner and a party in somebody's house. If it was in one guy's field, he would make the party. So that's how they socialized. They couldn't socialize with people who'd talk about tomatoes or clothes or something; they had to talk to people who understood the values they had, which were mathematical values. They're attracted to mathematics like a drug addict is attracted to drugs. They can't stay away from it.;
--Fagi Levinson ... Read more

Customer Reviews (2)

Well done book of interviews of MIT mathematicians
If you are interested in people who do math, the history of mathematics, or have taken any math at MIT, this book is probably for you.

An intriguing look at mathematics and the men behind it
Though never in the eye of popular culture, these men kept society advancing with their minds. "Recountings: Conversations with MIT Mathematicians" is a collection of interviews and anecdotes from the geniuses of MIT who have pursued mathematics as their life's careers and obsessions. These men have been responsible for major scientific advances throughout history and picking their minds in a volume that's more interesting than one could think math class could ever be. "Recountings" is an intriguing look at mathematics and the men behind it.
... Read more

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In the 17th century, the small but culturally and intellectually eminent city of Basel was the home of one of the most prominent mathematical families of all time, the Bernoullis, and their friend, protege, and master Leonhard Euler. The author chronicles their lives and achievements at a time when modern analysis and its applications to physics burst on the scene and created a methodological framework for modern science. Written for young adults, this book conveys the excitement of a scientific culture that impacts our life to this day and will serve as an inspiration to gifted young people to devote their lives to scientific pursuits. ... Read more