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Infinitesimal Hardcover – 1 May 2014

3.5 out of 5 stars 4 customer reviews

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Product details

  • Hardcover: 368 pages
  • Publisher: Farrar, Straus and Giroux (1 May 2014)
  • Language: English
  • ISBN-10: 0374176817
  • ISBN-13: 978-0374176815
  • Product Dimensions: 16 x 3.1 x 23.4 cm
  • Average Customer Review: 3.5 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Bestsellers Rank: 1,754,005 in Books (See Top 100 in Books)

Product Description


You probably don't think of the development of calculus as ripe material for a political thriller, but Amir Alexander has given us just that in "Infinitesimal." "Jordan Ellenberg, The Wall Street Journal"

Packed with vivid detail and founded on solid scholarship, ["Infinitesimal"] is both a rich history and a gripping page turner. "Jennifer Ouellette, The New York Times Book Review"

[A] finely detailed, dramatic story. "John Allen Paulos, The New York Times"

Alexander pulls off the impressive feat of putting a subtle mathematical concept centre stage in a ripping historical narrative . . . this is a complex story told with skill and verve, and overall Alexander does an excellent job . . . There is much in this fascinating book. "Times Higher Education"

A triumph. "Nature"

Every page of this book displays Alexander's passionate love of the history of mathematics. He helps readers refigure problems from over the centuries with him, creating pleasurable excursions through Euclid, Archimedes, Galileo, Cavalieri, Torricelli, Hobbes, and Wallis while explaining how seemingly timeless and abstract problems were deeply rooted in different worldviews. "Infinitesimal "captures beautifully a world on the cusp of inventing calculus but not quite there, struggling with what might be lost in the process of rendering mathematics less certain and familiar. "Paula E. Findlen, The Chronicle of Higher Education"

With a sure hand, Mr. Alexander links mathematical principles to seminal events in Western cultural history, and has produced a vibrant account of a disputatious era of human thought, propelled in no small part by the smallest part there is. "Alan Hirshfeld, The Wall Street Journal"

"Infinitesimal" is a gripping and thorough history of the ultimate triumph of [a] mathematical tool . . . If you are fascinated by numbers, "Infinitesimal "will inspire you to dig deeper into the implications of the philosophy of mathematics and of knowledge. "New Scientist"

Brilliantly documented . . . Alexander shines . . . the story of the infinitesimals is fascinating. "Owen Gingerich, The American Scholar"

Back in the 17th century, the unorthodox idea [of infinitesimals], which dared to suggest the universe was an imperfect place full of mathematical paradoxes, was considered dangerous and even heretical . . . Alexander puts readers in the middle of European intellectuals' public and widespread battles over the theory, filling the book's pages with both formulas and juicy character development. "Bill Andrews, Discover"

In "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World," Amir Alexander successfully weaves a gripping narrative of the historical struggle over the seemingly innocuous topic of infinitesimals. He does an excellent job exploring the links between the contrasting religious and political motivations that lead to acceptance or refusal of the mathematical theory, skillfully breathing life into a potentially dry subject. "Infinitesimal "will certainly leave its readers with a newfound appreciation for the simple line, occasion for such controversy in the emergence of modern Europe. "Emilie Robert Wong, The Harvard Book Review"

Fluent and richly informative "Jonathan Ree, Literary Review (UK)"

Alexander tells this story of intellectual strife with the high drama and thrilling tension it deserves, weaving a history of mathematics through the social and religious upheavals that marked much of the era . . .The author navigates even the most abstract mathematical concepts as deftly as he does the layered social history, and the result is a book about math that is actually fun to read. A fast-paced history of the singular idea that shaped a multitude of modern achievements. "Kirkus (starred review)"

[Infinitesimal] gives readers insight into a real-world Da Vinci Code like intrigue with this look at the history of a simple, yet pivotal, mathematical concept . . . Alexander explores [a] war of ideas in the context of a world seething with political and social unrest. This in-depth history offers a unique view into the mathematical idea that became the foundation of our open, modern world. "Publishers Weekly"

A bracing reminder of the human drama behind mathematical formulas. "Bryce Christensen, Booklist"

A gripping account of the power of a mathematical idea to change the world. Amir Alexander writes with elegance and verve about how passion, politics, and the pursuit of knowledge collided in the arena of mathematics to shape the face of modernity. A page-turner full of fascinating stories about remarkable individuals and ideas, Infinitesimal will help you understand the world at a deeper level. "Edward Frenkel, Professor of Mathematics, University of California, Berkeley, and author of Love and Math"

In this fascinating book, Amir Alexander vividly re-creates a wonderfully strange chapter of scientific history, when fine-grained arguments about the foundations of mathematical analysis were literally matters of life and death, and fanatical Jesuits and English philosophers battled over the nature of geometry, with the fate of their societies hanging in the balance. You will never look at calculus the same way again. "Jordan Ellenberg, Professor of Mathematics, University of Wisconsin Madison, and author of How Not to Be Wrong"

You may find it hard to believe that illustrious mathematicians, philosophers, and religious thinkers would engage in a bitter dispute over infinitely small quantities. Yet this is precisely what happened in the seventeenth century. In Infinitesimal, Amir Alexander puts this fascinating battle in historical and intellectual context. "Mario Livio, astrophysicist, Space Telescope Science Institute, and author of Brilliant Blunders"

With considerable wit and unusual energy, Amir Alexander charts the great debate about whether mathematics could be reduced to a rigorous pattern of logical and orderly deductions or whether, instead, it could be an open-ended and exciting endeavor to explore the world's mysteries. Infinitesimal shows why the lessons of mathematics count so much in the modern world. "Simon Schaffer, Professor of the History of Science, University of Cambridge"

In Infinitesimal, Amir Alexander offers a new reading of the beginning of the modern period in which mathematics plays a starring role. He brings to life the protagonists of the battle over infinitesimals as if they were our contemporaries, while preserving historical authenticity. The result is a seamless synthesis of cultural history and storytelling in which mathematical concepts and personalities emerge in parallel. The history of mathematics has rarely been so readable. "Michael Harris, Professor of Mathematics, Columbia University and Universite Paris Diderot"

We thought we knew the whole story: Copernicus, Galileo, the sun in the center, the Church rushing to condemn. Now this remarkable book puts the deeply subversive doctrine of atomism and its accompanying mathematics at the heart of modern science. "Margaret C. Jacob, Distinguished Professor of History, University of California, Los Angeles""

About the Author

Amir Alexander teaches history at UCLA. He is the author of "Geometrical Landscapes "and "Duel at Dawn." His work has been featured in "Nature," "The Guardian," and other publications. He lives in Los Angeles, California.

Customer Reviews

3.5 out of 5 stars
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Top Customer Reviews

Format: Hardcover
While some books have obscure titles, a combination of the title and the subtitle will usually make it plain what the book is about. But I can pretty much guarantee that most readers, seeing Infinitesimal - how a dangerous mathematical theory shaped the modern world would leap to an incorrect conclusion as I did. The dangerous aspect of infinitesimals was surely going to be related in some way to calculus, but I expected it to be about the great priority debate between Newton and Leibniz, where in fact the book concentrates on the precursors to their work that would make the use of infinitesimals - quantities that are vanishingly close to zero - acceptable in mathematics.

The book is in two distinct sections. The first focuses on the history of the Jesuits, from their founding to their weighing into the mathematical debate against those who wanted to use infinitesimals in maths. For the Jesuits, everything was cut and dried, and where Aristotle's view and the geometry of Euclid had an unchanging nature that made them acceptable, the use of infinitesimals was far too redolent of change and rebellion. This was interesting, particularly in the way that the history gave background on Galileo's rise and fall seen from a different viewpoint (as he was in the ascendancy, the Jesuits were temporarily losing power, and vice versa). However, this part goes on far too long and says the same thing pretty much over and over again. This is, I can't help but feel, a fairly small book, trying to look bigger and more important than it is by being padded.

The second section I found considerably more interesting, though this was mostly as a pure history text. I was fairly ignorant about the origins of the civil war and the impact of its outcome, and Amir Alexander lays this out well.
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Format: Hardcover
The story or the origins of the integral calculus in the era of the Reformation and post-reformation with the possibility to align a period of great social and political upheaval with the names and personalities of the great Italian and (later) northern European mathematicians has the potential to be an absorbing thriller for those inclined and less inclined to mathematics. Amir Alexander makes an interesting story boring by repetitive repetition. He badly needed a vigorous editor who would have cut two-thirds of the length without losing anything. The aversion of the Jesuits to the new mathematics and their adherence to Euclidian maths as a bulwark against the threatening social chaos of the protestant ethos is maybe not surprising and the lengths to which they went in maligning our early mathematical heroes (Galileo et al.) might be an analogy for the current obsession of the American administration with Al Quaeda but the story loses all its potential excitement in Infinitesimal through repetitiveness. Again, the battle between Hobbes and Wallis provides new insights to the obsessive mind of the author of Leviathan but why tell the same story again and again page after page in part 2 of Infinitesimal. And why not tell us more of the parts that other greats of the era made of all this hateful to-and fro. Finally an epilogue carrying the story of the new maths forward to Newton and Leibnittz would given some better context to the story for the modern reader.

I will definitely not recommend this book to my friends - even the maths inclined among them. Not even 1 star
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Format: Paperback Verified Purchase
Very good.
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Format: Kindle Edition Verified Purchase
Extremely interesting and unusual story, very well told, with plenty of background and historical relevance
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Most Helpful Customer Reviews on (beta) 4.1 out of 5 stars 91 reviews
67 of 72 people found the following review helpful
3.0 out of 5 stars Misleading... 3 July 2014
By Timothy Haugh - Published on
Format: Hardcover
As a math teacher, I’m often on the lookout for books that will help my students make connections between math and its importance, whether that be practical or historical. As I was teaching AP Calculus this year, Prof. Alexander’s book drew my attention. I was hoping for something that would really make a strong case for the importance of infinitesimal mathematics. Unfortunately, this book turned out to be something other than what I was looking for.

Essentially, there was considerably less discussion of math than I expected. Though there are some nice forays into some important basics, the touches on the foundational ideas here are quite brief. Primarily, this is a book of history. And yet, even the focus of the history is not mainly on mathematical ideas. This is a history of conflict where mathematics played a small part.

Infinitesimal is divided into two parts, each of which covers a major historical conflict. Part I deals with the Reformation and Counter-reformation. Our primary characters here are the Galileans and the Jesuits. In fact, there is a rather extensive history of the Jesuits and Prof. Alexander does a nice job of showing their developing educational philosophy. He describes how the Jesuits rejected the concept of the infinitesimal in favor of Euclidean geometry more for reasons of philosophy than general mathematics. In describing this conflict, however, Prof. Alexander deserves credit for being less hostile towards the Jesuits than one often finds in these descriptions, even if he overreaches a bit at the end, claiming that this rejection of the new math held back the development of math and science in Italy for centuries whereas the Protestant areas of Europe made the great leaps forward. This is not quite as true or as simple as Prof. Alexander tries to make it out to be.

Part II deals with the English Civil War. Here, the focus is almost entirely on Thomas Hobbes (for Euclidean geometry) and John Wallis (for infinitesimals). Once again, these men’s difference in mathematical technique was somewhat of a sideshow in their political differences—Hobbes and his Leviathan for a strong monarchy and Wallis a beneficiary of the Commonwealth. Somehow, both men managed to survive the chaos of their time with heads intact, though it could be argued that both men’s mathematical development suffered because of the need to achieve political ends. Still, Prof. Alexander seems to argue that it is the rise of Wallis and the slow decline of Hobbes that leads to Newton and the rise of England as the birthplace of much of the new physics which, once again, is not quite as true or simple as may be implied in this book.

In the end, I felt a bit misled. Though there is some very nice biography and history here, the math seems really to be secondary to the conflicts presented, however much Prof. Alexander wants to bring them to the fore. And his overall arguments about this mathematical theory shaping the modern world; well, this theory might have played a small role in this world of high intellectual ferment but, as much as I believe in the importance of math, there’s a lot more going on here than that. Prof. Alexander seems to know that, if his thesis doesn’t quite allow him to admit it.
51 of 57 people found the following review helpful
3.0 out of 5 stars A Brief History of the Jesuits and Infinitesimals That Needs Editing 7 May 2014
By Steven R. Staton - Published on
Format: Kindle Edition Verified Purchase
This book is a quick read, and it seriously could be pared down with some editing. I found the repetition of the thesis to be tedious (e.g. the Jesuits suppressed the mathematics of the Infinitesimal because it clashed with their dogma) and really wish this book be edited to get to the point (pun intended) without having to circle around and repeat ideas that were introduced and examined with a fine tooth comb already. The details are fine, but the reiteration of the information was really off-putting.

I was also surprised that the story ends abruptly *before* Newton, when the mathematical world really took off thanks to the math of the infinite and continuous (i.e. the Calculus). There is a lot of mathematical history that would have added meat to this story starting in the late 1670's that simply isn't there (never mind the epic battle between Newton and Leibniz).

This story is richer than the book eludes to and I would strongly recommend that the author consider a second edition that had less repetition of plot and more history (especially post 1660) of this branch of mathematics. It's a shame that the e-book cannot include interactive diagrams of the key geometric proofs from Hobbes and the Italians, too.
16 of 17 people found the following review helpful
2.0 out of 5 stars Same Mistake As Hobbes 23 Sept. 2014
By bonnie_blu - Published on
Format: Hardcover
I was quite disappointed in this book for a number of reasons. First let me state that I am not a mathematician, although I have studied math through calculus. However, I am well versed in history.

1) The author makes the same mistake as Hobbes (and numerous others in the past and present) by attempting to make events fit his thesis. He states that the battle over infinitesimals was a key player in the massive social changes that took place in the 16th and 17th centuries in Europe and England. Anyone familiar with this period in history can argue that the mathematical battles were but one symptom of the very complex social stressors that swept Europe and England during that time. However, the author cherry picks events to support his thesis while almost ignoring the numerous other forces, characters, etc. that were in play.
2) He credits the success of Wallis's approach over Hobbes with all successive mathematical and technological advances into modernity. As history has shown over and over again, it is never this simple when determining the causes and courses of human history.
3) The book is in serious need of editing. It is mind-numbingly redundant and often wanders off into tangents that add little to nothing to the information.

What should have been an interesting book on this period in the history of mathematics is seriously flawed (in my opinion) by the author's attempt to shoehorn events and people to fit his thesis, and his attempt to make the outcome of the mathematical conflict responsible for our modern world.
73 of 94 people found the following review helpful
5.0 out of 5 stars Not exactly war of all againts all, but still a very damaging fight 10 April 2014
By Mariano Apuya Jr - Published on
Format: Hardcover
The opening chapters of "Infinitesimal" are about a board of Jesuits in the 17th century ruling on legitimacy of a mathematical topic. That which would be binding on every university of the Jesuit order-the most prestigious of the time. So I thought this was a book about intolerant and unscientific clergy oppressing reason or a hagiography for secularism. Expect none of that.
I have had come across this topic somewhere but I can't recall chapter in a book I don't remember. The point of that passage was the hostility of clergy to science
Roughly the present book is the history of mathematical thought of the infinitesimal. Halfway through I had the suspicion that it was the history of the limit concept, it is but partly so. This book is in fact a history book. There is substantial coverage of topics on social and historical subjects such as the reformation, the formation and fortunes of the Jesuits, even an entire chapter that is a précis of Thomas Hobbes' philosophy. And all of this is related to the subject of how the concept of infinitesimals took hold in what I think is an original analysis. Up until halfway, I was skeptical of the various arguments and even doubted its veracity-I was looking for holes in Mr. Alexander's account of the whole thing.

How it could be that geometrical thinking suppressed the infinitesimal concept from taking hold is fascinating (not solely geometry). It also had a lot to do with groupthink between several groups and some of these groups can be thought of as coteries. Geometric thinking introduced to humanity the concept of the Proof, should there be differences between Geometric proof and the kind students study in a `transition to abstract mathematics' course today? "Infinitesimal" maps the course it took.
There is a joke in mathematics publishing that for every equation in a book, half the readers go away. There really are not a lot of equations in this book, mostly it is geometric figures with the accompanying explanations-my rough count is ten. My reading of the book confirms my disability to follow geometric arguments but for the vast majority of readers this would be welcome as they are clearly written by Mr. Alexander. I'm hesitant to give this book the full five stars, clicked on five stars anyways because the reduction is infinitesimal. Has a place in my book collection.
7 of 7 people found the following review helpful
4.0 out of 5 stars A Deeply Divisive Debate about Reality 31 Aug. 2014
By H. J. Spencer PhD, renegade-Physicist - Published on
Format: Hardcover Verified Purchase
This book is much more than an esoteric history of an area of mathematics. It tracks the ancient rivalry between ‘rationalists’ and ‘empiricists’. The dominant rationalists have always believed that human minds (at least those possessed by educated intellectuals) are capable of understanding the world purely by thought alone. The empiricists acknowledge that reality is far too complicated for humans to just guess its detailed structures. This is not simply an esoteric philosophical distinction but the difference in fundamental world-views that have deeply influenced the evolution of western civilization. In fact, rationalist intellectuals have usually looked to the logical perfection of mathematics as a justification for the preservation of religion and hierarchical social structures. In particular, the rationalists have raised the timeless, unchanging mathematical knowledge, represented by Euclidean geometry, as not just the only valid form of symbolic knowledge but as the only valid model of the logic of “proof”.

In particular, this book focuses on the battle between the reactionaries (e.g. Jesuits and Hobbes), who needed a model of timeless perfection to preserve their class-based religious and social privileges and reality-driven modernists, like Galileo and Bacon. The core of the disagreement was over the nature of the continuum, which was based on Euclid’s definition of a line as an infinite number of points. This intellectual argument implicitly links back to reality: is matter made of distinct atoms with empty space between them or are there no gaps between continuous matter? Although the model of the reactionaries was always Euclid's geometry, they never recognized they were only dealing with unreal definitions, as they faked out their arguments with appeals to 'real' lines etc. As such, they vigorously rejected the new concept of "indivisibles" (or "infinitesimals", the roots of calculus) and all ideas that were grounded in empirical studies of reality (like physics and the atomic hypothesis). Failure to admit debate about reality led Italy back into the Dark Ages while Northern Europe set off on the course of modernism.

As other reviewers have noted, this book would have benefitted quite a bit by including the story of the rivalry between Leibniz and Newton, who are usually credited with the invention of the calculus. As this book shows, this 17th Century rivalry had much older roots. Indeed, the book could also have been improved by establishing this acrimonious debate back in Classical Greece, where the atomic model, first proposed by Democritus, was immediately seen as an atheistic proposal that threatened traditional religion. The modern reader might assume that science has now firmly voted for the atomic model but the extensive use of the calculus embedded in Quantum Physics has preserved the conceptual features of the continuum advocates, so that we are now faced with the paradox of waves and particles. None-the-less, even readers with minimal competence in mathematics will enjoy discovering how this tiny idea of the infinitely small punctured an ancient dream: that the world is a perfectly rational place that is governed by strict mathematical rules.
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