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Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World von [Alexander, Amir]
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Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Kindle Edition

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'Fascinating. Amir Alexander vividly recreates a wonderfully strange chapter of scientific history... You will never look at calculus the same way again.' -- Jordan Ellenberg, Professor of Mathematics, University of Wisconsin-Madison 'Clever and enthralling.' -- Simon Schaffer, Professor of the History of Science, University of Cambridge 'We thought we knew the whole story: Copernicus, Galileo, the sun in the centre, 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 'A seamless synthesis of cultural history and storytelling... The history of mathematics has rarely been so readable.' -- Michael Harris, Professor of Mathematics, Columbia University and Universite Paris Diderot '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: Fr 'Gripping... Amir Alexander writes with elegance and verve... A page-turner full of fascinating stories about the struggles of remarkable individuals and ideas, Infinitesimal will help you understand the world at a deeper level.' -- Edward Frenkel, Professor, University of California at Berkeley, and author of Love and Math 'A real-world Da Vinci Code' Publishers Weekly "[Told with] high drama and thrilling tension." Kirkus Reviews (starred review) 'A gripping tale of mathematical, philosophical, and theological controversies in the run-up to calculus.' Ian Stewart, author of Professor Stewart's Cabinet of Mathematical Curiosities "Bertrand Russell once wrote that mathematics had a 'beauty cold and austere'... Amir Alexander shows that mathematics can also become entangled in ugliness hot and messy... [a] fascinating narrative." New York Times 'A gripping and thorough history of the ultimate triumph of the mathematical tool... Infinitesimal will inspire you to dig deeper into the implications of the philosophy of mathematics and knowledge' New Scientist 'A complex story told with skill and verve... Alexander does an excellent job of presenting both sides of the debate.' THES Book of the Week 'Amir Alexander's enthralling book presents a controversial mathematical breakthrough, vividly describing the players and showing exactly what was at stake.' Tony Mann, Director of the Maths Centre, University of Greenwich and Former President of the British 'A well-spun yarn, a cracking read... engaging...unique' History Today 'fluent and richly informative' Literary Review


Pulsing with drama and excitement, Infinitesimal celebrates the spirit of discovery, innovation, and intellectual achievement-and it will forever change the way you look at a simple line.

On August 10, 1632, five men in flowing black robes convened in a somber Roman palazzo to pass judgment on a deceptively simple proposition: that a continuous line is composed of distinct and infinitely tiny parts. With the stroke of a pen the Jesuit fathers banned the doctrine of infinitesimals, announcing that it could never be taught or even mentioned. The concept was deemed dangerous and subversive, a threat to the belief that the world was an orderly place, governed by a strict and unchanging set of rules. If infinitesimals were ever accepted, the Jesuits feared, the entire world would be plunged into chaos.

In Infinitesimal, the award-winning historian Amir Alexander exposes the deep-seated reasons behind the rulings of the Jesuits and shows how the doctrine persisted, becoming the foundation of calculus and much of modern mathematics and technology. Indeed, not everyone agreed with the Jesuits. Philosophers, scientists, and mathematicians across Europe embraced infinitesimals as the key to scientific progress, freedom of thought, and a more tolerant society. As Alexander reveals, it wasn't long before the two camps set off on a war that pitted Europe's forces of hierarchy and order against those of pluralism and change.

The story takes us from the bloody battlefields of Europe's religious wars and the English Civil War and into the lives of the greatest mathematicians and philosophers of the day, including Galileo and Isaac Newton, Cardinal Bellarmine and Thomas Hobbes, and Christopher Clavius and John Wallis. In Italy, the defeat of the infinitely small signaled an end to that land's reign as the cultural heart of Europe, and in England, the triumph of infinitesimals helped launch the island nation on a course that would make it the world's first modern state.

From the imperial cities of Germany to the green hills of Surrey, from the papal palace in Rome to the halls of the Royal Society of London, Alexander demonstrates how a disagreement over a mathematical concept became a contest over the heavens and the earth. The legitimacy of popes and kings, as well as our beliefs in human liberty and progressive science, were at stake-the soul of the modern world hinged on the infinitesimal.


  • Format: Kindle Edition
  • Dateigröße: 4009 KB
  • Seitenzahl der Print-Ausgabe: 369 Seiten
  • Verlag: Scientific American / Farrar, Straus and Giroux; Auflage: Reprint (8. April 2014)
  • Verkauf durch: Amazon Media EU S.à r.l.
  • Sprache: Englisch
  • ASIN: B00FIL9B7E
  • Text-to-Speech (Vorlesemodus): Aktiviert
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Die hilfreichsten Kundenrezensionen auf (beta) 4.1 von 5 Sternen 91 Rezensionen
67 von 72 Kunden fanden die folgende Rezension hilfreich
3.0 von 5 Sternen Misleading... 4. Juli 2014
Von Timothy Haugh - Veröffentlicht auf
Format: Gebundene Ausgabe
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 von 57 Kunden fanden die folgende Rezension hilfreich
3.0 von 5 Sternen A Brief History of the Jesuits and Infinitesimals That Needs Editing 7. Mai 2014
Von Steven R. Staton - Veröffentlicht auf
Format: Kindle Edition Verifizierter Kauf
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 von 17 Kunden fanden die folgende Rezension hilfreich
2.0 von 5 Sternen Same Mistake As Hobbes 23. September 2014
Von bonnie_blu - Veröffentlicht auf
Format: Gebundene Ausgabe
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 von 94 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Not exactly war of all againts all, but still a very damaging fight 10. April 2014
Von Mariano Apuya Jr - Veröffentlicht auf
Format: Gebundene Ausgabe
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 von 7 Kunden fanden die folgende Rezension hilfreich
4.0 von 5 Sternen A Deeply Divisive Debate about Reality 31. August 2014
Von H. J. Spencer PhD, renegade-Physicist - Veröffentlicht auf
Format: Gebundene Ausgabe Verifizierter Kauf
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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