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Infinitesimal [Englisch] [Gebundene Ausgabe]

Amir Alexander

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3. Juli 2014
On August 10, 1632, five leading Jesuits convened in a sombre Roman palazzo to pass judgment on a simple idea: that a continuous line is composed of distinct and limitlessly tiny parts. The doctrine would become the foundation of calculus, but on that fateful day the judges ruled that it was forbidden. With the stroke of a pen they set off a war for the soul of the modern world. Amir Alexander takes us from the bloody religious strife of the sixteenth century to the battlefields of the English civil war and the fierce confrontations between leading thinkers like Galileo and Hobbes. The legitimacy of popes and kings, as well as our modern beliefs in human liberty and progressive science, hung in the balance; the answer hinged on the infinitesimal. Pulsing with drama and excitement, Infinitesimal will forever change the way you look at a simple line.

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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

Über den Autor und weitere Mitwirkende

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, among others. He lives in Los Angeles, California.

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Die hilfreichsten Kundenrezensionen auf (beta) 4.0 von 5 Sternen  49 Rezensionen
25 von 28 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.
58 von 73 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.
15 von 18 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen The battle between hierarchical and egalitarian worldviews 24. April 2014
Von Fred Erling Wenstøp - Veröffentlicht auf
Format:Kindle Edition|Verifizierter Kauf
This is one of the most interesting books I have read. It gives an insight into the battle of world views -- hierarchical versus egalitarian. And if it really is true that a sophisticated mathematical question could be a key point in this battle, it is really remarkable. The author makes you believe it with his lucid and entertaining style which makes you read the book without interruption.
6 von 6 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Fascinating and fun to read 20. Mai 2014
Von Third Norn - Veröffentlicht auf
Format:Gebundene Ausgabe|Verifizierter Kauf
Hard to put down. I wonder if it would be as much fun for someone who had not been a math geek in high school, as I was. That said, the math is not hard to follow. The book is more a social history than a scientific one.
104 von 141 Kunden fanden die folgende Rezension hilfreich
2.0 von 5 Sternen Fundamentally confused 14. April 2014
Von Polymath - Veröffentlicht auf
Format:Gebundene Ausgabe
There are two different debates in the foundations of mathematics that are confused here.

The first debate is whether the continuous line is made up of infinitely small points. This was settled by the development of the real number system which allowed each point on the line to be expressed algebraically by a decimal expansion.

The second question is whether the real numbers were enough or whether the line had additional points on it so that the corresponding number system failed to satisfy the Archimedan axiom (in other words, points greater than 0 but less than ALL the numbers 1, 1/2, 1/3, 1/4 , 1/5, ... ).

The second view was used by many mathematicians in the 17th through 19th centuries to develop calculus in a way that was NOT logically rigorous, and which was CORRECTLY criticized as incoherent by Bishop Berkeley and others, culminating in the logically rigorous and irreproachable treatment of Cauchy in the 1830's, further developed and modernized by Weierstrass and Dedekind a generation later, which used limits to avoid infinitesimals and allowed completely convincing proofs to be given.

It was not until Abraham Robinson in the 1950's demonstrated that infinitesimals could be consistent if one was much more careful than the 17th-19th century mathematicians had been, that alternatives to the standard real number system were taken seriously again.

Points infinitely small? Always ok. Numbers as intervals between points infinitely small? Properly rejected as non-rigorous and unnecessary prior to 1960; now accepted as capable of being made rigorous but still not necessary.

Modern physics casts doubt on the scientific necessity of the earlier view of infinitely small points on the line represented by the standard real numbers, because of the existence of a tiny fundamental length scale, so the argument is not settled by an appeal to physical reality.

The version of the real numbers we use, given by decimal expansions or rational approximations with no infinitesimals, really is logically better, as was first recognized by Archimedes two millennia before Calculus (which he invented) was rediscovered by Newton and Leibniz. Archimedes used infinitesimals as a mental shortcut to find results that he later made rigorous by using perfectly valid limit arguments of the kind Cauchy later generalized.

Confusing these two issues in order to contrast modern irreligious freethinkers with dogmatic old Christian fuddy-duddies and bask in political self-satisfaction is tendentious and fundamentally wrong.
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