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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Commentaires client les plus utiles sur Amazon.com (beta)
34 internautes sur 38 ont trouvé ce commentaire utile
A Brief History of the Jesuits and Infinitesimals That Needs Editing7 mai 2014
Steven R. Staton
- Publié sur Amazon.com
Format: Format Kindle
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.
20 internautes sur 22 ont trouvé ce commentaire utile
Misleading...4 juillet 2014
- Publié sur Amazon.com
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.
64 internautes sur 81 ont trouvé ce commentaire utile
Not exactly war of all againts all, but still a very damaging fight10 avril 2014
Mariano Apuya Jr
- Publié sur Amazon.com
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.
18 internautes sur 22 ont trouvé ce commentaire utile
The battle between hierarchical and egalitarian worldviews24 avril 2014
Fred Erling Wenstøp
- Publié sur Amazon.com
Format: Format Kindle
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.
105 internautes sur 143 ont trouvé ce commentaire utile
Fundamentally confused14 avril 2014
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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.