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Infinitesimal: How a Dangerous Mathematical…

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Amir R. Alexander

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230578,394 (3.92)10
"The epic battle over a mathematical concept that shook the old order and shaped the world as we know it. On August 10, 1632, five leaders of the Society of Jesus convened in a somber 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's Infinitesimal is the story of the struggle that pitted Europe's entrenched powers against voices for tolerance and change. It 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. We see how a small mathematical disagreement became a contest over the nature of the heavens and the earth: Was the world entirely known and ruled by a divinely sanctioned rationality and hierarchy? Or was it a vast and mysterious place, ripe for exploration? 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--and celebrates the spirit of discovery, innovation, and intellectual achievement"--… (more)



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Infinitesimal is a quirky little book. Its basic thesis is that various responses to an arcane mathematical concept, infinitesimals, or the infinite amount of parts into which a line can be divided, somehow accounts for the political struggles of 16th century Italy and of 17th century Britain. To say the least, this is a bold thesis, and I remained unconvinced at the end, despite enjoying the author’s almost heroic efforts.

On a first level, the book is a political history of (a) the Catholic Church’s response to the Protestant Reformation and (b) Britain in the 1600’s, contrasting the two historical periods by their different approaches to mathematics. In the process, the author takes us through the founding and early days of the Society of Jesus (the Jesuit order) in the Counter-reformation of the 16th century, particularly on the Italian peninsula (there being no “Italy” as such in those days). He then moves on to the political and religious struggles in Britain in the next century.

Alexander’s retelling of the history is interesting, if very unconventional. In particular, his relation of the debate between Thomas Hobbes (as author of Leviathan) and John Wallis is told from an angle I had never seen before. Hobbes, who fancied himself as a great geometer (he thought he had “squared the circle”—later proved to be impossible), crossed intellectual swords with Wallis, who approached mathematics as an experimental pursuit. Their respective approaches to mathematics probably influenced their political philosophies in the way Alexander describes. Nevertheless, neither actually solved the problem of the infinitesimals—and that story is not included in this book.

Alexander has presented a history through a highly unorthodox, but stimulating, prism. He is probably correct that the concept of infinitesimals, which was expanded to become calculus, really did “shape” the modern world. After all, classical physics is expressed in the language of calculus. But the “shaping “ caused by that concept was scientific, mathematical, and intellectual—not political. Alexander’s retelling of the story leaves untold the final rigorous and elegant explication and derivation of calculus by Newton, Leibnitz, and Euler. And that explication and derivation had little or no political consequences.

This book is worth reading just for the history even if its attribution of causation is farfetched.

(JAB) ( )
  nbmars | Oct 19, 2015 |
I finished Infinitesimal: How a dangerous mathematical theory shaped the modern world by Amir Alexander, with mixed, if mostly positive feelings. Despite the title, the preponderance of diagrams, and the extended trips into the nature of number theory, this is not a math book. It's a history book, with math as one of the main characters in the drama that spans nearly three hundred years. Not being a math person myself, I still had no trouble following along Alexander's explanations and summations of the various heated, do-or-die "proofs" that were flung back and forth by the various proponents of this or that theory. But this is not a history of the discovery of a new kind of math (the theory of the infinitely small having laid the groundwork for the development of calculus). This is a history of why the discovery of a new kind of math was such a big deal, and so...well, "dangerous."

Basically it is the story of the struggle between those who ascribed to a Euclidean description of reality, and those who did not. Or rather, those who began with theory and applied it to reality, and those who began with the real world, and tried to deduce the theory that would describe it. The former is based in the creation of abstract proofs. The latter on deductions based on observation and experimentation. The former appealed to the authoritarian and conservative elements of the Catholic Church, which in the 15th century was running a rear guard action in the struggle for souls in the face of the rise of Lutheranism. Euclidean geometry might be called "the official math" of the church -- it was the only math taught, and more to the point, the only math allowed, in all those Jesuit-run schools that spread throughout Europe and ended up the church's best defense against Protestantism.

The problem, of course, with demanding that the real world be interpreted in terms of an officially sanctioned theory is that you run into trouble when something happens that doesn't fit that theory. In the case of Euclidean Geometry, that included any number of uncomfortable paradoxes (Zeno's arrow, etc) and the inability to accurately calculate things that would be really useful -- say, the volume of a spiral.

The theory of infinitesimals, of "the infinitely small" was developed in part to overcome these challenges -- which it does, beautifully, but only at the expense of Euclid's entire notion of perfect abstract form. The argument over infinitesimals...indeed, over the primacy of observation over abstract truth...is much of what fueled the dispute between the Church and Galileo and his circle. Galileo was no mathematician, but he was able to use the theory of the infinitely small to make sense of Copernicus's heliocentric model and his own astronomical observations. Kepler, too, used the method to help calculate the elliptical orbits that improved on the Copernican model of the motion of the planets. And in the end, the church's objections to Galileo and those in his circle were as much about those methods as they were about any given theory Galileo put forth.

Alexander makes a complex subject entertaining and interesting -- certainly it will appeal to people who like Dava Sobel, or Richard Holmes' The Age of Wonder. The book is especially strong in the second half, which is dominated by an account of the creation of the Royal Society, and the rivalry between the polymath Thomas Hobbes and his bete noir, John Wallis -- which reached a fever pitch that would have delighted the tabloids, if any had existed at the time. Apparently, the ability (or not) to back one's claim to be able to square the circle -- or double the cube -- was serious stuff, with wide-ranging political repercussions.

It's only here that I felt a twinge of skepticism about Alexander's central premise. That mathematical theories could become stand-ins and justifications for political philosophies and theological truths is easily understood. But that the fall of Italy from it's intellectual primacy during the Renaissance, and England's subsequent rise as an industrial power can be put down to Rome's refusal to embrace the theory of the infinitely small seems...well...rather infinitely stretched. Political powers rise and fall for a myriad of reasons, usually acting in concert. And in the end it isn't really clear whether the efforts to suppress a troublesome mathematical method was part of the cause, or simply a symptom of Italy's faltering vitality and its descent from center of the intellectual universe to a hidebound backwater left behind by the rest of Europe.
2 vote southernbooklady | Nov 25, 2014 |
Shows how the modern world was balanced between tyranny and chaos. ( )
  jefware | Nov 12, 2014 |
A fascinating book that illustrates the impact of politics and religion on mathematics. Very interesting how the rudiments of Calculus were believed by some long before Lenitz and Newton. ( )
  MathMaverick | Nov 1, 2014 |
My "reading" of this book was largely page skipping, because it's mostly [yechh] religion-oriented history (ROH). By my reckoning, only 85 of the 290 pages are actually about mathematics, discussing the form of calculus-like manipulations in the decades before Newton and Leibniz. It never gets to the dy/dx notation or the integral symbol, let alone the modern ideas about infinitesimals in nonstandard analysis. The ROH parts might be beneficial for readers who do not yet understand how harmful religion was (and is) in hindering knowledge and progress.
  fpagan | Jun 17, 2014 |
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No continuous thing is divisible into things without parts. - Aristotle
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On August 10, 1632, five men in flowing black robes came together in a somber Roman palazzo on the left bank of the Tiber River.
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