Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Amir Alexander

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.

Genres: History, Science, Mathematics, Nonfiction, Philosophy, History Of Science, Religion

353 pages, Paperback

First published March 1, 2014

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

Reviews

[Simon Clark · Rating 4 out of 5]

Infinitesimal is, at first glance a history of a mathematical idea. But it is much more than that.

The book is really an examination of authoritarianism in England and Italy in the 17th century, and how the state and the church, respectively, responded to a paradigm-changing idea. That idea was that a smoothly varying curve is actually composed of an infinite number of infinitely small straight lines. These days this is hardly a revolutionary idea, but it represented a radical departure from the already ancient mathematics of Euclid for 17th century mathematicians.

In Italy Galileo and his disciples embraced the idea, and faced fierce opposition from the Jesuits, a hardcore corps of the Catholic church. The church, reeling from the Lutheran doctrines and the rise of Protestantism, declared the idea heretical. Meanwhile in England the mathematician John Wallis similarly embraced the idea and was met head-on by Thomas Hobbes. In the aftermath of the English civil war the state had to answer similar questions of authority to the Catholic church, but reached radically different outcomes.

In many ways Infinitesimal is an account of the decline of one great power of Europe and the rise of another, with the seeds of the Industrial Revolution and the era of colonialism sown 200 years prior. It touches on many seemingly disparate strands of history and ties them together to form a compelling narratives with heroes, villains, humour, and real weight.

It should be noted that this isn't some frothy frolic through one aspect of the history of maths - this is a hefty book, both in page count and in density. While I enjoyed it a lot, and my appreciation of it has only grown in time, it sometimes can be a bit dense for its own good. The large list of references reinforces this - Infinitesimal is a meticulously researched book, almost with the feel of an academic publication rather than a popular account, and while it may get a bit carried away with restating the same point several times with different references, it would serve as a good model in the future for books dealing with similar subject matter to follow.

Aside from this, and a slightly confusing pseudo-chronological narrative in the first half dealing with Italy, I can highly recommend this book to anyone interested in the history of mathematics and its interaction with society as a whole.

[Katie · Rating 3 out of 5]

Let's all imagine a finite line. See it? Good.

Now, let's imagine that this line is made up of indivisible points. How many of them are there?

Option A: a LOT. But this runs us into a problem: even there are a billion points on a line, what's to stop someone from dividing them into two billion? Then four billion?
Option B: If we're going to divide forever as suggested in option A, perhaps there are a infinite number of indivisible points that make up the line.
Paradox: If we assume that these infinite points have any physical magnitude (even a tiny one!) the line would be infinite in length. If we assume that they don't have a physical magnitude, then the line shouldn't exist at all.

Hmmm.

Amir Alexander's Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World deals with the resurgence of this paradox at the start of the 17th century, after it had laid dormant for about 2,000 years. I'm a bit out of my element here (the 17th century is after the time I tend to read about, and math deeply intimidates me) but my overall impression is that Alexander has written a really fun an engaging book, but not a terribly nuanced one.

The central conceit is that in 17th century Europe, technical mathematical debates were inextricably intertwined with contemporary concerns about politics, religion, and social stability. This summation of Thomas Hobbes's attitude towards mathematical sums up the book's thesis and tone fairly well:

Mathematics, Hobbes insisted, must begin with first principles and proceed deductively, step by step, to ever-more-complex but equally certain truths... In this manner, Hobbes believed, an entire world could be constructed - perfectly rational, absolutely transparent, and fully known, a world that held no secrets and whose rules were as simple and absolute as the principles of geometry. It was, when all was said and done, the world of the Leviathan, the supreme sovereigns whose decrees have the power of indisputable truth. Any attempt to tinker with the perfect rational reasoning of mathematics would undermine the perfect rational order of the state and lead to discord, factionalism, and civil war. (286).

Mathematics, for all its abstractions, dealt with the principles that underpinned the world. Topple those, you topple everything else. Alexander explores these ideas in two main sections. The first deals with the Jesuit support of traditional Euclidean geometry and its remarkably successful silencing campaign against Italian mathematicians attempting to introduce the concept of infinitesimals. The second section focuses on the decades-long feud between Thomas Hobbes (proponent of traditional geometric proofs - mostly - and of a state in which all authority was ceded to a sovereign) and John Wallis (Parliamentarian and promoter of a new form of "inductive" mathematics).

Part One - after a truly lengthy diversion into the history of the Reformation and the Jesuit Order - first takes a look at Christopher Clavius, the man who brought mathematics to the Jesuits. As Jesuit schools spread all over the European continent (at least those that still maintained Catholic footholds), they had a very particular curriculum and a very particular hierarchy of subjects. Mathematics was not really a part of it. Christopher Clavius aimed to change that. He was a genuine fan of math, and saw it as an intrinsic part of the Jesuit mission and theology in general. Euclidean proofs were perfect, beautiful, ordered: they were demonstrative of God's plan in the world, the secret but certain underpinnings of the universe. This was particularly attractive in a Post-Reformation society often rent by uncertainty. Math was clear and certain proof of order, and Clavius used this spin to make mathematics a primary part of Jesuit education.

Right around this same time, though, other mathematicians in Italy and abroad were discovering the advantages of the use of infinitesimals. These mathematicians were led by, among others, Bonaventura Cavalieri. The use of infinitesimals began to arise in Italian mathematics as a convenient way of determining areas and comparing figures that would otherwise be difficult to calculate in accordance with traditional mathematical methods. Essentially, Cavalieri introduced the ideas that the area of a figure could be equated to "all its lines" and then that these lines could be compared to each other. It took a few additional steps from there (that I don't really understand because I'm functionally illiterate in terms of mathematics) to reach the calculus created by Leibniz and Newton. This approach, however, was too filled with paradox and uncertainty for the Jesuits and the order systematically shut down all its proponents. In its most extreme example, they were even able to convince Clement IX to shut down the entire Jesuat Order, which had existed for 300 years, because of the mathematical view of Cavalieri and several other members.

Part Two swings over to 17th century England, during and after the political upheavals of the English Civil War. This mathematical dispute, between John Wallis and Thomas Hobbes, is the more interesting of the two. Alexander paints part one as a fairly standard "religion vs. science" narrative, for better and worse. In England, though, matters are trickier and more secular.

Thomas Hobbes became a proponent of a slightly-altered version of Euclidean geometry fairly late in his life, and Alexander paints it as an outgrowth and bolster to his political theory. Hobbes's conception of the Leviathan state - that the populace cedes its sovereignty to a single centralized ruler in order to prevent the mayhem and chaos of the state of nature - was one that believed in a strictly ordered universe. Order and certainty were bulwarks to disaster. Alexander posits that Hobbes saw mathematics, especially geometry, as a kind of kindred spirit, a realm of logic, proof, and certain outcomes. It was because of this that Hobbes became somewhat obsessed in the last decades of life to solve by Euclidean means a handful of problems that had remained unsolvable, the chief of which was squaring the circle. Alexander's Hobbes is a kind of eccentric perfectionist, believing that a successful squaring of the circle would somehow save England from the Diggers.

Unsurprisingly, then, the man had little room for those weird, paradoxical infinitesimals. Enter John Wallis, who managed to become a professor of mathematics at Oxford somehow after being taught accounting by his little brother and working for a while as a government code-breaker (it was probably a political appointment). Wallis's math was abhorrent to Hobbes: he played around in paradoxes, he divided things by infinity. What a mess. But it was on purpose: Wallis's view of what mathematics should be was drastically different from Hobbes's. Instead of a beautiful bulwark of order, Wallis's math was investigative. It aimed to stir up controversies and push at the borders of what was known to try to discover something new. It wasn't a coincidence that Wallis was heavily involved with the emerging Royal Society in London: just as the Society would set up public experiments and debate the meaning of its results, Wallis would play with infinitesimals, experiment with different mathematical relationships and methodologies, in the hope that he would discover something new. It was inductive mathematics, producing results that were likely or probable. Hobbes never accepted it.

It's a fascinating book, and I learned a lot from it. I toyed with giving it four stars just out of enjoyment, but the history stickler in me wound up insisting on three. Alexander often states his case far too strongly and bluntly: the relationship between mathematics and the State or the Church is central, but it's rarely backed up with explicit examples from the time. And the books conclusions take things rather far, by suggesting that because the Jesuits shut down Italian exploration of infinitesimal mathematics they essentially murdered modernity in Italy and turned into a poor, sad, backwater. He then turns around and says that because infinitesimals were accepted in England, a place that had previously been a cultural backwater (!!!) became a leading engine of modernity. There's some truth in all that, of course, but it's drastically oversimplified. A little nuance could have gone a long way.

[Jamie Smith · Rating 4 out of 5]

If you are a mathematician, the word infinitesimal may conjure up a range of topics, but if you are a layman like me, calculus is what comes to mind. And so, I picked this book up thinking that it would be a history of the development of calculus and the rival claims of primacy made by Newton and Leibniz. Not so; calculus receives only a passing mention at the very end of the book.

As it happens, questions about infinitesimals go back to antiquity, and rival approaches to interpreting the concept in the seventeenth century would divide Europe and change the course of intellectual history. Consider a line segment of length x. How many slices can you divide it into? An infinite number, perhaps? Now take another segment, of length 2x. Does it divide into 2 times infinity, or are the infinite slices from the first segment somehow stretched? And if they can be stretched then why couldn’t additional infinitely small slices can be placed between them? The word “infinite” starts to be hard to pin down.

Questions like this may seem interesting, even mildly amusing, but at one time they were the center of a contentious and high stakes philosophical and theological debate. The Catholic church was gravely concerned that open ended concepts like infinity might break down the well ordered Aristotelian foundation by which the world was believed to run. The consequences of this debate would re-order the mathematical world, shifting its center of gravity to the north, where future advancements would take place. The Catholic south, once the center of mathematics, would become a scientific backwater.

This book addresses its subject in the broader context of the Reformation, which was already sundering Christendom. The Jesuit order became the Catholic church’s bulwark in defense of the faith, and they did a magnificent job, reclaiming much of Germany, Hungary, Poland, and Romania from Protestantism. They were famous for their schools, the finest in Europe, and even noblemen and rich merchants who were not Catholic sent their sons to be educated in the Jesuit tradition. Their pedagogical traditions, however, followed narrow and rigidly defined criteria. As the author says, “The Jesuits believed that the purpose of education was not to encourage the free exchange of ideas, but to inculcate certain truths.” (p. 55)

Mathematics was not initially seen as an important part of the Jesuit schools. It was thought that students needed only enough to do simple arithmetic, but it was a plebeian field of study, useful for merchants doing their accounts, of little interest to those training to be society’s future leaders. It was Christopher Clavius who brought math into the Jesuit tradition, convincing the order that it could be used to positively prove god’s presence and orderly benevolence.

This leads the book to a discussion of inductive versus deductive reasoning, which the author handles well. Deductive logic starts from the whole and breaks it down into its components, and can lead to certainty. After all, with Euclid’s geometry proofs you can, for example, prove to anyone capable of understanding reason that the sum of the interior angles of a triangle equal 180 degrees. It is an airtight argument, and Euclid, starting from the simplest of principles, deduced one theorem after another, explaining the world in a way that is still taught 2300 years later.

Not all questions can be answered deductively, however. Sometimes you don’t know the final form, and instead have only scattered observations and data, which you are trying to fit into a coherent whole. This requires induction, which does not give certainty because it is entirely possible to put the pieces together in such a way that creates a solution which is elegant, comprehensive, enlightening – and wrong. This is not a flaw in the inductive approach, which is, after all, the way modern science works, but it can never reach the level of certainty which deduction can provide.

Galileo strongly supported the idea of using infinitesimals as the best way to understand the world as it actually works, but this method involved inductive logic, so naturally the Jesuits abhorred it, unable to accept the idea that god may have left unresolvable gaps in creation, which could lead people to start wondering if there was a creator at all, or if perhaps other explanations for time and space needed to be explored.

The Jesuits insisted that truth must be one, and in Euclidean geometry they believed they had found the perfect demonstration of the power of such a system to mold the world and prevent dissent. The Galileans also sought truth, but their approach was the reverse of that of the Jesuits: instead of imposing a unified order upon the world, they attempted to study the world as given, and to find the order within. And whereas the Jesuits sought to eliminate mysteries and ambiguities in order to arrive at a crystal-clear, unified truth, the Galileans were willing to accept a certain level of ambiguity and even paradox, as long as it led to a deeper understanding of the question at hand.(p. 176-177)

Initially, none of this mattered to anyone except mathematicians and philosophers. The Church was confident enough in its hold over hearts and minds that it could ignore, or even show bemused tolerance toward these kinds of idea, and people could write and publish as they wished. As the Protestant revolt took hold, however, enforcement of orthodoxy became a grim business, with transgressions punished by fines, imprisonment, and even the dreaded Inquisition. Galileo, who had also been a strong supporter of the new infinitesimal approach, was hauled before the Inquisition based on his writings about the Copernican system. He was an old man, in poor health, but was shown the instruments of torture which would be used on him unless he recanted, so he did (E pur si muove!).

Anyone who wanted to write on the new science of infinitesimals either needed to find someone to publish his work anonymously, or have it printed in the north, and risk his name getting back to Church authorities. Not surprisingly, Catholic controlled Europe settled into dogmatic orthodoxy, and left new ideas about science and math to the Protestant north, where they flourished.

The scene then shifted to England, and two of the most remarkable self-taught mathematicians of the age. One of them was Thomas Hobbes, known today as the author of Leviathan, but a brilliant polymath, and the book takes time to examine him and his life. Leviathan shocks and appalls modern readers, because the twentieth century showed how repressive and murderous totalitarian rule could be. It was not like this in Hobbes’ time. England had recently emerged from civil war, where chaos and sectarianism had swept the land. Hobbes considered what he knew of history and of human nature, and concluded that only an absolute sovereign, even a bad one, could keep order, and only order could provide the protection needed for a stable society. His book lays out his premises one by one, building to a conclusion that many philosophers still find troubling but unassailable. We can abhor the reality of totalitarianism, but don’t bother trying to find logical inconsistencies in Hobbes’ support for it – better men than you and I have tried and failed.

His foil was John Wallis, who was a completely unqualified political appointee when he was given a post at Oxford to teach mathematics, but who threw himself into its study and became one of the finest mathematicians of his age. Wallis was a strong supporter of the use of infinitesimals while Hobbes, not surprisingly, hated them. The two men spent half a century sniping at each other in print, but it was ultimately Wallis who had the final say. Hobbes believed, as did the Jesuits, that deductive logic as applied to geometry could solve any problem, and show the workings of god to man. This appealed to his need for order and regularity, but unfortunately, the real world doesn’t work that way, and he made himself ridiculous publishing proposed solutions to things like squaring the circle.

The book ends, not surprisingly, with the introduction of Newton, who would take the concept of infinitesimals (which he called fluxions) and create derivative calculus. I will not get into the controversy with Leibniz other than to remark that it was a fairly sordid affair on the part of both men, but it is Leibniz’s symbology that is used for the most part in modern calculus. The real value of this book is not that it is about calculus, but about how great thinkers laid the foundation for its development. Newton did not pluck the concept out of thin air. When he said that he had stood on the shoulders of giants, these were the giants he was referring to.

[James Swenson · Rating 3 out of 5]

Background: I'm a professional mathematician, but not a historian of mathematics.

The topic is fascinating: who would have guessed that a culture war could have raged for a century over whether lines were or were not made up of points?

I enjoyed the author's analysis of how various mathematicians took sides on this issue that corresponded to their ideas about social structures and government, and the political and religious upheavals through which they lived. The claim is that mathematicians who supported kings and popes insisted on mathematical ideas that could be traced to Aristotle (but, unfortunately, are now understood to be wrong), while the forebears of the Enlightenment adopted powerful, but radical, new techniques (though their logical foundations were not yet fully in place). By the end, however, those arguments became frustatingly repetitive; I think it would have been fairly easy, and beneficial, to cut about fifty pages.

It is annoying, too, in a book with this title, that the terms "infinitesimal" and "indivisible" have been consistently conflated. I would love to see the author address the concrete question: Is a point infinitesimal? [For what it's worth: According to the modern technical usage, the answer is no.] I don't know enough history to decide if I should simply blame the author for this lack of clarity, or if it reflects a general confusion among the mathematicians of the time, studying concepts that would not be fully distinguishable for another century.

The unforgivable shock was that a chapter on the dispute between philosopher Thomas Hobbes and Newton's teacher John Wallis was followed by the word: "Epilogue." It's as if the author takes it for granted that everyone knows how this story ends. I think, though, that a book that so successfully draws the battle lines across which Aristotle and the Jesuits faced Galileo and the Royal Society is incomplete without some description of the ideas of Weierstrass, Cauchy, and others, which gave us our modern conception of the continuum.

[Bonnie_blu · Rating 2 out of 5]

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.

[Avery · Rating 3 out of 5]

There was a great opportunity here to describe the real thinking of Enlightenment mathematics. Alas, it was missed. The writer tells us history with the smug hindsight of someone who knows how math "really" works, and the result is a pretty bog-standard Galileo narrative. To really understand what the debate was all about, I recommend René Guénon's book The Metaphysical Principles of the Infinitesimal Calculus.

[Amelia · Rating 4 out of 5]

A detailed, sometimes scathing, and occasionally hilarious account of the tumultuous shift in mathematical thinking from classical geometry to modern calculus, this book focuses on the period from Martin Luther to the end of the 18th century. Furthermore, the tale mostly focuses on events in Italy (including great Italian mathematicians, the Jesuits, the Jesuats and the Pope) and events in England (including the Glorious Revolution and the founding of the Royal Society).

The intellectual battle, across Europe and time, about the controversial concept of the infinitesimals has shaped much of humankind's future, and certainly deeply affected the development of Italy, Great Britain, mathematics as a discipline, and the progress of science and industry in general.

This book is well worth reading for anyone interested in history, mathematics, science, or progress, political or religious institutions, or the European intellectual climate of the 16th and 17th centuries. While the actual mathematics discussed is not difficult (I'm certainly no mathematician), the breadth of history and ther personalities described are fascinating. The writing is detailed, smooth, and captivating.

[Brian Clegg · Rating 3 out of 5]

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. He also portrays the mental battle between philosopher Thomas Hobbes and mathematician John Wallis in a very interesting fashion. I knew, for example, that Wallis had been the first to use the lemniscate, the symbol for infinity used in calculus, but wasn't aware how much he was a self-taught mathematician who took an approach to maths that would horrify any modern maths professional, treating it more as an experimental science where induction was key, than a pure discipline where everything has to be proved.

Hobbes, I only really knew as a name, associated with that horrible frontispiece of his 'masterpiece' Leviathan, which seems to the modern eye a work of madness, envisaging a state where the monarch's word is so supreme that the people are more like automata, cells in a body or bees in a hive rather than individual, thinking humans. What I hadn't realised is that Hobbes was also an enthusiastic mathematician who believed it was possible to derive all his philosophy from geometry - and geometry alone, with none of Wallis' cheating little infinitesimals. The pair attacked each other in print for many years, though Hobbes' campaign foundered to some extent on his inability to see that geometry was not capable of everything (he regularly claimed he had worked out how to square the circle, a geometrically impossible task).

Although I enjoyed finding out more about the historical context it's perhaps unfortunate that Alexander is a historian, rather than someone with an eye to modern science, as I felt the first two sections, which effectively described the winning of the war by induction and experimentation over a view that expected mathematics to be a pure predictor of reality, would have benefited hugely from being contrasted with modern physics, where some would argue that far too much depends on starting with mathematics and predicting outcomes, rather than starting with observation and experiment. An interesting book without doubt, but not quite what it could have been.

[Mark · Rating 5 out of 5]

I’m trying to make up my mind whether Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World by Amir Alexander is a religious, political, and social history with a mathematics theme, or a focused history of an important branch of mathematics irresistibly gift-wrapped in colorful religious, political, and social packaging. Ultimately, it is of no consequence in this deeply-absorbing and tantalizing tale of a world on the brink of discovering calculus.

Alexander explains: “In its simplest form the doctrine states that every line is composed of a string of points, or “indivisibles,” which are the line’s building blocks, and which cannot themselves be divided. This seems intuitively plausible, but it also leaves much unanswered. For instance, if a line is composed of indivisibles, how many and how big are they?” Even if we say there are a billion billion, it still leaves us with indivisibles of positive magnitude, meaning they can be further divided.

The other possibility, says Alexander, “…is that there is not a ‘very large number’ of indivisibles in a line, but actually an infinite number of them.” But then, laying these infinite indivisibles side by side would create a line of infinite length, contradicting that the original line was finite in length!

The battle lines were drawn for two schools of support: those who accepted the concept of the infinitely small and those who didn’t. In part 1 of the book, where the battleground in 16th-century Italy, Alexander describes the fight between the Society of Jesus, or the Jesuits, who are the anti-infinitesimalists, and Galileo and his followers, the infinitesimalists. In part 2, England is the battleground and the combatants are two 17th-century mathematicians, Thomas Hobbes and John Wallis. Numerous other participants enter the fray, for and against: other mathematicians; popes such as Leo X, Paul III, and Clement IX; rulers and influentials such as Charles V, Swedish King Gustavus Adolphus, Oliver Cromwell, and Charles II; and “other reformers, revolutionaries, and courtiers” such as Martin Luther, Charles Cavendish, and Samuel Sorbière.

How Alexander chronicles the debates, the political and religious skirmishes and their outcomes makes for stirring reading. His research is meticulous and credible, his writing elegant and lucid despite a knotty and subtle mathematical concept. Along the way, readers are treated to a dense yet concise and enormously interesting history of such organizations as the Society of Jesus and The Royal Society of London, both critical players in the high-stakes conflict. As well as passion and drama surrounding the mathematical dispute at large, Alexander’s recounting of this story is not without style and humor.

Two thoughtful and valuable sections included in the book are: a list of the dramatis personae, with a short biography of each and with whom they sided; and a detailed time line placing all events in their proper sequence and context. The author has blended history and science into a fast-paced, informative page-turner. With this book as a commercial, I would love to be a fly on the wall—or, better still—a student in the classroom of one of Professor Alexander’s lectures!

[❀ Diana ❀ · Rating 3 out of 5]

Pentru cineva care a fugit de matematică (în special de geometrie) pe tot parcursul școlarității, Amir Alexander a reușit, relativ, să-mi atragă atenția asupra subiectului legat de infinitul mic.

Sec. XVI și XVII au fost pentru Europa un start cam anevoios în cursa pentru supremație în materie de știință; italienii erau în vârful muntelui, germanicii se delectau cu de-ale lor, englezii își dădeau cu stângul în dreptul până la Hobbes, iar restul continentului era aparent în repaus. Până la conflictul de proporții dintre Hobbes și Wallis cu privire la cuadratura cercului, înaintea lor au fost iezuiții și iezuații care se „atacau” între ei: Aristotel era depășit și mulți nu puteau accepta o reformă a matematicii (cu toții știm CINE avea un cuvânt de spus cu privire la orice).

În anul 1651 apare capodopera lui Hobbes, Leviatanul, și de aici începe Anglia să pună stăpânire pe matematică și să-i lase pe italieni să-și vadă de treaba lor în peninsula lor de cizmă. Odată ce tânărul Wallis intră în concurența pentru noi idealuri matematice și-l umilește pe Hobbes după publicarea cărții De corpore, englezii își asigură contribuția la schimbarea ireversibilă a lumii.

[William Bies · Rating 4 out of 5]

The science writer Amir Alexander has a knack for portraying the intellectual controversies pertaining to the science and mathematics of by-gone eras so as to bring out the significance to the larger cultural matrix they would have had for the contemporary participants, which we, at our distant remove, may well not be that sensitive to. Not only that, he seems to have enough of a synthetic overview of the historical record to fasten his attention on things that might not be familiar to the interested layman, even one who is fairly knowledgeable. A good example of this is his treatment of the Galois legend—the basics of which everyone has heard of, but from which he draws out aspects of the affair and its context that are hardly common knowledge—in his Duel at Dawn: Heroes, Martyrs and the Rise of Modern Mathematics from 2010. Alexander’s more recent Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World from 2014 does not disappoint. This work concerns itself with the role played by infinitesimals in the astonishing rise of mathematics in western Europe during the seventeenth century, the time of the so-called scientific revolution in which the discovery of the calculus figured as an instrumental part.

Now, what was the controversy about? Isn’t mathematics the model discipline in which, supposedly and unlike every other field, indubitable consensus about the rightness of a novel result can be reached, at least among the experts? Not quite so, as Alexander deftly demonstrates. The question at issue is whether the continuum can be analyzed into atomic constituents, or indivisibles. If mathematicians had been content to follow the precedent set by Archimedes back in antiquity, there would have been nothing to fight about. Archimedes’ method of exhaustion—known to us from a palimpsest recovered only as late as 1910—consists in, say, breaking down the volume of a frustum into layers that are, nominally, infinitely thin and then adding up their areas to arrive at the volume of the solid figure one starts with. This procedure is questionable, on the face of it: for isn’t the outer surface one calculates with a series of jagged steps instead of smooth, as it should be? With the tools available to the ancient Greeks, the rigor of this approach can certainly be debated, and was. Yet, Archimedes’ brilliant contribution was to side-step the legitimate question as to the rigor of employing infinitesimals. Rather, he surmises what the result has to be based on this heuristic procedure, but then proves it with unquestionable rigor (assuming the law of the excluded middle as given). For, he shows that, if the volume were either larger or smaller (by any arbitrarily small amount) than the postulated answer, it would lead to a contradiction; ergo, the answer has to be right!

In Italy at the juncture Alexander considers, the late Renaissance, the Greek mathematical tradition had long since been entirely recovered and innovative new work was in progress. Thus, everyone would have known very well about Archimedes and his method of exhaustion—despite the paucity of extant manuscript evidence. To be sure, the method of infinitesimals as used by Archimedes had an air of disreputability to it, but there was no disagreement about the correctness of the answers it yields. So: something else new must have happened, or Alexander would not have the materials about which to write a book.

Why, then, were the protagonists locked in such a bitter dispute? First, a little background. To set the stage for the later conflict, Alexander relates how mathematics won a place previously denied it in the curriculum at the honored Jesuit colleges, as a result of the sponsorship of the Jesuit mathematician Clavius. The view before then had been that only natural philosophy is truly scientific and demonstrative, because it deals with the causes of the phenomena, whereas the mathematical sciences such as astronomy had, traditionally, been concerned only with calculation and prediction according to an unproved—this is the grave point—formulaic model of the phenomenon. Clavius’ motivation for promoting the educational status of mathematics lay in the certainty of its procedures, on which all could agree, as opposed to the interminable strife among competing opinions characteristic of the natural sciences. In Alexander’s retelling, and he is persuasive in this, that Clavius’ attitude garnered acceptance among his superiors must be attributed to the confessional situation then prevailing in Europe, in the age of the Counter-Reformation. Roman Catholics, among whom above all the Jesuit order took the intellectual lead, were concerned to restore order in response to the societal chaos that had been unleashed over the course of the previous century by the Protestants, first of all by the magisterial Reformers and later, even more destructively, by the multitudinous sects that had cropped up. A mathematical curriculum modeled on Euclid’s Elements promised to inculcate in young minds a respect for order and intellectual rigor that could only benefit the Catholic position. That is why, by the acme of Clavius’ career, mathematics had established itself as a full equal of the other sciences, with institutional privileges for its teachers to go with such an improved standing.

To return to the controversy over infinitesimals. The method of exhaustion always had the drawback that one has first to have a putative answer to the problem, gained by whatever heuristic means or even by guessing, before one can apply Archimedes’ reductio ad absurdum in order to obtain a theorem. Now, at the time, most of the leading mathematical minds were Italian. At the prompting of Cavalieri, Torricelli and Galileo, a more robust approach to the indivisibles became popular in the early decades of the seventeenth century. They succeeded in solving quadrature problems that had eluded Archimedes, with this difference: they no longer insisted upon an absolutely rigorous demonstration to verify the answer suggested by the heuristic method. In consequence, they promoted a philosophy of mathematics in which an heuristic approach, complemented with careful criticism of the potentially paradoxical results it can generate if applied incautiously, was sufficient. This novel take on what the very mathematical enterprise consists in could be justified in light of the spectacular results it yielded, in the decades during which the groundwork of what was to become the calculus was being elaborated.

Yet, a cavalier and freewheeling approach such as this was unacceptable to the establishment Jesuit mathematicians, because it tends to render dubitable the absolutely reliable order they prized in the mathematical endeavor as a paragon of all correct and responsible thought. Hence, the stage was set for the great controversy that forms the subject of Alexander’s investigations, what he calls the ‘battle of the mathematicians’ or the ‘war on the infinitely small’. His account of the history of the controversy, its major players on both sides and the vicissitudes endured by the proponents of the indivisibles, who ultimately lost out, is gripping. What makes the tale tragic is that each side was in possession of understandable motives, and the stakes were high. Alexander does a very good job of describing the perspectives of the respective parties to the dispute, in view of the intellectual, societal, religious and political context. Based solely on Alexander’s account, it does look as if the Jesuits engaged in some fairly nasty academic politics, which turns out to have had deleterious consequences for the vitality of intellectual life across Italy for centuries.

The second half of the present work changes the scene and examines how the very same controversy played out in England, with a divergent outcome. Here, the immediate context was not just the Counter-Reformation, but also the upheavals associated with the English revolution and restoration and its aftermath. Thomas Hobbes was the main protagonist on the side corresponding to the Jesuits in Italy. Though known today as a political philosopher, he was, in fact, a reputable mathematician in his day, who saw in Euclid much the same advantage that Clavius did, as pertains to its promise to uphold societal order, or what he calls Leviathan, which, as everyone knows, was Hobbes’ primary concern. On the opposing side, the defender of infinitesimals was John Wallis, who started out his career as a politician but somehow secured a tenured position as a professor at Oxford, whereupon he felt driven to prove his credentials as a mathematician. And, indeed, so he did, marvelously well; everyone remembers Wallis today as an immediate forerunner of Newton. As befits his background—he learned his mathematics not at the university but from tradesmen, who needed it for strictly practical applications like accounting and actuarial estimates—Wallis’ philosophy of mathematics was of the rough-and-ready, heuristic kind, in which proof was not even deemed necessary. For him, mathematics (much like string theory today) involves probable or what could be called experimental knowledge, not demonstrable theorems, in analogy with the natural sciences on Francis Bacon’s reckoning. Alexander’s sketch of the contrasting careers and mathematical philosophies of Hobbes and Wallis is excellent. In the end, after years of animated controversy, Hobbes disgraced himself with some inept claims and Wallis triumphed. Newton and Leibniz are his intellectual heirs, who brought forth the calculus as we know it today.

Alexander’s extensive, though digressive, treatment throughout the book of social, political and military developments that impinge upon the controversy is competent, as are as well his pocket biographies of the personalities involved, and makes for interesting reading. There are no glaring errors or misrepresentations that this reviewer noted (admittedly not an expert historian); one feels one is in the company of a scholar raconteur who knows his field and its wider significance very well, even if he may not be a professional historian. Nothing too visibly amateurish or dilettantish, unlike what one finds in, say Strogatz or most popular writers who attempt to describe the history surrounding a scientific discovery (q.v. this recensionist’s review of the latter’s Infinite Powers on the origins of the calculus during the same period treated here).

Two lessons may be drawn from Alexander’s narrative. First, the crucial importance of academic freedom. Although Alexander goes too far in putting forward the Jesuits’ defeat of their indivisibilist opponents as nearly a monocausal explanation for the societal and cultural backwardness of Italy for centuries during the modern period, after its star had shone so brightly during the Renaissance, he does make a persuasive case that it did retard the standing of the country in the sphere of mathematics and the natural sciences, at least, in comparison to the leap taken in northern Europe shortly thereafter. Not until the second half of the nineteenth century, with few exceptions, did Italians begin to distinguish themselves again in the upper echelons of the mathematical world. Nevertheless, the question of the proper fate of rigor was not decided in England, either (in the negative, that would be); not until the nineteenth century did the calculus, which as we have seen lay on a rather shaky, heuristic foundation at the time of its discovery, receive a rigorous formulation that could satisfy the exacting demands of classic mathematical practice according to the antique Hellenic ideal at the hands of, inter alia, Cauchy, Dedekind and Weierstrass (it would not be altogether amiss to characterize Wallis, Newton, Leibniz, Euler and so forth as neo-Babylonians or neo-Egyptians as far as their methods are concerned). This observation should suffice to make clear, what to many less discerning might seem counterintuitive, that the freewheeling style of the calculus’ founders is not by any means the only way to do creative mathematics, or creative work in general (en garde, Richard Feynman!). The prodigious advances that took place during the twentieth century, in all fields, would not have been possible but for the absolutely rigorous foundation erected during the previous century (cf. also the twentieth-century Bourbaki school, which has done so much in the spirit of Cauchy and Weierstrass for present-day research practice, if not even more; q.v. the forthcoming companion review of Jeremy Gray’s Plato’s Ghost).

Rather than suppress their indivisibilist opponents, the Jesuits ought to have been prepared to wait two hundred years or, if more impatient and enterprising, to promote teachings and to foster an educational atmosphere that would accelerate progress along the lines they favored. Surely, if Cauchy and Weierstrass had existed in that earlier epoch, they would have been powerful supports to the overall vision held by the Jesuits for the philosophy of mathematics.

Second, we encounter here a telltale warning about the limits of popular expositions of intellectual disputes, whatever they may be. Alexander signally fails to engage the properly mathematical issue at hand in the controversy over infinitesimals, apart from his descriptions of the arguments pro and con as they stood in the seventeenth century. Can we comprehend the continuum at all, or: is Russell right after all, contra Aristotle? In the Aristotelian view, it makes no sense to speak of atomic constituents of the continuum, as that would mean an actual infinity. Instead, the continuum must be pictured as indefinitely divisible, what would be a mollified, potential infinity. Russell, and with him the overwhelming majority of current-day mathematicians, would counter that, after Cantor, we can indeed envision an uncountable set of discrete points, which we can arrange into an ordered field without any gaps, a.k.a. the real number line. True (and this is testimony to the greatness of the human spirit), but these logical formalists are not necessarily right to demote the place of spatial intuition. For, if nothing else, Robinson’s non-standard analysis and Conway’s surreal numbers show that we can quite well imagine the continuum in other ways than Dedekind’s canonical arithmetization, on alternative axiomatic foundations (non-Archimedean). Therefore, this reviewer will propose a continued neo-Kantian role for intuition in suggesting hitherto unimagined possibilities. Yet, Russell and his camp are partly right; once duly formalized, these new perspectives will be brought once again under the logical formalist umbrella. Nothing about these matters to be found in Alexander’s text; speculation on the future as opposed to scholarly reconstruction of the past falls outside his sphere of competence.

[Michael Burnam-Fink · Rating 4 out of 5]

Infinitesimal tells an interesting story about the genesis of the scientific revolution through the rather esoteric creation of the indefinitely small and indivisible. The idea of infinitesimals had been around for ages: Zeno's paradoxes are based on the idea that space can be sliced smaller and smaller. And for us, they're literally high school math. dx/dy and all that.

In the 16th and 17th century, disorder was at the top of mind for many elites. The Catholic Church had been rocked by the Protestant Reformation, and only belatedly offered theological counters. The elite intellectual shock troops of Catholicism was the new Jesuit order, which combined rigorous philosophical learning with absolute obedience to hierarchy, going up to the Jesuit Governor General and then the Pope. The Jesuits relied on a strict censorship regime to maintain this order, and sought an intellectual underpinning in Euclid's geometry. Geometry was both old (and therefore safe) and promised a perfectly rational and ordered system. In England, Thomas Hobbes, a tutor to aristocratic, political philosopher, and amateur mathematician, pursued a similar vision of absolute order in his Leviathan. Hobbes was also fascinated by the promise of geometry to create a perfect order.

Against absolute order, a few mathematicians postulated another way of thinking. Perhaps lines were made up of an infinite series of points. Planes were made of lines next to each other, like a sheet of paper. Solids were like a book of many sheets. The Italian branch of this school included Galileo, Torricelli, and Bonaventura Francesco Cavalieri, a mathematician of the Jesuat (note the "a") order. In England, Hobbes' main opponent was John Wallis, a member of the nascent Royal Society.

As Alexander discusses, the stability of Euclid's geometry was intellectual tied to political and theological stability. In Italy, the Jesuits had enough authority to have Galileo sentenced to house arrest. Torrecelli died before 40 of a fever. Cavalieri, who wrote the first major book on infinitesimals, was dealt with by having the entire Jesuat order dissolved.

Events in England followed a very different course. Hobbes was successfully baited by Wallis, as Hobbes erroneous claimed he'd "squared the circle", a problem which was later found to be impossible via Euclidian means. Wallis was a decent mathematician and a consummate political operator, who over decades saw Hobbes sidelined and ridiculed.

While it may be much to ascribe single causes (and Alexander is careful not to), Italy stagnated under Jesuit intellectual rigidity, becoming a poor backwater. England birthed the scientific and industrial revolutions, developments which would have been impossible without calculus.

[Dave · Rating 2 out of 5]

I had high hopes for this one. I'm a retired math teacher who taught calculus for decades, and infinitesimals form the basis of the calculus, as well as much of modern mathematics. I did enjoy the history involved in the book, from the geopolitical history of Renaissance Europe, to the institutional history of the Roman Catholic Church, to the English Civil War and the Restoration, but I kept hoping for more on the mathematics. I don't think the author did a very good job clearly explaining either the nature of infinitesimals, or the deep reasons why the concept was truly controversial, especially for a lay (non-mathematical) audience. Also, and this is a pet peeve of mine, throughout the book, the author uses the expression "straight lines", which I was always taught, and which I, myself, taught, should be avoided as redundant. "Straight line" sounds as silly to me as "round circle". All lines are straight.

[William Schram · Rating 4 out of 5]

As a person from the modern era, it is difficult to understand how people thought of things five hundred years ago. It is also difficult to understand how something as innocuous as the idea of the infinitesimal could be on the chopping block of anyone, much less the Catholic Church.

Amir Alexander does a wonderful job of explaining how the idea of infinitesimals led to the world of modernity. While it wasn’t the only contributing factor it certainly affected some parts of the world. He expertly weaves history and mathematics together to demonstrate his thesis.

Alexander opens Infinitesimal by discussing the Jesuit scholars responsible for what was taught in Jesuit schools of the era. He goes into the meeting where a few strokes of a pen led to the idea of ‘indivisibles’ being an anathema to the entire System. When we properly get into the book it is split into two main parts. The first part discusses the Jesuits; their history and how they came to power is discussed in some detail. It describes the uncertainty of the era with skill and aplomb. Then Ignatius of Loyola came to be and captivated the world with his dedication and moral compass. Granted, he wasn’t always like that, but an event in his life changed that and he was transformed. Why were the Jesuits important at the time? Well, Martin Luther had pinned his 95 theses to a church in Germany and almost single-handedly divided the Roman Catholic Church into several competing factions. The Protestant movement seemed unstoppable, and the Papacy was weak and ineffectual due to their love of worldly goods. The background provided makes an excellent backdrop to the Jesuits stepping in and creating colleges that the common man wanted to attend. As I mentioned earlier in this review, there was a ruling body that decided what was acceptable to be taught all over the schools. The Jesuits did not think highly of mathematics at all; not until a single event changed their tune a bit. This was the development of the Gregorian Calendar.

We all know what a calendar is, it’s that thing that tells you what day it is. However, to the Church, it was something much more. How were they to have Easter and their other feasts on the correct day if the Julian Calendar had slipped so much? For centuries, the calendar was the Julian calendar, and it seemed to work well. It had a flaw though, in how it was off by a tad per year. Throughout a millennium the little mistakes added up and the calendar was off by 11 days. Finally, in 1582, Pope Gregory XIII decided to react. He eliminated 11 days from that year, making the date jump from October 4th to October 15th. Even with all of this, the Jesuits only liked Euclidean Geometry. Every other technique was held with suspicion. So when a series of Italian Mathematicians came up with a proto-calculus the ruling body of the Jesuits clamped down hard on them. This is unfortunate since the Italians had been at the forefront of so many developments since the Renaissance. Italy became an intellectual backwater, where no one who was studying mathematics wanted to go. Even one of the most famous mathematicians of all time is more associated with France than his native Italy, Joseph-Louis Lagrange.

In the second part of the book, we go to the Protestant sections of Europe at the time, and we find them to be far more trusting and accepting of the idea. We first introduce the ideas of Thomas Hobbes and his Leviathan. In Thomas Hobbes’ time, England was embroiled in Civil War. Hobbes saw first hand the terror and horror that led from such ideas and created the idea of the Leviathan to counter it. However, one of his lesser-known books got him into a battle with John Wallis, a mathematician.

There is more to the book, but I don’t want to spoil all of it, and I don’t like typing too much in a review. So in conclusion, this book is marvelous. As I said, it does a great job of weaving together history and mathematics.

[Raquel · Rating 3 out of 5]

During the times of the Jesuits and Hobbes science was in its preliminary form. The classics, and with it theology, latin and greek, were considered the pinnacle of knowledge. It gives me the impression that at that time mathematics was synonym of Euclidean Geometry (Was not there any statistics? was statistics part of EG?) . People saw EG as a very strict discipline and wanted to derive political arguments from it. Some people also thought that all mathematics must come from a list of axioms as in EG. Thus, when another type of math appeared it was hard for them to think of it as math. This new type of math is called calculus. Calculus's proofs at that time lacked the formality of EG. This caused people to disregard it or fight over its validity.

Anyway, calculus has been formalised now, it has been very useful to humanity and there are now many branches of mathematics. I was only left with one doubt, did not mathematicians use induction (as coined by Bacon) before Bacon?

* The book is very repetitive. Sections take little account of previous sections.

[Peter · Rating 3 out of 5]

Entertaining review of some of the intellectual conundrums of the 16th and 17th centuries, along with their political and religious overlaps. I do think, however, that the struggle over 'infinitesimals' is but a subset of the more profound shift from deductive to inductive reasoning (or at least the inclusion of the latter into intellectual pursuit).

[Peter Flom · Rating 5 out of 5]

Why did the Catholic Church (and particularly the Jesuits) object to the use of infinitesimals? How did they enforce their objections? What does this have to do with Thomas Hobbes? And how did it lead to the scientific stagnation of Italy and the rise of England?

All these questions are addressed in this excellent book.

[Frank · Rating 3 out of 5]

One may reasonably complain of being misled by the title since the first half of he book is concerned more a history of Roman Catholicism around the time of the Reformation than about anything to do with mathematics. Apart from that, there is some useful material, especially about the attitude of several protagonists that the upholding of what they regarded as a complete and perfect system of plane geometry would imply perfection in their favourite philosophical system.

I also enjoyed reading of Jesuit efforts to oppose (then) unorthodox views by injunctions against thinking about them 😆. OTOH, this approach was far from shared by all Jesuits, and we can admire their sincerity and conscientiousness in taking the intellect seriously.

I remember that Arthur Koestler went further, speculating that the reason the church treated the recalcitrant Galileo so gently was because the Jesuits had arrived at the same conclusions previously, and were struggling at the time with how to move forward. Something of this showed thru in the attitude of Gulden, who was ready to accept the potential usefulness the method of indivisibles, while accurately pointing out that it had defects in need of adjustment.

[Mel · Rating 3 out of 5]

The history of the Jesuits and the Catholic church who blocked progress of mathematics and what is now known as infinitesimal. They were stuck in their dogma and couldn't fathom that things could be so tiny, infinitesimally tiny. If you went against the catholic church they made sure your life was ruined if you managed to escape being locked up.
the history was interesting. The math was confusing, and the information of John Wallis creating the infinity symbol.
If you love Math, this is a must read. If you hate Math, it's a fascinating read. If you enjoy catholic church history, this is also a must read.

[Ozan · Rating 4 out of 5]

"Continuumul este format din infinitezimale?" asa se incheie (un pic abrupt, fara continuitatea contributiei lui Newton/Leibniz), o carte ce prezinta disputa asupra unei idei matematice care a schimbat lumea (disputa intre ordinul iezuit/Societatea lui Isus si "infinitezimalisti").
Disputa a dus la franarea progresului in stiinte matematice in peninsula Italica, dar l-a sporit in tarile mai nordice, cu precadere in Anglia.

[Austin · Rating 4 out of 5]

Very interesting story, especially if you enjoy the history of math. The two main shortcomings are that the author always says everything (at least) twice, and ultimately it feels like he places more importance on the fight over infinitesimals than it really deserves—though the book does a great job of explaining why that seemingly academic issue did have way more significance than you would think.

Fun read and very informative.

[Yb · Rating 5 out of 5]

the book not only talks about math, it covers politics, religions and many subjects. It's about a academic war occurred in 17c and how did it greatly influenced the modern science &social system. highly recommended.
it's a war between liberals and totalitarian, and liberals always wins in the end.
本书并不是对数学的纯粹学术讨论，它讲述了一场17世纪在欧洲发生的的学术战争，以及它是怎样对现代科学和社会制度产生了深远而重要的影响。强推。
自由与极权的抗争。自由总是笑到最后

[Adriano Gil · Rating 5 out of 5]

Mais de 300 páginas detalhando como os Jesuítas com o aval da Igreja Católica perseguiram a idéia dos números infinitesimais. Evitados desde os paradoxos encontrados por Zenão de Eleia, são enfim abraçados pelos matemáticos dos séculos XVII e XVIII e se tornam a base da matemática moderna. Além de muito contexto histórico, o livro também traz alguns exemplos matemáticos interessantes. Recomendo!

[Ari · Rating 2 out of 5]

About a third of the way in, I realized I was hate-reading. The book is purportedly about the reception of infinitesimal techniques in mathematics in the 17th century and the struggles around them. But the book is the worst kind of history-of-science, the kind where there are heroes and villains and the main question is who the author admires, rather than who was right about the math and the science. It is a book that claims to be about mathematics where the actual math is quite unimportant.

I am concerned the author does not fully understand the math or is being careless with it. At one point he says that you can only construct algebraic numbers or ratios. This is not right. You can only construct _quadratic_ ones. This is why it's impossible to construct a cube with twice the volume of the given -- cube roots are not constructible even though algebraic.

There is a long interesting section about Hobbes. The author shows you that Hobbes (the villain) was a complete crank who kept going on about the purity and inexorability of geometric methods, but also insisting that he had squared the circle. This got Hobbes into an incredibly nasty public dispute with John Wallace, professor of geometry at Oxford. (The hero) Wallace was a moderates parliamentarian clergyman who thought that Hobbes was both a sinister atheist and also a crackpot.

Meanwhile, Wallace thought that proof was unnecessary, and if you just showed that it worked for N equals 1…4, that was just as good as proof. The author quotes Fermat explaining that this is not a proof and in math you need proofs. The author attributes this to Catholic inflexibility and does not acknowledge the fact that Fermat is entirely correct. Anybody on the continent must’ve been appalled by both sides of the debate.

But this account leaves me thinking worse of Hobbes, Wallace, and the author of the book I’m reading.

[Mykyta Kuzmenko · Rating 4 out of 5]

Отличное сравнение раннего этапа развития анализа бесконечно малых на примере Италии и Англии. Жаль ничего не написано о параллельных процессах во Франции и Германии. Автор часто возвращается к одним и тем же тезисам, но в целом книга открыла для меня много не известных ранее деталей истории.

[Iris · Rating 4 out of 5]

The ending felt a bit abrupt, would have liked to continue the journey to Newton/Leibniz at least.

Overall a very interesting read, the book presented some very interesting points on the direct link between people's view on mathematics and their views on society/politics/religion.

[Ady ZYN · Rating 5 out of 5]

Highly recommended!

[Al Bità · Rating 5 out of 5]

In the first paragraph of his Acknowledgements section at the end of this book, Amir Alexander tells us: “The roots of this book go far back, to my first year as a graduate student at Stanford, when I wrote a paper arguing that infinitesimals were politically subversive in seventeenth-century Europe.” He intended to develop this idea further; the intervening years precluded but did not diminish his desire. Now, many years later, this book (published in 2014) is the result.

Potential readers should not be put off by the title and sub-title of this book by mistakenly thinking that it is all (and only) about some obscure mathematical theory. Certainly, the latter is the starting point, but this is used more as a conceit used to elaborate on more universal themes concerning power and ideology and their authoritarian applications within European society at large.

In is simplest form, the mathematical issue in question deals with whether one considers a line as consisting of a continuous extension of a point (i.e. a continuum of a point), or whether it is made up of an infinite number of points along the line (i.e. made up of infinitesimals). A philosophical problem stems from the definition of a point as something which has neither length nor breadth (i.e. it is essentially “nothing”); so what exactly is the meaning either of extending a “nothing”, or by adding lots of “nothings” together, to form a “length”? As corollaries, the definitions of a plane and a solid are “contaminated” depending on which definition is “approved”.

Alexander has chosen the 17th-c as a major turning point on this matter: dozens of centuries before this time, education in mathematic concentrated on the “continuum” as the preferred approach; after this time, it was the “infinitesimal” approach that ultimately dominated with the development and flourishing of the calculus and the many associated mathematics it has spawned. For those who might want to follow up on a brief history relating to the mathematics of the continuum versus the infinitesimal Alexander provides a Time Table among the information at the back of the book.

The book comes in two parts: the first set mostly in Italy, and with the power wielded by the Jesuits (the militant Counter-Reformation vanguard of the Catholics) in their authoritarian battle to institute one and only one way of approaching and teaching mathematics in their extensive schools; the second part shifts the focus to northern Europe, in particular to the philosopher/geometer Thomas Hobbes, and specifically to his battle against John Wallis, who would end up being one of the founders of the Royal Society. There is a particular irony in this transposition. The north was the region lost by the Catholics to the Protestant Reformation; Hobbes hated the Catholics, and in particular the Jesuits, yet in mathematics he was the main proponent of the “continuum” approach preferred by the Jesuits.

Indeed, ironies abound throughout this entertaining book, and one can find numerous examples, based on the different levels and structures of society (political, social, religious, etc.) where “battles” are found on any number of grounds, and with shifting allegiances adding to the complexity of who holds what power in relation to what and/or whom. Alexander has cobbled together a fascinating group of individuals all in one way or another “involved” in this “problem”. His Dramatis Personae section found at the back of the book lists 38 personages, most of whom one has never heard of. All have roles to play, some more important than others; all together one finds a rather wonderful and informative bunch of protagonists well worth becoming acquainted with!

Books which re-examine a certain period, but from an unexpected starting point, can be rather revealing and informative, especially since most of the characters and incidences thus revealed rarely enter general cultural discourse — and when combined with Alexander’s smooth, eminently readable style, what we have here is something as intriguing and fascinating as anything one might care to read. I found the book exciting and rather marvellous (this sort of history is right up my alley) and I thoroughly enjoyed the journey.

[Mathman101 · Rating 4 out of 5]

If you want a compelling read about the clash of religion, science and society, this is an excellent read.

[Jeroen · Rating 2 out of 5]

Doesn't know what it wants to be. Popular title but the danger still eludes me. Besides this, large parts of the book suddenly dive into Jesuit or English history in much distracting detail. Meh.