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 d 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.

# Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

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 d 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.

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5out of 5Simon Clark–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 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.

5out of 5Katie–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 in 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.

5out of 5James Swenson–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 s 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.

5out of 5Avery–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. 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.

5out of 5Amelia–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, 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.

5out of 5Bonnie_blu–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. An 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.

4out of 5Riya James–It's strange and wonderful to see how mathematicians and philosophers and theologians of the day looked at the same mathematical theories and methods of reasoning we all look at (in the stage of development that it was in in the 16th and 17th centuries) and saw in them parallels of potential social and political orders. And whole philosophies. And saw it clearly enough that they dedicated entire careers to waging intellectual wars (that weren't always above childish pettiness) with those they sa It's strange and wonderful to see how mathematicians and philosophers and theologians of the day looked at the same mathematical theories and methods of reasoning we all look at (in the stage of development that it was in in the 16th and 17th centuries) and saw in them parallels of potential social and political orders. And whole philosophies. And saw it clearly enough that they dedicated entire careers to waging intellectual wars (that weren't always above childish pettiness) with those they saw as threats. It's always nice to follow the beginnings of a subject that is so foundational to so many innovations, and to see that it wasn't revolutionary and that it didn't happen all in a day. And in following along its progress, to see that maybe it was the logical next step a lot of people would've taken. This is history told as it should be, though I thought it more repetitive (at times) than it needed to be. Also, the irreverent smugness that is inspired by seeing the Church on the wrong side of the history of calculus is a somewhat-bonus.

5out of 5Brian Clegg–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 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.

4out of 5Dave–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 eith 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.

4out of 5Mark–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 it 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!

4out of 5Raquel–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 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.

4out of 5Peter–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). 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).

5out of 5Peter Flom–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.

5out of 5William Bies–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 inter 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.

4out of 5William Schram–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 w 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.

4out of 5Val Dusek–This book starts with a fascinating anecdote of the Jesuits' banning of infinitesimal lengths to defend Aquinas' support of Aristotle, who denied that a line is made up of points. Alexander does not really discuss Aristotle's claim that a continuous line is not made of an actual infinity points, but the individual points, except for endpoints and divisions of the line only exist potentially. The author then goes into detail for 50+ pages about the origins and development of the Jesuit order. Thi This book starts with a fascinating anecdote of the Jesuits' banning of infinitesimal lengths to defend Aquinas' support of Aristotle, who denied that a line is made up of points. Alexander does not really discuss Aristotle's claim that a continuous line is not made of an actual infinity points, but the individual points, except for endpoints and divisions of the line only exist potentially. The author then goes into detail for 50+ pages about the origins and development of the Jesuit order. This too is interesting, but is mostly not needed to launch the book and its theme. One thing that surprised and disappointed me was that the author devotes only a few pages to Leibniz on infinitesimals. Leibniz is the person who in the early modern period spent the most time and pages of writing struggling with the status of infinitesimals. T. W. Arthur has produced a large book collecting Leibniz's passages on the "Labyrinth of the Continuum," which serves as the title of the book. Arthur also supplies a 70-page introduction on the issue. (Yale, 2001) Another surprising lacuna in Alexander's book is that there are only a few mentions devoted to Bishop George Berkeley's criticisms of Newton's use of infinitesimals. Alexander apparently thinks Berkeley's claims in "The Analyst" are not of scientific value when Berkeley did find a number of inconsistencies in Newton's treatment, such as Newton claiming that zero squared is less than zero in order to distinguish the infinite smallness of a derivative from the infinite smallness of a second derivative. Douglas Jesseph in "Berkeley's Philosophy of Mathematics" (Chicago 1993) gives a thorough account that the Alexander might have used. Alexander does discuss Thomas Hobbes and the dispute with Wallace over the squaring of the circle, though Hobbes view is much less subtle than Berkeley. The author does end with a section on the late twentieth century rehabilitation of Leibniz's infinitesimals in Abraham Robinson's non-standard analysis, neglecting Lawvere's category theoretical topoi and its development for calculus in Anders Kock's synthetic differential geometry, which gives a different treatment of real infinitesimals as well as L. E. J. Brouwer's "intuitionist" account of the continuum, which is more like Aristotle's than Leibniz's in that the continuum is not made up of an actual infinity of points, but rather individual points can be picked out as needed from a genuine homogeneous continuum, an idea that Hermann Weyl also developed in his work "The Continuum." After narrating the Jesuits' banning of infinitesimals, the work does not subsequently follow up with a thorough or complete treatment of the issue .

4out of 5Al Bità–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 shoul 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.

5out of 5Mathman101–This review has been hidden because it contains spoilers. To view it, click here. If you want a compelling read about the clash of religion, science and society, this is an excellent read.

4out of 5Charity Moore–One of the most interesting books I've ever read. Great narrator for the audio version. One of the most interesting books I've ever read. Great narrator for the audio version.

5out of 5Robert Spillman–Amazing story of how the concept of small numbers, especially "0," were considered impossible by some of history's greatest thinkers. Even the church gets involved, since there always has to be "something." This background provides additional insight, and respect, for those scientists and mathematicians who kept working on what they thought was correct, in spite of dangerous opposition to their ideas. After all, every math student knows that "you can't divide by zero." For many of us, the answer Amazing story of how the concept of small numbers, especially "0," were considered impossible by some of history's greatest thinkers. Even the church gets involved, since there always has to be "something." This background provides additional insight, and respect, for those scientists and mathematicians who kept working on what they thought was correct, in spite of dangerous opposition to their ideas. After all, every math student knows that "you can't divide by zero." For many of us, the answer is "Yes, it is equal to infinity." But infinity is the realm of religion, so be careful what you say.

4out of 5Brad Eastman–Every now and then you read a book about a subject about which you never thought or never thought you cared ad you are blown away by the new vistas opened. Infinitesimal is such a book. I don't know whether to classify this book as the history of math, the theology of math or the math of political philosophy, but it is a very engaging read that makes you think about how our own views of math mirror our political and religious views. Me. Alexander writes about the history of mathematics in the se Every now and then you read a book about a subject about which you never thought or never thought you cared ad you are blown away by the new vistas opened. Infinitesimal is such a book. I don't know whether to classify this book as the history of math, the theology of math or the math of political philosophy, but it is a very engaging read that makes you think about how our own views of math mirror our political and religious views. Me. Alexander writes about the history of mathematics in the seventeenth century, specifically the controversy between classical Euclidian geometry and the newer experimental infinitesimal geometry. However, that history of geometry was very wound up in the history of the reformation, the Catholic counter reformation and the debate about absolutism vs. representative government, Euclidian geometry starts with simple, agreed definitions of lines, planes, etc. and then deduces by logic theorems from those definitions. However, Euclidian geometry could not solve certain real world problems, like calculating the areas and volumes of anything but simple shapes. Infinitesimals posit that lines are made up of an infinite amount of indivisible points and then proceeds from induction to solve real world problems that Euclidian geometry could not solve. However, the method of infinitesimals leads to some logical paradoxes. For instance, if points are indivisible, how can a line be divided? If infinitesimals have no area, then adding them together should lead to a line or plane with no length or a plane with no area. The Jesuits were champions of Euclidian geometry. They believed the rational, incontrovertible logic of Euclidian geometry showed God's plan for the world was structured and universal, which mirrored the Catholic church. The Jesuits therefore banned the teaching of infinitesimals which they felt encouraged speculation without the guide of authority, which mirrored Protestantism. Actually, there main mathematical opponents were other monks and Galileo, but the eventually decalred the method of infinitesimals to be banned. Similarly, Thomas Hobbes (who was an atheist) found order and logic in Euclidean geometry to be comforting in the chaos of the English civil war. Hobbes, a royalist, felt the order implied by Euclidian geometry needed to be mirrored in civil society, with all authority emanating from an all powerful ruler, the Leviathan. He was opposed by the Royal Society founders who believed in free debate, experimentation and results versus first principles. Many of these men were religious reformists supporters of Parliament. There belief in experimentation and results over logic mirrored their belief in democratic debate and compromise. The chief proponent of infinitesimals in the Royal Society was John Wallis. The method of infinitesimals laid the foundations for calculus. I enjoyed getting out of my comfort zone and thinking about structures of thought across mathematics, history, theology and politics. Mr. Alexander does a great job of weaving all of these strands together. While he appears to have some sympathy for the proponents of infinitesimals, he treats Hobbes and the Jesuits with much respect for their accomplishments. My only complaint with his writing is he tended to repeat his main point several times. However, this distracted slightly from the reading, but did not diminish my enthusiasm for a really unique work. In the interest of full disclosure, I should note that the publisher sent me the work for free and asked me to review it on Goodreads. I don't believe this affected my judgment.

5out of 5Troy Blackford–To start with, this book sounded like it had an interesting premise. I've read a couple of books dealing with the gradual acceptance of so called 'irrational' numbers, which really just means decimals that have no 'ratio,' or, in plainer speak, cannot be expressed as fractions. Pi, for example, is such a number. I thought this would be a book along those lines, about the idea that there is an infinity of fractional numbers between each 'rational' number (at least, I *think* that's what the topic To start with, this book sounded like it had an interesting premise. I've read a couple of books dealing with the gradual acceptance of so called 'irrational' numbers, which really just means decimals that have no 'ratio,' or, in plainer speak, cannot be expressed as fractions. Pi, for example, is such a number. I thought this would be a book along those lines, about the idea that there is an infinity of fractional numbers between each 'rational' number (at least, I *think* that's what the topic of this book is supposed to be. Not a good sign that I read the whole book and still don't know). But a quick glance at the subtitle shows that this is actually going to be a look at how the concept affected history. Okay, that still sounds cool. I love that sort of thing. So far, so good. But the actual content of the book is somewhat perplexing. We hear a lot about 'the infinitely small,' and how mathematicians throughout history used the concept in various geometric proofs. But... This book could better be called 'A Lot of Stuff About the Jesuits and Then a Lot of Stuff About John Hobbes.' At one point, the author is stretching to prove the relevance of this book's subject matter so much, he blames the decline of Italy in terms of the world stage on the success of the Jesuits at banning the teaching of 'the infinitely small' in the papal lands... It's all... very stretchy. A lot of cool history was on display here, and I learned a lot. But I learned a lot about, for example, the Diggers of St. George's Hill, which were an anti-land holding protest group in the mid 1600's. I learned a lot about English history. I learned a lot about... a lot of things that I wouldn't file under the heading 'pertaining to numbers like 1.3289583058938092309234986098234...' I am glad I read this book (and I was also glad when it was over!) but I don't think I could recommend it in good conscience. Interesting, but muddled and all over the place.

4out of 5Jonathan Gnagy–This book was interesting and eye-opening on many levels; I learned a lot about 16th and 17th century Europe and how the religious views of the time (and those responsible for them) shaped -- usually for the worse -- the free exchange of ideas. Infinitesimal described, in some detail, the connection between the views of the Jesuits and their abhorrence for mathematical theorems involving infinitesimals. The story was told as a war over an idea that could have been lost; that despite the utility This book was interesting and eye-opening on many levels; I learned a lot about 16th and 17th century Europe and how the religious views of the time (and those responsible for them) shaped -- usually for the worse -- the free exchange of ideas. Infinitesimal described, in some detail, the connection between the views of the Jesuits and their abhorrence for mathematical theorems involving infinitesimals. The story was told as a war over an idea that could have been lost; that despite the utility of theorems based on infinitesimals, the church banned and nearly obliterated the concept and could have for all-time. While I found nearly everything in the book fascinating and informative, I found most of the conclusions flimsy. Obviously, the idea of infinitesimals had merit as it flourished into calculus under Leibniz and Newton, which changed the world. That said, I was unconvinced by the book that the Jesuits were singularly annoyed with infinitesimals instead of nearly any idea that conflicted with their views. I was unconvinced that Hobbes' obsession with geometry extended to his views on politics or philosophy in any meaningful way (other than providing a paradigm for approaching them as a science). This is to say that if Hobbes had instead been enamored with some later mathematical approach instead of geometry, it would not have materially changed Leviathan in any material way. I could be wrong on this, but I was at least not convinced by this book that it was so. In short, I think the "dangerous mathematical theory" had less impact on shaping 16th and 17th-century clergy than the clergy had on delaying its formal acceptance. I found the history interesting, but I found the overall lack of exploration of the actual mathematics frustrating and I found the thesis unconvincing.

5out of 5loafingcactus–Calculus, if the author is to believed (and I see no reason why he shouldn't, but I have also done nothing to confirm the tale) is at least as if not more important in the history of ideas and reconciliation of historic Christianity with modern science as the Copernican Revolution. It is simply much less discussed and paid attention to because the arguments of mathematics, theology and philosophy are enough to test the attention span of any student of history, either recalcitrant or willing. Othe Calculus, if the author is to believed (and I see no reason why he shouldn't, but I have also done nothing to confirm the tale) is at least as if not more important in the history of ideas and reconciliation of historic Christianity with modern science as the Copernican Revolution. It is simply much less discussed and paid attention to because the arguments of mathematics, theology and philosophy are enough to test the attention span of any student of history, either recalcitrant or willing. Other reviewers have claimed the author is given to melodrama and perhaps the real story is so incredibly boring that this is the melodramatic version, but my take was that the author neither was unnecessarily dull nor exhibited any unusual skill in making the topic interesting. This is an "eat your vegetables" book that is adequately written and I have done my duty in reading it.

4out of 5Mary–Discusses the history of the mathematical theory of the infinitesimal before its development into calculus. The author's focus is more on the political/religious implications than the mathematics (though the math is explained). Part 1 focuses on the Jesuits' fight against mostly Italian mathematicians starting roughly with Galileo. Part 2 discusses the battle between Hobbes and Wallis in 17th century (civil war era) England. There are some odd repetitions in part 2 that make it feel like the cha Discusses the history of the mathematical theory of the infinitesimal before its development into calculus. The author's focus is more on the political/religious implications than the mathematics (though the math is explained). Part 1 focuses on the Jesuits' fight against mostly Italian mathematicians starting roughly with Galileo. Part 2 discusses the battle between Hobbes and Wallis in 17th century (civil war era) England. There are some odd repetitions in part 2 that make it feel like the chapters were not written or edited as one entity. I would have liked to see the story carried a bit further into the development of modern mathematics, but perhaps that does not fit so neatly into the author's focus on political/religious implications.

4out of 5Tlaura–Somewhat torn between the well-written informative general history and the totally ludicrous premise that the (exceedingly reasonable) dispute among 17th century mathematicians over whether to accept infinitesimal methods was the real crucible of modernity. This is the sort of book that presents Galileo as more important to 17th century mathematics than Veita or Descartes, glosses over Newton's and Barrow's Euclid-fundamentalism, and presents Fermat as an anti-modernist in mathematics in the ser Somewhat torn between the well-written informative general history and the totally ludicrous premise that the (exceedingly reasonable) dispute among 17th century mathematicians over whether to accept infinitesimal methods was the real crucible of modernity. This is the sort of book that presents Galileo as more important to 17th century mathematics than Veita or Descartes, glosses over Newton's and Barrow's Euclid-fundamentalism, and presents Fermat as an anti-modernist in mathematics in the service of its crazy premise. On the other hand, I learned a lot about Clavius* and Hobbes (the third star is entirely for the fact that the author seems to genuinely like Hobbes even though he's on Team Black in the narrative.) * It's also crazy that "Clavius" is not accepted by the spell-check.

4out of 5John Igo–This was basically two books: Infinitesimal's in Italy, and Infinitesimal's in England. Infinitesimal's in Italy - The history of how the Jesuits killed the idea, in favor of Euclid's geometry. This took them from leading the intellectual Renaissance, to being spectators. Infinitesimal's in England - This history of how infinitesimals were embraced and ushered in an era of "Experimental Mathematics." Thus, how England came to dominate the intellectual Renaissance and eventually the world. My qua This was basically two books: Infinitesimal's in Italy, and Infinitesimal's in England. Infinitesimal's in Italy - The history of how the Jesuits killed the idea, in favor of Euclid's geometry. This took them from leading the intellectual Renaissance, to being spectators. Infinitesimal's in England - This history of how infinitesimals were embraced and ushered in an era of "Experimental Mathematics." Thus, how England came to dominate the intellectual Renaissance and eventually the world. My qualms are with the way the Author paints this as basically the biggest factor in the intellectual stagnation/resurgence of a country. And with all the religious history in this book - it's half math history, half religion history.

4out of 5Matt McCormick–Whether this book needs more or less I’m still not sure. Too often repetitive, it overstates the influence and impact of a debate largely held by a handful of natural philosophers and mathematicians. The author digresses into an overly long description of the English Civil War which, although having a bearing on the book’s premise, seems to have been attached to core writing to add pages to the completed work. Infinitesimal has the advantage of being accessible to those of us without formal train Whether this book needs more or less I’m still not sure. Too often repetitive, it overstates the influence and impact of a debate largely held by a handful of natural philosophers and mathematicians. The author digresses into an overly long description of the English Civil War which, although having a bearing on the book’s premise, seems to have been attached to core writing to add pages to the completed work. Infinitesimal has the advantage of being accessible to those of us without formal training in math and reflects a passion clearly held by the author. I found its value came more from the description of how the political/religious institutions of the 17th century stifled or promoted the accumulation of knowledge and the development of governing institutions.

5out of 5Eric–Really two books put together, one on developments in Italy, the other on England. Either topic could have been expanded into its own book, but this one does as good as possible without being twice as long.

5out of 5Michael Cornish–This review has been hidden because it contains spoilers. To view it, click here. It was a well written book and clearly quite researched. The central thesis is that certain people made mathematics a core of their philosophy, so philosophical disputes inevitably became mathematical ones. Add to that the new method of infinitesimals (precursor to differential and integral calculus), which was an area ripe for debate. This led to heated debates on the fundamentals of mathematics, the stakes of which were moral, religious, and political philosophies. The Jesuits made Euclidean g It was a well written book and clearly quite researched. The central thesis is that certain people made mathematics a core of their philosophy, so philosophical disputes inevitably became mathematical ones. Add to that the new method of infinitesimals (precursor to differential and integral calculus), which was an area ripe for debate. This led to heated debates on the fundamentals of mathematics, the stakes of which were moral, religious, and political philosophies. The Jesuits made Euclidean geometry a central point in their desire for a hierarchical structure as well as an exemplar of perfect knowledge. The introduction of a new method implied Euclidean geometry was just one method among several to be used and improved upon by anyone, with no clear hierarchy. Moreover, the method of infinitesimals reinterpreted the fundamental objects of geometry (points, lines, etc). On the other hand, the proponents of the new method were finding many new results and so admired the method for its ability to allow innovation (something apparently not permitted by the Jesuits). One thing that was interesting was to note the power struggle within the Catholic Church. The Jesuits are not 'the Catholics'. The Catholic Church endorsed new ideas, even from Galileo, but the interpretation and leniency of such endorsements ebbed and flowed with the powers of the factions within the church. Thomas Hobbes modelled his philosophy after Euclid's Book of Elements. He believed that Euclidean geometry could contain all knowledge of mathematics and so if there were problems that Euclidean geometry could not solve then the entire edifice of his political philosophy would crumble. His political philosophy was briefly discussed and, from what I gathered, amounted to saying that there is a natural, inevitable structure of society which is hierarchical and aligns perfectly with everyone's incentives. John Wallis wants plurality and uses a Baconian inspired method of induction for mathematics. This amounts to the use of infinitesimals. No doubt, both were incorrect in methodology, but since Hobbes put such stake in his structure and Euclidean geometry's ability to solve every problem, he repeated made a fool of himself by claiming to have squared the circle. It was quite interesting to see the personal dispute between these two individuals, as well as all of the personal relations which led each person to their positions within various power structures. The world was far from a meritocracy. An interesting feature of the book is that it shows that these debates only really mattered when the powers at be were in competition. The Jesuits had reasonable concerns about the method of infinitesimals but resorted to legal bans on the method when they had the authority. Hobbes had no such authority. However, due to the pluralistic view point of Wallis, no such ban would take place from Wallis's end (Hobbes had no political power). Hobbes lost the battle, according to the book, because he repeatedly made a fool of himself with regards to squaring the circle. I would not be surprised if Wallis and his cronies also blocked promotions within institutions from Hobbesian types. The battle was academic, but the methods used to fight were a mixture of argument and authority. Another interesting feature of the book is that it shows the issues with the method. It wasn't that the method of indivisibles made perfect sense at the outset and the old fuddy-duddies just didn't want to be bothered with it. There were legitimate problems that the proponents of the method repeatedly ignored. Mathematical progress, it seems, is a very human affair. If you know a bit about the history of Mathematics, it isn't until the 19th century that people like Cauchy and Weierstrass set out to make the differential and integral calculus much more rigorous. Historically, the method's utility was first shown and then the debate of how to rationalise it came second. I don't want to say this is the rule, but certainly it has happened time and again.