Oil month Automotive HPC Rebound Introducing Prime Wardrobe nav_sap_plcc_ascpsc Unlimited Music. Always ad-free. Learn more. GNO for Samsung S9 Starting at $39.99 Grocery Handmade Mother's Day gifts Book a house cleaner for 2 or more hours on Amazon Fifty Shades Freed available to buy Fifty Shades Freed available to buy Fifty Shades Freed available to buy  Kids Edition Echo Dot Fire tablets: Designed for entertainment Kindle Paperwhite GNO Shop now Start your Baby Registry



This book is much more than an esoteric history of an area of mathematics. It tracks the ancient rivalry between ‘rationalists’ and ‘empiricists’. The dominant rationalists have always believed that human minds (at least those possessed by educated intellectuals) are capable of understanding the world purely by thought alone. The empiricists acknowledge that reality is far too complicated for humans to just guess its detailed structures. This is not simply an esoteric philosophical distinction but the difference in fundamental world-views that have deeply influenced the evolution of western civilization. In fact, rationalist intellectuals have usually looked to the logical perfection of mathematics as a justification for the preservation of religion and hierarchical social structures. In particular, the rationalists have raised the timeless, unchanging mathematical knowledge, represented by Euclidean geometry, as not just the only valid form of symbolic knowledge but as the only valid model of the logic of “proof”.

In particular, this book focuses on the battle between the reactionaries (e.g. Jesuits and Hobbes), who needed a model of timeless perfection to preserve their class-based religious and social privileges and reality-driven modernists, like Galileo and Bacon. The core of the disagreement was over the nature of the continuum, which was based on Euclid’s definition of a line as an infinite number of points. This intellectual argument implicitly links back to reality: is matter made of distinct atoms with empty space between them or are there no gaps between continuous matter? Although the model of the reactionaries was always Euclid's geometry, they never recognized they were only dealing with unreal definitions, as they faked out their arguments with appeals to 'real' lines etc. As such, they vigorously rejected the new concept of "indivisibles" (or "infinitesimals", the roots of calculus) and all ideas that were grounded in empirical studies of reality (like physics and the atomic hypothesis). Failure to admit debate about reality led Italy back into the Dark Ages while Northern Europe set off on the course of modernism.

As other reviewers have noted, this book would have benefitted quite a bit by including the story of the rivalry between Leibniz and Newton, who are usually credited with the invention of the calculus. As this book shows, this 17th Century rivalry had much older roots. Indeed, the book could also have been improved by establishing this acrimonious debate back in Classical Greece, where the atomic model, first proposed by Democritus, was immediately seen as an atheistic proposal that threatened traditional religion. The modern reader might assume that science has now firmly voted for the atomic model but the extensive use of the calculus embedded in Quantum Physics has preserved the conceptual features of the continuum advocates, so that we are now faced with the paradox of waves and particles. None-the-less, even readers with minimal competence in mathematics will enjoy discovering how this tiny idea of the infinitely small punctured an ancient dream: that the world is a perfectly rational place that is governed by strict mathematical rules.
0Comment| 13 people found this helpful. Was this review helpful to you? Report abuse
on June 22, 2016
I like the book, though I found it a bit overlong and sometimes redundant. Further, I suspect that the author may have the cart before the horse in thinking that failure to study infinitesimals stultified Italy, rather than the other way round. Also, I sometimes found the going hard because the author failed to distinguish between an infinitesimal and an indivisible. Nevertheless, it was fascinating to learn that the Jesuits opposed the study of infinitesimals on theological grounds whereas in northern Europe – Protestant countries – the concept was generally accepted and led to development of calculus. I had no idea that Thomas (nasty, brutish, and short) Hobbes had so vigorously opposed the concept of infinitesimals, and perhaps more surprisingly I had never heard of John Wallis, who was sort of the hero of the book and vigorously defended the concept of infinitesimals in England (and invented the symbol for infinity). Indeed, unless Alexander is exaggerating, it appears that without Wallis, Newton would not have developed the calculus. Nevertheless, I find it very hard to believe that England prospered and Italy stagnated simply because England developed mathematics and Italy did not.
0Comment| 6 people found this helpful. Was this review helpful to you? Report abuse
on July 6, 2016
Interesting, fascinating, enlightening. Infinitesimal introduced me to concepts and characters I had never encountered and showed me how a long forgotten series of catfights among snooty-nosed intellectuals led us to the world we live in. The closest equivalent concept I have at hand is the struggle between Keynesian and Hayekian economics.

At the same time, I got the feeling that the author had been paid for x pages, but his thesis only required 4x/5. He filled the remaining 20% with a never-ending re-explanation of the basic struggle. It was helpful the first few times; by page 200 (out of almost 300), I started to bleep over them.

Still: an awesome book that helps explain where we came from and why we're here. Isn't that what we're all looking for?
0Comment| One person found this helpful. Was this review helpful to you? Report abuse
on December 18, 2016
The author writes very interestingly about the religious and political positions with regard to---of all things!--the nature of the continuum that in 17th century Italy and England impeded the development of the calculus, and about the oddly-motivated positions taken by Gallileo and John Wallis that ultimately broke that barrier. But, though a portrait of Newton adorns the dust jacket, he has strangely little to say about the final chapter (Fermat, Newton, Leibniz) of his story, or about any of the other important mathematics that was gong on during the centuries in question. Promotes the view that mathematical developments are culturally determined, which is surely only part of the story.
0Comment| One person found this helpful. Was this review helpful to you? Report abuse
on June 27, 2014
The author has succeeded in writing a compelling account of how the work of brilliant 17th century mathematicians provoked conflicts of great cultural significance within the Catholic Church. He also explains with clarity how these conflicts had relevance to the reformation and the evolution of European political entities. The profiles of individual actors in mathematics and the Church are fascinating although it can be challenging to follow the large number of Italian names and places.

The underlying mathematical theory of indivisibles, which was the cornerstone of the conflict, the forerunner of calculus and perhaps even atomic physics, is explained with enough clarity that most readers without a background in math will readily understand it. I enjoyed reading Infinitesimal as a brilliant history of religion, science and philosophy as they interacted 350 years ago, a glimpse of mathematical genius and a multifaceted biography of extraordinary people.
0Comment| 2 people found this helpful. Was this review helpful to you? Report abuse
on July 24, 2014
I am enjoying reading this book, although I still have about the last 25% to finish. The book is a scholarly history of Mathematical (and, to a lesser extent, Physics) developments from roughly the time of Galileo through to (I am anticipating) the 17th or 18th Century. For those who have studied the Calculus, the general discussions will be riveting as a narration of the historical developments (and controversies) that led up to the concept of the infinitesimal, accepted in the modern notation as, for example, "dx". Entwined with the somewhat clumsy and rambling development preceding the Calculus is the politicized environment fostered by the Jesuit Order of the Church. The Jesuits, as it turned out, were on the wrong side of developments.

People today tend to be critical of the Church as having hindered scientific and mathematical developments in the roughly 1300 to 1700 time frame, but one should keep in mind that the Church was the dominant and most extensive learned community, at least in the European theatre, at that time. There really was no other comparable organization to play the role of cultural integrator. True, there were individuals (such as Galileo) who illuminated the way, but then, as now, organized progress was the off-spring of large organizations.

The Jesuits were on the wrong side of the philosophy of the infinitesimal, which later was developed by Leibnitz (in Germany) and Newton (in England) to create the most powerful mathematical philosophy known to date. If the resulting Calculus had been known, for example, to Edmund Halley, his tour de force of demonstrating the periodicity of his comet's orbit would have been enormously less tedious. Incidentally, Newton's and Leibnits' work, although very similar, were not identical to each other ... but each accused the other of plagiarism!

We tend to think of historical developments as being linear. But for those who live those developments, they are not. This book illustrates the often meandering and error filled missteps associated with human progress, and how human pride, obstinacy, and sometimes, downright dishonesty and raw politics play a role.

The Calculus was a concept whose time had come, we recognize today, but we sometimes fail to realize that the development was via human stumbling, intrigue, politics, pride and even, it seems, stubborn orneriness at times! Human affairs do not follow clean and neat orderliness: the point is very clearly illustrated by this book.

If you take the time to read this book, your patience will be richly rewarded.
11 comment| 6 people found this helpful. Was this review helpful to you? Report abuse
on March 5, 2018
an amazing return into the world , when everything was united. May be e return to past math studies but sure to intrigue all who wish to visit a time when what you think about arcane studies matter.
0Comment|Was this review helpful to you? Report abuse
on July 7, 2017
Even non-mathematicians should profit from this, lots of good information about how science, and politics, developed at a crucial time. BTW, 19th century mathematicians figured out how to eliminate most of the stuff about infinitesimals from formal mathematics. We have retained the notions of dimensionless points, 1-dimensional lines, etc., through set-theoretic interpretations of everything.
0Comment| One person found this helpful. Was this review helpful to you? Report abuse
on October 9, 2014
Wide-ranging, well-researched, waving a great tapestry as a aback ground,mand totally leaving in thrall.
If we in the west ever felt smug with respect to other parts of the word with reference to our "objective" methodologies, we should remember that we were no different at all not very long ago. (Going off on a tangent, we can quite easily add the atrocities we in the west committed in the 1930s through Hitler and Stalin, the 1990s in the former Yugoslavia .... We are NO different)
Terrific author.
0Comment|Was this review helpful to you? Report abuse
on November 21, 2015
Coming from a weak mathematical background, I found it almost necessary to make myself more interested in math in one way or another. I had taken Calc 1, and wanted to do better in Calc 2. Reading this book made Calculus seem much more interesting than when I was trying to memorize derivations. It put everything into contact, which I really appreciated. I also learned way more about Catholicism than I had expected to, considering the subject matter.
0Comment|Was this review helpful to you? Report abuse