Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions:

(1) In what sense are these dangerous?

(2) The ban on infinitesimals and the trial against Galileo's alleged endorsement of heliocentrism date from the same year: 1632 (and in fact occurred within a month of each other). Is there any reason for such a coincidence?

What I find particularly interesting is Alexander's comment that infinitesimals were officially declared forbidden by catholic clerics on 10 august 1632. The reason this is interesting is because the date 1632 falls precisely in a critical period in Fermat's mathematical activity. Fermat originally introduced his techique of adequality in 1629, but it was first made known to a wider audience in the late 1630s. In the meantime infinitesimals have been declared persona non-grata. This may explain Fermat's legendary reluctance to talk about infinitesimals. In this he may have been more affected than for instance Wallis who spoke freely about infinitesimals. Wallis was not catholic but a presbyterian.

Note 1. I have edited the question to address the concern of critics. Interested readers are invited to click on the "reopen" button below.

Note 2. Wiki reports that the original heliocentric ban dates from 1615. Furthermore, In September 1632, Galileo was ordered to come to Rome to stand trial. He finally arrived in February 1633 and was brought before inquisitor Vincenzo Maculani to be charged. Thus the infinitesimal ban from august 1632 seems to be a separate development.

Note 3. Here is Amir Alexander's own description of his historical work: I am currently working on a new book, provisionally entitled Infinitely Small, which examines the interconnections between mathematics and political and social order. Mathematics, at its most abstract, is the science of order, and it follows that different conceptions of mathematics have been associated with different views of proper social arrangements. In particular, the book will examine a sequence of historical instances in which mathematical infinitesimals acquired political significance, showing that even the purest mathematics can at times serve to buttress or undermine a political order. See here.

Note 4. Paulos provides a hint of an answer in the following terms: To the Jesuits, tradition, resoluteness and authority seemed bound up with Euclid and Catholicism; chaos, confusion and paradoxes were associated with infinitesimals and the motley array of proliferating Protestant sects. See here.

Note 5. See also this NPR review.

Note 6. The latest review is in the Notices of the American Mathematical Society by Slava Gerovitch.

Mikhail Katz
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    The book is actually a novel. I guess the author is here a bit ironic: since infinitesimal calculus opened an era of great scientific progress, it represented a danger for the old orders, especially the religious powers. – Tom-Tom Feb 03 '14 at 13:24
  • @V.Rossetto, I am quite sure this book is not a novel but rather a historical work. – Mikhail Katz Feb 03 '14 at 13:28
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    -1 asking question based on something that hasn't not been released to the public. How can anyone make objective judgement/opinion! – achille hui Feb 03 '14 at 13:38
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    @achillehui, I labeled this "soft-question" and was curious about reasonable guesses of what a respected historian might have in mind here. – Mikhail Katz Feb 03 '14 at 13:39
  • @user72694 Reading the description, this is nowhere near a question about mathematics. – AlexR Feb 03 '14 at 13:52
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    @AlexR, it is a question about the history of mathematics. I will elaborate the question a bit to illustrate this. – Mikhail Katz Feb 03 '14 at 13:53
  • This may not be what the author meant, but mathematicians often use "dangerous" to refer to something non-rigorous or ambiguous that may lead to false results. – Jack M Feb 03 '14 at 14:17
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    @JackM, that's just it, infinitesimals were no more or less rigorous or ambiguous than other mathematical concepts at the time. Nor are they today. – Mikhail Katz Feb 03 '14 at 14:22
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    August 1632 is when Galileo's book was banned. I guess we have to wait until this book comes out to find what is up with it... As it stands, this post looks like advertising for the book. – GEdgar Feb 03 '14 at 14:43
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    So the question is something like: Why were inventions such as infinitesimals considered dangerous by clerics? Or, in what way could a new mathematical concept be a threat to the catholic regime? Well I think it's a great question, and as for the answer I have no idea. – GPerez Feb 03 '14 at 14:54
  • @GEdgar, I have no personal, business, or professional relationship with Amir Alexander, though we did exchange an email or two a few years ago. I think he raises some intriguing questions. I am not sure why it is necessary to imply wrongdoing in raising such an issue. – Mikhail Katz Feb 03 '14 at 14:55
  • @GPerez, thanks for your comment, feel free to edit the question to clarify the issues, and don't forget to click on "reopen". – Mikhail Katz Feb 03 '14 at 14:57
  • @GEdgar, do you have some more details of on Galileo and 1632? Is this the geocentric model thing? I didn't know this was related to infinitesimals. Fascinating. – Mikhail Katz Feb 03 '14 at 14:58
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    One assumes that Galileo's book was banned because it says the earth moves around the sun, and not because it has infinitesimal methods in it. But (until Alexander's book comes out) we cannot tell if this or some other "ban" is the one referred to there. – GEdgar Feb 03 '14 at 15:03
  • @GEdgar, thanks. Do you have month/date for the heliocentric ban? Perhaps it involved the same "hearing". – Mikhail Katz Feb 03 '14 at 15:07
  • Wiki reports that the original heliocentric ban dates from 1615. Furthermore, *In September 1632, Galileo was ordered to come to Rome to stand trial. He finally arrived in February 1633 and was brought before inquisitor Vincenzo Maculani to be charged.* Thus the infinitesimal ban from august 1632 seems to be a separate development. – Mikhail Katz Feb 03 '14 at 15:14
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    I didn't read the book but from the Amazon summary, it looks like a mathematical Da Vinci code, not an historical book. With its numerous symbols, maths has some mystic appearance for the non-mathematician. In the real world, the use of $0$ has been forbidden for a while by Christians since it was considered as th Devil's number. – Taladris Feb 03 '14 at 15:18
  • @Taladris, I added Alexander's own description of his historical work as note 3 (see question). – Mikhail Katz Feb 03 '14 at 15:29
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    @Taladris The Jesuits were not mathematical ignorami. See [this list](http://en.wikipedia.org/wiki/List_of_Jesuit_scientists), where there should be several familiar names, some also prior to 1632. – Per Manne Feb 03 '14 at 21:16
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    I don't know the details of this story, but I believe the meeting referred to by the OP could only affect Jesuit teaching, not Catholic teaching in general. Fermat was not a Jesuit (neither was Mersenne), and should not be affected. – Per Manne Feb 03 '14 at 21:34
  • @PerManne, I believe a few decades ago the Vatican retracted its ruling against Galileo and apologized for a mistake. I am not sure where the Jesuits come in. – Mikhail Katz Feb 04 '14 at 13:30
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    @user72694 The Society of Jesus, mentioned in the [book description](http://www.amazon.com/Infinitesimal-Dangerous-Mathematical-Theory-Shaped/dp/0374176817/ref=sr_1_1?ie=UTF8&qid=1391554643&sr=8-1&keywords=amir+alexander), is just the official name of the Jesuits. – Per Manne Feb 04 '14 at 23:10
  • @PerManne, which body was behind the ruling against heliocentrism? – Mikhail Katz Feb 05 '14 at 13:03
  • I don't know anything about this, but according to Wikipedia Newton was born in 1642 and Leibniz in 1946. Both invented calculus independently around 1660, how could infinitesimals be banned before their birth? – Mark Fantini Feb 05 '14 at 13:42
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    @Fantini, excellent question. The beasts were referred to in Latin as "infinite parva", or *infinitely small*. The term "infinitesimal" itself was not introduced until around 1670. If you check Alexander's comments above, you will notice that the title originally proposed was "infinitely small", not "infinitesimal", which would have been more accurate historically. But the publisher apparently chose to go for a more glamorous title. – Mikhail Katz Feb 05 '14 at 14:06
  • Even if infinitely small, I don't see the connection to Galileo. I don't think he knew how to integrate/differentiate before Newton/Leibniz, so what good could come out of this "infinitely small" concept to attack his current problems? – Mark Fantini Feb 05 '14 at 15:40
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    @Fantini, Kepler, Galileo, Fermat, and others used infinitesimals in their mathematical work *before* Newton and Leibniz. – Mikhail Katz Feb 05 '14 at 16:04
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    @user72694 The doctrine of heliocentrism was forbidden by [the Congregation of the Index](http://en.wikipedia.org/wiki/Catholic_index#Sacred_Congregation_of_the_Index_.281571.E2.80.931917.29) in 1616. It was a congregation of cardinals, similar to the Congregation of the Inquisition, which convicted Galileo in 1633. – Per Manne Feb 07 '14 at 17:16
  • @PerManne, thanks very much for the clarification. I am still curious about the near-coincidence of dates: The Jesuit body ruled against infinitesimals in august 1632, and Galileo was put on trial by the congregation you mentioned in september 1632, only a month later. Apparently something was in the air in the early 1630s. Do you have any thoughts on this? Perhaps you can recommend a source. – Mikhail Katz Feb 09 '14 at 08:24
  • This exists: https://hsm.stackexchange.com/ – AJY May 05 '17 at 01:17
  • @AJY, Mathematics StackExchange has an extensive [tag:math-history] tag and this *question* fits within the parameters of that tag. The HSM panel is much smaller and there is less chance of getting information there. – Mikhail Katz May 05 '17 at 06:47
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    Dangerous in a different sense: Infinitesimals -------> Calculus ------> Physics and Engineering -------> Thermonuclear Weapons. All that in just 300 years while modern humans have existed for 200,000 years. – Count Iblis May 12 '17 at 01:34
  • @GEdgar. Galileo's booked was also banned because it said the Earth rotated about a polar axis. – DanielWainfleet May 12 '17 at 02:44
  • @MarkFantini. Wallis, who was Newton's predecessor in the Lucas "Chair", discovered the infinite product for $\pi$ well before Newton's time, by methods equivalent to calculus, but without the full generality of it. Fermat had, in modern notation, $x^{n+1}/(n+1)=\int_0^x t^ndt.$ The notion of a limit of a sequence was certainly around. – DanielWainfleet May 12 '17 at 02:56
  • Infinitesimals are still dangerous to many modern students of calculus, who often have not been taught the logical foundations of $\mathbb R,$ and have vague or confused ideas about it. For example, they may feel confident that no positive real is less than every member of $\{10^{-n}:n\in \mathbb N\}$ but be unsure whether there is a member of $\mathbb R$ that's less than $1$ but greater than every member of $\{1-10^{-n}: n\in \mathbb N\}. $ – DanielWainfleet May 12 '17 at 03:04
  • @DanielWainfleet, most freshman calculus students are not familiar with what you describe as "logical foundations of $\mathbb R$" just because they have not taken real analysis yet. A controlled study from the 1970s indicated that students learning calculus with infinitesimals actually have a slight advantage over their peers following a traditional approach as far as understanding of key concepts of the calculus is concerned. As far as the problem of .9 is concerned, you can consult my answer [here](https://math.stackexchange.com/a/1613121/72694) :-) – Mikhail Katz Jul 02 '17 at 15:34

3 Answers3


I believe this has to do with Jesuit opposition to atomism, rather than their position on infinitesimals. Of course, the two are linked and evolved together in the early 17th century. Today we consider atomism a physical theory, but at the time there was no distinction between a mathematical continuum and the physical continuum, just as there was no distinction between Euclidean geometry and the geometry of the space around us.

Aristotelian physics maintained that time, space, and matter were infinitely divisible, and the Jesuits had sided with this idea. They kept records over various ideas which they had debated and found to be flawed, and atomistic ideas appear here several times throughout the first half of the 17th century.

The idea that the continuum consisted of finitely many indivisible particles, each with some physical extension, was considered to be contrary to dogmas about the Holy Communion, and hence particularly offensive. It could be taken to imply that Christ was present in the bread and wine only to a limited degree, corresponding to the number of indivisibles present. This idea was explicitly forbidden in 1608, and in the following years the Jesuit doctrine was refined to forbid atomism also in the case when there were considered to be infinitely many indivisibles.

Galileo used some atomistic ideas to explain his new physics. When his Dialogue was published in February 1632, it would be natural to examine these ideas again, and presumably this is what happened in the meeting in August 1632 mentioned by Alexander.

(For some more details, see the chapter by Palmerino in The New Science and Jesuit Science: Seventeenth Century Perspectives. She does not mention the meeting in 1632, though.)

Per Manne
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  • Thanks for a thoughful answer. You mentioned that Galileo used atomistic ideas to explain his new physics. Are you referring to heliocentrism or other aspects of his new physics? Wiki says he was put on trial for heliocentrism but wiki may well be wrong and this was only part of the deliberations. – Mikhail Katz Feb 09 '14 at 08:28
  • I notice that the wiki article you linked on the Dialogue does not mention atomism. Do you have more details on this? – Mikhail Katz Feb 09 '14 at 09:39
  • @user72694 The main reference to atomism in Galileo is [Galileo, Heretic](http://www.amazon.com/Galileo-Heretic-Pietro-Redondi/dp/069102426X/ref=sr_1_1?s=books&ie=UTF8&qid=1391973306&sr=1-1&keywords=galileo+redondi) by Redondi (which I haven't read). There is some debate to whether atomism played a part in the trial of Galileo, raised by the book of Redondi. The atomism is distinct form heliocentrism. For example, Galileo suggests in the Dialogue that matter is kept together by the vacuum between the indivisible particles. – Per Manne Feb 09 '14 at 19:32
  • "This idea was explicitly forbidden in 1608" I'm skeptical. What kind of prohibition could that be? Can you provide some reference? – leonbloy Apr 08 '14 at 16:54
  • @leonbloy, this is exactly what the amazon summary says at the link I provided: *The doctrine would become the foundation of calculus, but on that fateful day the judges ruled that it was forbidden.* – Mikhail Katz Apr 08 '14 at 18:15
  • @leonbloy The idea was put on an official list of censored opinions, kept by the Jesuits. It can be compared to the [index of prohibited books](http://en.wikipedia.org/wiki/Index_Librorum_Prohibitorum), but with a more limited scope. Still, you'd better not teach atomism in science classes of any Jesuit college, if you cared about keeping your job. – Per Manne Apr 09 '14 at 11:40
  • @leonbloy, the issue in some cases was not that of keeping or not keeping your job but rather being burned at the stake; see Redondi's book for some interesting examples. – Mikhail Katz Jan 21 '16 at 11:40
  • Sorry to bring this up 3 years later, but on the face of it, this seems backwards. The idea of Infinitesimals align with things being infinitely divisible. Allowing only divisions of finite length is closer to the atomic idea (though not quite the same since the divisions can be any size). – Paul Sinclair Feb 21 '17 at 05:02

I got the book. An interesting read. It is historical, not a novel.

The "prohibition" mentioned was a prohibition by the Jesuits of what could be taught at Jesuit schools and colleges. Their educational system, probably the best in the world at that time, was based on a unified curriculum. The idea of "indivisibles" was banned, being contrary to Aristotle. "Indivisibles" was the early form of what became the integral calculus years later.

Amir concentrates on two places the contest of whether "indivisibles" should be allowed in mathematics was played out. In Italy, where the two sides were the Jesuits and Galileo's followers. And in England, where the two sides were Hobbes and Wallis. The book contains a lot of interesting background material on those protagonists (the Jesuits, Thomas Hobbes, John Wallis).

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A belated addition: from various reading, which I cannot recall precisely, the Jesuits were unhappy far more about "atomism" than heliocentrism, because "atomism" would seem to strongly indicate that "transubstantiation" (the alleged conversion of wine into Jesus' blood, or whatever) was impossible.

paul garrett
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    The references are there in Alexander but they are somewhat camouflaged. The relevant authors are Festa, Redondi, and a few others. Predictably, I can send you a more detailed text :-) – Mikhail Katz Apr 30 '17 at 12:40