It is well known among historians of Fermat that, while his technique of adequality prepared the ground for the general framework later developed by Leibniz and Newton, Fermat himself gave very little in the way of explanation of his technique exploiting a symbol $E$ that appears to us to behave like an infinitesimal.

There is an interesting issue concerning Fermat's opponents and whether he may have felt a need to be watchful in his actions correspondingly, and specifically with regard to how he presented his method of adequality that could potentially run the risk of being found heretical because of its connection to the infinitely small, indivisibles and the decomposition of the continuum into indivisibles, and atomism. The latter were thought to be at tension with established canon (specifically Trent 13.2 and the problems atomism constitutes for transubstantiation and eucharist similar to Grassi's criticism of Galileo, as discussed by both Redondi and Festa) and were repeatedly banned; for details see the books by Redondi and Alexander.

(1) Fermat was a judge on the Parlement de Toulouse. He must have aggravated some people at the Parlement de Toulouse by seeking out help in Paris to get the appointment on the Chambre d'Edit in Castres. The king overruled the decision of the Parlement and named Fermat instead.

(2) The president of the Parlement seems to have been an opponent of Fermat. There is a document that was once secret that contains a character report on Fermat describing him as "interessé" and implying that he was unreliable. The damaging secret report was written by a crony of the President's.

(3) Descartes attacked Fermat's technique of adequality. This must have been well known at the time because Descartes was famous. It would be interesting to determine whether Fermat's enemies ever used this against him.

(4) According to Barner's interpretation of a capital case at the Parlement de Toulouse involving a priest named Delpoy, the latter was executed apparently over helping a Calvinist girl hide after she escaped from a re-education school. The execution apparently went against Fermat's recommendations. Fermat was reporting on this case but was not the one who ruled to execute the priest.

(5) According to Barner, Fermat was shocked by the Delpoy thing and couldn't concentrate on his math for at least a month, according to Kenelm Digby. There was a correspondence between Wallis and Fermat at the time through Digby that came to a standstill following the Delpoy affair.

(6) Barner also documents a sharp drop in Fermat's productivity at the Parlement for a month following the execution of Delpoy.

(7) As far as Weil's comment with regard to Mahoney's claims concerning Digby's letter to Wallis: Weil wrote: "an inquisitive historian might do worse than try to find out whether the above story was not a figment of [DIGBY'S] lively imagination." While Mahoney did not put in such an effort, Barner did, and found archival evidence that Fermat was in distress over the Delpoy affair.

I am seeking such information about Fermat's possible opponents and their possible actions against him in the context of sensitive events such as the indictment and execution of the priest Delpoy in order to gauge whether Fermat may have also been hesitant to be too frank about the foundations of his method of adequality because of possible theological risks involved and the possible use of this by his opponents. Do we have evidence of such actions by his opponents beyond the president's attempt to block Fermat's appointment at Castres, Descartes' attack against adequality, and disagreements over Delpoy?

Note 1. See this related thread on Redondi, Galileo, and atomism.

Note 2. See a related 2018 publication in Foundations of Science.

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    $\begingroup$ This strikes me as rampant speculation unless we actually examine the intellectual history of the time without reading our own preconceptions into it. How do we know points (1)-(5) are not just cherry-picking? $\endgroup$ – Yemon Choi Aug 8 '16 at 14:47
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    $\begingroup$ I'm voting to close this question as belonging much better on hsm.stackexchange.com $\endgroup$ – Yemon Choi Aug 8 '16 at 14:47
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    $\begingroup$ The idea that questions on history belong to hsm is a joke, as is hsm. $\endgroup$ – Franz Lemmermeyer Aug 8 '16 at 22:53
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    $\begingroup$ @JoelReyesNoche, I think what Franz is trying to say, and I agree with him, is that the HSM site is pretty moribund. I would not describe it as a failure but at any rate there is very little traffic and few sustained users. $\endgroup$ – Mikhail Katz Aug 9 '16 at 11:52
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    $\begingroup$ Since the question has been re-opened, let me clarify my earlier comments. To evaluate whether "Fermat may have also been hesitant to be too frank... because of possible theological risks involved and the possible use of this by his enemies" we need to do more than read about things that happened to Fermat. We need an idea of the times in which he lived, the mores and the culture and the politics in which he moved. The five things that Mikhail mentions could be set against all the many instances where nothing happened: my comment about cherry-picking was a warning against sample bias. $\endgroup$ – Yemon Choi Aug 9 '16 at 15:47

Maybe the following article http://arxiv.org/abs/1306.5973 (Is mathematical history written by the victors?) and references therein will be useful :-).

Fermat's life and work is carefully investigated in the book "The Mathematical Career of Pierre de Fermat, 1601-1665" by Michael Sean Mahoney: http://press.princeton.edu/titles/5449.html See, however, Gerry Myerson's comment below and https://mathematicswithoutapologies.wordpress.com/2015/03/25/andre-weil-vs-history-of-mathematics/

P.S. I have just found two papers by Klaus Barner (in French) which might be useful (in case you are not already aware of them):

https://eudml.org/doc/10129 (Pierre Fermat Sa vie privée et professionnelle)

http://www.zeitschrift-rechtsgeschichte.de/de/article_id/581 (Fermat et l'affaire Delpoy)

P.P.S. Two more papers by Klaus Barner:

http://www.degruyter.com/view/j/dmvm.2001.9.issue-3/dmvm-2001-0070/dmvm-2001-0070.xml (Das Leben Fermats)

http://www.math-in-europe.eu/images/information_pdf/hist_phil_pdf/hist_pdf/barner.pdf (Pierre de Fermat (1601? - 1665) His life beside mathematics)

Russian science fiction writer Alexander Kazantsev wrote a historical novel about Fermat's life: http://www.litmir.me/bd/?b=261604 (Острее шпаги). A German translation of it does exist: https://www.amazon.de/Alexander-Kasanzew-Sch%C3%A4rfer-als-Degen/dp/3355003638 (Schärfer als Degen).

From the more scientific point of view, maybe the following book will be useful: https://www.amazon.co.uk/Public-Life-Toulouse-1463-1789-Cosmopolitan/dp/0801421918 (Public Life in Toulouse, 1463-1789: From Municipal Republic to Cosmopolitan City, by R.~A.~Schneider).

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    $\begingroup$ I like that article a lot :-) $\endgroup$ – Mikhail Katz Aug 9 '16 at 13:32
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    $\begingroup$ The second paragraph looks useful. Perhaps I am being slow, but could you explain why the first paragraph is relevant to understanding the intellectual and sociological history of Fermat and his intellectual/ mathematical milieu? $\endgroup$ – Yemon Choi Aug 9 '16 at 15:51
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    $\begingroup$ Weil wrote a scathing review of Mahoney's book, Bull. Amer. Math. Soc. Volume 79, Number 6 (1973), 1138-1149, also available from projecteuclid.org/euclid.bams/1183535132 $\endgroup$ – Gerry Myerson Aug 9 '16 at 22:52
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    $\begingroup$ Weil's review of Mahoneys book on Fermat 's excellent and worth reading fro anyone interested in these questions. Michael Harris sharply criticizes it (and Weil's general attitude about historians of mathematics) in his blog post linked by Zurab, but he does not really give any argument against it. $\endgroup$ – Joël Aug 10 '16 at 14:39
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    $\begingroup$ Historians are fond of accusing each other of whiggish history so this aspect is best ignored. Weil's criticism of Mahoney's take on Fermat's adequality is well-taken. $\endgroup$ – Mikhail Katz Aug 10 '16 at 15:03

The point (5) is at most doubtful. It was made by Mahoney in his book on Fermat, but Mahoney's interpretation of what Digby writes to Wallis does not correspond to what Digby actually writes, as quoted by Weil in his review cited in comments:

I have had nothing from him [Fermat] but excuses . . . . It is true it came to him upon the nick of his removing his seat of Judicature from Castres to Tholose; where he is supreme Judge in the soveraign Court of Parliament. And since that, he hath been taken up with some Capital causes of great importance; in which in the end he hath given a famous and much applauded sentence for the burning of a Priest that had abused his function; which is but newly finished; and execution done accordingly. But this which might be an excuse to many other, is none to Mons. Fermat, who is incredibly quick and smart in any thing he taketh in hand.

So Fermat did not answer with his usual celerity to mathematical questions, but there is nothing but Mahoney's imagination that indicates that it was because he was shocked by a judicial decision.

The point (3) might be valid, but remember that Descartes had much more serious problems with (ecclesiastic and otherwise) authorities than Fermat ever had (if indeed he had any), so it seems unlikely that his "ennemies" would use Descartes against him.

(1), (2) and (4) are unrelated to the mathematical work of Fermat.

Given the evidence provided, I don't see any reason to think hat Fermat was afraid of being found heretical for some part of his mathemamatical work.

I understand you are asking for more evidence concerning this supposition, which I have not, pointing in one direction or te other, but that supposition seems to me rather arbitrary at this point.

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    $\begingroup$ Joël, with regard to Fermat's reaction to the Delpoy thing I was basing myself not on Mahoney who was severely criticized by Weil himself but rather on Barner, who provides a detailed analysis of this. Barner provides much evidence that Fermat was in shock. $\endgroup$ – Mikhail Katz Aug 10 '16 at 14:23
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    $\begingroup$ Concerning (1), (2) and (4), to a modern reader items concerning religious persecution at the time may seem unrelated to mathematics or science. Note however that in the 17th century science was not viewed as a separate intellectual endeavor and some of its conclusions have been found heretical and ended with serious trouble for the scientist in question, a situation Fermat may have wanted to avoid. $\endgroup$ – Mikhail Katz Aug 10 '16 at 14:26
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    $\begingroup$ I just checked my question and found that I did not use the term "afraid" contrary to your comment above. Fermat may have refrained from describing his $E$ as infinitely small and from mentioning the infinitely small (as did his contemporaries Kepler and Galileo in one way or another) out of respect for his contemporary catholic theologians. Note that the matter of atomism was not merely controversial but actually an item of hot contention between catholics and protestants. Fermat in his role as mediator between the two communities... $\endgroup$ – Mikhail Katz Aug 10 '16 at 14:36
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    $\begingroup$ ...(in his judicial role at Castres) may have found it difficult to get into such disputed territory. $\endgroup$ – Mikhail Katz Aug 10 '16 at 14:48
  • $\begingroup$ Hi Joël, I had to award the bounty so it went to the highest-scoring answer even though neither of the answers is definitive at this point. $\endgroup$ – Mikhail Katz Aug 18 '16 at 6:41

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