# Seeking sources: Catholic church and the development of mathematics

I teach mathematics at a Catholic liberal arts college. This semester, I am teaching Real Analysis and I want students to complete a semester-long research project that extends beyond the course content and into relevant issues of history and philosophy. One idea I had is for students to investigate the relationship between the Catholic church and mathematics, perhaps including (but not limited to):

• Some historical figure from the church who also contributed significantly to the development of mathematics
• Examples of mathematical concepts or theorems that were interpreted divinely
• Some significant mathematician who was also a practicing Catholic and wrote about the relationship between these two aspects of their life
• Examples of the Catholic church commenting on, or intervening in, the development of mathematics

Googling has not yielded anything particularly fruitful. I am not really interested in the interpretation of axiomatic mathematics as a reflection of divinity, because this is not especially unique to Catholicism. And I do want to stick to mathematics and not science, in general (e.g. the Galileo affair), because this is an upper-level course that is required for mathematics majors.

Main question: Can anyone suggest some examples of sources that I can suggest to any of my students who want to work on such a project? This may include books, films, blog posts, scholarly articles, any format and depth, really. What matters most is that it is highly relevant to the historical development of mathematics (not just interpreting mathematics retrospectively) and the Catholic church. (Bonus points if it's somehow related to the development of mathematical analysis and calculus, specifically!)

(Meta comment: I would like to tag this with "religion" but do not yet have the reputation. I will leave it to more frequent site-users to determine if this should be a new, valid tag, or if something more specific like "Catholicism" would be warranted, as well.)

• Grégoire de Saint-Vincent (quadrature of hyperbola) was a Jesuit, Mancosu's Measuring the Size of Infinite Collections touches on infinities in theology pre-Cantor. On Cantor's musings on the "absolute infinite" of God vs the transfinite see Ternullo's Gödel’s Cantorianism – Conifold Jan 4 '18 at 4:38
• I had no idea of the existence of this lady until I did a search after seeing your post. She is certainly interesting, Maria Gaetana Agnesi, born 1718. Wikipedia has an article on her. – Gordon Jan 4 '18 at 7:35
• en.m.wikipedia.org/wiki/Maria_Gaetana_Agnesi – Gordon Jan 4 '18 at 14:12
• I would think a major problem with this question is that a great deal of mathematics either predates Christianity or arises from parts of the world where it is not the dominant religion. Maths is also a tool of science and the study of the physical world and cosmology, something which you acknowledge has not always sat easy with the Catholic Church. It imposes a lot of constraints to require that a person was prominent as a cleric in the church hierarchy (not merely a Catholic believer), prominent as theoretical mathematician, but excluding any involved in the physical sciences. – Steve Jan 4 '18 at 18:27
• I should clarify that Cantor himself was a Lutheran, but he took inspiration from medieval scholastics, who were of course Catholics. Another such example is Leibniz. His muses for logic and combinatorics were Catholic mystics Lull and Nicholas of Cusa, both known for mixing mathematics with theology. As ecumenist, he was also actively involved with Catholic establishment and Jesuits, in 1689 he was offered to manage the Vatican library, conditional on conversion to Catholicism (he declined). Then he befriended one of five Jesuit mathematicians sent with a diplomatic mission to China, Bouvet. – Conifold Jan 5 '18 at 8:34

"Examples of the Catholic church commenting on, or intervening in, the development of mathematics:" Counter-reformation issues in the 17th century around transubstantiation/consubstantiation interpretation of the eucharist directly influenced attitudes toward the techniques of indivisibles that were seen as closely related to atomism and therefore opposed on doctrinal grounds by many catholic theologians based on canon 2 of the 13th Session of the Council of Trent in the preceding century.

The said canon was a particularly doctrinaire endorsement of the pagan Aristotelian doctrine of hylomorphism. The doctrine postulates a pseudoscientific substratum ("hylo") underpinning all forms (that's the "morph" part), or a kind of primordial undifferentiated Play-Do. Such a speculative scheme is somewhat analogous to ether, similarly rejected by scientists in the 19th century. Meanwhile, protestants envisioned less literal interpretations (such as consubstantiation) that tolerated elements of atomism.

The result was that in catholic Italy, developments in indivisibles and the techniques leading to the emerging infinitesimal calculus came to a virtual stop whereas mostly protestant lands tolerated such mathematical activity leading to the development of the infinitesimal analysis:

en 1700, c'est le vide intégral en ce qui concerne la pratique des mathématiques nouvelles en Italie (page 183 in Robinet).

The reference is Robinet, A. (1991) La conqu\^ete de la chaire de mathématiques de Padoue par les leibniziens. Revue d'Histoire des Sciences, 44(2), 181--201.

le grand nombre des mathématiciens de [l'Ordre] resta jusqu'a la fin du XVIII$^e$ siecle profondément attaché aux méthodes euclidiennes (page 77 in Bosmans).

The reference is Bosmans, H. (1927) André Tacquet (S. J.) et son traité d''Arithmétique théorique et pratique.' Isis, 9(1), 66--82.

These and related issues are discussed in detail in this 2018 publication in Foundations of Science which is also available on the arxiv.

I would add, as example, that can be a bit investigated the work of Padre Girolamo Saccheri, a Jesuit priest, and his famous "Euclides ab omni naevo vindicatus" (Euclid Vindicated from Every Blemish) in which he first discovered many theorems of what will then be called hyperbolic geometry. He however ended his books by saying that such theorems were evidently wrong and concluded, erroneously, that hyperbolic geometry was not possible, thus failing to anticipate Non Euclidean Geometry.

There were many discussion whether this false statement reflected theological considerations (there was and should be only one geometry allowed by God) and whether it would have been possible for him to print his book if his conclusions would have been different (the book had to be approved by Inquisition).

It certainly contributed the fact, correctly cited above by Mikhail Katz, that he rejected completely the theory of infinitesimals and the analytic approach to geometry. Not by chance his major mistake in the book appears when he tries to extend at infinity some properties of hyperbolic lines.

Cauchy, who gave calculus its modern formalization (cf. Grabiner's The Origins of Cauchy's Rigorous Calculus), was a "significant mathematician who was also a practicing Catholic." From Belhoste's biography of him, p. viii:

"Truth," he wrote in 1842, "is a priceless treasure which, whenever we manage to acquire it, cannot bring us remorse and sorrow; it cannot disquiet and distress our soul. The mere thought of its heavenly attributes, of its divine beauty suffices to replenish us for all the sacrifices we may have made in discovering it. Indeed, the joy of heaven itself is but the full and complete possession of immortal truth."

Vol. 1, ch. 12 & 13 of Valson's biography of Cauchy are on his Christian thoughts and works. Cauchy's profession of faith is on vol. 1, pp. 173-4.

Bishop Nicole Oresme (1320-1382) (cf. his MacTutor mathematical biography) studied fractional powers, invented x-y coordinate geometry, and is part of the pre-history of calculus (cf. "III. Medieval Contributions" of Boyer's The History of the Calculus and Its Conceptual Development and ch. 2 "The Dream of Oresme" of Peter Pesic's Music and the Making of Modern Science).

Bishop M.-L. Guérard des Lauriers, O.P., was a 20th century Dominican, mathematician, and philosopher of mathematics; he did his dissertation under Cartan and was classmates with Levi-Civita and André Weil.

• While it is true that Cauchy was Catholic it is incorrect that he gave the calculus its modern form, pace Grabiner; this piece of Weierstrassian hagiography was refuted in this 2017 article in Mat.Stud. and elsewhere. As far as Valson's biography goes, all modern Cauchy scholars argree that it is a piece of Cauchy hagiography of little historical value. – Mikhail Katz Jan 18 '18 at 9:45
• To the down-voter: Why the down-vote? – Geremia Jan 19 '18 at 17:41

You must tell them about the book Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander. It's about, among other things, how the Jesuit priests stopped the progress of Infinitesimal Calculus in Italy because the foundations of the subject were shaky at the time (to say the least).

Perhaps that you might also mention the fact that the Big Bang theory is due to a catholic priest, who was also a physicist, Georges Lemaître.

• As I mentioned in my answer they opposed indivisibles not because the foundations were shaky (Cavalieri's foundations were perfectly fine) but rather they were convinced that indivisibles and atomism were contrary to canon 2 of Session 13 of the Council of Trent and related doctrines. – Mikhail Katz Jan 4 '18 at 10:55

An interesting story is the one of Matteo Ricci, a missionary jesuit in China. He wrote the first Chinese translation of Euclid's Elements. He was also a cartographer.

Regarding real analysis, Pietro Mengoli proposed one of the most influential problem in the early days of calculus: the Basel problem, famously solved by Euler. Also, Francesco Faà di Bruno was an avid mathematical researcher as well as a devoted priest and theologian.

Look up Bernard Bolzano, who should be known to all students of analysis through the Bolzano-Weierstrass theorem. His work became known largely after his death and after others had already rediscovered his results independently.

• The question of how "independently" they rediscovered Bolzano's results merits closer scrutiny. People around Weierstrass knew about Bolzano's papers. Heine may well have seen Bolzano's work before publishing his seminal contributions. – Mikhail Katz Jan 4 '18 at 12:07

There are gigantic lists (Wikipedia) of men of the cloth in Mathematics and Science: