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Does Calculus Represent Reality? The Early Fights Over Indivisibles

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.


Indivisible talk has its roots as far back as Archimedes, but the basic Jesuit position of indivisibles of the 16th century was definitely against their existence for if they were real then the logic of the Universe – and therefore the Jesuit’s work – would be called into question. Without the Euclidean geometry as a gold standard, what would be the point of doing math? Indivisibles brought chaos, not order. They were intuition based as opposed to derived from solid physical, resulting in questionable paradoxes. Indivisibles needed to be eliminated for the Jesuit order to ensure the integrity of reality (Amir 119-120).

One of the first public stances from the Jesuits of the time was advanced by Benito Pereira, who in 1576 wrote a natural philosophy book that discusses geometrical concepts like points, lines, and so on. Using these, he built an argument for anything being infinitely divisible and therefore not being composed of indivisibles. In 1597, Francisco Suarez wrote Disputation on Metaphysics in which Aristolian physics are used to also show the infinite splitting of things, but unlike Pereira who denounced indivisibles, Suarez instead feels its unlikely that they would be how our reality is (120-122).

For most Jesuit scholars of the time, the pro/con groups for indivisibles were roughly the same in number. No one really felt they were a big deal, and without an official direction for the Order, each was left to develop their own ideas on it. Claudio Acquaviva, the general superior of the Order, changed that. After seeing the widespread opinions on the subject, he knew the Order had to be consistent in its teachings. And so, in 1601 he had a group of 5 to act as Revisionists, finding out what needed to be censored, and amongst the topics for that discussion was infinitesimals. In 1606, the first statement on the official position on them was released, prohibiting talks on them, but it didn’t seem to stop the rise in interest on the topic from such notables as Galileo and Valerio, both sharing their insights in 1604 (122-4).

Another notable person who held an interest on the topic was Kepler, who in 1609 wrote Astronomia Nova (The New Astronomy), which talked about much of his work with his mentor, Tycho Brahe. Other topics broached in the book included infinitesimal ideas pertaining to elliptical arcs, finding volumes of wine casks, and a sphere is made up of infinite cones with their points at the center of the sphere. Not too surprisingly, the Revionists were not pleased with the work and in 1613 they condemned it, claiming it did not represent reality (Amir 124, Bell).

With the increased public attention to indivisibles gathering, the Revisionists in 1615 make it clear that the topic was no longer to be taught in any Jesuit school. This put Luca Valerio, a former associate of the Jesuit Order, in a tight spot because he was friends with Galileo, someone on the opposite viewpoint as the Jesuits. As Galileo began to earn the spotlight from several religious orders for his controversial works, Valerio had no choice but to separate himself from his friend and rejoin the ranks of the Jesuits in 1616, abandoning his post at the Lycian Academy. He abandoned his work on indivisibles and never did anything mathematically significant again (Amir 125-7).

With all this talk of ranks forming along the indivisibles, were there any Jesuits for indivisibles? Yes, like Gregory St. Vincent, who in 1625 discovered several methods for finding areas and volumes of geometrical figures. Among that work was a solution to squaring the circle, or that given a circle’s area can I construct a square that is equivalent in area to it. Using indivisible methods known as “’Inductus lani in planum”, he found a solution and sent the work to Rome for approval. It made it to the top general of the Jesuit Order, Mirtio Vitelleschi, who noted the similarities to indivisibles. He did not give the work any approval. It wouldn’t be until 1647, after Mirtio died, that the work finally saw his work released (128-9).

From 1616 to 1632, much upheaval was in the Jesuit Order as anew Pope came into power and their own ranks saw some power struggles, plus the antics of Galileo kept many members engaged in fights. But on August 10, 1632 Rensus Geneal gathered the Jesuits to begin the battle against infinitesimals. Their first target was on of their own: Rodrigo de Arriaga of Prague. In his Cursus philisophicus much of the Jesuit philosophy was discussed and was used as a template for others in the Order, but a section of the book talked about our reality being composed of indivisibles (possibly as an homage to his friend St. Vincent). Rensus could not let it stand, and so formally bans all works pertaining to indivisibles. This didn’t stop Jesuits from releasing their work, however (138-140).

Cavalieri vs. Guldin

Obviously being unable to stop people from publishing their work grated on the order, and several personal fights resulted in it, whether they were intentional or not. Take as a case in point the conflict between Paul Guldin and Cavalieri. In 1635 Cavalieri publishes Geometria indivisibilius, which as its title implies talked about geometrical uses for indivisibles with regards to having thing 2-D sheets stacking up to make a 3-D cube. In 1641 Paul wrote a lengthy letter entitled De Centro Gravitatus critiquing Cavalieri’s work, saying that the proofs were not scientific, which at the time meant that they were not found in the Euclidean manner of a compass and a ruler. At the time, anything claiming to be math that did not result from these tools was not accepted and rejected as fancy (Amir 82, 152; Boyd, Bell).

Paul also had a problem with the idea of a plane being made of an infinite number of lines and even less happy with the infinite number of planes that exist. After all, it was nonsense to think about such shapes that could not be made and thus had no basis in reality, he argued. But if one digs deeper into Paul’s background, we find that he was brought up in the Jesuit tradition (Amir 84).

This school of thought not only required the aforementioned Euclidean methods but that all proofs built up from simplicity to complexity and that logic led to clarity of the Universe. They held “certainty, hierarchy, and order” higher up than many of their colleagues. You see, Paul wasn’t trying to pick a fight with Cavalieri: he was following his faith and what he felt was the correct approach to rationality and not fantasy. Indivisibles were constructs of the mind and as good as fiction as far as he was concerned. For Paul, to build planes from infinite lines and solids from infinite planes was just nonsense, none of them would have any width. If this was the new state of math, then what as the point of any rigor that had been previously established? Guldin couldn’t see it with these indivisibles (84,152-4).



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Cavalieri knew he had a good theory and was not going to take that rebuttal lightly. He was going to utilize what we may call the Galileo method of a counter-argument, which is generating fictional characters debating the viewpoints as to render any outside parties less sensitive to direct attack. However, his friend Giannantonio Rocca recommended against it because that idea could alternatively be seen as belittling to Paul by not directly addressing it (84-5).

In 1647, Cavalieri finally published his rebuke in Exercitationis Geometricae Sex. In it under the section On Guldin, Cavalieri make up surfaces and as a whole act like one. He is able to demonstrate how his theory can work on all surfaces and that they can be that unit. However, he still avoids many geometrical techniques of the time because he feels a mental construction services more than some geometrical construct. He even goes on to mention that indivisibles may not even be real but instead are possibly a tool only. Even if so, the applications of the tool were not to be disputed (85, 155).

Of course, to a Jesuit of the time none of that would have been seen as logical. In fact, it violates one of the principles of the faith: that the Universe is the same as always and never changing, for the order and hierarchy of God’s work must go on endlessly. Any paradoxes that would arise, such as an indivisible, can eventually be explained. But in Cavalieri’s case, he went with his intuition that the idea existed, and why go against something that is so clear to a person? Of course, this isn’t a good position to justify one’s beliefs, and goes to the heart of truth vs. extrapolation. Guldan needed to see the justification, not be told it was true because it was, for Cavalieri would have simply pointed to the shapes and said they exist so the method must be sound. Both died before their dispute was resolved, but it does hint at the need to proof the ideas if new followers were to join the indivisible movement (85, 156-7).

The Fight Moves On

And that is what happened. For the next 50 years, more authors came forward with their indivisible ideas and not many won recognition because of politics, lack of reason, or suppression. But a select few did show the proof that was desired, and their names are forever solidified in the math annals of history: Newton and Leibniz. The foundation had been set by many before them, but they built the house with all the material they found lying around.

Works Cited

Amir, Alexander. Infinitesimal. Scientific American: New York, 2014. Print. 118-129, 138-140, 152-7.

---. “The Secret Spiritual History of Calculus.” Scientific American Apr. 2015. Print. 82, 84-5.

Bell, John L. “” Stanford, 06 Sept. 2013. Web. 20 Jun. 2018.

Boyd, Andy. “No. 3114: Indivisibles.” The Engines of Our Ingenuity, 09 Mar. 2017. Web. 20 Jun. 2018.

© 2018 Leonard Kelley


Leonard Kelley (author) on October 02, 2018:

Thanks, Tim! Glad you liked it!

Tim Truzy from U.S.A. on October 02, 2018:

Excellent article. This piece provides good information about the history of calculus. Thanks a lot. I enjoyed reading it.



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