 Hi, I'm Peter Adamson and you're listening to the History of Philosophy podcast brought to you with the support of the Philosophy Department at King's College London and the LMU in Munich online at history of philosophy.net. Today's episode will be an interview about Peter Ramis and mathematics with Robert Golding, who is associate professor in the program of liberal studies and director of the program in the history and philosophy of science at the University of Notre Dame. Hi Robert. Hey, hey Peter. Thank you very much for coming on the podcast. Thanks for having me. Can you just start by reminding our listeners who Peter Ramis was like, when did he live? Where did he live? What sort of things did he write about? Sure. So the simple answer is that he was a French logician. More complicated, he was someone who tried to range their interests over the entire curriculum of the liberal arts in a way that was controversial at the time. He was born in 1515, a kind of poor provincial kid who made his way to the University of Paris, where he excelled. He pretty much stayed there for the rest of his life. He came into the public scene in 1543, when he published a couple of books that were extremely critical of Aristotle. And in the wake of that, he was banned for several years from teaching philosophy at the university. So when he came back to philosophical teaching, it was as a member of the Collège Royale, now the Collège de France, where he stayed the rest of his life. He was a Protestant, or at least became a Protestant later in his life, and in 1572 was killed in the massacre on Saint de Bartholomew's Day when Catholics in Paris turned against their Protestant neighbors. You said that he came back to teaching after this hiatus of being banned. Was he more polite about Aristotle once he returned? No, if anything, he considered that his safe space in the Collège Royale gave him the license to be even more outspoken against Aristotle. And he used it as a kind of a safe chair from which to try and reform the whole curriculum of the University of Paris, particularly along mathematical lines. In the eight years or so that he was banned from teaching philosophy, he turned to mathematics instead and gave lectures in mathematics, and on his return tried to come up with a way of reforming the curriculum that would be based around the quadrivium as much as the trivium. Okay, so that brings us on to our main topic here, which is mathematics and his thought. But before we get into what Remus thought about mathematics, can you say something about how mathematics was seen in the time leading up to him, and what the main issues in philosophy of mathematics were up to his time? I think there's two parts to the answer. First of all, how was it treated at the University of Paris in particular, where Remus was a professor, and then the larger question of what was going on in the philosophy of mathematics at the time. At the university, as Remus was very outspoken in indicating, mathematics was very much a second class subject. It was not taught very seriously at the university. In fact, two professorships of the Collège Royale had been reserved for mathematicians specifically because the university was felt not to be living up to its mission of actually teaching the whole of the liberal arts, mathematics in particular. So, Remus came into a system in which mathematics was neglected. In terms of interest in mathematics, there was a whole lot going on, I guess, in the philosophy of mathematics in this period as well. One question that was being debated at the time was a so-called question on the certainty of mathematics, whether mathematics lived up to the standards of Aristotelian demonstration. It wasn't a question that Remus himself was very interested in because he wasn't very interested in Aristotle. At least, he didn't believe that Aristotle's strictures on these things were always worth following. But that was a question that was being debated at the universities at the time. In more humanistic circles, but also at the universities, the debate over the philosophy of mathematics had been changed by the publication of Proclus' commentary on the first book of Eucalyptus' Elements, which does set out a Platonist philosophy of mathematics. People were wrestling with that. They didn't quite understand it until the 17th century. But insofar as people were thinking about what are mathematical objects, they were either looking at Aristotle or they were trying to make sense of Proclus' account. Can I just ask you about the first of those problems you mentioned, which is the thing about whether mathematics is certain. So, there's a couple of things that I think are striking about that. One is that still today, we tend to think of mathematics as a kind of paradigm of certainty. And that's been true pretty much throughout the history of philosophy. And second of all, Aristotle keeps giving mathematical examples himself in the philosophy of analytics, right? So, why would anyone doubt in this period that mathematics rose to the appropriate level of certainty for science? So, one of the questions they raised was whether the proofs that you read, let's say, in Eucalyptus' Elements, were causal proofs. And it's very difficult to discern exactly what a causal proof would look like in mathematics. But since mathematics doesn't seem to talk about the causes of mathematical objects, it seems that you can't have demonstrative syllogisms in the way that Aristotle thinks. Because he thinks that the middle term of the syllogism has to identify the cause that links the... Exactly. Exactly. If you can't do that, if you're proving something about isosceles triangles, what's the cause of a triangle? What's the cause of its angle sum? Yeah. Okay, I see. Speaking of the history of mathematics, this is actually something Remus has ideas about, right? So, he has very strong views about how mathematics developed in antiquity already. What are these ideas? How true are they? So, he writes a book in 1567, which is the first comprehensive history of mathematics. It's written within a polemic about the nature of mathematics itself that he's having with some of his Aristotelian opponents. He tries to draw an account of the history of mathematics, which traces it back to the Garden of Eden, as many other historians, histories of mathematics did at the time as well, but then draws it all the way through to the modern period. The story he's trying to tell is essentially there was a natural kind of reasoning at one time in human history. And by natural reasoning, he meant reasoning based on practical experience, on actual familiarity with measuring things and counting things and so on, which gradually gets obscured by people like Plato and Aristotle and even Euclid. So, on the one hand, he writes some works in which he tries to pull apart their mathematical examples and mathematical proofs, say they're just obfuscating the simple truth about mathematics. Mathematics kind of lies on the surface of the world, as Remus thinks, and also structures the human mind in some sense. If a proof is difficult, let's say, it means it must be wrong in some way, because mathematics should be the most evident thing of all. So, his history of mathematics is a history of people gradually obscuring what was once a clear truth. That's why he gets very excited about German mathematics in particular, because as he sees that these people are doing real things, these people are actually measuring things, or with engineers in Italy as well, people who are actually finding mathematics in the real world, they seem to be doing real mathematics. Yeah. So, when does he think the obscurantism came in? I mean, when was the... when is there a turn away from kind of pure, natural, nice, easy to understand mathematics to this more difficult mathematics? Do we even have any texts? Yes, well, so he likes some of the reports of how mathematics was done. So, he uses, for instance, again, Proclus's commentary on Euclid, which has a little potted history of mathematics drawn from the Aristotelian eudemus. The person he really blames for it all going badly wrong is Plato. And he blames Plato because Plato had the opportunity, through his travels to Egypt, to southern Italy, and so on, to draw together all of the strands of mathematics that had been separated after the fall. And Plato was the one human being who had those all united in his own person. But because of his vanity, and that's the word that Remus uses, because of his vanity, Plato decided to obscure it and make mathematics difficult and only approachable through the means that Plato would allow people to approach it. So, actually, that means he would contrast Plato to earlier Pythagoreanism, whereas some Renaissance figures saw Plato as a kind of culminating figure in this way. Yeah. And one of the things, you know, Remus's own history of philosophy changes over his life. In his earlier works, he's very much a Socratic philosopher. He, or he at least, claims to be a follower of Socrates. He admires Plato. He believes that Plato is an anti-metaphysical writer. He reads Plato in a particular way, particularly the beginning of the Parmenides. He thinks that's Plato's last word on the theory of forms, a kind of destructive approach to them. By his late work, he's been into Tyra Plato as well. And at that point, Plato becomes the reason that things are going wrong in mathematics as well. Okay. So you mentioned already that Remus wanted to bring in a mathematical method in philosophy and also in teaching. And he's very famous, to the extent that he's famous, for his systematic approach to philosophy and science, for using lots of tables and diagrams. So on the face of it, that already looks kind of mathematical. How exactly does he see his method as mathematical? Well, that's a really good question. First, on the tables and diagrams, they're more associated with Remus's followers than with Remus himself. Remus uses these famous bracketed tables in his dialectic occasionally, but those wonderful things in which you see whole sciences divided up into these page after page long tables, those are really his German followers that do that kind of thing. It's implicit in the writing though, Remus is thinking in this kind of dichotomous way, dividing subjects up into further and further pieces. It is mathematical, at least as he wants to imagine mathematics should be. Ideally, he thinks mathematics should, like all natural things, be able to be divided up dichotomously, neatly into all of its parts so that one can proceed from the most general down to the most special. And the mind, when it sees the vision, will immediately ascend to it because it conforms to nature. In fact, when he actually turned to Euclid and Archimedes and other texts, he discovered mathematics was nothing like that. So he actually publishes his own versions of geometry and arithmetic, which do attempt to conform to that. I think it's true though, when he thinks of his method, when his method is developed, which is simply a way of arranging knowledge, he really does at the time imagine that it is in some way related to mathematical reasoning. So the idea is not that actual mathematical proofs or use of numbers or something, that that's going to turn up in every branch of science, but rather that the structure of mathematics is replicated in the structure of other sciences. To some extent, I just want to say two things to that. First of all, the method itself isn't supposed to be a means of discovering anything. It's supposed to be a means of arranging ideas, and insofar as mathematics had this reputation of being very well arranged, that's why things should look a bit mathematical, all the other sciences as well. But there was another sense as well in which Remus really thought that the arts in general could be made more mathematical. And this particularly comes out in a couple of speeches that he gave. In 1551, when he first is allowed to teach philosophy again, he's immediately attacked by some of his opponents who decide, well, if we can't kick him out of the university altogether, we'll just make sure that none of his students can graduate. So they accuse him of not teaching the curriculum correctly. And it looks kind of fishy. He turns up at the College Royale and says, I'm now the professor of philosophy and eloquence, and I will teach philosophy through literary texts. So the Aristotelians freak out, and they say, well, he obviously can't be teaching the things he's supposed to teach. So he gives the speech on philosophical teaching, very defensive speech, in which he goes through the curriculum as it's set out in the statutes and says, yeah, I'm doing this, I'm doing this, I'm checking all the boxes for the students. A few years later in 1557, when he's much more well established and he's actually been spending the last couple of years working with mathematicians, editing and publishing mathematical texts, he reissues the speech, reprints the speech, doesn't tell anyone that he's edited it, but it's actually a completely different speech. And if you turn to that section, he's now saying, not my students are doing this, but he offers an ideal curriculum for the University of Paris, which is mathematical. And it ends up the last year is going to be devoted, he says, to physics based on mathematical principles, not the physics of Aristotle's physics, a kind of natural philosophy without mathematics, but something which starts from mathematics. That sounds actually very congenial in a way to the way we now think about physics. Yeah. We imagine physicists doing lots and lots of mathematics. Yeah. Is that really what he has in mind? I mean, does he think that concrete mathematical methods should in fact be used in physics? It's in some ways a difficult question, because if you actually look at the works he published on physics, they don't look very mathematical at all. So the book that he claims is going to take the place of natural philosophy in his curriculum is essentially Virgil's Georgics, which he systematizes and comments on. But he also, and there's some evidence that in about the same time that he gave this renewed version of the speech, he was lecturing to his students on atomism. And he was giving a version of atomism, which talked about geometrical objects as atomic objects, which were directly reflected in the world. So the world is made up of atoms because geometry itself was made up of atoms. And there's still traces of that in his later writings. He gradually tries to face it because he gets so much criticism from some of his Aristotelian opponents when they hear he's lecturing on this. But if you look at early versions of his works published in the 1550s, these lectures are still there. And that's what he seemed to have in mind when he's talking about a physics based on mathematical principles. So does he imagine that you could give a geometrical analysis of what's going on at the atomic level? So is this something like what Plato has in mind in the Timaeus? Yeah, so he's inspired by Plato. He's inspired by the traditions of the early academy that seem to embrace a kind of geometrical atomism, the idea that not just physical objects but geometrical objects themselves are made up of indivisible lines, smallest geometrical units. And he finds that as he sifts through the late antique commentaries on Aristotle, written by Neoplatonists, who all reject that view. People like Proclus, Simplicius, Philoponus, and so on, they all reject that kind of geometrical atomism as being incoherent. But Remus reads between the lines and says there was a theory here. It was an original theory that Plato was teaching the academy. Aristotle buried it. But for a while Remus saw it as part of his remit to revive this theory, which would put nature and mathematics back in conformity with each other. And how does he think that we would know this? I mean, is the idea that we can just posit as an obvious truth that there must be a mathematical structure at the basis of physical reality? Or is it somehow based on empirical evidence? Not the kind of empirical evidence that we would imagine. It really comes down to what Remus thinks geometry is. Remus begins his work on geometry like he begins many of his works by defining it and defining it in a very characteristically Remus way. Geometry is the art of measuring well. And insofar as something can't be measured, it's not geometrical. So he has this wonderful passage in one of his works in which he says, you know, what is a geometer? A geometer isn't someone who plays with objects inside his head. He's a busy doer of things. He's actually out there with a ruler and a plumb line and he's measuring things and so on. And he said that people who actually measure things tell you there's a limit to which you can measure things. There's a smallest possible measurement. And he says, so we can't posit anything below that smallest possible measurement. Geometry itself breaks down into these kind of indivisible chunks because geometry is limited by the capacity of measurement. Oh, that's really interesting. So it's almost like he thinks that we have to posit an underlying physical structure that matches the limits of our own knowledge. Yeah, yeah. So it's a kind of assumption that the world must line up with human structures of knowing. Exactly. And that's a constant through, even from his very earliest book, work on dialectic. He, you know, he insists in that work that his dialectic is natural and it reflects the structure of nature and it reflects the structure of the human mind. And his philosophy changed in various ways over the next decades, but he never really lost that fundamental idea that there has to be a kind of mirroring of those three things in each other. Okay, so something that you've mentioned a few times now is that he really doesn't like Aristotelianism. Yeah. And I think now we've maybe heard enough about Remus to maybe approach this in a more concrete way. Sure. So what is it about Aristotelianism that he doesn't like? How is it failing to carry out the project that you've been describing? So there was a myth that was put around by some of his earliest biographers that he wrote a master's thesis called Everything Written by Aristotle is a Lie. And the word he used for lie is commentitium, which is actually a Remus term of art. It means something that is confected rather than natural. That's probably not true, but it pretty much reflects what Remus thinks about Aristotle. It should be true. It should be true. It really should be true because that's his approach that Aristotle is constantly papering over the nature of reality with abstract concepts. He writes several books that are just devoted to pulling apart Aristotelian works of the scholae physicae. The Lectures on the Physics is just a kind of a tedious chapter by chapter, line by line, lampooning of the physics. He does a similar thing for the metaphysics as well. The irony is that when you look closely at what Remus actually believes, he often looks kind of Aristotelian. So when he's talking about the philosophy of mathematics, in general he can't quite understand Proclus' philosophy. He certainly doesn't like the theory of forms or the idea of kind of abstract mathematical objects. He comes down to a theory of abstraction that's pretty much the same as Aristotle's. That's what the mathematical objects are in our mind. But he nevertheless relentlessly attacks Aristotle for presenting it badly, for using poor arguments, for relying on logical sophisms to get where he wants to go, and above all for not organizing things well. And that's Remus's real problem that, again, physics should reflect the structure of the human mind, should reflect the structure of the world, it should be easy, it should be straightforward, it should flow according to the famous method, because the method is a natural way that art should work and it doesn't. So a lot of what he ends up doing, if you look at his own works, his works on mathematics and dialectic, he's often taking the ideas of others and his own ideas and then rearranging them into these little atomic units that are all strung together in the correct order at last. That's kind of nice actually, because it means that his writings mirror the nature of reality themselves by having these atomic... Absolutely, yeah, yeah. You know, I think that's a very good observation that he really does think arts are made out of documenta, by which he means kind of irreducible bits of knowledge. How much do we know about how he actually went about teaching his students? I mean, you mentioned these speeches he gave about what he was planning to do and whether it fulfilled the requirements of the curriculum, but do we have more concrete evidence about what actually happened in Remus's classroom? Yeah, we do. He gave public lectures on the one hand, at least in his early years, those public lectures were incredibly popular at the University of Paris. He was probably the most sought-after lecturer there. Because he kept making fun of Aristotle? Yeah, he made fun of Aristotle, he made relentless fun of his colleagues, that's why they kept taking him to court and trying to get him kicked out of the university. So, you know, there's these descriptions of him kind of pacing the lecture theatre, ranting about Aristotle, ranting about his colleagues. There's more than a thousand students packed into the room, standing in the aisle, sitting on desks, just to witness this phenomenon. He was also reputed to have a perfect Latin style. So, even his opponents said, yeah, the guy sounds like Cicero. He's a jerk, but he sounds like Cicero. So, he was very much sought after for that. He also taught on a smaller scale in the two colleges where he worked at the Collège Ave Maria and the Collège de Préle, where he was principal for many years. So, he had these kind of small study groups as well. He taught mathematics. He wasn't actually a professor of mathematics, he was, as I mentioned, the professor of philosophy and eloquence, though you'll often find, especially after his death, portraits of Remus that describe him as being a professor of mathematics because he became so associated with mathematics. He wasn't very good at mathematics. That's kind of, you know, one of the ironies that although he loved it and he loved the idea of mathematics, he really struggled with it. So, one of his biographers, one of his students, described how Remus would prepare for a lecture in mathematics. He would spend the entire morning surrounded by this kind of little commission of mathematicians, young mathematicians that he had put together around him, being drilled in the subject on which he would have to lecture that afternoon and just kind of regurgitate what it was that they had finally got into his head. Didn't always go well. So, there's a very hostile pamphlet written against Remus called On the Disastrous Deanship of Peter Remus, and it describes him going into a class, having been prepped by students in the morning and standing at the board trying to prove a mathematical proposition and staring at the board dumber than a fish, absolutely unable to figure out how the proof should go ahead. That's very sad. Yeah. Yeah. And did Remus also change the range of texts that he had his students read, or is he mostly just using a new approach but to the same text? He really does change the texts, and that's what gets people upset. He might have been able to argue that the students are getting the same experience, they're being exposed to the same ideas, but not in a way that anybody at the university recognizes being legitimate. So, he would teach natural philosophy from Virgil's Georgics. He would in general try and teach all the subjects of the curriculum from poets or orators on the assumption that they too were natural reasons. And if we're able to analyze their texts well enough, break it down into its kind of atomic pieces and put them together in the right order, we'll discover that they actually have a natural account of the world that's hidden in the texts somewhere. So, he changes the texts in that sense. He's simply not reading, for a large part, Aristotle to the students. He's critiquing Aristotle to them and then giving them literary alternatives to read to the same end. He also really does set about publishing new mathematical texts. So, this little committee of mathematicians that he assembles around himself in the 1550s, they're advising him on what to say in his lectures, but they're also editing Greek mathematical texts, works by Euclid, works by Archimedes, translating them, writing very, very influential prefaces to them, which set them into some kind of context. And those works get republished again and again. The prefaces to those works get published long after Remus is dead. Okay, so one last question, which is about his legacy and influence. We already mentioned the famous use of diagrams in texts by other authors who are obviously inspired by Remus. Does his influence go deeper than that? Is it really just a kind of, here's how you could go about setting up graphical representations of the sciences, or do they really take over his whole program of mathematization and so on? It's a good question. Remus, of course, has a vogue for at least the next half century or more. After Remus dies, it becomes huge in Protestant countries in particular, as Remus is recognized as a Protestant martyr. It does have some effect, I think, on the way people think about the sciences, and there's some particular people you could pick out as being deeply influenced by Remus. So Bill Abroad Snell, who goes on to discover the law of refraction, is a Remus and works out of Remus textbooks and is really thinking through mathematics in a very structured Remus way. And he writes his own manuscript on refraction, which is certainly influenced by Remus. Francois Piet, the great French algebraist, it's difficult to tell exactly how deeply the influence goes, but Remus himself, in some of his mathematical work, says there's books in Euclid where you can just throw away the impossible to follow reasoning in book 10 or in book 2 of the Euclid's Elements, let's say, and replace it with algebra. And Remus is not the first, but certainly one of the most popular authors to say there's a way of reading the ancients that turns what they're saying into algebraic reasoning. And that's pretty much what Piet arguably is doing as well. He says he wants means to be able to solve any problem. And what that means really is approaching the problems that the ancients set and recasting them in algebraic terms. There's one other example as well. Remus, in all of his writings, insists that people have to reason naturally. He's very upset with the fact that astronomy uses so-called hypotheses, that they observe how the planets are moving and then they imagine that there are some circles moving up there. He says in some of his works that he would like to see an astronomy without hypotheses. He even writes to Reticus, Copernicus's kind of right-hand guy, and says it's such a shame that Copernicus couldn't come up with a way of having an astronomy without hypotheses. And Reticus writes back and basically tells me he doesn't know what he's talking about. But in one of his works he says, I promise a professorship to anybody who can come up with an astronomy without any hypotheses. Finally, Kepler, long after Remus dies, semi-ironically claims the professorship. And he says, I've now built what Remus wanted, which is an astronomy which doesn't make hypothetical circles in the sky, but in fact relies on the natural causes of things. Okay, thank you very much for that wide-ranging and interesting look at mathematics and Remus, and for coming on the podcast. Great, thank you very much. And please join me next time as we continue to look at philosophy during the Reformation, here on The History of Philosophy Without Any Gaps.