The Impossible and the Imaginable

The Impossible and the Imaginable in Medieval Logic: An interview with Graziana Ciola

By Guido Alt

Summer 2024 – In this issue of Philosophy and Academy we are happy to have Graziana Ciola (RU Nijmegen). In this interview (recorded in April 2024), Graziana talks about Marsilius of Inghen, the medieval logic of consequences, how inconceivable concepts become thinkable, and the medieval roots of renaissance mathematical conceptualizations of imaginary numbers.

*Due to a connection issue, my (Guido’s) video and audio quality in this interview are suboptimal – but fortunately our guest Graziana’s segment turned out perfectly fine. Check out also the transcription of the interview below.

About Graziana Ciola

Graziana Ciola works at the Radboud University in Nijmegen. She has studied for her PhD at the Scuola Normale Superiore and at UCLA. After that she worked as a postdoc at UCLA and at Durham, before landing in Nijmegen. She has published on medieval philosophy and on the history and philosophy of logic and is an expert on the logical works of Marsilius of Inghen. Right now, she is leading an ERC project titled “i² – The Impossible and the Imaginable: Late-Medieval Semantics of Impossibility and the Roots of Complex Mathematics”.

Transcript of the interview

Guido Alt: Thank you so much for joining us here at the IPM Graziana! I’ll briefly introduce you and then we’ll just move on to the questions. Graziana Ciola teaches at the Radboud University in Nijmegen and she runs an ERC project right now at the same institution. We’ll hear more about this project soon. I’m particularly happy to have Graziana for this interview today, so thank you very much for your time.

Graziana Ciola: Thank you very much for having me, it’s really a pleasure!

GA: Your project that I’ve just announced explores a shift in the history of mathematics looking at medieval logic. Can you tell us first about the latter, namely medieval logic? What research projects on medieval logic have you been developing, and what are the ones that you are currently engaged in?

GC: Yes, so my relationship with medieval logic kind of emerged from my undergrad study actually. I always wanted to do something that had to do with Latin and to do with Mathematics. Somehow I ended up doing Philosophy and there I got stuck: I fell in love with medieval things, so I’ve developed my Master level thesis for the Institute for Advanced Studies in Pavia – where I did my undergrad and master studies – on Gilbert of Poitiers and twelfth century theories of modalities, for the most part trying to figure out what is going on with Gilbert and whether he has a notion of logical possibility. So that’s how I got into this. I was already a bit on the semantic side of things, I was interested in possibility, necessity, impossibility, and in stuff like counterfactual identity – whether you can be you and not someone else across different possible worlds -, that type of stuff. The medievals are kind of very rich to explore these things, so I had a historical interest, but I also had a theoretical interest, and logic was my gateway drug to these matters.

Then I went to do my PhD, and I knew I wanted to do history and philosophy of logic. I ended up at the Scuola Normale in Pisa, where there were people doing logic at the time, like Gabriele Lolli and Massimo Mugnai, so logic, set theory, and history of logic, and Marsilius of Inghen kind of happened to me, meaning that I got in with a project on Abaelard and on twelfth century logic, and then someone was around who had this reproduction of a manuscript of Marsilius’ Consequences. I just happened to receive it and someone said, ‘see what you can do about it, I don’t want to work on it.’ I was like ‘yeah, this is cool stuff,’ and that became my PhD thesis. I started straightforwardly with 14^th century nominalism. I had already worked on the 14^th century at university. I wanted to switch back to the 12^th, but I got stuck in the 14^th, I used to work on other matters. Franciscans and that kind of thing. If you look at my list of publications, you see (John of) Ruperscissa coming up time and again, he will be forever with me. But I managed to actually switch to actual logic and mostly to nominalist authors through the theory of consequences.

Theories of consequences are those developments and systematizations, especially in the fourteenth century, of whatever relation subsists from an assumption, premise or set of premises and a conclusion, and trying to figure out how you get from one to the other in broadest terms. Of course we can there in several ways, we can get there syllogistically – a syllogism is a consequence -, you can get there by asserting the premises – so you stand by them, you believe them -, you can get there by, well, if the premises were true – they might not be – then the consequence would be true, you would still be drawn there. I was interested in these types of theories, and Marsilius of Inghen had a treatise on this. It was unedited like many of Marsilius of Inghen’s works. So I’ve started to do the edition and historical-philosophical commentary for my PhD thesis. At the end of the day, I just continued working on these themes, I was at UCLA for a year and a half afterwards – I did my PhD in co-supervision in UCLA, so it was not getting far from home -, Still working on theories of consequences and later reception, so medieval logic broadly construed, also chronologically.

Then I went to Durham for a year, developing a little project on Richard de Bury, (Walter) Burley, and the new physics, so the connection between this type of reasoning, reasoning by consequences, trying to figure out the relationship between premises and conclusions, and their applications to the reception of Aristotle’s Physics in the late 13^th century. So, this is the conceptual background for the later movement of the calculatores in the English tradition. So, trying to figure out how these two concerns, doing logic and figuring out the physical world, in a generically Aristotelian picture, came together in this specific set of authors and concerns in Northern England around this Bishop of Durham, Richard de Bury. That’s what I did for a while. Then I just ended up in Nijmegen with a postdoc, again working on Marsilius again, because I realized that it was, not so much where my heart lied, but where a new way of conceiving consequences was.

If you look at the history of how we think about these matters, if you think about logical consequence, this following of something from something else – a conclusion from a premise, a consequent from an antecedent -, and what can be drawn from what, is at the heart of what a logic is. At the end of the day, what you want a logical system to give you is a way from going to A to B and know that you’re actually at B. That you can reach B and you’re not at C and not at D, but you’re exactly at the conclusion you want to be at, without any doubt. That is what a logic does. At the end of the day it is just an instrument for reasoning. So far, so good.

In the way in which modern logicians, so logicians from around the time of De Morgan, Frege, and 19^th century logicians, looked back at these enterprises in the Middle Ages, they started criticizing the idea – that’s almost a fair criticism – that the medievals cannot really distinguish what the different iterations of this following are. What I mean is this: according to this people, and that’s partly true, the medievals can’t account for what is a syntactic consequence – whatever I can draw from a set of axioms syntactically and formally -, what is a consequence in meaning, so what we would call a material consequence – what I can draw as a conclusion from the meaning of the term or the premise or whatever you have at hand, like ‘you are man, therefore you are necessarily a mortal’; there’s nothing formal in that, but it still seems to pretty much follow that if you’re a human you’ll die eventually -, and what is a conditional or what is an inference. These are formally distinguished notions, and of course you can start distinguishing them formally at some point, but there seem to be pretty much a big meatloaf in the medieval tradition (not only in the medieval tradition). That’s a criticism that was moved, and I was really interested in this problem.

I was really interested in this problem in a particular set of authors, and those are the 14^th century nominalists, because the 14^th century nominalists, whoever or whatever they might be, seem to maintain – I’m talking about people like Ockham, Buridan, Marsilius of Inghen, Peter of Ailly, Albert of Saxony, all these folks – more or less the same semantics, according to which we are basically talking individual things. When you refer to individual things you kind of need to be able to point at them. Why wouldn’t these people with very elegant, complex and recognizably well-built logic systems – we still think of Ockham as a great logician, we still think of Buridan as a great logician – wouldn’t be able to distinguish these notions? This was the way I got into this matter.

Of course, when you work on consequences, on a theory of inference, of entailment, or however you want to call it, you end up having to deal with what the terms refer to, for a series of reasons. Firstly, medieval theories of consequences are not theories of formal consequence like we would define them nowadays, even though partly they are. The Buridanian definition that’s endorsed by all these 14^th century nominalist authors followers of Buridan, is that a consequence is formal when it still holds under any uniform permutation of the terms – categorematic terms -, which is more or less how we would define something formal. But Buridan had plenty of material consequences which are just as valid. When you work on these things, you have to start looking at the reference of the terms, so not only at what the terms mean – ‘human’ means all the humans –, but actually at the whole set of humans that these terms pick up, for a series of technical reasons. So, starting to look at the semantics and at the modal semantics is a bit unavoidable. That was what I was working on and this idea of trying to figure out the relationship between medieval logic and mathematics had always been a bit in the background since I started working on these things. I’m answering the other questions at the same time, I’m sorry!

GA: But that’s actually great! It laid down the medieval logic bit. For my next question I want to ask about Marsilius of Inghen, on whose thought you’re an expert. What areas of his thought both in logic, that you mentioned a bit, and perhaps in other domains, do you find more interesting or promising for research now?

GC: I’m definitely into Marsilius’ logic, I don’t think I can make a mystery of that or that there’s any ambiguity about that. I think Marsilius was just a nice person, he was a very interesting author and a very unusual nominalist. Overall, you look at this philosopher at the bulk of his work, that is still for the vast majority unedited, so definitely there’s plenty of research to do just using the objective texts. That’s step one. He has been very influential as a natural philosopher. I think that the work of Marsilius with the most and longest circulation besides the treatises on supposition and ampliation, that are a central part of my project right now, are his questions on Aristotle’s De Generatione et Corruptione (On Generation and Corruption). As far as I know there are more than 300 manuscripts, I think that’s 372, but I don’t remember on the top of my head the exact number. But more than 300 copies is a lot, without counting the printed editions. It’s text that has been commented on and criticized still by people like Gallileo, so it has a long tradition. It’s hugely influential. So, definitely Marsilius’ natural philosophy needs a lot more work to be carried out, just for the brute amount of material to work on.

He’s an interesting theologian, and Maarten Hoenen has been carrying out work on Marsilius as a theologian. But the overall impression from Marsilius, is that he tries to be coherent, but as a game. He’s a very playful philosopher, which makes him a very good philosopher. He doesn’t seem like the type of person who stands by a principle just because he believes in it as a preconception. You see him go to the extreme consequences of any assumption, and just build a coherent systematic picture around it, and try to make it fit with everything else he’s maintaining. Overall, he’s very much a possibilist: he’ll say, ‘if you assume this, then this is the consequence, do you what want, I think this might work.’ He’s kind of noncommittal, which I enjoy, I find it personally a good and healthy way of handling these matters. And apparently, he was a very pious man and very generous. Monica Brînzei, who’s concluding if she didn’t already just recently conclude an ERC project studying the influences between masters in central and eastern European Universities, actually found sources according to which Marsilius loaned money for students in need across Europe. He was a nice fellow. Of course he was asking for money back eventually, but I imagine he offered more than most. So yes, nothing changes, the more you look at medieval sources the more you realize that nothing has changed! But I really think he was a nice guy; I like to study him and to spend time with him.

I think his logic is extremely interesting, and what is being tested right now by people in my team, is that what Marsilius does in modal semantics is a precursor of stuff like Modal Meinongianism, so Impossible Worlds semantics, which is a weird position to have if you’re a nominalist. But if you’re playful and you get to the extreme consequences, then you’re not particularly committed to the purism of not having anything else but individuals. Of course, if you don’t have anything else but individuals in the world, that doesn’t mean that you don’t need the powerful analytical tools where you put in other stuff.

So, that’s my fondness for Marsilius. I think his logic is interesting, there’s a lot of stuff to do both historical and philosophical on his logic and metaphysics. And definitely historians of science should look at Marsilius’ natural philosophy, so physics and On Generation and Corruption. He has a very weird and eccentric version of nominalism. As you know and as your readers probably know, nominalism to 14^th century authors is an anachronistic label. That’s how 15^th century philosophers started identifying the school masters and the tendencies they wanted to embody. If Marsilius is a nominalist and if there’s any such thing as 14^th century nominalism, definitely he is a cool and interesting one to be seen in the context, for historical and conceptual reasons. A cool guy.

GA: Yes, I’m glad you mentioned Marsilius’ views on modal semantics because it plays a central part in one of the subprojects of your current ERC. I was wondering if you could tell a little bit about that project. What are the guiding ideas that started it, where it is right now, and where you see the project going ahead?

GC: Yes, so we are just a few months in [the interview was recorded in April 2024], so we’re still setting things up. There’s an autobiographical bit on how this project came up basically. I mentioned that I was thinking about this since I was doing all the things I have been doing at the time, and then I got two fundamental concerns. One is about the way in which I was taught mathematics in the last year of middle school, actually the first year of high school, and that’s the following way. You start doing equations, and at some point, they give you square roots, and they tell you that you can’t do the square root of a negative number, that is impossible.

Okay, so far so good, makes sense. I mean, if minus per minus is plus, nothing negative is the square of anything, so you can’t do the square, because that’s a reverse operation. Then five months or three months later, depending on how fast you go in your middle school program, you get to high school, and the first thing they teach you is complex numbers and imaginary mathematics, and I was like, ‘what?’ I’m trying to reconcile myself with that ‘what.’ If you look at what happens in history, up to basically the 16^th century, but overall, up to the 19^th century when you can give a complete algebra for complex numbers, people have had the same worry, widespread among fine mathematicians. You can’t do the square root of a negative number, that doesn’t make any sense, that’s just impossible. That was in the background of my ‘childhood trauma’ if you want. On the other hand, one of the things that was fashionable when I was a PhD – so not that many years ago I hope -, the issue when you wanted to start doing medieval logic, it was already falling out of fashion to do medieval logic by formalizing medieval theories to contemporary language. I started (the PhD) in 2012, Terrence Parsons’ book Articulating Medieval Logic came out in 2014, and that’s a book that shows you how you can formalize medieval logic from within, without imposing any logical formalism that’s superimposed. Of course you can formalize anything however you want, but you need to make the theory you want to formalize speak for itself and not lose anything essential.

At that time, the question had become why is medieval logic not mathematized? I still think it’s a big question for certain circles. These people had a model of axiomatic theory, because they had the first nine books of Euclid’s Elements. So that’s an axiomatic theory. Why in the world they didn’t think they could do this with less ambiguous instruments than a regimented version of medieval Latin? It might be an artificial regimented language, but still not a pure logical language, it’s not aseptic, it is a language that brings semantical and metaphysical dirty concerns within it. On the one hand because, of course, medieval logic is a reflection on language itself, on what you can do with it up to a certain degree of artificiality. I also started thinking about what if the relation between logic and mathematics is not of dependence on the end of logic, like didactical, pedagogical and conceptual dependence, what if these people see actually mathematics as a subordinated science to what logic is doing? It seemed to me more conceptually fruitful.

Then I started thinking this. There are empty terms, those are a big problem in medieval logic, or in any vaguely Aristotelian system, because for a series of technical reasons, a vaguely Aristotelian logic and the medieval for the most part, do not want to have or can’t afford to have true affirmative statements about subjects that are not instantiated in reality, about what we call empty terms. So, for example: ‘All dodos are birds’ – a medieval logician would say that’s false. Of course, dodos are birds even if there are no dodos, because what else could they possibly be? But think of ideal gasses, an ideal gas is a gas that’s infinitely compressible. Of course there is no ideal gas in nature, and nothing physical and concrete is infinitely compressible, so why would you want that statement to be true? So that is a problem about affirmatives in medieval logic. Of course, you might want to have true statements about things that are not out there to make true general statements about natural classes that happen to be empty. Medieval logicians usually talk about roses in winter, thunder on a clear day, but you have to think it in general terms within certain restrictions, that’s what the problem is about.

At some point, and the theory seems to have been formalized by Marsilius, they start talking about referents that are not in the real world, but are in an imaginable modal domain, and those are the imaginabilia. These imaginabilia for Marsilius can include stuff that’s intrinsically contradictory. I’m going to the technicality of this, but I can say something true at the outskirts of ampliation theory, affirmatively true, about the round square, the standard medieval example is the chimera. Things that are made up of conceptual bits that can’t stay together, that are contradictory to each other. And still I can say something true about it, which is convenient if you want to talk about space with no dimension, which is a mathematical point, or space with no matter. It’s a good tool to have in your arsenal, let’s put it like that. This is basically Marsilius’ new addition to modal semantics, and it gets picked up very influentially by the tradition.

So, we have my trauma with square roots of negative numbers, and we have all this very nice picture of 14^th century modal semantics. But look at what happens. A square root of a negative number is a self-contradictory operation, because it is a reversible operation of nothing, it is the reverse operation of a square that is not there, right? So, it seems to be intrinsically contradictory. But so is a chimera, so is a round square, it is something just as contradictory and as inconceivable as a square root of a negative number. But guess what, this is the same problem. So, 14^th century logic and up to 16^th century mathematics, or premodern mathematics up to a certain point, had the fundamentally same problem. They didn’t have the conceptual tools to think about what such a thing as a chimera or the square root of a negative number could possibly mean, so that you just could not conceive it. And the hypothesis that’s moving this project was that what if instead of trying to figure out why mathematics is not something in medieval logic, we figure out whether these developments in medieval logic actually influenced the history of mathematics. Because once 14^th century logicians start to think about chimeras positively and you become able to do stuff about chimeras, you can possibly trace a line of continuity up to the first person who actually writes conceptually about doing the square root of the negative numbers as something that’s not completely inconceivable, and that’s Girolamo Cardano in 16^th century Pavia. The historical guess is that people who received Marsilius, who were partly transmitted in Pavia, have been read by Cardano, so that the logic of Marsilius has given people like Cardano the conceptual tools to actually think that this operation is not as meaningless as was previously deemed by the tradition.

Sorry if I’m putting it confusedly, but that’s how it came up, a bunch of suggestions and worries coming together. And the simple observation of a historical fact – there is a massive transmission of Marsilius in Pavia. People like Anneliese Maier once upon a time made the mistake of thinking that this passage in Giovanni Marliani, where he calls Marsilius of Inghen Marsilius noster, witnessed Marsilius actually going to Pavia at some point in 1370s, when he kind of falls off the map. Nobody believes that that’s the case anymore. What it does witness is that Marsilius is read in Pavia and there are discussions about Marsilius in Pavia, particularly in Marliani and in these authors, that are authors that on matters of conceivability, impossibility, imaginability are punctually quoted by Cardano. So, if there is an actual transmission, it’s most likely through these authors. And that’s what this project historically is trying to test, besides trying to figure out the conceptual relationships between bits and pieces and what Marsilius’ modal semantics actually does. That’s the picture.

GA: This fascinating hypothesis the project sort of runs on is very original because the connection as you just mentioned between medieval logic and mathematics is sort of far from clear yet. I want to ask one more question about this, which concerns the texts on mathematics that the project and the team is looking at currently. Can you describe a bit what kind of mathematical practitioners or mathematical works are relevant for the project, you already mentioned Cardano of course.

GC: Yes, we are actually working mostly on later mathematical works. The mathematical part of the project will be given to a postdoc, who ideally will be a historian of Renaissance mathematics. Because very intentionally we have no medieval mathematics in the project per se. We only have mathematical developments from Cardano onwards, and we are going to trace the reflection on the vocabulary. Because Cardano calls these numbers as impossible numbers or sophistical numbers. The first person to give you a calculus for these things is Raffaele Bombelli in his algebra, and he calls them impossible numbers. The first to use the term imaginary as far as I know is Descartes. So, part of the issue will be figuring out where the vocabulary with which we talk about these numbers comes from, how it evolves up to Descartes, to Mendelssohn, to the 16^th up to the early 17^th century period. But we’re not looking at any medieval mathematical texts, for the simple reason that this type of operation is not carried out, working on the square root of negative numbers. Cardano is not the first one to do it. [Nicolo] Tartaglia does it, [Scipione] del Fero does it. He is the first one to actually write about what he is doing and to reflect on it. But it’s an issue that comes up within some technical problems with solving cubic equations for the most part, and some square equations. If you think about it, it’s really easy to stumble upon the square root of a negative number. People like Heron of Alexandria, so ancient Greek mathematicians, had already done so. If I say x² + 1=0, how do you solve it? x²= -1, so x x equals the square root of -1. Ancient mathematicians stumble upon this and freak out, saying that there must be a mistake somewhere, this doesn’t make any sense. So, the techniques and the mathematical problems in the Latin West were definitely tackled a bit later. As far as I know, there is no systematic philosophical reflection on this particular issue in medieval mathematics proper.

Overall, we are looking at conceptual developments. What I really care about is how a concept becomes thinkable. Once the concept becomes thinkable, you can have a series of applications of that concept. The most obvious manipulable and traceable example seems to me to be figuring out how an inconceivable impossibility in mathematics becomes conceivable and positive, because you can obtain results by going through these impossible nonexistent numbers. That’s the move. Lots of renaissance mathematics, not a lot of medieval mathematics, but that’s on purpose. Because the claim is that you should look for the roots of the problem – pardon the play on concepts -, for the roots of negative numbers in medieval logic, rather than in mathematical practices.

GA: I have to go to our last question, about the ERC Grant itself and the experience of getting it, can you tell us a little bit about how that played out?

GC: Yes, that was a lot of work. I think there is a myth that these grants have a bit of a mystique if you want. They’re really hard to get, they’re impossible to get, and for the few elects – that’s all not true. What you really need is a good idea. A good idea that’s financeable by the ERC Grant, which is something that’s slightly crazy, but not too crazy, so crazy enough. I think that this project, trying to figure out whether really these medieval chimeras are the conceptual and actually genetical grandmothers of complex mathematics, of imaginary numbers, it’s actually the right kind of crazy. But you don’t need an extraordinary profile, you just need to be credible for the type of work you want to carry out. The guidelines that the ERC does and the support tools for writing a good grant are freely accessible online, and I strongly encourage everybody to use them. Statistically speaking these grants are not impossible to get, part of the issue is that not as many people apply as they could, because they think it’s very hard. It’s a lot of work, and it’s a bit of a bet. Of course, it can really change the situation for a young scholar like I was – I hope I still am -, it can give you your own team, it can help you develop big complex research with many interconnected parts that you wouldn’t be able to develop by yourself. So, your idea needs to fit the scheme.

But if you have a big idea, and if you need a lot of money and a lot of people to test whether it works, and it’s just a bit off the beaten path, the ERC is the scheme for you. Of course, I was a bit stressed through the process, because the mystique was there, but looking backwards I shouldn’t have been, not as I was. It’s feasible, of course, since I did it, and I’m no great genius, I don’t have a thousand publications, I just like to think about these little things and these logical games. I think that for this project schemewise and fundingwise you don’t necessarily need to be talking about non-European, non-white, or non-male philosophers, in that you don’t need any particularly socially relevant things. But of course, imaginary numbers are very socially relevant. We are talking right now because we have a lot of equations with j, that’s how the physicists write it. Without the imaginary numbers you wouldn’t have Maxwell equations, you wouldn’t have a screen, we wouldn’t be talking. So, that’s their relevance. Certainly, is not going to help you build bridges. What’s going to help you is trying to figure out how we think about things that are hard to conceive, it’s going to help you figure out whether these medieval theories can still speak to contemporary enterprises in logic, especially in non-standard modal semantics. Let’s not forget that all developments in modal and temporal logics have been carried out in the past century looking at what the medievals were doing. Contemporary logic might not remember it very well, and it’s not it’s job to remember it very well, but it was kind of born – by direct filiation and not by remote tradition – from suggestions from the Middle Ages. So, this is a conversation that can be fruitful and that projects like this can contribute to. But overall, I’m trying to figure out whether my hypothesis is right or not – big risk, big reward. So that’s my career path for now, just doing the work, and then we’ll see. I think there are many doors that can be opened or closed depending on the research. Certainly, we have a big open question, the question that’s hanging in the background of everything I’ve been saying today, and that’s what is late-medieval nominalism. I’m not going to answer it now and I don’t think anybody is going to answer anytime soon by themselves, but these things are a little bits and pieces that can help with figuring it out.

GA: With that I’d like to thank you again Graziana for the interview, it was amazing, and I love hearing you about these things. I’m sure that a lot of our readers will also like to hear about this.

GC: Thanks for having me! I’ve been rambling a bit, but I hope some of it will be interesting and not too messy. But really thank you for having me. I hope this project can be suggestive at least for people who are not technicians. I think that working on technicalities should have more general answers, or more general features of interest than the merely technical result. So, thank you, I appreciate it.