In practice, mathematicians have been 'proving' their results in many ways, in many places, for thousands of years. In principle, however, what is a proof? Usually, we look to geometry, specifically the geometry of Euclid. But what are the fundamental building blocks of a Euclidean proof?
Until quite recently, the Renaissance, this question remained open—due to uncertainties about who Euclid was, the structure of his arguments, and even the layout of his pages.
This lecture looks at how the language and practices that we now associate with Euclid hardened into our dominant idea of proof in the 1570s.
A lecture by Dr Richard Oosterhoff
The transcript and downloadable versions of the lecture are available from the Gresham College website:
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- I'm going to speak a lot about Euclid and perhaps the very first word here ought to be an invention, as you'll see from some of the other talks today. However, one of the ideas that we have about proof and about demonstration and how these things fit together tends to come up to us via Euclid. And so how Euclid is received turns out to be enormously significant. I'm going to talk first about a medieval version of Euclid, some several medieval versions of Euclid, and then how that coalesces into the kind of Euclid we recognize and gain a lot of inspiration today over the course of the 16th century. And I'm going to suggest in a way that means we should have a looser vision of proof than sometimes we bring. Now what's a proof? I think that's a valuable thing to start with. I don't want to get too technical. What I want you to do is cast your minds back to the times when you yourself have felt the power of going through a proof. Your mind has been drawn, given a few things, but then using those things, building towards a statement where everything locks into place and you feel like the conclusion you've reached is balanced precisely on a set of givens that you started out with. Now that vision of proof is what gives us the sensation of certainty of a universality, perhaps even. And that's something like what we can see when we turn to Aristotle as he's describing the process of proof. Reviel Netz when he's talking about ancient mathematics focuses a little bit on this sensation. The issue is not what made Greek mathematics valid. The question is rather what made it felt to be valid, for felt to be valid it certainly was. So this idea of the sensation of certainty that comes to us from talking through a certain kind of narrative, that's what I want to think about a little bit today. Aristotle gave Renaissance folk, medieval folk the standard definition of what counts as a geometrical proof. He's basing this a little bit on something like Euclid, and I think you could recognize some of what I've just described there, The sense that you start out with axioms and then you move by inexorable steps towards a conclusion that feels like it has been proven. And Geoffrey Lloyd, for example, points out that even Aristotle, that's what he means by the Organon here. And Euclid, even when we read these sorts of figures, we're skewed a little bit in how we approach demonstration. We need a bigger version of demonstration in order even to understand the Greek materials we kind of think leads us towards our notion of demonstration. This is actually a point that Descartes expanded upon at different points as well. So Descartes is someone who thought he had a very good, strong definition of demonstration, of proving things, but he complained about certain sorts of people who use the narrative practices of proof, who use the language of proof. And this is what he thought was the language of proof. It's actually quite close to what we might see. You have some definitions, you have some postulates, axioms or common notions, and then you use those to prove, let's say, annunciations problems, theorems, demonstrations and corollaries and so on. So that kind of anatomy, that language of describing what's going on in geometrical narration, he says, also can lead us astray so that students, when they see something parsed out as axioms and so on, think that they're be being shown a proof, but maybe not always. So that's all of this is just a way, a preamble, a way to say to you, the way things are laid out on the page and the narrative practices that we use to lead us from one point to another in the context of mathematics, often we perceive as a very particular form of proof. I want to trouble that a little bit by thinking a little bit about who Euclid was in the medieval period and how Euclid turned into something particular at two different moments in the Renaissance. So the long view is the medieval period, and then we'll move to different ideas about invention that are linked to proof in the Renaissance. So let's step back. How was Euclid actually encountered by medieval readers? I mean, this is one way, if you were Greek speaking, you might en encounter Euclid in the ninth century. And something that to notice here is what I'm calling the meson page, the way the page is organized and how it's very hard to tell the difference between the enunciation or thing to be proved, the statement, that you're working with and the rest of the matter that you're using to make that proof work. There are a few other things to say about it, but this picture, you can't see the details. We don't need to read the Greek together anyways to get the point, isn't divided. Other early manuscripts we have aren't divided very carefully around annunciations. And there's this other bit of dense text that's around here, which should be quite interesting to us, which turns out to be commentary or descriptions or alternative ways of thinking about what's going on within the text itself. Now, the question I want you to think about is, which of this is Euclid? Who counts when you're looking at this text, which bits come from Euclid directly? Well, if you were a medieval reader, you might think that Euclid is the author of only very specific parts of the text, partly because one of the versions of Euclid that's transmitted is the translation from Boethius. And in it he suggests to us that only the annunciations, only the corollaries, the theorems, all those sorts of things are in fact Euclid. The rest of it is optional extra because it's a gloss. And if you read the versions of Euclid that come to us in this form, you often either skip proofs entirely. I love this example because you have images that are supposed to show you what's going on in the statement, or you have alternative glosses from medieval commentators that make this up. This turns out, as Euclid by the 12th and 13th century turns into other sorts of translations in the Latin tradition, to be the way that people approach the full text of Euclid when it becomes available. So if you look at, for example, the most significant and the most widely read and the most influential version of Euclid in the Middle Ages, "Campanus of Novara", he's taken the whole lot of what we would think of as the proof. And he said, I've got new versions of that in many cases, or very often, he'll come around and he'll just tweak the original because he sees the annunciations here as the main bit. That's the bit that's actually Euclid, and the rest of it is commentary functionally, it's a gloss, it's a thing that's used to explain what's really going on in the main text. Some of that shifts then, you can see some interesting things happen with the very first editions in print of Euclid beginning in the 15th century. So, first, this is the first version of "Campanus" that's printed, that's the medieval version. There's another version, Zamberti, that is produced, which is retranslated from the Greek. The "Campanus", remember lots of bits that aren't actually transmitted from antiquity but are considered glosses. Zamberti offers a counter perspective. He's going to suggest to us that in fact, Euclid's authored also a whole bunch of demonstrations, the things that we think of as proofs. When he does so, he translates, I mean, there's a number of issues with his translation, one of which is that he hasn't always understood the Greek very well, and he always doesn't always understand the maths very well. And as a result, it turns out, as someone like Luca Pacioli points out that "Campanus" is actually good maths, Zamberti is good Latin, but bad maths, right? So you can see the problems that are starting to arise here. Authorship or your assumptions about what Euclid means, what bits of the text belong to Euclid are going to start having implications for how you actually start to work out the maths. Not only that, you start to have plural versions of what shall we say are Euclid, right? You have different, not only translations, but different textual traditions in which some of the bits aren't Euclid at all. And to sort out that sort of problem, you get something like this version which sets Pacioli, well "Campanus", sorry, alongside Zamberti next to each other, so that you as the reader can productively work your way back and forth between these things in order to sort out for yourself what's really going on in the text. Now, the overriding message here is that mathematical narrative, the thing that you really need to follow for these people is what happens in the annunciations, not what's happening below the line, not what's happening in the demonstration. That proof demonstration is explanatory and it's additional, and it's even optional. So if you're doing geometry, you can do away with that. And you get a whole lot of textbooks in the period where that idea of geometry is actually put to work, where you remove all the demonstrations and you just move with the sensation of being able to follow a lot of descriptions of mathematical contents. Turns out that's actually quite productive because it allows you to do kind of undisciplined things with the maths, you can start applying maths to all kinds of practical problems in ways that maybe the proof distracted you from or didn't let you do. So this is just one example of a set of authors, Lefevre d'Etaples, Clichtove, Bovelles, and there's also others involved in making this volume where they're basically trying to come up with a mathematical curriculum that allows you to do precisely that. Furthermore, that sort of turns into a kind of a physics. So Gregor Reisch's "Margarita Philosophica" is a really, it's a best selling textbook. He starts to do some similar bits, or rather, he includes appendices where they're doing that kind of math. So every student going through universities learning how to do math without the demonstrations. And I was thinking that I had a picture of a musical text one way, one implication of this way of thinking about demonstration as an optional thing is that you can start to reason between different mathematical disciplines in new ways. So this becomes a fertile sort of way of thinking about what's going on in mathematics. One implication I want you to then draw based on what I've just said, is that for figures like Oronce Fine, proof becomes interchangeable with practice. So when someone like Oronce Fine gives you a manual for learning how to do mathematics, sometimes he's not going to say, now you need to apply the proof so that you can understand the enunciation. Instead, what he's going to do is give you some homework and give you a lot of constructions to do and ask you to make an instrument. And in so doing, he's asking you to embody what's going on in that thing that you're supposed to prove and not think it through, but work it through. Now I'm making an overly fine distinction, right? Maybe working it through as well here probably later has something to do with thinking it through as well. But here, I just want you to see what this allows you to do when you start thinking through the idea of proof as a gloss or as an explanation. A second implication is one that I find really, really evocative, which is that instead we tend to think of proofs as elegant when they're cut down, when they're spare. But this creates a different mathematical style, and it's a mathematics of abundance, of copia. If you can do one thing three ways, do it three ways instead of one way, right, which is runs counters to some of the intuitions that we often bring to things like geometrical proof. It means that if you can offer a different modality alongside the first modality, you're going to do that. So for example, when Lefevre d'Etaples produces this sort of textbook, this is the one you've seen already where he's set multiple additions next to each other in order to understand Euclid, the best option for him is to provide all the versions of Euclid, all the extra demonstrations, because that's what's going to benefit you best as a reader. When he writes about music, this is the sheet I was looking for earlier, he doesn't give you, he gives you an explanation of how to make the instrument in certain places as well as an explanation for how to reason mathematically through the relationship between different sorts of tones. And I think that's significant enough to pause over because this matches a set of habits that if we look to renaissance literature are everywhere in the culture, right? So one thing that we've learned over the last 40 years is how much copia, Latin word for abundance, is one of the cardinal virtues of learning how to write well and how to think well in renaissance kind of learned culture. And the thing that I want to point out is, by seeing mathematics as in its kind of narrow tidal form in which demonstration becomes not just an exercise, it's stripping things away until you can finally see the narrow truth of the matter, but instead building up a whole lot of material so that you can grow and make different kinds of connections between disciplines that fits with a vision of literary invention that was rampant in the Renaissance. And Terrence Cave would be a starting point, his book on a Erasmus and the Cornucopian text for thinking through that sort of method, that narrative style, shall we say. So I've promised to describe a little bit of how that shifts in, again, maybe the best title is an invention of proof or something like that. What I mean by that is how do we narrow this world of multiple options, if you like, to the kind of more stripped down version of thinking about mathematical narration that we associate with Descartes, and that we associate with geometry when we read it now, in which Euclid still bears a lot of responsibility for looking like an exemplar of demonstrative reasoning. And there I want to think a little bit about what two authors, but the main one, and the first one is Federico Commandino. So, he is a scholar in the household of the Duke of Verbenone in Italy. I talked earlier about a bunch of collaborators around someone like Jacques Lefevre d'Etaples, Clichtove, Bovelles, the circle. They're university scholars, they're interested in getting students through a curriculum and using mathematics as a way of training people to think a particular way, building the habits of copia so that they can become productive in other domains as well. Federico Commandino has a very different sort of set of goals, on the one hand, so he is not working in a university context, he's working in the context of a court that is interested in elegance, in a court that is interested in painting, sculpture and mathematics in particular. So the Duke of Verbenone trained by Commandino is interested in recovering visions of symmetry, the kinds of things that we think of when we think of renaissance art, history and maths, right? And Commandino's the guy for that, partly because of the many additions that he produces of ancient texts. So high prestige texts, engineering texts, and the new translations that he's making out of Greek especially. And the most significant of the texts that he produces, and he himself sees this, is his translation of Euclid. So I mentioned the medieval tradition, I mentioned that first sort of somewhat troubling, not terribly great mathematically, but interesting Latin-y version of Zamberti. Everyone knows that Zamberti's problematic. So Commandino has decided to step into the gap. And when he produces his version, he's very self-consciously touching the central nerve of mathematical culture in his world. Now, one of the things that's really striking about the Prolegomenon that he attaches to his addition is that he has to sort out who Euclid is, right? So the Euclid that I've tried to represent to you, it isn't actually clear which bits of the text are Euclid. And in fact, when you have to try to make a decision about that, you have to wade through thickets of different possibilities on the assumption that many of what looked like demonstrations are in fact commentaries, medieval commentaries, or so on. Zamberti had suggested that in fact some of this was to do with, well first of all, he wasn't actually sure himself who Euclid was. He thought that Euclid of Megara and Euclid of the elements were the same person. Turns out they're not. He also thought that Theon of Alexandria bore particular responsibility from some of the writing. And one of the key points that Commandino makes based on a careful reading of an ancient commentary of the first book of Euclid, is that in fact these two aren't dissimilar. So he is already interested in the authorship of Euclid in one particular way. He's going to clear up who that is. And then secondly, he makes a deliberate and lengthy argument for why in fact the proofs that he's translated don't belong to just anyone and don't come out of a commentary tradition generally, but in fact have been tweaked and edited, but responsibly used by Theon of Alexandria in the fourth century. providing the reader now with something like an authoritative text in which you can actually see who the real Euclid is. Now there's a third thing that he's also done based on this commentary by Proclus. He decides that the language for describing the parts of a geometrical argument isn't adequate and teases out of this ancient philosopher language for distinguishing different parts. Because what you could, in order to be able to say more clearly what it is that Euclid's doing when he constructs a demonstration. And one of the striking things that I find as a reader of additions of this text is that very often this is the bit that early modern readers get a kick out of this is the bit that they're underscoring or as this particular reader has done renaming in the margins. So all of a sudden we right around because of this edition, we start to have a language for talking about a proof anatomizing a proof if you like. And you can't read the details here, so you're going to have to take my word for it a little bit. But one of the things that happens throughout the actual text that Commandino leaves us is a clear distinction between what in fact belongs to Euclid, and that's the authoritative proof and then his own commentary where he deals with various troubles in the text or mathematical problems. So notice what he's done, he's managed to scrape together what is actually Euclid. And then he's very carefully separated out his own particular sort of glosses, his commentaries, visually on the page. And then he goes further. And in that commentary, he spends a lot of time distinguishing all the very bits, what counts as a theorem and so on when he's explaining what Euclid is doing in the text. So there's a sense, right, that the first implication here is that for the first time, what we have is something like a canonical Euclid that matches kind of the Euclid that we would recognize. And that has big implications because that's the starting point for say, 17th century mathematicians when they're trying to explain how to do maths and then tuck it into other parts of natural philosophy and so on. So it's that Euclid that becomes canonical and that Euclid that offers a particular vocabulary of proof. Or I think anatomy of proof is kind of the term I'd prefer. - There's another one. Remember, I've gotten excited about the possibility that there's a nice overlap between these ideals for thinking through mathematical narration and larger literary ideals in the period. And I think it's not too far to reach when we say that Commandino represents, unlike our school men that we talked about earlier, a different persona both in his own life, right? So he's focused, he belongs to this context where elegance and kind of spare ease in which just a few words can do a huge amount of work in courtly life. That sprezzatura according to the Italians, right? He's representing that not only in the way he thinks he would like to come across as a court here, but also the way he wants to think as a mathematician. And that's quite interesting because it matches a little more closely what we think of characterizing a good mathematical proof. What we think of as characterizing Euclid himself, if you like, this prizing not of copia, of abundance, of kind of a fertile mass of ideas, but rather the thread that holds it all together and everything else being paired apart. That kind of an economy. I want to just spend a moment thinking about Christoff Clavius, who is a Jesuit who in a way wrote the most popular Euclid, teaching Euclid, of the early modern period. It went through a number of editions after 1574. So 1572 is when Commandino produces his Euclid, Clavius builds on what has now seen to be the state of the art Euclid, but when he does it, he adds masses more material. So look what's happening with a pedagogue such as Clavius, who takes the anatomy from Propos, but then from Commandino, but then applies that anatomy in his own commentary in a much more kind of direct and thorough going way than even Commandino had done. The point I want you to get from this is not, you know, is not very subtle, it's simply that what Commandino accomplishes with that Euclid turns into the teaching tool, right? This turns into the starting point for every other version of Euclid that is used within classrooms and in other additions. And when indeed Euclid is rejigged in different forms over the next couple hundred years, Clavius is usually the starting point for that. But Clavius as interpreting Commandino. So that's just a point to say this moment that I kind of playfully am identifying in the 1570s as a really key kind of turning moment. But there's, you know, you can always make that line fuzzier if you like, nevertheless does bear some real responsibility for kind of the ongoing sense of who Euclid is. And as I see it also for our ideas of proof. So I want to suggest that, you know, what we think Euclid is is in itself a question of history, right? We get it from somewhere and we don't get it directly from, you know, the fourth, the third century BC, we get it mediated via these kinds of additions. And so what we experience when we are working through the first book of Euclid, or indeed a lot of other forms of proof, is the prompting of a lot of forms of narration, that are pressuring our own reading. One of the things I want to suggest is how significant that moment is. So I've dwelt on that long enough, but the most interesting bit I think is what this implies about what happens before the 1570s, right? So if the Euclid we sort of work with only comes into being in the 1570s, what's happening in that kind of wild west? That's a overly open term, perhaps, that very puriform set of possibilities for thinking through a mathematical proof, a mathematical narration before the grid of the Proclian anatomy has been put onto Euclid via Commandino and Clavius. And if we can open that up, we can get to a much richer, much more interesting sort of understanding of what's actually going on, say in medieval or early Renaissance mathematics. But in a second move we can also make much more interesting connections to other traditions of doing mathematics that stand outside of the Euclidean one. And I think we'll probably hear a bit more about that later on today. Thank you. (audience applauds) - You spoke a lot about Euclidean proof. Was there any discussion in the medieval period about Euclidean truth? Because in the 19th century lots of other geometries developed of course, and in modern times with relativity and so on then, you know, Euclid in geometry is true for most practical purposes, but not for all. - You talked about the fact that the way the medieval scholars presented Euclid, at least at a certain period, they emphasize less on the proofs, at least not as Euclid's proves, more on a certain set of truths which can be listed, whereas subsequently all the way till now mathematics is very much emphasized on proof. That's the main thing mathematicians concentrate on. So it seems as though we've gone like one level deeper from truths to proof. And I was wondering, do you think there would be something maybe another level deeper, more fundamental? Because, so it seems that at some point we're going from truth to truth, now we're going from proof to proof. You think there's maybe a third level deeper than that? I'm not sure if that's clear, but. - Yeah, yeah. So I mean it's possible that these questions are related somewhere deep down at a fourth level. So let me think about questions of truth and Euclidean truth. What I don't think you have in the Middle Ages are kind of the pressure put on the fifth postulate, you know, the parallel postulate, that turns into kind of non-Euclidean geometries. And so Euclid is generally taken to be pretty good at explaining how geometrical objects behave and so on throughout the Middle Ages. I don't think I'm going wrong by telling you that. What is interesting is that a lot of emphasis is put on being able to see with what's called the light of nature, those mathematical objects. And the way one would do that very often is suggest that Euclid isn't the whole story, that there is a deeper method, a math thesis that might have like be related to theology in a particular way, perhaps. So in the same way that mathematical objects are completely separated from the world of things, but accessible by your intellect. So the assumption goes, metaphysics and theology allow you access to really deep truths by a mode of intuition that's purely intellectual. And you can find that, for example, in Boethius, one of the philosophers I was talking about, who gave us this kind of set of annunciations only. And when Boethius does that, he gives us kind of a way of talking about mathematics as gesture towards a deeper, what we might call a math thesis universalis, which becomes a hot topic of discussion and debate leading up and beyond Descartes. And I think that's where your question overlaps actually with this question about a deeper sort of thing. So the division that Boethius leaves us with is between four forms of mathematical reasoning of which basically two are the most fundamental. The the two forms of pure maths are geometry and number theory are arithmetic. And the claim is that if you can, there must be a deeper kind of reasoning that allows you to make connections between those two sorts of things. But nobody quite knows how to do that. And maybe if you keep playing with annunciations, and maybe if you keep kind of commenting on this, you can give people the tools with which to achieve the internal insight that's going to allow them to do that. And I wonder if someone who would be open to that kind of math thesis universalis actually might be interested in kind of querying whether Euclid's got it all right. So that might feed back into your question again. - I was wondering, you were talking about Euclid, maybe his main importance was the axiomatic method that he produced. And that's reproduced much later. You mentioned Spinoza there for instance, who used that and then there was Piago of course later still, and possibly Wittgenstein as well. So is it in fact that the Euclidean method when it's taken away from physical things is really part of formal logic and not part of mathematics anyway? You see what I mean? - Right. So that's precisely the question that people are thinking of actually when they're thinking about this math thesis universalis, right? So Descartes in his rules for the direction of the mind kind of says maths and sorts of forms of something like axiomatic reasoning is going to give us a logic that we can apply to everything. But of course what's really hard is to spell out precisely how that works and the claim that there is an underlying logic in a way that originates in this Propos book that I mentioned by this fourth century neo-platonic philosopher. He claims that there is something that underlies the connection between the geometry and arithmetic and that it is just basically logic, at least that's one reading of how he, it's a complicated little sentence in which he offers this. And I can't repeat it just off the tongue right now, but that is the issue at root. Now the question of making that stick turns out to be much trickier. So I would suggest that at least in this period, people would be keen on asking you, okay, show me then, is axiomatics just logic? And it turns out that's harder to do than than said, even though it looks quite promising. - Thank you. Okay, I'm going to ask my question. So it's about diagrams. - Yeah, yeah. - To what extent is proof by diagram trusted, accepted? You know, if you show a diagram, does this convince you, and is that valid? 'Cause if you look in Euclid, you don't get that. So how does this change or does it in the Renaissance? - Yes. I think this is one of those questions that people have intuitions about, historians of maths have intuitions about, but don't have a fully worked up narrative for. So I hope that we spend a lot more time on this question. One of the things that I find fascinating about reading the people who are, so I have often looked at lots of textbooks of the kind that I've alluded to, the ones with just the annunciations where people are being asked to intuit their way through a mathematical narrative. And one of the things that people write often or do in the margins is do things with compasses. They're writing things out. And so in a way, Michael Wardhaugh, sorry Benjamin Wardhaugh has made the claim that in some ways a lot of what's going on here is, you know, recipes for diagram making, right? So the goal is that you spend time doing things on your own. And so even when there's not a diagram in front of you, you're being asked to produce one as a reader and you're expected to read much more actively than we might think and therefore in some cases in fact extra space is left, you know, in the margin and so on for that. Partly because that's a habit that books from this period are much more scribbled up than we tend to have in our libraries, or at least, you know, than I've tended to do in the past for sure. And that habit of writing as you're reading tends to bleed over into precisely this issue. So that's one way of thinking through the question of diagrams here. Now, the in principle question, it depends on who you're reading. Some people think it's great just to use diagrams and no questions asked. That's part of the demonstration and some people are are concerned about that. Jacques Peletier du Mans, for example is a renaissance geometer who's raising questions, how valid is this? There's more to say of course, but. - There always is, there always is. Thank you. Okay, so I'm going to put Richard on the spot with this question, he thought he'd finished for the day, you know, back on duty. So we had a question online that just came in a minute too late for the questions last time from Marcus who asks, if medieval mathematicians dealt with the fact that Euclid's geometry has axioms, but the number theory doesn't. So he says how do you deal with the fact that Euclid's geometry but not his number theory has postulates? His question. - I don't know if this is speaking, should I stand up? So that's a really interesting question, and there's a bit of that I don't know. So there's an empirical thing of having to go and look at all those commentaries. On the one hand, one thing that is fascinating though is that when people take that number theory that's found there and then they start to write independent treatises on it, they do supply postulates and they scrape around and they try to find them together. So Jordanis de la Moray, for example, is an example of someone who writes 10 books on number theory and he begins with the same practice that he's seen Euclid do in the geometry. And you can look at some of what I've been saying about this kind of suspicion that in fact proof constantly allows you to kind of flexibly bring together different kind of materials into new recipes, I suppose we should call them too, in the various versions of this number theory that is sort of sourced there. And in other words, it does the same thing that I've been arguing for Euclid, which is to say there's more work needed done, I think on book five and its commentary tradition of Euclid in the Middle Ages.