 Hi, I'm Peter Adamson, and you're listening to the History of Philosophy podcast, brought to you with the support of King's College London and the Liberhulme Trust, online at www.historyoffilosophy.net. Today's episode will be an interview about ancient mathematics with Serafina Cuomo, who is a reader in Roman history at Birkbeck College in London. Hi, Serafina. Hello, Peter. Thanks for coming on. I'd like to start by asking you a very basic question. What did the ancients understand by mathematics? Obviously, it would have included arithmetic and geometry, both of which are words that come from Greek, in fact. What else would it have included? It would have included a lot of other things that we don't necessarily consider to be mathematics today. Like everything, it depended on whom you asked. For some people, mathematics could really be taken to concern knowledge in general. So someone who was a mathematic horse was someone who was interested in learning about pretty much everything. But more specifically, alongside arithmetic and geometry, which you would expect, I think that many ancient Greeks and perhaps Romans would have included astronomy, music, perhaps even things like astrology. In fact, in late antiquity, the word mathematicos often defined an astrologer rather than just a mathematician in general. And on the Roman side, the word geometris often indicated a land surveyor rather than just a geometrician. So even the usage of the words connected to mathematics covered a much wider range than today. That mention of land surveying brings up something else, which is that although if I just say the word mathematics, I guess that listeners will have in their mind something very abstract, the sort of things they did in high school maybe, ancient mathematics would have included or at least related to a lot of very practical disciplines and enterprises like land surveying and maybe engineering and so on. So could you say something about that? Yeah, definitely. In fact, a lot of authors report the story that perhaps is familiar about the origin of geometry. Geometry means land measurement. And they say that it started in Egypt. When the Nile flooded, it was difficult to recognise, remember where your plot of land ended and someone else's plot of land started. So geometry was invented so that people after the Nile floods could find where their piece of land was again. The very origin of mathematics and geometry was seen as practical. It's still a matter for debate to which extent sophisticated, we could say, mathematics was used in things like building machines, architecture and other applied sciences. But we definitely have a lot of textual evidence and in some cases archaeological evidence to indicate that mathematics wasn't just an abstract pursuit, but it was seen as something that had to do with everyday life. And they were capable of really astonishing feats of engineering as well. Yeah, definitely. Perhaps one of the best areas to look at is that of military engineering and what they could do with catapults. We're lucky in that when it comes to catapults, we both have good textual sources and a lot of archaeological remains. Enthusiasts, alongside with scholars, and obviously there can be scholarly enthusiasts and enthusiastic scholars, today have engaged in reconstructing ancient catapults on the basis of ancient evidence using materials that could have been used at the time and so on. And they've verified that they could build catapults which could shoot both accurately and powerfully. Really no less a mathematical person than Archimedes was known for his work on war engines. There's something about a hook that he built that comes down and grabs boats and lifts them into the water and then drops them back down and destroys them, right? Yes, yes. Archimedes in antiquity may almost have been more famous for his military devices than for his mathematical achievements in the field of pure geometry. He held the Romans at bay for two years when they were laying siege to his city and they only managed to defeat Syracuse, Archimedes' city, through deceit because someone betrayed the Syracusans. So Archimedes was a very good example of someone who could do both very abstract, sophisticated, advanced geometry and build machines which were all the more accurate and effective because of their mathematical foundations. Obviously then mathematics covers a very wide terrain. So what exactly is it that unifies this together into one discipline? I mean if we talk about ancient mathematics, what's the overarching notion involved here? That's a very difficult question. So I'm just going to throw a couple of things at you rather than give an answer. My first guess would be that mathematics, the various disciplines that constituted mathematics, were unified by the idea of number. The ancients were aware that there are limitations to what mathematics can do. So they were aware that even though number is involved in all the various disciplines, you can't for instance measure everything perfectly. Sometimes numbers can only express a certain proportion or a certain multitude up to a certain point. So number insofar as it underlies arithmetic, geometry, and in the form of proportions, it underlies a lot of building, both building machines and building houses. Number would be a good candidate for a unifying principle for all these disciplines. Okay, well then that raises the question I guess of what numbers are. So I'll ask you another very difficult question. What did the ancients think that numbers are? I think it's clear from our evidence that numbers for the ancients were much more concrete than perhaps the idea of number we have today. Some ancients, philosophers in particular, Plato comes to mind, probably had a more abstract idea of number as something that is not necessarily attached to an object but can be detached and then studied in its own right. But I think your average Greek or Roman in the street often thought of number as something that is totally concrete. When they counted, they counted with pebbles or tokens on a counting board or on an abacus. In a sense that's what numbers were for them, concrete objects that you could manipulate and touch. In some forms of mathematical notation in classical Greece, you don't just have a sign for five, say, the sign for five embeds another little sign that indicates this is five coins of a certain denomination. So it seems to me that if one had to start answering that question and we could devote a whole series just to that, one of the things to bear in mind is that most people probably would have thought that numbers were concrete, real, to do with objects rather than abstract. Does that relate to something that at least is often said about Greek mathematics, which is that they have a kind of spatial understanding of number in the sense that they think of numbers as geometrical, so like square numbers and so on. We might think of square numbers as just an example, so the area of a square is going to be the square of the side. But did they actually think about, and I guess now I'm asking more about people who were doing mathematical advanced research, did they actually think that numbers were spatial extensions sometimes, or is that an oversimplification? I think that's a good description of what is going on in some authors. It's interesting, for instance, that the Pythagoreans are attributed views that fit with what you're saying. Or later mathematicians, whom we could call neo-Pythagoreans, such as Nicomachus, especially spent a lot of time talking about square numbers and cube numbers and even pentagonal numbers, so visualising numbers in geometrical shapes. It's a way, if you like, going back to our previous question, it's a way of establishing a unifying principle, which is number. So if there is a connection between the two, you've also found a way of unifying arithmetic and geometry. It's also probably significant that even in Euclid's Elements, in the books devoted to arithmetic, there are diagrams. Numbers are represented by line segments. Speaking of Euclid, something else I wanted to ask you is about mathematics and methodology. So I guess that one reason that mathematics was seen as philosophically important by ancient philosophers is that it gives you a model for how to build knowledge, and obviously Euclid's Elements would be a really good example of that. Could you just tell us something about how Euclid's Elements works methodologically, first of all, and then maybe tell us to what extent that's representative of ancient mathematics? Yes, probably most people will know this. My generation, I won't reveal all the time, but my generation and people older than me actually studied Euclid's Elements when they did mathematics at school. Euclid's starts from undemonstrated premises. So book one of Euclid's Elements contains definitions, postulates and common notions, which are also known as axioms. None of these propositions is demonstrated. You have to accept them in order to carry on. Definitions are things like what a line is, what a point is, and so on. Postulates are more complex statements that you are asked to accept, at least for the time being. The most famous one is probably postulate number five about parallel lines. And not accepting that postulate leads then to the creation of non-Euclidean geometries. Common notions are things that seem self-evident, such as if you add equals to equals, then equals will result. On the basis of that, Euclid then builds a whole edifice of geometry, which is axiomatic in that it starts from this undemonstrated premises and deductive in that it goes from general principles to particular specific results, specifically on the triangle constructed in the diagram is laid out in front of you. So these proofs are not just about a specific triangle whose sides are three, four and five but about any right-angled triangle. In that sense it is deductive. Euclid didn't invent this method. We find something that looks a lot like it in, for instance, Aristotle's posterior analytics. A lot of ink has been spent discussing what the connection is between Euclid and, say, previous philosophy. The method is used definitely after him. We find a similar method in Archimedes, we find it in Apollonius, to the point where some historians of mathematics identify a whole tradition of mathematics in the line of Euclid, and the big names that belong to it are Euclid, Archimedes and Apollonius. I'm interested in something you mentioned there, which is that when you're trying to prove something you draw a diagram. How do you get the universality bit? So if I draw a diagram and I show you that what I'm arguing for works with the triangle that I've just drawn, for example, how is it that I'm allowed to infer that it will work for any triangle? Why couldn't someone just say, well, right, but you didn't show it for acute triangles, so now draw me an acute triangle and do the same, or do a similar demonstration with that? Well, in fact, sometimes we do find cases like that, not in Euclid so much, but in later geometers. What you describe, which is a kind of analysis of sub-cases, is one of the features of late ancient geometry. Euclid doesn't have that unless there is an actual need to provide a different proposition for a different kind of triangle. In a sense, the diagram is an opportunity for the reader to go and do another diagram and verify that it works. But in itself, the proposition works precisely because even if you draw a different kind of triangle, which is not, you know, whose sides are not 3, 4, 5, as long as it's right angle, you'll see that the proposition still works. And I guess that the way that the text actually works is it describes a diagram, right? And then there are many, in fact, infinitely many triangles you could draw that would satisfy the diagram. Yes. And the thought is, well, the proof will work for any triangle that you can draw on these instructions. Exactly. But does he ever explicitly say that, though? I mean, does he call attention to what he's doing in terms of the methodology and why it's a proof? Not really. Euclid, unlike other mathematicians, never steps out of the text and gives us a statement about what he's doing. We'd like it to have that with, for instance, Archimedes, Apollonius to some extent, but the authorial voice of Euclid is completely absent, to the point where some scholars think that there was no Euclid. Oh, he's like Homer. He's like Homer, yeah. So the work almost came together, and there were later additions and accretions, which we know there were, but as an individual, Euclid actually remains quite opaque. Which is one reason it's hard to place him relative to, say, influences from the philosophical tradition, I assume. Yes. What about the principles, the definitions and the postulates and so on? If these are just given to us and not demonstrated, I guess the thought might be, these are obviously true, or the thought might be, if you thought these were true, then you should accept the following things which will follow from them. And I guess that, given you just said that Euclid doesn't really reflect explicitly on his method, maybe we don't know what he had in mind, but do later mathematicians talk about the status of these starting points and why you should accept them? They do. One very interesting case in point is, again, a late ancient author called Proclus. Many people wouldn't even say that Proclus was a mathematician. He's more famous for being a philosopher. But he wrote one of our most extensive commentaries on Book One of the Elements, and he goes at enormous lengths to discuss exactly what status every bit in Euclid's Book One. Exactly what he does with Plato when he comments on Plato. But before getting into Proclus, who I'll be covering in a later episode anyway, let's talk a little bit about Roman mathematics. What happens in the transition of mathematics from the Greek world to the Roman world? Obviously, I guess that Roman scholars and intellectuals would have been reading Greek and engaging with the Greek tradition, so the two things would be very closely connected. So what kind of changes do we have once we get into the Roman period? The usual story about the transition from Greek to Roman mathematics is that the Greeks were more theoretical, more philosophically inclined, more abstract in their way of thinking, and that was reflected in their mathematics, where we find works which we could describe as pure geometry. On the other hand, the Romans were pragmatic, practical, concrete, and so that's reflected in their mathematics, which is mostly about measuring land, counting taxes, and so on. The usual story is founded on some pieces of evidence. One can think, for instance, of some phrases in Cicero. Cicero himself says that the Greeks were praising pure geometry, whereas the Romans were more interested in calculation and measurement. And Cicero obviously was a key figure in the appropriation of Greek knowledge, including philosophy, on the part of the Romans. I think, however, that we should try and go beyond the usual story, and I'd like to point to a few facts. One is that several Greek mathematicians actually operated within the Greco-Roman world. We could mention Ptolemy, we could mention Diophantus, obviously all the late ancient mathematicians, hero of Alexandria. We know that they were as aware of the presence of the Romans as the Romans must have been of them. That's definitely true in the case of hero of Alexandria. The second thing to bear in mind is that the Greeks didn't just have pure geometry, they also had practical mathematics. The fact that we identify them more with one tradition of mathematics is to some extent a reflection of the view that Cicero espouses and that has become very, very influential centuries after, even down to our day. So the Roman view then was that the Greeks are more abstract, they're more interested in theory, and the Romans are kind of down to earth. They're more interested in getting things done and winning wars and building things. Yeah, I wouldn't even say the Romans view, we could just say Cicero's view, but to the extent to which that was shared by several Romans, yes. Obviously, I don't need to cash out the political implications of all that. It's kind of convenient to have the Greeks in this abstract head in the clouds position and the Romans in the dominant politically powerful position. The fact that this is not necessarily a reflection of what was going on could be seen in the very figure of Archimedes. Actually, that brings up something that I think in a way has been running through a lot of what we've said, which is that mathematics often seems to have this connection to political power. So we've talked about using mathematics in warfare, we've talked about using it to measure land, and presumably the government, for lack of a better word, is in charge of figuring out whose measurement is correct. And we might also think about, for example, the use of mathematics in voting in Athenian democracy and so on. And it almost sounds like ancient mathematics needs to be understood as a kind of tool or even weapon that was used by certain people in society to gain a political advantage over other people. Or is that too radical a proposal? No, I think the reasoning is radical at all. Insofar as mathematics is connected to ideas of accuracy, fairness, transparency even, we find it as a thread running through political discourse. One way of looking at this, for instance, which is what I'm working on at the moment, is looking at accounts. You mean like bookkeeping? Like bookkeeping, but accounts that were sometimes inscribed on stone and displayed in public places. The fact that this was done obviously sent a certain message about accountability. If you're ready to publish your accounts, that also means that you're not afraid of people going through them to see if you've embezzled money. So it sends a message about transparency, good government, fairness and so on. And it's a practice that we find possibly in its strongest form in 5th and 4th century BC Athens, supposedly the cradle of democracy. It's very interesting that in the Roman Republic, for instance, generals who won big victories and were awarded the triumph had to display the account for the campaign during the triumphal procession. That's a practice that ends with the empire also because generals really no longer are allowed triumphs. Only the emperor has to be seen as the triumphant one. One of the last forms of public accounts we find in Rome is probably in the big bilingual inscription set in various copies all over the empire by the Emperor Augustus, the Res Gestae inscription, where he details all the things that is done and all the money that is spent out of his own pocket is came to add. And at the end, in some version, there was an account adding up the money and giving you a sum total that represented Augustus' generosity in a very concrete form. If I could maybe finish by asking you about something rather different, which is about Pythagoreanism. So it's obviously a big issue, but I'm just about to get to Neoplatonism. And I've already been talking in previous episodes about how the so-called middle Platonists fused Pythagorean ideas about mathematics with Platonist philosophy. So I was just curious whether you could say something about whether Pythagorean philosophers and mathematicians actually contributed anything to the history of mathematics. So someone like Nicomachus, who you mentioned earlier, would he have been a really serious mathematician who proved new things in mathematics? Or were they just fooling around with numbers? It all depends on your definition of mathematician and of serious mathematician. I'm not sure that Nicomachus proved new things. There is a theorem that goes under the name of Theorem of Nicomachus. But to be honest, I was looking again today. He states the theorem. We call it a theorem, but he doesn't prove it. What's the theorem? All the terms in the odd places in a series in double or triple ratio are squares. So if you take the series in double ratio, 1, 2, 4, 8, 16, 32, 64, and so on, all the series in triple ratio, 1, 3, 9, 27, 81, 243, and so on, all the numbers in odd places are squares. And you can verify that, but it doesn't give us a proof. So was he a serious mathematician? I think he was. He spends a lot of time to tell us how important mathematics is, how mathematics is the key to everything, how numbers, in the form sometimes of squares, cubic pentagonal numbers, can even help us understand geometrical figures and so on. So his contribution and that of Pythagoreanism in general is, I think, this strong belief that mathematics is the key to understanding reality. If it's true that Plato in the time years was being Pythagorean, and you look at the influence that the time uses had on the key figures in the scientific revolution, such as Galilei, then you see where the true importance of the Pythagoreans for mathematics and science really is. And Plato, of course, says that we should get into philosophy by doing mathematics, so I'm sure he'd be very pleased that we spent this episode doing mathematics before getting to Neo-Platonism. Thanks, Serafina, very much for coming on. And I hope you'll join me next time when I'll start to look at the works of Plotinus, here on the History of Philosophy, without any gaps.