# 86 - Serafina Cuomo on Ancient Mathematics

How did the mathematics of figures like Euclid and Archimides relate to ancient philosophy? Peter finds out in an interview with Serafina Cuomo.

Themes:

• S. Cuomo, *Pappus of Alexandria and the Mathematics of Late Antiquity *(Cambridge: 2000).

*• *S. Cuomo, *Ancient Mathematics* (London: 2001).

• S. Cuomo, *Technology and Culture in Greek and Roman Antiquity* (Cambridge: 2007).

Note: this transcription was produced by automatic voice recognition software. It has been corrected by hand, but may still contain errors. We are very grateful to Tim Wittenborg for his production of the automated transcripts and for the efforts of a team of volunteer listeners who corrected the texts.

Peter Adamson: 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?

Serafina Cuomo: 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 "mathematikos" 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 mathematikos often defined an astrologer rather than just a mathematician in general. And on the Roman side, the word "geometres" 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.

Peter Adamson: 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?

Serafina Cuomo: Yeah, definitely. In fact, a lot of authors report the story that perhaps is familiar about the origin of geometry. Geometry means land measurements, and they say that it started in Egypt. When the Nile flooded, it was difficult to recognize or 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 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.

Peter Adamson: And they were capable of really astonishing feats of engineering as well.

Serafina Cuomo: 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 have both good textual sources and a lot of archaeological remains. Enthusiasts alongside with scholars - and obviously they 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.

Peter Adamson: Actually 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?

Serafina Cuomo: 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.

Peter Adamson: 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?

Serafina Cuomo: 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 constitute 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.

Peter Adamson: 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?

Serafina Cuomo: 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, 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 abstractions.

Peter Adamson: 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. So that, I mean, 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, I mean, 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?

Serafina Cuomo: 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, spent a lot of time talking about square numbers and cube numbers and even pentagonal numbers, so visualizing 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.

Peter Adamson: 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?

Serafina Cuomo: Yes, probably most people will know this. My generation and people older than me actually studied Euclid's Elements when they did mathematics at school. Euclid starts from undemonstrated premises. So Book 1 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 the line is, what the 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 5 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 these undemonstrated premises, and deductive in that it goes from general principles to particular specific results - specifically on the triangle constructed in the diagram that's laying out in front of you. So his proofs are not just about a specific triangle whose sides are 3, 4 and 5 but about any right angled triangle. In that sense it's 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 and 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.

Peter Adamson: 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? I mean 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 a similar demonstration with that?

Serafina Cuomo: 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 - whose sides are not 3, 4, and 5 - as long as it's right angle you'll see that the proposition still works.

Peter Adamson: I guess that the way that the text actually works is it describes a diagram and then there are many in fact infinitely many triangles you could draw that would satisfy the diagram, and the thought as well, the proof will work for any triangle that you can draw on these instructions. 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?

Serafina Cuomo: Not really. Euclid unlike other mathematicians never steps out of the text and gives us a statement about what he's doing. We have that with Archimedes, and 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.

Peter Adamson: He's like Homer.

Serafina Cuomo: It's like Homer, yeah. So you know the work almost came together and you know there were later additions and accretions which we know there were, but as an individual Euclid actually remains quite opaque.

Peter Adamson: So that's one reason it's hard to place him relative to influences from the philosophical tradition I assume. 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?

Serafina Cuomo: 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 on at enormous lengths to discuss exactly what status every bit in Euclid's book one has.

Peter Adamson: 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? I mean 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?

Serafina Cuomo: 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 are 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.

Peter Adamson: 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.

Serafina Cuomo: Yeah, I wouldn't even say 'the Roman's view,' we could just say Cicero's view - but to the extent to which that was shared by several Romans, yes. Actually I don't need to cache 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.

Peter Adamson: 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?

Serafina Cuomo: I don't think it's 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.

Peter Adamson: You mean like bookkeeping? That kind of accounts?

Serafina Cuomo: 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 fifth and fourth 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 a 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 he has done and all the money that he has spent - out of his own pocket, he's keen to add. And at the end, in some versions, there was an account adding up the money and giving you a sum total that represented Augustus' generosity in a very concrete form.

Peter Adamson: 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 Neo-platonism, 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?

Serafina Cuomo: 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. It states the theorem. We call it a theorem, but it doesn't prove it.

Peter Adamson: What's the theorem?

Serafina Cuomo: 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 the 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 square, 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 Timaeus was being Pythagorean, and you look at the influence that the Timaeus has had on 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.

## Comments

Mathematical ObjectsVery interesting talk, and I particularly enjoyed the connections between ancient mathematics and practical activities. One thing that came up here was the problem of mathematical ontology -- whether there are such things as independently existing mathematical objects and so on -- and Cuomo's response to this was that mathematical objects were then thought of as being much more physical than we think of them now. But even back then, things weren't so simple: if we take square numbers, then we can think of them as groups of physical objects arranged in squares. But still: we know that, for example, every other square number is even. And we can exhibit that by taking the physical objects in a suitable square number and rearranging them in two lines, with objects in one line corresponding to objects in the other line. So, if we really want to say that "ancient mathematics is all about collections of objects, suitably arranged", we also have to say that we are allowed to rearrange the collections. So we are really saying that "ancient mathematics is all about collections of objects, up to rearrangement". (And if you think that this argument sounds familiar, it's a rerun of Benacerraf's paper "What Numbers Could Not Be", where he argues that numbers cannot be identified with sets with a certain structure: rather they are sets with a certain structure up to suitable rearrangement of the structures.)

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