Revealing the secrets of Sanskrit mathematics with Prof Clemency Montelle - Episode #74

We talk with Professor Clemency Montelle all about her work decoding the mathematics of ancient manuscripts. In a wide-ranging discussion, we cover some of the concepts that first arose in India, hear about a mentor who hosted meals based on ancient recipes, and learn how important cultural and historical context is for the questions that mathematicians ask.

While she was at the INI, Professor Montelle delivered the Kirk Lecture for the Modern History of Mathematics Programme, which you can watch here: https://www.youtube.com/watch?v=sa2kN-li984 

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This episode was supported by the quantitative research firm G-Research, which has launched a new series of mathematical puzzles called G-Riddles. It's free to try and could win you a cash prize: https://www.gresearch.com/griddles/

Living Proof is the podcast of the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge. 

The Isaac Newton Institute is a national and international visitor research institute. It runs research programmes on selected themes in mathematics and the mathematical sciences with applications over a wide range of science and technology. It attracts leading mathematical scientists from the UK and overseas to interact in research over an extended period.

Produced by Jon Farrow.

Edited by Keerthi Raj.

Music: 'Origami' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au

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**Correction** At 5:26, the date of the earliest numerical tables is the beginning of the second millenium before the Common Era.

Transcript

Prof Clemency Montelle

Context is huge for mathematics. It's driving people to figure out what questions they want to ask and what problems they want to solve.

Jon Farrow (INI)

Welcome to the Living Proof podcast. I'm Jon Farrow, the Communications Manager here at the Isaac Newton Institute. Every year, thousands of mathematical scientists walk through our doors. In some cases, they come for just a day to give a lecture or to come to a workshop. Sometimes they stay for six months on one of our long-term research programs, and they all have stories to tell. 

With this podcast, we want to highlight some of those amazing mathematical stories and share our love of mathematics with the world. This episode, we're talking all about the history of mathematics. 

We're joined by Professor Clemency Montell, who is the head of the School of Mathematics and Statistics at the University of Canterbury in New Zealand. Clemency is a historian of mathematics with an impressive resume. Having completed her PhD as a Fulbright Scholar at Brown University, she went on to hold the Rutherford Discovery Fellowship and is now the head of her department. Not only has she learned to read ancient Latin, Greek, Sanskrit, and even cuneiform, she combines this linguistic ability with mathematics and history to discover how mathematical ideas arise and move from culture to culture. 

As the distinguished Kirk Fellow for last year's Modern History of Mathematics program, she's now in Cambridge for a month and is delivering lectures and workshops and visiting libraries to decode manuscripts. Hello, Professor Montelle. Welcome to the Living Proof Podcast and to the Isaac Newton Institute. I hope you've been enjoying your stay so far.

Prof Clemency Montelle

It's been wonderful. Thank you.

Jon Farrow (INI)

I feel like when we learn about mathematics in school, it's a lot of men from Greece, like Pythagoras, and England, like Newton, Germany, France. Why isn't that the full story?

Prof Clemency Montelle

It's such an important point. And it's not that story that we tell or that we get told is wrong. It's just that it's a very small thread of what is a much richer tapestry, I think, of stories. And of course, no one's going to deny the incredible achievements of Gauss and Newton and Pythagoras and all of those other wonderful influences that we had throughout the history of mathematics. But it's so much richer story that we've got to tell. 

Because mathematics, of course, is such a deeply global human activity. And in fact, really interestingly, we find that our systems of numeration emerged before we even had words to describe them. So if you look back into our earliest records in the ancient Near East, for instance, around the 4th millennium BCE, so we're going way back, the very first tablets or documents that we have are numbers or numerical and then writing came soon afterwards. 

So really the bigger picture of course is everybody else who sustained and promoted an advanced mathematical activity. So you've got merchants who travelled and had amazing systems of reckoning and arithmetical schemes. You had scribes who compiled astronomical documents and copied them for generations to be able to read and be inspired by. You've got astronomers, you've got calendar makers, you've got these amazing women computers that did this sort of complex calculation, sometimes in observatories or scientific institutions before computers actually took over, like electronic computers. 

All of this has this really important legacy of what it was for humans to be mathematically literate. So it's not that the story we tell is wrong, it's just it's really far too small.

Jon Farrow (INI)

A lot of your recent work focuses specifically on Sanskrit and the history of mathematics in India. Can you tell us a bit more about what you found?

Prof Clemency Montelle

Oh gosh, what haven't I found in Sanskrit sources? Really, there's such a breadth of mathematically interesting material in the Sanskrit sources, primarily because we're talking about... Gosh, 3,000 years of Sanskrit mathematical activity, and that's not even counting the material that was written in the vernacular. All the stuff that we have that are numerical tables or other sorts of records that we don't distinctly class as mathematical first-hand. So, gosh, some of the things I've found in Sanskrit mathematical texts I've been delighted by some of the geometrical diagrams that they include, the emerging forms of symbolic styles of reasoning with the algebraic techniques and some of the 2nd millennium sources. I've been overwhelmed by the number of numerical tables that we see in manuscript archives. We've just got lifetimes. lifetimes full of work to work through this.

Jon Farrow (INI)

So what are these tables about?

Prof Clemency Montelle

So mostly they're tables for astronomical phenomena. So they're just, this is like data science before it was even a thing, right? And so if people wanted to answer questions like, where will the planet be tomorrow? Or when my son is born, what are the, you know, where are all the configurations of all the important things? How would you answer that question? Well, early, earliest astronomy, you would have to be really expert and know all the algorithms and perform these long, laborious calculations. But once tables emerged on the scene, it's almost like sort of this ready reckoner or a computer, you could simply look them up. So it relieved some of the complexity around the mathematics to embedding that complexity in tabular form. So it really was a very cool invention, if you like.

Jon Farrow (INI)

And when about was that?

Prof Clemency Montelle

So tables, of course, started really, really early in the ancient Near East, and we see them in Greece and other places, but in India, they were relatively late on the scene. The earliest tables we have around about the beginning of the first millennium of the Common Era. We do see, I mean, it depends really what you want to call a table. There was tabulated data that was encoded in metrical verse. We can talk about metrical verse later. Sign tables, for instance, that might be memorized and brought to light. But yes, there's so much wonderful stuff. There's trigonometry, which I didn't realize when I really first came to the discipline that emerged in India most properly with the sign. We see this really beautiful arithmetic of positive and negative numbers and zeros. So a lot of the ideas that is foundational for mathematics today, we really see the beginnings of in India.

Jon Farrow (INI)

Do you think that they sort of emerged first in India and then moved through the world to the rest of the world, or did they emerge independently?

Prof Clemency Montelle

See, it just depends, really. A nice example, perhaps, is Pythagoras's theorem as a common theorem that, you know, we all learn at school pretty early on. really useful theorem about the length of the diagonal of a right-angled triangle. So we see this first popping up in the ancient Naries, somewhat indirectly, but we see lists of numbers which are separated which have a relationship of so-called Pythagorean triples. But just the interest in finding the length of diagonals of rectangles, we see them also in India in about 800 BCE and the context of ritual geometry. So altar-building priests needed to make bricks that were made out of squares or rectangles or triangles or the like. And they have encoded in their more religiously oriented texts an expression for what we now know as Pythagoras's theorem and the Baudhayana Sulba Sutra. So that came much earlier than Pythagoras. Pythagoras, of course, talks about it as well. 

And it's not really to say that there's necessarily transmission that, you know, Pythagoras learnt of this in particular places necessarily. Some of the records are really too scanty for us to say, but actually really useful mathematical ideas often just pop up independently. So it's not a question of priority or preeminence. It's just perhaps testament to something that's very universal about the nature of mathematics, but also that it's very culturally determined depending on what questions various cultures of inquiry have and what they want to do with their mathematics.

Jon Farrow (INI)

In these Sanskrit texts, the language, you know, is thousands of years old in some cases. How can you even read something that's that old?

Prof Clemency Montelle

So there's the wonderful thing about Sanskrit is that it's remained relatively stable, unlike many other languages. You know, we often joke, you know, perhaps going back to Chaucer or some Old English stuff, it's starting to become hard for us to understand because the language has evolved so quickly. In Sanskrit, there's this amazing grammarian called Panani, and we think he's probably around the 5th or the 4th century BCE. And he produced what was an amazing rules-based generative grammar. And in fact, Sanskrit is anatomically organized. It really is based on the physiology of your mouth. So it starts with the gutturals right at the back of our mouth, so and the corresponding nasal, and then it goes. goes to our palatals, to our retroflexes, to our dentals, and to our labials. So moving closer and closer towards the front of our mouth. And so there's a beautiful mathematic or scientific organization to the language, which I think is perhaps singular amongst all languages that we see. And it's just so rich and diverse. So every noun has three numbers and eight different cases, and there are three genders and, you know, one of the amazing dictionaries we have of Sanskrit has 180,000 headwords and there's 2,000 different principal verb formats. And so it's both an absolutely beautiful, aesthetically aligned language, but it's also got the scientific precision. And because of Panini codifying it in a way that almost seems mathematical or scientific, it's meant that the language hasn't changed over the last 2,500 years, I suppose. Yeah.

Jon Farrow (INI)

That's incredible. When you go into, these libraries, these archives, and you're reading something that's, 2,000 years old, or maybe it was, reproduced, but the words are original from 2,500 years ago, what does that feel like? How does that, you know, tell us about that feeling of sort of uncovering something that no one has read or seen for a long time?

Prof Clemency Montelle

Yeah, it's really quite special. And I used to joke with some of my colleagues, particularly my PhD supervisor, David Pingrey, it's like, well, we're this person's only friend in the sense that we are now their voice and you have to take that responsibility really seriously. And the minute you humanize that activity, you stop looking at the text as some abstract expression of a mathematical idea that you might recognize to be familiar to something in modern mathematics, but you're actually saying, I'm the voice of this person. 

What were they trying to express? Why did they try and express it? You know, what sort of language did they use and how is that significant? 

So there's something deeply human about this aspect, but there's also something deeply rewarding because it's mathematics as well. And unlike many other disciplines that we might study, philosophy, law texts, literature, There's an underlying logic in mathematics, which we can rely on most of the time, that helps us when the text is a bit damaged or difficult to understand, we can appeal to this author knew what he or she was talking about. They were convinced of some sort of logic or underlying structure. There is something to be found, you know, which we need to uncover, which makes mathematical sense. And that's sort of a sustaining, particularly when you're breaking your head over some text, which you just can't quite understand the meaning. 

A lot of these words aren't even in the dictionary because, you know, as we know, mathematicians have very precise or very specialized vocabulary that hasn't changed in history. And so, and sometimes it can be particular to a particular author. And so understanding what that is, you can't look up the word in the dictionary and say, oh, this means, I don't know, the hypotenuse. What you have to do is figure out its lexicographical meaning from the context. And so you're juggling all these balls in the air, but the wonderful satisfaction of the aha moment where you've figured out what the algorithm is or why it makes sense or some sort of appeal to something deeper is really, really special.

Jon Farrow (INI)

Can you give us an example of what kind of math we're talking about here?

Prof Clemency Montelle

Let me see. So, for instance, trigonometry is a really interesting one in the history of Indian sources, and of course it was indispensable for astronomers. It helped determine the elevation or the altitude of celestial bodies. It provided sort of the way to measure things within the grid of the celestial sphere. What was remarkable about this, because of its tie to astronomy, is that trigonometry was always a branch of astronomy. It took quite a long time for it to transverse into more mathematical contexts and be studied as sort of more interesting mathematical set of functions or the like.

Jon Farrow (INI)

That's incredible. And I guess, yeah, tell me a little bit about how important poetry was to mathematics in this time.

Prof Clemency Montelle

Yeah, so this is the other wonderful aspect of the Sanskrit intellectual sciences that really captured my imagination and meant that I've focused on them. And that is that in Sanskrit, there's a real veneration of the spoken word, and it goes back to broader social religious traditions. So things were meant to be recited and encoded orally and memorized. 

And so what happens is that all disciplines, mathematics included and astronomy, were composed in metrically definite patterns, or mathematical verse, or as we commonly say, mathematical poetry. So that means that puts some really interesting constraints on the astronomers, who also have to be poets in a sense that they have to encode their rules and algorithms and numerical data into these metrically definite patterns. And so decoding them or reading them, because you have to work within the meter, so this meter takes priority over meaning, means that you have to do a bit of work to be able to uncover the meaning. 

And we see really some exceptional, prodigious feats of memory. both in terms of memorizing many, many verses, a whole entire astronomical system if you like, but also this incredible parsimony of the astronomer composers who really have to think about every syllable that they're using because somebody's got to memorize it and carry it down. 

It's also quite cute because it's a form, this sort of metrical pattern, it's a bit of an error correcting code if you like. It makes sure that the text gets really preserved because it's very easy to find a mistake if the syllable's wrong or there are too many. And so that means that's another reason, in addition to the fact that Sanskrit has been preserved so beautifully grammatically, that all of the texts have been preserved in such intact to us today. Simply because of this very strict format in which it was composed.

And so there are some lovely verses, mathematical poetry, which embody some of these trigonometric principles or trigonometric values. For instance, you can memorize a sign table, or at least there's a verb in a sign table from a wonderful mathematician astronomer called Brahmagupta, who wrote in 628 CE, the work called the Brahmas Buddha Siddhanta, and also another one called the Khandaka Kadika several decades later in 665 CE. 

And so his sign table, the Kandakadika was supposed to be this wonderful handbook, which made everything a little bit easier and so abbreviated some of the rules and simplified some of them without too much loss of accuracy. So unlike in his really big theoretical exposition, the Brahmas Buddha Siddhanta, which has this very big, precise sign table, he has a sign table with six values, which is encapsulated in just a handful of the balls and it goes trim shat sanavarasen du jinatiti vishaya so there's your sign table right there in this sort of very metrically interesting pattern and so it literally goes the radius of this circle on which these sine values are to be understood is 150. So it actually gives us the sine differences for 15 degree intervals of increasing arc. 39, 36, 31, let me see, I'm just trying to remember the verse, 24, which is the number of Jinnas or saints, Jinnah, Titti, Vishaya. See, I just repeat this lovely mnemonic to help remember things. So Jinnah is saints, 24, Tittis, there are 15, these are lunar days, and Vishaya is the number. So there's the sine table encapsulated in a little verse that you can remember. So even somebody a millennia and a half afterwards can encode it and remember it.

Jon Farrow (INI)

That's incredible. And we've talked about trigonometry and sine tables and such, but is there one concept that people might be surprised to learn was actually discussed in India before anywhere else?

Prof Clemency Montelle

Yeah. So I think a really amazing one, and this sort of is just sort of blew my mind when I first thought of it, and it's because of the certain ubiquitous of it that is the decimal place value system which emerged in India. We have evidence for it in around the middle of the 1st millennium of the Common Era. But at that stage, it seemed quite sophisticated, so it suggests a mature tradition had been going on. Of course, we can only infer from the sources that we have what was happening. Things could have been happening, but just not documented any earlier. So we have to be a bit careful with our dating. And I think Laplace put it beautifully when he said, this is an invention that is just so simple, he said, that this very simplicity is the very reason for our not being sufficiently aware how much admiration it deserves. And that's because we use numbers every day, right? I mean, and the idea that you can just have these 10 little glyphs, these 10 little symbols to represent whatsoever number that you want, with this idea of this place holding zero, which can differentiate the number 11 from 101 for a million and one, is really quite novel. And it took us a long time as humans to get well sorted out about that number system. Earlier systems of numeration were really ingenious, but didn't have the same arithmetic facility, or you would run out of symbols for bigger numbers, or just there were other features of the system that made arithmetic a little bit harder, or just these numerical processes, so really this ingeniousness. of these 10 figures, these 10 symbols to represent a number system, the way they work so well for long multiplication and all other things that we've got is really quite brilliant.

Jon Farrow (INI)

By comparison, like if you didn't have this place value system, What might you have done?

Prof Clemency Montelle

Yeah, so the Greeks, for instance, used the letters of the Greek alphabet. Alpha was 1, beta was 2, gamma was 3. And then when they got to 10, they just chose another letter, which was iota. And when they got to 20, they chose another one, which is kappa. And so you can see what's going to happen here. Once we get to the end of the Greek alphabet, you've run out of letters and you can only get to a certain. So famously, the Greeks could get up to a myriad, which was a capital M for them, which was 10,000, but then they couldn't go any further. That's got to have some limitations with it, right? And in fact, that explains the very famous work that Archimedes wrote called the Sand Reckoner. So Archimedes posed the question, how many grains of sand would it take to fill up the entire universe? And he thought, actually, that's not enough, right till the edge of the, you know, the known celestial sphere and beyond. And the reason that this question was so interesting for him is that he had to devise a system of numeration, a way to express that with the biggest value or the biggest symbol that was 10,000. So he talks about myriads of myriads to the power of myriads and myriads, and it's just, it's really this clunky, gorgeous system. very much an artifact of the Greek system of numeration, but shows what the limitations were around our counting.

Jon Farrow (INI)

These cultures, while sort of happening in different parts of the world and a little bit independent, they also spoke to each other, right? So how did these ideas move through the world?

Prof Clemency Montelle

Yeah, it's fascinating. So, you know, diplomatic trips, when kings and other Important people did trips to other places in the world, of course. They would take scientists and scholars with them to document things. Wonderful things happened through these diplomatic missions, sharing of ideas and the like. Jesuit missionaries went to India with amazing European works and sat down with the Sanskrit pundits, and wonderful conversations ensued about the sorts of particular features of this new mathematics and astronomy they were bringing. The spread of the Islamic Empire, for instance, meant that trigonometry reached to Europe via Spain, up through there, as did these decimal place value number system that we see. That's how it got to Europe. So conquest, diplomatic envoys, and sometimes ways that we don't know. For instance, and that's the lovely thing about studying the history of mathematics as well, is that if you see one number pop up in one culture and you see it pop up somewhere else, and I'm talking about, a precise number, a good one is sort of mean synodic, the length of the mean synodic month. And in Babylonian sources, this was set, or we have evidence that they set this at the sexagesimal number, 29, 31, 50. So it's a really precise number. We see that number pop up in Greek sources as well. So even if we don't understand the ways in which they shared that very directly, although hopefully we'll uncover some new sources that might shed better light on that, we can be sure that those two cultures had some sort of contact with each other and understood and shared this knowledge and this beautiful sort of collaboration. which we, collaboration we really value today and of our ancestors did as well.

Jon Farrow (INI)

On Living Proof, we've spoken to a lot of mathematicians, but this kind of work connecting the history, mathematics, language, anthropology, that's quite different, I think, from our normal fare. So how did you get into this field of study?

Prof Clemency Montelle

Yes, so I, yes, there's a wonderful story, which I've told before, but loosely it really goes on. I studied mathematics and classical studies with a focus on Greek and Latin. So it was very much the C.P. Snow, who's a famed Cambridge scholar, two cultures type idea of the humanities and the scientific disciplines being brought together. And I realized after several years that I could read ancient mathematical texts and I could bring my skills of mathematics to bear on that as well as the fact that I could read them in Greek or Latin and then learnt Sanskrit and the like. And so I really brought to bear this humanistic approach to mathematics.

Jon Farrow (INI)

In the pursuit of this study, what challenges did you encounter?

Prof Clemency Montelle

So when you're combining two fields as broad as humanities-based, you know, literature and language, and you're also looking at mathematics, and we're talking about the whole range of mathematics, you know, we've talked about quantitative stuff, we've talked about trigonometry, we've talked about the applications to spherical astronomy. It means you need to be keep up with all of those disciplines on equal footing. So I might find myself at a conference with classicists and need to speak to them in terms of, really singular focus on the Greek language in a way that they would understand. I also might need to talk to mathematicians and make sure that really... There's an alignment between the level of mathematics that I'm looking at and speaking it in a language that they understand and keeping myself up with, all the corners of mathematics as I need to successfully read these texts. So I think those are the challenges with interdisciplinary stuff. It's being an expert in everything and needing to bring that to bear in ways which you just can't possibly anticipate at the beginning.

Jon Farrow (INI)

To actually kind of do this work, walk us through what it's like. So you sort of after weeks and months of correspondence, you know, you've organized a visit, you show up to the library, they open the door, and what happens next?

Prof Clemency Montelle

You have a bit of a hunch or you have a moment of clarity or a serendipitous moment and you think, I'm just gonna study this or that, or I've fallen upon this author, or I've a hunch that there's something in this manuscript that might be interesting, or this particular set of sources. And then you start that process of tracking down the sources, particularly for those that haven't got modern-day editions with them, and that's frequently the case, and that's often the more exciting one. It's a real contribution to scholarship when you can look at something that nobody else has looked at before. So we'll start with looking at these catalogues of manuscripts to find out where they are. And they might be in the Bodleian, or they might be in a manuscript library or India, or even my university in New Zealand has a few Sanskrit manuscripts as well. And then getting copies of them, and more and more, of course, libraries are jumping on board with digitizing these things. Sometimes I've had to travel to these libraries to request copies, sometimes it's harder to get them than other times, and the like. Then it's to sit down and so what will happen, particularly if we're studying a text with my colleagues, we will sit down and often we'll have, I don't know, 3 to 10 to 20 copies of the same text. So we talk about building a critical edition there and that means looking at all of these 10 texts which have become different because there's a different scribe has copied them. And of course, as we all know, if you sat down and had to copy something for 8 hours a day, 365 days a year, you're going to make mistakes. And so we look at the divergence between all of these manuscripts to create what we think the original might have looked like. And there's this wonderful mantra that we have, beware the intelligent scribe. So sometimes scribes come along and think they know better and they add all this material and then we have to document that patiently as well. But that's all really important because it helps us get inside the heads of these people who were who were copying and who valued and engaged with the material. So once we've got that edition, then we start reading it in whatever language, and as I said, really complex issues around how are they expressing themselves, what language are they using, and I mean the precise words to do so, and all of those other things. And then there's the really complicated work of trying to maintain the authenticity and integrity of the text, but transmitting that to a modern audience who are used to symbolic styles like algebraic forms of framing things and quite a modern notation. So getting that balance, because we obviously want modern people to read and appreciate them and for them not to be put off or for that to be a barrier. and then to locate them in this wider tradition of history of mathematics. So it's really exciting. So right from that very hunch of the text we should read up until how do we convey this sense of what this mathematics meant to these people and how it was used to a modern audience, it's a real, real gift.

Jon Farrow (INI)

I guess if you're trying to sort of translate it into language that a modern audience might understand, You can't ask the author, you can't check. How do you arrive at that sort of...

Prof Clemency Montelle

But it's really hard and that's, but that's one of the wonderful things why this is not, the sort of the untranslatability and there's various... philosophies of translation, which isn't just restricted to mathematics, of course, this is something lots of thinkers have grappled with over the years. How do we translate something, and you know, it's not just poetry, or it's not just philosophy, or how would we translate, you know, Dostoevsky into English and really get a sense of what it meant to read that if you were reading the Russian. So you can either go, but mathematics poses a particular challenge, in that sometimes there's a universal principle, like it might be there, I don't know, you can recognize that an ancient Near Eastern tablet is solving what we would call solving a quadratic equation. And so the real temptation is to write down x squared plus whatever in modern notation, say, wow, they're doing algebra and really they've got the quadratic formula, negative b plus or minus the square root, you know, all of that. And, but that of course does a fair bit of Or at least that's problematic for our text, because it doesn't really, what it has is it's flattened the significance. And usually it's the case that they might be solving a quadratic equation geometrically, or they don't have symbolic styles of algebra, which was much later to come. And so when you flatten it by using modern symbolism to translate it, you're losing a whole lot of that insight. So there are other scholars who the philosophy is try and preserve that special cultural feel of what it was and try and make it feel like you're definitely reading mathematics, which is foreign and definitely not modern mathematics. That can often be quite impenetrable because it's quite hard to read. So it's where you find somewhere in the middle of where you feel comfortable with as a translator. But yeah, mathematical translations really pose all these sorts of challenges.

Jon Farrow (INI)

Well, thank you for that training and coming up in this world from, you know, sort of Greek and Latin and mathematics. And did you have any inspirational mentors that kind of helped you along the way?

Prof Clemency Montelle

Oh, gosh, I've got so many more to mention. But perhaps, I guess, sort of pivotal to me was traveling to Brown University at the then History of Mathematics department, the only named one at the time in the United States. And the exceptional privilege I had to work with the late Professor Pingree, for whom I read these mathematical texts in different languages every morning and every afternoon. So it was enough to sort of do your head in on some days. Also, Professor Kim Plofka, who is now delighted to call a colleague, who gave me my first lesson in Sanskrit mathematics, and it involved talking about prosody and learning how to chant. And I was like, where's the maths? But perhaps what I'll do is dwell upon another member at Brown. Brown had 1 1/2 faculty members at the time. The half was Professor Alice Slotsky and a seriologist who would drive up from Yale for 2 1/2 days a week. And the reason I choose her is she was, she came to do her PhD later in life, so that was already pretty inspiring. but she really tried to convey the living tradition of our historical people. So one event that she had, for instance, was she'd throw a party at the end of the year where she would cook dishes from cuneiform tablets, from recipes. The kicker was these recipes had no quantities. And some of the things they would say, oh, it's got kaskuda in it. And like a syriologist had no idea what that was. Was it mustard? Was it some sort of basil? You know. And so She was like, this is a living tradition. This is really important. We're going to experiment and bring this, translate this culinary experience of what it would have been for our ancient peoples to have this feast and bring it into modern audiences. And I think that's an analogy or a metaphor at least for what we are doing historically. We actually want to translate this experience. of these ancient peoples to our modern audiences because it reminds us of what's distinctly human about our endeavor.

Jon Farrow (INI)

Yeah, no, that's really nice. Thinking about the endeavor and the sort of the context of mathematics, I feel like mathematicians often think of themselves as kind of immune from their context, right? There's this idea that, you know, the mass is mass and it's objective and it's quite separate from the culture. Is that true?

Prof Clemency Montelle

Oh goodness, do you suppose I'm going to agree with you? I really think, and maybe we've got Plato to blame for that one. So Plato's philosophy, which has been surprisingly durable, of course, really posited a mathematics which was abstract, free from the messiness of human culture and really outside of human experience. And this has persisted, I think, up until the modern day. And this aspect of mathematics is really quite beautiful. and compelling to many of us, but mathematics is so much more than that. As I hope I've documented in this lovely conversation we've been having, context is huge for mathematics. It's driving people to figure out what questions they want to ask and what problems they want to solve. The whole broader context of how we express ourselves be it in the form of metrical verse, or in a syllabic language, or in a logographic language, or our number system play a part in what insights we're going to have. Our symbolic systems of how we represent things. There's a lot of different branches of mathematics that all have their own symbolic systems, and depending on what you use, you see different insights into how you represent it. So I think mathematics is absolutely not context or culture free, and that's one of the beautiful things about it. And history has really shown that there's been such a diversity of activity, that why wouldn't we expect this diversity to continue in ways which are really beautiful? Because mathematics is addressing problems, we haven't even figured out what they are yet, and who knows what mathematics we will need to develop to look at some of the problems. and opportunities that we have in the future. So now I'm a very strong advocate of the cultural, the cultural contingency of mathematics and shaping the very thing. It's a human activity after all.

Jon Farrow (INI)

Is there an example from the Sanskrit mathematics, for example, that because of the way the culture was that they, you know, were asking certain questions that weren't being asked elsewhere?

Prof Clemency Montelle

Yeah. I think an interesting one, maybe not certain questions, but certain ways in which mathematics presented itself. We talked about the oral environment. Well, when you have to memorize stuff, long and complicated proofs or symbol systems, right, because you're actually orally reciting something, you don't need to symbolically represent. So there's no room for sort of an X or a necessarily, at least in early times. And so the concept of proof was quite different in India. That doesn't mean they didn't deeply validate their results, but they had much more of a broad range of ways in which they might convince themselves of the validity of a result. And this is a very broad and quite rich system as well. It included things like authority, direct perception, inference, analogy, and other things. So there's a lot of logical tools at their disposal which are quite distinct, and part of that was because of the medium in which mathematics had to be encoded.

Jon Farrow (INI)

If there was one thing that you wish more mathematicians knew about the history of mathematics, what would that be?

Prof Clemency Montelle

I just think it's diversity, really. Yeah, we go back to the men of mathematics, you know, the familiar names that we come across that are household names for mathematicians, the Gausses, the Newtons, the Leibnizs. the Koshy's and et cetera, that these are really important and wonderful achievements and many of them have been so durable that we use them today, which is why they know the names. But mathematics is so much more a human and broad endeavor and can be found in all pockets. And so to understand that diversity and that color is not only really... a wonderful boost and the source of pride for the discipline, but indeed it's useful for the ways in which we perhaps attract more people to the discipline or break down some of the barriers for the perceptions of mathematics about being a little bit cold or dry or inaccessible. It seems to me this is just this wonderful untapped wealth of colour that we could do better promoting.

Jon Farrow (INI)

I think that's a great place to leave it, actually. That was fantastic. Thank you.

Prof Clemency Montelle

Thank you.

Jon Farrow (INI)

If you're the kind of person who hears a puzzle and can't resist trying to solve it, you're going to love this. G-Research, the quantitative research and technology firm, is launching a brand-new monthly series of mathematical riddles designed for people who enjoy thinking deeply. Every month, the riddles get a little bit tougher. And if you crack them, you'll be in with a chance to win real cash prizes. It's free to enter, fun to try, and perfect for anybody who likes stretching their brain. So if you're ready for a challenge, head to gresearch.com/griddles That's gresearch.com slash griddles, and go give it a try. Thank you, Clemency, for a fantastic conversation. I learned so much about Sanskrit and the beauty of ancient mathematics, and I'm sure our listeners did too. If you enjoyed the podcast, please make sure you subscribe to get notifications when the next episode is out. Until then, see you next time on Living Proof, the podcast of the Isaac Newton Institute for Mathematical Sciences.