What is a “valid mathematical proof”? To inquire into such a hotly debated question we might want to look at how past mathematicians tackled this question.
This lecture will provide examples outside of what has been called a “colonial library”, using in particular Sanskrit sources, to argue that mathematical texts from all over the world contained not only proofs but also many other types of mathematical reasoning whose stories still need to be documented.
A lecture by Professor Agathe Keller
The transcript and downloadable versions of the lecture are available from the Gresham College website: https://www.gresham.ac.uk/watch-now/maths-proof
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- We're going to talk about the serious problems that exist with the way the history of proofs in mathematics has been written, and reflect on the all the stories that can and still need to be written by diving into ancient Sanskrit mathematical sources to look at interesting and surprising ways in which different authors reasoned on and with mathematical objects. Indeed, standard histories of proofs in mathematics often say nothing of mathematical texts outside of Europe, as if they didn't exist, and if by chance they do evoke in other languages than European ones, they evoke text in other languages than European ones, we can encounter, for instance, the kind of statements given by Morris Kline, a teacher and mathematics, popularizer in his classic "Mathematical Thoughts From Ancient to Modern Times", which was published in 1972. So he writes, "Perhaps most interesting is the Hindu's and Arab's self contradictory concept of mathematics. Both worked freely in arithmetic and algebra, and yet did not concern themselves at all with the notion of proof. That the Egyptians and Babylonians were content to accept their few arithmetic and geometric rules on an empirical basis is not surprising. This is a natural basis for almost all human knowledge, but the Hindus and Arabs were aware of the totally new concept of mathematical proof promulgated by the Greeks. Both civilizations were, on the whole, uncritical, despite the Arabic commentaries on Euclid, hence they may have been content to take mathematics as they found it." So such stands are all the more surprising that, as I will show in what follows, all sorts of mathematical texts in Arabic, Sanskrit, but also in Chinese, containing proofs and sometimes quite elaborate reasonings have been known and studied in Europe from the beginning of the 19th century. In other words, but you probably already know this, the history of mathematical proofs has been and continues to be a very political history. So in what follows, my aim will be to give a display of the diversity of mathematical reasonings that can be found in Sanskrit mathematical texts, to show that as we decolonize the library from which we choose the texts with which we study mathematical reasonings, we might also question the focus we have on the notion of proof, that is on the will to find rigorous, irrevocable reasonings dealing with the truth of a mathematical assessment. I think that we are all aware that ideas of what makes a valid reasoning in general and in mathematics in particular have undergone radical changes in the late 20th and beginning of the 21st century, and as algorithmics and statistical approaches to proofs have gained momentum, and humans in the past too, actually, had many different ways of reasoning in mathematics, and many different ways of thinking of what made a valid reasoning, and this diversity, I think, I want to argue, is worth studying too. So exactly 10 years ago, Karine Chemla edited a book that made quite a sensation in the discipline of the history of ancient mathematics, and it's called "The History of Mathematical Proof in Ancient Traditions". The studies in this book continue to inspire historians of ancient mathematics today, notably the study of mathematical diagrams and the way they have been edited. So in the introduction, Chemla first establishes what from the end of the 19th century would've been the stabilized standard history of proof, which the previous speakers actually have already talked about, but, so she says that it's in the end of the 19th century that people have decided, or that's where most historians and mathematicians actually also would write that we owed the idea of mathematical proof to mathematicians of Greek antiquity, especially to Euclid's elements, which itself would have been inspired by Aristotle's Posterior Analytics, and that central to the idea of proof would've been the idea of the certitude of the veracity of a statement, and so that the correlate would have been the importance of rigor, and in this idea was that before classical Greece, there would've been no evidence of proof, and furthermore, this unique idea of proof would later have been developed only in Europe, and in other words, this idea of proof is part of what would have made classical Greece and modern Europe exceptional. So we can amend the standard idea of the history of mathematical proof in many ways. We know from the previous speakers that such a history and such a definition should be infinitely more complex, and Chemla remarks that mathematicians of today read proofs for many reasons, not just to be convinced by the truth or the correctness of a statement, and we could also think that this might have also been the case in the past, and she also shows that such a model of proof was used for all sorts of things that have nothing to do with mathematics, like for instance, to be a model of rigorous reasoning for philosophers or thought of as a tool by Jesuits for conversions to Christianism, and so, as I think it's already obvious, such a definition has been then used as a landmark by which reasonings in past mathematical texts were judged. It serves until today as a criteria by which certain corpus of sources are accepted as containing proofs, and others not, but truly, when we read mathematical documents of the past, some mathematical text may adhere more or less to this standard, but many do not, so that this way of valuing text conveniently selects some text and discards others. In other words, Kline's point of view has a long history, and we can go back to the middle of the 19th century, for instance, and to see what it actually echoes into Kline's 1972 statement, sort of echoes into what Jean-Baptiste Biot said a hundred years before him, who was a French mathematician, astronomer and physicist, who wrote in 1841, "This peculiar habit of mind, following which the Arabs, as the Chinese and the Hindus, limited their scientific writings to the statement of a series of rules, which once given are only to be verified by their applications without requiring any logical demonstration or connections between them. This gives those oriental nations a remarkable character of dissimilarity, I would even add of intellectual inferiority, comparatively to the Greeks, with whom any proposition is established by reasoning and generates logically deduced consequences. But actually, what the contributors to Chemla's book show is that during the first three quarters of the 19th century, there were many different ideas of proofs that were discussed by mathematicians and historians of mathematics. It was possible to imagine algebraical proofs, notably, which implied not to dismiss the history of computation or algebra as secondary or of no interest for the history of mathematical proof, and so the adoption of a standard shape and history of proof took hold in the history and philosophy of mathematics at the end of the 19th century, and it led to discarding all sorts of mathematical texts, not only those in Chinese and in Sanskrit, but also those in Greek or in Latin that did not follow these norms. So it's this creation of a limited corpus of ancient texts of mathematical proofs that I call a colonial library, and in part because it was constructed at the height of the European imperial expansion, and it's the whole constitution of such a library that I think we should reconsider. So in what follows, first I would like to underline that part of the problem in the history of mathematical proofs, which is a wider historiographical problem, is that some discourses have attached modes of reasoning to peoples, nations or religions in a homogeneous way, and that this fuels identity politics in the history of mathematics, and second, that we can document the mathematical reasonings we find in texts, studying together the practices of reasonings, the testify of the aims of these reasonings, and the kind of text that were shaped for this aim. This means going beyond the idea of proof, because past mathematicians have written reasonings whose aims might not have been limited to establishing the truth of a statement or the correctness of an algorithm. For example, they could aim at providing generalizations or want to show that a mathematical object or a procedure can be interpreted in different mathematical and non-mathematical contexts, for instance. So I'll first show how the history of proof was important in the writing of the history of mathematics in South Asia, and underline the dangers of nationalist historiographies, even when they focus on actors' categories, and then provide examples of the diversity of mathematical reasonings that can be found in Sanskrit mathematical sources. So at the end of the 18th century, a Scottish mathematician and philosopher, John Playfair, officially called for a search for mathematical texts from South Asia, and one of the interesting things about his call is that he had an equal interest in geometry, arithmetic and astronomy, and this is in contrast to the usual focus on geometry as being, you know, the great tradition of mathematics. Henry Thomas Colbrooke, who is known as the father of Indology, and who knew Playfair and read him, took an early interest in collecting manuscripts of mathematics and astronomy in Sanskrit during his long stay in India, first as a member of the East Indian Company, and then at as the head of the Asiatic Society of Bengal and Calcutta, and after spending more than 30 years in India, he composed the most influential and enduring translation of Sanskrit mathematical text in English of the early 19th century, which is this book that came out in 1817 called "Algebra with Arithmetic and Mensuration". So in particular, he worked on Bhaskara II, which we call him to distinguish him from another Bhaskara we're going to see, who is a younger Bhaskara I, mathematical texts devoted to arithmetic, Lilavati, which is actually the name of the young woman to whom, supposedly, part of the problems are addressed, but Lilavati means also, "With fun," and in his algebra, bijaganita. So his translation starts with a preliminary dissertation in which he declares that, indeed, Sanskrit mathematical text contain proofs. So this is what he writes. "On the subject of demonstrations, it's to be remarked that the Hindu mathematicians proved propositions both algebraically and geometrically, as is particularly noticed by Bhaskara himself toward the close of his algebra, where he gives both modes of proof of a remarkable method for the solution of indeterminate problems, which in involve a factum of two unknown quantities." So he, to claim that ancient Hindu mathematicians had both algebraic and geometrical demonstrations, he's using some of Bhaskara's statements that are in a very localized place of his algebra, in which if I read Colbrooke's translation of the text, at some point Bhaskara writes, "The demonstration follows, it is twofold in every case, one geometrical and the other algebraic," and he writes a little bit later, "The algebraic demonstration must be exhibited to those who do not comprehend the geometric one." The term that Colbrooke translates as demonstration from the Sanskrit is upapatti, and we'll be interested in this term, you can see it here, which will become the standard term translated from the Sanskrit, into English, as proof. So we can come back to what Colbrooke is doing here. He's actually taking some assertions made by Bhaskara in one part of his text, and I have to say something about, just to explain exactly the nature of these assertions, Sanskrit mathematical treatises are written with sutras that are often versified and aphoristic, and so they're written to come with a commentary, and Bhaskara himself wrote a prose commentary on his own treatises, and so these are notations from his prose commentary, and so anyway, there's two notations in the whole of the algebra, and Colbrooke takes them out and he generalizes it, not only to all of Bhaskara's text, but to the whole of Hindu mathematicians. So we can understand what Colbrooke is doing here. He's not trying, actually, to make an identity of all Hindu mathematicians. Colbrooke's aim when he's doing this is that he wants to make comparisons, he wants to be able to compare what authors that were writing in Sanskrit were doing with authors that were writing in Arabic and Latin. So for him, it's in this process that he creates this, but he is creating a homogenous idea of Hindu mathematicians and a homogeneous idea of proof along the way, and Colbrooke's publication was very influential in Europe and in Asia during all the 19th century. Mathematicians in France, like Schal in Germany, read the proofs his translation contained. Discussions on notions of proof in mathematical education were thus developed using Colbrooke's translation, and notably Hermann Hankel, for instance, developed the idea that an ideal education should combine together two types of proofs, the analytical, which was attributed to the Greeks, and the intuitive, attributed to the Indians, and something else happened was that, as you can see here, Colbrooke took proofs he found in all sorts of different commentaries and added them as footnotes to his translation, giving the feeling that proofs only belong to commentaries and could only be... So yeah, there's two things. First of all, the only proofs we see in the text he gives are in commentaries, and second, he's not translating the whole of a commentary but giving parts that he considers interesting, and so visually in the text we can see that they're made of bits and pieces, and by the beginning of the... So this gave the idea, and this is something that it became very standard that people say in Sanskrit. Actually, they don't usually say in Sanskrit mathematical texts, but in Sanskrit mathematical texts, treatises have no proofs. They're just a set of rules, and if there are some, they're in bits and pieces, in common terms. So, but by the beginning of the 20th century, the existence of such proofs seems to have been forgotten and standard histories overlooked them altogether, but Colbrooke's claim has been taken up at the end of the 20th and at the beginning of the 21st century by a certain number of historians of mathematics in South Asia, aiming at having the history of mathematics in Sanskrit enter a global conversation on the history of mathematical proofs, and so both the two authors I'm quoting here, is MD Srinivas and Ramasubramanian, both quote the quotation of Colbrooke I gave you before, to say that in the Sanskrit mathematical tradition, as they say, the kind of proofs that were given were either algebraic or geometrical, and well, MD Srinivas adds also, "Clearly the tradition of exposition of upapattis," so upapattis, remember, it's the word that was used by Bhaskara. "The exposition of upapattis is much older, and Bhaskaracharya," so Bhaskaracharya is one of the ways of calling Bhaskara II. "... and the later mathematicians and astronomers are merely following the traditional practice of providing detailed upapattis in their commentaries to earlier or their own works. So here's what he's saying here is actually that it's an immemorial tradition. It's not only in Bhaskara, it existed before, and it exists after, and then, "The notion of upapatti is significantly different from the notion of proof as understood in the Greek as well as the modern Western traditions of mathematics." So it's really the construction of an immemorial tradition of Sanskrit proofs, and opposing it to a Western, what would have been a Western tradition, as if that was homogeneous as well, and Ramasubramanian also adds, having quoted Colbrooke, "The upapattis of Indian mathematics, unlike the Western tradition, are not formulated with reference to a formal axiomatic deductive system." So he's also then opposing this tradition with the West. "One often finds the statement iyam atra vasana, when the commentator is about to begin to explain or demonstrate something. Meaning wise, this statement, iyam atra vasana is equivalent to atropapattih. Both the forms being equivalent, there is hardly any consideration for choosing one over the other." So what he is saying basically is that whatever the name, there can be different names for the reasonings that are used, they're the same. It's the same tradition of proof. Okay. So behind what looks like a true interest in actors' categories lies what is in today's India truly a very strong claim, that all Sanskrit mathematical texts, written over several centuries in very different contexts, all functioned in the same way, and here then, a mode of reasoning is harnessed to a language. If it is not the geographical space of South Asia, to a religion, Hinduism, or to a caste, Brahmanism. Such claims, then, I think you can see resonate with what is a wider political use of history of science by the current nationalist Hindu government in India. It constructs a non-Western but homogenous mathematical identity for Hindus. So I'd like to argue that Sanskrit mathematical texts had very diverse ways of reasoning that do not always correspond to such a portrait of the mathematical proofs they contain, and actually they're more surprising than that. So I would like to show, notably, that proofs need not be algebraic or geometric in Sanskrit texts, and that they don't necessarily belong to commentaries, and that some reasonings might be larger or different than proofs in as much as they might not be aiming at establishing correctness or truth. So I'm going to just go through a certain number of examples, and the first one I wanted to give to you is an example taken from a treatise that was written in the seventh century by Brahmagupta, and who wrote a treatise called the "Corrected Astronomical Treatise of Brahma". And in this treatise, in the chapter two, Brahmagupta provides a table of signs with 24 values, and in chapter 21, he provides mathematical procedures that derive this table, and others, actually. So actually, the earliest sine tables in Sanskrit sources date from the fifth century, and for Brahmagupta, the sine was both a geometrical, and, I mean, for most of the authors, astronomical and mathematical authors who worked in Sanskrit with the signs, they considered it both as something numerical and geometrical at the same time, and the sine was defined actually in a figure that's called the bow-field, and so it was defined from the idea of the cord of an arc, and you would take half the cord of half the arc, and that would be what is the sine, and this enabled astronomers and mathematicians to construct some of the first trigonometrical circles and with which you could use very easily the Pythagorean theorem to derive geometrically and numerically half cords, which were the sines. Okay. So this is to say, I'm making this point just because I wanted to tell you that in what follows the reasoning of that Brahmagupta is going to follow is both numerical and geometrical, and I don't know, I wouldn't know it how to make a difference in it, but I don't think it's algebraical either. So what happens is that, so Brahmagupta gives 24 values of signs, and when he comes back in chapter 21 to this table, he gives rules that actually, so he gives a first set of rules which provides modes to compute three initial sines which correspond to the sine of 30 degrees, 45 degrees and 60 degrees, once you know the radius of your circle, and then from these rules, from these three sines, then he provides a rule to derive all other sines. Okay? So this is already, in doing this, Brahmagupta is doing something. So I don't know what it is. Is he showing how he derives his table, but he's doing something more, because actually his rules are general, therefore for any radius of a circle, so he's giving rules on how to derive 24 sines, and in particular his own sine table, but he's doing something more, which is that he doesn't give one way to compute the sine of 30 degrees, 45 degrees and 60 degrees, he gives two ways, and it's the same thing, if you consider the rules to compute all the other signs from this, he doesn't give one rule, he gives two rules to do this. So what is he doing? So here we have a, this is typical of the kind of problems we have. He's not telling us what he's doing, and even his commentator is not commenting on that, he comments on something else, and he's giving his own take on all of this, but he's not reflecting on what Brahmagupta is doing. So there's many, there's different ways that we can approach the fact that he's doubling things. Of course you can say, well, he's giving two rules because well, if one is not easy for you, you can use the other. If you're really making practical computations, maybe it's better to have one rule and another, but as we, as you'll agree with me, there's something theoretical in the rules he's giving, because he is establishing his several rules, and another thing that we can wonder is, does he think that one of the rules actually explains the other, that one is the more sort of an explanation of the other, or is it that he believes that by giving two independent rules, he's actually showing something? So this we don't know, but actually we have other, so the fact of doubling rules is something that we find very often in Sanskrit astronomical treatises, but we also find this kind of mode of reasoning using two rules to read one rule and then doubling it up with another rule, in a contemporary of Brahmagupta, but who is a commentator who does write in prose, and who's called Bhaskara. So this is Bhaskara I, it's the Bhaskara of the seventh century, not like the 12th century one we talked about before, and Bhaskara, in a commentary he makes to a fifth century treatises, is it comments on a rule that is given by Aryabhata, and which concerns a gnomon. So we have a gnomon, and a source of light, and we have, we know this, this distance, and this distance, and this distance, and we want to compute the shadow, we want to know the shadow of the gnomon, and so Aryabhata gives a rule that I'm not going to read out, but it basically, the rule is a multiplication followed by a division, and what Bhaskara is going to say is, well, this rule actually is a rule of three. So the rule of three was the way that Sanskrit authors would talk about equal proportions, or to talk about similar triangles, but what I want to underline here is that, to say this, to show that he's using one procedure, he's going to reinterpret it as another procedure. Bhaskara does something specific. He turns it into a question. So this computation is a rule of three. How, if, from the top of the base, which is greater than the gnomon, the size of the space between the gnomon and the base, which is a shadow, is obtained, then what is obtained with the gnomon? The shadow is obtained. So what I want to just, the only thing I want to underline is that there's a way of doing this which is by asking a question, but what's interesting for me here is also that he's rereading the rule with another rule, and he could do this also, I mean, in another part of his commentary, he's going to also talk about the Pythagorean theorem, and to do that, there's another way. So you can take any rule where there's squares that are added or subtracted, and that give another square, and to show that the rule is about the Pythagorean theorem, you're going to show that it's dealing with a right triangle, and this is done by actually renaming segments. So there's the segments of a right triangle in Sanskrit are called, so there's the hypotenuse is called karna, and the upright side is called koti, and so when you see a koti, or bhuja, or a karna, appear in the text, then it's a way for the commentator to say, you can apply, you can reread the set of arbitrary rules as being something dealing with the Pythagorean theorem. So here once again, we have a reasoning that has no name, but where what Bhaskara is doing is that he's taking what looks like procedures, because I forgot to tell you that, it's so obvious for me, that the sutras of mathematical text more often than not are just giving procedures of mathematical problems, and general procedures that look like an arbitrary set of steps. When Bhaskara interprets them, they become a set of steps, which step by step have a meaning, have a mathematical meaning, so that the rereading with another rule seems to be a way of giving an interpretation, or explaining the steps one by one. But, okay, actually in Bhaskara's commentary, there are many discussions about many different reasonings. So there's reasonings he does to which he gives no names, and then there's all sorts of references to all sorts of reasonings, sometimes, that he names and does. So among them, so he does talk, he does use the term upapatti, that we saw before, but it's used to be contrasted with tradition, agama, and he uses it to say, yeah, you know, you can't take something just as a tradition, you have to give a proof for it, but he uses also other terms, like he actually gives examples of verifications, which is, if you have a procedure, and if you turn it upside down, and you do each of, if you invert it, this is would be what he calls a verification. Then there are moments in the text where he talks about explaining, and when he does this, he uses, once again, all sorts of vocabulary. So some that could be that of the function of any commentary to explain an other, which might mean to establish, and other, which might mean to show or to teach. So there's this whole variety of words and, well, we still have to understand what it means, but in particular he uses... okay, I'll come back. Actually, there's a slide that I probably changed place, but I'll come back to some of Bhaskara's modes of explanation on diagrams later on. So there's also another commentator I want to talk to you about, whose name is Prthudhaka, and so Prthudhaka is making a commentary on Brahmagupta, that we saw, and he, Prthudhaka, has read Aryabhata and Bhaskara's commentaries and he quotes them. Okay, so this is, I'm telling you all of this because I want to show you. So we saw a mode of reasoning with Brahmagupta, then we saw a contemporary of Brahmagupta, Bhaskara, who has sort of other ways of reasoning, maybe, or something similar, and then we have somebody who's from the ninth century and who knows all of these guys, and he's going to do other sorts of reasonings again. Actually, Prthudhaka writes the name of his commentary, it's called "Commentary with Explanation", and the idea is what, what is, for Prthudhaka, vasana? So actually, I'm still trying to figure out what is a vasana, and I'm currently editing Prthudhaka's commentary, so I'm still trying to figure out what is a vasana for Prthudhaka, but there's something that he does, so I wanted to take an example of his work on arithmetical sequences, and there's, when he gives, for him giving an explanation starts by sort of extending the realm in which the rule can be applied. So Brahmagupta gives a general rule for summing the terms of an arithmetical sequence, and Prthudhaka first suggests to consider this as he considers a stack of bricks. Then he suggests to extend the point of view on the sequences and to consider it as a capital that's being invested once more and more, and then he considers, if you're into capital, then as something that is as a debt. So this, yeah, or some money that you're lending, and so you can use negative and positive numbers, and then he starts giving examples. So these are in the form of solved examples, he starts playing with the examples, just with the numbers, so it becomes a numerical problem, and then he considers, he gets into another explanation, which is considering that, actually, a sequence can be thought of as a sum of areas of rectangles of side one, and with each value that you're stacked, and along the side, he's going to say that, actually, then, the sum of the areas of all these rectangles is equal to the area of a rectangle. So he's saying this by, for him, this moment, which we'd say it's a kind of proof, you know, he would be proving that the area of this rectangle is equal to the sum of this, but for him it seems that the explanation has to do with adding more and more context, different contexts in which the rules make sense, each step of the rules make sense, and actually, he goes on in his commentary, looking at, he gives a rule to sum the terms of a geometric sequence, and he explains it by understanding this rule as being part of a discipline that's combinatorics, but that is used in diversification in Sanskrit. So there's, in Brahmagupta's astronomical treatise, there's a whole chapter on these combinatorics of diversification, and Prthudhaka is going to explain the geometrical sequence with this, and he's going to also use algebra, and in another rule, he's going to interpret these geometrical sequences as being linked to algebra, or being that you can understand them in algebra. So in other words, I'm not sure yet what is an explanation for Prthudhaka. What he, for him is something that's so important that he's giving the title to his commentary about this, but it seems that what he's interested in is looking, if you take a procedure at looking at all the different contexts in which it is, it can be more than applied, in which it has a meaning, and that these contexts need not be mathematical. They can be to, because for Brahmagupta, actually, algebra is not part of mathematics, it's part of astronomy, for instance. Okay, so this was an example. So there was a whole tradition of, so this is where I get back to my diagrams, there was a whole tradition of explanation on diagrams that we find in Sanskrit mathematical commentaries, and the problem with these explanations is that a lot of them were oral, or so very typically, Bhaskara, the one who comments on Aryabhata, will say, this is when he says, okay, an explanation should be given on a diagram, and then the diagram is drawn, or sometimes it's not even drawn, and you're supposed to see from the diagram what is doing, what are the rules that are given, and actually I think that the diagrams, we see, the diagrams themselves, we see them circulating from text to text and commentary to commentary, even though, as you see from commentary to commentary, the way of explaining the values given to how you explain change all the time, but it seems that these diagrams were also a way of a repository of giving proofs and committing them to memory. I see them as small libraries of proofs as they circulate. So this is the famous trigonometrical circle, and this one is actually a diagram that has to do with cyclic quadrilaterals, which was one topic on which Brahmagupta reasoned a lot, and this is that interested so much German mathematicians at the end of the 19th century, and Prthudhaka gave a diagram of one of these cyclic quadrilaterals, and even though it might not look like one, it looks like a triangle here, but, and that that is, that has been, that was circulated very much. Okay. So I want to end maybe of all my examples of all these different types of modes of reasoning that for our authors were seen as validating what they were doing, to talk about another commentator, Sankara Variyar, who was from Kerala, from the Kerala school. So in the Kerala, from the 15th century onwards, there were a certain number of mathematicians based in the southWest of India, really developed some incredible mathematical procedures, notably to approximate pi, and the sines, with what looks like something like infinite series, and so they're held sometimes as being the prehistorians of calculus. Some people even think that Jesuits could have stolen some of the text and transmitted it to Newton, secretly or something, and so, indeed actually, in the 15th, what happens is that a lot of, so a lot of these rules are attributed to Madhava, from the 15th century, but we only have them through quotations of commentators and treatise writers, who are going to write on them. So in one of his commentary to the Lilavati of Bhaskara, Sankara Variyar comes up with, I dunno if I have other... yeah, comes up with a proof for this approximation of the circumference of a circle, which is another way of giving an approximation of pi, and in one, so Sankara Variyar, actually, doesn't talk about upapatti or vasana. He really is talking about establishing the truth. He's using sadh, and he's using expressions about this, and something else is going on is that when he uses the rule of three, when he wants to show that things are proportionate, well sometimes he uses the question mode, but sometimes he uses, you know, reasonings about the nature of the things that are compared that really, mm, makes us think of the kind of reasonings that were done in Euclidean geometry. So all thought this needs to be more elaborated. It seems to me that we have here somebody who is writing in Sanskrit, but seems to be doing something that is a hybrid, probably of a Sanskrit tradition, but of a Euclidean standard proof of some sort, and so, yeah, these modes, modes of reasoning, traveled also and hybrided themselves. Okay, and so to come to my conclusion, there is something a bit arresting in the persistence of this ignorance on traditions of proof outside of the Euclidean norm, which has to do maybe in part with what the philosopher Achille Mbembe calls late European-centrism, that is of the persistence of Eurocentric and colonial structures of thought in a world where everybody knows that Europe and the United States has never been the unique place in which worthy intellectual productions were composed. So I have entitled this presentation, "Let's Decolonize the History of Mathematical Proofs" because I believe that decolonizing the history is possible only through a collective critical effort, that as professional historians of mathematics, but also as teachers, as an educated public, such as the one in Gresham, we should not let's spread a discourse which closes the doors of what mathematics is or not. So mathematical reasonings were, and they still are, very diverse. They are not necessarily about truth. They were certainly not homogenous across a geographical area, in the same language, or by people who had the same religion. So let us remember this when we hear people speaking of Western science, British mathematics, or Islamic proofs. The good news is that we have resources to write other new histories that are more stimulating, histories that testify to how just like today, people practicing mathematics had many different ways of reasoning, and that there were whole ecologies of argumentation that still need to be investigated, and I don't think that these modes of reasoning necessarily will give anything to mathematicians today. I don't think they're going to bring new mathematical ways of reasoning, but I think that they are testimony of a part of our common humanity in its culture, like its cultures of reasoning, in the same way that we're interested in beautiful pieces of art, or beautiful pieces of literature, wherever they come from and whatever their time, so there's beautiful pieces of reasoning and they're interesting, just like that, and for in themselves. Thank you. (audience applauding) - [Audience] So you gave an example through sine values about using two different methods, two different ways to find the same answer, and instead of just using one way to find one answer, he went and interpreted it in a different way. Do you think that finding two ways to find the same answer is important, and why? - So actually, as a historian, I'm not going to make any statements about if it's important or if it's good, or if it's the truth. I'm interested in describing how it is, but I find it interesting, and actually, one of my points that maybe wasn't clear because I'm still trying to find for myself how to interpret the fact that many authors in Sanskrit wrote two different rules, what could appear as two different rules for a different... but why did they do it? Why, what was behind it? And actually, I think there were different interpretations. Sometimes they really thought that there were some fundamental algorithms, like the rule of three, like the Pythagorean theorem, that was thought of as a procedure, and that these were like the basis from which all the other algorithms could be understood and get a mathematical meaning, but sometimes, maybe it was really thought that if you arrive at the same place, from two different ways, then you're gathering a sort of strength and proving something that's correct, but we need to know what, the idea is to understand what the actors themselves, the authors themselves thought, so we're lucky when they really tell us, but if not, we have to use all sorts of clues to try to guess. - [Audience] I'm not really sure if this makes sense, but would you say that the rules and findings made by Hindu, Chinese or Arab mathematicians, which may be different to findings by Western mathematicians but still accurate, could be more relevant today or even taught at schools than the findings made by Western mathematicians, if reoccurring Western dominance wasn't to exist? - First of all, I want to just say that it's, I don't understand, I honestly don't understand why we just don't integrate, you know, different mathematical texts in the histories we write. So it could be in the kind of histories, that the preliminary stories that we give in math textbooks, or just in the general public. So there's something which is what I call, what actually Achille Mbembe calls the persistence of ignorance. So it's really this idea, and which I think has to come with, in a way, to be more direct. It's a very, it's the legacy we have of racist and white supremacist modes of thought that come from us from the 19th century, or the late 19th century, and that has, that is our legacy, and what I want to say is, okay, then after that, mathematics is part of our humanity. As soon as we have writing, we have mathematics, and as soon as we have mathematical writings, we have things like the Pythagorean theorem. So it's called the Pythagorean theorem, but actually it's something that's common to us as human beings, I'd say. And then, but of course there's different people who did different things, at different times, but one thing I want to say is not, like, even if you were, were a Hindu... So first of all you could write in Sanskrit and not be a Hindu, but even if you were a Hindu writing in Sanskrit in the seventh century, you could be doing some two different, even if you were neighbors, you could be doing two different things, two different things. So it doesn't make sense to say that, you know, to make a huge bag and say, "Oh, they had the same religion, they had the same language," And of course, yes, they did incredible things. They did incredible things sometimes before Europeans did something that was similar, but that's not, my point is we're not in a race of trying to show who was the first. I think it's also interesting in itself and in understanding how a seventh century Sanskrit commentator thought about mathematics, and what he thought a reasoning that was interesting was, and maybe it wasn't about truth, maybe it was about something else. - [Audience] So I was wondering, if in these various texts, of mathematics in India, if there is a different conceptualization between the different parts of mathematics, between algebra and geometry, arithmetic, is there something, a different idea of the relation between them, as opposed to the Greek legacy of mathematics? - Okay, thank you for this nice question. Actually, so there's, of course, what each and every author thought about mathematics is still something we need to... we're still editing these texts and trying to understand them, but certainly authors, first of all, distinguished in between different topics of mathematics. So they did have a distinguished, they did talk about arithmetics, geometry and algebra, and, well, Bhaskara II, Bhaskaracharya, is very famous, because he's the one, he actually saw a continuity in between arithmetics and algebra, so for him, there's something about, I don't think it's the quantities, it's the operands, it's the kind of things that you can operate on, so there would be something like a common structure, and so Colbrooke, who knew a little bit the algebraists in England at the time, forced a little bit of Bhaskara's, in his translation of Bhaskara's algebra and arithmetic, the sort of the structural link that would make that algebra would be sort of like a fundamental structure that could explain also arithmetic, but there are elements of this in Bhaskara's text, and okay, so, and if you take the... so that's Bhaskara, the canonical Bhaskara of the 12th century. The first century Bhaskara has, is different in his conception of mathematics in the sense where he gives several definitions of how you can think of mathematics and he doesn't decide which one he wants. One of one of his definitions of mathematics is to say actually, mathematics is the science of algorithms, of computing, and so they could be what then you can distinguish the different types of objects that go in it, so it can be arithmetical objects or geometrical objects, but what counts is the algorithm, but then he says, then you can always give a geometrical interpretation to an arithmetical problem, and vice versa, but he never goes back to this, so it's just like a statement that's there, and in his commentary, algebra is not so important. - [Audience] One example of where East meets West in mathematics, I think, is Ramanujan, the Indian mathematician who was working with Hardy in about a hundred years ago, and my understanding of the situation there is that Hardy's initial reaction was that Ramanujan had no idea of proof whatsoever, but nonetheless, he respected his great mathematical intuition, and it's subsequently been found, as I understand it, that many of Ramanujan's statements were actually true, some were false, and some, I think, are still unproven. Do you have any comments on that? Was Ramanujan influenced by the Indian traditions of mathematics? - Well, of course, I don't know. I haven't worked on Ramanujan's work, and I wouldn't dare to say so, but it would be, wouldn't it be just natural that he had some... I don't, actually, I don't even know his biography exactly. He probably, at the end of the 19th century, probably was trained in a kind of curriculum that was available for Indian students, though that would've been in English mathematical manuals, but he could have also had access to other South Indian traditions where he was, but I just wanted underline that East meets West is all along the history of the South Asia, and so, like for instance in Kerala, it seems to me that when we see these proofs coming up with these rules of proportion that are completely different, it gives the feeling that there was some kind of trans, there was some kind of knowledge of other types of mathematical and astronomical traditions that were there, and they probably came from, well, it was during the Mogul Empire, so there was a lot of Persian texts going on, but even before that, you know, in the fifth century, actually some of the first astrological and astronomical texts that we have in Sanskrit bear Hellenistic influence. So on the one hand, they think that in India, all the time, there was always all sorts of texts, modes of doing mathematics and thinking about it that came in and were absorbed into all sorts of different ways of doing, and reciprocally, as you know, because, like the idea of trigonometry or part of the arithmetic that was developed in South Asia also traveled. So I think we, this is my point, is, does East and West really have a meaning? Isn't it something that we constructed, and that if we hold onto it, we're creating a divided world, by holding down to these categories. - Okay, thanks. So I'm afraid we do have to stop promptly, but I just want to flag up, if you have enjoyed today, the next sort of mathematical lecture at Gresham will be by the Gresham professor of geometry on the 22nd November, which, so you can look on the Gresham website for that. The next BSHN meeting will be our, I think our Christmas meeting, on the 3rd of December. Check out our website for that. I just want to thank everybody who's been involved in organizing this event today, and finally of course, let's have a gigantic round of applause for our speakers. Thank you. (audience applauding)