{"version":"1.0.0","segments":[{"startTime":0.09,"endTime":2.43,"body":"- We're going to talk"},{"startTime":0.09,"endTime":2.43,"body":"about the serious problems"},{"startTime":2.43,"endTime":6.42,"body":"that exist with the way the"},{"startTime":2.43,"endTime":6.42,"body":"history of proofs in mathematics"},{"startTime":6.42,"endTime":9.973,"body":"has been written, and reflect"},{"startTime":6.42,"endTime":9.973,"body":"on the all the stories"},{"startTime":9.973,"endTime":12.99,"body":"that can and still need to be written"},{"startTime":12.99,"endTime":16.95,"body":"by diving into ancient"},{"startTime":12.99,"endTime":16.95,"body":"Sanskrit mathematical sources"},{"startTime":16.95,"endTime":20.55,"body":"to look at interesting and surprising ways"},{"startTime":20.55,"endTime":21.87,"body":"in which different authors"},{"startTime":21.87,"endTime":26.22,"body":"reasoned on and with mathematical objects."},{"startTime":26.22,"endTime":29.754,"body":"Indeed, standard histories"},{"startTime":26.22,"endTime":29.754,"body":"of proofs in mathematics"},{"startTime":29.754,"endTime":33.84,"body":"often say nothing of mathematical texts"},{"startTime":33.84,"endTime":37.133,"body":"outside of Europe, as"},{"startTime":33.84,"endTime":37.133,"body":"if they didn't exist,"},{"startTime":37.133,"endTime":40.2,"body":"and if by chance they do evoke"},{"startTime":40.2,"endTime":42.671,"body":"in other languages than European ones,"},{"startTime":42.671,"endTime":46.71,"body":"they evoke text in other"},{"startTime":42.671,"endTime":46.71,"body":"languages than European ones,"},{"startTime":46.71,"endTime":48.72,"body":"we can encounter, for instance,"},{"startTime":48.72,"endTime":52.41,"body":"the kind of statements"},{"startTime":48.72,"endTime":52.41,"body":"given by Morris Kline,"},{"startTime":52.41,"endTime":56.977,"body":"a teacher and mathematics,"},{"startTime":52.41,"endTime":56.977,"body":"popularizer in his classic"},{"startTime":56.977,"endTime":60.51,"body":"\"Mathematical Thoughts From"},{"startTime":56.977,"endTime":60.51,"body":"Ancient to Modern Times\","},{"startTime":60.51,"endTime":62.55,"body":"which was published in 1972."},{"startTime":62.55,"endTime":64.267,"body":"So he writes,"},{"startTime":64.267,"endTime":68.25,"body":"\"Perhaps most interesting"},{"startTime":64.267,"endTime":68.25,"body":"is the Hindu's and Arab's"},{"startTime":68.25,"endTime":71.133,"body":"self contradictory concept of mathematics."},{"startTime":72.36,"endTime":74.94,"body":"Both worked freely in"},{"startTime":72.36,"endTime":74.94,"body":"arithmetic and algebra,"},{"startTime":74.94,"endTime":77.13,"body":"and yet did not concern themselves at all"},{"startTime":77.13,"endTime":79.08,"body":"with the notion of proof."},{"startTime":79.08,"endTime":81.84,"body":"That the Egyptians and"},{"startTime":79.08,"endTime":81.84,"body":"Babylonians were content"},{"startTime":81.84,"endTime":85.29,"body":"to accept their few"},{"startTime":81.84,"endTime":85.29,"body":"arithmetic and geometric rules"},{"startTime":85.29,"endTime":88.44,"body":"on an empirical basis is not surprising."},{"startTime":88.44,"endTime":92.28,"body":"This is a natural basis for"},{"startTime":88.44,"endTime":92.28,"body":"almost all human knowledge,"},{"startTime":92.28,"endTime":94.26,"body":"but the Hindus and Arabs were aware"},{"startTime":94.26,"endTime":97.05,"body":"of the totally new concept"},{"startTime":94.26,"endTime":97.05,"body":"of mathematical proof"},{"startTime":97.05,"endTime":98.793,"body":"promulgated by the Greeks."},{"startTime":102.39,"endTime":105.81,"body":"Both civilizations were,"},{"startTime":102.39,"endTime":105.81,"body":"on the whole, uncritical,"},{"startTime":105.81,"endTime":108.81,"body":"despite the Arabic commentaries on Euclid,"},{"startTime":108.81,"endTime":111.48,"body":"hence they may have been"},{"startTime":108.81,"endTime":111.48,"body":"content to take mathematics"},{"startTime":111.48,"endTime":112.497,"body":"as they found it.\""},{"startTime":113.67,"endTime":117.1,"body":"So such stands are all the more surprising"},{"startTime":117.1,"endTime":120.39,"body":"that, as I will show in what follows,"},{"startTime":120.39,"endTime":124.26,"body":"all sorts of mathematical"},{"startTime":120.39,"endTime":124.26,"body":"texts in Arabic, Sanskrit,"},{"startTime":124.26,"endTime":126.27,"body":"but also in Chinese,"},{"startTime":126.27,"endTime":130.29,"body":"containing proofs and sometimes"},{"startTime":126.27,"endTime":130.29,"body":"quite elaborate reasonings"},{"startTime":130.29,"endTime":132.6,"body":"have been known and studied in Europe"},{"startTime":132.6,"endTime":135.27,"body":"from the beginning of the 19th century."},{"startTime":135.27,"endTime":138.33,"body":"In other words, but you"},{"startTime":135.27,"endTime":138.33,"body":"probably already know this,"},{"startTime":138.33,"endTime":140.64,"body":"the history of mathematical"},{"startTime":138.33,"endTime":140.64,"body":"proofs has been"},{"startTime":140.64,"endTime":144.153,"body":"and continues to be a"},{"startTime":140.64,"endTime":144.153,"body":"very political history."},{"startTime":145.14,"endTime":149.16,"body":"So in what follows, my aim will be to give"},{"startTime":149.16,"endTime":152.73,"body":"a display of the diversity"},{"startTime":149.16,"endTime":152.73,"body":"of mathematical reasonings"},{"startTime":152.73,"endTime":155.803,"body":"that can be found in"},{"startTime":152.73,"endTime":155.803,"body":"Sanskrit mathematical texts,"},{"startTime":155.803,"endTime":159.6,"body":"to show that as we decolonize the library"},{"startTime":159.6,"endTime":161.82,"body":"from which we choose the texts"},{"startTime":161.82,"endTime":164.76,"body":"with which we study"},{"startTime":161.82,"endTime":164.76,"body":"mathematical reasonings,"},{"startTime":164.76,"endTime":167.13,"body":"we might also question the focus we have"},{"startTime":167.13,"endTime":169.38,"body":"on the notion of proof, that is"},{"startTime":169.38,"endTime":173.04,"body":"on the will to find rigorous,"},{"startTime":169.38,"endTime":173.04,"body":"irrevocable reasonings"},{"startTime":173.04,"endTime":176.133,"body":"dealing with the truth of"},{"startTime":173.04,"endTime":176.133,"body":"a mathematical assessment."},{"startTime":178.02,"endTime":179.49,"body":"I think that we are all aware"},{"startTime":179.49,"endTime":183.21,"body":"that ideas of what makes a"},{"startTime":179.49,"endTime":183.21,"body":"valid reasoning in general"},{"startTime":183.21,"endTime":184.8,"body":"and in mathematics in particular"},{"startTime":184.8,"endTime":186.99,"body":"have undergone radical changes"},{"startTime":186.99,"endTime":190.56,"body":"in the late 20th and"},{"startTime":186.99,"endTime":190.56,"body":"beginning of the 21st century,"},{"startTime":190.56,"endTime":193.74,"body":"and as algorithmics and"},{"startTime":190.56,"endTime":193.74,"body":"statistical approaches to proofs"},{"startTime":193.74,"endTime":195.303,"body":"have gained momentum,"},{"startTime":196.35,"endTime":198.84,"body":"and humans in the past too, actually,"},{"startTime":198.84,"endTime":201.9,"body":"had many different ways of"},{"startTime":198.84,"endTime":201.9,"body":"reasoning in mathematics,"},{"startTime":201.9,"endTime":203.49,"body":"and many different ways of thinking"},{"startTime":203.49,"endTime":205.44,"body":"of what made a valid reasoning,"},{"startTime":205.44,"endTime":208.98,"body":"and this diversity, I"},{"startTime":205.44,"endTime":208.98,"body":"think, I want to argue,"},{"startTime":208.98,"endTime":210.423,"body":"is worth studying too."},{"startTime":211.5,"endTime":216.45,"body":"So exactly 10 years ago,"},{"startTime":211.5,"endTime":216.45,"body":"Karine Chemla edited a book"},{"startTime":216.45,"endTime":219.33,"body":"that made quite a"},{"startTime":216.45,"endTime":219.33,"body":"sensation in the discipline"},{"startTime":219.33,"endTime":221.857,"body":"of the history of ancient"},{"startTime":219.33,"endTime":221.857,"body":"mathematics, and it's called"},{"startTime":221.857,"endTime":225.117,"body":"\"The History of Mathematical"},{"startTime":221.857,"endTime":225.117,"body":"Proof in Ancient Traditions\"."},{"startTime":226.11,"endTime":228.42,"body":"The studies in this"},{"startTime":226.11,"endTime":228.42,"body":"book continue to inspire"},{"startTime":228.42,"endTime":230.694,"body":"historians of ancient mathematics today,"},{"startTime":230.694,"endTime":234.18,"body":"notably the study of mathematical diagrams"},{"startTime":234.18,"endTime":236.373,"body":"and the way they have been edited."},{"startTime":237.99,"endTime":240.815,"body":"So in the introduction,"},{"startTime":237.99,"endTime":240.815,"body":"Chemla first establishes"},{"startTime":240.815,"endTime":243.54,"body":"what from the end of the 19th century"},{"startTime":243.54,"endTime":247.71,"body":"would've been the stabilized"},{"startTime":243.54,"endTime":247.71,"body":"standard history of proof,"},{"startTime":247.71,"endTime":250.17,"body":"which the previous speakers actually"},{"startTime":250.17,"endTime":251.7,"body":"have already talked about,"},{"startTime":251.7,"endTime":254.85,"body":"but, so she says that it's in"},{"startTime":251.7,"endTime":254.85,"body":"the end of the 19th century"},{"startTime":254.85,"endTime":256.98,"body":"that people have decided,"},{"startTime":256.98,"endTime":261.12,"body":"or that's where most"},{"startTime":256.98,"endTime":261.12,"body":"historians and mathematicians"},{"startTime":261.12,"endTime":262.68,"body":"actually also would write"},{"startTime":262.68,"endTime":265.29,"body":"that we owed the idea"},{"startTime":262.68,"endTime":265.29,"body":"of mathematical proof"},{"startTime":265.29,"endTime":267.99,"body":"to mathematicians of Greek antiquity,"},{"startTime":267.99,"endTime":270.09,"body":"especially to Euclid's elements,"},{"startTime":270.09,"endTime":272.1,"body":"which itself would have been inspired"},{"startTime":272.1,"endTime":275.85,"body":"by Aristotle's Posterior Analytics,"},{"startTime":275.85,"endTime":279.72,"body":"and that central to the idea"},{"startTime":275.85,"endTime":279.72,"body":"of proof would've been the idea"},{"startTime":279.72,"endTime":283.14,"body":"of the certitude of the"},{"startTime":279.72,"endTime":283.14,"body":"veracity of a statement,"},{"startTime":283.14,"endTime":286.26,"body":"and so that the correlate would have been"},{"startTime":286.26,"endTime":288.18,"body":"the importance of rigor,"},{"startTime":288.18,"endTime":293.18,"body":"and in this idea was that"},{"startTime":288.18,"endTime":293.18,"body":"before classical Greece,"},{"startTime":294.0,"endTime":296.73,"body":"there would've been no evidence of proof,"},{"startTime":296.73,"endTime":299.73,"body":"and furthermore, this unique idea of proof"},{"startTime":299.73,"endTime":303.146,"body":"would later have been"},{"startTime":299.73,"endTime":303.146,"body":"developed only in Europe,"},{"startTime":303.146,"endTime":304.92,"body":"and in other words,"},{"startTime":304.92,"endTime":307.53,"body":"this idea of proof is part"},{"startTime":304.92,"endTime":307.53,"body":"of what would have made"},{"startTime":307.53,"endTime":310.863,"body":"classical Greece and"},{"startTime":307.53,"endTime":310.863,"body":"modern Europe exceptional."},{"startTime":313.86,"endTime":316.86,"body":"So we can amend the standard idea"},{"startTime":316.86,"endTime":320.85,"body":"of the history of mathematical"},{"startTime":316.86,"endTime":320.85,"body":"proof in many ways."},{"startTime":320.85,"endTime":322.71,"body":"We know from the previous speakers"},{"startTime":322.71,"endTime":325.2,"body":"that such a history and such a definition"},{"startTime":325.2,"endTime":327.57,"body":"should be infinitely more complex,"},{"startTime":327.57,"endTime":330.24,"body":"and Chemla remarks that"},{"startTime":327.57,"endTime":330.24,"body":"mathematicians of today"},{"startTime":330.24,"endTime":331.86,"body":"read proofs for many reasons,"},{"startTime":331.86,"endTime":334.29,"body":"not just to be convinced by the truth"},{"startTime":334.29,"endTime":336.75,"body":"or the correctness of a statement,"},{"startTime":336.75,"endTime":339.42,"body":"and we could also think"},{"startTime":336.75,"endTime":339.42,"body":"that this might have also"},{"startTime":339.42,"endTime":342.063,"body":"been the case in the past,"},{"startTime":345.3,"endTime":348.51,"body":"and she also shows that"},{"startTime":345.3,"endTime":348.51,"body":"such a model of proof"},{"startTime":348.51,"endTime":350.31,"body":"was used for all sorts of things"},{"startTime":350.31,"endTime":353.79,"body":"that have nothing to do with"},{"startTime":350.31,"endTime":353.79,"body":"mathematics, like for instance,"},{"startTime":353.79,"endTime":357.36,"body":"to be a model of rigorous"},{"startTime":353.79,"endTime":357.36,"body":"reasoning for philosophers"},{"startTime":357.36,"endTime":360.03,"body":"or thought of as a tool by Jesuits"},{"startTime":360.03,"endTime":361.953,"body":"for conversions to Christianism,"},{"startTime":364.08,"endTime":367.26,"body":"and so, as I think it's already obvious,"},{"startTime":367.26,"endTime":370.89,"body":"such a definition has been"},{"startTime":367.26,"endTime":370.89,"body":"then used as a landmark"},{"startTime":370.89,"endTime":374.58,"body":"by which reasonings in past"},{"startTime":370.89,"endTime":374.58,"body":"mathematical texts were judged."},{"startTime":374.58,"endTime":377.13,"body":"It serves until today as a criteria"},{"startTime":377.13,"endTime":379.8,"body":"by which certain corpus"},{"startTime":377.13,"endTime":379.8,"body":"of sources are accepted"},{"startTime":379.8,"endTime":383.16,"body":"as containing proofs, and others not,"},{"startTime":383.16,"endTime":386.31,"body":"but truly, when we read"},{"startTime":383.16,"endTime":386.31,"body":"mathematical documents of the past,"},{"startTime":386.31,"endTime":388.92,"body":"some mathematical text may adhere"},{"startTime":388.92,"endTime":392.22,"body":"more or less to this"},{"startTime":388.92,"endTime":392.22,"body":"standard, but many do not,"},{"startTime":392.22,"endTime":395.46,"body":"so that this way of valuing text"},{"startTime":395.46,"endTime":399.093,"body":"conveniently selects some"},{"startTime":395.46,"endTime":399.093,"body":"text and discards others."},{"startTime":400.112,"endTime":405.06,"body":"In other words, Kline's point"},{"startTime":400.112,"endTime":405.06,"body":"of view has a long history,"},{"startTime":405.06,"endTime":408.24,"body":"and we can go back to the"},{"startTime":405.06,"endTime":408.24,"body":"middle of the 19th century,"},{"startTime":408.24,"endTime":411.42,"body":"for instance, and to see"},{"startTime":408.24,"endTime":411.42,"body":"what it actually echoes"},{"startTime":411.42,"endTime":415.71,"body":"into Kline's 1972 statement,"},{"startTime":415.71,"endTime":418.62,"body":"sort of echoes into what"},{"startTime":415.71,"endTime":418.62,"body":"Jean-Baptiste Biot said"},{"startTime":418.62,"endTime":420.54,"body":"a hundred years before him,"},{"startTime":420.54,"endTime":424.77,"body":"who was a French mathematician,"},{"startTime":420.54,"endTime":424.77,"body":"astronomer and physicist,"},{"startTime":424.77,"endTime":427.507,"body":"who wrote in 1841,"},{"startTime":427.507,"endTime":429.87,"body":"\"This peculiar habit of mind,"},{"startTime":429.87,"endTime":433.59,"body":"following which the Arabs, as"},{"startTime":429.87,"endTime":433.59,"body":"the Chinese and the Hindus,"},{"startTime":433.59,"endTime":435.51,"body":"limited their scientific writings"},{"startTime":435.51,"endTime":439.56,"body":"to the statement of a series"},{"startTime":435.51,"endTime":439.56,"body":"of rules, which once given"},{"startTime":439.56,"endTime":442.56,"body":"are only to be verified"},{"startTime":439.56,"endTime":442.56,"body":"by their applications"},{"startTime":442.56,"endTime":445.961,"body":"without requiring any"},{"startTime":442.56,"endTime":445.961,"body":"logical demonstration"},{"startTime":445.961,"endTime":448.44,"body":"or connections between them."},{"startTime":448.44,"endTime":450.42,"body":"This gives those oriental nations"},{"startTime":450.42,"endTime":453.3,"body":"a remarkable character of dissimilarity,"},{"startTime":453.3,"endTime":456.3,"body":"I would even add of"},{"startTime":453.3,"endTime":456.3,"body":"intellectual inferiority,"},{"startTime":456.3,"endTime":458.22,"body":"comparatively to the Greeks,"},{"startTime":458.22,"endTime":461.55,"body":"with whom any proposition"},{"startTime":458.22,"endTime":461.55,"body":"is established by reasoning"},{"startTime":461.55,"endTime":464.733,"body":"and generates logically"},{"startTime":461.55,"endTime":464.733,"body":"deduced consequences."},{"startTime":466.601,"endTime":471.6,"body":"But actually, what the"},{"startTime":466.601,"endTime":471.6,"body":"contributors to Chemla's book show"},{"startTime":471.6,"endTime":475.71,"body":"is that during the first three"},{"startTime":471.6,"endTime":475.71,"body":"quarters of the 19th century,"},{"startTime":475.71,"endTime":477.51,"body":"there were many different ideas of proofs"},{"startTime":477.51,"endTime":479.52,"body":"that were discussed by mathematicians"},{"startTime":479.52,"endTime":481.53,"body":"and historians of mathematics."},{"startTime":481.53,"endTime":485.52,"body":"It was possible to imagine"},{"startTime":481.53,"endTime":485.52,"body":"algebraical proofs, notably,"},{"startTime":485.52,"endTime":487.14,"body":"which implied not to dismiss"},{"startTime":487.14,"endTime":490.11,"body":"the history of computation or algebra"},{"startTime":490.11,"endTime":492.27,"body":"as secondary or of no interest"},{"startTime":492.27,"endTime":495.221,"body":"for the history of mathematical proof,"},{"startTime":495.221,"endTime":499.2,"body":"and so the adoption of a standard"},{"startTime":495.221,"endTime":499.2,"body":"shape and history of proof"},{"startTime":499.2,"endTime":502.17,"body":"took hold in the history and"},{"startTime":499.2,"endTime":502.17,"body":"philosophy of mathematics"},{"startTime":502.17,"endTime":504.24,"body":"at the end of the 19th century,"},{"startTime":504.24,"endTime":508.2,"body":"and it led to discarding all"},{"startTime":504.24,"endTime":508.2,"body":"sorts of mathematical texts,"},{"startTime":508.2,"endTime":511.38,"body":"not only those in Chinese and in Sanskrit,"},{"startTime":511.38,"endTime":513.57,"body":"but also those in Greek or in Latin"},{"startTime":513.57,"endTime":515.403,"body":"that did not follow these norms."},{"startTime":518.31,"endTime":520.86,"body":"So it's this creation of a limited corpus"},{"startTime":520.86,"endTime":522.94,"body":"of ancient texts of mathematical proofs"},{"startTime":523.8,"endTime":526.26,"body":"that I call a colonial library,"},{"startTime":526.26,"endTime":529.422,"body":"and in part because it was constructed"},{"startTime":529.422,"endTime":532.653,"body":"at the height of the"},{"startTime":529.422,"endTime":532.653,"body":"European imperial expansion,"},{"startTime":533.7,"endTime":536.7,"body":"and it's the whole"},{"startTime":533.7,"endTime":536.7,"body":"constitution of such a library"},{"startTime":536.7,"endTime":539.01,"body":"that I think we should reconsider."},{"startTime":539.01,"endTime":544.01,"body":"So in what follows, first"},{"startTime":539.01,"endTime":544.01,"body":"I would like to underline"},{"startTime":546.54,"endTime":547.65,"body":"that part of the problem"},{"startTime":547.65,"endTime":549.78,"body":"in the history of mathematical proofs,"},{"startTime":549.78,"endTime":552.45,"body":"which is a wider"},{"startTime":549.78,"endTime":552.45,"body":"historiographical problem,"},{"startTime":552.45,"endTime":555.69,"body":"is that some discourses have"},{"startTime":552.45,"endTime":555.69,"body":"attached modes of reasoning"},{"startTime":555.69,"endTime":559.77,"body":"to peoples, nations or"},{"startTime":555.69,"endTime":559.77,"body":"religions in a homogeneous way,"},{"startTime":559.77,"endTime":562.47,"body":"and that this fuels identity politics"},{"startTime":562.47,"endTime":564.42,"body":"in the history of mathematics,"},{"startTime":564.42,"endTime":568.2,"body":"and second, that we can document"},{"startTime":564.42,"endTime":568.2,"body":"the mathematical reasonings"},{"startTime":568.2,"endTime":569.28,"body":"we find in texts,"},{"startTime":569.28,"endTime":572.25,"body":"studying together the"},{"startTime":569.28,"endTime":572.25,"body":"practices of reasonings,"},{"startTime":572.25,"endTime":575.97,"body":"the testify of the aims"},{"startTime":572.25,"endTime":575.97,"body":"of these reasonings,"},{"startTime":575.97,"endTime":579.483,"body":"and the kind of text that"},{"startTime":575.97,"endTime":579.483,"body":"were shaped for this aim."},{"startTime":580.86,"endTime":583.08,"body":"This means going beyond the idea of proof,"},{"startTime":583.08,"endTime":585.9,"body":"because past mathematicians"},{"startTime":583.08,"endTime":585.9,"body":"have written reasonings"},{"startTime":585.9,"endTime":587.79,"body":"whose aims might not have been limited"},{"startTime":587.79,"endTime":592.32,"body":"to establishing the truth of a statement"},{"startTime":593.61,"endTime":595.593,"body":"or the correctness of an algorithm."},{"startTime":596.58,"endTime":601.08,"body":"For example, they could aim"},{"startTime":596.58,"endTime":601.08,"body":"at providing generalizations"},{"startTime":601.08,"endTime":604.23,"body":"or want to show that a"},{"startTime":601.08,"endTime":604.23,"body":"mathematical object or a procedure"},{"startTime":604.23,"endTime":606.57,"body":"can be interpreted in"},{"startTime":604.23,"endTime":606.57,"body":"different mathematical"},{"startTime":606.57,"endTime":609.153,"body":"and non-mathematical"},{"startTime":606.57,"endTime":609.153,"body":"contexts, for instance."},{"startTime":610.05,"endTime":613.5,"body":"So I'll first show how the"},{"startTime":610.05,"endTime":613.5,"body":"history of proof was important"},{"startTime":613.5,"endTime":616.79,"body":"in the writing of the history"},{"startTime":613.5,"endTime":616.79,"body":"of mathematics in South Asia,"},{"startTime":616.79,"endTime":621.51,"body":"and underline the dangers of"},{"startTime":616.79,"endTime":621.51,"body":"nationalist historiographies,"},{"startTime":621.51,"endTime":625.08,"body":"even when they focus"},{"startTime":621.51,"endTime":625.08,"body":"on actors' categories,"},{"startTime":625.08,"endTime":627.27,"body":"and then provide examples of the diversity"},{"startTime":627.27,"endTime":629.43,"body":"of mathematical reasonings"},{"startTime":627.27,"endTime":629.43,"body":"that can be found"},{"startTime":629.43,"endTime":632.07,"body":"in Sanskrit mathematical sources."},{"startTime":632.07,"endTime":634.713,"body":"So at the end of the 18th century,"},{"startTime":636.33,"endTime":639.54,"body":"a Scottish mathematician and"},{"startTime":636.33,"endTime":639.54,"body":"philosopher, John Playfair,"},{"startTime":639.54,"endTime":641.73,"body":"officially called for a search"},{"startTime":641.73,"endTime":644.01,"body":"for mathematical texts from South Asia,"},{"startTime":644.01,"endTime":646.2,"body":"and one of the interesting"},{"startTime":644.01,"endTime":646.2,"body":"things about his call"},{"startTime":646.2,"endTime":649.47,"body":"is that he had an equal interest"},{"startTime":649.47,"endTime":653.4,"body":"in geometry, arithmetic and astronomy,"},{"startTime":653.4,"endTime":657.84,"body":"and this is in contrast to"},{"startTime":653.4,"endTime":657.84,"body":"the usual focus on geometry"},{"startTime":657.84,"endTime":661.383,"body":"as being, you know, the great"},{"startTime":657.84,"endTime":661.383,"body":"tradition of mathematics."},{"startTime":662.67,"endTime":664.323,"body":"Henry Thomas Colbrooke,"},{"startTime":665.22,"endTime":668.009,"body":"who is known as the father of Indology,"},{"startTime":668.009,"endTime":671.07,"body":"and who knew Playfair and read him,"},{"startTime":671.07,"endTime":673.83,"body":"took an early interest"},{"startTime":671.07,"endTime":673.83,"body":"in collecting manuscripts"},{"startTime":673.83,"endTime":676.12,"body":"of mathematics and astronomy in Sanskrit"},{"startTime":677.1,"endTime":678.824,"body":"during his long stay in India,"},{"startTime":678.824,"endTime":681.93,"body":"first as a member of"},{"startTime":678.824,"endTime":681.93,"body":"the East Indian Company,"},{"startTime":681.93,"endTime":683.61,"body":"and then at as the head"},{"startTime":683.61,"endTime":686.343,"body":"of the Asiatic Society"},{"startTime":683.61,"endTime":686.343,"body":"of Bengal and Calcutta,"},{"startTime":687.27,"endTime":690.57,"body":"and after spending more"},{"startTime":687.27,"endTime":690.57,"body":"than 30 years in India,"},{"startTime":690.57,"endTime":694.11,"body":"he composed the most influential"},{"startTime":690.57,"endTime":694.11,"body":"and enduring translation"},{"startTime":694.11,"endTime":696.36,"body":"of Sanskrit mathematical text in English"},{"startTime":696.36,"endTime":698.7,"body":"of the early 19th century,"},{"startTime":698.7,"endTime":702.09,"body":"which is this book that came out in 1817"},{"startTime":702.09,"endTime":705.837,"body":"called \"Algebra with"},{"startTime":702.09,"endTime":705.837,"body":"Arithmetic and Mensuration\"."},{"startTime":707.52,"endTime":711.643,"body":"So in particular, he"},{"startTime":707.52,"endTime":711.643,"body":"worked on Bhaskara II,"},{"startTime":713.52,"endTime":715.11,"body":"which we call him to distinguish him"},{"startTime":715.11,"endTime":717.254,"body":"from another Bhaskara we're going to see,"},{"startTime":717.254,"endTime":719.25,"body":"who is a younger Bhaskara I,"},{"startTime":719.25,"endTime":723.03,"body":"mathematical texts devoted"},{"startTime":719.25,"endTime":723.03,"body":"to arithmetic, Lilavati,"},{"startTime":723.03,"endTime":725.67,"body":"which is actually the"},{"startTime":723.03,"endTime":725.67,"body":"name of the young woman"},{"startTime":725.67,"endTime":730.67,"body":"to whom, supposedly, part of"},{"startTime":725.67,"endTime":730.67,"body":"the problems are addressed,"},{"startTime":732.54,"endTime":735.45,"body":"but Lilavati means also, \"With fun,\""},{"startTime":735.45,"endTime":739.71,"body":"and in his algebra, bijaganita."},{"startTime":739.71,"endTime":743.991,"body":"So his translation starts with"},{"startTime":739.71,"endTime":743.991,"body":"a preliminary dissertation"},{"startTime":743.991,"endTime":746.46,"body":"in which he declares that, indeed,"},{"startTime":746.46,"endTime":750.12,"body":"Sanskrit mathematical text contain proofs."},{"startTime":750.12,"endTime":752.403,"body":"So this is what he writes."},{"startTime":754.447,"endTime":756.6,"body":"\"On the subject of demonstrations,"},{"startTime":756.6,"endTime":759.06,"body":"it's to be remarked that"},{"startTime":756.6,"endTime":759.06,"body":"the Hindu mathematicians"},{"startTime":759.06,"endTime":763.35,"body":"proved propositions both"},{"startTime":759.06,"endTime":763.35,"body":"algebraically and geometrically,"},{"startTime":763.35,"endTime":766.35,"body":"as is particularly noticed"},{"startTime":763.35,"endTime":766.35,"body":"by Bhaskara himself"},{"startTime":766.35,"endTime":768.39,"body":"toward the close of his algebra,"},{"startTime":768.39,"endTime":771.9,"body":"where he gives both modes of"},{"startTime":768.39,"endTime":771.9,"body":"proof of a remarkable method"},{"startTime":771.9,"endTime":774.21,"body":"for the solution of"},{"startTime":771.9,"endTime":774.21,"body":"indeterminate problems,"},{"startTime":774.21,"endTime":778.047,"body":"which in involve a factum"},{"startTime":774.21,"endTime":778.047,"body":"of two unknown quantities.\""},{"startTime":779.34,"endTime":783.657,"body":"So he, to claim that"},{"startTime":779.34,"endTime":783.657,"body":"ancient Hindu mathematicians"},{"startTime":790.14,"endTime":793.71,"body":"had both algebraic and"},{"startTime":790.14,"endTime":793.71,"body":"geometrical demonstrations,"},{"startTime":793.71,"endTime":798.09,"body":"he's using some of Bhaskara's statements"},{"startTime":798.09,"endTime":801.3,"body":"that are in a very localized"},{"startTime":798.09,"endTime":801.3,"body":"place of his algebra,"},{"startTime":801.3,"endTime":805.71,"body":"in which if I read Colbrooke's"},{"startTime":801.3,"endTime":805.71,"body":"translation of the text,"},{"startTime":805.71,"endTime":808.657,"body":"at some point Bhaskara writes,"},{"startTime":808.657,"endTime":812.04,"body":"\"The demonstration follows,"},{"startTime":808.657,"endTime":812.04,"body":"it is twofold in every case,"},{"startTime":812.04,"endTime":814.71,"body":"one geometrical and the other algebraic,\""},{"startTime":814.71,"endTime":816.337,"body":"and he writes a little bit later,"},{"startTime":816.337,"endTime":819.6,"body":"\"The algebraic demonstration"},{"startTime":816.337,"endTime":819.6,"body":"must be exhibited to those"},{"startTime":819.6,"endTime":822.117,"body":"who do not comprehend the geometric one.\""},{"startTime":823.373,"endTime":826.47,"body":"The term that Colbrooke translates"},{"startTime":826.47,"endTime":829.41,"body":"as demonstration from"},{"startTime":826.47,"endTime":829.41,"body":"the Sanskrit is upapatti,"},{"startTime":829.41,"endTime":832.733,"body":"and we'll be interested in"},{"startTime":829.41,"endTime":832.733,"body":"this term, you can see it here,"},{"startTime":832.733,"endTime":835.99,"body":"which will become the standard term"},{"startTime":835.99,"endTime":840.99,"body":"translated from the Sanskrit,"},{"startTime":835.99,"endTime":840.99,"body":"into English, as proof."},{"startTime":842.79,"endTime":847.47,"body":"So we can come back to what"},{"startTime":842.79,"endTime":847.47,"body":"Colbrooke is doing here."},{"startTime":847.47,"endTime":852.47,"body":"He's actually taking some"},{"startTime":847.47,"endTime":852.47,"body":"assertions made by Bhaskara"},{"startTime":855.45,"endTime":859.62,"body":"in one part of his text, and"},{"startTime":855.45,"endTime":859.62,"body":"I have to say something about,"},{"startTime":859.62,"endTime":863.64,"body":"just to explain exactly the"},{"startTime":859.62,"endTime":863.64,"body":"nature of these assertions,"},{"startTime":863.64,"endTime":867.99,"body":"Sanskrit mathematical treatises"},{"startTime":863.64,"endTime":867.99,"body":"are written with sutras"},{"startTime":867.99,"endTime":872.22,"body":"that are often versified and aphoristic,"},{"startTime":872.22,"endTime":876.03,"body":"and so they're written to"},{"startTime":872.22,"endTime":876.03,"body":"come with a commentary,"},{"startTime":876.03,"endTime":880.44,"body":"and Bhaskara himself"},{"startTime":876.03,"endTime":880.44,"body":"wrote a prose commentary"},{"startTime":880.44,"endTime":882.605,"body":"on his own treatises,"},{"startTime":882.605,"endTime":886.617,"body":"and so these are notations"},{"startTime":882.605,"endTime":886.617,"body":"from his prose commentary,"},{"startTime":886.617,"endTime":888.63,"body":"and so anyway, there's two notations"},{"startTime":888.63,"endTime":890.34,"body":"in the whole of the algebra,"},{"startTime":890.34,"endTime":894.663,"body":"and Colbrooke takes them"},{"startTime":890.34,"endTime":894.663,"body":"out and he generalizes it,"},{"startTime":896.43,"endTime":899.16,"body":"not only to all of Bhaskara's text,"},{"startTime":899.16,"endTime":902.43,"body":"but to the whole of Hindu mathematicians."},{"startTime":902.43,"endTime":906.78,"body":"So we can understand what"},{"startTime":902.43,"endTime":906.78,"body":"Colbrooke is doing here."},{"startTime":906.78,"endTime":908.28,"body":"He's not trying, actually,"},{"startTime":908.28,"endTime":912.259,"body":"to make an identity of"},{"startTime":908.28,"endTime":912.259,"body":"all Hindu mathematicians."},{"startTime":912.259,"endTime":914.61,"body":"Colbrooke's aim when he's doing this"},{"startTime":914.61,"endTime":917.52,"body":"is that he wants to make comparisons,"},{"startTime":917.52,"endTime":919.41,"body":"he wants to be able to compare"},{"startTime":919.41,"endTime":921.84,"body":"what authors that were writing in Sanskrit"},{"startTime":921.84,"endTime":924.54,"body":"were doing with authors that were writing"},{"startTime":924.54,"endTime":925.89,"body":"in Arabic and Latin."},{"startTime":925.89,"endTime":930.483,"body":"So for him, it's in this"},{"startTime":925.89,"endTime":930.483,"body":"process that he creates this,"},{"startTime":931.71,"endTime":936.71,"body":"but he is creating a homogenous"},{"startTime":931.71,"endTime":936.71,"body":"idea of Hindu mathematicians"},{"startTime":937.68,"endTime":942.68,"body":"and a homogeneous idea"},{"startTime":937.68,"endTime":942.68,"body":"of proof along the way,"},{"startTime":943.32,"endTime":947.49,"body":"and Colbrooke's publication"},{"startTime":943.32,"endTime":947.49,"body":"was very influential"},{"startTime":947.49,"endTime":951.123,"body":"in Europe and in Asia"},{"startTime":947.49,"endTime":951.123,"body":"during all the 19th century."},{"startTime":952.17,"endTime":956.539,"body":"Mathematicians in France,"},{"startTime":952.17,"endTime":956.539,"body":"like Schal in Germany,"},{"startTime":956.539,"endTime":959.97,"body":"read the proofs his translation contained."},{"startTime":959.97,"endTime":963.78,"body":"Discussions on notions of"},{"startTime":959.97,"endTime":963.78,"body":"proof in mathematical education"},{"startTime":963.78,"endTime":966.69,"body":"were thus developed using"},{"startTime":963.78,"endTime":966.69,"body":"Colbrooke's translation,"},{"startTime":966.69,"endTime":969.66,"body":"and notably Hermann Hankel, for instance,"},{"startTime":969.66,"endTime":972.0,"body":"developed the idea that an ideal education"},{"startTime":972.0,"endTime":975.39,"body":"should combine together"},{"startTime":972.0,"endTime":975.39,"body":"two types of proofs,"},{"startTime":975.39,"endTime":979.02,"body":"the analytical, which was"},{"startTime":975.39,"endTime":979.02,"body":"attributed to the Greeks,"},{"startTime":979.02,"endTime":982.863,"body":"and the intuitive,"},{"startTime":979.02,"endTime":982.863,"body":"attributed to the Indians,"},{"startTime":986.34,"endTime":990.06,"body":"and something else happened"},{"startTime":986.34,"endTime":990.06,"body":"was that, as you can see here,"},{"startTime":990.06,"endTime":991.59,"body":"Colbrooke took proofs he found"},{"startTime":991.59,"endTime":993.3,"body":"in all sorts of different commentaries"},{"startTime":993.3,"endTime":996.45,"body":"and added them as footnotes"},{"startTime":993.3,"endTime":996.45,"body":"to his translation,"},{"startTime":996.45,"endTime":1000.2,"body":"giving the feeling that proofs"},{"startTime":996.45,"endTime":1000.2,"body":"only belong to commentaries"},{"startTime":1000.2,"endTime":1001.67,"body":"and could only be..."},{"startTime":1001.67,"endTime":1002.93,"body":"So yeah, there's two things."},{"startTime":1002.93,"endTime":1007.1,"body":"First of all, the only proofs"},{"startTime":1002.93,"endTime":1007.1,"body":"we see in the text he gives"},{"startTime":1007.1,"endTime":1009.35,"body":"are in commentaries, and second,"},{"startTime":1009.35,"endTime":1014.35,"body":"he's not translating the"},{"startTime":1009.35,"endTime":1014.35,"body":"whole of a commentary"},{"startTime":1014.39,"endTime":1017.66,"body":"but giving parts that he"},{"startTime":1014.39,"endTime":1017.66,"body":"considers interesting,"},{"startTime":1017.66,"endTime":1019.82,"body":"and so visually in the text"},{"startTime":1019.82,"endTime":1023.15,"body":"we can see that they're"},{"startTime":1019.82,"endTime":1023.15,"body":"made of bits and pieces,"},{"startTime":1023.15,"endTime":1026.0,"body":"and by the beginning of the..."},{"startTime":1026.0,"endTime":1027.95,"body":"So this gave the idea,"},{"startTime":1027.95,"endTime":1030.56,"body":"and this is something that"},{"startTime":1027.95,"endTime":1030.56,"body":"it became very standard"},{"startTime":1030.56,"endTime":1034.01,"body":"that people say in Sanskrit."},{"startTime":1034.01,"endTime":1035.39,"body":"Actually, they don't usually say"},{"startTime":1035.39,"endTime":1036.8,"body":"in Sanskrit mathematical texts,"},{"startTime":1036.8,"endTime":1038.93,"body":"but in Sanskrit mathematical texts,"},{"startTime":1038.93,"endTime":1040.4,"body":"treatises have no proofs."},{"startTime":1040.4,"endTime":1042.77,"body":"They're just a set of rules,"},{"startTime":1040.4,"endTime":1042.77,"body":"and if there are some,"},{"startTime":1042.77,"endTime":1044.97,"body":"they're in bits and"},{"startTime":1042.77,"endTime":1044.97,"body":"pieces, in common terms."},{"startTime":1046.16,"endTime":1050.6,"body":"So, but by the beginning"},{"startTime":1046.16,"endTime":1050.6,"body":"of the 20th century,"},{"startTime":1050.6,"endTime":1053.51,"body":"the existence of such proofs"},{"startTime":1050.6,"endTime":1053.51,"body":"seems to have been forgotten"},{"startTime":1053.51,"endTime":1057.08,"body":"and standard histories"},{"startTime":1053.51,"endTime":1057.08,"body":"overlooked them altogether,"},{"startTime":1057.08,"endTime":1059.33,"body":"but Colbrooke's claim has been taken up"},{"startTime":1059.33,"endTime":1060.337,"body":"at the end of the 20th"},{"startTime":1060.337,"endTime":1062.69,"body":"and at the beginning of the 21st century"},{"startTime":1062.69,"endTime":1065.33,"body":"by a certain number of"},{"startTime":1062.69,"endTime":1065.33,"body":"historians of mathematics"},{"startTime":1065.33,"endTime":1066.59,"body":"in South Asia,"},{"startTime":1066.59,"endTime":1070.79,"body":"aiming at having the history"},{"startTime":1066.59,"endTime":1070.79,"body":"of mathematics in Sanskrit"},{"startTime":1070.79,"endTime":1072.98,"body":"enter a global conversation"},{"startTime":1072.98,"endTime":1075.77,"body":"on the history of mathematical proofs,"},{"startTime":1075.77,"endTime":1079.49,"body":"and so both the two"},{"startTime":1075.77,"endTime":1079.49,"body":"authors I'm quoting here,"},{"startTime":1079.49,"endTime":1083.78,"body":"is MD Srinivas and Ramasubramanian,"},{"startTime":1083.78,"endTime":1088.78,"body":"both quote the quotation of"},{"startTime":1083.78,"endTime":1088.78,"body":"Colbrooke I gave you before,"},{"startTime":1089.75,"endTime":1094.75,"body":"to say that in the Sanskrit"},{"startTime":1089.75,"endTime":1094.75,"body":"mathematical tradition,"},{"startTime":1095.03,"endTime":1098.63,"body":"as they say, the kind of"},{"startTime":1095.03,"endTime":1098.63,"body":"proofs that were given"},{"startTime":1098.63,"endTime":1103.4,"body":"were either algebraic or geometrical,"},{"startTime":1103.4,"endTime":1108.4,"body":"and well, MD Srinivas adds also,"},{"startTime":1108.867,"endTime":1112.67,"body":"\"Clearly the tradition of"},{"startTime":1108.867,"endTime":1112.67,"body":"exposition of upapattis,\""},{"startTime":1112.67,"endTime":1113.93,"body":"so upapattis, remember,"},{"startTime":1113.93,"endTime":1117.157,"body":"it's the word that was used by Bhaskara."},{"startTime":1117.157,"endTime":1120.71,"body":"\"The exposition of"},{"startTime":1117.157,"endTime":1120.71,"body":"upapattis is much older,"},{"startTime":1120.71,"endTime":1122.59,"body":"and Bhaskaracharya,\""},{"startTime":1122.59,"endTime":1127.59,"body":"so Bhaskaracharya is one of the"},{"startTime":1122.59,"endTime":1127.59,"body":"ways of calling Bhaskara II."},{"startTime":1127.883,"endTime":1130.85,"body":"\"... and the later"},{"startTime":1127.883,"endTime":1130.85,"body":"mathematicians and astronomers"},{"startTime":1130.85,"endTime":1134.39,"body":"are merely following"},{"startTime":1130.85,"endTime":1134.39,"body":"the traditional practice"},{"startTime":1134.39,"endTime":1138.05,"body":"of providing detailed"},{"startTime":1134.39,"endTime":1138.05,"body":"upapattis in their commentaries"},{"startTime":1138.05,"endTime":1140.483,"body":"to earlier or their own works."},{"startTime":1141.56,"endTime":1143.69,"body":"So here's what he's saying here"},{"startTime":1143.69,"endTime":1146.3,"body":"is actually that it's"},{"startTime":1143.69,"endTime":1146.3,"body":"an immemorial tradition."},{"startTime":1146.3,"endTime":1149.06,"body":"It's not only in Bhaskara,"},{"startTime":1146.3,"endTime":1149.06,"body":"it existed before,"},{"startTime":1149.06,"endTime":1152.127,"body":"and it exists after, and then,"},{"startTime":1152.127,"endTime":1155.6,"body":"\"The notion of upapatti"},{"startTime":1152.127,"endTime":1155.6,"body":"is significantly different"},{"startTime":1155.6,"endTime":1157.94,"body":"from the notion of proof"},{"startTime":1155.6,"endTime":1157.94,"body":"as understood in the Greek"},{"startTime":1157.94,"endTime":1161.51,"body":"as well as the modern Western"},{"startTime":1157.94,"endTime":1161.51,"body":"traditions of mathematics.\""},{"startTime":1161.51,"endTime":1165.02,"body":"So it's really the construction"},{"startTime":1161.51,"endTime":1165.02,"body":"of an immemorial tradition"},{"startTime":1165.02,"endTime":1169.15,"body":"of Sanskrit proofs, and opposing it"},{"startTime":1171.5,"endTime":1174.83,"body":"to a Western, what would have"},{"startTime":1171.5,"endTime":1174.83,"body":"been a Western tradition,"},{"startTime":1174.83,"endTime":1177.65,"body":"as if that was homogeneous as well,"},{"startTime":1177.65,"endTime":1182.65,"body":"and Ramasubramanian also"},{"startTime":1177.65,"endTime":1182.65,"body":"adds, having quoted Colbrooke,"},{"startTime":1183.927,"endTime":1186.26,"body":"\"The upapattis of Indian mathematics,"},{"startTime":1186.26,"endTime":1187.88,"body":"unlike the Western tradition,"},{"startTime":1187.88,"endTime":1189.74,"body":"are not formulated with reference"},{"startTime":1189.74,"endTime":1193.029,"body":"to a formal axiomatic deductive system.\""},{"startTime":1193.029,"endTime":1198.029,"body":"So he's also then opposing"},{"startTime":1193.029,"endTime":1198.029,"body":"this tradition with the West."},{"startTime":1201.297,"endTime":1205.52,"body":"\"One often finds the"},{"startTime":1201.297,"endTime":1205.52,"body":"statement iyam atra vasana,"},{"startTime":1205.52,"endTime":1208.46,"body":"when the commentator is"},{"startTime":1205.52,"endTime":1208.46,"body":"about to begin to explain"},{"startTime":1208.46,"endTime":1210.711,"body":"or demonstrate something."},{"startTime":1210.711,"endTime":1215.111,"body":"Meaning wise, this"},{"startTime":1210.711,"endTime":1215.111,"body":"statement, iyam atra vasana"},{"startTime":1215.111,"endTime":1219.65,"body":"is equivalent to atropapattih."},{"startTime":1219.65,"endTime":1221.66,"body":"Both the forms being equivalent,"},{"startTime":1221.66,"endTime":1223.34,"body":"there is hardly any consideration"},{"startTime":1223.34,"endTime":1225.14,"body":"for choosing one over the other.\""},{"startTime":1225.14,"endTime":1228.56,"body":"So what he is saying basically"},{"startTime":1225.14,"endTime":1228.56,"body":"is that whatever the name,"},{"startTime":1228.56,"endTime":1230.12,"body":"there can be different names"},{"startTime":1230.12,"endTime":1233.48,"body":"for the reasonings that"},{"startTime":1230.12,"endTime":1233.48,"body":"are used, they're the same."},{"startTime":1233.48,"endTime":1236.14,"body":"It's the same tradition of proof."},{"startTime":1236.14,"endTime":1237.342,"body":"Okay."},{"startTime":1237.342,"endTime":1242.342,"body":"So behind what looks like a true interest"},{"startTime":1244.37,"endTime":1249.05,"body":"in actors' categories lies"},{"startTime":1244.37,"endTime":1249.05,"body":"what is in today's India"},{"startTime":1249.05,"endTime":1251.03,"body":"truly a very strong claim,"},{"startTime":1251.03,"endTime":1253.46,"body":"that all Sanskrit mathematical texts,"},{"startTime":1253.46,"endTime":1257.24,"body":"written over several centuries"},{"startTime":1253.46,"endTime":1257.24,"body":"in very different contexts,"},{"startTime":1257.24,"endTime":1259.94,"body":"all functioned in the same way,"},{"startTime":1259.94,"endTime":1262.67,"body":"and here then, a mode of"},{"startTime":1259.94,"endTime":1262.67,"body":"reasoning is harnessed"},{"startTime":1262.67,"endTime":1263.99,"body":"to a language."},{"startTime":1263.99,"endTime":1268.99,"body":"If it is not the geographical"},{"startTime":1263.99,"endTime":1268.99,"body":"space of South Asia,"},{"startTime":1269.09,"endTime":1273.41,"body":"to a religion, Hinduism,"},{"startTime":1269.09,"endTime":1273.41,"body":"or to a caste, Brahmanism."},{"startTime":1273.41,"endTime":1276.26,"body":"Such claims, then, I"},{"startTime":1273.41,"endTime":1276.26,"body":"think you can see resonate"},{"startTime":1276.26,"endTime":1279.5,"body":"with what is a wider political"},{"startTime":1276.26,"endTime":1279.5,"body":"use of history of science"},{"startTime":1279.5,"endTime":1283.25,"body":"by the current nationalist"},{"startTime":1279.5,"endTime":1283.25,"body":"Hindu government in India."},{"startTime":1283.25,"endTime":1285.05,"body":"It constructs a non-Western"},{"startTime":1285.05,"endTime":1288.473,"body":"but homogenous mathematical"},{"startTime":1285.05,"endTime":1288.473,"body":"identity for Hindus."},{"startTime":1289.85,"endTime":1293.42,"body":"So I'd like to argue that"},{"startTime":1289.85,"endTime":1293.42,"body":"Sanskrit mathematical texts"},{"startTime":1293.42,"endTime":1295.37,"body":"had very diverse ways of reasoning"},{"startTime":1295.37,"endTime":1297.789,"body":"that do not always"},{"startTime":1295.37,"endTime":1297.789,"body":"correspond to such a portrait"},{"startTime":1297.789,"endTime":1301.64,"body":"of the mathematical proofs they contain,"},{"startTime":1301.64,"endTime":1305.03,"body":"and actually they're more"},{"startTime":1301.64,"endTime":1305.03,"body":"surprising than that."},{"startTime":1305.03,"endTime":1309.95,"body":"So I would like to show, notably,"},{"startTime":1309.95,"endTime":1313.19,"body":"that proofs need not be"},{"startTime":1309.95,"endTime":1313.19,"body":"algebraic or geometric"},{"startTime":1313.19,"endTime":1316.88,"body":"in Sanskrit texts, and"},{"startTime":1313.19,"endTime":1316.88,"body":"that they don't necessarily"},{"startTime":1316.88,"endTime":1319.88,"body":"belong to commentaries,"},{"startTime":1316.88,"endTime":1319.88,"body":"and that some reasonings"},{"startTime":1319.88,"endTime":1322.25,"body":"might be larger or different than proofs"},{"startTime":1322.25,"endTime":1324.65,"body":"in as much as they might not be aiming"},{"startTime":1324.65,"endTime":1328.823,"body":"at establishing correctness or truth."},{"startTime":1329.81,"endTime":1334.81,"body":"So I'm going to just go through"},{"startTime":1329.81,"endTime":1334.81,"body":"a certain number of examples,"},{"startTime":1335.15,"endTime":1337.88,"body":"and the first one I wanted to give to you"},{"startTime":1337.88,"endTime":1342.35,"body":"is an example taken from a treatise"},{"startTime":1342.35,"endTime":1346.58,"body":"that was written in the"},{"startTime":1342.35,"endTime":1346.58,"body":"seventh century by Brahmagupta,"},{"startTime":1346.58,"endTime":1347.93,"body":"and who wrote a treatise"},{"startTime":1347.93,"endTime":1352.25,"body":"called the \"Corrected"},{"startTime":1347.93,"endTime":1352.25,"body":"Astronomical Treatise of Brahma\"."},{"startTime":1352.25,"endTime":1356.873,"body":"And in this treatise, in the chapter two,"},{"startTime":1358.01,"endTime":1363.01,"body":"Brahmagupta provides a table"},{"startTime":1358.01,"endTime":1363.01,"body":"of signs with 24 values,"},{"startTime":1363.47,"endTime":1368.47,"body":"and in chapter 21, he provides"},{"startTime":1363.47,"endTime":1368.47,"body":"mathematical procedures"},{"startTime":1368.78,"endTime":1372.053,"body":"that derive this table,"},{"startTime":1368.78,"endTime":1372.053,"body":"and others, actually."},{"startTime":1373.58,"endTime":1378.58,"body":"So actually, the earliest sine"},{"startTime":1373.58,"endTime":1378.58,"body":"tables in Sanskrit sources"},{"startTime":1379.22,"endTime":1384.22,"body":"date from the fifth century,"},{"startTime":1379.22,"endTime":1384.22,"body":"and for Brahmagupta,"},{"startTime":1384.26,"endTime":1387.32,"body":"the sine was both a"},{"startTime":1384.26,"endTime":1387.32,"body":"geometrical, and, I mean,"},{"startTime":1387.32,"endTime":1390.23,"body":"for most of the authors,"},{"startTime":1390.23,"endTime":1392.39,"body":"astronomical and mathematical authors"},{"startTime":1392.39,"endTime":1394.73,"body":"who worked in Sanskrit with the signs,"},{"startTime":1394.73,"endTime":1397.58,"body":"they considered it both"},{"startTime":1394.73,"endTime":1397.58,"body":"as something numerical"},{"startTime":1397.58,"endTime":1399.323,"body":"and geometrical at the same time,"},{"startTime":1400.25,"endTime":1404.93,"body":"and the sine was defined"},{"startTime":1400.25,"endTime":1404.93,"body":"actually in a figure"},{"startTime":1404.93,"endTime":1406.94,"body":"that's called the bow-field,"},{"startTime":1406.94,"endTime":1411.94,"body":"and so it was defined from the"},{"startTime":1406.94,"endTime":1411.94,"body":"idea of the cord of an arc,"},{"startTime":1412.82,"endTime":1415.88,"body":"and you would take half"},{"startTime":1412.82,"endTime":1415.88,"body":"the cord of half the arc,"},{"startTime":1415.88,"endTime":1418.01,"body":"and that would be what is the sine,"},{"startTime":1418.01,"endTime":1423.01,"body":"and this enabled astronomers"},{"startTime":1418.01,"endTime":1423.01,"body":"and mathematicians"},{"startTime":1423.47,"endTime":1426.71,"body":"to construct some of the"},{"startTime":1423.47,"endTime":1426.71,"body":"first trigonometrical circles"},{"startTime":1426.71,"endTime":1429.62,"body":"and with which you could use very easily"},{"startTime":1429.62,"endTime":1431.42,"body":"the Pythagorean theorem"},{"startTime":1431.42,"endTime":1436.42,"body":"to derive geometrically"},{"startTime":1431.42,"endTime":1436.42,"body":"and numerically half cords,"},{"startTime":1437.12,"endTime":1438.95,"body":"which were the sines."},{"startTime":1438.95,"endTime":1440.09,"body":"Okay."},{"startTime":1440.09,"endTime":1442.88,"body":"So this is to say, I'm making this point"},{"startTime":1442.88,"endTime":1445.97,"body":"just because I wanted to"},{"startTime":1442.88,"endTime":1445.97,"body":"tell you that in what follows"},{"startTime":1445.97,"endTime":1449.78,"body":"the reasoning of that"},{"startTime":1445.97,"endTime":1449.78,"body":"Brahmagupta is going to follow"},{"startTime":1449.78,"endTime":1452.36,"body":"is both numerical and geometrical,"},{"startTime":1452.36,"endTime":1454.31,"body":"and I don't know, I wouldn't know it"},{"startTime":1454.31,"endTime":1456.56,"body":"how to make a difference in it,"},{"startTime":1456.56,"endTime":1459.233,"body":"but I don't think it's algebraical either."},{"startTime":1460.58,"endTime":1462.323,"body":"So what happens is that,"},{"startTime":1463.697,"endTime":1468.697,"body":"so Brahmagupta gives 24 values of signs,"},{"startTime":1469.76,"endTime":1474.76,"body":"and when he comes back in"},{"startTime":1469.76,"endTime":1474.76,"body":"chapter 21 to this table,"},{"startTime":1477.26,"endTime":1480.71,"body":"he gives rules that actually,"},{"startTime":1480.71,"endTime":1482.93,"body":"so he gives a first set of rules"},{"startTime":1482.93,"endTime":1487.93,"body":"which provides modes to"},{"startTime":1482.93,"endTime":1487.93,"body":"compute three initial sines"},{"startTime":1488.27,"endTime":1489.74,"body":"which correspond to the sine"},{"startTime":1489.74,"endTime":1493.55,"body":"of 30 degrees, 45 degrees and 60 degrees,"},{"startTime":1493.55,"endTime":1496.04,"body":"once you know the radius of your circle,"},{"startTime":1496.04,"endTime":1499.22,"body":"and then from these rules,"},{"startTime":1496.04,"endTime":1499.22,"body":"from these three sines,"},{"startTime":1499.22,"endTime":1502.986,"body":"then he provides a rule"},{"startTime":1499.22,"endTime":1502.986,"body":"to derive all other sines."},{"startTime":1502.986,"endTime":1504.05,"body":"Okay?"},{"startTime":1504.05,"endTime":1507.14,"body":"So this is already, in doing this,"},{"startTime":1507.14,"endTime":1508.73,"body":"Brahmagupta is doing something."},{"startTime":1508.73,"endTime":1510.08,"body":"So I don't know what it is."},{"startTime":1510.08,"endTime":1513.56,"body":"Is he showing how he derives his table,"},{"startTime":1513.56,"endTime":1515.24,"body":"but he's doing something more,"},{"startTime":1515.24,"endTime":1517.46,"body":"because actually his rules are general,"},{"startTime":1517.46,"endTime":1522.46,"body":"therefore for any radius of a circle,"},{"startTime":1522.59,"endTime":1526.433,"body":"so he's giving rules on"},{"startTime":1522.59,"endTime":1526.433,"body":"how to derive 24 sines,"},{"startTime":1527.63,"endTime":1532.28,"body":"and in particular his own sine table,"},{"startTime":1532.28,"endTime":1534.473,"body":"but he's doing something more,"},{"startTime":1535.79,"endTime":1538.707,"body":"which is that he doesn't"},{"startTime":1535.79,"endTime":1538.707,"body":"give one way to compute"},{"startTime":1538.707,"endTime":1542.72,"body":"the sine of 30 degrees,"},{"startTime":1538.707,"endTime":1542.72,"body":"45 degrees and 60 degrees,"},{"startTime":1542.72,"endTime":1544.49,"body":"he gives two ways,"},{"startTime":1544.49,"endTime":1547.46,"body":"and it's the same thing,"},{"startTime":1544.49,"endTime":1547.46,"body":"if you consider the rules"},{"startTime":1547.46,"endTime":1550.04,"body":"to compute all the other signs from this,"},{"startTime":1550.04,"endTime":1553.673,"body":"he doesn't give one rule, he"},{"startTime":1550.04,"endTime":1553.673,"body":"gives two rules to do this."},{"startTime":1555.32,"endTime":1557.393,"body":"So what is he doing?"},{"startTime":1558.26,"endTime":1560.48,"body":"So here we have a,"},{"startTime":1560.48,"endTime":1562.91,"body":"this is typical of the"},{"startTime":1560.48,"endTime":1562.91,"body":"kind of problems we have."},{"startTime":1562.91,"endTime":1564.852,"body":"He's not telling us what he's doing,"},{"startTime":1564.852,"endTime":1568.25,"body":"and even his commentator"},{"startTime":1564.852,"endTime":1568.25,"body":"is not commenting on that,"},{"startTime":1568.25,"endTime":1570.14,"body":"he comments on something else,"},{"startTime":1570.14,"endTime":1573.44,"body":"and he's giving his own"},{"startTime":1570.14,"endTime":1573.44,"body":"take on all of this,"},{"startTime":1573.44,"endTime":1577.13,"body":"but he's not reflecting on"},{"startTime":1573.44,"endTime":1577.13,"body":"what Brahmagupta is doing."},{"startTime":1577.13,"endTime":1578.93,"body":"So there's many, there's different ways"},{"startTime":1578.93,"endTime":1581.78,"body":"that we can approach the fact"},{"startTime":1578.93,"endTime":1581.78,"body":"that he's doubling things."},{"startTime":1581.78,"endTime":1584.27,"body":"Of course you can say,"},{"startTime":1581.78,"endTime":1584.27,"body":"well, he's giving two rules"},{"startTime":1584.27,"endTime":1587.39,"body":"because well, if one is not easy for you,"},{"startTime":1587.39,"endTime":1588.23,"body":"you can use the other."},{"startTime":1588.23,"endTime":1590.42,"body":"If you're really making"},{"startTime":1588.23,"endTime":1590.42,"body":"practical computations,"},{"startTime":1590.42,"endTime":1593.72,"body":"maybe it's better to have"},{"startTime":1590.42,"endTime":1593.72,"body":"one rule and another,"},{"startTime":1593.72,"endTime":1596.54,"body":"but as we, as you'll agree with me,"},{"startTime":1596.54,"endTime":1599.57,"body":"there's something theoretical"},{"startTime":1596.54,"endTime":1599.57,"body":"in the rules he's giving,"},{"startTime":1599.57,"endTime":1603.89,"body":"because he is establishing"},{"startTime":1599.57,"endTime":1603.89,"body":"his several rules,"},{"startTime":1603.89,"endTime":1606.77,"body":"and another thing that we can wonder is,"},{"startTime":1606.77,"endTime":1608.6,"body":"does he think that one of the rules"},{"startTime":1608.6,"endTime":1610.46,"body":"actually explains the other,"},{"startTime":1610.46,"endTime":1615.17,"body":"that one is the more sort of"},{"startTime":1610.46,"endTime":1615.17,"body":"an explanation of the other,"},{"startTime":1615.17,"endTime":1617.31,"body":"or is it that he believes that"},{"startTime":1618.47,"endTime":1623.27,"body":"by giving two independent rules,"},{"startTime":1623.27,"endTime":1625.25,"body":"he's actually showing something?"},{"startTime":1625.25,"endTime":1626.582,"body":"So this we don't know,"},{"startTime":1626.582,"endTime":1629.72,"body":"but actually we have other,"},{"startTime":1629.72,"endTime":1632.03,"body":"so the fact of doubling rules is something"},{"startTime":1632.03,"endTime":1637.03,"body":"that we find very often in"},{"startTime":1632.03,"endTime":1637.03,"body":"Sanskrit astronomical treatises,"},{"startTime":1641.54,"endTime":1646.54,"body":"but we also find this"},{"startTime":1641.54,"endTime":1646.54,"body":"kind of mode of reasoning"},{"startTime":1647.06,"endTime":1650.15,"body":"using two rules to read one rule"},{"startTime":1650.15,"endTime":1652.433,"body":"and then doubling it up with another rule,"},{"startTime":1653.33,"endTime":1656.66,"body":"in a contemporary of Brahmagupta,"},{"startTime":1656.66,"endTime":1659.72,"body":"but who is a commentator"},{"startTime":1656.66,"endTime":1659.72,"body":"who does write in prose,"},{"startTime":1659.72,"endTime":1661.7,"body":"and who's called Bhaskara."},{"startTime":1661.7,"endTime":1663.053,"body":"So this is Bhaskara I,"},{"startTime":1663.053,"endTime":1666.38,"body":"it's the Bhaskara of the seventh century,"},{"startTime":1666.38,"endTime":1670.04,"body":"not like the 12th century"},{"startTime":1666.38,"endTime":1670.04,"body":"one we talked about before,"},{"startTime":1670.04,"endTime":1675.04,"body":"and Bhaskara, in a commentary he makes"},{"startTime":1675.47,"endTime":1677.603,"body":"to a fifth century treatises,"},{"startTime":1678.83,"endTime":1682.43,"body":"is it comments on a rule"},{"startTime":1678.83,"endTime":1682.43,"body":"that is given by Aryabhata,"},{"startTime":1682.43,"endTime":1685.4,"body":"and which concerns a gnomon."},{"startTime":1685.4,"endTime":1690.4,"body":"So we have a gnomon,"},{"startTime":1685.4,"endTime":1690.4,"body":"and a source of light,"},{"startTime":1690.86,"endTime":1695.86,"body":"and we have, we know this, this distance,"},{"startTime":1697.04,"endTime":1702.04,"body":"and this distance, and this distance,"},{"startTime":1702.08,"endTime":1704.51,"body":"and we want to compute the shadow,"},{"startTime":1704.51,"endTime":1707.15,"body":"we want to know the shadow of the gnomon,"},{"startTime":1707.15,"endTime":1712.15,"body":"and so Aryabhata gives a rule"},{"startTime":1707.15,"endTime":1712.15,"body":"that I'm not going to read out,"},{"startTime":1712.31,"endTime":1716.0,"body":"but it basically, the"},{"startTime":1712.31,"endTime":1716.0,"body":"rule is a multiplication"},{"startTime":1716.0,"endTime":1718.22,"body":"followed by a division,"},{"startTime":1718.22,"endTime":1722.84,"body":"and what Bhaskara is going to say is,"},{"startTime":1722.84,"endTime":1727.84,"body":"well, this rule actually"},{"startTime":1722.84,"endTime":1727.84,"body":"is a rule of three."},{"startTime":1728.48,"endTime":1732.05,"body":"So the rule of three was the"},{"startTime":1728.48,"endTime":1732.05,"body":"way that Sanskrit authors"},{"startTime":1732.05,"endTime":1733.97,"body":"would talk about equal proportions,"},{"startTime":1733.97,"endTime":1736.643,"body":"or to talk about similar triangles,"},{"startTime":1737.69,"endTime":1742.69,"body":"but what I want to underline"},{"startTime":1737.69,"endTime":1742.69,"body":"here is that, to say this,"},{"startTime":1744.35,"endTime":1749.35,"body":"to show that he's using one procedure,"},{"startTime":1749.45,"endTime":1752.693,"body":"he's going to reinterpret"},{"startTime":1749.45,"endTime":1752.693,"body":"it as another procedure."},{"startTime":1754.22,"endTime":1755.96,"body":"Bhaskara does something specific."},{"startTime":1755.96,"endTime":1758.6,"body":"He turns it into a question."},{"startTime":1758.6,"endTime":1761.12,"body":"So this computation is a rule of three."},{"startTime":1761.12,"endTime":1763.91,"body":"How, if, from the top of the base,"},{"startTime":1763.91,"endTime":1766.37,"body":"which is greater than the gnomon,"},{"startTime":1766.37,"endTime":1769.07,"body":"the size of the space between"},{"startTime":1766.37,"endTime":1769.07,"body":"the gnomon and the base,"},{"startTime":1769.07,"endTime":1770.84,"body":"which is a shadow, is obtained,"},{"startTime":1770.84,"endTime":1773.24,"body":"then what is obtained with the gnomon?"},{"startTime":1773.24,"endTime":1774.41,"body":"The shadow is obtained."},{"startTime":1774.41,"endTime":1778.85,"body":"So what I want to just, the"},{"startTime":1774.41,"endTime":1778.85,"body":"only thing I want to underline"},{"startTime":1778.85,"endTime":1780.71,"body":"is that there's a way of doing this"},{"startTime":1780.71,"endTime":1782.93,"body":"which is by asking a question,"},{"startTime":1782.93,"endTime":1785.27,"body":"but what's interesting for me here is also"},{"startTime":1785.27,"endTime":1788.51,"body":"that he's rereading the"},{"startTime":1785.27,"endTime":1788.51,"body":"rule with another rule,"},{"startTime":1788.51,"endTime":1790.37,"body":"and he could do this also, I mean,"},{"startTime":1790.37,"endTime":1795.37,"body":"in another part of his commentary,"},{"startTime":1795.92,"endTime":1800.27,"body":"he's going to also talk about"},{"startTime":1795.92,"endTime":1800.27,"body":"the Pythagorean theorem,"},{"startTime":1800.27,"endTime":1801.92,"body":"and to do that, there's another way."},{"startTime":1801.92,"endTime":1805.01,"body":"So you can take any rule"},{"startTime":1801.92,"endTime":1805.01,"body":"where there's squares"},{"startTime":1805.01,"endTime":1808.67,"body":"that are added or subtracted,"},{"startTime":1805.01,"endTime":1808.67,"body":"and that give another square,"},{"startTime":1808.67,"endTime":1812.87,"body":"and to show that the rule is"},{"startTime":1808.67,"endTime":1812.87,"body":"about the Pythagorean theorem,"},{"startTime":1812.87,"endTime":1816.95,"body":"you're going to show that it's"},{"startTime":1812.87,"endTime":1816.95,"body":"dealing with a right triangle,"},{"startTime":1816.95,"endTime":1820.82,"body":"and this is done by"},{"startTime":1816.95,"endTime":1820.82,"body":"actually renaming segments."},{"startTime":1820.82,"endTime":1825.82,"body":"So there's the segments of"},{"startTime":1820.82,"endTime":1825.82,"body":"a right triangle in Sanskrit"},{"startTime":1826.19,"endTime":1829.757,"body":"are called, so there's the"},{"startTime":1826.19,"endTime":1829.757,"body":"hypotenuse is called karna,"},{"startTime":1829.757,"endTime":1831.89,"body":"and the upright side is called koti,"},{"startTime":1831.89,"endTime":1836.89,"body":"and so when you see a"},{"startTime":1831.89,"endTime":1836.89,"body":"koti, or bhuja, or a karna,"},{"startTime":1837.3,"endTime":1838.793,"body":"appear in the text,"},{"startTime":1838.793,"endTime":1841.981,"body":"then it's a way for"},{"startTime":1838.793,"endTime":1841.981,"body":"the commentator to say,"},{"startTime":1841.981,"endTime":1846.981,"body":"you can apply, you can reread"},{"startTime":1841.981,"endTime":1846.981,"body":"the set of arbitrary rules"},{"startTime":1848.12,"endTime":1852.56,"body":"as being something dealing"},{"startTime":1848.12,"endTime":1852.56,"body":"with the Pythagorean theorem."},{"startTime":1852.56,"endTime":1856.64,"body":"So here once again, we have"},{"startTime":1852.56,"endTime":1856.64,"body":"a reasoning that has no name,"},{"startTime":1856.64,"endTime":1860.21,"body":"but where what Bhaskara is doing"},{"startTime":1860.21,"endTime":1862.82,"body":"is that he's taking what"},{"startTime":1860.21,"endTime":1862.82,"body":"looks like procedures,"},{"startTime":1862.82,"endTime":1865.76,"body":"because I forgot to tell you"},{"startTime":1862.82,"endTime":1865.76,"body":"that, it's so obvious for me,"},{"startTime":1865.76,"endTime":1870.2,"body":"that the sutras of mathematical"},{"startTime":1865.76,"endTime":1870.2,"body":"text more often than not"},{"startTime":1870.2,"endTime":1875.2,"body":"are just giving procedures"},{"startTime":1870.2,"endTime":1875.2,"body":"of mathematical problems,"},{"startTime":1875.63,"endTime":1880.63,"body":"and general procedures that look like"},{"startTime":1881.443,"endTime":1884.45,"body":"an arbitrary set of steps."},{"startTime":1884.45,"endTime":1889.45,"body":"When Bhaskara interprets them,"},{"startTime":1884.45,"endTime":1889.45,"body":"they become a set of steps,"},{"startTime":1890.33,"endTime":1892.64,"body":"which step by step have a meaning,"},{"startTime":1892.64,"endTime":1894.35,"body":"have a mathematical meaning,"},{"startTime":1894.35,"endTime":1897.883,"body":"so that the rereading with another rule"},{"startTime":1897.883,"endTime":1901.37,"body":"seems to be a way of"},{"startTime":1897.883,"endTime":1901.37,"body":"giving an interpretation,"},{"startTime":1901.37,"endTime":1903.803,"body":"or explaining the steps one by one."},{"startTime":1905.36,"endTime":1910.36,"body":"But, okay, actually in"},{"startTime":1905.36,"endTime":1910.36,"body":"Bhaskara's commentary,"},{"startTime":1912.99,"endTime":1916.88,"body":"there are many discussions"},{"startTime":1912.99,"endTime":1916.88,"body":"about many different reasonings."},{"startTime":1916.88,"endTime":1920.9,"body":"So there's reasonings he does"},{"startTime":1916.88,"endTime":1920.9,"body":"to which he gives no names,"},{"startTime":1920.9,"endTime":1923.0,"body":"and then there's all sorts of references"},{"startTime":1923.0,"endTime":1925.223,"body":"to all sorts of reasonings, sometimes,"},{"startTime":1926.39,"endTime":1927.74,"body":"that he names and does."},{"startTime":1927.74,"endTime":1930.41,"body":"So among them, so he does talk,"},{"startTime":1930.41,"endTime":1933.5,"body":"he does use the term"},{"startTime":1930.41,"endTime":1933.5,"body":"upapatti, that we saw before,"},{"startTime":1933.5,"endTime":1938.09,"body":"but it's used to be contrasted"},{"startTime":1933.5,"endTime":1938.09,"body":"with tradition, agama,"},{"startTime":1938.09,"endTime":1941.09,"body":"and he uses it to say, yeah, you know,"},{"startTime":1941.09,"endTime":1943.34,"body":"you can't take something"},{"startTime":1941.09,"endTime":1943.34,"body":"just as a tradition,"},{"startTime":1943.34,"endTime":1945.26,"body":"you have to give a proof for it,"},{"startTime":1945.26,"endTime":1947.0,"body":"but he uses also other terms,"},{"startTime":1947.0,"endTime":1951.59,"body":"like he actually gives"},{"startTime":1947.0,"endTime":1951.59,"body":"examples of verifications,"},{"startTime":1951.59,"endTime":1953.93,"body":"which is, if you have a procedure,"},{"startTime":1953.93,"endTime":1956.54,"body":"and if you turn it upside"},{"startTime":1953.93,"endTime":1956.54,"body":"down, and you do each of,"},{"startTime":1956.54,"endTime":1957.78,"body":"if you invert it,"},{"startTime":1957.78,"endTime":1961.34,"body":"this is would be what"},{"startTime":1957.78,"endTime":1961.34,"body":"he calls a verification."},{"startTime":1961.34,"endTime":1964.25,"body":"Then there are moments in the text"},{"startTime":1964.25,"endTime":1967.85,"body":"where he talks about explaining,"},{"startTime":1964.25,"endTime":1967.85,"body":"and when he does this,"},{"startTime":1967.85,"endTime":1971.33,"body":"he uses, once again,"},{"startTime":1967.85,"endTime":1971.33,"body":"all sorts of vocabulary."},{"startTime":1971.33,"endTime":1976.33,"body":"So some that could be that of the function"},{"startTime":1976.67,"endTime":1979.07,"body":"of any commentary to explain an other,"},{"startTime":1979.07,"endTime":1980.96,"body":"which might mean to establish,"},{"startTime":1980.96,"endTime":1983.84,"body":"and other, which might"},{"startTime":1980.96,"endTime":1983.84,"body":"mean to show or to teach."},{"startTime":1983.84,"endTime":1988.84,"body":"So there's this whole"},{"startTime":1983.84,"endTime":1988.84,"body":"variety of words and, well,"},{"startTime":1989.57,"endTime":1993.35,"body":"we still have to understand what it means,"},{"startTime":1993.35,"endTime":1995.53,"body":"but in particular he uses..."},{"startTime":1999.41,"endTime":2001.33,"body":"okay, I'll come back."},{"startTime":2001.33,"endTime":2006.33,"body":"Actually, there's a slide"},{"startTime":2001.33,"endTime":2006.33,"body":"that I probably changed place,"},{"startTime":2006.34,"endTime":2008.96,"body":"but I'll come back to some of Bhaskara's"},{"startTime":2010.99,"endTime":2013.633,"body":"modes of explanation on diagrams later on."},{"startTime":2014.89,"endTime":2019.54,"body":"So there's also another commentator"},{"startTime":2019.54,"endTime":2021.538,"body":"I want to talk to you about,"},{"startTime":2021.538,"endTime":2024.22,"body":"whose name is Prthudhaka,"},{"startTime":2024.22,"endTime":2028.27,"body":"and so Prthudhaka is making"},{"startTime":2024.22,"endTime":2028.27,"body":"a commentary on Brahmagupta,"},{"startTime":2028.27,"endTime":2032.02,"body":"that we saw, and he, Prthudhaka,"},{"startTime":2032.02,"endTime":2037.02,"body":"has read Aryabhata and"},{"startTime":2032.02,"endTime":2037.02,"body":"Bhaskara's commentaries"},{"startTime":2038.56,"endTime":2039.58,"body":"and he quotes them."},{"startTime":2039.58,"endTime":2041.71,"body":"Okay, so this is, I'm"},{"startTime":2039.58,"endTime":2041.71,"body":"telling you all of this"},{"startTime":2041.71,"endTime":2042.7,"body":"because I want to show you."},{"startTime":2042.7,"endTime":2045.34,"body":"So we saw a mode of"},{"startTime":2042.7,"endTime":2045.34,"body":"reasoning with Brahmagupta,"},{"startTime":2045.34,"endTime":2048.7,"body":"then we saw a contemporary"},{"startTime":2045.34,"endTime":2048.7,"body":"of Brahmagupta, Bhaskara,"},{"startTime":2048.7,"endTime":2051.73,"body":"who has sort of other"},{"startTime":2048.7,"endTime":2051.73,"body":"ways of reasoning, maybe,"},{"startTime":2051.73,"endTime":2053.89,"body":"or something similar,"},{"startTime":2053.89,"endTime":2057.04,"body":"and then we have somebody"},{"startTime":2053.89,"endTime":2057.04,"body":"who's from the ninth century"},{"startTime":2057.04,"endTime":2059.59,"body":"and who knows all of these guys,"},{"startTime":2059.59,"endTime":2063.193,"body":"and he's going to do other"},{"startTime":2059.59,"endTime":2063.193,"body":"sorts of reasonings again."},{"startTime":2064.753,"endTime":2069.55,"body":"Actually, Prthudhaka writes"},{"startTime":2064.753,"endTime":2069.55,"body":"the name of his commentary,"},{"startTime":2069.55,"endTime":2072.25,"body":"it's called \"Commentary with Explanation\","},{"startTime":2072.25,"endTime":2076.93,"body":"and the idea is what, what"},{"startTime":2072.25,"endTime":2076.93,"body":"is, for Prthudhaka, vasana?"},{"startTime":2076.93,"endTime":2081.93,"body":"So actually, I'm still"},{"startTime":2076.93,"endTime":2081.93,"body":"trying to figure out"},{"startTime":2082.21,"endTime":2084.61,"body":"what is a vasana, and"},{"startTime":2082.21,"endTime":2084.61,"body":"I'm currently editing"},{"startTime":2084.61,"endTime":2087.76,"body":"Prthudhaka's commentary, so"},{"startTime":2084.61,"endTime":2087.76,"body":"I'm still trying to figure out"},{"startTime":2087.76,"endTime":2090.4,"body":"what is a vasana for Prthudhaka,"},{"startTime":2090.4,"endTime":2091.66,"body":"but there's something that he does,"},{"startTime":2091.66,"endTime":2094.48,"body":"so I wanted to take an example of his work"},{"startTime":2094.48,"endTime":2096.76,"body":"on arithmetical sequences,"},{"startTime":2101.11,"endTime":2104.41,"body":"and there's, when he gives,"},{"startTime":2104.41,"endTime":2106.24,"body":"for him giving an explanation"},{"startTime":2106.24,"endTime":2110.02,"body":"starts by sort of extending the realm"},{"startTime":2110.02,"endTime":2112.75,"body":"in which the rule can be applied."},{"startTime":2112.75,"endTime":2115.09,"body":"So Brahmagupta gives a general rule"},{"startTime":2115.09,"endTime":2119.8,"body":"for summing the terms of"},{"startTime":2115.09,"endTime":2119.8,"body":"an arithmetical sequence,"},{"startTime":2119.8,"endTime":2124.65,"body":"and Prthudhaka first"},{"startTime":2119.8,"endTime":2124.65,"body":"suggests to consider this as"},{"startTime":2125.62,"endTime":2128.32,"body":"he considers a stack of bricks."},{"startTime":2128.32,"endTime":2133.32,"body":"Then he suggests to"},{"startTime":2128.32,"endTime":2133.32,"body":"extend the point of view"},{"startTime":2133.39,"endTime":2135.25,"body":"on the sequences"},{"startTime":2135.25,"endTime":2139.18,"body":"and to consider it as a"},{"startTime":2135.25,"endTime":2139.18,"body":"capital that's being invested"},{"startTime":2139.18,"endTime":2140.62,"body":"once more and more,"},{"startTime":2140.62,"endTime":2143.68,"body":"and then he considers,"},{"startTime":2140.62,"endTime":2143.68,"body":"if you're into capital,"},{"startTime":2143.68,"endTime":2146.14,"body":"then as something that is as a debt."},{"startTime":2146.14,"endTime":2151.14,"body":"So this, yeah, or some"},{"startTime":2146.14,"endTime":2151.14,"body":"money that you're lending,"},{"startTime":2153.55,"endTime":2156.91,"body":"and so you can use negative"},{"startTime":2153.55,"endTime":2156.91,"body":"and positive numbers,"},{"startTime":2156.91,"endTime":2158.77,"body":"and then he starts giving examples."},{"startTime":2158.77,"endTime":2161.68,"body":"So these are in the"},{"startTime":2158.77,"endTime":2161.68,"body":"form of solved examples,"},{"startTime":2161.68,"endTime":2164.74,"body":"he starts playing with the"},{"startTime":2161.68,"endTime":2164.74,"body":"examples, just with the numbers,"},{"startTime":2164.74,"endTime":2168.283,"body":"so it becomes a numerical problem,"},{"startTime":2169.21,"endTime":2173.14,"body":"and then he considers, he"},{"startTime":2169.21,"endTime":2173.14,"body":"gets into another explanation,"},{"startTime":2173.14,"endTime":2177.553,"body":"which is considering that, actually,"},{"startTime":2179.38,"endTime":2184.38,"body":"a sequence can be thought of"},{"startTime":2179.38,"endTime":2184.38,"body":"as a sum of areas of rectangles"},{"startTime":2185.017,"endTime":2188.89,"body":"of side one, and with each"},{"startTime":2185.017,"endTime":2188.89,"body":"value that you're stacked,"},{"startTime":2188.89,"endTime":2191.35,"body":"and along the side,"},{"startTime":2191.35,"endTime":2193.87,"body":"he's going to say that, actually, then,"},{"startTime":2193.87,"endTime":2198.43,"body":"the sum of the areas"},{"startTime":2193.87,"endTime":2198.43,"body":"of all these rectangles"},{"startTime":2198.43,"endTime":2201.16,"body":"is equal to the area of a rectangle."},{"startTime":2201.16,"endTime":2202.63,"body":"So he's saying this by,"},{"startTime":2202.63,"endTime":2207.16,"body":"for him, this moment, which"},{"startTime":2202.63,"endTime":2207.16,"body":"we'd say it's a kind of proof,"},{"startTime":2207.16,"endTime":2208.57,"body":"you know, he would be proving"},{"startTime":2208.57,"endTime":2210.58,"body":"that the area of this rectangle"},{"startTime":2210.58,"endTime":2212.17,"body":"is equal to the sum of this,"},{"startTime":2212.17,"endTime":2215.53,"body":"but for him it seems that the explanation"},{"startTime":2215.53,"endTime":2220.21,"body":"has to do with adding"},{"startTime":2215.53,"endTime":2220.21,"body":"more and more context,"},{"startTime":2220.21,"endTime":2225.21,"body":"different contexts in"},{"startTime":2220.21,"endTime":2225.21,"body":"which the rules make sense,"},{"startTime":2225.49,"endTime":2228.25,"body":"each step of the rules make sense,"},{"startTime":2228.25,"endTime":2233.123,"body":"and actually, he goes on in"},{"startTime":2228.25,"endTime":2233.123,"body":"his commentary, looking at,"},{"startTime":2234.52,"endTime":2239.52,"body":"he gives a rule to sum the"},{"startTime":2234.52,"endTime":2239.52,"body":"terms of a geometric sequence,"},{"startTime":2243.79,"endTime":2248.47,"body":"and he explains it by"},{"startTime":2243.79,"endTime":2248.47,"body":"understanding this rule"},{"startTime":2248.47,"endTime":2252.64,"body":"as being part of a discipline"},{"startTime":2248.47,"endTime":2252.64,"body":"that's combinatorics,"},{"startTime":2252.64,"endTime":2256.6,"body":"but that is used in"},{"startTime":2252.64,"endTime":2256.6,"body":"diversification in Sanskrit."},{"startTime":2256.6,"endTime":2259.57,"body":"So there's, in Brahmagupta's"},{"startTime":2256.6,"endTime":2259.57,"body":"astronomical treatise,"},{"startTime":2259.57,"endTime":2260.59,"body":"there's a whole chapter"},{"startTime":2260.59,"endTime":2265.0,"body":"on these combinatorics of diversification,"},{"startTime":2265.0,"endTime":2266.9,"body":"and Prthudhaka is going to explain"},{"startTime":2269.525,"endTime":2272.11,"body":"the geometrical sequence with this,"},{"startTime":2272.11,"endTime":2277.11,"body":"and he's going to also use"},{"startTime":2272.11,"endTime":2277.11,"body":"algebra, and in another rule,"},{"startTime":2277.33,"endTime":2282.33,"body":"he's going to interpret"},{"startTime":2277.33,"endTime":2282.33,"body":"these geometrical sequences"},{"startTime":2284.83,"endTime":2288.88,"body":"as being linked to algebra,"},{"startTime":2288.88,"endTime":2293.29,"body":"or being that you can"},{"startTime":2288.88,"endTime":2293.29,"body":"understand them in algebra."},{"startTime":2293.29,"endTime":2296.11,"body":"So in other words, I'm not sure yet"},{"startTime":2296.11,"endTime":2299.77,"body":"what is an explanation for Prthudhaka."},{"startTime":2299.77,"endTime":2302.59,"body":"What he, for him is"},{"startTime":2299.77,"endTime":2302.59,"body":"something that's so important"},{"startTime":2302.59,"endTime":2306.28,"body":"that he's giving the title"},{"startTime":2302.59,"endTime":2306.28,"body":"to his commentary about this,"},{"startTime":2306.28,"endTime":2309.91,"body":"but it seems that what he's"},{"startTime":2306.28,"endTime":2309.91,"body":"interested in is looking,"},{"startTime":2309.91,"endTime":2311.71,"body":"if you take a procedure at looking"},{"startTime":2311.71,"endTime":2315.43,"body":"at all the different"},{"startTime":2311.71,"endTime":2315.43,"body":"contexts in which it is,"},{"startTime":2315.43,"endTime":2318.97,"body":"it can be more than applied,"},{"startTime":2315.43,"endTime":2318.97,"body":"in which it has a meaning,"},{"startTime":2318.97,"endTime":2323.14,"body":"and that these contexts"},{"startTime":2318.97,"endTime":2323.14,"body":"need not be mathematical."},{"startTime":2323.14,"endTime":2324.19,"body":"They can be to,"},{"startTime":2324.19,"endTime":2327.16,"body":"because for Brahmagupta, actually,"},{"startTime":2327.16,"endTime":2329.17,"body":"algebra is not part of mathematics,"},{"startTime":2329.17,"endTime":2331.393,"body":"it's part of astronomy, for instance."},{"startTime":2332.59,"endTime":2334.663,"body":"Okay, so this was an example."},{"startTime":2335.513,"endTime":2338.95,"body":"So there was a whole tradition of,"},{"startTime":2338.95,"endTime":2341.32,"body":"so this is where I get"},{"startTime":2338.95,"endTime":2341.32,"body":"back to my diagrams,"},{"startTime":2341.32,"endTime":2344.451,"body":"there was a whole tradition"},{"startTime":2341.32,"endTime":2344.451,"body":"of explanation on diagrams"},{"startTime":2344.451,"endTime":2349.451,"body":"that we find in Sanskrit"},{"startTime":2344.451,"endTime":2349.451,"body":"mathematical commentaries,"},{"startTime":2349.93,"endTime":2352.51,"body":"and the problem with these explanations"},{"startTime":2352.51,"endTime":2357.223,"body":"is that a lot of them were"},{"startTime":2352.51,"endTime":2357.223,"body":"oral, or so very typically,"},{"startTime":2358.36,"endTime":2363.36,"body":"Bhaskara, the one who comments"},{"startTime":2358.36,"endTime":2363.36,"body":"on Aryabhata, will say,"},{"startTime":2364.48,"endTime":2366.433,"body":"this is when he says,"},{"startTime":2368.74,"endTime":2372.82,"body":"okay, an explanation should"},{"startTime":2368.74,"endTime":2372.82,"body":"be given on a diagram,"},{"startTime":2372.82,"endTime":2374.68,"body":"and then the diagram is drawn,"},{"startTime":2374.68,"endTime":2376.21,"body":"or sometimes it's not even drawn,"},{"startTime":2376.21,"endTime":2380.673,"body":"and you're supposed to see"},{"startTime":2376.21,"endTime":2380.673,"body":"from the diagram what is doing,"},{"startTime":2382.249,"endTime":2385.36,"body":"what are the rules that are given,"},{"startTime":2385.36,"endTime":2388.81,"body":"and actually I think that the diagrams,"},{"startTime":2388.81,"endTime":2391.3,"body":"we see, the diagrams themselves,"},{"startTime":2391.3,"endTime":2394.69,"body":"we see them circulating from text to text"},{"startTime":2394.69,"endTime":2396.52,"body":"and commentary to commentary,"},{"startTime":2396.52,"endTime":2400.45,"body":"even though, as you see from"},{"startTime":2396.52,"endTime":2400.45,"body":"commentary to commentary,"},{"startTime":2400.45,"endTime":2405.45,"body":"the way of explaining the"},{"startTime":2400.45,"endTime":2405.45,"body":"values given to how you explain"},{"startTime":2405.52,"endTime":2410.52,"body":"change all the time, but it"},{"startTime":2405.52,"endTime":2410.52,"body":"seems that these diagrams"},{"startTime":2410.89,"endTime":2415.89,"body":"were also a way of a"},{"startTime":2410.89,"endTime":2415.89,"body":"repository of giving proofs"},{"startTime":2417.111,"endTime":2420.46,"body":"and committing them to memory."},{"startTime":2420.46,"endTime":2424.39,"body":"I see them as small libraries"},{"startTime":2420.46,"endTime":2424.39,"body":"of proofs as they circulate."},{"startTime":2424.39,"endTime":2428.29,"body":"So this is the famous"},{"startTime":2424.39,"endTime":2428.29,"body":"trigonometrical circle,"},{"startTime":2428.29,"endTime":2431.588,"body":"and this one is actually a diagram"},{"startTime":2431.588,"endTime":2435.94,"body":"that has to do with cyclic quadrilaterals,"},{"startTime":2435.94,"endTime":2439.87,"body":"which was one topic on which"},{"startTime":2435.94,"endTime":2439.87,"body":"Brahmagupta reasoned a lot,"},{"startTime":2439.87,"endTime":2443.08,"body":"and this is that interested"},{"startTime":2439.87,"endTime":2443.08,"body":"so much German mathematicians"},{"startTime":2443.08,"endTime":2444.91,"body":"at the end of the 19th century,"},{"startTime":2444.91,"endTime":2447.58,"body":"and Prthudhaka gave a diagram"},{"startTime":2447.58,"endTime":2450.43,"body":"of one of these cyclic quadrilaterals,"},{"startTime":2450.43,"endTime":2452.65,"body":"and even though it"},{"startTime":2450.43,"endTime":2452.65,"body":"might not look like one,"},{"startTime":2452.65,"endTime":2454.18,"body":"it looks like a triangle here,"},{"startTime":2454.18,"endTime":2457.9,"body":"but, and that that is, that has been,"},{"startTime":2457.9,"endTime":2461.653,"body":"that was circulated very much."},{"startTime":2462.94,"endTime":2464.5,"body":"Okay."},{"startTime":2464.5,"endTime":2468.64,"body":"So I want to end maybe of all my examples"},{"startTime":2468.64,"endTime":2473.465,"body":"of all these different"},{"startTime":2468.64,"endTime":2473.465,"body":"types of modes of reasoning"},{"startTime":2473.465,"endTime":2478.465,"body":"that for our authors"},{"startTime":2473.465,"endTime":2478.465,"body":"were seen as validating"},{"startTime":2479.23,"endTime":2480.79,"body":"what they were doing,"},{"startTime":2480.79,"endTime":2485.79,"body":"to talk about another"},{"startTime":2480.79,"endTime":2485.79,"body":"commentator, Sankara Variyar,"},{"startTime":2487.84,"endTime":2490.54,"body":"who was from Kerala,"},{"startTime":2487.84,"endTime":2490.54,"body":"from the Kerala school."},{"startTime":2490.54,"endTime":2495.54,"body":"So in the Kerala, from"},{"startTime":2490.54,"endTime":2495.54,"body":"the 15th century onwards,"},{"startTime":2495.683,"endTime":2498.28,"body":"there were a certain"},{"startTime":2495.683,"endTime":2498.28,"body":"number of mathematicians"},{"startTime":2498.28,"endTime":2502.27,"body":"based in the southWest of India,"},{"startTime":2502.27,"endTime":2505.69,"body":"really developed some incredible"},{"startTime":2502.27,"endTime":2505.69,"body":"mathematical procedures,"},{"startTime":2505.69,"endTime":2509.77,"body":"notably to approximate pi, and the sines,"},{"startTime":2509.77,"endTime":2513.64,"body":"with what looks like something"},{"startTime":2509.77,"endTime":2513.64,"body":"like infinite series,"},{"startTime":2513.64,"endTime":2516.31,"body":"and so they're held sometimes as being"},{"startTime":2516.31,"endTime":2519.76,"body":"the prehistorians of calculus."},{"startTime":2519.76,"endTime":2522.16,"body":"Some people even think that Jesuits"},{"startTime":2522.16,"endTime":2526.077,"body":"could have stolen some of the text"},{"startTime":2526.077,"endTime":2530.983,"body":"and transmitted it to Newton,"},{"startTime":2526.077,"endTime":2530.983,"body":"secretly or something,"},{"startTime":2532.02,"endTime":2537.02,"body":"and so, indeed actually, in the 15th,"},{"startTime":2537.43,"endTime":2540.16,"body":"what happens is that a lot"},{"startTime":2537.43,"endTime":2540.16,"body":"of, so a lot of these rules"},{"startTime":2540.16,"endTime":2543.94,"body":"are attributed to Madhava,"},{"startTime":2540.16,"endTime":2543.94,"body":"from the 15th century,"},{"startTime":2543.94,"endTime":2548.74,"body":"but we only have them through"},{"startTime":2543.94,"endTime":2548.74,"body":"quotations of commentators"},{"startTime":2548.74,"endTime":2550.96,"body":"and treatise writers, who"},{"startTime":2548.74,"endTime":2550.96,"body":"are going to write on them."},{"startTime":2550.96,"endTime":2555.96,"body":"So in one of his commentary"},{"startTime":2550.96,"endTime":2555.96,"body":"to the Lilavati of Bhaskara,"},{"startTime":2558.108,"endTime":2563.108,"body":"Sankara Variyar comes up with,"},{"startTime":2558.108,"endTime":2563.108,"body":"I dunno if I have other..."},{"startTime":2563.47,"endTime":2568.47,"body":"yeah, comes up with a proof"},{"startTime":2563.47,"endTime":2568.47,"body":"for this approximation"},{"startTime":2569.71,"endTime":2571.51,"body":"of the circumference of a circle,"},{"startTime":2571.51,"endTime":2575.32,"body":"which is another way of"},{"startTime":2571.51,"endTime":2575.32,"body":"giving an approximation of pi,"},{"startTime":2575.32,"endTime":2580.32,"body":"and in one, so Sankara Variyar, actually,"},{"startTime":2581.23,"endTime":2585.13,"body":"doesn't talk about upapatti or vasana."},{"startTime":2585.13,"endTime":2589.24,"body":"He really is talking about"},{"startTime":2585.13,"endTime":2589.24,"body":"establishing the truth."},{"startTime":2589.24,"endTime":2594.22,"body":"He's using sadh, and he's"},{"startTime":2589.24,"endTime":2594.22,"body":"using expressions about this,"},{"startTime":2594.22,"endTime":2597.64,"body":"and something else is going on is that"},{"startTime":2597.64,"endTime":2599.44,"body":"when he uses the rule of three,"},{"startTime":2599.44,"endTime":2602.44,"body":"when he wants to show that"},{"startTime":2599.44,"endTime":2602.44,"body":"things are proportionate,"},{"startTime":2602.44,"endTime":2606.58,"body":"well sometimes he uses the question mode,"},{"startTime":2606.58,"endTime":2609.28,"body":"but sometimes he uses, you know,"},{"startTime":2609.28,"endTime":2614.2,"body":"reasonings about the nature of"},{"startTime":2609.28,"endTime":2614.2,"body":"the things that are compared"},{"startTime":2614.2,"endTime":2619.2,"body":"that really, mm, makes us"},{"startTime":2614.2,"endTime":2619.2,"body":"think of the kind of reasonings"},{"startTime":2619.21,"endTime":2622.39,"body":"that were done in Euclidean geometry."},{"startTime":2622.39,"endTime":2625.9,"body":"So all thought this needs"},{"startTime":2622.39,"endTime":2625.9,"body":"to be more elaborated."},{"startTime":2625.9,"endTime":2628.78,"body":"It seems to me that we have here somebody"},{"startTime":2628.78,"endTime":2632.17,"body":"who is writing in Sanskrit,"},{"startTime":2632.17,"endTime":2635.47,"body":"but seems to be doing"},{"startTime":2632.17,"endTime":2635.47,"body":"something that is a hybrid,"},{"startTime":2635.47,"endTime":2637.27,"body":"probably of a Sanskrit tradition,"},{"startTime":2637.27,"endTime":2642.27,"body":"but of a Euclidean standard"},{"startTime":2637.27,"endTime":2642.27,"body":"proof of some sort,"},{"startTime":2644.029,"endTime":2647.02,"body":"and so, yeah, these"},{"startTime":2644.029,"endTime":2647.02,"body":"modes, modes of reasoning,"},{"startTime":2647.02,"endTime":2650.773,"body":"traveled also and hybrided themselves."},{"startTime":2651.91,"endTime":2656.713,"body":"Okay, and so to come to my conclusion,"},{"startTime":2657.79,"endTime":2659.92,"body":"there is something a bit arresting"},{"startTime":2659.92,"endTime":2664.18,"body":"in the persistence of this"},{"startTime":2659.92,"endTime":2664.18,"body":"ignorance on traditions of proof"},{"startTime":2664.18,"endTime":2668.74,"body":"outside of the Euclidean norm,"},{"startTime":2664.18,"endTime":2668.74,"body":"which has to do maybe in part"},{"startTime":2668.74,"endTime":2671.47,"body":"with what the philosopher Achille Mbembe"},{"startTime":2671.47,"endTime":2674.89,"body":"calls late European-centrism,"},{"startTime":2674.89,"endTime":2678.4,"body":"that is of the persistence of Eurocentric"},{"startTime":2678.4,"endTime":2680.68,"body":"and colonial structures of thought"},{"startTime":2680.68,"endTime":2682.537,"body":"in a world where everybody knows"},{"startTime":2682.537,"endTime":2685.18,"body":"that Europe and the United States"},{"startTime":2685.18,"endTime":2687.19,"body":"has never been the unique place"},{"startTime":2687.19,"endTime":2690.793,"body":"in which worthy intellectual"},{"startTime":2687.19,"endTime":2690.793,"body":"productions were composed."},{"startTime":2692.11,"endTime":2694.547,"body":"So I have entitled this presentation,"},{"startTime":2694.547,"endTime":2698.11,"body":"\"Let's Decolonize the History"},{"startTime":2694.547,"endTime":2698.11,"body":"of Mathematical Proofs\""},{"startTime":2698.11,"endTime":2701.71,"body":"because I believe that decolonizing"},{"startTime":2698.11,"endTime":2701.71,"body":"the history is possible"},{"startTime":2701.71,"endTime":2705.22,"body":"only through a collective critical effort,"},{"startTime":2705.22,"endTime":2708.28,"body":"that as professional"},{"startTime":2705.22,"endTime":2708.28,"body":"historians of mathematics,"},{"startTime":2708.28,"endTime":2713.17,"body":"but also as teachers,"},{"startTime":2708.28,"endTime":2713.17,"body":"as an educated public,"},{"startTime":2713.17,"endTime":2715.063,"body":"such as the one in Gresham,"},{"startTime":2717.25,"endTime":2719.8,"body":"we should not let's spread a discourse"},{"startTime":2719.8,"endTime":2723.883,"body":"which closes the doors of"},{"startTime":2719.8,"endTime":2723.883,"body":"what mathematics is or not."},{"startTime":2726.97,"endTime":2730.629,"body":"So mathematical reasonings were,"},{"startTime":2730.629,"endTime":2732.31,"body":"and they still are, very diverse."},{"startTime":2732.31,"endTime":2734.62,"body":"They are not necessarily about truth."},{"startTime":2734.62,"endTime":2736.51,"body":"They were certainly not homogenous"},{"startTime":2736.51,"endTime":2739.93,"body":"across a geographical"},{"startTime":2736.51,"endTime":2739.93,"body":"area, in the same language,"},{"startTime":2739.93,"endTime":2742.42,"body":"or by people who had the same religion."},{"startTime":2742.42,"endTime":2745.78,"body":"So let us remember this"},{"startTime":2742.42,"endTime":2745.78,"body":"when we hear people speaking"},{"startTime":2745.78,"endTime":2750.583,"body":"of Western science, British"},{"startTime":2745.78,"endTime":2750.583,"body":"mathematics, or Islamic proofs."},{"startTime":2751.57,"endTime":2753.91,"body":"The good news is that we have resources"},{"startTime":2753.91,"endTime":2757.54,"body":"to write other new histories"},{"startTime":2753.91,"endTime":2757.54,"body":"that are more stimulating,"},{"startTime":2757.54,"endTime":2760.42,"body":"histories that testify"},{"startTime":2757.54,"endTime":2760.42,"body":"to how just like today,"},{"startTime":2760.42,"endTime":2763.15,"body":"people practicing mathematics"},{"startTime":2763.15,"endTime":2765.49,"body":"had many different ways of reasoning,"},{"startTime":2765.49,"endTime":2768.64,"body":"and that there were whole"},{"startTime":2765.49,"endTime":2768.64,"body":"ecologies of argumentation"},{"startTime":2768.64,"endTime":2771.13,"body":"that still need to be investigated,"},{"startTime":2771.13,"endTime":2775.09,"body":"and I don't think that"},{"startTime":2771.13,"endTime":2775.09,"body":"these modes of reasoning"},{"startTime":2775.09,"endTime":2780.04,"body":"necessarily will give anything"},{"startTime":2775.09,"endTime":2780.04,"body":"to mathematicians today."},{"startTime":2780.04,"endTime":2781.27,"body":"I don't think they're going to bring"},{"startTime":2781.27,"endTime":2784.66,"body":"new mathematical ways of reasoning,"},{"startTime":2784.66,"endTime":2787.09,"body":"but I think that they are testimony"},{"startTime":2787.09,"endTime":2791.44,"body":"of a part of our common"},{"startTime":2787.09,"endTime":2791.44,"body":"humanity in its culture,"},{"startTime":2791.44,"endTime":2793.0,"body":"like its cultures of reasoning,"},{"startTime":2793.0,"endTime":2794.71,"body":"in the same way that we're interested"},{"startTime":2794.71,"endTime":2796.48,"body":"in beautiful pieces of art,"},{"startTime":2796.48,"endTime":2800.17,"body":"or beautiful pieces of literature,"},{"startTime":2796.48,"endTime":2800.17,"body":"wherever they come from"},{"startTime":2800.17,"endTime":2801.4,"body":"and whatever their time,"},{"startTime":2801.4,"endTime":2804.13,"body":"so there's beautiful pieces of reasoning"},{"startTime":2804.13,"endTime":2807.52,"body":"and they're interesting, just like that,"},{"startTime":2807.52,"endTime":2809.44,"body":"and for in themselves."},{"startTime":2809.44,"endTime":2810.698,"body":"Thank you."},{"startTime":2810.698,"endTime":2813.865,"body":"(audience applauding)"},{"startTime":2815.62,"endTime":2819.13,"body":"- [Audience] So you gave an"},{"startTime":2815.62,"endTime":2819.13,"body":"example through sine values"},{"startTime":2819.13,"endTime":2822.73,"body":"about using two different methods,"},{"startTime":2822.73,"endTime":2825.16,"body":"two different ways to"},{"startTime":2822.73,"endTime":2825.16,"body":"find the same answer,"},{"startTime":2825.16,"endTime":2830.02,"body":"and instead of just using"},{"startTime":2825.16,"endTime":2830.02,"body":"one way to find one answer,"},{"startTime":2830.02,"endTime":2833.44,"body":"he went and interpreted"},{"startTime":2830.02,"endTime":2833.44,"body":"it in a different way."},{"startTime":2833.44,"endTime":2836.83,"body":"Do you think that finding two"},{"startTime":2833.44,"endTime":2836.83,"body":"ways to find the same answer"},{"startTime":2836.83,"endTime":2839.053,"body":"is important, and why?"},{"startTime":2840.31,"endTime":2842.86,"body":"- So actually, as a historian,"},{"startTime":2842.86,"endTime":2846.82,"body":"I'm not going to make any"},{"startTime":2842.86,"endTime":2846.82,"body":"statements about if it's important"},{"startTime":2846.82,"endTime":2848.98,"body":"or if it's good, or if it's the truth."},{"startTime":2848.98,"endTime":2851.83,"body":"I'm interested in describing how it is,"},{"startTime":2851.83,"endTime":2854.53,"body":"but I find it interesting, and actually,"},{"startTime":2854.53,"endTime":2856.78,"body":"one of my points that maybe wasn't clear"},{"startTime":2856.78,"endTime":2860.32,"body":"because I'm still trying"},{"startTime":2856.78,"endTime":2860.32,"body":"to find for myself"},{"startTime":2860.32,"endTime":2864.28,"body":"how to interpret the fact"},{"startTime":2860.32,"endTime":2864.28,"body":"that many authors in Sanskrit"},{"startTime":2864.28,"endTime":2866.23,"body":"wrote two different rules,"},{"startTime":2866.23,"endTime":2869.23,"body":"what could appear as two"},{"startTime":2866.23,"endTime":2869.23,"body":"different rules for a different..."},{"startTime":2869.23,"endTime":2870.79,"body":"but why did they do it?"},{"startTime":2870.79,"endTime":2872.86,"body":"Why, what was behind it?"},{"startTime":2872.86,"endTime":2876.31,"body":"And actually, I think there"},{"startTime":2872.86,"endTime":2876.31,"body":"were different interpretations."},{"startTime":2876.31,"endTime":2878.74,"body":"Sometimes they really"},{"startTime":2876.31,"endTime":2878.74,"body":"thought that there were"},{"startTime":2878.74,"endTime":2882.64,"body":"some fundamental algorithms,"},{"startTime":2878.74,"endTime":2882.64,"body":"like the rule of three,"},{"startTime":2882.64,"endTime":2884.29,"body":"like the Pythagorean theorem,"},{"startTime":2884.29,"endTime":2886.33,"body":"that was thought of as a procedure,"},{"startTime":2886.33,"endTime":2888.73,"body":"and that these were like the basis"},{"startTime":2888.73,"endTime":2891.7,"body":"from which all the other"},{"startTime":2888.73,"endTime":2891.7,"body":"algorithms could be understood"},{"startTime":2891.7,"endTime":2893.95,"body":"and get a mathematical meaning,"},{"startTime":2893.95,"endTime":2895.99,"body":"but sometimes, maybe it was really thought"},{"startTime":2895.99,"endTime":2899.89,"body":"that if you arrive at the same place,"},{"startTime":2899.89,"endTime":2901.24,"body":"from two different ways,"},{"startTime":2901.24,"endTime":2903.91,"body":"then you're gathering a sort of strength"},{"startTime":2903.91,"endTime":2906.19,"body":"and proving something that's correct,"},{"startTime":2906.19,"endTime":2908.89,"body":"but we need to know what,"},{"startTime":2908.89,"endTime":2911.77,"body":"the idea is to understand"},{"startTime":2908.89,"endTime":2911.77,"body":"what the actors themselves,"},{"startTime":2911.77,"endTime":2913.63,"body":"the authors themselves thought,"},{"startTime":2913.63,"endTime":2916.84,"body":"so we're lucky when they"},{"startTime":2913.63,"endTime":2916.84,"body":"really tell us, but if not,"},{"startTime":2916.84,"endTime":2919.783,"body":"we have to use all sorts"},{"startTime":2916.84,"endTime":2919.783,"body":"of clues to try to guess."},{"startTime":2921.64,"endTime":2923.35,"body":"- [Audience] I'm not really"},{"startTime":2921.64,"endTime":2923.35,"body":"sure if this makes sense,"},{"startTime":2923.35,"endTime":2926.23,"body":"but would you say that"},{"startTime":2923.35,"endTime":2926.23,"body":"the rules and findings"},{"startTime":2926.23,"endTime":2929.44,"body":"made by Hindu, Chinese"},{"startTime":2926.23,"endTime":2929.44,"body":"or Arab mathematicians,"},{"startTime":2929.44,"endTime":2933.31,"body":"which may be different to"},{"startTime":2929.44,"endTime":2933.31,"body":"findings by Western mathematicians"},{"startTime":2933.31,"endTime":2934.69,"body":"but still accurate,"},{"startTime":2934.69,"endTime":2938.2,"body":"could be more relevant today"},{"startTime":2934.69,"endTime":2938.2,"body":"or even taught at schools"},{"startTime":2938.2,"endTime":2941.26,"body":"than the findings made by"},{"startTime":2938.2,"endTime":2941.26,"body":"Western mathematicians,"},{"startTime":2941.26,"endTime":2944.92,"body":"if reoccurring Western"},{"startTime":2941.26,"endTime":2944.92,"body":"dominance wasn't to exist?"},{"startTime":2944.92,"endTime":2947.02,"body":"- First of all, I want"},{"startTime":2944.92,"endTime":2947.02,"body":"to just say that it's,"},{"startTime":2947.02,"endTime":2950.68,"body":"I don't understand, I"},{"startTime":2947.02,"endTime":2950.68,"body":"honestly don't understand"},{"startTime":2950.68,"endTime":2953.02,"body":"why we just don't integrate, you know,"},{"startTime":2953.02,"endTime":2956.56,"body":"different mathematical texts"},{"startTime":2953.02,"endTime":2956.56,"body":"in the histories we write."},{"startTime":2956.56,"endTime":2959.35,"body":"So it could be in the kind of histories,"},{"startTime":2959.35,"endTime":2964.35,"body":"that the preliminary stories"},{"startTime":2959.35,"endTime":2964.35,"body":"that we give in math textbooks,"},{"startTime":2964.6,"endTime":2966.28,"body":"or just in the general public."},{"startTime":2966.28,"endTime":2969.55,"body":"So there's something which is what I call,"},{"startTime":2969.55,"endTime":2971.83,"body":"what actually Achille Mbembe"},{"startTime":2971.83,"endTime":2974.2,"body":"calls the persistence of ignorance."},{"startTime":2974.2,"endTime":2975.85,"body":"So it's really this idea,"},{"startTime":2975.85,"endTime":2979.33,"body":"and which I think has"},{"startTime":2975.85,"endTime":2979.33,"body":"to come with, in a way,"},{"startTime":2979.33,"endTime":2980.38,"body":"to be more direct."},{"startTime":2980.38,"endTime":2982.72,"body":"It's a very, it's the legacy we have"},{"startTime":2982.72,"endTime":2986.702,"body":"of racist and white"},{"startTime":2982.72,"endTime":2986.702,"body":"supremacist modes of thought"},{"startTime":2986.702,"endTime":2990.28,"body":"that come from us from the 19th century,"},{"startTime":2990.28,"endTime":2992.77,"body":"or the late 19th century, and that has,"},{"startTime":2992.77,"endTime":2994.683,"body":"that is our legacy,"},{"startTime":2994.683,"endTime":2999.683,"body":"and what I want to say"},{"startTime":2994.683,"endTime":2999.683,"body":"is, okay, then after that,"},{"startTime":2999.718,"endTime":3002.28,"body":"mathematics is part of our humanity."},{"startTime":3002.28,"endTime":3005.667,"body":"As soon as we have writing,"},{"startTime":3002.28,"endTime":3005.667,"body":"we have mathematics,"},{"startTime":3005.667,"endTime":3008.16,"body":"and as soon as we have"},{"startTime":3005.667,"endTime":3008.16,"body":"mathematical writings,"},{"startTime":3008.16,"endTime":3010.35,"body":"we have things like the"},{"startTime":3008.16,"endTime":3010.35,"body":"Pythagorean theorem."},{"startTime":3010.35,"endTime":3013.29,"body":"So it's called the Pythagorean theorem,"},{"startTime":3013.29,"endTime":3016.59,"body":"but actually it's something that's common"},{"startTime":3016.59,"endTime":3018.36,"body":"to us as human beings, I'd say."},{"startTime":3018.36,"endTime":3020.97,"body":"And then, but of course"},{"startTime":3018.36,"endTime":3020.97,"body":"there's different people"},{"startTime":3020.97,"endTime":3023.22,"body":"who did different things,"},{"startTime":3020.97,"endTime":3023.22,"body":"at different times,"},{"startTime":3023.22,"endTime":3026.13,"body":"but one thing I want to say is not, like,"},{"startTime":3026.13,"endTime":3027.81,"body":"even if you were, were a Hindu..."},{"startTime":3027.81,"endTime":3029.67,"body":"So first of all you"},{"startTime":3027.81,"endTime":3029.67,"body":"could write in Sanskrit"},{"startTime":3029.67,"endTime":3031.26,"body":"and not be a Hindu,"},{"startTime":3031.26,"endTime":3033.06,"body":"but even if you were a Hindu"},{"startTime":3033.06,"endTime":3036.0,"body":"writing in Sanskrit in"},{"startTime":3033.06,"endTime":3036.0,"body":"the seventh century,"},{"startTime":3036.0,"endTime":3037.95,"body":"you could be doing some two different,"},{"startTime":3037.95,"endTime":3039.66,"body":"even if you were neighbors,"},{"startTime":3039.66,"endTime":3041.73,"body":"you could be doing two different things,"},{"startTime":3041.73,"endTime":3042.72,"body":"two different things."},{"startTime":3042.72,"endTime":3047.72,"body":"So it doesn't make sense"},{"startTime":3042.72,"endTime":3047.72,"body":"to say that, you know,"},{"startTime":3048.636,"endTime":3050.557,"body":"to make a huge bag and say,"},{"startTime":3050.557,"endTime":3052.71,"body":"\"Oh, they had the same religion,"},{"startTime":3052.71,"endTime":3055.135,"body":"they had the same language,\""},{"startTime":3055.135,"endTime":3059.43,"body":"And of course, yes, they"},{"startTime":3055.135,"endTime":3059.43,"body":"did incredible things."},{"startTime":3059.43,"endTime":3060.6,"body":"They did incredible things"},{"startTime":3060.6,"endTime":3065.43,"body":"sometimes before Europeans did"},{"startTime":3060.6,"endTime":3065.43,"body":"something that was similar,"},{"startTime":3065.43,"endTime":3068.67,"body":"but that's not, my point"},{"startTime":3065.43,"endTime":3068.67,"body":"is we're not in a race"},{"startTime":3068.67,"endTime":3071.85,"body":"of trying to show who was the first."},{"startTime":3071.85,"endTime":3076.62,"body":"I think it's also interesting"},{"startTime":3071.85,"endTime":3076.62,"body":"in itself and in understanding"},{"startTime":3076.62,"endTime":3080.22,"body":"how a seventh century Sanskrit commentator"},{"startTime":3080.22,"endTime":3082.08,"body":"thought about mathematics,"},{"startTime":3082.08,"endTime":3087.08,"body":"and what he thought a reasoning"},{"startTime":3082.08,"endTime":3087.08,"body":"that was interesting was,"},{"startTime":3087.235,"endTime":3090.63,"body":"and maybe it wasn't about truth,"},{"startTime":3090.63,"endTime":3092.33,"body":"maybe it was about something else."},{"startTime":3093.52,"endTime":3098.52,"body":"- [Audience] So I was wondering,"},{"startTime":3093.52,"endTime":3098.52,"body":"if in these various texts,"},{"startTime":3100.53,"endTime":3102.09,"body":"of mathematics in India,"},{"startTime":3102.09,"endTime":3106.29,"body":"if there is a different conceptualization"},{"startTime":3106.29,"endTime":3109.02,"body":"between the different"},{"startTime":3106.29,"endTime":3109.02,"body":"parts of mathematics,"},{"startTime":3109.02,"endTime":3112.26,"body":"between algebra and geometry, arithmetic,"},{"startTime":3112.26,"endTime":3114.15,"body":"is there something, a different idea"},{"startTime":3114.15,"endTime":3116.157,"body":"of the relation between them,"},{"startTime":3116.157,"endTime":3120.18,"body":"as opposed to the Greek"},{"startTime":3116.157,"endTime":3120.18,"body":"legacy of mathematics?"},{"startTime":3120.18,"endTime":3122.52,"body":"- Okay, thank you for this nice question."},{"startTime":3122.52,"endTime":3125.19,"body":"Actually, so there's, of course,"},{"startTime":3125.19,"endTime":3129.66,"body":"what each and every author"},{"startTime":3125.19,"endTime":3129.66,"body":"thought about mathematics"},{"startTime":3129.66,"endTime":3131.46,"body":"is still something we need to..."},{"startTime":3131.46,"endTime":3133.05,"body":"we're still editing these texts"},{"startTime":3133.05,"endTime":3134.679,"body":"and trying to understand them,"},{"startTime":3134.679,"endTime":3138.06,"body":"but certainly authors, first of all,"},{"startTime":3138.06,"endTime":3141.33,"body":"distinguished in between"},{"startTime":3138.06,"endTime":3141.33,"body":"different topics of mathematics."},{"startTime":3141.33,"endTime":3143.01,"body":"So they did have a distinguished,"},{"startTime":3143.01,"endTime":3146.58,"body":"they did talk about arithmetics,"},{"startTime":3143.01,"endTime":3146.58,"body":"geometry and algebra,"},{"startTime":3146.58,"endTime":3151.58,"body":"and, well, Bhaskara II,"},{"startTime":3146.58,"endTime":3151.58,"body":"Bhaskaracharya, is very famous,"},{"startTime":3152.91,"endTime":3154.17,"body":"because he's the one,"},{"startTime":3154.17,"endTime":3158.13,"body":"he actually saw a continuity"},{"startTime":3158.13,"endTime":3160.68,"body":"in between arithmetics and algebra,"},{"startTime":3160.68,"endTime":3162.9,"body":"so for him, there's something about,"},{"startTime":3162.9,"endTime":3165.72,"body":"I don't think it's the"},{"startTime":3162.9,"endTime":3165.72,"body":"quantities, it's the operands,"},{"startTime":3165.72,"endTime":3168.3,"body":"it's the kind of things"},{"startTime":3165.72,"endTime":3168.3,"body":"that you can operate on,"},{"startTime":3168.3,"endTime":3171.45,"body":"so there would be something"},{"startTime":3168.3,"endTime":3171.45,"body":"like a common structure,"},{"startTime":3171.45,"endTime":3175.86,"body":"and so Colbrooke, who knew a little bit"},{"startTime":3175.86,"endTime":3178.23,"body":"the algebraists in England at the time,"},{"startTime":3178.23,"endTime":3181.955,"body":"forced a little bit of Bhaskara's,"},{"startTime":3181.955,"endTime":3185.37,"body":"in his translation of Bhaskara's"},{"startTime":3181.955,"endTime":3185.37,"body":"algebra and arithmetic,"},{"startTime":3185.37,"endTime":3188.43,"body":"the sort of the structural link"},{"startTime":3188.43,"endTime":3191.1,"body":"that would make that algebra"},{"startTime":3188.43,"endTime":3191.1,"body":"would be sort of like"},{"startTime":3191.1,"endTime":3194.58,"body":"a fundamental structure that"},{"startTime":3191.1,"endTime":3194.58,"body":"could explain also arithmetic,"},{"startTime":3194.58,"endTime":3199.08,"body":"but there are elements of"},{"startTime":3194.58,"endTime":3199.08,"body":"this in Bhaskara's text,"},{"startTime":3199.08,"endTime":3202.53,"body":"and okay, so, and if you take the..."},{"startTime":3202.53,"endTime":3203.76,"body":"so that's Bhaskara,"},{"startTime":3203.76,"endTime":3206.16,"body":"the canonical Bhaskara"},{"startTime":3203.76,"endTime":3206.16,"body":"of the 12th century."},{"startTime":3206.16,"endTime":3208.593,"body":"The first century Bhaskara has,"},{"startTime":3209.756,"endTime":3213.78,"body":"is different in his"},{"startTime":3209.756,"endTime":3213.78,"body":"conception of mathematics"},{"startTime":3213.78,"endTime":3217.02,"body":"in the sense where he"},{"startTime":3213.78,"endTime":3217.02,"body":"gives several definitions"},{"startTime":3217.02,"endTime":3218.94,"body":"of how you can think of mathematics"},{"startTime":3218.94,"endTime":3222.223,"body":"and he doesn't decide which one he wants."},{"startTime":3222.223,"endTime":3225.9,"body":"One of one of his definitions"},{"startTime":3222.223,"endTime":3225.9,"body":"of mathematics is to say"},{"startTime":3225.9,"endTime":3229.86,"body":"actually, mathematics is"},{"startTime":3225.9,"endTime":3229.86,"body":"the science of algorithms,"},{"startTime":3229.86,"endTime":3233.16,"body":"of computing, and so they could be what"},{"startTime":3233.16,"endTime":3236.58,"body":"then you can distinguish the"},{"startTime":3233.16,"endTime":3236.58,"body":"different types of objects"},{"startTime":3236.58,"endTime":3239.19,"body":"that go in it, so it can"},{"startTime":3236.58,"endTime":3239.19,"body":"be arithmetical objects"},{"startTime":3239.19,"endTime":3240.87,"body":"or geometrical objects,"},{"startTime":3240.87,"endTime":3242.85,"body":"but what counts is the algorithm,"},{"startTime":3242.85,"endTime":3244.11,"body":"but then he says,"},{"startTime":3244.11,"endTime":3247.86,"body":"then you can always give a"},{"startTime":3244.11,"endTime":3247.86,"body":"geometrical interpretation"},{"startTime":3247.86,"endTime":3251.103,"body":"to an arithmetical"},{"startTime":3247.86,"endTime":3251.103,"body":"problem, and vice versa,"},{"startTime":3253.14,"endTime":3254.7,"body":"but he never goes back to this,"},{"startTime":3254.7,"endTime":3257.49,"body":"so it's just like a"},{"startTime":3254.7,"endTime":3257.49,"body":"statement that's there,"},{"startTime":3257.49,"endTime":3262.197,"body":"and in his commentary,"},{"startTime":3257.49,"endTime":3262.197,"body":"algebra is not so important."},{"startTime":3263.562,"endTime":3264.395,"body":"- [Audience] One example"},{"startTime":3264.395,"endTime":3266.79,"body":"of where East meets West in mathematics,"},{"startTime":3266.79,"endTime":3270.42,"body":"I think, is Ramanujan,"},{"startTime":3266.79,"endTime":3270.42,"body":"the Indian mathematician"},{"startTime":3270.42,"endTime":3275.42,"body":"who was working with Hardy"},{"startTime":3270.42,"endTime":3275.42,"body":"in about a hundred years ago,"},{"startTime":3276.42,"endTime":3278.88,"body":"and my understanding"},{"startTime":3276.42,"endTime":3278.88,"body":"of the situation there"},{"startTime":3278.88,"endTime":3283.23,"body":"is that Hardy's initial"},{"startTime":3278.88,"endTime":3283.23,"body":"reaction was that Ramanujan"},{"startTime":3283.23,"endTime":3288.09,"body":"had no idea of proof"},{"startTime":3283.23,"endTime":3288.09,"body":"whatsoever, but nonetheless,"},{"startTime":3288.09,"endTime":3291.845,"body":"he respected his great"},{"startTime":3288.09,"endTime":3291.845,"body":"mathematical intuition,"},{"startTime":3291.845,"endTime":3295.11,"body":"and it's subsequently been"},{"startTime":3291.845,"endTime":3295.11,"body":"found, as I understand it,"},{"startTime":3295.11,"endTime":3299.07,"body":"that many of Ramanujan's"},{"startTime":3295.11,"endTime":3299.07,"body":"statements were actually true,"},{"startTime":3299.07,"endTime":3303.06,"body":"some were false, and some,"},{"startTime":3299.07,"endTime":3303.06,"body":"I think, are still unproven."},{"startTime":3303.06,"endTime":3304.917,"body":"Do you have any comments on that?"},{"startTime":3304.917,"endTime":3306.57,"body":"Was Ramanujan influenced"},{"startTime":3306.57,"endTime":3310.24,"body":"by the Indian traditions of mathematics?"},{"startTime":3310.24,"endTime":3312.133,"body":"- Well, of course, I don't know."},{"startTime":3312.133,"endTime":3314.43,"body":"I haven't worked on Ramanujan's work,"},{"startTime":3314.43,"endTime":3318.33,"body":"and I wouldn't dare to"},{"startTime":3314.43,"endTime":3318.33,"body":"say so, but it would be,"},{"startTime":3318.33,"endTime":3321.511,"body":"wouldn't it be just"},{"startTime":3318.33,"endTime":3321.511,"body":"natural that he had some..."},{"startTime":3321.511,"endTime":3325.65,"body":"I don't, actually, I don't even"},{"startTime":3321.511,"endTime":3325.65,"body":"know his biography exactly."},{"startTime":3325.65,"endTime":3328.05,"body":"He probably, at the end"},{"startTime":3325.65,"endTime":3328.05,"body":"of the 19th century,"},{"startTime":3328.05,"endTime":3331.62,"body":"probably was trained"},{"startTime":3328.05,"endTime":3331.62,"body":"in a kind of curriculum"},{"startTime":3331.62,"endTime":3335.34,"body":"that was available for Indian students,"},{"startTime":3335.34,"endTime":3340.34,"body":"though that would've been in"},{"startTime":3335.34,"endTime":3340.34,"body":"English mathematical manuals,"},{"startTime":3340.53,"endTime":3343.02,"body":"but he could have also had access"},{"startTime":3343.02,"endTime":3346.92,"body":"to other South Indian"},{"startTime":3343.02,"endTime":3346.92,"body":"traditions where he was,"},{"startTime":3346.92,"endTime":3351.92,"body":"but I just wanted underline"},{"startTime":3346.92,"endTime":3351.92,"body":"that East meets West"},{"startTime":3353.34,"endTime":3356.67,"body":"is all along the history"},{"startTime":3353.34,"endTime":3356.67,"body":"of the South Asia,"},{"startTime":3356.67,"endTime":3358.95,"body":"and so, like for instance in Kerala,"},{"startTime":3358.95,"endTime":3363.95,"body":"it seems to me that when we"},{"startTime":3358.95,"endTime":3363.95,"body":"see these proofs coming up"},{"startTime":3364.56,"endTime":3366.81,"body":"with these rules of proportion"},{"startTime":3366.81,"endTime":3368.131,"body":"that are completely different,"},{"startTime":3368.131,"endTime":3372.09,"body":"it gives the feeling that"},{"startTime":3368.131,"endTime":3372.09,"body":"there was some kind of trans,"},{"startTime":3372.09,"endTime":3373.68,"body":"there was some kind of knowledge"},{"startTime":3373.68,"endTime":3377.67,"body":"of other types of mathematical"},{"startTime":3373.68,"endTime":3377.67,"body":"and astronomical traditions"},{"startTime":3377.67,"endTime":3381.3,"body":"that were there, and"},{"startTime":3377.67,"endTime":3381.3,"body":"they probably came from,"},{"startTime":3381.3,"endTime":3383.67,"body":"well, it was during the Mogul Empire,"},{"startTime":3383.67,"endTime":3386.28,"body":"so there was a lot of"},{"startTime":3383.67,"endTime":3386.28,"body":"Persian texts going on,"},{"startTime":3386.28,"endTime":3390.36,"body":"but even before that, you"},{"startTime":3386.28,"endTime":3390.36,"body":"know, in the fifth century,"},{"startTime":3390.36,"endTime":3392.595,"body":"actually some of the first"},{"startTime":3392.595,"endTime":3397.2,"body":"astrological and astronomical"},{"startTime":3392.595,"endTime":3397.2,"body":"texts that we have in Sanskrit"},{"startTime":3397.2,"endTime":3400.08,"body":"bear Hellenistic influence."},{"startTime":3400.08,"endTime":3404.07,"body":"So on the one hand, they think"},{"startTime":3400.08,"endTime":3404.07,"body":"that in India, all the time,"},{"startTime":3404.07,"endTime":3407.293,"body":"there was always all sorts of texts,"},{"startTime":3407.293,"endTime":3411.63,"body":"modes of doing mathematics"},{"startTime":3407.293,"endTime":3411.63,"body":"and thinking about it"},{"startTime":3411.63,"endTime":3414.36,"body":"that came in and were absorbed"},{"startTime":3414.36,"endTime":3417.69,"body":"into all sorts of different ways of doing,"},{"startTime":3417.69,"endTime":3419.22,"body":"and reciprocally, as you know,"},{"startTime":3419.22,"endTime":3421.53,"body":"because, like the idea of trigonometry"},{"startTime":3421.53,"endTime":3425.22,"body":"or part of the arithmetic that"},{"startTime":3421.53,"endTime":3425.22,"body":"was developed in South Asia"},{"startTime":3425.22,"endTime":3426.27,"body":"also traveled."},{"startTime":3426.27,"endTime":3431.163,"body":"So I think we, this is my point, is,"},{"startTime":3432.48,"endTime":3436.14,"body":"does East and West really have a meaning?"},{"startTime":3436.14,"endTime":3438.24,"body":"Isn't it something that we constructed,"},{"startTime":3438.24,"endTime":3443.24,"body":"and that if we hold onto it,"},{"startTime":3438.24,"endTime":3443.24,"body":"we're creating a divided world,"},{"startTime":3444.24,"endTime":3447.033,"body":"by holding down to these categories."},{"startTime":3448.56,"endTime":3449.393,"body":"- Okay, thanks."},{"startTime":3449.393,"endTime":3451.86,"body":"So I'm afraid we do have to stop promptly,"},{"startTime":3451.86,"endTime":3456.27,"body":"but I just want to flag up,"},{"startTime":3451.86,"endTime":3456.27,"body":"if you have enjoyed today,"},{"startTime":3456.27,"endTime":3458.55,"body":"the next sort of mathematical"},{"startTime":3456.27,"endTime":3458.55,"body":"lecture at Gresham"},{"startTime":3458.55,"endTime":3461.1,"body":"will be by the Gresham"},{"startTime":3458.55,"endTime":3461.1,"body":"professor of geometry"},{"startTime":3461.1,"endTime":3462.87,"body":"on the 22nd November,"},{"startTime":3462.87,"endTime":3465.206,"body":"which, so you can look on"},{"startTime":3462.87,"endTime":3465.206,"body":"the Gresham website for that."},{"startTime":3465.206,"endTime":3468.9,"body":"The next BSHN meeting will be our,"},{"startTime":3468.9,"endTime":3471.27,"body":"I think our Christmas meeting,"},{"startTime":3468.9,"endTime":3471.27,"body":"on the 3rd of December."},{"startTime":3471.27,"endTime":3472.833,"body":"Check out our website for that."},{"startTime":3473.85,"endTime":3475.35,"body":"I just want to thank everybody"},{"startTime":3475.35,"endTime":3478.23,"body":"who's been involved in"},{"startTime":3475.35,"endTime":3478.23,"body":"organizing this event today,"},{"startTime":3478.23,"endTime":3479.063,"body":"and finally of course,"},{"startTime":3479.063,"endTime":3481.32,"body":"let's have a gigantic round"},{"startTime":3479.063,"endTime":3481.32,"body":"of applause for our speakers."},{"startTime":3481.32,"endTime":3482.496,"body":"Thank you."},{"startTime":3482.496,"endTime":3485.663,"body":"(audience applauding)"}]}