{"version":"1.0.0","segments":[{"startTime":0.75,"endTime":4.11,"body":"- I'm going to speak a lot"},{"startTime":0.75,"endTime":4.11,"body":"about Euclid and perhaps"},{"startTime":4.11,"endTime":6.42,"body":"the very first word here"},{"startTime":4.11,"endTime":6.42,"body":"ought to be an invention,"},{"startTime":6.42,"endTime":9.09,"body":"as you'll see from some"},{"startTime":6.42,"endTime":9.09,"body":"of the other talks today."},{"startTime":9.09,"endTime":12.787,"body":"However, one of the ideas"},{"startTime":9.09,"endTime":12.787,"body":"that we have about proof"},{"startTime":12.787,"endTime":14.4,"body":"and about demonstration"},{"startTime":14.4,"endTime":16.44,"body":"and how these things fit together tends"},{"startTime":16.44,"endTime":18.48,"body":"to come up to us via Euclid."},{"startTime":18.48,"endTime":21.87,"body":"And so how Euclid is received turns out"},{"startTime":21.87,"endTime":23.13,"body":"to be enormously significant."},{"startTime":23.13,"endTime":26.97,"body":"I'm going to talk first about"},{"startTime":23.13,"endTime":26.97,"body":"a medieval version of Euclid,"},{"startTime":26.97,"endTime":28.89,"body":"some several medieval versions of Euclid,"},{"startTime":28.89,"endTime":31.83,"body":"and then how that coalesces"},{"startTime":28.89,"endTime":31.83,"body":"into the kind of Euclid"},{"startTime":31.83,"endTime":35.04,"body":"we recognize and gain a"},{"startTime":31.83,"endTime":35.04,"body":"lot of inspiration today"},{"startTime":35.04,"endTime":37.02,"body":"over the course of the 16th century."},{"startTime":37.02,"endTime":38.67,"body":"And I'm going to suggest"},{"startTime":37.02,"endTime":38.67,"body":"in a way that means"},{"startTime":38.67,"endTime":39.801,"body":"we should have a looser vision"},{"startTime":39.801,"endTime":43.8,"body":"of proof than sometimes we bring."},{"startTime":43.8,"endTime":45.51,"body":"Now what's a proof?"},{"startTime":45.51,"endTime":47.81,"body":"I think that's a valuable"},{"startTime":45.51,"endTime":47.81,"body":"thing to start with."},{"startTime":49.02,"endTime":50.16,"body":"I don't want to get too technical."},{"startTime":50.16,"endTime":52.633,"body":"What I want you to do is cast"},{"startTime":50.16,"endTime":52.633,"body":"your minds back to the times"},{"startTime":52.633,"endTime":56.01,"body":"when you yourself have felt"},{"startTime":56.01,"endTime":58.47,"body":"the power of going through a proof."},{"startTime":58.47,"endTime":62.73,"body":"Your mind has been drawn,"},{"startTime":58.47,"endTime":62.73,"body":"given a few things,"},{"startTime":62.73,"endTime":64.38,"body":"but then using those things,"},{"startTime":64.38,"endTime":68.13,"body":"building towards a statement"},{"startTime":64.38,"endTime":68.13,"body":"where everything locks"},{"startTime":68.13,"endTime":69.42,"body":"into place and you feel like"},{"startTime":69.42,"endTime":71.64,"body":"the conclusion you've reached"},{"startTime":71.64,"endTime":74.34,"body":"is balanced precisely on a set"},{"startTime":74.34,"endTime":76.623,"body":"of givens that you started out with."},{"startTime":77.82,"endTime":82.74,"body":"Now that vision of proof is what gives us"},{"startTime":82.74,"endTime":87.74,"body":"the sensation of certainty of"},{"startTime":82.74,"endTime":87.74,"body":"a universality, perhaps even."},{"startTime":88.14,"endTime":89.73,"body":"And that's something like"},{"startTime":89.73,"endTime":91.71,"body":"what we can see when we turn to Aristotle"},{"startTime":91.71,"endTime":93.903,"body":"as he's describing the process of proof."},{"startTime":95.31,"endTime":97.786,"body":"Reviel Netz when he's talking"},{"startTime":95.31,"endTime":97.786,"body":"about ancient mathematics"},{"startTime":97.786,"endTime":100.92,"body":"focuses a little bit on this sensation."},{"startTime":100.92,"endTime":103.89,"body":"The issue is not what made"},{"startTime":100.92,"endTime":103.89,"body":"Greek mathematics valid."},{"startTime":103.89,"endTime":106.29,"body":"The question is rather what made it felt"},{"startTime":106.29,"endTime":109.89,"body":"to be valid, for felt to"},{"startTime":106.29,"endTime":109.89,"body":"be valid it certainly was."},{"startTime":109.89,"endTime":113.175,"body":"So this idea of the sensation"},{"startTime":109.89,"endTime":113.175,"body":"of certainty that comes"},{"startTime":113.175,"endTime":117.69,"body":"to us from talking through"},{"startTime":113.175,"endTime":117.69,"body":"a certain kind of narrative,"},{"startTime":117.69,"endTime":120.693,"body":"that's what I want to think"},{"startTime":117.69,"endTime":120.693,"body":"about a little bit today."},{"startTime":123.06,"endTime":126.3,"body":"Aristotle gave Renaissance folk,"},{"startTime":126.3,"endTime":128.28,"body":"medieval folk the standard definition"},{"startTime":128.28,"endTime":130.56,"body":"of what counts as a geometrical proof."},{"startTime":130.56,"endTime":133.335,"body":"He's basing this a little"},{"startTime":130.56,"endTime":133.335,"body":"bit on something like Euclid,"},{"startTime":133.335,"endTime":135.45,"body":"and I think you could recognize some"},{"startTime":135.45,"endTime":137.4,"body":"of what I've just described there,"},{"startTime":137.4,"endTime":139.65,"body":"The sense that you start out with axioms"},{"startTime":139.65,"endTime":142.118,"body":"and then you move by"},{"startTime":139.65,"endTime":142.118,"body":"inexorable steps towards"},{"startTime":142.118,"endTime":145.233,"body":"a conclusion that feels"},{"startTime":142.118,"endTime":145.233,"body":"like it has been proven."},{"startTime":147.27,"endTime":149.957,"body":"And Geoffrey Lloyd, for example,"},{"startTime":149.957,"endTime":153.36,"body":"points out that even Aristotle,"},{"startTime":153.36,"endTime":155.73,"body":"that's what he means by the Organon here."},{"startTime":155.73,"endTime":160.2,"body":"And Euclid, even when we"},{"startTime":155.73,"endTime":160.2,"body":"read these sorts of figures,"},{"startTime":160.2,"endTime":163.5,"body":"we're skewed a little bit in"},{"startTime":160.2,"endTime":163.5,"body":"how we approach demonstration."},{"startTime":163.5,"endTime":166.92,"body":"We need a bigger version of"},{"startTime":163.5,"endTime":166.92,"body":"demonstration in order even"},{"startTime":166.92,"endTime":169.95,"body":"to understand the Greek materials we kind"},{"startTime":169.95,"endTime":173.283,"body":"of think leads us towards"},{"startTime":169.95,"endTime":173.283,"body":"our notion of demonstration."},{"startTime":176.25,"endTime":178.56,"body":"This is actually a point"},{"startTime":176.25,"endTime":178.56,"body":"that Descartes expanded"},{"startTime":178.56,"endTime":180.78,"body":"upon at different points as well."},{"startTime":180.78,"endTime":184.02,"body":"So Descartes is someone who"},{"startTime":180.78,"endTime":184.02,"body":"thought he had a very good,"},{"startTime":184.02,"endTime":188.163,"body":"strong definition of"},{"startTime":184.02,"endTime":188.163,"body":"demonstration, of proving things,"},{"startTime":190.075,"endTime":195.075,"body":"but he complained about"},{"startTime":190.075,"endTime":195.075,"body":"certain sorts of people who use"},{"startTime":195.166,"endTime":198.18,"body":"the narrative practices of proof,"},{"startTime":198.18,"endTime":200.73,"body":"who use the language of proof."},{"startTime":200.73,"endTime":203.31,"body":"And this is what he thought"},{"startTime":200.73,"endTime":203.31,"body":"was the language of proof."},{"startTime":203.31,"endTime":205.2,"body":"It's actually quite close"},{"startTime":203.31,"endTime":205.2,"body":"to what we might see."},{"startTime":205.2,"endTime":206.31,"body":"You have some definitions,"},{"startTime":206.31,"endTime":209.007,"body":"you have some postulates,"},{"startTime":206.31,"endTime":209.007,"body":"axioms or common notions,"},{"startTime":209.007,"endTime":212.7,"body":"and then you use those"},{"startTime":209.007,"endTime":212.7,"body":"to prove, let's say,"},{"startTime":212.7,"endTime":215.46,"body":"annunciations problems, theorems,"},{"startTime":215.46,"endTime":217.5,"body":"demonstrations and corollaries and so on."},{"startTime":217.5,"endTime":218.733,"body":"So that kind of anatomy,"},{"startTime":219.66,"endTime":224.66,"body":"that language of describing"},{"startTime":219.66,"endTime":224.66,"body":"what's going on in geometrical"},{"startTime":225.18,"endTime":226.766,"body":"narration, he says,"},{"startTime":226.766,"endTime":230.31,"body":"also can lead us astray so that students,"},{"startTime":230.31,"endTime":234.254,"body":"when they see something parsed"},{"startTime":230.31,"endTime":234.254,"body":"out as axioms and so on,"},{"startTime":234.254,"endTime":237.18,"body":"think that they're be being shown a proof,"},{"startTime":237.18,"endTime":238.563,"body":"but maybe not always."},{"startTime":239.52,"endTime":242.22,"body":"So that's all of this is just a way,"},{"startTime":242.22,"endTime":244.288,"body":"a preamble, a way to say to you,"},{"startTime":244.288,"endTime":247.83,"body":"the way things are laid out"},{"startTime":244.288,"endTime":247.83,"body":"on the page and the narrative"},{"startTime":247.83,"endTime":250.89,"body":"practices that we use to"},{"startTime":247.83,"endTime":250.89,"body":"lead us from one point"},{"startTime":250.89,"endTime":253.59,"body":"to another in the context of mathematics,"},{"startTime":253.59,"endTime":257.7,"body":"often we perceive as a very"},{"startTime":253.59,"endTime":257.7,"body":"particular form of proof."},{"startTime":257.7,"endTime":259.62,"body":"I want to trouble that"},{"startTime":257.7,"endTime":259.62,"body":"a little bit by thinking"},{"startTime":259.62,"endTime":263.13,"body":"a little bit about who Euclid was"},{"startTime":263.13,"endTime":265.5,"body":"in the medieval period and how Euclid"},{"startTime":265.5,"endTime":267.9,"body":"turned into something particular"},{"startTime":267.9,"endTime":269.94,"body":"at two different moments"},{"startTime":267.9,"endTime":269.94,"body":"in the Renaissance."},{"startTime":269.94,"endTime":271.77,"body":"So the long view is the medieval period,"},{"startTime":271.77,"endTime":275.19,"body":"and then we'll move to"},{"startTime":271.77,"endTime":275.19,"body":"different ideas about invention"},{"startTime":275.19,"endTime":278.343,"body":"that are linked to proof"},{"startTime":275.19,"endTime":278.343,"body":"in the Renaissance."},{"startTime":279.51,"endTime":281.043,"body":"So let's step back."},{"startTime":286.98,"endTime":291.36,"body":"How was Euclid actually"},{"startTime":286.98,"endTime":291.36,"body":"encountered by medieval readers?"},{"startTime":291.36,"endTime":295.83,"body":"I mean, this is one way,"},{"startTime":291.36,"endTime":295.83,"body":"if you were Greek speaking,"},{"startTime":295.83,"endTime":300.54,"body":"you might en encounter"},{"startTime":295.83,"endTime":300.54,"body":"Euclid in the ninth century."},{"startTime":300.54,"endTime":303.72,"body":"And something that to notice here is"},{"startTime":303.72,"endTime":306.372,"body":"what I'm calling the meson page,"},{"startTime":306.372,"endTime":309.15,"body":"the way the page is organized"},{"startTime":306.372,"endTime":309.15,"body":"and how it's very hard"},{"startTime":309.15,"endTime":312.42,"body":"to tell the difference"},{"startTime":309.15,"endTime":312.42,"body":"between the enunciation"},{"startTime":312.42,"endTime":316.86,"body":"or thing to be proved, the statement,"},{"startTime":316.86,"endTime":319.23,"body":"that you're working with and the rest"},{"startTime":319.23,"endTime":322.803,"body":"of the matter that you're"},{"startTime":319.23,"endTime":322.803,"body":"using to make that proof work."},{"startTime":324.99,"endTime":326.55,"body":"There are a few other"},{"startTime":324.99,"endTime":326.55,"body":"things to say about it,"},{"startTime":326.55,"endTime":328.805,"body":"but this picture, you"},{"startTime":326.55,"endTime":328.805,"body":"can't see the details."},{"startTime":328.805,"endTime":330.93,"body":"We don't need to read the"},{"startTime":328.805,"endTime":330.93,"body":"Greek together anyways"},{"startTime":330.93,"endTime":333.433,"body":"to get the point, isn't divided."},{"startTime":333.433,"endTime":336.0,"body":"Other early manuscripts"},{"startTime":333.433,"endTime":336.0,"body":"we have aren't divided"},{"startTime":336.0,"endTime":338.309,"body":"very carefully around annunciations."},{"startTime":338.309,"endTime":341.49,"body":"And there's this other bit of"},{"startTime":338.309,"endTime":341.49,"body":"dense text that's around here,"},{"startTime":341.49,"endTime":343.8,"body":"which should be quite interesting to us,"},{"startTime":343.8,"endTime":347.88,"body":"which turns out to be"},{"startTime":343.8,"endTime":347.88,"body":"commentary or descriptions"},{"startTime":347.88,"endTime":349.62,"body":"or alternative ways of thinking"},{"startTime":349.62,"endTime":353.49,"body":"about what's going on"},{"startTime":349.62,"endTime":353.49,"body":"within the text itself."},{"startTime":353.49,"endTime":357.03,"body":"Now, the question I want"},{"startTime":353.49,"endTime":357.03,"body":"you to think about is,"},{"startTime":357.03,"endTime":359.88,"body":"which of this is Euclid?"},{"startTime":359.88,"endTime":362.76,"body":"Who counts when you're"},{"startTime":359.88,"endTime":362.76,"body":"looking at this text,"},{"startTime":362.76,"endTime":364.803,"body":"which bits come from Euclid directly?"},{"startTime":366.18,"endTime":367.83,"body":"Well, if you were a medieval reader,"},{"startTime":367.83,"endTime":370.35,"body":"you might think that Euclid is the author"},{"startTime":370.35,"endTime":372.36,"body":"of only very specific parts of the text,"},{"startTime":372.36,"endTime":374.94,"body":"partly because one of"},{"startTime":372.36,"endTime":374.94,"body":"the versions of Euclid"},{"startTime":374.94,"endTime":379.59,"body":"that's transmitted is the"},{"startTime":374.94,"endTime":379.59,"body":"translation from Boethius."},{"startTime":379.59,"endTime":384.59,"body":"And in it he suggests to us"},{"startTime":379.59,"endTime":384.59,"body":"that only the annunciations,"},{"startTime":384.6,"endTime":386.7,"body":"only the corollaries, the theorems,"},{"startTime":386.7,"endTime":390.51,"body":"all those sorts of things"},{"startTime":386.7,"endTime":390.51,"body":"are in fact Euclid."},{"startTime":390.51,"endTime":394.17,"body":"The rest of it is optional"},{"startTime":390.51,"endTime":394.17,"body":"extra because it's a gloss."},{"startTime":394.17,"endTime":397.02,"body":"And if you read the versions of Euclid"},{"startTime":397.02,"endTime":398.793,"body":"that come to us in this form,"},{"startTime":400.65,"endTime":403.628,"body":"you often either skip proofs entirely."},{"startTime":403.628,"endTime":405.66,"body":"I love this example"},{"startTime":403.628,"endTime":405.66,"body":"because you have images"},{"startTime":405.66,"endTime":407.73,"body":"that are supposed to show you"},{"startTime":407.73,"endTime":409.38,"body":"what's going on in the statement,"},{"startTime":410.638,"endTime":413.88,"body":"or you have alternative glosses"},{"startTime":413.88,"endTime":416.85,"body":"from medieval commentators"},{"startTime":413.88,"endTime":416.85,"body":"that make this up."},{"startTime":416.85,"endTime":419.417,"body":"This turns out, as Euclid by the 12th"},{"startTime":419.417,"endTime":422.34,"body":"and 13th century turns into"},{"startTime":419.417,"endTime":422.34,"body":"other sorts of translations"},{"startTime":422.34,"endTime":425.43,"body":"in the Latin tradition, to be the way"},{"startTime":425.43,"endTime":427.8,"body":"that people approach"},{"startTime":425.43,"endTime":427.8,"body":"the full text of Euclid"},{"startTime":427.8,"endTime":429.78,"body":"when it becomes available."},{"startTime":429.78,"endTime":431.52,"body":"So if you look at, for example,"},{"startTime":431.52,"endTime":434.28,"body":"the most significant"},{"startTime":431.52,"endTime":434.28,"body":"and the most widely read"},{"startTime":434.28,"endTime":436.8,"body":"and the most influential version of Euclid"},{"startTime":436.8,"endTime":439.437,"body":"in the Middle Ages, \"Campanus of Novara\","},{"startTime":440.34,"endTime":441.87,"body":"he's taken the whole lot of"},{"startTime":441.87,"endTime":444.72,"body":"what we would think of as the proof."},{"startTime":444.72,"endTime":448.71,"body":"And he said, I've got new"},{"startTime":444.72,"endTime":448.71,"body":"versions of that in many cases,"},{"startTime":448.71,"endTime":449.82,"body":"or very often,"},{"startTime":449.82,"endTime":452.52,"body":"he'll come around and he'll"},{"startTime":449.82,"endTime":452.52,"body":"just tweak the original"},{"startTime":452.52,"endTime":457.08,"body":"because he sees the annunciations"},{"startTime":452.52,"endTime":457.08,"body":"here as the main bit."},{"startTime":457.08,"endTime":458.94,"body":"That's the bit that's actually Euclid,"},{"startTime":458.94,"endTime":462.63,"body":"and the rest of it is commentary"},{"startTime":458.94,"endTime":462.63,"body":"functionally, it's a gloss,"},{"startTime":462.63,"endTime":464.34,"body":"it's a thing that's used to explain"},{"startTime":464.34,"endTime":466.383,"body":"what's really going on in the main text."},{"startTime":469.02,"endTime":471.15,"body":"Some of that shifts then,"},{"startTime":471.15,"endTime":472.41,"body":"you can see some interesting things"},{"startTime":472.41,"endTime":475.77,"body":"happen with the very"},{"startTime":472.41,"endTime":475.77,"body":"first editions in print"},{"startTime":475.77,"endTime":478.233,"body":"of Euclid beginning in the 15th century."},{"startTime":479.82,"endTime":480.873,"body":"So,"},{"startTime":482.64,"endTime":484.05,"body":"first, this is the first version"},{"startTime":484.05,"endTime":485.703,"body":"of \"Campanus\" that's printed,"},{"startTime":487.26,"endTime":488.673,"body":"that's the medieval version."},{"startTime":490.8,"endTime":494.91,"body":"There's another version,"},{"startTime":490.8,"endTime":494.91,"body":"Zamberti, that is produced,"},{"startTime":494.91,"endTime":496.923,"body":"which is retranslated from the Greek."},{"startTime":499.86,"endTime":500.693,"body":"The \"Campanus\","},{"startTime":500.693,"endTime":503.64,"body":"remember lots of bits that"},{"startTime":500.693,"endTime":503.64,"body":"aren't actually transmitted"},{"startTime":503.64,"endTime":505.92,"body":"from antiquity but are considered glosses."},{"startTime":505.92,"endTime":508.464,"body":"Zamberti offers a counter perspective."},{"startTime":508.464,"endTime":512.948,"body":"He's going to suggest to us that in fact,"},{"startTime":512.948,"endTime":516.75,"body":"Euclid's authored also a"},{"startTime":512.948,"endTime":516.75,"body":"whole bunch of demonstrations,"},{"startTime":516.75,"endTime":519.6,"body":"the things that we think of as proofs."},{"startTime":519.6,"endTime":523.8,"body":"When he does so, he translates, I mean,"},{"startTime":523.8,"endTime":525.69,"body":"there's a number of issues"},{"startTime":523.8,"endTime":525.69,"body":"with his translation,"},{"startTime":525.69,"endTime":527.22,"body":"one of which is that he hasn't always"},{"startTime":527.22,"endTime":528.33,"body":"understood the Greek very well,"},{"startTime":528.33,"endTime":531.417,"body":"and he always doesn't always"},{"startTime":528.33,"endTime":531.417,"body":"understand the maths very well."},{"startTime":531.417,"endTime":534.12,"body":"And as a result, it turns out,"},{"startTime":534.12,"endTime":537.03,"body":"as someone like Luca Pacioli"},{"startTime":534.12,"endTime":537.03,"body":"points out that \"Campanus\""},{"startTime":537.03,"endTime":541.803,"body":"is actually good maths,"},{"startTime":537.03,"endTime":541.803,"body":"Zamberti is good Latin,"},{"startTime":542.76,"endTime":545.46,"body":"but bad maths, right?"},{"startTime":545.46,"endTime":548.94,"body":"So you can see the problems"},{"startTime":545.46,"endTime":548.94,"body":"that are starting to arise here."},{"startTime":548.94,"endTime":551.1,"body":"Authorship or your assumptions"},{"startTime":551.1,"endTime":554.61,"body":"about what Euclid means,"},{"startTime":554.61,"endTime":557.7,"body":"what bits of the text"},{"startTime":554.61,"endTime":557.7,"body":"belong to Euclid are going"},{"startTime":557.7,"endTime":559.44,"body":"to start having implications"},{"startTime":559.44,"endTime":562.293,"body":"for how you actually start"},{"startTime":559.44,"endTime":562.293,"body":"to work out the maths."},{"startTime":568.53,"endTime":569.37,"body":"Not only that,"},{"startTime":569.37,"endTime":571.71,"body":"you start to have plural versions"},{"startTime":571.71,"endTime":573.75,"body":"of what shall we say are Euclid, right?"},{"startTime":573.75,"endTime":576.15,"body":"You have different, not only translations,"},{"startTime":576.15,"endTime":577.65,"body":"but different textual traditions"},{"startTime":577.65,"endTime":579.72,"body":"in which some of the bits"},{"startTime":577.65,"endTime":579.72,"body":"aren't Euclid at all."},{"startTime":579.72,"endTime":581.85,"body":"And to sort out that sort of problem,"},{"startTime":581.85,"endTime":584.25,"body":"you get something like this version"},{"startTime":584.25,"endTime":588.81,"body":"which sets Pacioli,"},{"startTime":584.25,"endTime":588.81,"body":"well \"Campanus\", sorry,"},{"startTime":588.81,"endTime":592.14,"body":"alongside Zamberti next to each other,"},{"startTime":592.14,"endTime":595.74,"body":"so that you as the reader can"},{"startTime":592.14,"endTime":595.74,"body":"productively work your way"},{"startTime":595.74,"endTime":598.23,"body":"back and forth between these"},{"startTime":595.74,"endTime":598.23,"body":"things in order to sort out"},{"startTime":598.23,"endTime":601.293,"body":"for yourself what's really"},{"startTime":598.23,"endTime":601.293,"body":"going on in the text."},{"startTime":604.17,"endTime":606.51,"body":"Now, the overriding message here"},{"startTime":606.51,"endTime":609.51,"body":"is that mathematical narrative,"},{"startTime":609.51,"endTime":611.76,"body":"the thing that you really need to follow"},{"startTime":611.76,"endTime":615.45,"body":"for these people is what"},{"startTime":611.76,"endTime":615.45,"body":"happens in the annunciations,"},{"startTime":615.45,"endTime":617.46,"body":"not what's happening below the line,"},{"startTime":617.46,"endTime":619.563,"body":"not what's happening in the demonstration."},{"startTime":620.398,"endTime":623.67,"body":"That proof demonstration is explanatory"},{"startTime":623.67,"endTime":626.61,"body":"and it's additional,"},{"startTime":623.67,"endTime":626.61,"body":"and it's even optional."},{"startTime":626.61,"endTime":629.49,"body":"So if you're doing geometry,"},{"startTime":626.61,"endTime":629.49,"body":"you can do away with that."},{"startTime":629.49,"endTime":632.37,"body":"And you get a whole lot"},{"startTime":629.49,"endTime":632.37,"body":"of textbooks in the period"},{"startTime":632.37,"endTime":635.91,"body":"where that idea of geometry"},{"startTime":632.37,"endTime":635.91,"body":"is actually put to work,"},{"startTime":635.91,"endTime":639.3,"body":"where you remove all the"},{"startTime":635.91,"endTime":639.3,"body":"demonstrations and you just move"},{"startTime":639.3,"endTime":642.09,"body":"with the sensation of being able to follow"},{"startTime":642.09,"endTime":644.883,"body":"a lot of descriptions of"},{"startTime":642.09,"endTime":644.883,"body":"mathematical contents."},{"startTime":646.11,"endTime":647.91,"body":"Turns out that's actually quite productive"},{"startTime":647.91,"endTime":649.32,"body":"because it allows you to do kind"},{"startTime":649.32,"endTime":651.3,"body":"of undisciplined things with the maths,"},{"startTime":651.3,"endTime":653.73,"body":"you can start applying maths"},{"startTime":651.3,"endTime":653.73,"body":"to all kinds of practical"},{"startTime":653.73,"endTime":655.56,"body":"problems in ways that maybe"},{"startTime":655.56,"endTime":658.413,"body":"the proof distracted you"},{"startTime":655.56,"endTime":658.413,"body":"from or didn't let you do."},{"startTime":659.46,"endTime":662.76,"body":"So this is just one example"},{"startTime":659.46,"endTime":662.76,"body":"of a set of authors,"},{"startTime":662.76,"endTime":665.763,"body":"Lefevre d'Etaples, Clichtove, Bovelles,"},{"startTime":666.658,"endTime":669.45,"body":"and there's also others"},{"startTime":666.658,"endTime":669.45,"body":"involved in making this volume"},{"startTime":669.45,"endTime":671.37,"body":"where they're basically trying to come up"},{"startTime":671.37,"endTime":672.66,"body":"with a mathematical curriculum"},{"startTime":672.66,"endTime":674.64,"body":"that allows you to do precisely that."},{"startTime":674.64,"endTime":678.84,"body":"Furthermore, that sort of"},{"startTime":674.64,"endTime":678.84,"body":"turns into a kind of a physics."},{"startTime":678.84,"endTime":682.11,"body":"So Gregor Reisch's \"Margarita"},{"startTime":678.84,"endTime":682.11,"body":"Philosophica\" is a really,"},{"startTime":682.11,"endTime":684.06,"body":"it's a best selling textbook."},{"startTime":684.06,"endTime":685.65,"body":"He starts to do some similar bits,"},{"startTime":685.65,"endTime":687.3,"body":"or rather, he includes appendices"},{"startTime":687.3,"endTime":688.98,"body":"where they're doing that kind of math."},{"startTime":688.98,"endTime":691.615,"body":"So every student going"},{"startTime":688.98,"endTime":691.615,"body":"through universities learning"},{"startTime":691.615,"endTime":694.443,"body":"how to do math without the demonstrations."},{"startTime":696.12,"endTime":699.3,"body":"And I was thinking that I had a picture"},{"startTime":699.3,"endTime":701.491,"body":"of a musical text one way,"},{"startTime":701.491,"endTime":705.48,"body":"one implication of this way of"},{"startTime":701.491,"endTime":705.48,"body":"thinking about demonstration"},{"startTime":705.48,"endTime":708.15,"body":"as an optional thing is"},{"startTime":705.48,"endTime":708.15,"body":"that you can start to reason"},{"startTime":708.15,"endTime":710.49,"body":"between different mathematical"},{"startTime":708.15,"endTime":710.49,"body":"disciplines in new ways."},{"startTime":710.49,"endTime":712.89,"body":"So this becomes a fertile"},{"startTime":710.49,"endTime":712.89,"body":"sort of way of thinking"},{"startTime":712.89,"endTime":715.083,"body":"about what's going on in mathematics."},{"startTime":716.31,"endTime":718.08,"body":"One implication I want you"},{"startTime":718.08,"endTime":722.07,"body":"to then draw based on what I've just said,"},{"startTime":722.07,"endTime":724.713,"body":"is that for figures like Oronce Fine,"},{"startTime":726.09,"endTime":729.81,"body":"proof becomes interchangeable"},{"startTime":726.09,"endTime":729.81,"body":"with practice."},{"startTime":729.81,"endTime":733.32,"body":"So when someone like Oronce"},{"startTime":729.81,"endTime":733.32,"body":"Fine gives you a manual"},{"startTime":733.32,"endTime":734.91,"body":"for learning how to do mathematics,"},{"startTime":734.91,"endTime":736.74,"body":"sometimes he's not going to say,"},{"startTime":736.74,"endTime":738.36,"body":"now you need to apply the proof so"},{"startTime":738.36,"endTime":739.98,"body":"that you can understand the enunciation."},{"startTime":739.98,"endTime":742.38,"body":"Instead, what he's going to do is give"},{"startTime":742.38,"endTime":744.42,"body":"you some homework and give you a lot"},{"startTime":744.42,"endTime":746.37,"body":"of constructions to do and ask you"},{"startTime":746.37,"endTime":747.573,"body":"to make an instrument."},{"startTime":748.8,"endTime":749.88,"body":"And in so doing,"},{"startTime":749.88,"endTime":753.78,"body":"he's asking you to embody"},{"startTime":749.88,"endTime":753.78,"body":"what's going on in that thing"},{"startTime":753.78,"endTime":756.45,"body":"that you're supposed to prove"},{"startTime":753.78,"endTime":756.45,"body":"and not think it through,"},{"startTime":756.45,"endTime":757.38,"body":"but work it through."},{"startTime":757.38,"endTime":760.41,"body":"Now I'm making an overly"},{"startTime":757.38,"endTime":760.41,"body":"fine distinction, right?"},{"startTime":760.41,"endTime":763.096,"body":"Maybe working it through"},{"startTime":760.41,"endTime":763.096,"body":"as well here probably later"},{"startTime":763.096,"endTime":765.24,"body":"has something to do with"},{"startTime":763.096,"endTime":765.24,"body":"thinking it through as well."},{"startTime":765.24,"endTime":767.85,"body":"But here, I just want you to see what"},{"startTime":767.85,"endTime":771.33,"body":"this allows you to do when you start"},{"startTime":771.33,"endTime":773.67,"body":"thinking through the idea of proof"},{"startTime":773.67,"endTime":776.043,"body":"as a gloss or as an explanation."},{"startTime":777.33,"endTime":782.33,"body":"A second implication is"},{"startTime":777.33,"endTime":782.33,"body":"one that I find really,"},{"startTime":783.03,"endTime":786.21,"body":"really evocative, which is that instead"},{"startTime":786.21,"endTime":789.27,"body":"we tend to think of proofs as elegant"},{"startTime":789.27,"endTime":792.36,"body":"when they're cut down, when they're spare."},{"startTime":792.36,"endTime":795.03,"body":"But this creates a different"},{"startTime":792.36,"endTime":795.03,"body":"mathematical style,"},{"startTime":795.03,"endTime":799.2,"body":"and it's a mathematics"},{"startTime":795.03,"endTime":799.2,"body":"of abundance, of copia."},{"startTime":799.2,"endTime":802.075,"body":"If you can do one thing three ways,"},{"startTime":802.075,"endTime":804.96,"body":"do it three ways instead"},{"startTime":802.075,"endTime":804.96,"body":"of one way, right,"},{"startTime":804.96,"endTime":807.33,"body":"which is runs counters"},{"startTime":804.96,"endTime":807.33,"body":"to some of the intuitions"},{"startTime":807.33,"endTime":810.243,"body":"that we often bring to things"},{"startTime":807.33,"endTime":810.243,"body":"like geometrical proof."},{"startTime":811.35,"endTime":815.55,"body":"It means that if you can"},{"startTime":811.35,"endTime":815.55,"body":"offer a different modality"},{"startTime":815.55,"endTime":818.91,"body":"alongside the first modality,"},{"startTime":815.55,"endTime":818.91,"body":"you're going to do that."},{"startTime":818.91,"endTime":823.77,"body":"So for example, when Lefevre d'Etaples"},{"startTime":823.77,"endTime":825.51,"body":"produces this sort of textbook,"},{"startTime":825.51,"endTime":827.597,"body":"this is the one you've seen"},{"startTime":825.51,"endTime":827.597,"body":"already where he's set multiple"},{"startTime":827.597,"endTime":831.09,"body":"additions next to each other"},{"startTime":827.597,"endTime":831.09,"body":"in order to understand Euclid,"},{"startTime":831.09,"endTime":833.37,"body":"the best option for him is to provide"},{"startTime":833.37,"endTime":837.771,"body":"all the versions of Euclid,"},{"startTime":833.37,"endTime":837.771,"body":"all the extra demonstrations,"},{"startTime":837.771,"endTime":841.233,"body":"because that's what's going to"},{"startTime":837.771,"endTime":841.233,"body":"benefit you best as a reader."},{"startTime":842.43,"endTime":844.62,"body":"When he writes about music,"},{"startTime":844.62,"endTime":847.26,"body":"this is the sheet I was"},{"startTime":844.62,"endTime":847.26,"body":"looking for earlier,"},{"startTime":847.26,"endTime":848.7,"body":"he doesn't give you,"},{"startTime":848.7,"endTime":851.05,"body":"he gives you an explanation"},{"startTime":848.7,"endTime":851.05,"body":"of how to make the instrument"},{"startTime":851.05,"endTime":853.837,"body":"in certain places as"},{"startTime":851.05,"endTime":853.837,"body":"well as an explanation"},{"startTime":853.837,"endTime":857.76,"body":"for how to reason mathematically"},{"startTime":857.76,"endTime":861.063,"body":"through the relationship between"},{"startTime":857.76,"endTime":861.063,"body":"different sorts of tones."},{"startTime":862.11,"endTime":864.15,"body":"And I think that's significant enough"},{"startTime":864.15,"endTime":867.564,"body":"to pause over because this matches"},{"startTime":867.564,"endTime":870.27,"body":"a set of habits that if"},{"startTime":867.564,"endTime":870.27,"body":"we look to renaissance"},{"startTime":870.27,"endTime":873.39,"body":"literature are everywhere"},{"startTime":870.27,"endTime":873.39,"body":"in the culture, right?"},{"startTime":873.39,"endTime":876.27,"body":"So one thing that we've"},{"startTime":873.39,"endTime":876.27,"body":"learned over the last 40 years"},{"startTime":876.27,"endTime":880.35,"body":"is how much copia, Latin"},{"startTime":876.27,"endTime":880.35,"body":"word for abundance,"},{"startTime":880.35,"endTime":883.29,"body":"is one of the cardinal virtues of learning"},{"startTime":883.29,"endTime":886.11,"body":"how to write well and how to think well"},{"startTime":886.11,"endTime":888.663,"body":"in renaissance kind of learned culture."},{"startTime":889.59,"endTime":892.53,"body":"And the thing that I want to point out is,"},{"startTime":892.53,"endTime":896.29,"body":"by seeing mathematics as in"},{"startTime":892.53,"endTime":896.29,"body":"its kind of narrow tidal form"},{"startTime":897.3,"endTime":900.66,"body":"in which demonstration"},{"startTime":897.3,"endTime":900.66,"body":"becomes not just an exercise,"},{"startTime":900.66,"endTime":902.67,"body":"it's stripping things"},{"startTime":900.66,"endTime":902.67,"body":"away until you can finally"},{"startTime":902.67,"endTime":904.35,"body":"see the narrow truth of the matter,"},{"startTime":904.35,"endTime":907.14,"body":"but instead building up"},{"startTime":904.35,"endTime":907.14,"body":"a whole lot of material"},{"startTime":907.14,"endTime":911.49,"body":"so that you can grow"},{"startTime":907.14,"endTime":911.49,"body":"and make different kinds"},{"startTime":911.49,"endTime":913.21,"body":"of connections between disciplines"},{"startTime":914.22,"endTime":918.21,"body":"that fits with a vision"},{"startTime":914.22,"endTime":918.21,"body":"of literary invention"},{"startTime":918.21,"endTime":920.25,"body":"that was rampant in the Renaissance."},{"startTime":920.25,"endTime":923.61,"body":"And Terrence Cave would"},{"startTime":920.25,"endTime":923.61,"body":"be a starting point,"},{"startTime":923.61,"endTime":926.47,"body":"his book on a Erasmus"},{"startTime":923.61,"endTime":926.47,"body":"and the Cornucopian text"},{"startTime":927.6,"endTime":930.718,"body":"for thinking through that sort of method,"},{"startTime":930.718,"endTime":932.673,"body":"that narrative style, shall we say."},{"startTime":942.556,"endTime":944.43,"body":"So I've promised to describe a little bit"},{"startTime":944.43,"endTime":946.155,"body":"of how that shifts in, again,"},{"startTime":946.155,"endTime":948.66,"body":"maybe the best title is an invention"},{"startTime":948.66,"endTime":950.763,"body":"of proof or something like that."},{"startTime":951.84,"endTime":953.58,"body":"What I mean by that is"},{"startTime":953.58,"endTime":956.22,"body":"how do we narrow this"},{"startTime":953.58,"endTime":956.22,"body":"world of multiple options,"},{"startTime":956.22,"endTime":957.245,"body":"if you like,"},{"startTime":957.245,"endTime":961.56,"body":"to the kind of more stripped"},{"startTime":957.245,"endTime":961.56,"body":"down version of thinking"},{"startTime":961.56,"endTime":963.27,"body":"about mathematical narration"},{"startTime":963.27,"endTime":965.73,"body":"that we associate with Descartes,"},{"startTime":965.73,"endTime":969.69,"body":"and that we associate with"},{"startTime":965.73,"endTime":969.69,"body":"geometry when we read it now,"},{"startTime":969.69,"endTime":973.62,"body":"in which Euclid still bears"},{"startTime":969.69,"endTime":973.62,"body":"a lot of responsibility"},{"startTime":973.62,"endTime":978.393,"body":"for looking like an exemplar"},{"startTime":973.62,"endTime":978.393,"body":"of demonstrative reasoning."},{"startTime":980.49,"endTime":983.02,"body":"And there I want to"},{"startTime":980.49,"endTime":983.02,"body":"think a little bit about"},{"startTime":984.0,"endTime":985.98,"body":"what two authors, but the main one,"},{"startTime":985.98,"endTime":988.173,"body":"and the first one is Federico Commandino."},{"startTime":989.64,"endTime":990.473,"body":"So,"},{"startTime":992.67,"endTime":994.53,"body":"he is a scholar in the household"},{"startTime":994.53,"endTime":996.693,"body":"of the Duke of Verbenone in Italy."},{"startTime":997.556,"endTime":1000.68,"body":"I talked earlier about"},{"startTime":997.556,"endTime":1000.68,"body":"a bunch of collaborators"},{"startTime":1000.68,"endTime":1003.306,"body":"around someone like"},{"startTime":1000.68,"endTime":1003.306,"body":"Jacques Lefevre d'Etaples,"},{"startTime":1003.306,"endTime":1006.199,"body":"Clichtove, Bovelles, the circle."},{"startTime":1006.199,"endTime":1009.23,"body":"They're university scholars,"},{"startTime":1009.23,"endTime":1011.949,"body":"they're interested in"},{"startTime":1009.23,"endTime":1011.949,"body":"getting students through"},{"startTime":1011.949,"endTime":1015.197,"body":"a curriculum and using mathematics"},{"startTime":1015.197,"endTime":1019.19,"body":"as a way of training people"},{"startTime":1015.197,"endTime":1019.19,"body":"to think a particular way,"},{"startTime":1019.19,"endTime":1022.28,"body":"building the habits of"},{"startTime":1019.19,"endTime":1022.28,"body":"copia so that they can"},{"startTime":1022.28,"endTime":1024.43,"body":"become productive in"},{"startTime":1022.28,"endTime":1024.43,"body":"other domains as well."},{"startTime":1025.97,"endTime":1028.1,"body":"Federico Commandino has a very different"},{"startTime":1028.1,"endTime":1032.54,"body":"sort of set of goals, on the one hand,"},{"startTime":1032.54,"endTime":1035.187,"body":"so he is not working in"},{"startTime":1032.54,"endTime":1035.187,"body":"a university context,"},{"startTime":1035.187,"endTime":1036.86,"body":"he's working in the context of a court"},{"startTime":1036.86,"endTime":1039.92,"body":"that is interested in elegance,"},{"startTime":1039.92,"endTime":1042.62,"body":"in a court that is interested in painting,"},{"startTime":1042.62,"endTime":1045.44,"body":"sculpture and mathematics in particular."},{"startTime":1045.44,"endTime":1050.04,"body":"So the Duke of Verbenone"},{"startTime":1045.44,"endTime":1050.04,"body":"trained by Commandino"},{"startTime":1051.556,"endTime":1056.12,"body":"is interested in recovering"},{"startTime":1051.556,"endTime":1056.12,"body":"visions of symmetry,"},{"startTime":1056.12,"endTime":1057.68,"body":"the kinds of things that"},{"startTime":1056.12,"endTime":1057.68,"body":"we think of when we think"},{"startTime":1057.68,"endTime":1060.353,"body":"of renaissance art,"},{"startTime":1057.68,"endTime":1060.353,"body":"history and maths, right?"},{"startTime":1061.187,"endTime":1063.29,"body":"And Commandino's the guy for that,"},{"startTime":1063.29,"endTime":1066.08,"body":"partly because of the many additions"},{"startTime":1066.08,"endTime":1070.25,"body":"that he produces of ancient texts."},{"startTime":1070.25,"endTime":1074.39,"body":"So high prestige texts, engineering texts,"},{"startTime":1074.39,"endTime":1077.31,"body":"and the new translations that he's making"},{"startTime":1078.584,"endTime":1080.573,"body":"out of Greek especially."},{"startTime":1086.15,"endTime":1089.15,"body":"And the most significant of"},{"startTime":1086.15,"endTime":1089.15,"body":"the texts that he produces,"},{"startTime":1089.15,"endTime":1093.74,"body":"and he himself sees this, is"},{"startTime":1089.15,"endTime":1093.74,"body":"his translation of Euclid."},{"startTime":1093.74,"endTime":1097.37,"body":"So I mentioned the medieval tradition,"},{"startTime":1097.37,"endTime":1102.11,"body":"I mentioned that first"},{"startTime":1097.37,"endTime":1102.11,"body":"sort of somewhat troubling,"},{"startTime":1102.11,"endTime":1103.81,"body":"not terribly great mathematically,"},{"startTime":1104.963,"endTime":1108.983,"body":"but interesting Latin-y"},{"startTime":1104.963,"endTime":1108.983,"body":"version of Zamberti."},{"startTime":1110.39,"endTime":1112.49,"body":"Everyone knows that"},{"startTime":1110.39,"endTime":1112.49,"body":"Zamberti's problematic."},{"startTime":1112.49,"endTime":1114.83,"body":"So Commandino has decided"},{"startTime":1112.49,"endTime":1114.83,"body":"to step into the gap."},{"startTime":1114.83,"endTime":1118.07,"body":"And when he produces his version,"},{"startTime":1118.07,"endTime":1121.88,"body":"he's very self-consciously touching"},{"startTime":1121.88,"endTime":1126.88,"body":"the central nerve of mathematical"},{"startTime":1121.88,"endTime":1126.88,"body":"culture in his world."},{"startTime":1129.23,"endTime":1131.09,"body":"Now, one of the things"},{"startTime":1129.23,"endTime":1131.09,"body":"that's really striking"},{"startTime":1131.09,"endTime":1133.52,"body":"about the Prolegomenon that he attaches"},{"startTime":1133.52,"endTime":1136.508,"body":"to his addition is that he has"},{"startTime":1136.508,"endTime":1139.4,"body":"to sort out who Euclid is, right?"},{"startTime":1139.4,"endTime":1142.4,"body":"So the Euclid that I've"},{"startTime":1139.4,"endTime":1142.4,"body":"tried to represent to you,"},{"startTime":1142.4,"endTime":1145.73,"body":"it isn't actually clear which"},{"startTime":1142.4,"endTime":1145.73,"body":"bits of the text are Euclid."},{"startTime":1145.73,"endTime":1148.73,"body":"And in fact, when you have to try"},{"startTime":1148.73,"endTime":1149.93,"body":"to make a decision about that,"},{"startTime":1149.93,"endTime":1153.02,"body":"you have to wade through thickets"},{"startTime":1149.93,"endTime":1153.02,"body":"of different possibilities"},{"startTime":1153.02,"endTime":1155.54,"body":"on the assumption that many of what looked"},{"startTime":1155.54,"endTime":1160.4,"body":"like demonstrations are"},{"startTime":1155.54,"endTime":1160.4,"body":"in fact commentaries,"},{"startTime":1160.4,"endTime":1162.286,"body":"medieval commentaries, or so on."},{"startTime":1162.286,"endTime":1165.62,"body":"Zamberti had suggested that in fact some"},{"startTime":1165.62,"endTime":1169.01,"body":"of this was to do with, well first of all,"},{"startTime":1169.01,"endTime":1171.556,"body":"he wasn't actually sure"},{"startTime":1169.01,"endTime":1171.556,"body":"himself who Euclid was."},{"startTime":1171.556,"endTime":1176.54,"body":"He thought that Euclid"},{"startTime":1171.556,"endTime":1176.54,"body":"of Megara and Euclid"},{"startTime":1176.54,"endTime":1177.8,"body":"of the elements were the same person."},{"startTime":1177.8,"endTime":1179.57,"body":"Turns out they're not."},{"startTime":1179.57,"endTime":1184.31,"body":"He also thought that Theon"},{"startTime":1179.57,"endTime":1184.31,"body":"of Alexandria bore particular"},{"startTime":1184.31,"endTime":1188.57,"body":"responsibility from some of the writing."},{"startTime":1188.57,"endTime":1192.89,"body":"And one of the key points"},{"startTime":1188.57,"endTime":1192.89,"body":"that Commandino makes"},{"startTime":1192.89,"endTime":1194.66,"body":"based on a careful reading"},{"startTime":1194.66,"endTime":1199.07,"body":"of an ancient commentary of"},{"startTime":1194.66,"endTime":1199.07,"body":"the first book of Euclid,"},{"startTime":1199.07,"endTime":1203.63,"body":"is that in fact these"},{"startTime":1199.07,"endTime":1203.63,"body":"two aren't dissimilar."},{"startTime":1203.63,"endTime":1205.34,"body":"So he is already interested"},{"startTime":1203.63,"endTime":1205.34,"body":"in the authorship"},{"startTime":1205.34,"endTime":1207.02,"body":"of Euclid in one particular way."},{"startTime":1207.02,"endTime":1208.85,"body":"He's going to clear up who that is."},{"startTime":1208.85,"endTime":1211.04,"body":"And then secondly,"},{"startTime":1211.04,"endTime":1213.26,"body":"he makes a deliberate"},{"startTime":1213.26,"endTime":1216.18,"body":"and lengthy argument for why in fact"},{"startTime":1217.16,"endTime":1222.16,"body":"the proofs that he's translated"},{"startTime":1217.16,"endTime":1222.16,"body":"don't belong to just anyone"},{"startTime":1222.23,"endTime":1224.96,"body":"and don't come out of a"},{"startTime":1222.23,"endTime":1224.96,"body":"commentary tradition generally,"},{"startTime":1224.96,"endTime":1228.14,"body":"but in fact have been tweaked and edited,"},{"startTime":1228.14,"endTime":1231.362,"body":"but responsibly used"},{"startTime":1228.14,"endTime":1231.362,"body":"by Theon of Alexandria"},{"startTime":1231.362,"endTime":1232.883,"body":"in the fourth century."},{"startTime":1233.9,"endTime":1236.6,"body":"providing the reader now with something"},{"startTime":1236.6,"endTime":1240.2,"body":"like an authoritative text in"},{"startTime":1240.2,"endTime":1243.26,"body":"which you can actually see"},{"startTime":1240.2,"endTime":1243.26,"body":"who the real Euclid is."},{"startTime":1243.26,"endTime":1244.25,"body":"Now there's a third thing that"},{"startTime":1244.25,"endTime":1248.663,"body":"he's also done based on"},{"startTime":1244.25,"endTime":1248.663,"body":"this commentary by Proclus."},{"startTime":1252.476,"endTime":1255.05,"body":"He decides that the language"},{"startTime":1255.05,"endTime":1256.64,"body":"for describing the parts of"},{"startTime":1256.64,"endTime":1259.35,"body":"a geometrical argument isn't adequate"},{"startTime":1260.48,"endTime":1265.48,"body":"and teases out of this"},{"startTime":1260.48,"endTime":1265.48,"body":"ancient philosopher language"},{"startTime":1265.67,"endTime":1268.1,"body":"for distinguishing different parts."},{"startTime":1268.1,"endTime":1269.25,"body":"Because what you could,"},{"startTime":1270.8,"endTime":1273.384,"body":"in order to be able to say more clearly"},{"startTime":1273.384,"endTime":1275.15,"body":"what it is that Euclid's doing"},{"startTime":1275.15,"endTime":1276.8,"body":"when he constructs a demonstration."},{"startTime":1276.8,"endTime":1279.29,"body":"And one of the striking"},{"startTime":1276.8,"endTime":1279.29,"body":"things that I find as a reader"},{"startTime":1279.29,"endTime":1283.61,"body":"of additions of this text is"},{"startTime":1279.29,"endTime":1283.61,"body":"that very often this is the bit"},{"startTime":1283.61,"endTime":1286.52,"body":"that early modern readers get"},{"startTime":1283.61,"endTime":1286.52,"body":"a kick out of this is the bit"},{"startTime":1286.52,"endTime":1287.72,"body":"that they're underscoring"},{"startTime":1287.72,"endTime":1289.64,"body":"or as this particular reader"},{"startTime":1289.64,"endTime":1294.64,"body":"has done renaming in the margins."},{"startTime":1296.3,"endTime":1299.45,"body":"So all of a sudden we right"},{"startTime":1296.3,"endTime":1299.45,"body":"around because of this edition,"},{"startTime":1299.45,"endTime":1302.513,"body":"we start to have a language"},{"startTime":1299.45,"endTime":1302.513,"body":"for talking about a proof"},{"startTime":1302.513,"endTime":1305.36,"body":"anatomizing a proof if you like."},{"startTime":1305.36,"endTime":1307.28,"body":"And you can't read the details here,"},{"startTime":1307.28,"endTime":1309.95,"body":"so you're going to have to take"},{"startTime":1307.28,"endTime":1309.95,"body":"my word for it a little bit."},{"startTime":1309.95,"endTime":1313.07,"body":"But one of the things that"},{"startTime":1309.95,"endTime":1313.07,"body":"happens throughout the actual"},{"startTime":1313.07,"endTime":1318.07,"body":"text that Commandino leaves"},{"startTime":1313.07,"endTime":1318.07,"body":"us is a clear distinction"},{"startTime":1318.2,"endTime":1322.31,"body":"between what in fact belongs to Euclid,"},{"startTime":1322.31,"endTime":1325.28,"body":"and that's the authoritative"},{"startTime":1322.31,"endTime":1325.28,"body":"proof and then his own"},{"startTime":1325.28,"endTime":1328.55,"body":"commentary where he deals"},{"startTime":1325.28,"endTime":1328.55,"body":"with various troubles"},{"startTime":1328.55,"endTime":1331.4,"body":"in the text or mathematical problems."},{"startTime":1331.4,"endTime":1333.62,"body":"So notice what he's done,"},{"startTime":1333.62,"endTime":1338.03,"body":"he's managed to scrape together"},{"startTime":1333.62,"endTime":1338.03,"body":"what is actually Euclid."},{"startTime":1338.03,"endTime":1340.7,"body":"And then he's very carefully"},{"startTime":1338.03,"endTime":1340.7,"body":"separated out his own"},{"startTime":1340.7,"endTime":1343.81,"body":"particular sort of"},{"startTime":1340.7,"endTime":1343.81,"body":"glosses, his commentaries,"},{"startTime":1343.81,"endTime":1347.72,"body":"visually on the page."},{"startTime":1347.72,"endTime":1349.67,"body":"And then he goes further."},{"startTime":1349.67,"endTime":1351.254,"body":"And in that commentary,"},{"startTime":1351.254,"endTime":1355.7,"body":"he spends a lot of time"},{"startTime":1351.254,"endTime":1355.7,"body":"distinguishing all the very bits,"},{"startTime":1355.7,"endTime":1357.57,"body":"what counts as a theorem and so on"},{"startTime":1358.64,"endTime":1362.093,"body":"when he's explaining what"},{"startTime":1358.64,"endTime":1362.093,"body":"Euclid is doing in the text."},{"startTime":1369.11,"endTime":1370.25,"body":"So there's a sense, right,"},{"startTime":1370.25,"endTime":1373.16,"body":"that the first implication here"},{"startTime":1370.25,"endTime":1373.16,"body":"is that for the first time,"},{"startTime":1373.16,"endTime":1375.59,"body":"what we have is something"},{"startTime":1373.16,"endTime":1375.59,"body":"like a canonical Euclid"},{"startTime":1375.59,"endTime":1379.01,"body":"that matches kind of the"},{"startTime":1375.59,"endTime":1379.01,"body":"Euclid that we would recognize."},{"startTime":1379.01,"endTime":1380.45,"body":"And that has big implications"},{"startTime":1380.45,"endTime":1382.22,"body":"because that's the starting point for say,"},{"startTime":1382.22,"endTime":1385.25,"body":"17th century mathematicians"},{"startTime":1382.22,"endTime":1385.25,"body":"when they're trying"},{"startTime":1385.25,"endTime":1388.28,"body":"to explain how to do"},{"startTime":1385.25,"endTime":1388.28,"body":"maths and then tuck it"},{"startTime":1388.28,"endTime":1391.34,"body":"into other parts of natural"},{"startTime":1388.28,"endTime":1391.34,"body":"philosophy and so on."},{"startTime":1391.34,"endTime":1394.38,"body":"So it's that Euclid that becomes canonical"},{"startTime":1395.3,"endTime":1396.77,"body":"and that Euclid that offers"},{"startTime":1396.77,"endTime":1398.963,"body":"a particular vocabulary of proof."},{"startTime":1402.319,"endTime":1405.269,"body":"Or I think anatomy of proof is"},{"startTime":1402.319,"endTime":1405.269,"body":"kind of the term I'd prefer."},{"startTime":1407.222,"endTime":1408.17,"body":"- There's another one."},{"startTime":1408.17,"endTime":1409.7,"body":"Remember, I've gotten excited"},{"startTime":1409.7,"endTime":1413.06,"body":"about the possibility that"},{"startTime":1409.7,"endTime":1413.06,"body":"there's a nice overlap"},{"startTime":1413.06,"endTime":1415.04,"body":"between these ideals for thinking"},{"startTime":1415.04,"endTime":1416.54,"body":"through mathematical narration"},{"startTime":1416.54,"endTime":1420.083,"body":"and larger literary ideals in the period."},{"startTime":1421.04,"endTime":1425.12,"body":"And I think it's not too far to reach"},{"startTime":1425.12,"endTime":1428.21,"body":"when we say that Commandino represents,"},{"startTime":1428.21,"endTime":1430.85,"body":"unlike our school men that"},{"startTime":1428.21,"endTime":1430.85,"body":"we talked about earlier,"},{"startTime":1430.85,"endTime":1435.14,"body":"a different persona both"},{"startTime":1430.85,"endTime":1435.14,"body":"in his own life, right?"},{"startTime":1435.14,"endTime":1439.488,"body":"So he's focused, he"},{"startTime":1435.14,"endTime":1439.488,"body":"belongs to this context"},{"startTime":1439.488,"endTime":1444.44,"body":"where elegance and kind"},{"startTime":1439.488,"endTime":1444.44,"body":"of spare ease in which"},{"startTime":1444.44,"endTime":1446.0,"body":"just a few words can do a huge"},{"startTime":1446.0,"endTime":1449.489,"body":"amount of work in courtly life."},{"startTime":1449.489,"endTime":1452.903,"body":"That sprezzatura according"},{"startTime":1449.489,"endTime":1452.903,"body":"to the Italians, right?"},{"startTime":1453.98,"endTime":1457.1,"body":"He's representing that not"},{"startTime":1453.98,"endTime":1457.1,"body":"only in the way he thinks"},{"startTime":1457.1,"endTime":1459.26,"body":"he would like to come"},{"startTime":1457.1,"endTime":1459.26,"body":"across as a court here,"},{"startTime":1459.26,"endTime":1462.8,"body":"but also the way he wants"},{"startTime":1459.26,"endTime":1462.8,"body":"to think as a mathematician."},{"startTime":1462.8,"endTime":1465.26,"body":"And that's quite interesting"},{"startTime":1462.8,"endTime":1465.26,"body":"because it matches"},{"startTime":1465.26,"endTime":1467.22,"body":"a little more closely what we think"},{"startTime":1468.147,"endTime":1470.81,"body":"of characterizing a"},{"startTime":1468.147,"endTime":1470.81,"body":"good mathematical proof."},{"startTime":1470.81,"endTime":1473.27,"body":"What we think of as"},{"startTime":1470.81,"endTime":1473.27,"body":"characterizing Euclid himself,"},{"startTime":1473.27,"endTime":1477.23,"body":"if you like, this prizing"},{"startTime":1473.27,"endTime":1477.23,"body":"not of copia, of abundance,"},{"startTime":1477.23,"endTime":1480.02,"body":"of kind of a fertile mass of ideas,"},{"startTime":1480.02,"endTime":1483.11,"body":"but rather the thread"},{"startTime":1480.02,"endTime":1483.11,"body":"that holds it all together"},{"startTime":1483.11,"endTime":1485.123,"body":"and everything else being paired apart."},{"startTime":1486.47,"endTime":1487.673,"body":"That kind of an economy."},{"startTime":1498.02,"endTime":1500.81,"body":"I want to just spend a moment"},{"startTime":1500.81,"endTime":1503.0,"body":"thinking about Christoff Clavius,"},{"startTime":1503.0,"endTime":1508.0,"body":"who is a Jesuit who in a"},{"startTime":1503.0,"endTime":1508.0,"body":"way wrote the most popular"},{"startTime":1510.17,"endTime":1514.493,"body":"Euclid, teaching Euclid,"},{"startTime":1510.17,"endTime":1514.493,"body":"of the early modern period."},{"startTime":1515.33,"endTime":1519.17,"body":"It went through a number"},{"startTime":1515.33,"endTime":1519.17,"body":"of editions after 1574."},{"startTime":1519.17,"endTime":1523.4,"body":"So 1572 is when Commandino"},{"startTime":1519.17,"endTime":1523.4,"body":"produces his Euclid,"},{"startTime":1523.4,"endTime":1527.48,"body":"Clavius builds on what has now seen"},{"startTime":1527.48,"endTime":1530.243,"body":"to be the state of the art Euclid,"},{"startTime":1531.26,"endTime":1534.77,"body":"but when he does it, he"},{"startTime":1531.26,"endTime":1534.77,"body":"adds masses more material."},{"startTime":1534.77,"endTime":1538.83,"body":"So look what's happening with"},{"startTime":1534.77,"endTime":1538.83,"body":"a pedagogue such as Clavius,"},{"startTime":1540.23,"endTime":1544.79,"body":"who takes the anatomy from Propos,"},{"startTime":1544.79,"endTime":1546.893,"body":"but then from Commandino,"},{"startTime":1548.72,"endTime":1552.9,"body":"but then applies that"},{"startTime":1548.72,"endTime":1552.9,"body":"anatomy in his own commentary"},{"startTime":1553.839,"endTime":1555.39,"body":"in a much more kind of direct"},{"startTime":1556.526,"endTime":1560.81,"body":"and thorough going way than"},{"startTime":1556.526,"endTime":1560.81,"body":"even Commandino had done."},{"startTime":1560.81,"endTime":1562.85,"body":"The point I want you to"},{"startTime":1560.81,"endTime":1562.85,"body":"get from this is not,"},{"startTime":1562.85,"endTime":1564.95,"body":"you know, is not very subtle,"},{"startTime":1564.95,"endTime":1568.64,"body":"it's simply that what"},{"startTime":1564.95,"endTime":1568.64,"body":"Commandino accomplishes"},{"startTime":1568.64,"endTime":1572.21,"body":"with that Euclid turns into"},{"startTime":1568.64,"endTime":1572.21,"body":"the teaching tool, right?"},{"startTime":1572.21,"endTime":1575.33,"body":"This turns into the starting"},{"startTime":1572.21,"endTime":1575.33,"body":"point for every other version"},{"startTime":1575.33,"endTime":1579.29,"body":"of Euclid that is used within classrooms"},{"startTime":1579.29,"endTime":1580.37,"body":"and in other additions."},{"startTime":1580.37,"endTime":1584.419,"body":"And when indeed Euclid is rejigged"},{"startTime":1584.419,"endTime":1587.18,"body":"in different forms over the"},{"startTime":1584.419,"endTime":1587.18,"body":"next couple hundred years,"},{"startTime":1587.18,"endTime":1589.963,"body":"Clavius is usually the"},{"startTime":1587.18,"endTime":1589.963,"body":"starting point for that."},{"startTime":1589.963,"endTime":1593.14,"body":"But Clavius as interpreting Commandino."},{"startTime":1598.401,"endTime":1602.401,"body":"So that's just a point"},{"startTime":1598.401,"endTime":1602.401,"body":"to say this moment that I"},{"startTime":1603.59,"endTime":1607.67,"body":"kind of playfully am identifying"},{"startTime":1603.59,"endTime":1607.67,"body":"in the 1570s as a really"},{"startTime":1607.67,"endTime":1608.96,"body":"key kind of turning moment."},{"startTime":1608.96,"endTime":1609.83,"body":"But there's, you know,"},{"startTime":1609.83,"endTime":1612.983,"body":"you can always make that"},{"startTime":1609.83,"endTime":1612.983,"body":"line fuzzier if you like,"},{"startTime":1614.03,"endTime":1617.03,"body":"nevertheless does bear"},{"startTime":1614.03,"endTime":1617.03,"body":"some real responsibility"},{"startTime":1617.03,"endTime":1621.26,"body":"for kind of the ongoing"},{"startTime":1617.03,"endTime":1621.26,"body":"sense of who Euclid is."},{"startTime":1621.26,"endTime":1626.183,"body":"And as I see it also"},{"startTime":1621.26,"endTime":1626.183,"body":"for our ideas of proof."},{"startTime":1627.83,"endTime":1630.053,"body":"So I want to suggest that, you know,"},{"startTime":1632.51,"endTime":1635.21,"body":"what we think Euclid is is in itself"},{"startTime":1635.21,"endTime":1636.66,"body":"a question of history, right?"},{"startTime":1638.39,"endTime":1639.59,"body":"We get it from somewhere"},{"startTime":1639.59,"endTime":1642.8,"body":"and we don't get it directly"},{"startTime":1639.59,"endTime":1642.8,"body":"from, you know, the fourth,"},{"startTime":1642.8,"endTime":1646.227,"body":"the third century BC, we get it mediated"},{"startTime":1646.227,"endTime":1648.71,"body":"via these kinds of additions."},{"startTime":1648.71,"endTime":1652.7,"body":"And so what we experience"},{"startTime":1648.71,"endTime":1652.7,"body":"when we are working"},{"startTime":1652.7,"endTime":1654.29,"body":"through the first book of Euclid,"},{"startTime":1654.29,"endTime":1656.96,"body":"or indeed a lot of other forms of proof,"},{"startTime":1656.96,"endTime":1661.793,"body":"is the prompting of a lot"},{"startTime":1656.96,"endTime":1661.793,"body":"of forms of narration,"},{"startTime":1665.57,"endTime":1667.943,"body":"that are pressuring our own reading."},{"startTime":1670.73,"endTime":1674.72,"body":"One of the things I want to suggest"},{"startTime":1674.72,"endTime":1677.06,"body":"is how significant that moment is."},{"startTime":1677.06,"endTime":1679.2,"body":"So I've dwelt on that long enough,"},{"startTime":1679.2,"endTime":1682.07,"body":"but the most interesting bit I think"},{"startTime":1682.07,"endTime":1684.35,"body":"is what this implies about"},{"startTime":1684.35,"endTime":1688.763,"body":"what happens before the 1570s, right?"},{"startTime":1689.75,"endTime":1693.41,"body":"So if the Euclid we sort of work with only"},{"startTime":1693.41,"endTime":1697.133,"body":"comes into being in the 1570s,"},{"startTime":1698.39,"endTime":1700.91,"body":"what's happening in"},{"startTime":1698.39,"endTime":1700.91,"body":"that kind of wild west?"},{"startTime":1700.91,"endTime":1704.18,"body":"That's a overly open term, perhaps,"},{"startTime":1704.18,"endTime":1707.18,"body":"that very puriform set of possibilities"},{"startTime":1707.18,"endTime":1709.82,"body":"for thinking through a mathematical proof,"},{"startTime":1709.82,"endTime":1711.45,"body":"a mathematical narration"},{"startTime":1713.09,"endTime":1716.58,"body":"before the grid of the Proclian anatomy"},{"startTime":1717.56,"endTime":1720.983,"body":"has been put onto Euclid"},{"startTime":1717.56,"endTime":1720.983,"body":"via Commandino and Clavius."},{"startTime":1722.75,"endTime":1724.55,"body":"And if we can open that up,"},{"startTime":1724.55,"endTime":1726.17,"body":"we can get to a much richer,"},{"startTime":1726.17,"endTime":1728.21,"body":"much more interesting"},{"startTime":1726.17,"endTime":1728.21,"body":"sort of understanding"},{"startTime":1728.21,"endTime":1729.35,"body":"of what's actually going on,"},{"startTime":1729.35,"endTime":1732.2,"body":"say in medieval or early"},{"startTime":1729.35,"endTime":1732.2,"body":"Renaissance mathematics."},{"startTime":1732.2,"endTime":1736.07,"body":"But in a second move we"},{"startTime":1732.2,"endTime":1736.07,"body":"can also make much more"},{"startTime":1736.07,"endTime":1738.29,"body":"interesting connections to"},{"startTime":1736.07,"endTime":1738.29,"body":"other traditions of doing"},{"startTime":1738.29,"endTime":1742.22,"body":"mathematics that stand"},{"startTime":1738.29,"endTime":1742.22,"body":"outside of the Euclidean one."},{"startTime":1742.22,"endTime":1743.21,"body":"And I think we'll probably hear"},{"startTime":1743.21,"endTime":1745.76,"body":"a bit more about that later on today."},{"startTime":1745.76,"endTime":1747.332,"body":"Thank you."},{"startTime":1747.332,"endTime":1750.332,"body":"(audience applauds)"},{"startTime":1751.52,"endTime":1755.193,"body":"- You spoke a lot about Euclidean proof."},{"startTime":1755.193,"endTime":1757.49,"body":"Was there any discussion"},{"startTime":1757.49,"endTime":1760.433,"body":"in the medieval period"},{"startTime":1757.49,"endTime":1760.433,"body":"about Euclidean truth?"},{"startTime":1761.39,"endTime":1762.95,"body":"Because in the 19th century lots"},{"startTime":1762.95,"endTime":1765.527,"body":"of other geometries developed of course,"},{"startTime":1765.527,"endTime":1767.63,"body":"and in modern times with relativity"},{"startTime":1767.63,"endTime":1771.86,"body":"and so on then, you"},{"startTime":1767.63,"endTime":1771.86,"body":"know, Euclid in geometry"},{"startTime":1771.86,"endTime":1773.84,"body":"is true for most practical purposes,"},{"startTime":1773.84,"endTime":1775.4,"body":"but not for all."},{"startTime":1775.4,"endTime":1777.92,"body":"- You talked about the fact that"},{"startTime":1777.92,"endTime":1781.49,"body":"the way the medieval"},{"startTime":1777.92,"endTime":1781.49,"body":"scholars presented Euclid,"},{"startTime":1781.49,"endTime":1782.81,"body":"at least at a certain period,"},{"startTime":1782.81,"endTime":1784.61,"body":"they emphasize less on the proofs,"},{"startTime":1784.61,"endTime":1787.31,"body":"at least not as Euclid's proves,"},{"startTime":1787.31,"endTime":1791.9,"body":"more on a certain set of"},{"startTime":1787.31,"endTime":1791.9,"body":"truths which can be listed,"},{"startTime":1791.9,"endTime":1795.35,"body":"whereas subsequently all the way till now"},{"startTime":1795.35,"endTime":1798.38,"body":"mathematics is very much"},{"startTime":1795.35,"endTime":1798.38,"body":"emphasized on proof."},{"startTime":1798.38,"endTime":1802.673,"body":"That's the main thing"},{"startTime":1798.38,"endTime":1802.673,"body":"mathematicians concentrate on."},{"startTime":1804.05,"endTime":1805.4,"body":"So it seems as though we've gone"},{"startTime":1805.4,"endTime":1810.2,"body":"like one level deeper"},{"startTime":1805.4,"endTime":1810.2,"body":"from truths to proof."},{"startTime":1810.2,"endTime":1811.033,"body":"And I was wondering,"},{"startTime":1811.033,"endTime":1812.63,"body":"do you think there would be something"},{"startTime":1813.59,"endTime":1816.473,"body":"maybe another level"},{"startTime":1813.59,"endTime":1816.473,"body":"deeper, more fundamental?"},{"startTime":1817.583,"endTime":1820.07,"body":"Because, so it seems that at some point"},{"startTime":1820.07,"endTime":1821.54,"body":"we're going from truth to truth,"},{"startTime":1821.54,"endTime":1823.37,"body":"now we're going from proof to proof."},{"startTime":1823.37,"endTime":1826.943,"body":"You think there's maybe a"},{"startTime":1823.37,"endTime":1826.943,"body":"third level deeper than that?"},{"startTime":1828.658,"endTime":1829.61,"body":"I'm not sure if that's clear, but."},{"startTime":1829.61,"endTime":1831.263,"body":"- Yeah, yeah."},{"startTime":1832.833,"endTime":1835.58,"body":"So I mean it's possible"},{"startTime":1832.833,"endTime":1835.58,"body":"that these questions"},{"startTime":1835.58,"endTime":1839.633,"body":"are related somewhere deep"},{"startTime":1835.58,"endTime":1839.633,"body":"down at a fourth level."},{"startTime":1842.0,"endTime":1844.49,"body":"So let me think about questions"},{"startTime":1844.49,"endTime":1848.87,"body":"of truth and Euclidean truth."},{"startTime":1848.87,"endTime":1851.87,"body":"What I don't think you have"},{"startTime":1848.87,"endTime":1851.87,"body":"in the Middle Ages are kind"},{"startTime":1851.87,"endTime":1854.75,"body":"of the pressure put on"},{"startTime":1851.87,"endTime":1854.75,"body":"the fifth postulate,"},{"startTime":1854.75,"endTime":1856.22,"body":"you know, the parallel postulate,"},{"startTime":1856.22,"endTime":1859.253,"body":"that turns into kind of"},{"startTime":1856.22,"endTime":1859.253,"body":"non-Euclidean geometries."},{"startTime":1860.12,"endTime":1863.21,"body":"And so Euclid is generally"},{"startTime":1860.12,"endTime":1863.21,"body":"taken to be pretty good"},{"startTime":1863.21,"endTime":1868.13,"body":"at explaining how"},{"startTime":1863.21,"endTime":1868.13,"body":"geometrical objects behave"},{"startTime":1868.13,"endTime":1870.638,"body":"and so on throughout the Middle Ages."},{"startTime":1870.638,"endTime":1874.76,"body":"I don't think I'm going"},{"startTime":1870.638,"endTime":1874.76,"body":"wrong by telling you that."},{"startTime":1874.76,"endTime":1878.55,"body":"What is interesting is that"},{"startTime":1874.76,"endTime":1878.55,"body":"a lot of emphasis is put"},{"startTime":1882.29,"endTime":1885.08,"body":"on being able to see with what's called"},{"startTime":1885.08,"endTime":1889.193,"body":"the light of nature, those"},{"startTime":1885.08,"endTime":1889.193,"body":"mathematical objects."},{"startTime":1890.36,"endTime":1894.29,"body":"And the way one would do"},{"startTime":1890.36,"endTime":1894.29,"body":"that very often is suggest"},{"startTime":1894.29,"endTime":1897.95,"body":"that Euclid isn't the whole story,"},{"startTime":1897.95,"endTime":1899.36,"body":"that there is a deeper method,"},{"startTime":1899.36,"endTime":1902.42,"body":"a math thesis that might"},{"startTime":1899.36,"endTime":1902.42,"body":"have like be related"},{"startTime":1902.42,"endTime":1904.31,"body":"to theology in a particular way, perhaps."},{"startTime":1904.31,"endTime":1907.07,"body":"So in the same way that"},{"startTime":1904.31,"endTime":1907.07,"body":"mathematical objects"},{"startTime":1907.07,"endTime":1909.74,"body":"are completely separated"},{"startTime":1907.07,"endTime":1909.74,"body":"from the world of things,"},{"startTime":1909.74,"endTime":1913.523,"body":"but accessible by your intellect."},{"startTime":1914.51,"endTime":1916.61,"body":"So the assumption goes,"},{"startTime":1916.61,"endTime":1918.734,"body":"metaphysics and theology allow you access"},{"startTime":1918.734,"endTime":1923.48,"body":"to really deep truths"},{"startTime":1918.734,"endTime":1923.48,"body":"by a mode of intuition"},{"startTime":1923.48,"endTime":1924.95,"body":"that's purely intellectual."},{"startTime":1924.95,"endTime":1926.9,"body":"And you can find that, for example,"},{"startTime":1926.9,"endTime":1930.74,"body":"in Boethius, one of the"},{"startTime":1926.9,"endTime":1930.74,"body":"philosophers I was talking about,"},{"startTime":1930.74,"endTime":1934.61,"body":"who gave us this kind of"},{"startTime":1930.74,"endTime":1934.61,"body":"set of annunciations only."},{"startTime":1934.61,"endTime":1936.86,"body":"And when Boethius does that,"},{"startTime":1936.86,"endTime":1938.964,"body":"he gives us kind of a way of talking"},{"startTime":1938.964,"endTime":1943.64,"body":"about mathematics as"},{"startTime":1938.964,"endTime":1943.64,"body":"gesture towards a deeper,"},{"startTime":1943.64,"endTime":1946.88,"body":"what we might call a"},{"startTime":1943.64,"endTime":1946.88,"body":"math thesis universalis,"},{"startTime":1946.88,"endTime":1949.041,"body":"which becomes a hot topic of discussion"},{"startTime":1949.041,"endTime":1953.6,"body":"and debate leading up"},{"startTime":1949.041,"endTime":1953.6,"body":"and beyond Descartes."},{"startTime":1953.6,"endTime":1955.55,"body":"And I think that's where"},{"startTime":1953.6,"endTime":1955.55,"body":"your question overlaps"},{"startTime":1955.55,"endTime":1958.13,"body":"actually with this question"},{"startTime":1955.55,"endTime":1958.13,"body":"about a deeper sort of thing."},{"startTime":1958.13,"endTime":1961.484,"body":"So the division that"},{"startTime":1958.13,"endTime":1961.484,"body":"Boethius leaves us with"},{"startTime":1961.484,"endTime":1965.54,"body":"is between four forms of"},{"startTime":1961.484,"endTime":1965.54,"body":"mathematical reasoning"},{"startTime":1965.54,"endTime":1969.59,"body":"of which basically two"},{"startTime":1965.54,"endTime":1969.59,"body":"are the most fundamental."},{"startTime":1969.59,"endTime":1971.25,"body":"The the two forms of pure maths"},{"startTime":1972.099,"endTime":1974.323,"body":"are geometry and number"},{"startTime":1972.099,"endTime":1974.323,"body":"theory are arithmetic."},{"startTime":1976.52,"endTime":1979.37,"body":"And the claim is that if you can,"},{"startTime":1979.37,"endTime":1982.01,"body":"there must be a deeper kind"},{"startTime":1979.37,"endTime":1982.01,"body":"of reasoning that allows you"},{"startTime":1982.01,"endTime":1985.43,"body":"to make connections between"},{"startTime":1982.01,"endTime":1985.43,"body":"those two sorts of things."},{"startTime":1985.43,"endTime":1988.16,"body":"But nobody quite knows how to do that."},{"startTime":1988.16,"endTime":1991.25,"body":"And maybe if you keep"},{"startTime":1988.16,"endTime":1991.25,"body":"playing with annunciations,"},{"startTime":1991.25,"endTime":1993.74,"body":"and maybe if you keep kind"},{"startTime":1991.25,"endTime":1993.74,"body":"of commenting on this,"},{"startTime":1993.74,"endTime":1996.08,"body":"you can give people the"},{"startTime":1993.74,"endTime":1996.08,"body":"tools with which to achieve"},{"startTime":1996.08,"endTime":1997.847,"body":"the internal insight that's"},{"startTime":1996.08,"endTime":1997.847,"body":"going to allow them to do that."},{"startTime":1997.847,"endTime":2000.82,"body":"And I wonder if someone who"},{"startTime":1997.847,"endTime":2000.82,"body":"would be open to that kind"},{"startTime":2000.82,"endTime":2002.71,"body":"of math thesis universalis actually"},{"startTime":2002.71,"endTime":2003.85,"body":"might be interested in kind of"},{"startTime":2003.85,"endTime":2006.01,"body":"querying whether Euclid's"},{"startTime":2003.85,"endTime":2006.01,"body":"got it all right."},{"startTime":2006.01,"endTime":2008.62,"body":"So that might feed back"},{"startTime":2006.01,"endTime":2008.62,"body":"into your question again."},{"startTime":2008.62,"endTime":2009.52,"body":"- I was wondering,"},{"startTime":2009.52,"endTime":2011.93,"body":"you were talking about Euclid,"},{"startTime":2011.93,"endTime":2012.763,"body":"maybe his main importance"},{"startTime":2012.763,"endTime":2016.09,"body":"was the axiomatic method that he produced."},{"startTime":2016.09,"endTime":2017.44,"body":"And that's reproduced much later."},{"startTime":2017.44,"endTime":2019.3,"body":"You mentioned Spinoza there for instance,"},{"startTime":2019.3,"endTime":2021.397,"body":"who used that and then there"},{"startTime":2021.397,"endTime":2023.41,"body":"was Piago of course later still,"},{"startTime":2023.41,"endTime":2025.903,"body":"and possibly Wittgenstein as well."},{"startTime":2027.059,"endTime":2031.6,"body":"So is it in fact that the"},{"startTime":2027.059,"endTime":2031.6,"body":"Euclidean method when it's taken"},{"startTime":2031.6,"endTime":2033.73,"body":"away from physical things is really part"},{"startTime":2033.73,"endTime":2037.15,"body":"of formal logic and not"},{"startTime":2033.73,"endTime":2037.15,"body":"part of mathematics anyway?"},{"startTime":2037.15,"endTime":2037.983,"body":"You see what I mean?"},{"startTime":2037.983,"endTime":2038.816,"body":"- Right."},{"startTime":2038.816,"endTime":2041.11,"body":"So that's precisely the question"},{"startTime":2038.816,"endTime":2041.11,"body":"that people are thinking"},{"startTime":2041.11,"endTime":2042.61,"body":"of actually when they're thinking"},{"startTime":2042.61,"endTime":2044.86,"body":"about this math thesis universalis, right?"},{"startTime":2044.86,"endTime":2047.11,"body":"So Descartes in his"},{"startTime":2044.86,"endTime":2047.11,"body":"rules for the direction"},{"startTime":2047.11,"endTime":2051.04,"body":"of the mind kind of says maths"},{"startTime":2051.04,"endTime":2053.68,"body":"and sorts of forms of"},{"startTime":2051.04,"endTime":2053.68,"body":"something like axiomatic"},{"startTime":2053.68,"endTime":2055.93,"body":"reasoning is going to give us a logic"},{"startTime":2055.93,"endTime":2057.55,"body":"that we can apply to everything."},{"startTime":2057.55,"endTime":2059.8,"body":"But of course what's really"},{"startTime":2057.55,"endTime":2059.8,"body":"hard is to spell out precisely"},{"startTime":2059.8,"endTime":2064.27,"body":"how that works and the claim"},{"startTime":2059.8,"endTime":2064.27,"body":"that there is an underlying"},{"startTime":2064.27,"endTime":2067.06,"body":"logic in a way that"},{"startTime":2064.27,"endTime":2067.06,"body":"originates in this Propos book"},{"startTime":2067.06,"endTime":2068.65,"body":"that I mentioned by this fourth"},{"startTime":2068.65,"endTime":2071.5,"body":"century neo-platonic philosopher."},{"startTime":2071.5,"endTime":2074.86,"body":"He claims that there is"},{"startTime":2071.5,"endTime":2074.86,"body":"something that underlies"},{"startTime":2074.86,"endTime":2078.19,"body":"the connection between the geometry"},{"startTime":2078.19,"endTime":2081.4,"body":"and arithmetic and that it"},{"startTime":2078.19,"endTime":2081.4,"body":"is just basically logic,"},{"startTime":2081.4,"endTime":2084.4,"body":"at least that's one reading of how he,"},{"startTime":2084.4,"endTime":2086.797,"body":"it's a complicated little"},{"startTime":2084.4,"endTime":2086.797,"body":"sentence in which he offers this."},{"startTime":2086.797,"endTime":2088.909,"body":"And I can't repeat it just"},{"startTime":2086.797,"endTime":2088.909,"body":"off the tongue right now,"},{"startTime":2088.909,"endTime":2092.233,"body":"but that is the issue at root."},{"startTime":2092.233,"endTime":2094.54,"body":"Now the question of"},{"startTime":2092.233,"endTime":2094.54,"body":"making that stick turns"},{"startTime":2094.54,"endTime":2095.65,"body":"out to be much trickier."},{"startTime":2095.65,"endTime":2099.293,"body":"So I would suggest that"},{"startTime":2095.65,"endTime":2099.293,"body":"at least in this period,"},{"startTime":2099.293,"endTime":2101.83,"body":"people would be keen on asking you,"},{"startTime":2101.83,"endTime":2106.83,"body":"okay, show me then, is"},{"startTime":2101.83,"endTime":2106.83,"body":"axiomatics just logic?"},{"startTime":2106.84,"endTime":2110.176,"body":"And it turns out that's"},{"startTime":2106.84,"endTime":2110.176,"body":"harder to do than than said,"},{"startTime":2110.176,"endTime":2113.173,"body":"even though it looks quite promising."},{"startTime":2115.63,"endTime":2116.463,"body":"- Thank you."},{"startTime":2116.463,"endTime":2118.15,"body":"Okay, I'm going to ask my question."},{"startTime":2118.15,"endTime":2119.621,"body":"So it's about diagrams."},{"startTime":2119.621,"endTime":2121.3,"body":"- Yeah, yeah."},{"startTime":2121.3,"endTime":2124.96,"body":"- To what extent is proof by"},{"startTime":2121.3,"endTime":2124.96,"body":"diagram trusted, accepted?"},{"startTime":2124.96,"endTime":2127.9,"body":"You know, if you show a"},{"startTime":2124.96,"endTime":2127.9,"body":"diagram, does this convince you,"},{"startTime":2127.9,"endTime":2129.19,"body":"and is that valid?"},{"startTime":2129.19,"endTime":2132.82,"body":"'Cause if you look in"},{"startTime":2129.19,"endTime":2132.82,"body":"Euclid, you don't get that."},{"startTime":2132.82,"endTime":2135.88,"body":"So how does this change or"},{"startTime":2132.82,"endTime":2135.88,"body":"does it in the Renaissance?"},{"startTime":2135.88,"endTime":2136.713,"body":"- Yes."},{"startTime":2137.98,"endTime":2140.53,"body":"I think this is one of those"},{"startTime":2137.98,"endTime":2140.53,"body":"questions that people have"},{"startTime":2140.53,"endTime":2143.38,"body":"intuitions about, historians"},{"startTime":2140.53,"endTime":2143.38,"body":"of maths have intuitions about,"},{"startTime":2143.38,"endTime":2146.56,"body":"but don't have a fully"},{"startTime":2143.38,"endTime":2146.56,"body":"worked up narrative for."},{"startTime":2146.56,"endTime":2150.19,"body":"So I hope that we spend a lot"},{"startTime":2146.56,"endTime":2150.19,"body":"more time on this question."},{"startTime":2150.19,"endTime":2151.99,"body":"One of the things that I find fascinating"},{"startTime":2151.99,"endTime":2155.26,"body":"about reading the people who are,"},{"startTime":2155.26,"endTime":2157.66,"body":"so I have often looked"},{"startTime":2155.26,"endTime":2157.66,"body":"at lots of textbooks"},{"startTime":2157.66,"endTime":2160.3,"body":"of the kind that I've alluded to,"},{"startTime":2160.3,"endTime":2161.17,"body":"the ones with just the annunciations"},{"startTime":2161.17,"endTime":2163.21,"body":"where people are being asked"},{"startTime":2163.21,"endTime":2166.393,"body":"to intuit their way through"},{"startTime":2163.21,"endTime":2166.393,"body":"a mathematical narrative."},{"startTime":2167.5,"endTime":2169.6,"body":"And one of the things"},{"startTime":2167.5,"endTime":2169.6,"body":"that people write often"},{"startTime":2169.6,"endTime":2173.26,"body":"or do in the margins is"},{"startTime":2169.6,"endTime":2173.26,"body":"do things with compasses."},{"startTime":2173.26,"endTime":2174.19,"body":"They're writing things out."},{"startTime":2174.19,"endTime":2177.01,"body":"And so in a way, Michael Wardhaugh,"},{"startTime":2177.01,"endTime":2179.56,"body":"sorry Benjamin Wardhaugh"},{"startTime":2177.01,"endTime":2179.56,"body":"has made the claim that"},{"startTime":2179.56,"endTime":2184.56,"body":"in some ways a lot of"},{"startTime":2179.56,"endTime":2184.56,"body":"what's going on here is,"},{"startTime":2184.63,"endTime":2187.42,"body":"you know, recipes for"},{"startTime":2184.63,"endTime":2187.42,"body":"diagram making, right?"},{"startTime":2187.42,"endTime":2191.41,"body":"So the goal is that you spend"},{"startTime":2187.42,"endTime":2191.41,"body":"time doing things on your own."},{"startTime":2191.41,"endTime":2194.62,"body":"And so even when there's not"},{"startTime":2191.41,"endTime":2194.62,"body":"a diagram in front of you,"},{"startTime":2194.62,"endTime":2197.32,"body":"you're being asked to produce"},{"startTime":2194.62,"endTime":2197.32,"body":"one as a reader and you're"},{"startTime":2197.32,"endTime":2200.32,"body":"expected to read much more"},{"startTime":2197.32,"endTime":2200.32,"body":"actively than we might think"},{"startTime":2200.32,"endTime":2203.617,"body":"and therefore in some cases"},{"startTime":2200.32,"endTime":2203.617,"body":"in fact extra space is left,"},{"startTime":2203.617,"endTime":2205.99,"body":"you know, in the margin"},{"startTime":2203.617,"endTime":2205.99,"body":"and so on for that."},{"startTime":2205.99,"endTime":2208.891,"body":"Partly because that's a habit"},{"startTime":2205.99,"endTime":2208.891,"body":"that books from this period"},{"startTime":2208.891,"endTime":2211.63,"body":"are much more scribbled up than we tend"},{"startTime":2211.63,"endTime":2213.983,"body":"to have in our libraries,"},{"startTime":2211.63,"endTime":2213.983,"body":"or at least, you know,"},{"startTime":2213.983,"endTime":2216.027,"body":"than I've tended to do"},{"startTime":2213.983,"endTime":2216.027,"body":"in the past for sure."},{"startTime":2216.027,"endTime":2219.94,"body":"And that habit of writing"},{"startTime":2216.027,"endTime":2219.94,"body":"as you're reading tends"},{"startTime":2219.94,"endTime":2221.89,"body":"to bleed over into precisely this issue."},{"startTime":2221.89,"endTime":2223.45,"body":"So that's one way of thinking"},{"startTime":2223.45,"endTime":2224.77,"body":"through the question of diagrams here."},{"startTime":2224.77,"endTime":2226.42,"body":"Now, the in principle question,"},{"startTime":2226.42,"endTime":2227.967,"body":"it depends on who you're reading."},{"startTime":2227.967,"endTime":2231.37,"body":"Some people think it's"},{"startTime":2227.967,"endTime":2231.37,"body":"great just to use diagrams"},{"startTime":2231.37,"endTime":2232.6,"body":"and no questions asked."},{"startTime":2232.6,"endTime":2234.2,"body":"That's part of the demonstration"},{"startTime":2235.69,"endTime":2238.09,"body":"and some people are are"},{"startTime":2235.69,"endTime":2238.09,"body":"concerned about that."},{"startTime":2238.09,"endTime":2239.35,"body":"Jacques Peletier du Mans,"},{"startTime":2239.35,"endTime":2241.537,"body":"for example is a renaissance geometer"},{"startTime":2241.537,"endTime":2243.79,"body":"who's raising questions,"},{"startTime":2243.79,"endTime":2244.963,"body":"how valid is this?"},{"startTime":2247.33,"endTime":2249.25,"body":"There's more to say of course, but."},{"startTime":2249.25,"endTime":2251.2,"body":"- There always is, there always is."},{"startTime":2251.2,"endTime":2252.033,"body":"Thank you."},{"startTime":2252.033,"endTime":2252.866,"body":"Okay, so I'm going to put Richard"},{"startTime":2252.866,"endTime":2254.02,"body":"on the spot with this question,"},{"startTime":2254.02,"endTime":2255.49,"body":"he thought he'd finished for the day,"},{"startTime":2255.49,"endTime":2257.68,"body":"you know, back on duty."},{"startTime":2257.68,"endTime":2259.99,"body":"So we had a question online"},{"startTime":2257.68,"endTime":2259.99,"body":"that just came in a minute too"},{"startTime":2259.99,"endTime":2263.684,"body":"late for the questions last"},{"startTime":2259.99,"endTime":2263.684,"body":"time from Marcus who asks,"},{"startTime":2263.684,"endTime":2268.684,"body":"if medieval mathematicians dealt"},{"startTime":2263.684,"endTime":2268.684,"body":"with the fact that Euclid's"},{"startTime":2268.87,"endTime":2271.63,"body":"geometry has axioms,"},{"startTime":2271.63,"endTime":2275.35,"body":"but the number theory doesn't."},{"startTime":2275.35,"endTime":2276.61,"body":"So he says how do you deal"},{"startTime":2276.61,"endTime":2277.78,"body":"with the fact that Euclid's geometry"},{"startTime":2277.78,"endTime":2279.76,"body":"but not his number theory has postulates?"},{"startTime":2279.76,"endTime":2281.777,"body":"His question."},{"startTime":2281.777,"endTime":2284.403,"body":"- I don't know if this is"},{"startTime":2281.777,"endTime":2284.403,"body":"speaking, should I stand up?"},{"startTime":2285.55,"endTime":2287.77,"body":"So that's a really interesting question,"},{"startTime":2287.77,"endTime":2289.96,"body":"and there's a bit of that I don't know."},{"startTime":2289.96,"endTime":2292.45,"body":"So there's an empirical"},{"startTime":2289.96,"endTime":2292.45,"body":"thing of having to go"},{"startTime":2292.45,"endTime":2294.1,"body":"and look at all those commentaries."},{"startTime":2294.1,"endTime":2297.73,"body":"On the one hand, one thing"},{"startTime":2294.1,"endTime":2297.73,"body":"that is fascinating though"},{"startTime":2297.73,"endTime":2300.76,"body":"is that when people"},{"startTime":2297.73,"endTime":2300.76,"body":"take that number theory"},{"startTime":2300.76,"endTime":2302.89,"body":"that's found there and then they"},{"startTime":2302.89,"endTime":2304.93,"body":"start to write independent"},{"startTime":2302.89,"endTime":2304.93,"body":"treatises on it,"},{"startTime":2304.93,"endTime":2308.68,"body":"they do supply postulates"},{"startTime":2304.93,"endTime":2308.68,"body":"and they scrape around"},{"startTime":2308.68,"endTime":2309.76,"body":"and they try to find them together."},{"startTime":2309.76,"endTime":2312.04,"body":"So Jordanis de la Moray, for example,"},{"startTime":2312.04,"endTime":2313.79,"body":"is an example of someone who writes"},{"startTime":2314.669,"endTime":2316.69,"body":"10 books on number theory"},{"startTime":2316.69,"endTime":2318.94,"body":"and he begins with the same practice"},{"startTime":2318.94,"endTime":2322.81,"body":"that he's seen Euclid do in the geometry."},{"startTime":2322.81,"endTime":2325.42,"body":"And you can look at some of"},{"startTime":2322.81,"endTime":2325.42,"body":"what I've been saying about this"},{"startTime":2325.42,"endTime":2329.59,"body":"kind of suspicion that in fact"},{"startTime":2325.42,"endTime":2329.59,"body":"proof constantly allows you"},{"startTime":2329.59,"endTime":2332.053,"body":"to kind of flexibly"},{"startTime":2329.59,"endTime":2332.053,"body":"bring together different"},{"startTime":2332.053,"endTime":2333.97,"body":"kind of materials into new recipes,"},{"startTime":2333.97,"endTime":2335.743,"body":"I suppose we should call them too,"},{"startTime":2337.81,"endTime":2341.07,"body":"in the various versions"},{"startTime":2337.81,"endTime":2341.07,"body":"of this number theory"},{"startTime":2341.07,"endTime":2343.81,"body":"that is sort of sourced there."},{"startTime":2343.81,"endTime":2345.07,"body":"And in other words,"},{"startTime":2345.07,"endTime":2347.92,"body":"it does the same thing that"},{"startTime":2345.07,"endTime":2347.92,"body":"I've been arguing for Euclid,"},{"startTime":2349.0,"endTime":2352.48,"body":"which is to say there's"},{"startTime":2349.0,"endTime":2352.48,"body":"more work needed done,"},{"startTime":2352.48,"endTime":2355.33,"body":"I think on book five and"},{"startTime":2352.48,"endTime":2355.33,"body":"its commentary tradition"},{"startTime":2355.33,"endTime":2357.553,"body":"of Euclid in the Middle Ages."}]}