{"version":"1.0.0","segments":[{"startTime":0.0,"endTime":3.48,"body":"(audience applauds)"},{"startTime":3.48,"endTime":6.6,"body":"- Hello, I am also going"},{"startTime":3.48,"endTime":6.6,"body":"to be thinking about"},{"startTime":6.6,"endTime":9.21,"body":"the question of what a"},{"startTime":6.6,"endTime":9.21,"body":"mathematical proof is today,"},{"startTime":9.21,"endTime":10.68,"body":"but I'm interested in modern proofs,"},{"startTime":10.68,"endTime":14.103,"body":"and modern mathematics"},{"startTime":10.68,"endTime":14.103,"body":"as you find it nowadays."},{"startTime":16.83,"endTime":18.27,"body":"Mathematical proof, now,"},{"startTime":18.27,"endTime":20.61,"body":"is still a central pillar"},{"startTime":20.61,"endTime":25.08,"body":"of the methodology of pure mathematics,"},{"startTime":25.08,"endTime":27.57,"body":"it contains a lot of the"},{"startTime":25.08,"endTime":27.57,"body":"techniques, and the ideas,"},{"startTime":27.57,"endTime":29.61,"body":"and the concepts, the insights,"},{"startTime":29.61,"endTime":31.71,"body":"and connections and interdependencies,"},{"startTime":31.71,"endTime":35.18,"body":"and really all of the"},{"startTime":31.71,"endTime":35.18,"body":"cunning of pure mathematics,"},{"startTime":35.18,"endTime":36.3,"body":"so in pure mathematics,"},{"startTime":36.3,"endTime":38.13,"body":"it does a lot of work."},{"startTime":38.13,"endTime":40.2,"body":"Proof is also found in"},{"startTime":38.13,"endTime":40.2,"body":"other parts of mathematics,"},{"startTime":40.2,"endTime":41.55,"body":"you find it in applied mathematics,"},{"startTime":41.55,"endTime":43.08,"body":"you find it in engineering,"},{"startTime":43.08,"endTime":46.35,"body":"you find proofs in"},{"startTime":43.08,"endTime":46.35,"body":"statistics at times as well,"},{"startTime":46.35,"endTime":49.2,"body":"so it's really an"},{"startTime":46.35,"endTime":49.2,"body":"incredibly important concept"},{"startTime":49.2,"endTime":50.19,"body":"in mathematics,"},{"startTime":50.19,"endTime":51.6,"body":"and today, I'm going to tell you about"},{"startTime":51.6,"endTime":54.0,"body":"why it's also quite a lot like"},{"startTime":54.0,"endTime":55.593,"body":"your favorite cooking recipes,"},{"startTime":56.7,"endTime":58.08,"body":"for making your favorite dessert,"},{"startTime":58.08,"endTime":59.98,"body":"like rhubarb crumble, would be my one."},{"startTime":62.97,"endTime":67.74,"body":"Like the physicists have a"},{"startTime":62.97,"endTime":67.74,"body":"standard model of physics,"},{"startTime":67.74,"endTime":70.92,"body":"I think mathematicians and"},{"startTime":67.74,"endTime":70.92,"body":"philosophers of mathematics"},{"startTime":70.92,"endTime":73.23,"body":"have, over the last century or so,"},{"startTime":73.23,"endTime":75.6,"body":"settled on a standard model of proof"},{"startTime":75.6,"endTime":76.68,"body":"that looks something very similar"},{"startTime":76.68,"endTime":78.48,"body":"to what Richard was describing,"},{"startTime":78.48,"endTime":81.6,"body":"about the Euclidean, and"},{"startTime":78.48,"endTime":81.6,"body":"Aristotelian before that,"},{"startTime":81.6,"endTime":84.72,"body":"idea of proof as a deductive argument."},{"startTime":84.72,"endTime":85.77,"body":"So the standard model"},{"startTime":85.77,"endTime":87.75,"body":"that a lot of people would subscribe to"},{"speaker":"looks something like this","startTime":89.07,"endTime":91.83,"body":"it says that a proof is"},{"speaker":"looks something like this","startTime":89.07,"endTime":91.83,"body":"a deductive argument,"},{"speaker":"looks something like this","startTime":91.83,"endTime":94.02,"body":"which means it's a logically"},{"speaker":"looks something like this","startTime":91.83,"endTime":94.02,"body":"structured sequence"},{"speaker":"looks something like this","startTime":94.02,"endTime":95.1,"body":"of assertions,"},{"speaker":"looks something like this","startTime":95.1,"endTime":97.8,"body":"and you start from accepted"},{"speaker":"looks something like this","startTime":95.1,"endTime":97.8,"body":"axioms, or premises,"},{"speaker":"looks something like this","startTime":97.8,"endTime":98.633,"body":"and then proceed"},{"speaker":"looks something like this","startTime":98.633,"endTime":100.77,"body":"by inference rules that"},{"speaker":"looks something like this","startTime":98.633,"endTime":100.77,"body":"everyone agrees upon,"},{"speaker":"looks something like this","startTime":100.77,"endTime":101.603,"body":"and at the end,"},{"speaker":"looks something like this","startTime":101.603,"endTime":103.41,"body":"you get to a conclusion,"},{"speaker":"looks something like this","startTime":103.41,"endTime":104.243,"body":"which is, in fact,"},{"speaker":"looks something like this","startTime":104.243,"endTime":105.63,"body":"the theorem you were proving all along,"},{"speaker":"looks something like this","startTime":105.63,"endTime":107.73,"body":"and that's the thing you"},{"speaker":"looks something like this","startTime":105.63,"endTime":107.73,"body":"stated before you started,"},{"speaker":"looks something like this","startTime":107.73,"endTime":108.73,"body":"something like that."},{"speaker":"looks something like this","startTime":109.71,"endTime":112.5,"body":"That standard model has"},{"speaker":"looks something like this","startTime":109.71,"endTime":112.5,"body":"been extremely successful"},{"speaker":"looks something like this","startTime":112.5,"endTime":113.6,"body":"in mathematics"},{"speaker":"looks something like this","startTime":113.6,"endTime":115.47,"body":"in a huge number of ways."},{"speaker":"looks something like this","startTime":115.47,"endTime":117.24,"body":"The two most important, to my mind,"},{"speaker":"looks something like this","startTime":117.24,"endTime":119.73,"body":"are in making it possible"},{"speaker":"looks something like this","startTime":117.24,"endTime":119.73,"body":"to do meta-mathematics,"},{"speaker":"looks something like this","startTime":119.73,"endTime":121.56,"body":"so you take these things,"},{"speaker":"looks something like this","startTime":121.56,"endTime":124.65,"body":"and you can start doing"},{"speaker":"looks something like this","startTime":121.56,"endTime":124.65,"body":"mathematics about mathematics,"},{"speaker":"looks something like this","startTime":124.65,"endTime":127.41,"body":"this leads to celebrated results"},{"speaker":"looks something like this","startTime":127.41,"endTime":130.47,"body":"like Godel's incompleteness theorems,"},{"speaker":"looks something like this","startTime":130.47,"endTime":132.57,"body":"and many more besides."},{"speaker":"looks something like this","startTime":132.57,"endTime":136.02,"body":"The second is the"},{"speaker":"looks something like this","startTime":132.57,"endTime":136.02,"body":"computerization of mathematics,"},{"speaker":"looks something like this","startTime":136.02,"endTime":137.07,"body":"so using computers"},{"speaker":"looks something like this","startTime":137.07,"endTime":139.8,"body":"both to check results"},{"speaker":"looks something like this","startTime":137.07,"endTime":139.8,"body":"that people have produced,"},{"speaker":"looks something like this","startTime":139.8,"endTime":142.773,"body":"but also to generate new theorems"},{"speaker":"looks something like this","startTime":139.8,"endTime":142.773,"body":"that people haven't done,"},{"speaker":"looks something like this","startTime":143.61,"endTime":144.93,"body":"both of those things rely"},{"speaker":"looks something like this","startTime":144.93,"endTime":148.503,"body":"on using some sort of"},{"speaker":"looks something like this","startTime":144.93,"endTime":148.503,"body":"model of proof like that."},{"speaker":"looks something like this","startTime":149.46,"endTime":153.81,"body":"However, this model doesn't do everything"},{"speaker":"looks something like this","startTime":153.81,"endTime":155.7,"body":"that you might want from a model of proof,"},{"speaker":"looks something like this","startTime":155.7,"endTime":156.87,"body":"and so I'm going to argue"},{"speaker":"looks something like this","startTime":156.87,"endTime":159.36,"body":"for a kind of alternative model today."},{"speaker":"looks something like this","startTime":159.36,"endTime":163.47,"body":"I think that this model doesn't"},{"speaker":"looks something like this","startTime":159.36,"endTime":163.47,"body":"really faithfully capture"},{"speaker":"looks something like this","startTime":163.47,"endTime":166.98,"body":"all of the features of"},{"speaker":"looks something like this","startTime":163.47,"endTime":166.98,"body":"actual proofs in practice"},{"speaker":"looks something like this","startTime":166.98,"endTime":168.33,"body":"in modern mathematics."},{"speaker":"looks something like this","startTime":168.33,"endTime":170.28,"body":"So it excludes picture proofs,"},{"speaker":"looks something like this","startTime":170.28,"endTime":172.02,"body":"that we've already discussed,"},{"speaker":"looks something like this","startTime":172.02,"endTime":173.31,"body":"they are quite controversial,"},{"speaker":"looks something like this","startTime":173.31,"endTime":175.441,"body":"so it excludes something like that,"},{"speaker":"looks something like this","startTime":175.441,"endTime":179.16,"body":"it suppresses the role"},{"speaker":"looks something like this","startTime":175.441,"endTime":179.16,"body":"of the mathematicians"},{"speaker":"looks something like this","startTime":179.16,"endTime":180.33,"body":"who are doing the proving,"},{"speaker":"looks something like this","startTime":180.33,"endTime":184.26,"body":"and who are reading"},{"speaker":"looks something like this","startTime":180.33,"endTime":184.26,"body":"proofs by other people,"},{"speaker":"looks something like this","startTime":184.26,"endTime":187.443,"body":"so it suppresses the"},{"speaker":"looks something like this","startTime":184.26,"endTime":187.443,"body":"social aspects of proving."},{"speaker":"looks something like this","startTime":190.08,"endTime":192.63,"body":"The final thing that I am going"},{"speaker":"looks something like this","startTime":190.08,"endTime":192.63,"body":"to take a lot of issue with"},{"speaker":"looks something like this","startTime":192.63,"endTime":194.84,"body":"is the emphasis on assertions,"},{"speaker":"looks something like this","startTime":194.84,"endTime":197.49,"body":"on sequences of assertions."},{"speaker":"looks something like this","startTime":197.49,"endTime":198.72,"body":"By emphasizing this,"},{"speaker":"looks something like this","startTime":198.72,"endTime":200.52,"body":"it rules out other types of language"},{"speaker":"looks something like this","startTime":200.52,"endTime":201.54,"body":"that might be used in proofs"},{"speaker":"looks something like this","startTime":201.54,"endTime":203.61,"body":"that don't take that"},{"speaker":"looks something like this","startTime":201.54,"endTime":203.61,"body":"sort of grammatical form,"},{"speaker":"looks something like this","startTime":203.61,"endTime":205.11,"body":"and I'm going to argue"},{"speaker":"looks something like this","startTime":205.11,"endTime":207.57,"body":"that this is more than"},{"speaker":"looks something like this","startTime":205.11,"endTime":207.57,"body":"just a superficial thing"},{"speaker":"looks something like this","startTime":207.57,"endTime":208.403,"body":"that goes wrong,"},{"speaker":"looks something like this","startTime":208.403,"endTime":210.33,"body":"I think that by looking at the grammar"},{"speaker":"looks something like this","startTime":210.33,"endTime":212.1,"body":"of certain aspects of proofs,"},{"speaker":"looks something like this","startTime":212.1,"endTime":214.68,"body":"we learn something about"},{"speaker":"looks something like this","startTime":212.1,"endTime":214.68,"body":"the nature of proof"},{"speaker":"looks something like this","startTime":214.68,"endTime":216.123,"body":"in modern mathematics."},{"speaker":"looks something like this","startTime":217.65,"endTime":219.99,"body":"So today, I'm going to present"},{"speaker":"looks something like this","startTime":219.99,"endTime":221.43,"body":"an alternative model of proofs"},{"speaker":"looks something like this","startTime":221.43,"endTime":224.13,"body":"and make a very simple argument for it,"},{"speaker":"looks something like this","startTime":224.13,"endTime":226.287,"body":"across multiple things."},{"speaker":"looks something like this","startTime":226.287,"endTime":227.76,"body":"I think this alternative"},{"speaker":"looks something like this","startTime":227.76,"endTime":229.89,"body":"is going to capture some"},{"speaker":"looks something like this","startTime":227.76,"endTime":229.89,"body":"of the interesting features"},{"speaker":"looks something like this","startTime":229.89,"endTime":230.79,"body":"of real proofs"},{"speaker":"looks something like this","startTime":230.79,"endTime":233.343,"body":"that embed some of the"},{"speaker":"looks something like this","startTime":230.79,"endTime":233.343,"body":"social aspects I mentioned,"},{"speaker":"looks something like this","startTime":234.21,"endTime":235.68,"body":"and the idea is simple,"},{"speaker":"looks something like this","startTime":235.68,"endTime":237.233,"body":"that we should think of"},{"speaker":"looks something like this","startTime":235.68,"endTime":237.233,"body":"proofs in mathematics"},{"speaker":"looks something like this","startTime":237.233,"endTime":241.77,"body":"through a nice analogy to"},{"speaker":"looks something like this","startTime":237.233,"endTime":241.77,"body":"recipes, and cooking recipes,"},{"speaker":"looks something like this","startTime":241.77,"endTime":243.393,"body":"that everyone's familiar with."},{"speaker":"looks something like this","startTime":246.42,"endTime":248.22,"body":"I've got four points"},{"speaker":"looks something like this","startTime":248.22,"endTime":251.76,"body":"that I think are crucial"},{"speaker":"looks something like this","startTime":248.22,"endTime":251.76,"body":"to this analogy for today."},{"speaker":"looks something like this","startTime":251.76,"endTime":252.72,"body":"First of all,"},{"speaker":"looks something like this","startTime":252.72,"endTime":254.88,"body":"and this is the main one"},{"speaker":"looks something like this","startTime":252.72,"endTime":254.88,"body":"I'm going to focus on,"},{"speaker":"looks something like this","startTime":254.88,"endTime":258.06,"body":"both recipes and proofs are"},{"speaker":"looks something like this","startTime":254.88,"endTime":258.06,"body":"used to give instructions,"},{"speaker":"looks something like this","startTime":258.06,"endTime":258.893,"body":"and in fact,"},{"speaker":"looks something like this","startTime":258.893,"endTime":261.42,"body":"we heard some of this already"},{"speaker":"looks something like this","startTime":258.893,"endTime":261.42,"body":"in Richard's talk just now."},{"speaker":"looks something like this","startTime":262.32,"endTime":263.7,"body":"In the recipe case,"},{"speaker":"looks something like this","startTime":263.7,"endTime":266.67,"body":"the instructions are for"},{"speaker":"looks something like this","startTime":263.7,"endTime":266.67,"body":"how to make something tasty,"},{"speaker":"looks something like this","startTime":266.67,"endTime":268.59,"body":"if you do it right,"},{"speaker":"looks something like this","startTime":268.59,"endTime":269.49,"body":"in the proof case,"},{"speaker":"looks something like this","startTime":269.49,"endTime":270.323,"body":"they're going to be instructions"},{"speaker":"looks something like this","startTime":270.323,"endTime":273.36,"body":"for carrying out a certain"},{"speaker":"looks something like this","startTime":270.323,"endTime":273.36,"body":"kind of mathematical reasoning,"},{"speaker":"looks something like this","startTime":273.36,"endTime":277.383,"body":"and you gain mathematical"},{"speaker":"looks something like this","startTime":273.36,"endTime":277.383,"body":"knowledge if you do it right."},{"speaker":"looks something like this","startTime":279.18,"endTime":280.53,"body":"And I think this is reflected"},{"speaker":"looks something like this","startTime":280.53,"endTime":283.347,"body":"in the kind of language"},{"speaker":"looks something like this","startTime":280.53,"endTime":283.347,"body":"that can be used in proofs,"},{"speaker":"looks something like this","startTime":283.347,"endTime":285.39,"body":"and that's something I'm going"},{"speaker":"looks something like this","startTime":283.347,"endTime":285.39,"body":"to talk about in a second,"},{"speaker":"looks something like this","startTime":285.39,"endTime":286.92,"body":"we've got some empirical results"},{"speaker":"looks something like this","startTime":286.92,"endTime":288.78,"body":"of studies of mathematical papers"},{"speaker":"looks something like this","startTime":288.78,"endTime":290.28,"body":"that I'm going to tell you about."},{"speaker":"looks something like this","startTime":291.18,"endTime":293.46,"body":"Secondly, the analogy to proofs"},{"speaker":"looks something like this","startTime":293.46,"endTime":297.39,"body":"emphasizes the real activity"},{"speaker":"looks something like this","startTime":293.46,"endTime":297.39,"body":"of doing mathematics,"},{"speaker":"looks something like this","startTime":297.39,"endTime":300.9,"body":"and that built-in social stuff,"},{"speaker":"looks something like this","startTime":300.9,"endTime":302.523,"body":"this real-life practice."},{"speaker":"looks something like this","startTime":303.69,"endTime":306.42,"body":"A written proof is like a recipe"},{"speaker":"looks something like this","startTime":306.42,"endTime":309.12,"body":"in that both are primarily interesting"},{"speaker":"looks something like this","startTime":309.12,"endTime":312.18,"body":"because of the actions that"},{"speaker":"looks something like this","startTime":309.12,"endTime":312.18,"body":"are associated with them."},{"speaker":"looks something like this","startTime":312.18,"endTime":315.603,"body":"You could sit down and read"},{"speaker":"looks something like this","startTime":312.18,"endTime":315.603,"body":"through a cookbook if you like,"},{"speaker":"looks something like this","startTime":315.603,"endTime":317.103,"body":"like a weirdo,"},{"speaker":"looks something like this","startTime":318.57,"endTime":321.27,"body":"but the main use of having a"},{"speaker":"looks something like this","startTime":318.57,"endTime":321.27,"body":"cookbook sat in your kitchen"},{"speaker":"looks something like this","startTime":321.27,"endTime":322.35,"body":"is that you want to open it up,"},{"speaker":"looks something like this","startTime":322.35,"endTime":323.79,"body":"and find a recipe you want to cook,"},{"speaker":"looks something like this","startTime":323.79,"endTime":325.683,"body":"and then actually making that recipe."},{"speaker":"looks something like this","startTime":329.01,"endTime":331.44,"body":"Likewise, I think that"},{"speaker":"looks something like this","startTime":329.01,"endTime":331.44,"body":"proofs by themselves"},{"speaker":"looks something like this","startTime":331.44,"endTime":334.44,"body":"can be a nice and important written record"},{"speaker":"looks something like this","startTime":334.44,"endTime":335.76,"body":"of how to go about doing"},{"speaker":"looks something like this","startTime":335.76,"endTime":338.793,"body":"some piece of mathematical reasoning,"},{"speaker":"looks something like this","startTime":339.75,"endTime":341.94,"body":"but the interesting thing is"},{"speaker":"looks something like this","startTime":339.75,"endTime":341.94,"body":"to actually do the reasoning,"},{"speaker":"looks something like this","startTime":341.94,"endTime":343.29,"body":"to actually carry out the reasoning,"},{"speaker":"looks something like this","startTime":343.29,"endTime":344.19,"body":"or repeat the reasoning"},{"speaker":"looks something like this","startTime":344.19,"endTime":346.35,"body":"that someone else has already"},{"speaker":"looks something like this","startTime":344.19,"endTime":346.35,"body":"figured out how to do,"},{"speaker":"looks something like this","startTime":346.35,"endTime":349.143,"body":"and realize why something"},{"speaker":"looks something like this","startTime":346.35,"endTime":349.143,"body":"is true for yourself."},{"speaker":"looks something like this","startTime":351.57,"endTime":354.54,"body":"The third feature of the"},{"speaker":"looks something like this","startTime":351.57,"endTime":354.54,"body":"analogy I'd like to bring out"},{"speaker":"looks something like this","startTime":354.54,"endTime":357.39,"body":"is that it really emphasizes"},{"speaker":"looks something like this","startTime":357.39,"endTime":360.54,"body":"the mathematicians"},{"speaker":"looks something like this","startTime":357.39,"endTime":360.54,"body":"involved in mathematics."},{"speaker":"looks something like this","startTime":360.54,"endTime":363.27,"body":"A recipe has an author,"},{"speaker":"looks something like this","startTime":360.54,"endTime":363.27,"body":"maybe multiple authors,"},{"speaker":"looks something like this","startTime":363.27,"endTime":364.14,"body":"and a reader,"},{"speaker":"looks something like this","startTime":364.14,"endTime":366.87,"body":"and so do proofs."},{"speaker":"looks something like this","startTime":366.87,"endTime":369.81,"body":"The mathematician, as an author,"},{"speaker":"looks something like this","startTime":369.81,"endTime":373.5,"body":"wants to communicate how"},{"speaker":"looks something like this","startTime":369.81,"endTime":373.5,"body":"to prove some theorem,"},{"speaker":"looks something like this","startTime":373.5,"endTime":375.45,"body":"the reader then wants to go"},{"speaker":"looks something like this","startTime":373.5,"endTime":375.45,"body":"through the reasoning too"},{"speaker":"looks something like this","startTime":375.45,"endTime":377.19,"body":"and understand how it works"},{"speaker":"looks something like this","startTime":377.19,"endTime":380.01,"body":"by following that"},{"speaker":"looks something like this","startTime":377.19,"endTime":380.01,"body":"reasoning in a certain way."},{"speaker":"looks something like this","startTime":380.01,"endTime":381.78,"body":"And thinking in these terms"},{"speaker":"looks something like this","startTime":381.78,"endTime":383.49,"body":"directs us to think about the ways"},{"speaker":"looks something like this","startTime":383.49,"endTime":386.64,"body":"in which the communication functions,"},{"speaker":"looks something like this","startTime":386.64,"endTime":388.47,"body":"so it makes us think questions like,"},{"speaker":"looks something like this","startTime":388.47,"endTime":392.13,"body":"how do I best write a proof"},{"speaker":"looks something like this","startTime":388.47,"endTime":392.13,"body":"to communicate how to do this,"},{"speaker":"looks something like this","startTime":392.13,"endTime":395.85,"body":"how do different audiences read proofs,"},{"speaker":"looks something like this","startTime":395.85,"endTime":398.85,"body":"and there's interesting"},{"speaker":"looks something like this","startTime":395.85,"endTime":398.85,"body":"empirical research out there"},{"speaker":"looks something like this","startTime":398.85,"endTime":402.48,"body":"by a lot of mathematics"},{"speaker":"looks something like this","startTime":398.85,"endTime":402.48,"body":"education researchers"},{"speaker":"looks something like this","startTime":402.48,"endTime":403.8,"body":"who study this,"},{"speaker":"looks something like this","startTime":403.8,"endTime":405.21,"body":"how mathematicians read proofs"},{"speaker":"looks something like this","startTime":405.21,"endTime":406.637,"body":"versus how students read proofs,"},{"speaker":"looks something like this","startTime":406.637,"endTime":410.31,"body":"versus how younger children"},{"speaker":"looks something like this","startTime":406.637,"endTime":410.31,"body":"start to read proofs"},{"speaker":"looks something like this","startTime":410.31,"endTime":415.31,"body":"when they're first introduced to them."},{"speaker":"looks something like this","startTime":415.5,"endTime":417.27,"body":"The final point of the analogy"},{"speaker":"looks something like this","startTime":417.27,"endTime":419.85,"body":"is that there's lots of"},{"speaker":"looks something like this","startTime":417.27,"endTime":419.85,"body":"different ways and styles"},{"speaker":"looks something like this","startTime":419.85,"endTime":421.833,"body":"of producing proofs."},{"speaker":"looks something like this","startTime":423.03,"endTime":424.11,"body":"If you think about recipes,"},{"speaker":"looks something like this","startTime":424.11,"endTime":424.943,"body":"think of the difference"},{"speaker":"looks something like this","startTime":424.943,"endTime":428.01,"body":"between a Nigella recipe"},{"speaker":"looks something like this","startTime":424.943,"endTime":428.01,"body":"and a Jamie Oliver recipe,"},{"speaker":"looks something like this","startTime":428.01,"endTime":430.23,"body":"or a Gordon Ramsay or Mary Berry,"},{"speaker":"looks something like this","startTime":430.23,"endTime":432.78,"body":"or you think of an Ottolenghi cookbook"},{"speaker":"looks something like this","startTime":432.78,"endTime":435.69,"body":"versus a hand-me-down recipe"},{"speaker":"looks something like this","startTime":432.78,"endTime":435.69,"body":"from your grandparents,"},{"speaker":"looks something like this","startTime":435.69,"endTime":437.46,"body":"or something like this,"},{"speaker":"looks something like this","startTime":437.46,"endTime":439.08,"body":"and also think of the difference"},{"speaker":"looks something like this","startTime":439.08,"endTime":441.78,"body":"between a recipe that"},{"speaker":"looks something like this","startTime":439.08,"endTime":441.78,"body":"you found in a cookbook"},{"speaker":"looks something like this","startTime":441.78,"endTime":443.61,"body":"versus something that's"},{"speaker":"looks something like this","startTime":441.78,"endTime":443.61,"body":"presented on YouTube,"},{"speaker":"looks something like this","startTime":443.61,"endTime":445.44,"body":"where you can skip ahead"},{"speaker":"looks something like this","startTime":443.61,"endTime":445.44,"body":"to the relevant bits"},{"speaker":"looks something like this","startTime":445.44,"endTime":447.84,"body":"and skip over the backstory"},{"speaker":"looks something like this","startTime":445.44,"endTime":447.84,"body":"that they weave in,"},{"speaker":"looks something like this","startTime":447.84,"endTime":448.84,"body":"something like that."},{"speaker":"looks something like this","startTime":450.18,"endTime":453.09,"body":"Similarly, different mathematicians"},{"speaker":"looks something like this","startTime":450.18,"endTime":453.09,"body":"write in different ways,"},{"speaker":"looks something like this","startTime":453.09,"endTime":454.83,"body":"and different purposes of proofs"},{"speaker":"looks something like this","startTime":454.83,"endTime":456.24,"body":"are presented in different ways,"},{"speaker":"looks something like this","startTime":456.24,"endTime":457.53,"body":"in different contexts,"},{"speaker":"looks something like this","startTime":457.53,"endTime":459.39,"body":"and different mathematicians"},{"speaker":"looks something like this","startTime":459.39,"endTime":461.28,"body":"have different styles of their writing,"},{"speaker":"looks something like this","startTime":461.28,"endTime":463.17,"body":"and all of these things add up"},{"speaker":"looks something like this","startTime":463.17,"endTime":466.38,"body":"to quite a diversity of how"},{"speaker":"looks something like this","startTime":463.17,"endTime":466.38,"body":"you might present proofs"},{"speaker":"looks something like this","startTime":466.38,"endTime":467.33,"body":"in different cases."},{"speaker":"looks something like this","startTime":468.21,"endTime":471.18,"body":"So that's the big idea of this analogy,"},{"speaker":"looks something like this","startTime":471.18,"endTime":473.28,"body":"of this cooking idea,"},{"speaker":"looks something like this","startTime":473.28,"endTime":474.36,"body":"I'm going to jump now"},{"speaker":"looks something like this","startTime":474.36,"endTime":475.8,"body":"into thinking about the specifics"},{"speaker":"looks something like this","startTime":475.8,"endTime":477.573,"body":"of the language of written proofs."},{"speaker":"looks something like this","startTime":479.79,"endTime":481.23,"body":"I didn't know until a few days ago"},{"speaker":"looks something like this","startTime":481.23,"endTime":483.24,"body":"that Wikipedia has its own cookbook,"},{"speaker":"looks something like this","startTime":483.24,"endTime":486.78,"body":"but here's an example of"},{"speaker":"looks something like this","startTime":483.24,"endTime":486.78,"body":"how to make a sunshine cake,"},{"speaker":"looks something like this","startTime":486.78,"endTime":488.293,"body":"I don't know what that is either,"},{"speaker":"looks something like this","startTime":488.293,"endTime":491.43,"body":"but I want to bring out"},{"speaker":"looks something like this","startTime":488.293,"endTime":491.43,"body":"some of the features"},{"speaker":"looks something like this","startTime":491.43,"endTime":493.323,"body":"of the language of written recipes."},{"speaker":"looks something like this","startTime":495.36,"endTime":497.82,"body":"If you look at the language"},{"speaker":"looks something like this","startTime":495.36,"endTime":497.82,"body":"used to present this cake,"},{"speaker":"looks something like this","startTime":497.82,"endTime":500.22,"body":"I think it's fairly typical,"},{"speaker":"looks something like this","startTime":500.22,"endTime":501.36,"body":"and what's typical about it"},{"speaker":"looks something like this","startTime":501.36,"endTime":504.21,"body":"is that it presents a"},{"speaker":"looks something like this","startTime":501.36,"endTime":504.21,"body":"list of instructions."},{"speaker":"looks something like this","startTime":504.21,"endTime":506.88,"body":"Each of the sentences here"},{"speaker":"looks something like this","startTime":504.21,"endTime":506.88,"body":"is in the imperative mood,"},{"speaker":"looks something like this","startTime":506.88,"endTime":510.99,"body":"which is the mood of a sentence"},{"speaker":"looks something like this","startTime":506.88,"endTime":510.99,"body":"that gives instructions,"},{"speaker":"looks something like this","startTime":510.99,"endTime":513.33,"body":"and you've got a whole list of verbs"},{"speaker":"looks something like this","startTime":513.33,"endTime":515.04,"body":"that are used to form these imperatives,"},{"speaker":"looks something like this","startTime":515.04,"endTime":516.78,"body":"these instructions,"},{"speaker":"looks something like this","startTime":516.78,"endTime":517.613,"body":"in different ways."},{"speaker":"looks something like this","startTime":517.613,"endTime":519.3,"body":"So you've got the separate,"},{"speaker":"looks something like this","startTime":517.613,"endTime":519.3,"body":"you've got the beat,"},{"speaker":"looks something like this","startTime":519.3,"endTime":520.89,"body":"you've got more beating,"},{"speaker":"looks something like this","startTime":520.89,"endTime":522.87,"body":"you've got addition, add this,"},{"speaker":"looks something like this","startTime":522.87,"endTime":526.02,"body":"continue, fold, fold, add, add, pour,"},{"speaker":"looks something like this","startTime":526.02,"endTime":526.98,"body":"and finally bake it,"},{"speaker":"looks something like this","startTime":526.98,"endTime":529.143,"body":"and then you've got your cake, fantastic."},{"speaker":"looks something like this","startTime":532.503,"endTime":534.72,"body":"That would be a very"},{"speaker":"looks something like this","startTime":532.503,"endTime":534.72,"body":"standard thing to think about"},{"speaker":"looks something like this","startTime":534.72,"endTime":537.213,"body":"when you're thinking of"},{"speaker":"looks something like this","startTime":534.72,"endTime":537.213,"body":"the language of recipes."},{"speaker":"looks something like this","startTime":538.05,"endTime":539.28,"body":"Now, actually, I'm going to show you"},{"speaker":"looks something like this","startTime":539.28,"endTime":542.64,"body":"that this is pretty common in"},{"speaker":"looks something like this","startTime":539.28,"endTime":542.64,"body":"mathematical texts as well."},{"speaker":"looks something like this","startTime":542.64,"endTime":545.22,"body":"I've taken here a proof"},{"speaker":"looks something like this","startTime":545.22,"endTime":549.15,"body":"from a well known real analysis textbook"},{"speaker":"looks something like this","startTime":549.15,"endTime":551.82,"body":"of a well known theorem, the"},{"speaker":"looks something like this","startTime":549.15,"endTime":551.82,"body":"Bolzano-Weierstrass theorem,"},{"speaker":"looks something like this","startTime":551.82,"endTime":553.41,"body":"one of the first theorems"},{"speaker":"looks something like this","startTime":551.82,"endTime":553.41,"body":"you'd come across"},{"speaker":"looks something like this","startTime":553.41,"endTime":555.333,"body":"as an undergraduate learning analysis."},{"speaker":"looks something like this","startTime":557.16,"endTime":559.02,"body":"Maybe it's no surprise I'm"},{"speaker":"looks something like this","startTime":557.16,"endTime":559.02,"body":"using it as an example,"},{"speaker":"looks something like this","startTime":559.02,"endTime":559.89,"body":"if you look here,"},{"speaker":"looks something like this","startTime":559.89,"endTime":562.8,"body":"you're going to find a lot of"},{"speaker":"looks something like this","startTime":559.89,"endTime":562.8,"body":"the similar kinds of language."},{"speaker":"looks something like this","startTime":562.8,"endTime":564.48,"body":"So you've got, again,"},{"speaker":"looks something like this","startTime":564.48,"endTime":568.11,"body":"suppose, let, define, divide,"},{"speaker":"looks something like this","startTime":564.48,"endTime":568.11,"body":"choose, divide, choose,"},{"speaker":"looks something like this","startTime":568.11,"endTime":568.943,"body":"and it goes on,"},{"speaker":"looks something like this","startTime":568.943,"endTime":570.573,"body":"it's got more of these as you go on."},{"speaker":"looks something like this","startTime":571.86,"endTime":574.38,"body":"I think, if you were to present"},{"speaker":"looks something like this","startTime":571.86,"endTime":574.38,"body":"this to a mathematician,"},{"speaker":"looks something like this","startTime":574.38,"endTime":575.4,"body":"they would, again, think"},{"speaker":"looks something like this","startTime":575.4,"endTime":578.442,"body":"that it's a very typical, textbook proof"},{"speaker":"looks something like this","startTime":578.442,"endTime":579.633,"body":"of a standard result."},{"speaker":"looks something like this","startTime":583.56,"endTime":585.6,"body":"So if this is a typical textbook result,"},{"speaker":"looks something like this","startTime":585.6,"endTime":587.49,"body":"we've actually got a problem already,"},{"speaker":"looks something like this","startTime":587.49,"endTime":588.51,"body":"because, if you'll recall,"},{"speaker":"looks something like this","startTime":588.51,"endTime":590.01,"body":"the standard model that I started with,"},{"speaker":"looks something like this","startTime":590.01,"endTime":592.44,"body":"it tells you that proofs"},{"speaker":"looks something like this","startTime":590.01,"endTime":592.44,"body":"are sequences of assertions,"},{"speaker":"looks something like this","startTime":592.44,"endTime":594.12,"body":"that just assert things."},{"speaker":"looks something like this","startTime":594.12,"endTime":595.23,"body":"This isn't asserting things,"},{"speaker":"looks something like this","startTime":595.23,"endTime":596.97,"body":"it's telling you to do things."},{"speaker":"looks something like this","startTime":596.97,"endTime":598.26,"body":"So there's a clash there"},{"speaker":"looks something like this","startTime":598.26,"endTime":600.39,"body":"between the standard"},{"speaker":"looks something like this","startTime":598.26,"endTime":600.39,"body":"idea of what proofs are"},{"speaker":"looks something like this","startTime":600.39,"endTime":602.64,"body":"and how they're actually"},{"speaker":"looks something like this","startTime":600.39,"endTime":602.64,"body":"written in practice."},{"speaker":"looks something like this","startTime":603.81,"endTime":605.19,"body":"The question, then, is,"},{"speaker":"looks something like this","startTime":605.19,"endTime":606.69,"body":"is this clash,"},{"speaker":"looks something like this","startTime":606.69,"endTime":607.77,"body":"is that a deep problem"},{"speaker":"looks something like this","startTime":607.77,"endTime":609.96,"body":"or is it just a superficial problem"},{"speaker":"looks something like this","startTime":609.96,"endTime":612.76,"body":"of an isolated example that"},{"speaker":"looks something like this","startTime":609.96,"endTime":612.76,"body":"I've pulled out of a textbook?"},{"speaker":"looks something like this","startTime":612.76,"endTime":614.19,"body":"And I want to try and make the case"},{"speaker":"looks something like this","startTime":614.19,"endTime":615.81,"body":"that it's a deep problem,"},{"speaker":"looks something like this","startTime":615.81,"endTime":620.073,"body":"it's something that's an"},{"speaker":"looks something like this","startTime":615.81,"endTime":620.073,"body":"interesting, important observation."},{"speaker":"looks something like this","startTime":621.36,"endTime":622.193,"body":"So to do that,"},{"speaker":"looks something like this","startTime":622.193,"endTime":623.76,"body":"I'm actually going to"},{"speaker":"looks something like this","startTime":622.193,"endTime":623.76,"body":"draw on some evidence"},{"speaker":"looks something like this","startTime":623.76,"endTime":626.13,"body":"from empirical research."},{"speaker":"looks something like this","startTime":626.13,"endTime":628.32,"body":"This was done together with"},{"speaker":"looks something like this","startTime":626.13,"endTime":628.32,"body":"my colleague Matthew Inglis,"},{"speaker":"looks something like this","startTime":628.32,"endTime":629.703,"body":"at Loughborough University,"},{"speaker":"looks something like this","startTime":630.78,"endTime":634.17,"body":"and what we did was a"},{"speaker":"looks something like this","startTime":630.78,"endTime":634.17,"body":"corpus linguistic study."},{"speaker":"looks something like this","startTime":634.17,"endTime":636.15,"body":"And if you don't know corpus linguistics,"},{"speaker":"looks something like this","startTime":636.15,"endTime":638.64,"body":"it's to use a large body of texts"},{"speaker":"looks something like this","startTime":638.64,"endTime":642.54,"body":"to study the patterns of language use"},{"speaker":"looks something like this","startTime":642.54,"endTime":645.333,"body":"across a large sample of texts."},{"speaker":"looks something like this","startTime":648.24,"endTime":649.62,"body":"The idea that we had"},{"speaker":"looks something like this","startTime":649.62,"endTime":653.28,"body":"was that we would use"},{"speaker":"looks something like this","startTime":649.62,"endTime":653.28,"body":"papers from the arXiv,"},{"speaker":"looks something like this","startTime":653.28,"endTime":656.07,"body":"and the arXiv is a repository,"},{"speaker":"looks something like this","startTime":656.07,"endTime":660.903,"body":"it's an online database of"},{"speaker":"looks something like this","startTime":656.07,"endTime":660.903,"body":"new papers by mathematicians."},{"speaker":"looks something like this","startTime":662.49,"endTime":663.323,"body":"I think it's fair to say"},{"speaker":"looks something like this","startTime":663.323,"endTime":664.44,"body":"most mathematicians"},{"speaker":"looks something like this","startTime":664.44,"endTime":667.65,"body":"working in English speaking"},{"speaker":"looks something like this","startTime":664.44,"endTime":667.65,"body":"mathematics departments"},{"speaker":"looks something like this","startTime":667.65,"endTime":670.71,"body":"across the UK, and a lot"},{"speaker":"looks something like this","startTime":667.65,"endTime":670.71,"body":"of Europe, and the US,"},{"speaker":"looks something like this","startTime":670.71,"endTime":674.1,"body":"and Australasia, quite a lot"},{"speaker":"looks something like this","startTime":670.71,"endTime":674.1,"body":"of Japanese mathematicians,"},{"speaker":"looks something like this","startTime":674.1,"endTime":675.45,"body":"when they finish writing their papers,"},{"speaker":"looks something like this","startTime":675.45,"endTime":678.84,"body":"they upload it immediately to the arXiv"},{"speaker":"looks something like this","startTime":678.84,"endTime":679.77,"body":"as a pre-print,"},{"speaker":"looks something like this","startTime":679.77,"endTime":682.47,"body":"before it's even gone through peer review."},{"speaker":"looks something like this","startTime":682.47,"endTime":686.58,"body":"So the arXiv forms a wonderful database"},{"speaker":"looks something like this","startTime":686.58,"endTime":690.03,"body":"of mathematics papers"},{"speaker":"looks something like this","startTime":690.03,"endTime":691.83,"body":"as they're written for"},{"speaker":"looks something like this","startTime":690.03,"endTime":691.83,"body":"mathematics journals"},{"speaker":"looks something like this","startTime":691.83,"endTime":693.063,"body":"by mathematicians."},{"speaker":"looks something like this","startTime":697.02,"endTime":700.47,"body":"When you pick out a body"},{"speaker":"looks something like this","startTime":697.02,"endTime":700.47,"body":"of texts to look at,"},{"speaker":"looks something like this","startTime":700.47,"endTime":701.73,"body":"it comes with some caveats,"},{"speaker":"looks something like this","startTime":701.73,"endTime":704.82,"body":"so I want to do my due"},{"speaker":"looks something like this","startTime":701.73,"endTime":704.82,"body":"diligence as a researcher"},{"speaker":"looks something like this","startTime":704.82,"endTime":706.37,"body":"and give some of these caveats,"},{"speaker":"looks something like this","startTime":707.4,"endTime":708.42,"body":"by using texts from here,"},{"speaker":"looks something like this","startTime":708.42,"endTime":711.48,"body":"we're looking at a specific"},{"speaker":"looks something like this","startTime":708.42,"endTime":711.48,"body":"selection of people,"},{"speaker":"looks something like this","startTime":711.48,"endTime":713.58,"body":"so we're looking at"},{"speaker":"looks something like this","startTime":711.48,"endTime":713.58,"body":"professional mathematicians"},{"speaker":"looks something like this","startTime":713.58,"endTime":715.11,"body":"writing for professional mathematicians,"},{"speaker":"looks something like this","startTime":715.11,"endTime":716.52,"body":"most of the time,"},{"speaker":"looks something like this","startTime":716.52,"endTime":720.33,"body":"and we're looking at a"},{"speaker":"looks something like this","startTime":716.52,"endTime":720.33,"body":"cross-section of mathematicians"},{"speaker":"looks something like this","startTime":720.33,"endTime":723.915,"body":"based on language and"},{"speaker":"looks something like this","startTime":720.33,"endTime":723.915,"body":"patterns of uploading things,"},{"speaker":"looks something like this","startTime":723.915,"endTime":726.42,"body":"so this tells us about"},{"speaker":"looks something like this","startTime":723.915,"endTime":726.42,"body":"certain kinds of language"},{"speaker":"looks something like this","startTime":726.42,"endTime":727.317,"body":"in mathematics,"},{"speaker":"looks something like this","startTime":727.317,"endTime":729.66,"body":"you have to be careful"},{"speaker":"looks something like this","startTime":727.317,"endTime":729.66,"body":"about generalizing from here"},{"speaker":"looks something like this","startTime":729.66,"endTime":732.48,"body":"to mathematics further,"},{"speaker":"looks something like this","startTime":732.48,"endTime":735.33,"body":"though I think the results do generalize,"},{"speaker":"looks something like this","startTime":735.33,"endTime":737.763,"body":"but the study only shows so much."},{"speaker":"looks something like this","startTime":739.89,"endTime":741.84,"body":"The specific papers that we used"},{"speaker":"looks something like this","startTime":741.84,"endTime":743.76,"body":"to form our corpus for the study"},{"speaker":"looks something like this","startTime":743.76,"endTime":747.02,"body":"were all of the mathematics"},{"speaker":"looks something like this","startTime":743.76,"endTime":747.02,"body":"papers uploaded to the arXiv"},{"speaker":"looks something like this","startTime":747.02,"endTime":749.283,"body":"in the first three months of 2009."},{"speaker":"looks something like this","startTime":750.87,"endTime":753.09,"body":"They were processed in various ways"},{"speaker":"looks something like this","startTime":753.09,"endTime":755.097,"body":"to make it possible to do"},{"speaker":"looks something like this","startTime":753.09,"endTime":755.097,"body":"corpus linguistics with them,"},{"speaker":"looks something like this","startTime":755.097,"endTime":756.27,"body":"and you can read the paper"},{"speaker":"looks something like this","startTime":756.27,"endTime":757.65,"body":"if you want to know the details of that,"},{"speaker":"looks something like this","startTime":757.65,"endTime":759.96,"body":"it's probably not that"},{"speaker":"looks something like this","startTime":757.65,"endTime":759.96,"body":"interesting for a talk like this"},{"speaker":"looks something like this","startTime":759.96,"endTime":761.853,"body":"so I'm going to skip them largely."},{"speaker":"looks something like this","startTime":763.02,"endTime":764.73,"body":"Importantly, one thing they let us do"},{"speaker":"looks something like this","startTime":764.73,"endTime":766.53,"body":"was to separate out the texts"},{"speaker":"looks something like this","startTime":766.53,"endTime":768.72,"body":"that were just the bits of proof"},{"speaker":"looks something like this","startTime":768.72,"endTime":770.07,"body":"that were used in these papers,"},{"speaker":"looks something like this","startTime":770.07,"endTime":772.74,"body":"and all of the other mathematical"},{"speaker":"looks something like this","startTime":770.07,"endTime":772.74,"body":"text that was around it,"},{"speaker":"looks something like this","startTime":772.74,"endTime":774.57,"body":"so we had a corpus of proofs"},{"speaker":"looks something like this","startTime":774.57,"endTime":777.27,"body":"and we had a corpus of the"},{"speaker":"looks something like this","startTime":774.57,"endTime":777.27,"body":"rest of the mathematical text,"},{"speaker":"looks something like this","startTime":777.27,"endTime":778.87,"body":"we called it the non-proof bits."},{"speaker":"looks something like this","startTime":781.53,"endTime":784.29,"body":"What we were interested in was to find out"},{"speaker":"looks something like this","startTime":784.29,"endTime":787.2,"body":"about the way in which"},{"speaker":"looks something like this","startTime":784.29,"endTime":787.2,"body":"instructions were used"},{"speaker":"looks something like this","startTime":787.2,"endTime":790.62,"body":"in written proofs in these"},{"speaker":"looks something like this","startTime":787.2,"endTime":790.62,"body":"papers we were looking at."},{"speaker":"looks something like this","startTime":790.62,"endTime":792.817,"body":"So we did this in several different ways,"},{"speaker":"looks something like this","startTime":792.817,"endTime":796.17,"body":"there are three key ways"},{"speaker":"looks something like this","startTime":792.817,"endTime":796.17,"body":"that I'm going to show you."},{"speaker":"looks something like this","startTime":796.17,"endTime":799.5,"body":"The very first was to"},{"speaker":"looks something like this","startTime":796.17,"endTime":799.5,"body":"look at the frequency"},{"speaker":"looks something like this","startTime":799.5,"endTime":801.45,"body":"with which different verbs were used"},{"speaker":"looks something like this","startTime":801.45,"endTime":803.46,"body":"to form these instructions"},{"speaker":"looks something like this","startTime":801.45,"endTime":803.46,"body":"that I showed you,"},{"speaker":"looks something like this","startTime":803.46,"endTime":806.763,"body":"these imperative phrases."},{"speaker":"looks something like this","startTime":808.56,"endTime":810.36,"body":"We pulled out some mathematical verbs,"},{"speaker":"looks something like this","startTime":810.36,"endTime":811.92,"body":"again, the details of how we found those"},{"speaker":"looks something like this","startTime":811.92,"endTime":813.6,"body":"doesn't really matter."},{"speaker":"looks something like this","startTime":813.6,"endTime":814.5,"body":"What I'm showing you here"},{"speaker":"looks something like this","startTime":814.5,"endTime":819.5,"body":"is the contrast between these"},{"speaker":"looks something like this","startTime":814.5,"endTime":819.5,"body":"verbs used in the proofs,"},{"speaker":"looks something like this","startTime":820.95,"endTime":821.783,"body":"just as proofs,"},{"speaker":"looks something like this","startTime":821.783,"endTime":823.983,"body":"and the maths text more generally."},{"speaker":"looks something like this","startTime":826.35,"endTime":827.76,"body":"There was a pretty cunning way"},{"speaker":"looks something like this","startTime":827.76,"endTime":829.74,"body":"of being able to pick out"},{"speaker":"looks something like this","startTime":829.74,"endTime":833.13,"body":"which times the words let and suppose"},{"speaker":"looks something like this","startTime":833.13,"endTime":835.83,"body":"were being used to form instructions,"},{"speaker":"looks something like this","startTime":835.83,"endTime":837.0,"body":"which is that, in English,"},{"speaker":"looks something like this","startTime":837.0,"endTime":841.23,"body":"when you put those words at"},{"speaker":"looks something like this","startTime":837.0,"endTime":841.23,"body":"the start of the sentence,"},{"speaker":"looks something like this","startTime":841.23,"endTime":842.52,"body":"that's how you form the imperative,"},{"speaker":"looks something like this","startTime":842.52,"endTime":844.98,"body":"so we could look at it case-sensitive,"},{"speaker":"looks something like this","startTime":844.98,"endTime":847.68,"body":"and that means that you"},{"speaker":"looks something like this","startTime":844.98,"endTime":847.68,"body":"were looking at Let,"},{"speaker":"looks something like this","startTime":847.68,"endTime":848.61,"body":"with a capital L,"},{"speaker":"looks something like this","startTime":848.61,"endTime":849.443,"body":"and that means it says,"},{"speaker":"looks something like this","startTime":849.443,"endTime":850.5,"body":"Let something, something, something,"},{"speaker":"looks something like this","startTime":850.5,"endTime":852.51,"body":"rather than let in some other sense,"},{"speaker":"looks something like this","startTime":852.51,"endTime":853.91,"body":"same with all of the others."},{"speaker":"looks something like this","startTime":858.15,"endTime":860.28,"body":"The findings were this,"},{"speaker":"looks something like this","startTime":860.28,"endTime":861.69,"body":"that, actually, within the proofs,"},{"speaker":"looks something like this","startTime":861.69,"endTime":864.93,"body":"you really did find quite"},{"speaker":"looks something like this","startTime":861.69,"endTime":864.93,"body":"a lot of instructions,"},{"speaker":"looks something like this","startTime":864.93,"endTime":867.87,"body":"especially what I like to call a core set,"},{"speaker":"looks something like this","startTime":867.87,"endTime":869.79,"body":"from let down to write,"},{"speaker":"looks something like this","startTime":869.79,"endTime":871.23,"body":"these other ones, not so much."},{"speaker":"looks something like this","startTime":871.23,"endTime":874.26,"body":"So we found sort of usual suspects"},{"speaker":"looks something like this","startTime":874.26,"endTime":878.16,"body":"of imperative phrases, of instructions"},{"speaker":"looks something like this","startTime":878.16,"endTime":881.58,"body":"that are used in mathematics"},{"speaker":"looks something like this","startTime":878.16,"endTime":881.58,"body":"texts all the time."},{"speaker":"looks something like this","startTime":881.58,"endTime":884.37,"body":"So already, that is a refutation"},{"speaker":"looks something like this","startTime":884.37,"endTime":887.22,"body":"of some aspects of the standard view"},{"speaker":"looks something like this","startTime":887.22,"endTime":889.44,"body":"that they're just sequences of assertions,"},{"speaker":"looks something like this","startTime":889.44,"endTime":890.97,"body":"we found a whole lot of these,"},{"speaker":"looks something like this","startTime":890.97,"endTime":892.2,"body":"especially the word let,"},{"speaker":"looks something like this","startTime":892.2,"endTime":895.11,"body":"was showing up 4,500"},{"speaker":"looks something like this","startTime":892.2,"endTime":895.11,"body":"times per million words,"},{"speaker":"looks something like this","startTime":895.11,"endTime":896.28,"body":"that's a huge amount,"},{"speaker":"looks something like this","startTime":896.28,"endTime":897.3,"body":"but even these other ones,"},{"speaker":"looks something like this","startTime":897.3,"endTime":900.6,"body":"suppose, note, consider,"},{"speaker":"looks something like this","startTime":897.3,"endTime":900.6,"body":"assume, these other ones,"},{"speaker":"looks something like this","startTime":900.6,"endTime":903.3,"body":"they're showing up an awful"},{"speaker":"looks something like this","startTime":900.6,"endTime":903.3,"body":"lot in mathematical texts."},{"speaker":"looks something like this","startTime":904.26,"endTime":905.46,"body":"I think that's exciting,"},{"speaker":"looks something like this","startTime":905.46,"endTime":907.59,"body":"to have a specific list"},{"speaker":"looks something like this","startTime":905.46,"endTime":907.59,"body":"of which ones there are,"},{"speaker":"looks something like this","startTime":907.59,"endTime":909.723,"body":"to a researcher, that's very cool."},{"speaker":"looks something like this","startTime":911.97,"endTime":913.56,"body":"So now we've found those frequencies,"},{"speaker":"looks something like this","startTime":913.56,"endTime":915.78,"body":"one concern you might have"},{"speaker":"looks something like this","startTime":915.78,"endTime":919.47,"body":"is that rather than"},{"speaker":"looks something like this","startTime":915.78,"endTime":919.47,"body":"these being very common,"},{"speaker":"looks something like this","startTime":919.47,"endTime":921.66,"body":"they're very common in a"},{"speaker":"looks something like this","startTime":919.47,"endTime":921.66,"body":"small number of papers,"},{"speaker":"looks something like this","startTime":921.66,"endTime":923.46,"body":"that there's a few maniacs"},{"speaker":"looks something like this","startTime":923.46,"endTime":925.59,"body":"who are just writing"},{"speaker":"looks something like this","startTime":923.46,"endTime":925.59,"body":"every sentence like this,"},{"speaker":"looks something like this","startTime":925.59,"endTime":927.24,"body":"and most mathematicians aren't,"},{"speaker":"looks something like this","startTime":927.24,"endTime":929.01,"body":"so the next thing to look at is"},{"speaker":"looks something like this","startTime":929.01,"endTime":931.65,"body":"how broadly and how often are they used"},{"speaker":"looks something like this","startTime":931.65,"endTime":933.27,"body":"across these things?"},{"speaker":"looks something like this","startTime":933.27,"endTime":934.35,"body":"So I've got big table here,"},{"speaker":"looks something like this","startTime":934.35,"endTime":936.063,"body":"don't be frightened by it."},{"speaker":"looks something like this","startTime":936.99,"endTime":938.1,"body":"On the left-hand side,"},{"speaker":"looks something like this","startTime":938.1,"endTime":942.603,"body":"we looked at the number of"},{"speaker":"looks something like this","startTime":938.1,"endTime":942.603,"body":"files that they showed up in,"},{"speaker":"looks something like this","startTime":943.74,"endTime":944.82,"body":"and so, once again,"},{"speaker":"looks something like this","startTime":944.82,"endTime":946.44,"body":"let is by far the most common,"},{"speaker":"looks something like this","startTime":946.44,"endTime":950.55,"body":"it shows up in 80% of the"},{"speaker":"looks something like this","startTime":946.44,"endTime":950.55,"body":"papers with proofs in it"},{"speaker":"looks something like this","startTime":950.55,"endTime":951.383,"body":"in the files,"},{"speaker":"looks something like this","startTime":951.383,"endTime":953.28,"body":"so almost every paper that uses proof"},{"speaker":"looks something like this","startTime":953.28,"endTime":955.53,"body":"uses a let somewhere along the way,"},{"speaker":"looks something like this","startTime":955.53,"endTime":957.12,"body":"almost half of them use note somewhere,"},{"speaker":"looks something like this","startTime":957.12,"endTime":960.273,"body":"over 1/3 use suppose, consider,"},{"speaker":"looks something like this","startTime":957.12,"endTime":960.273,"body":"or assume somewhere there,"},{"speaker":"looks something like this","startTime":961.2,"endTime":962.76,"body":"and even further down,"},{"speaker":"looks something like this","startTime":962.76,"endTime":964.983,"body":"down to, say, 13% for observe."},{"speaker":"looks something like this","startTime":966.75,"endTime":968.28,"body":"However, in isolation,"},{"speaker":"looks something like this","startTime":968.28,"endTime":970.41,"body":"these percentages are"},{"speaker":"looks something like this","startTime":968.28,"endTime":970.41,"body":"hard to make sense of,"},{"speaker":"looks something like this","startTime":970.41,"endTime":972.6,"body":"so what I've done on the right-hand side"},{"speaker":"looks something like this","startTime":972.6,"endTime":976.38,"body":"is pick out a word that"},{"speaker":"looks something like this","startTime":972.6,"endTime":976.38,"body":"shows up a similar span"},{"speaker":"looks something like this","startTime":976.38,"endTime":977.7,"body":"across mathematics papers."},{"speaker":"looks something like this","startTime":977.7,"endTime":979.83,"body":"So what we see is that let shows up"},{"speaker":"looks something like this","startTime":979.83,"endTime":982.11,"body":"about as often as then in texts,"},{"speaker":"looks something like this","startTime":982.11,"endTime":984.27,"body":"note shows up about as often as function,"},{"speaker":"looks something like this","startTime":984.27,"endTime":986.28,"body":"suppose about as often as thus,"},{"speaker":"looks something like this","startTime":986.28,"endTime":988.23,"body":"and what this is going to show you"},{"speaker":"looks something like this","startTime":988.23,"endTime":992.43,"body":"is that even the ones that"},{"speaker":"looks something like this","startTime":988.23,"endTime":992.43,"body":"show up, say, 13%, 14%,"},{"speaker":"looks something like this","startTime":992.43,"endTime":995.22,"body":"show up as often as very"},{"speaker":"looks something like this","startTime":992.43,"endTime":995.22,"body":"common mathematical words"},{"speaker":"looks something like this","startTime":995.22,"endTime":997.56,"body":"like precisely, always, less,"},{"speaker":"looks something like this","startTime":997.56,"endTime":1000.5,"body":"these are very standard"},{"speaker":"looks something like this","startTime":997.56,"endTime":1000.5,"body":"mathematical words."},{"speaker":"looks something like this","startTime":1000.5,"endTime":1002.0,"body":"So what we took this to mean"},{"speaker":"looks something like this","startTime":1002.0,"endTime":1005.36,"body":"is that these imperative phrases"},{"speaker":"looks something like this","startTime":1005.36,"endTime":1006.41,"body":"are showing up quite broadly"},{"speaker":"looks something like this","startTime":1006.41,"endTime":1008.18,"body":"across mathematical phrases as well."},{"speaker":"looks something like this","startTime":1008.18,"endTime":1011.09,"body":"So even though sometimes"},{"speaker":"looks something like this","startTime":1008.18,"endTime":1011.09,"body":"it's only 13%, 14%,"},{"speaker":"looks something like this","startTime":1011.09,"endTime":1011.923,"body":"this is still"},{"speaker":"looks something like this","startTime":1011.923,"endTime":1015.353,"body":"fairly broad, normal use"},{"speaker":"looks something like this","startTime":1011.923,"endTime":1015.353,"body":"of mathematical language."},{"speaker":"looks something like this","startTime":1023.6,"endTime":1025.73,"body":"Finally, what I've been showing you"},{"speaker":"looks something like this","startTime":1025.73,"endTime":1030.17,"body":"are results for this usual"},{"speaker":"looks something like this","startTime":1025.73,"endTime":1030.17,"body":"suspects, core set of verbs,"},{"speaker":"looks something like this","startTime":1030.17,"endTime":1031.22,"body":"but we might wonder"},{"speaker":"looks something like this","startTime":1031.22,"endTime":1034.49,"body":"what other words can be"},{"speaker":"looks something like this","startTime":1031.22,"endTime":1034.49,"body":"used to form imperatives,"},{"speaker":"looks something like this","startTime":1034.49,"endTime":1038.84,"body":"so we went around looking for"},{"speaker":"looks something like this","startTime":1034.49,"endTime":1038.84,"body":"other kinds of instructions"},{"speaker":"looks something like this","startTime":1038.84,"endTime":1039.92,"body":"across the corpus."},{"speaker":"looks something like this","startTime":1039.92,"endTime":1042.89,"body":"I've got a nice, colorful word cloud here,"},{"speaker":"looks something like this","startTime":1042.89,"endTime":1045.08,"body":"these are all other verbs"},{"speaker":"looks something like this","startTime":1045.08,"endTime":1047.21,"body":"that can be used to form imperatives,"},{"speaker":"looks something like this","startTime":1047.21,"endTime":1048.59,"body":"and I've put all the ones here"},{"speaker":"looks something like this","startTime":1048.59,"endTime":1051.107,"body":"that we found examples for,"},{"speaker":"looks something like this","startTime":1051.107,"endTime":1052.1,"body":"and basically,"},{"speaker":"looks something like this","startTime":1052.1,"endTime":1054.65,"body":"these are almost all the verbs"},{"speaker":"looks something like this","startTime":1052.1,"endTime":1054.65,"body":"that we started looking for,"},{"speaker":"looks something like this","startTime":1054.65,"endTime":1056.9,"body":"you could find one or"},{"speaker":"looks something like this","startTime":1054.65,"endTime":1056.9,"body":"two examples at least"},{"speaker":"looks something like this","startTime":1056.9,"endTime":1059.483,"body":"of imperatives being formed of them."},{"speaker":"looks something like this","startTime":1063.17,"endTime":1064.82,"body":"Actually, I've got some examples."},{"speaker":"looks something like this","startTime":1066.92,"endTime":1068.39,"body":"Here's some typical maths sentences"},{"speaker":"I've pulled out of the corpus","startTime":1071.157,"endTime":1073.647,"body":"\"Set the total degree equal"},{"speaker":"I've pulled out of the corpus","startTime":1071.157,"endTime":1073.647,"body":"to the sum of the bi-degrees,\""},{"speaker":"I've pulled out of the corpus","startTime":1073.647,"endTime":1075.747,"body":"\"Form the commutative cube"},{"speaker":"I've pulled out of the corpus","startTime":1073.647,"endTime":1075.747,"body":"in which,\" blah, blah, blah,"},{"speaker":"I've pulled out of the corpus","startTime":1075.747,"endTime":1078.357,"body":"\"Sum the estimates in"},{"speaker":"I've pulled out of the corpus","startTime":1075.747,"endTime":1078.357,"body":"the previous corollary,\""},{"speaker":"I've pulled out of the corpus","startTime":1078.357,"endTime":1080.51,"body":"\"Estimate the difference"},{"speaker":"I've pulled out of the corpus","startTime":1078.357,"endTime":1080.51,"body":"on the right-hand side of,\""},{"speaker":"I've pulled out of the corpus","startTime":1080.51,"endTime":1081.413,"body":"blah, blah, blah."},{"speaker":"I've pulled out of the corpus","startTime":1082.64,"endTime":1084.8,"body":"These are the typical"},{"speaker":"I've pulled out of the corpus","startTime":1082.64,"endTime":1084.8,"body":"mathematical sentences"},{"speaker":"I've pulled out of the corpus","startTime":1084.8,"endTime":1085.94,"body":"you can find across things,"},{"speaker":"I've pulled out of the corpus","startTime":1085.94,"endTime":1090.94,"body":"these are less usual, less"},{"speaker":"I've pulled out of the corpus","startTime":1085.94,"endTime":1090.94,"body":"common imperative forms."},{"speaker":"I've pulled out of the corpus","startTime":1091.73,"endTime":1093.11,"body":"Nonetheless, we found examples"},{"speaker":"I've pulled out of the corpus","startTime":1093.11,"endTime":1094.7,"body":"of all kinds of different ones"},{"speaker":"I've pulled out of the corpus","startTime":1094.7,"endTime":1096.2,"body":"once we were looking for them."},{"speaker":"I've pulled out of the corpus","startTime":1102.41,"endTime":1106.97,"body":"So here's some conclusions from"},{"speaker":"I've pulled out of the corpus","startTime":1102.41,"endTime":1106.97,"body":"that little empirical study,"},{"speaker":"I've pulled out of the corpus","startTime":1106.97,"endTime":1109.61,"body":"is that we've got a"},{"speaker":"I've pulled out of the corpus","startTime":1106.97,"endTime":1109.61,"body":"core set of instructions"},{"speaker":"I've pulled out of the corpus","startTime":1109.61,"endTime":1111.92,"body":"that are used frequently in proofs,"},{"speaker":"I've pulled out of the corpus","startTime":1111.92,"endTime":1115.4,"body":"and they appear pretty broadly"},{"speaker":"I've pulled out of the corpus","startTime":1111.92,"endTime":1115.4,"body":"across mathematics papers"},{"speaker":"I've pulled out of the corpus","startTime":1115.4,"endTime":1116.87,"body":"on the arXiv,"},{"speaker":"I've pulled out of the corpus","startTime":1116.87,"endTime":1117.703,"body":"and finally,"},{"speaker":"I've pulled out of the corpus","startTime":1117.703,"endTime":1119.69,"body":"that there's also quite"},{"speaker":"I've pulled out of the corpus","startTime":1117.703,"endTime":1119.69,"body":"a diversity of proofs,"},{"speaker":"I've pulled out of the corpus","startTime":1119.69,"endTime":1120.59,"body":"that you're actually free"},{"speaker":"I've pulled out of the corpus","startTime":1120.59,"endTime":1123.683,"body":"to use a lot of different"},{"speaker":"I've pulled out of the corpus","startTime":1120.59,"endTime":1123.683,"body":"instructions in forming a proof,"},{"speaker":"I've pulled out of the corpus","startTime":1124.55,"endTime":1127.04,"body":"and that that will still"},{"speaker":"I've pulled out of the corpus","startTime":1124.55,"endTime":1127.04,"body":"be standard writing"},{"speaker":"I've pulled out of the corpus","startTime":1127.04,"endTime":1128.69,"body":"for mathematics."},{"speaker":"I've pulled out of the corpus","startTime":1128.69,"endTime":1131.51,"body":"So that's the end of the empirical study,"},{"speaker":"I've pulled out of the corpus","startTime":1131.51,"endTime":1134.798,"body":"let me move on to talking about pictures."},{"speaker":"I've pulled out of the corpus","startTime":1134.798,"endTime":1138.71,"body":"We've already heard about the"},{"speaker":"I've pulled out of the corpus","startTime":1134.798,"endTime":1138.71,"body":"controversy of picture proofs,"},{"speaker":"I've pulled out of the corpus","startTime":1138.71,"endTime":1140.84,"body":"so now we've thought about the language,"},{"speaker":"I've pulled out of the corpus","startTime":1140.84,"endTime":1142.01,"body":"let me move on to pictures"},{"speaker":"I've pulled out of the corpus","startTime":1142.01,"endTime":1146.72,"body":"and think a bit about the way"},{"speaker":"I've pulled out of the corpus","startTime":1142.01,"endTime":1146.72,"body":"in which the recipe idea works"},{"speaker":"I've pulled out of the corpus","startTime":1146.72,"endTime":1149.06,"body":"when you're dealing with pictures."},{"speaker":"I've pulled out of the corpus","startTime":1149.06,"endTime":1149.893,"body":"What I've got here"},{"speaker":"I've pulled out of the corpus","startTime":1149.893,"endTime":1154.13,"body":"is probably one of the most"},{"speaker":"I've pulled out of the corpus","startTime":1149.893,"endTime":1154.13,"body":"well known picture proofs"},{"speaker":"I've pulled out of the corpus","startTime":1154.13,"endTime":1155.9,"body":"outside of Euclid,"},{"speaker":"I've pulled out of the corpus","startTime":1155.9,"endTime":1157.51,"body":"it's attributed to Nicomachus,"},{"speaker":"I've pulled out of the corpus","startTime":1157.51,"endTime":1159.74,"body":"in this wonderful book by Roger Nelson"},{"speaker":"I've pulled out of the corpus","startTime":1159.74,"endTime":1161.267,"body":"called \"Proofs Without Words.\""},{"speaker":"I've pulled out of the corpus","startTime":1162.26,"endTime":1165.02,"body":"It's showing that the sum"},{"speaker":"I've pulled out of the corpus","startTime":1162.26,"endTime":1165.02,"body":"of the first n odd integers,"},{"speaker":"I've pulled out of the corpus","startTime":1165.02,"endTime":1166.91,"body":"starting from one, is n-squared,"},{"speaker":"I've pulled out of the corpus","startTime":1166.91,"endTime":1168.11,"body":"and the idea of the proof"},{"speaker":"I've pulled out of the corpus","startTime":1168.11,"endTime":1171.11,"body":"is to start from the"},{"speaker":"I've pulled out of the corpus","startTime":1168.11,"endTime":1171.11,"body":"bottom left-hand corner,"},{"speaker":"I've pulled out of the corpus","startTime":1171.11,"endTime":1172.7,"body":"you've got a base case there,"},{"speaker":"I've pulled out of the corpus","startTime":1172.7,"endTime":1174.29,"body":"and so you start with one,"},{"speaker":"I've pulled out of the corpus","startTime":1174.29,"endTime":1175.25,"body":"you add three,"},{"speaker":"I've pulled out of the corpus","startTime":1175.25,"endTime":1176.54,"body":"well it forms an extra layer"},{"speaker":"I've pulled out of the corpus","startTime":1176.54,"endTime":1177.74,"body":"and there's still a square,"},{"speaker":"I've pulled out of the corpus","startTime":1177.74,"endTime":1179.24,"body":"you add five,"},{"speaker":"I've pulled out of the corpus","startTime":1179.24,"endTime":1180.382,"body":"it forms the next layer,"},{"speaker":"I've pulled out of the corpus","startTime":1180.382,"endTime":1181.215,"body":"and you've still got a square,"},{"speaker":"I've pulled out of the corpus","startTime":1181.215,"endTime":1182.048,"body":"you add seven,"},{"speaker":"I've pulled out of the corpus","startTime":1182.048,"endTime":1183.29,"body":"you form the next layer,"},{"speaker":"I've pulled out of the corpus","startTime":1183.29,"endTime":1185.72,"body":"all the way up to infinity,"},{"speaker":"I've pulled out of the corpus","startTime":1185.72,"endTime":1186.553,"body":"that's the idea."},{"speaker":"I've pulled out of the corpus","startTime":1188.12,"endTime":1192.17,"body":"We already touched on picture"},{"speaker":"I've pulled out of the corpus","startTime":1188.12,"endTime":1192.17,"body":"proofs being controversial"},{"speaker":"I've pulled out of the corpus","startTime":1192.17,"endTime":1193.7,"body":"amongst philosophers,"},{"speaker":"I've pulled out of the corpus","startTime":1193.7,"endTime":1194.99,"body":"here's two of the main reasons"},{"speaker":"I've pulled out of the corpus","startTime":1194.99,"endTime":1197.57,"body":"that people do not like picture proofs."},{"speaker":"I've pulled out of the corpus","startTime":1197.57,"endTime":1201.74,"body":"The first one is to do with"},{"speaker":"I've pulled out of the corpus","startTime":1197.57,"endTime":1201.74,"body":"the standard model of proofs,"},{"speaker":"I've pulled out of the corpus","startTime":1201.74,"endTime":1205.673,"body":"which is that pictures are"},{"speaker":"I've pulled out of the corpus","startTime":1201.74,"endTime":1205.673,"body":"not sequences of assertions,"},{"speaker":"I've pulled out of the corpus","startTime":1206.54,"endTime":1208.333,"body":"so they're not really proofs by that idea,"},{"speaker":"I've pulled out of the corpus","startTime":1208.333,"endTime":1209.75,"body":"and the idea is dominant,"},{"speaker":"I've pulled out of the corpus","startTime":1209.75,"endTime":1211.94,"body":"so this is a real problem."},{"speaker":"I've pulled out of the corpus","startTime":1211.94,"endTime":1214.16,"body":"Furthermore, if you try"},{"speaker":"I've pulled out of the corpus","startTime":1211.94,"endTime":1214.16,"body":"and rescue picture proofs"},{"speaker":"I've pulled out of the corpus","startTime":1214.16,"endTime":1217.433,"body":"by extracting some assertions"},{"speaker":"I've pulled out of the corpus","startTime":1214.16,"endTime":1217.433,"body":"or propositions from them,"},{"speaker":"I've pulled out of the corpus","startTime":1218.3,"endTime":1220.88,"body":"it's undetermined what those should be,"},{"speaker":"I've pulled out of the corpus","startTime":1220.88,"endTime":1222.14,"body":"and the order in which they are,"},{"speaker":"I've pulled out of the corpus","startTime":1222.14,"endTime":1223.61,"body":"what's the logic of these things,"},{"speaker":"I've pulled out of the corpus","startTime":1223.61,"endTime":1224.54,"body":"oh, I'm so confused,"},{"speaker":"I've pulled out of the corpus","startTime":1224.54,"endTime":1225.99,"body":"get rid of them all entirely,"},{"speaker":"I've pulled out of the corpus","startTime":1227.03,"endTime":1229.1,"body":"so people don't like them for that reason."},{"speaker":"I've pulled out of the corpus","startTime":1229.1,"endTime":1230.12,"body":"The other problem"},{"speaker":"I've pulled out of the corpus","startTime":1230.12,"endTime":1233.78,"body":"is that pictures can"},{"speaker":"I've pulled out of the corpus","startTime":1230.12,"endTime":1233.78,"body":"only show a single case,"},{"speaker":"I've pulled out of the corpus","startTime":1233.78,"endTime":1235.703,"body":"rather than proving a general theorem."},{"speaker":"I've pulled out of the corpus","startTime":1237.83,"endTime":1239.48,"body":"If we go back to this one here,"},{"speaker":"I've pulled out of the corpus","startTime":1239.48,"endTime":1241.07,"body":"I've not shown you the general case,"},{"speaker":"I've pulled out of the corpus","startTime":1241.07,"endTime":1243.89,"body":"I've only shown you the"},{"speaker":"I've pulled out of the corpus","startTime":1241.07,"endTime":1243.89,"body":"case for n equals eight,"},{"speaker":"I've pulled out of the corpus","startTime":1243.89,"endTime":1245.243,"body":"the first eight steps,"},{"speaker":"I've pulled out of the corpus","startTime":1246.2,"endTime":1247.28,"body":"that's not a proof,"},{"speaker":"I've pulled out of the corpus","startTime":1247.28,"endTime":1249.17,"body":"that's a demonstration of one case,"},{"speaker":"I've pulled out of the corpus","startTime":1249.17,"endTime":1251.723,"body":"maybe eight cases if"},{"speaker":"I've pulled out of the corpus","startTime":1249.17,"endTime":1251.723,"body":"you're feeling generous."},{"speaker":"I've pulled out of the corpus","startTime":1253.52,"endTime":1257.723,"body":"Now, if we approach this from"},{"speaker":"I've pulled out of the corpus","startTime":1253.52,"endTime":1257.723,"body":"a more recipe perspective,"},{"speaker":"I've pulled out of the corpus","startTime":1259.725,"endTime":1263.06,"body":"I don't find these concerns"},{"speaker":"I've pulled out of the corpus","startTime":1259.725,"endTime":1263.06,"body":"about picture proofs"},{"speaker":"I've pulled out of the corpus","startTime":1263.06,"endTime":1264.68,"body":"all that convincing,"},{"speaker":"I've pulled out of the corpus","startTime":1264.68,"endTime":1267.74,"body":"and I think the recipe"},{"speaker":"I've pulled out of the corpus","startTime":1264.68,"endTime":1267.74,"body":"model helps to understand"},{"speaker":"I've pulled out of the corpus","startTime":1267.74,"endTime":1270.04,"body":"why I don't find these"},{"speaker":"I've pulled out of the corpus","startTime":1267.74,"endTime":1270.04,"body":"things very convincing."},{"speaker":"I've pulled out of the corpus","startTime":1271.37,"endTime":1272.48,"body":"First of all,"},{"speaker":"I've pulled out of the corpus","startTime":1272.48,"endTime":1275.0,"body":"when you take proofs to be recipes,"},{"speaker":"I've pulled out of the corpus","startTime":1275.0,"endTime":1276.11,"body":"you don't need to figure out"},{"speaker":"I've pulled out of the corpus","startTime":1276.11,"endTime":1279.35,"body":"which propositions, which"},{"speaker":"I've pulled out of the corpus","startTime":1276.11,"endTime":1279.35,"body":"assertions are being given,"},{"speaker":"I've pulled out of the corpus","startTime":1279.35,"endTime":1281.09,"body":"you can understand instructions"},{"speaker":"I've pulled out of the corpus","startTime":1281.09,"endTime":1283.49,"body":"without having to turn"},{"speaker":"I've pulled out of the corpus","startTime":1281.09,"endTime":1283.49,"body":"them into language at all."},{"speaker":"I've pulled out of the corpus","startTime":1283.49,"endTime":1287.9,"body":"Here is a page from"},{"speaker":"I've pulled out of the corpus","startTime":1283.49,"endTime":1287.9,"body":"some LEGO instructions,"},{"speaker":"I've pulled out of the corpus","startTime":1287.9,"endTime":1289.85,"body":"to build, I think, a"},{"speaker":"I've pulled out of the corpus","startTime":1287.9,"endTime":1289.85,"body":"Star Wars Snowspeeder,"},{"speaker":"I've pulled out of the corpus","startTime":1289.85,"endTime":1290.873,"body":"it's very cool,"},{"speaker":"I've pulled out of the corpus","startTime":1292.07,"endTime":1294.74,"body":"and you can give this to a six"},{"speaker":"I've pulled out of the corpus","startTime":1292.07,"endTime":1294.74,"body":"year-old, a seven year-old,"},{"speaker":"I've pulled out of the corpus","startTime":1294.74,"endTime":1296.93,"body":"and they'll be able to spend all afternoon"},{"speaker":"I've pulled out of the corpus","startTime":1296.93,"endTime":1299.75,"body":"building your LEGO Star Wars Snowspeeder"},{"speaker":"I've pulled out of the corpus","startTime":1299.75,"endTime":1301.73,"body":"without too many problems at all."},{"speaker":"I've pulled out of the corpus","startTime":1301.73,"endTime":1303.08,"body":"The way in which instructions are given"},{"speaker":"I've pulled out of the corpus","startTime":1303.08,"endTime":1304.97,"body":"are relatively complicated,"},{"speaker":"I've pulled out of the corpus","startTime":1304.97,"endTime":1306.77,"body":"and nonetheless entirely clear."},{"speaker":"I've pulled out of the corpus","startTime":1306.77,"endTime":1308.6,"body":"A page like this, we're 21 pages in,"},{"speaker":"I've pulled out of the corpus","startTime":1308.6,"endTime":1311.06,"body":"it's telling you to take"},{"speaker":"I've pulled out of the corpus","startTime":1308.6,"endTime":1311.06,"body":"these items that you've got,"},{"speaker":"I've pulled out of the corpus","startTime":1311.06,"endTime":1313.73,"body":"it tells you to combine"},{"speaker":"I've pulled out of the corpus","startTime":1311.06,"endTime":1313.73,"body":"them in certain ways,"},{"speaker":"I've pulled out of the corpus","startTime":1313.73,"endTime":1315.26,"body":"combine this one into that thing,"},{"speaker":"I've pulled out of the corpus","startTime":1315.26,"endTime":1317.78,"body":"then combine it in"},{"speaker":"I've pulled out of the corpus","startTime":1315.26,"endTime":1317.78,"body":"there, put it down here,"},{"speaker":"I've pulled out of the corpus","startTime":1317.78,"endTime":1319.79,"body":"there's a red arrow telling"},{"speaker":"I've pulled out of the corpus","startTime":1317.78,"endTime":1319.79,"body":"you to put things in there,"},{"speaker":"I've pulled out of the corpus","startTime":1319.79,"endTime":1321.92,"body":"and then the whole thing"},{"speaker":"I've pulled out of the corpus","startTime":1319.79,"endTime":1321.92,"body":"is put onto the back,"},{"speaker":"I've pulled out of the corpus","startTime":1321.92,"endTime":1323.27,"body":"there's a bit cropped off at the bottom"},{"speaker":"I've pulled out of the corpus","startTime":1323.27,"endTime":1325.07,"body":"with another piece that comes on,"},{"speaker":"I've pulled out of the corpus","startTime":1325.07,"endTime":1327.683,"body":"this is one page of 60 or so,"},{"speaker":"I've pulled out of the corpus","startTime":1329.57,"endTime":1332.603,"body":"it's very straightforward"},{"speaker":"I've pulled out of the corpus","startTime":1329.57,"endTime":1332.603,"body":"to me how these things work."},{"speaker":"I've pulled out of the corpus","startTime":1335.51,"endTime":1337.343,"body":"So to me,"},{"speaker":"I've pulled out of the corpus","startTime":1338.75,"endTime":1340.7,"body":"the pragmatics of using, say,"},{"speaker":"I've pulled out of the corpus","startTime":1340.7,"endTime":1343.26,"body":"arrows, and the boxes, and the colors"},{"speaker":"I've pulled out of the corpus","startTime":1344.45,"endTime":1346.64,"body":"is very easy to understand"},{"speaker":"I've pulled out of the corpus","startTime":1346.64,"endTime":1349.373,"body":"for basically anyone who"},{"speaker":"I've pulled out of the corpus","startTime":1346.64,"endTime":1349.373,"body":"has any practice doing so."},{"speaker":"I've pulled out of the corpus","startTime":1351.59,"endTime":1355.16,"body":"Even though the"},{"speaker":"I've pulled out of the corpus","startTime":1351.59,"endTime":1355.16,"body":"propositions these represent"},{"speaker":"I've pulled out of the corpus","startTime":1355.16,"endTime":1356.99,"body":"might be obscure and difficult,"},{"speaker":"I've pulled out of the corpus","startTime":1356.99,"endTime":1358.58,"body":"nonetheless, I think the"},{"speaker":"I've pulled out of the corpus","startTime":1356.99,"endTime":1358.58,"body":"instructions are clear,"},{"speaker":"I've pulled out of the corpus","startTime":1358.58,"endTime":1360.28,"body":"and what you're meant to"},{"speaker":"I've pulled out of the corpus","startTime":1358.58,"endTime":1360.28,"body":"be doing to enact this"},{"speaker":"I've pulled out of the corpus","startTime":1360.28,"endTime":1361.97,"body":"is very obvious."},{"speaker":"I've pulled out of the corpus","startTime":1361.97,"endTime":1364.433,"body":"Now, I've got a more"},{"speaker":"I've pulled out of the corpus","startTime":1361.97,"endTime":1364.433,"body":"controversial example."},{"speaker":"I've pulled out of the corpus","startTime":1365.411,"endTime":1368.244,"body":"(audience laughs)"},{"speaker":"I've pulled out of the corpus","startTime":1370.31,"endTime":1372.47,"body":"Anyone that's built"},{"speaker":"I've pulled out of the corpus","startTime":1370.31,"endTime":1372.47,"body":"some furniture from IKEA"},{"speaker":"I've pulled out of the corpus","startTime":1372.47,"endTime":1374.393,"body":"will recognize instructions like this."},{"speaker":"I've pulled out of the corpus","startTime":1376.22,"endTime":1378.5,"body":"People like to complain about them,"},{"speaker":"I've pulled out of the corpus","startTime":1378.5,"endTime":1380.69,"body":"but I disagree with anyone that does that,"},{"speaker":"I've pulled out of the corpus","startTime":1380.69,"endTime":1383.0,"body":"I think the instructions are, once again,"},{"speaker":"I've pulled out of the corpus","startTime":1383.0,"endTime":1386.3,"body":"wonderfully clear and super fun to follow,"},{"speaker":"I've pulled out of the corpus","startTime":1386.3,"endTime":1388.04,"body":"I think its fantastic."},{"speaker":"I've pulled out of the corpus","startTime":1388.04,"endTime":1389.0,"body":"Now let me tell you a bit about"},{"speaker":"I've pulled out of the corpus","startTime":1389.0,"endTime":1390.35,"body":"the pragmatics of these diagrams,"},{"speaker":"I've pulled out of the corpus","startTime":1390.35,"endTime":1391.64,"body":"because they're amazing."},{"speaker":"I've pulled out of the corpus","startTime":1391.64,"endTime":1392.75,"body":"A simple picture like this,"},{"speaker":"I've pulled out of the corpus","startTime":1392.75,"endTime":1394.76,"body":"where you've got no language at all,"},{"speaker":"I've pulled out of the corpus","startTime":1394.76,"endTime":1396.833,"body":"no language being used nonetheless,"},{"speaker":"I've pulled out of the corpus","startTime":1398.03,"endTime":1400.28,"body":"you've been told not"},{"speaker":"I've pulled out of the corpus","startTime":1398.03,"endTime":1400.28,"body":"to use a certain tool,"},{"speaker":"I've pulled out of the corpus","startTime":1400.28,"endTime":1401.93,"body":"so it's like, put that thing away,"},{"speaker":"I've pulled out of the corpus","startTime":1401.93,"endTime":1403.31,"body":"you need to put some screws in here,"},{"speaker":"I've pulled out of the corpus","startTime":1403.31,"endTime":1404.42,"body":"turn them round this way,"},{"speaker":"I've pulled out of the corpus","startTime":1404.42,"endTime":1406.07,"body":"you do that twice for"},{"speaker":"I've pulled out of the corpus","startTime":1404.42,"endTime":1406.07,"body":"two different things,"},{"speaker":"I've pulled out of the corpus","startTime":1406.07,"endTime":1406.903,"body":"at both ends,"},{"speaker":"I've pulled out of the corpus","startTime":1406.903,"endTime":1408.62,"body":"you've got the two things on both ends,"},{"speaker":"I've pulled out of the corpus","startTime":1408.62,"endTime":1410.03,"body":"you've got two people working together"},{"speaker":"I've pulled out of the corpus","startTime":1410.03,"endTime":1412.1,"body":"to make sure it doesn't"},{"speaker":"I've pulled out of the corpus","startTime":1410.03,"endTime":1412.1,"body":"collapse while you're doing it,"},{"speaker":"I've pulled out of the corpus","startTime":1412.1,"endTime":1414.02,"body":"it's telling you a huge"},{"speaker":"I've pulled out of the corpus","startTime":1412.1,"endTime":1414.02,"body":"amount of information"},{"speaker":"I've pulled out of the corpus","startTime":1414.02,"endTime":1415.94,"body":"in a relatively simple diagram."},{"speaker":"I've pulled out of the corpus","startTime":1415.94,"endTime":1417.41,"body":"I love building IKEA stuff,"},{"speaker":"I've pulled out of the corpus","startTime":1417.41,"endTime":1418.985,"body":"and you should too."},{"speaker":"I've pulled out of the corpus","startTime":1418.985,"endTime":1421.818,"body":"(audience laughs)"},{"speaker":"I've pulled out of the corpus","startTime":1425.99,"endTime":1427.64,"body":"Maybe the even more controversial thing"},{"speaker":"I've pulled out of the corpus","startTime":1427.64,"endTime":1429.92,"body":"is that I think that"},{"speaker":"I've pulled out of the corpus","startTime":1427.64,"endTime":1429.92,"body":"instructions from LEGO"},{"speaker":"I've pulled out of the corpus","startTime":1429.92,"endTime":1431.54,"body":"and instructions from IKEA"},{"speaker":"I've pulled out of the corpus","startTime":1431.54,"endTime":1435.44,"body":"are maybe a paradigm example"},{"speaker":"I've pulled out of the corpus","startTime":1431.54,"endTime":1435.44,"body":"of rigorous diagrams,"},{"speaker":"I've pulled out of the corpus","startTime":1435.44,"endTime":1437.33,"body":"I think they're fantastic,"},{"speaker":"I've pulled out of the corpus","startTime":1437.33,"endTime":1438.8,"body":"for telling you how to do things"},{"speaker":"I've pulled out of the corpus","startTime":1438.8,"endTime":1440.48,"body":"without any involvement of language"},{"speaker":"I've pulled out of the corpus","startTime":1440.48,"endTime":1442.493,"body":"or sequences of assertions whatsoever."},{"speaker":"I've pulled out of the corpus","startTime":1443.48,"endTime":1446.81,"body":"So I think that picture"},{"speaker":"I've pulled out of the corpus","startTime":1443.48,"endTime":1446.81,"body":"proofs can take advantage"},{"speaker":"I've pulled out of the corpus","startTime":1446.81,"endTime":1450.653,"body":"of many pragmatic features"},{"speaker":"I've pulled out of the corpus","startTime":1446.81,"endTime":1450.653,"body":"of pictorial communications."},{"speaker":"I've pulled out of the corpus","startTime":1452.63,"endTime":1454.58,"body":"This proof we saw already,"},{"speaker":"I've pulled out of the corpus","startTime":1454.58,"endTime":1456.89,"body":"it's implicitly an induction proof,"},{"speaker":"I've pulled out of the corpus","startTime":1456.89,"endTime":1459.35,"body":"and what you're learning here"},{"speaker":"I've pulled out of the corpus","startTime":1459.35,"endTime":1461.45,"body":"isn't the case of n equals eight,"},{"speaker":"I've pulled out of the corpus","startTime":1461.45,"endTime":1462.95,"body":"it's telling you how to go about"},{"speaker":"I've pulled out of the corpus","startTime":1462.95,"endTime":1464.78,"body":"undertaking a certain piece of action,"},{"speaker":"I've pulled out of the corpus","startTime":1464.78,"endTime":1465.613,"body":"it's telling you,"},{"speaker":"I've pulled out of the corpus","startTime":1465.613,"endTime":1467.78,"body":"okay, you start from the"},{"speaker":"I've pulled out of the corpus","startTime":1465.613,"endTime":1467.78,"body":"bottom left-hand corner,"},{"speaker":"I've pulled out of the corpus","startTime":1467.78,"endTime":1469.73,"body":"but however far you go,"},{"speaker":"I've pulled out of the corpus","startTime":1469.73,"endTime":1470.87,"body":"if you've understood the picture,"},{"speaker":"I've pulled out of the corpus","startTime":1470.87,"endTime":1472.64,"body":"you understand why you"},{"speaker":"I've pulled out of the corpus","startTime":1470.87,"endTime":1472.64,"body":"can add another layer,"},{"speaker":"I've pulled out of the corpus","startTime":1472.64,"endTime":1474.92,"body":"you can understand the induction step."},{"speaker":"I've pulled out of the corpus","startTime":1474.92,"endTime":1476.18,"body":"So to my mind,"},{"speaker":"I've pulled out of the corpus","startTime":1476.18,"endTime":1477.98,"body":"it isn't just showing a single case,"},{"speaker":"I've pulled out of the corpus","startTime":1477.98,"endTime":1479.69,"body":"it is showing a general recipe"},{"speaker":"I've pulled out of the corpus","startTime":1479.69,"endTime":1482.57,"body":"for how to go about doing"},{"speaker":"I've pulled out of the corpus","startTime":1479.69,"endTime":1482.57,"body":"something at any level,"},{"speaker":"I've pulled out of the corpus","startTime":1482.57,"endTime":1485.0,"body":"and it shows you not"},{"speaker":"I've pulled out of the corpus","startTime":1482.57,"endTime":1485.0,"body":"just how to do each step,"},{"speaker":"I've pulled out of the corpus","startTime":1485.0,"endTime":1486.6,"body":"but why it holds all the way up."},{"speaker":"I've pulled out of the corpus","startTime":1488.72,"endTime":1490.85,"body":"I've got another, Pythagoras,"},{"speaker":"I've pulled out of the corpus","startTime":1490.85,"endTime":1492.74,"body":"'cause we love doing that all the way,"},{"speaker":"I've pulled out of the corpus","startTime":1492.74,"endTime":1494.75,"body":"this is a proof, again,"},{"speaker":"I've pulled out of the corpus","startTime":1494.75,"endTime":1496.88,"body":"from the \"Proofs Without"},{"speaker":"I've pulled out of the corpus","startTime":1494.75,"endTime":1496.88,"body":"Words\" volume I mentioned,"},{"speaker":"I've pulled out of the corpus","startTime":1496.88,"endTime":1498.89,"body":"by Roger B. Nelson,"},{"speaker":"I've pulled out of the corpus","startTime":1498.89,"endTime":1501.44,"body":"this one's credited to Frank Burk,"},{"speaker":"I've pulled out of the corpus","startTime":1501.44,"endTime":1502.58,"body":"this is doing something similar again."},{"speaker":"I've pulled out of the corpus","startTime":1502.58,"endTime":1504.263,"body":"So it's a proof of Pythagoras' theorem,"},{"speaker":"I've pulled out of the corpus","startTime":1504.263,"endTime":1507.23,"body":"that a-squared plus b-squared"},{"speaker":"I've pulled out of the corpus","startTime":1504.263,"endTime":1507.23,"body":"is equal to c-squared."},{"speaker":"I've pulled out of the corpus","startTime":1507.23,"endTime":1509.63,"body":"The idea is that you take"},{"speaker":"I've pulled out of the corpus","startTime":1507.23,"endTime":1509.63,"body":"your original triangle"},{"speaker":"I've pulled out of the corpus","startTime":1509.63,"endTime":1510.463,"body":"and you inflate it,"},{"speaker":"I've pulled out of the corpus","startTime":1510.463,"endTime":1512.45,"body":"you times it b once to"},{"speaker":"I've pulled out of the corpus","startTime":1510.463,"endTime":1512.45,"body":"get the triangle here,"},{"speaker":"I've pulled out of the corpus","startTime":1512.45,"endTime":1514.7,"body":"you times it by a once"},{"speaker":"I've pulled out of the corpus","startTime":1512.45,"endTime":1514.7,"body":"to get the triangle here,"},{"speaker":"I've pulled out of the corpus","startTime":1514.7,"endTime":1516.59,"body":"you can put those two together"},{"speaker":"I've pulled out of the corpus","startTime":1516.59,"endTime":1518.72,"body":"to form a bigger triangle there."},{"speaker":"I've pulled out of the corpus","startTime":1518.72,"endTime":1520.01,"body":"If you multiply it by c,"},{"speaker":"I've pulled out of the corpus","startTime":1520.01,"endTime":1521.84,"body":"you get this triangle over here,"},{"speaker":"I've pulled out of the corpus","startTime":1521.84,"endTime":1522.673,"body":"and it turns out"},{"speaker":"I've pulled out of the corpus","startTime":1522.673,"endTime":1524.996,"body":"those two triangles are"},{"speaker":"I've pulled out of the corpus","startTime":1522.673,"endTime":1524.996,"body":"mirror images of each other,"},{"speaker":"I've pulled out of the corpus","startTime":1524.996,"endTime":1526.91,"body":"they're the same triangle,"},{"speaker":"I've pulled out of the corpus","startTime":1526.91,"endTime":1528.11,"body":"they're right-angled triangles"},{"speaker":"I've pulled out of the corpus","startTime":1528.11,"endTime":1530.81,"body":"with a-c and b-c on both those,"},{"speaker":"I've pulled out of the corpus","startTime":1530.81,"endTime":1533.03,"body":"but the hypotenuse for one"},{"speaker":"I've pulled out of the corpus","startTime":1530.81,"endTime":1533.03,"body":"is a-squared plus b-squared,"},{"speaker":"I've pulled out of the corpus","startTime":1533.03,"endTime":1534.74,"body":"the hypotenuse for the other is c-squared,"},{"speaker":"I've pulled out of the corpus","startTime":1534.74,"endTime":1535.573,"body":"so they're equal,"},{"speaker":"I've pulled out of the corpus","startTime":1535.573,"endTime":1537.41,"body":"bah, brilliant picture."},{"speaker":"I've pulled out of the corpus","startTime":1537.41,"endTime":1540.23,"body":"Notice how similar it"},{"speaker":"I've pulled out of the corpus","startTime":1537.41,"endTime":1540.23,"body":"is to this IKEA diagram,"},{"speaker":"I've pulled out of the corpus","startTime":1540.23,"endTime":1542.06,"body":"you've got arrows"},{"speaker":"I've pulled out of the corpus","startTime":1540.23,"endTime":1542.06,"body":"telling you to do things,"},{"speaker":"I've pulled out of the corpus","startTime":1542.06,"endTime":1543.77,"body":"you're moving this triangle,"},{"speaker":"I've pulled out of the corpus","startTime":1543.77,"endTime":1546.59,"body":"and inflating it, and"},{"speaker":"I've pulled out of the corpus","startTime":1543.77,"endTime":1546.59,"body":"twisting it over here,"},{"speaker":"I've pulled out of the corpus","startTime":1546.59,"endTime":1548.63,"body":"and then you're comparing"},{"speaker":"I've pulled out of the corpus","startTime":1546.59,"endTime":1548.63,"body":"two mirror images"},{"speaker":"I've pulled out of the corpus","startTime":1548.63,"endTime":1551.303,"body":"and seeing that they're"},{"speaker":"I've pulled out of the corpus","startTime":1548.63,"endTime":1551.303,"body":"actually the same idea,"},{"speaker":"I've pulled out of the corpus","startTime":1552.14,"endTime":1553.4,"body":"it's fantastic,"},{"speaker":"I've pulled out of the corpus","startTime":1553.4,"endTime":1557.09,"body":"I think it uses visual means"},{"speaker":"I've pulled out of the corpus","startTime":1553.4,"endTime":1557.09,"body":"to communicate instructions."},{"speaker":"I've pulled out of the corpus","startTime":1557.09,"endTime":1560.24,"body":"And it might be that some"},{"speaker":"I've pulled out of the corpus","startTime":1557.09,"endTime":1560.24,"body":"of these picture proofs"},{"speaker":"I've pulled out of the corpus","startTime":1560.24,"endTime":1563.06,"body":"require a certain level"},{"speaker":"I've pulled out of the corpus","startTime":1560.24,"endTime":1563.06,"body":"of mathematical training"},{"speaker":"I've pulled out of the corpus","startTime":1563.06,"endTime":1565.1,"body":"to understand how to do it,"},{"speaker":"I've pulled out of the corpus","startTime":1565.1,"endTime":1567.53,"body":"but I argue that they're not as mysterious"},{"speaker":"I've pulled out of the corpus","startTime":1567.53,"endTime":1570.083,"body":"as that standard view"},{"speaker":"I've pulled out of the corpus","startTime":1567.53,"endTime":1570.083,"body":"would have us believe."},{"speaker":"I've pulled out of the corpus","startTime":1572.3,"endTime":1574.49,"body":"I'm done now with picture proofs,"},{"speaker":"I've pulled out of the corpus","startTime":1574.49,"endTime":1576.65,"body":"let me say a last few minutes of things"},{"speaker":"I've pulled out of the corpus","startTime":1576.65,"endTime":1580.73,"body":"about the implications for education."},{"speaker":"I've pulled out of the corpus","startTime":1580.73,"endTime":1584.0,"body":"This is based on a paper with"},{"speaker":"I've pulled out of the corpus","startTime":1580.73,"endTime":1584.0,"body":"my colleague Keith Weber,"},{"speaker":"I've pulled out of the corpus","startTime":1584.0,"endTime":1586.31,"body":"from Rutgers University."},{"speaker":"I've pulled out of the corpus","startTime":1586.31,"endTime":1588.44,"body":"He looks very beautiful"},{"speaker":"I've pulled out of the corpus","startTime":1586.31,"endTime":1588.44,"body":"and clean shaven there,"},{"speaker":"I've pulled out of the corpus","startTime":1588.44,"endTime":1590.481,"body":"now he looks like a hobo."},{"speaker":"I've pulled out of the corpus","startTime":1590.481,"endTime":1591.35,"body":"(audience laughs)"},{"speaker":"I've pulled out of the corpus","startTime":1591.35,"endTime":1593.025,"body":"I hope he's watching."},{"speaker":"I've pulled out of the corpus","startTime":1593.025,"endTime":1595.358,"body":"(he laughs)"},{"speaker":"I've pulled out of the corpus","startTime":1597.2,"endTime":1599.0,"body":"We've seen that the recipe model"},{"speaker":"I've pulled out of the corpus","startTime":1599.0,"endTime":1600.41,"body":"that I've been talking about"},{"speaker":"I've pulled out of the corpus","startTime":1600.41,"endTime":1601.91,"body":"provides quite a different perspective"},{"speaker":"I've pulled out of the corpus","startTime":1601.91,"endTime":1603.62,"body":"than the standard view of proofs"},{"speaker":"I've pulled out of the corpus","startTime":1603.62,"endTime":1605.933,"body":"as logical sequences of assertions,"},{"speaker":"I've pulled out of the corpus","startTime":1606.95,"endTime":1610.19,"body":"so let me finish on"},{"speaker":"I've pulled out of the corpus","startTime":1606.95,"endTime":1610.19,"body":"telling you what that means"},{"speaker":"I've pulled out of the corpus","startTime":1610.19,"endTime":1611.453,"body":"for teaching proofs."},{"speaker":"I've pulled out of the corpus","startTime":1613.67,"endTime":1618.02,"body":"One common model of proof teaching"},{"speaker":"I've pulled out of the corpus","startTime":1618.02,"endTime":1620.75,"body":"amongst mathematics education specialists"},{"speaker":"I've pulled out of the corpus","startTime":1620.75,"endTime":1623.42,"body":"is often called something"},{"speaker":"I've pulled out of the corpus","startTime":1620.75,"endTime":1623.42,"body":"like the apprenticeship model,"},{"speaker":"I've pulled out of the corpus","startTime":1623.42,"endTime":1625.91,"body":"that you've got a little junior apprentice"},{"speaker":"I've pulled out of the corpus","startTime":1625.91,"endTime":1627.76,"body":"who you're teaching how to do things."},{"speaker":"I've pulled out of the corpus","startTime":1628.73,"endTime":1630.56,"body":"Without going into the"},{"speaker":"I've pulled out of the corpus","startTime":1628.73,"endTime":1630.56,"body":"strengths and weaknesses"},{"speaker":"I've pulled out of the corpus","startTime":1630.56,"endTime":1632.24,"body":"of an idea like that,"},{"speaker":"one thing is obvious","startTime":1633.5,"endTime":1635.96,"body":"if you want something like this to work,"},{"speaker":"one thing is obvious","startTime":1635.96,"endTime":1638.9,"body":"our conception of what"},{"speaker":"one thing is obvious","startTime":1635.96,"endTime":1638.9,"body":"mathematicians are doing,"},{"speaker":"one thing is obvious","startTime":1638.9,"endTime":1641.21,"body":"what you're apprenticing them to,"},{"speaker":"one thing is obvious","startTime":1641.21,"endTime":1642.313,"body":"has to be at least somewhat accurate"},{"speaker":"one thing is obvious","startTime":1642.313,"endTime":1644.45,"body":"to what they're actually doing."},{"speaker":"one thing is obvious","startTime":1644.45,"endTime":1647.15,"body":"If mathematicians are"},{"speaker":"one thing is obvious","startTime":1644.45,"endTime":1647.15,"body":"writing instructions,"},{"speaker":"one thing is obvious","startTime":1647.15,"endTime":1649.43,"body":"are using instructions in certain ways,"},{"speaker":"one thing is obvious","startTime":1649.43,"endTime":1651.05,"body":"but say that they aren't,"},{"speaker":"one thing is obvious","startTime":1651.05,"endTime":1653.72,"body":"then this presents a"},{"speaker":"one thing is obvious","startTime":1651.05,"endTime":1653.72,"body":"very confusing situation"},{"speaker":"one thing is obvious","startTime":1653.72,"endTime":1656.66,"body":"for a mathematics"},{"speaker":"one thing is obvious","startTime":1653.72,"endTime":1656.66,"body":"undergraduate or a school pupil"},{"speaker":"one thing is obvious","startTime":1656.66,"endTime":1658.553,"body":"trying to learn how to write proofs."},{"speaker":"one thing is obvious","startTime":1660.02,"endTime":1661.85,"body":"So if we conceptualize proofs"},{"speaker":"one thing is obvious","startTime":1661.85,"endTime":1663.263,"body":"as sequences of assertions,"},{"speaker":"one thing is obvious","startTime":1663.263,"endTime":1666.83,"body":"then we miss out on an"},{"speaker":"one thing is obvious","startTime":1663.263,"endTime":1666.83,"body":"important aspect of proving"},{"speaker":"one thing is obvious","startTime":1666.83,"endTime":1668.183,"body":"that should just be taught."},{"speaker":"one thing is obvious","startTime":1670.28,"endTime":1673.64,"body":"For example, think of some"},{"speaker":"one thing is obvious","startTime":1670.28,"endTime":1673.64,"body":"of the instructions I listed"},{"speaker":"one thing is obvious","startTime":1673.64,"endTime":1675.56,"body":"in the corpus linguistics study,"},{"speaker":"one thing is obvious","startTime":1675.56,"endTime":1677.99,"body":"what's the difference between"},{"speaker":"one thing is obvious","startTime":1675.56,"endTime":1677.99,"body":"an instruction to let,"},{"speaker":"one thing is obvious","startTime":1677.99,"endTime":1682.553,"body":"or to suppose, assume,"},{"speaker":"one thing is obvious","startTime":1677.99,"endTime":1682.553,"body":"take, fix, or consider?"},{"speaker":"one thing is obvious","startTime":1683.69,"endTime":1685.34,"body":"All of those things do very similar,"},{"speaker":"one thing is obvious","startTime":1685.34,"endTime":1687.35,"body":"but subtly different things"},{"speaker":"one thing is obvious","startTime":1687.35,"endTime":1688.73,"body":"in a mathematical proof."},{"speaker":"one thing is obvious","startTime":1688.73,"endTime":1691.25,"body":"The difference between"},{"speaker":"one thing is obvious","startTime":1688.73,"endTime":1691.25,"body":"let, or assume, or suppose,"},{"speaker":"one thing is obvious","startTime":1691.25,"endTime":1692.63,"body":"they're stylistic differences"},{"speaker":"one thing is obvious","startTime":1692.63,"endTime":1695.84,"body":"and differences in what they"},{"speaker":"one thing is obvious","startTime":1692.63,"endTime":1695.84,"body":"exactly mean for a proof,"},{"speaker":"one thing is obvious","startTime":1695.84,"endTime":1699.68,"body":"that mathematicians have"},{"speaker":"one thing is obvious","startTime":1695.84,"endTime":1699.68,"body":"to get quite competent"},{"speaker":"one thing is obvious","startTime":1699.68,"endTime":1701.66,"body":"at distinguishing between,"},{"speaker":"one thing is obvious","startTime":1701.66,"endTime":1703.643,"body":"so proof writing is very important."},{"speaker":"one thing is obvious","startTime":1705.68,"endTime":1708.8,"body":"I don't think I've ever heard"},{"speaker":"one thing is obvious","startTime":1705.68,"endTime":1708.8,"body":"of students being taught"},{"speaker":"one thing is obvious","startTime":1708.8,"endTime":1711.11,"body":"how to distinguish between those things,"},{"speaker":"one thing is obvious","startTime":1711.11,"endTime":1712.91,"body":"but clearly, it's something"},{"speaker":"one thing is obvious","startTime":1711.11,"endTime":1712.91,"body":"that could be taught"},{"speaker":"one thing is obvious","startTime":1712.91,"endTime":1714.923,"body":"as part of your introduction to proving."},{"speaker":"one thing is obvious","startTime":1718.13,"endTime":1722.9,"body":"A final lesson for teaching proofs,"},{"speaker":"one thing is obvious","startTime":1722.9,"endTime":1724.13,"body":"or something like this,"},{"speaker":"one thing is obvious","startTime":1724.13,"endTime":1729.13,"body":"is that the recipe model"},{"speaker":"one thing is obvious","startTime":1724.13,"endTime":1729.13,"body":"says something different"},{"speaker":"one thing is obvious","startTime":1729.17,"endTime":1731.723,"body":"about how we read and evaluate proofs."},{"speaker":"one thing is obvious","startTime":1732.65,"endTime":1734.33,"body":"If you've got a sequence of assertions,"},{"speaker":"one thing is obvious","startTime":1734.33,"endTime":1735.29,"body":"the kind of questions"},{"speaker":"one thing is obvious","startTime":1735.29,"endTime":1736.79,"body":"that you're asking about your assertions"},{"speaker":"one thing is obvious","startTime":1736.79,"endTime":1737.75,"body":"is something like,"},{"speaker":"one thing is obvious","startTime":1737.75,"endTime":1739.4,"body":"okay, is this assertion true,"},{"speaker":"one thing is obvious","startTime":1739.4,"endTime":1741.05,"body":"does it follow from the previous things"},{"speaker":"one thing is obvious","startTime":1741.05,"endTime":1743.15,"body":"by the accepted canons of logic?"},{"speaker":"one thing is obvious","startTime":1743.15,"endTime":1744.8,"body":"And those are great questions"},{"speaker":"one thing is obvious","startTime":1744.8,"endTime":1747.65,"body":"for some bits of mathematical reasoning,"},{"speaker":"one thing is obvious","startTime":1747.65,"endTime":1750.44,"body":"but if you're being given instructions,"},{"speaker":"one thing is obvious","startTime":1750.44,"endTime":1751.4,"body":"the kind of questions"},{"speaker":"one thing is obvious","startTime":1751.4,"endTime":1752.63,"body":"you want to ask about those instructions"},{"speaker":"one thing is obvious","startTime":1752.63,"endTime":1754.04,"body":"are more like,"},{"speaker":"one thing is obvious","startTime":1754.04,"endTime":1755.96,"body":"what action am I being asked to carry out,"},{"speaker":"one thing is obvious","startTime":1755.96,"endTime":1757.49,"body":"what does this mean exactly,"},{"speaker":"one thing is obvious","startTime":1757.49,"endTime":1760.28,"body":"can I do it, do I know how to do it,"},{"speaker":"one thing is obvious","startTime":1760.28,"endTime":1761.96,"body":"does it produce the right outcome,"},{"speaker":"one thing is obvious","startTime":1761.96,"endTime":1762.793,"body":"does it produce"},{"speaker":"one thing is obvious","startTime":1762.793,"endTime":1765.86,"body":"what the mathematician has"},{"speaker":"one thing is obvious","startTime":1762.793,"endTime":1765.86,"body":"told me it's going to produce,"},{"speaker":"one thing is obvious","startTime":1765.86,"endTime":1768.74,"body":"in what way does it guarantee"},{"speaker":"one thing is obvious","startTime":1765.86,"endTime":1768.74,"body":"the right kind of properties"},{"speaker":"one thing is obvious","startTime":1768.74,"endTime":1770.813,"body":"for what I'm expecting it to do?"},{"speaker":"one thing is obvious","startTime":1772.61,"endTime":1774.29,"body":"We believe that proof pedagogy"},{"speaker":"one thing is obvious","startTime":1774.29,"endTime":1776.33,"body":"should explicitly teach students"},{"speaker":"one thing is obvious","startTime":1776.33,"endTime":1780.14,"body":"about how to use these instructions"},{"speaker":"one thing is obvious","startTime":1776.33,"endTime":1780.14,"body":"appropriately, correctly"},{"speaker":"one thing is obvious","startTime":1780.14,"endTime":1783.29,"body":"and how to read and"},{"speaker":"one thing is obvious","startTime":1780.14,"endTime":1783.29,"body":"evaluate them correctly,"},{"speaker":"one thing is obvious","startTime":1783.29,"endTime":1786.647,"body":"alongside how to reason about assertions,"},{"speaker":"one thing is obvious","startTime":1786.647,"endTime":1787.797,"body":"and the logic of those."},{"speaker":"one thing is obvious","startTime":1788.69,"endTime":1791.09,"body":"So in a final slogan,"},{"speaker":"one thing is obvious","startTime":1791.09,"endTime":1793.46,"body":"if you want students to"},{"speaker":"one thing is obvious","startTime":1791.09,"endTime":1793.46,"body":"learn how proofs work,"},{"speaker":"one thing is obvious","startTime":1793.46,"endTime":1795.2,"body":"maybe you should teach"},{"speaker":"one thing is obvious","startTime":1793.46,"endTime":1795.2,"body":"them how proofs work"},{"speaker":"one thing is obvious","startTime":1795.2,"endTime":1796.223,"body":"in the actual sense."},{"speaker":"one thing is obvious","startTime":1797.33,"endTime":1799.78,"body":"Thank you very much, that's me done."},{"speaker":"one thing is obvious","startTime":1799.78,"endTime":1802.78,"body":"(audience applauds)"},{"speaker":"one thing is obvious","startTime":1806.323,"endTime":1807.77,"body":"- I'm going to start, if I may,"},{"speaker":"one thing is obvious","startTime":1807.77,"endTime":1810.53,"body":"with a couple of questions"},{"speaker":"one thing is obvious","startTime":1807.77,"endTime":1810.53,"body":"from the online audience,"},{"speaker":"one thing is obvious","startTime":1810.53,"endTime":1811.363,"body":"while people in here"},{"speaker":"one thing is obvious","startTime":1811.363,"endTime":1814.4,"body":"are thinking whether"},{"speaker":"one thing is obvious","startTime":1811.363,"endTime":1814.4,"body":"they want to ask things,"},{"speaker":"one thing is obvious","startTime":1814.4,"endTime":1817.1,"body":"so I'll give you two"},{"speaker":"one thing is obvious","startTime":1814.4,"endTime":1817.1,"body":"questions to get you going."},{"speaker":"one thing is obvious","startTime":1817.1,"endTime":1819.83,"body":"The first one, from Jeremy Gibbons,"},{"speaker":"one thing is obvious","startTime":1819.83,"endTime":1823.25,"body":"asks about the distinction between"},{"speaker":"one thing is obvious","startTime":1823.25,"endTime":1825.53,"body":"what he calls classical"},{"speaker":"one thing is obvious","startTime":1823.25,"endTime":1825.53,"body":"and constructive proofs,"},{"speaker":"one thing is obvious","startTime":1825.53,"endTime":1828.2,"body":"so many proofs of the"},{"speaker":"one thing is obvious","startTime":1825.53,"endTime":1828.2,"body":"existence of something,"},{"speaker":"one thing is obvious","startTime":1828.2,"endTime":1831.83,"body":"can be expressed as the recipe"},{"speaker":"one thing is obvious","startTime":1828.2,"endTime":1831.83,"body":"for constructing the thing,"},{"speaker":"one thing is obvious","startTime":1831.83,"endTime":1834.293,"body":"so he's asking about"},{"speaker":"one thing is obvious","startTime":1831.83,"endTime":1834.293,"body":"the distinction there."},{"speaker":"one thing is obvious","startTime":1835.19,"endTime":1838.587,"body":"And then, actually, it's"},{"speaker":"one thing is obvious","startTime":1835.19,"endTime":1838.587,"body":"the same person who adds,"},{"speaker":"one thing is obvious","startTime":1838.587,"endTime":1839.757,"body":"\"Apropos a proof has to be"},{"speaker":"one thing is obvious","startTime":1839.757,"endTime":1841.197,"body":"\"a logical sequence of assertions,"},{"speaker":"one thing is obvious","startTime":1841.197,"endTime":1844.017,"body":"\"shouldn't we distinguish"},{"speaker":"one thing is obvious","startTime":1841.197,"endTime":1844.017,"body":"between actual proofs"},{"speaker":"one thing is obvious","startTime":1844.017,"endTime":1846.227,"body":"\"and logical models of proofs?\""},{"speaker":"one thing is obvious","startTime":1847.37,"endTime":1848.203,"body":"In other words,"},{"speaker":"one thing is obvious","startTime":1848.203,"endTime":1849.47,"body":"we don't want to confuse"},{"speaker":"one thing is obvious","startTime":1848.203,"endTime":1849.47,"body":"the map with the territory"},{"speaker":"one thing is obvious","startTime":1849.47,"endTime":1850.52,"body":"is what he's,"},{"speaker":"one thing is obvious","startTime":1850.52,"endTime":1853.37,"body":"so yeah, what do you think?"},{"speaker":"one thing is obvious","startTime":1853.37,"endTime":1855.56,"body":"- Brilliant, those are great questions."},{"speaker":"one thing is obvious","startTime":1855.56,"endTime":1857.36,"body":"The idea of constructive mathematics"},{"speaker":"one thing is obvious","startTime":1857.36,"endTime":1861.65,"body":"is that you have to be"},{"speaker":"one thing is obvious","startTime":1857.36,"endTime":1861.65,"body":"able to give real items"},{"speaker":"one thing is obvious","startTime":1861.65,"endTime":1862.483,"body":"that you're talking about,"},{"speaker":"one thing is obvious","startTime":1862.483,"endTime":1864.92,"body":"and you can't use various techniques"},{"speaker":"one thing is obvious","startTime":1864.92,"endTime":1869.92,"body":"that are common in normal,"},{"speaker":"one thing is obvious","startTime":1864.92,"endTime":1869.92,"body":"classical mathematics out there,"},{"speaker":"one thing is obvious","startTime":1870.44,"endTime":1872.06,"body":"and that was a very big debate"},{"speaker":"one thing is obvious","startTime":1872.06,"endTime":1874.22,"body":"in the philosophy of mathematics,"},{"speaker":"one thing is obvious","startTime":1874.22,"endTime":1875.78,"body":"and amongst mathematicians for a while,"},{"speaker":"one thing is obvious","startTime":1875.78,"endTime":1878.78,"body":"and there's very famous"},{"speaker":"one thing is obvious","startTime":1875.78,"endTime":1878.78,"body":"constructivist mathematicians"},{"speaker":"one thing is obvious","startTime":1878.78,"endTime":1881.03,"body":"who are trying to restrict themselves"},{"speaker":"one thing is obvious","startTime":1881.03,"endTime":1882.95,"body":"from not using various principles"},{"speaker":"one thing is obvious","startTime":1882.95,"endTime":1884.361,"body":"that other mathematicians use,"},{"speaker":"one thing is obvious","startTime":1884.361,"endTime":1887.303,"body":"like proof by contradiction,"},{"speaker":"one thing is obvious","startTime":1884.361,"endTime":1887.303,"body":"or something like that."},{"speaker":"one thing is obvious","startTime":1888.35,"endTime":1889.7,"body":"For the constructivists,"},{"speaker":"one thing is obvious","startTime":1889.7,"endTime":1891.92,"body":"what I'm saying would be very welcome,"},{"speaker":"one thing is obvious","startTime":1891.92,"endTime":1894.17,"body":"they would love something"},{"speaker":"one thing is obvious","startTime":1891.92,"endTime":1894.17,"body":"like this, and say,"},{"speaker":"one thing is obvious","startTime":1894.17,"endTime":1895.76,"body":"oh yeah, that's what I'm talking about,"},{"speaker":"one thing is obvious","startTime":1895.76,"endTime":1898.28,"body":"I'm really glad you've"},{"speaker":"one thing is obvious","startTime":1895.76,"endTime":1898.28,"body":"shown this corpus study,"},{"speaker":"one thing is obvious","startTime":1898.28,"endTime":1899.81,"body":"that's fantastic."},{"speaker":"one thing is obvious","startTime":1899.81,"endTime":1901.34,"body":"But they might not be so happy"},{"speaker":"one thing is obvious","startTime":1901.34,"endTime":1903.98,"body":"with how bold I am with"},{"speaker":"one thing is obvious","startTime":1901.34,"endTime":1903.98,"body":"some of the other things,"},{"speaker":"one thing is obvious","startTime":1903.98,"endTime":1905.33,"body":"because I think classical mathematics"},{"speaker":"one thing is obvious","startTime":1905.33,"endTime":1906.713,"body":"looks a lot like this too."},{"speaker":"one thing is obvious","startTime":1908.21,"endTime":1912.11,"body":"The study we've done is of"},{"speaker":"one thing is obvious","startTime":1908.21,"endTime":1912.11,"body":"mathematics papers of all kinds,"},{"speaker":"one thing is obvious","startTime":1912.11,"endTime":1915.01,"body":"we've not just picked out"},{"speaker":"one thing is obvious","startTime":1912.11,"endTime":1915.01,"body":"constructive mathematics papers,"},{"speaker":"one thing is obvious","startTime":1916.76,"endTime":1918.71,"body":"so I think you can give recipe proofs"},{"speaker":"one thing is obvious","startTime":1918.71,"endTime":1919.94,"body":"using some of this language,"},{"speaker":"one thing is obvious","startTime":1919.94,"endTime":1921.8,"body":"say, doing a proof by contradiction"},{"speaker":"one thing is obvious","startTime":1921.8,"endTime":1922.633,"body":"just as well,"},{"speaker":"one thing is obvious","startTime":1922.633,"endTime":1924.44,"body":"you can suppose this is true,"},{"speaker":"one thing is obvious","startTime":1924.44,"endTime":1925.64,"body":"derive a contradiction,"},{"speaker":"one thing is obvious","startTime":1925.64,"endTime":1929.78,"body":"and then observe that it must"},{"speaker":"one thing is obvious","startTime":1925.64,"endTime":1929.78,"body":"have been false all along."},{"speaker":"one thing is obvious","startTime":1929.78,"endTime":1932.0,"body":"So I think what I'm describing"},{"speaker":"one thing is obvious","startTime":1932.0,"endTime":1934.75,"body":"is a model of how proofs"},{"speaker":"one thing is obvious","startTime":1932.0,"endTime":1934.75,"body":"are written in various ways,"},{"speaker":"one thing is obvious","startTime":1934.75,"endTime":1937.44,"body":"or how proofs are used"},{"speaker":"one thing is obvious","startTime":1934.75,"endTime":1937.44,"body":"to function cognitively"},{"speaker":"one thing is obvious","startTime":1938.36,"endTime":1939.62,"body":"that is somewhat different"},{"speaker":"one thing is obvious","startTime":1939.62,"endTime":1941.42,"body":"from the metaphysical debates"},{"speaker":"one thing is obvious","startTime":1941.42,"endTime":1943.73,"body":"that were motivating"},{"speaker":"one thing is obvious","startTime":1943.73,"endTime":1947.75,"body":"the intuitionist, constructivist tradition"},{"speaker":"one thing is obvious","startTime":1947.75,"endTime":1949.91,"body":"to break away from the"},{"speaker":"one thing is obvious","startTime":1947.75,"endTime":1949.91,"body":"classical mathematicians,"},{"speaker":"one thing is obvious","startTime":1949.91,"endTime":1951.06,"body":"and that sort of thing."},{"speaker":"one thing is obvious","startTime":1952.37,"endTime":1953.203,"body":"On the second question,"},{"speaker":"one thing is obvious","startTime":1953.203,"endTime":1954.036,"body":"I really like that question,"},{"speaker":"one thing is obvious","startTime":1954.036,"endTime":1956.84,"body":"about why I'm talking about models,"},{"speaker":"one thing is obvious","startTime":1956.84,"endTime":1958.82,"body":"is essentially the thing,"},{"speaker":"one thing is obvious","startTime":1958.82,"endTime":1962.93,"body":"which is that the standard"},{"speaker":"one thing is obvious","startTime":1958.82,"endTime":1962.93,"body":"view I've been talking about"},{"speaker":"one thing is obvious","startTime":1962.93,"endTime":1965.36,"body":"takes itself not even to"},{"speaker":"one thing is obvious","startTime":1962.93,"endTime":1965.36,"body":"be a model of reasoning,"},{"speaker":"one thing is obvious","startTime":1965.36,"endTime":1966.41,"body":"it just takes itself"},{"speaker":"one thing is obvious","startTime":1966.41,"endTime":1969.71,"body":"to be describing the reality"},{"speaker":"one thing is obvious","startTime":1966.41,"endTime":1969.71,"body":"of what mathematics is."},{"speaker":"one thing is obvious","startTime":1969.71,"endTime":1970.61,"body":"I'm calling it a model"},{"speaker":"one thing is obvious","startTime":1970.61,"endTime":1973.01,"body":"because I think it's great"},{"speaker":"one thing is obvious","startTime":1970.61,"endTime":1973.01,"body":"and has lots of successes,"},{"speaker":"one thing is obvious","startTime":1973.01,"endTime":1975.5,"body":"but is actually not always accurate"},{"speaker":"one thing is obvious","startTime":1975.5,"endTime":1977.51,"body":"and not always faithful to how proofs are."},{"speaker":"one thing is obvious","startTime":1977.51,"endTime":1978.83,"body":"So I'm presenting a rival model"},{"speaker":"one thing is obvious","startTime":1978.83,"endTime":1980.66,"body":"because I actually think"},{"speaker":"one thing is obvious","startTime":1978.83,"endTime":1980.66,"body":"both models are useful"},{"speaker":"one thing is obvious","startTime":1980.66,"endTime":1982.07,"body":"for different things,"},{"speaker":"one thing is obvious","startTime":1982.07,"endTime":1984.743,"body":"so that's why I'm talking"},{"speaker":"one thing is obvious","startTime":1982.07,"endTime":1984.743,"body":"about these two things."},{"speaker":"one thing is obvious","startTime":1985.79,"endTime":1986.623,"body":"- Thank you."},{"speaker":"one thing is obvious","startTime":1986.623,"endTime":1988.1,"body":"If I could ask,"},{"speaker":"one thing is obvious","startTime":1988.1,"endTime":1990.41,"body":"what's the translation of the phrase,"},{"speaker":"one thing is obvious","startTime":1990.41,"endTime":1992.0,"body":"without loss of generality,"},{"speaker":"one thing is obvious","startTime":1992.0,"endTime":1992.933,"body":"to a recipe?"},{"speaker":"one thing is obvious","startTime":1993.86,"endTime":1996.41,"body":"That's a very common phrase"},{"speaker":"one thing is obvious","startTime":1993.86,"endTime":1996.41,"body":"in mathematical proofs."},{"speaker":"one thing is obvious","startTime":1998.981,"endTime":2001.231,"body":"(he sighs)"},{"speaker":"one thing is obvious","startTime":2002.35,"endTime":2003.183,"body":"Sorry to put you on the spot like that."},{"speaker":"one thing is obvious","startTime":2003.183,"endTime":2005.74,"body":"- No, no, it's a lovely question."},{"speaker":"one thing is obvious","startTime":2005.74,"endTime":2006.88,"body":"Without loss of generality"},{"speaker":"one thing is obvious","startTime":2006.88,"endTime":2010.06,"body":"means that you've started"},{"speaker":"one thing is obvious","startTime":2006.88,"endTime":2010.06,"body":"talking about a particular thing,"},{"speaker":"one thing is obvious","startTime":2010.06,"endTime":2012.67,"body":"but you can just ignore some"},{"speaker":"one thing is obvious","startTime":2010.06,"endTime":2012.67,"body":"of the special features of it,"},{"speaker":"one thing is obvious","startTime":2012.67,"endTime":2014.053,"body":"that's the idea of it."},{"speaker":"one thing is obvious","startTime":2016.09,"endTime":2016.99,"body":"It's like if a recipe"},{"speaker":"one thing is obvious","startTime":2016.99,"endTime":2019.3,"body":"told you how to decorate one biscuit,"},{"speaker":"one thing is obvious","startTime":2019.3,"endTime":2022.06,"body":"but actually, all of the"},{"speaker":"one thing is obvious","startTime":2019.3,"endTime":2022.06,"body":"biscuits need decorating,"},{"speaker":"one thing is obvious","startTime":2022.06,"endTime":2023.8,"body":"so without loss of generality,"},{"speaker":"one thing is obvious","startTime":2023.8,"endTime":2026.05,"body":"start drawing a Christmas"},{"speaker":"one thing is obvious","startTime":2023.8,"endTime":2026.05,"body":"tree on this biscuit,"},{"speaker":"one thing is obvious","startTime":2026.05,"endTime":2027.73,"body":"but notice you could draw a reindeer,"},{"speaker":"one thing is obvious","startTime":2027.73,"endTime":2031.54,"body":"or a snowflake, or something else,"},{"speaker":"one thing is obvious","startTime":2031.54,"endTime":2033.28,"body":"there's a whole range of Christmas things,"},{"speaker":"one thing is obvious","startTime":2033.28,"endTime":2034.78,"body":"you still want to use the same icing,"},{"speaker":"one thing is obvious","startTime":2034.78,"endTime":2036.94,"body":"you still want to use this,"},{"speaker":"one thing is obvious","startTime":2034.78,"endTime":2036.94,"body":"you still want to do that,"},{"speaker":"one thing is obvious","startTime":2036.94,"endTime":2037.94,"body":"something like this."},{"speaker":"one thing is obvious","startTime":2039.589,"endTime":2040.422,"body":"- [First Questioner] Thank you for that"},{"speaker":"one thing is obvious","startTime":2040.422,"endTime":2042.343,"body":"very interesting talk you just gave."},{"speaker":"one thing is obvious","startTime":2043.63,"endTime":2046.175,"body":"I found the last slide you"},{"speaker":"one thing is obvious","startTime":2043.63,"endTime":2046.175,"body":"had shown very interesting,"},{"speaker":"one thing is obvious","startTime":2046.175,"endTime":2050.08,"body":"to contrast the two models of proofs,"},{"speaker":"one thing is obvious","startTime":2050.08,"endTime":2051.37,"body":"as you've just said,"},{"speaker":"one thing is obvious","startTime":2051.37,"endTime":2056.37,"body":"between the assertions"},{"speaker":"one thing is obvious","startTime":2051.37,"endTime":2056.37,"body":"and the recipe model."},{"speaker":"one thing is obvious","startTime":2057.64,"endTime":2058.473,"body":"Wouldn't it be possible"},{"speaker":"one thing is obvious","startTime":2058.473,"endTime":2061.06,"body":"that they're just two"},{"speaker":"one thing is obvious","startTime":2058.473,"endTime":2061.06,"body":"sides of the same coin?"},{"speaker":"one thing is obvious","startTime":2061.06,"endTime":2063.22,"body":"For example, in the last slide,"},{"speaker":"one thing is obvious","startTime":2063.22,"endTime":2065.05,"body":"you had shown,"},{"speaker":"one thing is obvious","startTime":2065.05,"endTime":2066.61,"body":"in the assertion model,"},{"speaker":"one thing is obvious","startTime":2066.61,"endTime":2067.99,"body":"it's, is this true,"},{"speaker":"one thing is obvious","startTime":2067.99,"endTime":2069.74,"body":"where in the other one it could be,"},{"speaker":"one thing is obvious","startTime":2070.87,"endTime":2073.12,"body":"do I see how this works?"},{"speaker":"one thing is obvious","startTime":2073.12,"endTime":2075.103,"body":"The statement in the assertion model,"},{"speaker":"one thing is obvious","startTime":2076.028,"endTime":2080.8,"body":"does this follow from the previous steps,"},{"speaker":"one thing is obvious","startTime":2080.8,"endTime":2082.27,"body":"in the other model could be,"},{"speaker":"one thing is obvious","startTime":2082.27,"endTime":2084.07,"body":"do I know how to do this?"},{"speaker":"one thing is obvious","startTime":2084.07,"endTime":2087.16,"body":"So could there be an"},{"speaker":"one thing is obvious","startTime":2084.07,"endTime":2087.16,"body":"equivalence between the two,"},{"speaker":"one thing is obvious","startTime":2087.16,"endTime":2090.73,"body":"instead of being two rival models"},{"speaker":"one thing is obvious","startTime":2090.73,"endTime":2092.803,"body":"of how to understand proofs?"},{"speaker":"one thing is obvious","startTime":2093.696,"endTime":2094.93,"body":"- That's a wonderful question,"},{"speaker":"one thing is obvious","startTime":2094.93,"endTime":2098.08,"body":"I've been bigging up the"},{"speaker":"one thing is obvious","startTime":2094.93,"endTime":2098.08,"body":"challenge between the two"},{"speaker":"one thing is obvious","startTime":2098.08,"endTime":2099.49,"body":"for dramatic effect,"},{"speaker":"one thing is obvious","startTime":2099.49,"endTime":2100.6,"body":"and I think you're totally right,"},{"speaker":"one thing is obvious","startTime":2100.6,"endTime":2103.03,"body":"that, really, you need to"},{"speaker":"one thing is obvious","startTime":2100.6,"endTime":2103.03,"body":"understand both of these things"},{"speaker":"one thing is obvious","startTime":2103.03,"endTime":2105.34,"body":"as going on at the same time."},{"speaker":"one thing is obvious","startTime":2105.34,"endTime":2106.6,"body":"If you read proofs,"},{"speaker":"one thing is obvious","startTime":2106.6,"endTime":2108.01,"body":"I've given you lots of data"},{"speaker":"one thing is obvious","startTime":2108.01,"endTime":2110.14,"body":"about how instructions are used in proofs,"},{"speaker":"one thing is obvious","startTime":2110.14,"endTime":2111.28,"body":"if you open up a textbook,"},{"speaker":"one thing is obvious","startTime":2111.28,"endTime":2112.18,"body":"you might find some of those,"},{"speaker":"one thing is obvious","startTime":2112.18,"endTime":2113.8,"body":"and you might find a proof"},{"speaker":"one thing is obvious","startTime":2112.18,"endTime":2113.8,"body":"that doesn't use any,"},{"speaker":"one thing is obvious","startTime":2113.8,"endTime":2115.0,"body":"it just uses those things."},{"speaker":"one thing is obvious","startTime":2115.0,"endTime":2117.22,"body":"And I think if you want"},{"speaker":"one thing is obvious","startTime":2115.0,"endTime":2117.22,"body":"to, say, teach mathematics,"},{"speaker":"one thing is obvious","startTime":2117.22,"endTime":2119.8,"body":"or understand the reasoning"},{"speaker":"one thing is obvious","startTime":2117.22,"endTime":2119.8,"body":"in all those things,"},{"speaker":"one thing is obvious","startTime":2119.8,"endTime":2122.92,"body":"both of those ways of"},{"speaker":"one thing is obvious","startTime":2119.8,"endTime":2122.92,"body":"doing things are essential"},{"speaker":"one thing is obvious","startTime":2122.92,"endTime":2124.543,"body":"in understanding proofs."},{"speaker":"one thing is obvious","startTime":2126.19,"endTime":2127.24,"body":"In fact, that's one of the reasons"},{"speaker":"one thing is obvious","startTime":2127.24,"endTime":2128.59,"body":"I like talking about them as models,"},{"speaker":"one thing is obvious","startTime":2128.59,"endTime":2132.22,"body":"is because different models are"},{"speaker":"one thing is obvious","startTime":2128.59,"endTime":2132.22,"body":"useful for different things,"},{"speaker":"one thing is obvious","startTime":2132.22,"endTime":2134.2,"body":"so I think that,"},{"speaker":"one thing is obvious","startTime":2134.2,"endTime":2136.75,"body":"yeah, you really want to"},{"speaker":"one thing is obvious","startTime":2134.2,"endTime":2136.75,"body":"have those in parallel."},{"speaker":"one thing is obvious","startTime":2136.75,"endTime":2138.49,"body":"In fact, I think there's"},{"speaker":"one thing is obvious","startTime":2136.75,"endTime":2138.49,"body":"various other models"},{"speaker":"one thing is obvious","startTime":2138.49,"endTime":2140.413,"body":"that also provide useful insights."},{"speaker":"one thing is obvious","startTime":2143.754,"endTime":2144.757,"body":"- [First Questioner] They both can be used"},{"speaker":"one thing is obvious","startTime":2144.757,"endTime":2145.93,"body":"for different things,"},{"speaker":"one thing is obvious","startTime":2145.93,"endTime":2148.42,"body":"or they are really the same thing"},{"speaker":"one thing is obvious","startTime":2148.42,"endTime":2151.96,"body":"but just can more easily be"},{"speaker":"one thing is obvious","startTime":2148.42,"endTime":2151.96,"body":"applied in different situations?"},{"speaker":"one thing is obvious","startTime":2151.96,"endTime":2154.63,"body":"- I don't think they're"},{"speaker":"one thing is obvious","startTime":2151.96,"endTime":2154.63,"body":"exactly the same thing."},{"speaker":"one thing is obvious","startTime":2154.63,"endTime":2159.63,"body":"In the same way as if you"},{"speaker":"one thing is obvious","startTime":2154.63,"endTime":2159.63,"body":"wanted to build a model, say,"},{"speaker":"one thing is obvious","startTime":2159.88,"endTime":2160.783,"body":"of a bridge,"},{"speaker":"one thing is obvious","startTime":2161.83,"endTime":2163.33,"body":"you might build a cardboard model"},{"speaker":"one thing is obvious","startTime":2163.33,"endTime":2164.62,"body":"to find out one thing about it,"},{"speaker":"one thing is obvious","startTime":2164.62,"endTime":2168.52,"body":"and you might do a detailed"},{"speaker":"one thing is obvious","startTime":2164.62,"endTime":2168.52,"body":"technical drawing of it"},{"speaker":"one thing is obvious","startTime":2168.52,"endTime":2170.68,"body":"to find out about its"},{"speaker":"one thing is obvious","startTime":2168.52,"endTime":2170.68,"body":"load-bearing properties"},{"speaker":"one thing is obvious","startTime":2170.68,"endTime":2171.67,"body":"to make sure it won't fall down,"},{"speaker":"one thing is obvious","startTime":2171.67,"endTime":2172.63,"body":"in another way."},{"speaker":"one thing is obvious","startTime":2172.63,"endTime":2173.5,"body":"These both tell you"},{"speaker":"one thing is obvious","startTime":2173.5,"endTime":2176.02,"body":"about the building of"},{"speaker":"one thing is obvious","startTime":2173.5,"endTime":2176.02,"body":"the bridge afterwards,"},{"speaker":"one thing is obvious","startTime":2176.02,"endTime":2177.07,"body":"but different features of it,"},{"speaker":"one thing is obvious","startTime":2177.07,"endTime":2179.98,"body":"and that's how I'm seeing the"},{"speaker":"one thing is obvious","startTime":2177.07,"endTime":2179.98,"body":"two different models working,"},{"speaker":"one thing is obvious","startTime":2179.98,"endTime":2182.23,"body":"so I think they're telling"},{"speaker":"one thing is obvious","startTime":2179.98,"endTime":2182.23,"body":"you about different,"},{"speaker":"one thing is obvious","startTime":2182.23,"endTime":2184.75,"body":"sometimes incompatible,"},{"speaker":"one thing is obvious","startTime":2182.23,"endTime":2184.75,"body":"features of the models."},{"speaker":"one thing is obvious","startTime":2184.75,"endTime":2186.28,"body":"Sometimes they'll tell you about things"},{"speaker":"one thing is obvious","startTime":2186.28,"endTime":2189.073,"body":"that the other one won't, and vice-versa."},{"speaker":"one thing is obvious","startTime":2190.99,"endTime":2193.0,"body":"- [Questioner] I'm going"},{"speaker":"one thing is obvious","startTime":2190.99,"endTime":2193.0,"body":"to pick up on one sentence"},{"speaker":"one thing is obvious","startTime":2193.0,"endTime":2193.99,"body":"in the middle of the talk,"},{"speaker":"one thing is obvious","startTime":2193.99,"endTime":2196.24,"body":"so apologies if you don't"},{"speaker":"one thing is obvious","startTime":2193.99,"endTime":2196.24,"body":"have anything to say"},{"speaker":"one thing is obvious","startTime":2196.24,"endTime":2197.863,"body":"about this one sentence,"},{"speaker":"one thing is obvious","startTime":2198.94,"endTime":2201.34,"body":"but yeah, you said that"},{"speaker":"one thing is obvious","startTime":2198.94,"endTime":2201.34,"body":"the picture diagrams"},{"speaker":"one thing is obvious","startTime":2201.34,"endTime":2204.1,"body":"are easy for people who have had practice,"},{"speaker":"one thing is obvious","startTime":2204.1,"endTime":2206.35,"body":"and I'm just wondering if"},{"speaker":"one thing is obvious","startTime":2204.1,"endTime":2206.35,"body":"you've thought anything"},{"speaker":"one thing is obvious","startTime":2206.35,"endTime":2210.04,"body":"about how that idea of"},{"speaker":"one thing is obvious","startTime":2206.35,"endTime":2210.04,"body":"practice plays out in maths,"},{"speaker":"one thing is obvious","startTime":2210.04,"endTime":2211.78,"body":"especially with resilience,"},{"speaker":"one thing is obvious","startTime":2211.78,"endTime":2213.94,"body":"and, we've all seen \"Bake Off,\""},{"speaker":"one thing is obvious","startTime":2213.94,"endTime":2215.23,"body":"sometimes it goes wrong,"},{"speaker":"one thing is obvious","startTime":2215.23,"endTime":2218.37,"body":"but you come back next week"},{"speaker":"one thing is obvious","startTime":2215.23,"endTime":2218.37,"body":"and everybody loves you,"},{"speaker":"one thing is obvious","startTime":2218.37,"endTime":2221.863,"body":"so how do we get that kind"},{"speaker":"one thing is obvious","startTime":2218.37,"endTime":2221.863,"body":"of mentality in maths?"},{"speaker":"one thing is obvious","startTime":2223.347,"endTime":2225.373,"body":"- That's a fantastic question."},{"speaker":"one thing is obvious","startTime":2233.8,"endTime":2238.45,"body":"I think that talking about"},{"speaker":"one thing is obvious","startTime":2233.8,"endTime":2238.45,"body":"needing experience and resilience"},{"speaker":"one thing is obvious","startTime":2238.45,"endTime":2243.45,"body":"is part of the reason that"},{"speaker":"one thing is obvious","startTime":2238.45,"endTime":2243.45,"body":"there are mathematicians"},{"speaker":"one thing is obvious","startTime":2243.94,"endTime":2245.86,"body":"who say things that no"},{"speaker":"one thing is obvious","startTime":2243.94,"endTime":2245.86,"body":"one else can understand,"},{"speaker":"one thing is obvious","startTime":2245.86,"endTime":2247.86,"body":"or very few other people can understand."},{"speaker":"one thing is obvious","startTime":2249.14,"endTime":2250.75,"body":"To learn how to do some of the things"},{"speaker":"one thing is obvious","startTime":2250.75,"endTime":2252.7,"body":"that you're being instructed to do"},{"speaker":"one thing is obvious","startTime":2252.7,"endTime":2254.23,"body":"requires quite a lot of experience"},{"speaker":"one thing is obvious","startTime":2254.23,"endTime":2255.943,"body":"of what's meant by things."},{"speaker":"one thing is obvious","startTime":2257.38,"endTime":2260.26,"body":"I've picked out some very"},{"speaker":"one thing is obvious","startTime":2257.38,"endTime":2260.26,"body":"straightforward instructions,"},{"speaker":"one thing is obvious","startTime":2260.26,"endTime":2262.48,"body":"those core set are things"},{"speaker":"one thing is obvious","startTime":2260.26,"endTime":2262.48,"body":"that everyone should learn,"},{"speaker":"one thing is obvious","startTime":2262.48,"endTime":2265.24,"body":"but if you are working"},{"speaker":"one thing is obvious","startTime":2262.48,"endTime":2265.24,"body":"in, say, combinatorics,"},{"speaker":"one thing is obvious","startTime":2265.24,"endTime":2268.9,"body":"you might be told to color"},{"speaker":"one thing is obvious","startTime":2265.24,"endTime":2268.9,"body":"these nodes in this way,"},{"speaker":"one thing is obvious","startTime":2268.9,"endTime":2271.45,"body":"and if you've not done"},{"speaker":"one thing is obvious","startTime":2268.9,"endTime":2271.45,"body":"combinatorics before,"},{"speaker":"one thing is obvious","startTime":2271.45,"endTime":2272.857,"body":"you might get our your"},{"speaker":"one thing is obvious","startTime":2271.45,"endTime":2272.857,"body":"coloring pencils, or something,"},{"speaker":"one thing is obvious","startTime":2272.857,"endTime":2274.09,"body":"but that's not the idea,"},{"speaker":"one thing is obvious","startTime":2274.09,"endTime":2275.35,"body":"you're doing something"},{"speaker":"one thing is obvious","startTime":2275.35,"endTime":2278.23,"body":"that's a technical"},{"speaker":"one thing is obvious","startTime":2275.35,"endTime":2278.23,"body":"understanding of combinatorics"},{"speaker":"one thing is obvious","startTime":2278.23,"endTime":2279.55,"body":"that's telling you what to do."},{"speaker":"one thing is obvious","startTime":2279.55,"endTime":2283.39,"body":"So I think that you need"},{"speaker":"one thing is obvious","startTime":2279.55,"endTime":2283.39,"body":"experience of these things"},{"speaker":"one thing is obvious","startTime":2283.39,"endTime":2285.52,"body":"because you need to understand"},{"speaker":"one thing is obvious","startTime":2285.52,"endTime":2287.47,"body":"what exactly you're being asked to do."},{"speaker":"one thing is obvious","startTime":2287.47,"endTime":2289.03,"body":"And in exactly the same way,"},{"speaker":"one thing is obvious","startTime":2289.03,"endTime":2290.74,"body":"what you're describing"},{"speaker":"one thing is obvious","startTime":2289.03,"endTime":2290.74,"body":"on \"Bake Off\" is amazing,"},{"speaker":"one thing is obvious","startTime":2290.74,"endTime":2292.24,"body":"because if you've watched \"Bake Off,\""},{"speaker":"one thing is obvious","startTime":2292.24,"endTime":2294.79,"body":"I do like a bit of \"Bake Off,\""},{"speaker":"one thing is obvious","startTime":2294.79,"endTime":2297.73,"body":"sometimes, they're given"},{"speaker":"one thing is obvious","startTime":2294.79,"endTime":2297.73,"body":"these very obscure recipes"},{"speaker":"one thing is obvious","startTime":2297.73,"endTime":2299.95,"body":"where they get, say, two instructions"},{"speaker":"one thing is obvious","startTime":2299.95,"endTime":2301.18,"body":"and they have to make this thing,"},{"speaker":"one thing is obvious","startTime":2301.18,"endTime":2302.83,"body":"and some of them don't know how to do it,"},{"speaker":"one thing is obvious","startTime":2302.83,"endTime":2305.05,"body":"they don't know how to make"},{"speaker":"one thing is obvious","startTime":2302.83,"endTime":2305.05,"body":"that particular pastry,"},{"speaker":"one thing is obvious","startTime":2305.05,"endTime":2306.55,"body":"or that particular icing,"},{"speaker":"one thing is obvious","startTime":2306.55,"endTime":2308.35,"body":"and they have to go and spy"},{"speaker":"one thing is obvious","startTime":2306.55,"endTime":2308.35,"body":"on what everyone else is doing"},{"speaker":"one thing is obvious","startTime":2308.35,"endTime":2309.49,"body":"to figure it out."},{"speaker":"one thing is obvious","startTime":2309.49,"endTime":2311.26,"body":"If you don't know what the"},{"speaker":"one thing is obvious","startTime":2309.49,"endTime":2311.26,"body":"instruction is telling you to do,"},{"speaker":"one thing is obvious","startTime":2311.26,"endTime":2313.783,"body":"you don't how to do these things."},{"speaker":"one thing is obvious","startTime":2315.25,"endTime":2316.15,"body":"Sorry, I don't have much to say"},{"speaker":"one thing is obvious","startTime":2316.15,"endTime":2317.98,"body":"on the resilience aspect of it,"},{"speaker":"one thing is obvious","startTime":2317.98,"endTime":2320.14,"body":"but it's certainly something"},{"speaker":"one thing is obvious","startTime":2317.98,"endTime":2320.14,"body":"that needs training"},{"speaker":"one thing is obvious","startTime":2320.14,"endTime":2321.853,"body":"to get to the bottom of."},{"speaker":"one thing is obvious","startTime":2322.78,"endTime":2325.57,"body":"- [Questioner] Thank you for"},{"speaker":"one thing is obvious","startTime":2322.78,"endTime":2325.57,"body":"a very interesting lecture."},{"speaker":"one thing is obvious","startTime":2325.57,"endTime":2329.533,"body":"We had a prize-giving"},{"speaker":"one thing is obvious","startTime":2325.57,"endTime":2329.533,"body":"here earlier this evening,"},{"speaker":"one thing is obvious","startTime":2330.91,"endTime":2335.91,"body":"is there a theorem that"},{"speaker":"one thing is obvious","startTime":2330.91,"endTime":2335.91,"body":"you would give a prize to,"},{"speaker":"one thing is obvious","startTime":2336.19,"endTime":2339.16,"body":"remembering that these are"},{"speaker":"one thing is obvious","startTime":2336.19,"endTime":2339.16,"body":"lectures for the general public,"},{"speaker":"one thing is obvious","startTime":2339.16,"endTime":2340.21,"body":"is there a theorem"},{"speaker":"one thing is obvious","startTime":2340.21,"endTime":2342.88,"body":"that the general public can get hold of"},{"speaker":"one thing is obvious","startTime":2342.88,"endTime":2345.52,"body":"that you would award a prize,"},{"speaker":"one thing is obvious","startTime":2345.52,"endTime":2347.72,"body":"and why would you give"},{"speaker":"one thing is obvious","startTime":2345.52,"endTime":2347.72,"body":"that theorem a prize?"},{"speaker":"one thing is obvious","startTime":2348.7,"endTime":2349.723,"body":"- Oh gosh."},{"speaker":"one thing is obvious","startTime":2354.544,"endTime":2359.544,"body":"Keeping in spirit with what"},{"speaker":"one thing is obvious","startTime":2354.544,"endTime":2359.544,"body":"I've been telling you about,"},{"speaker":"one thing is obvious","startTime":2359.71,"endTime":2361.03,"body":"let me, rather than picking a theorem,"},{"speaker":"one thing is obvious","startTime":2361.03,"endTime":2361.87,"body":"if you don't mind,"},{"speaker":"one thing is obvious","startTime":2361.87,"endTime":2363.01,"body":"pick a book of theorems,"},{"speaker":"one thing is obvious","startTime":2363.01,"endTime":2365.05,"body":"because something I was quoting here,"},{"speaker":"one thing is obvious","startTime":2365.05,"endTime":2366.07,"body":"and mentioned in passing,"},{"speaker":"one thing is obvious","startTime":2366.07,"endTime":2368.89,"body":"but I really love to tell people about"},{"speaker":"one thing is obvious","startTime":2368.89,"endTime":2372.64,"body":"is these books, volumes of picture proofs"},{"speaker":"one thing is obvious","startTime":2372.64,"endTime":2373.773,"body":"by Roger B. Nelson,"},{"speaker":"one thing is obvious","startTime":2373.773,"endTime":2375.28,"body":"and they're called \"Proofs Without Words,\""},{"speaker":"one thing is obvious","startTime":2375.28,"endTime":2376.87,"body":"there's three volumes now,"},{"speaker":"one thing is obvious","startTime":2376.87,"endTime":2378.94,"body":"and these are, to my mind,"},{"speaker":"one thing is obvious","startTime":2378.94,"endTime":2380.69,"body":"the most wonderful thing out there."},{"speaker":"one thing is obvious","startTime":2382.45,"endTime":2386.2,"body":"What it contains is a whole"},{"speaker":"one thing is obvious","startTime":2382.45,"endTime":2386.2,"body":"load of pictorial proofs"},{"speaker":"one thing is obvious","startTime":2386.2,"endTime":2387.82,"body":"of many different results,"},{"speaker":"one thing is obvious","startTime":2387.82,"endTime":2390.37,"body":"and I can't understand half of them,"},{"speaker":"one thing is obvious","startTime":2390.37,"endTime":2391.42,"body":"I can't figure them out,"},{"speaker":"one thing is obvious","startTime":2391.42,"endTime":2393.4,"body":"and that's what makes them so brilliant,"},{"speaker":"one thing is obvious","startTime":2393.4,"endTime":2397.81,"body":"is because there's clearly"},{"speaker":"one thing is obvious","startTime":2393.4,"endTime":2397.81,"body":"some bits of thinking in there"},{"speaker":"one thing is obvious","startTime":2397.81,"endTime":2400.18,"body":"that you need to spend some time"},{"speaker":"one thing is obvious","startTime":2400.18,"endTime":2402.783,"body":"really understanding what's"},{"speaker":"one thing is obvious","startTime":2400.18,"endTime":2402.783,"body":"going on in these pictures."},{"speaker":"one thing is obvious","startTime":2404.53,"endTime":2406.45,"body":"In some sense, they're proofs,"},{"speaker":"one thing is obvious","startTime":2406.45,"endTime":2408.07,"body":"whether or not they're really proofs"},{"speaker":"one thing is obvious","startTime":2408.07,"endTime":2410.68,"body":"if you have to really spend so"},{"speaker":"one thing is obvious","startTime":2408.07,"endTime":2410.68,"body":"much time figuring them out,"},{"speaker":"one thing is obvious","startTime":2410.68,"endTime":2411.513,"body":"I don't know."},{"speaker":"one thing is obvious","startTime":2411.513,"endTime":2413.83,"body":"So apologies if that's not one theorem,"},{"speaker":"one thing is obvious","startTime":2413.83,"endTime":2414.663,"body":"but many theorems,"},{"speaker":"one thing is obvious","startTime":2414.663,"endTime":2418.84,"body":"but if you want to go and"},{"speaker":"one thing is obvious","startTime":2414.663,"endTime":2418.84,"body":"look at something that I love,"},{"speaker":"one thing is obvious","startTime":2418.84,"endTime":2420.76,"body":"pick \"Proofs Without Words.\""},{"speaker":"one thing is obvious","startTime":2420.76,"endTime":2422.89,"body":"- [Questioner] Hi, thank"},{"speaker":"one thing is obvious","startTime":2420.76,"endTime":2422.89,"body":"you for the lovely talk."},{"speaker":"one thing is obvious","startTime":2422.89,"endTime":2423.76,"body":"I was just wondering"},{"speaker":"one thing is obvious","startTime":2423.76,"endTime":2427.57,"body":"about how much the language"},{"speaker":"one thing is obvious","startTime":2423.76,"endTime":2427.57,"body":"of how we write proofs"},{"speaker":"one thing is obvious","startTime":2427.57,"endTime":2428.403,"body":"has changed,"},{"speaker":"one thing is obvious","startTime":2428.403,"endTime":2429.73,"body":"or not so much changed, but developed,"},{"speaker":"one thing is obvious","startTime":2429.73,"endTime":2431.23,"body":"because we are writing proofs"},{"speaker":"one thing is obvious","startTime":2431.23,"endTime":2433.57,"body":"for other people to understand,"},{"speaker":"one thing is obvious","startTime":2433.57,"endTime":2437.77,"body":"so we're writing for"},{"speaker":"one thing is obvious","startTime":2433.57,"endTime":2437.77,"body":"the reader in a sense,"},{"speaker":"one thing is obvious","startTime":2437.77,"endTime":2438.73,"body":"as opposed to, say,"},{"speaker":"one thing is obvious","startTime":2438.73,"endTime":2441.07,"body":"if we were trying to make"},{"speaker":"one thing is obvious","startTime":2438.73,"endTime":2441.07,"body":"a computer prove something,"},{"speaker":"one thing is obvious","startTime":2441.07,"endTime":2443.86,"body":"and you do all the"},{"speaker":"one thing is obvious","startTime":2441.07,"endTime":2443.86,"body":"constructions in a side panel,"},{"speaker":"one thing is obvious","startTime":2443.86,"endTime":2445.24,"body":"and then the proof itself"},{"speaker":"one thing is obvious","startTime":2445.24,"endTime":2447.49,"body":"would be more of a one, two, three"},{"speaker":"one thing is obvious","startTime":2447.49,"endTime":2451.42,"body":"in the A implies B,"},{"speaker":"one thing is obvious","startTime":2447.49,"endTime":2451.42,"body":"implies C, implies done,"},{"speaker":"one thing is obvious","startTime":2451.42,"endTime":2452.253,"body":"kind of thing,"},{"speaker":"one thing is obvious","startTime":2452.253,"endTime":2454.72,"body":"so how much the language of proof,"},{"speaker":"one thing is obvious","startTime":2454.72,"endTime":2456.88,"body":"or the style of writing proofs"},{"speaker":"one thing is obvious","startTime":2456.88,"endTime":2459.22,"body":"means that we are trying to communicate,"},{"speaker":"one thing is obvious","startTime":2459.22,"endTime":2461.65,"body":"as opposed to prove the maths, in a sense,"},{"speaker":"one thing is obvious","startTime":2461.65,"endTime":2463.303,"body":"or alongside it somehow?"},{"speaker":"one thing is obvious","startTime":2464.89,"endTime":2469.45,"body":"- Wonderful, I think it"},{"speaker":"one thing is obvious","startTime":2464.89,"endTime":2469.45,"body":"has to be totally essential"},{"speaker":"one thing is obvious","startTime":2469.45,"endTime":2470.283,"body":"when you're describing them,"},{"speaker":"one thing is obvious","startTime":2470.283,"endTime":2472.0,"body":"it's about communication."},{"speaker":"one thing is obvious","startTime":2472.0,"endTime":2472.87,"body":"So what I've shown you here"},{"speaker":"one thing is obvious","startTime":2472.87,"endTime":2474.97,"body":"is mathematicians writing"},{"speaker":"one thing is obvious","startTime":2472.87,"endTime":2474.97,"body":"for mathematicians,"},{"speaker":"one thing is obvious","startTime":2474.97,"endTime":2479.11,"body":"a study of the language of"},{"speaker":"one thing is obvious","startTime":2474.97,"endTime":2479.11,"body":"professional maths papers."},{"speaker":"one thing is obvious","startTime":2479.11,"endTime":2481.06,"body":"My experience looking for these things"},{"speaker":"one thing is obvious","startTime":2481.06,"endTime":2483.28,"body":"is that if you open up textbooks"},{"speaker":"one thing is obvious","startTime":2483.28,"endTime":2485.5,"body":"aimed at undergraduate students,"},{"speaker":"one thing is obvious","startTime":2485.5,"endTime":2487.15,"body":"you'll find even more of"},{"speaker":"one thing is obvious","startTime":2485.5,"endTime":2487.15,"body":"this sort of language,"},{"speaker":"one thing is obvious","startTime":2487.15,"endTime":2488.89,"body":"that it's actually more common there,"},{"speaker":"one thing is obvious","startTime":2488.89,"endTime":2493.33,"body":"and there, the entire style"},{"speaker":"one thing is obvious","startTime":2493.33,"endTime":2495.31,"body":"is intended to be understandable,"},{"speaker":"one thing is obvious","startTime":2495.31,"endTime":2497.08,"body":"most of the time, for most textbooks,"},{"speaker":"one thing is obvious","startTime":2497.08,"endTime":2499.153,"body":"though it's not always true."},{"speaker":"one thing is obvious","startTime":2500.35,"endTime":2502.36,"body":"So that must be true,"},{"speaker":"one thing is obvious","startTime":2502.36,"endTime":2504.46,"body":"that they're trying to write"},{"speaker":"one thing is obvious","startTime":2502.36,"endTime":2504.46,"body":"in a more accessible style,"},{"speaker":"one thing is obvious","startTime":2504.46,"endTime":2505.293,"body":"and they've landed on this"},{"speaker":"one thing is obvious","startTime":2505.293,"endTime":2508.42,"body":"as a better way of communicating"},{"speaker":"one thing is obvious","startTime":2505.293,"endTime":2508.42,"body":"what they're doing."},{"speaker":"one thing is obvious","startTime":2508.42,"endTime":2510.79,"body":"Now, in contrast to a computer proof,"},{"speaker":"one thing is obvious","startTime":2510.79,"endTime":2513.7,"body":"it depends on the underlying"},{"speaker":"one thing is obvious","startTime":2510.79,"endTime":2513.7,"body":"algebra, and the theory,"},{"speaker":"one thing is obvious","startTime":2513.7,"endTime":2514.54,"body":"whether you're doing set theory,"},{"speaker":"one thing is obvious","startTime":2514.54,"endTime":2516.957,"body":"or higher order-type theory,"},{"speaker":"one thing is obvious","startTime":2514.54,"endTime":2516.957,"body":"or something like that,"},{"speaker":"one thing is obvious","startTime":2516.957,"endTime":2521.803,"body":"as what your computer is working on."},{"speaker":"one thing is obvious","startTime":2523.63,"endTime":2525.04,"body":"I've also spent some time"},{"speaker":"one thing is obvious","startTime":2525.04,"endTime":2527.38,"body":"looking at what instructions"},{"speaker":"one thing is obvious","startTime":2525.04,"endTime":2527.38,"body":"they're using in these things,"},{"speaker":"one thing is obvious","startTime":2527.38,"endTime":2530.83,"body":"and mostly, the language"},{"speaker":"one thing is obvious","startTime":2527.38,"endTime":2530.83,"body":"is more assertoric,"},{"speaker":"one thing is obvious","startTime":2530.83,"endTime":2532.24,"body":"it's more standard,"},{"speaker":"one thing is obvious","startTime":2532.24,"endTime":2535.09,"body":"because it's trying to convert mathematics"},{"speaker":"one thing is obvious","startTime":2535.09,"endTime":2538.21,"body":"closer to that logical"},{"speaker":"one thing is obvious","startTime":2535.09,"endTime":2538.21,"body":"ideal of the standard model."},{"speaker":"one thing is obvious","startTime":2538.21,"endTime":2540.223,"body":"Now, it's not actually always the case,"},{"speaker":"one thing is obvious","startTime":2541.12,"endTime":2544.09,"body":"the most popular theorem"},{"speaker":"one thing is obvious","startTime":2541.12,"endTime":2544.09,"body":"checker at the moment,"},{"speaker":"one thing is obvious","startTime":2544.09,"endTime":2545.68,"body":"I would say, is Lean,"},{"speaker":"one thing is obvious","startTime":2545.68,"endTime":2546.52,"body":"it's very popular,"},{"speaker":"one thing is obvious","startTime":2546.52,"endTime":2548.17,"body":"especially here in London, I believe,"},{"speaker":"one thing is obvious","startTime":2548.17,"endTime":2552.07,"body":"and Lean has several"},{"speaker":"one thing is obvious","startTime":2548.17,"endTime":2552.07,"body":"instructions still inside it."},{"speaker":"one thing is obvious","startTime":2552.07,"endTime":2554.02,"body":"So if you look at the coding in Lean,"},{"speaker":"one thing is obvious","startTime":2554.02,"endTime":2556.27,"body":"you'll find some instructions still there,"},{"speaker":"one thing is obvious","startTime":2556.27,"endTime":2557.71,"body":"so they're not entirely absent,"},{"speaker":"one thing is obvious","startTime":2557.71,"endTime":2561.01,"body":"so it's not like that is the"},{"speaker":"one thing is obvious","startTime":2557.71,"endTime":2561.01,"body":"pinnacle of the standard view,"},{"speaker":"one thing is obvious","startTime":2561.01,"endTime":2562.93,"body":"but it's less so."},{"speaker":"one thing is obvious","startTime":2562.93,"endTime":2566.2,"body":"So communicating to students,"},{"speaker":"one thing is obvious","startTime":2562.93,"endTime":2566.2,"body":"other mathematicians,"},{"speaker":"one thing is obvious","startTime":2566.2,"endTime":2567.37,"body":"or a computer"},{"speaker":"one thing is obvious","startTime":2567.37,"endTime":2569.98,"body":"will have different ways of"},{"speaker":"one thing is obvious","startTime":2567.37,"endTime":2569.98,"body":"writing associated with it,"},{"speaker":"one thing is obvious","startTime":2569.98,"endTime":2571.533,"body":"so you must be absolutely right."},{"speaker":"one thing is obvious","startTime":2572.47,"endTime":2573.64,"body":"- [Questioner] Thank you."},{"speaker":"one thing is obvious","startTime":2573.64,"endTime":2576.22,"body":"One word I haven't heard"},{"speaker":"one thing is obvious","startTime":2573.64,"endTime":2576.22,"body":"you use is algorithm."},{"speaker":"one thing is obvious","startTime":2576.22,"endTime":2578.62,"body":"To me, recipe equals algorithm,"},{"speaker":"one thing is obvious","startTime":2578.62,"endTime":2581.743,"body":"do this, do this, do this, and then stop."},{"speaker":"one thing is obvious","startTime":2582.67,"endTime":2585.97,"body":"Euclid, book one, proposition"},{"speaker":"one thing is obvious","startTime":2582.67,"endTime":2585.97,"body":"one is in two parts:"},{"speaker":"one thing is obvious","startTime":2585.97,"endTime":2588.79,"body":"how do you construct an"},{"speaker":"one thing is obvious","startTime":2585.97,"endTime":2588.79,"body":"equilateral triangle?"},{"speaker":"one thing is obvious","startTime":2588.79,"endTime":2590.38,"body":"Do this, do this, do this,"},{"speaker":"one thing is obvious","startTime":2590.38,"endTime":2591.91,"body":"and then, separately,"},{"speaker":"one thing is obvious","startTime":2591.91,"endTime":2595.06,"body":"he proves that the algorithm works."},{"speaker":"one thing is obvious","startTime":2595.06,"endTime":2599.14,"body":"So for a recipe for your rhubarb crumble,"},{"speaker":"one thing is obvious","startTime":2599.14,"endTime":2601.57,"body":"you do this, do this, do this,"},{"speaker":"one thing is obvious","startTime":2601.57,"endTime":2602.86,"body":"but the proof is different,"},{"speaker":"one thing is obvious","startTime":2602.86,"endTime":2605.543,"body":"the proof of the pudding is in the eating."},{"speaker":"one thing is obvious","startTime":2605.543,"endTime":2607.642,"body":"(audience laughs)"},{"speaker":"one thing is obvious","startTime":2607.642,"endTime":2611.203,"body":"So to me, recipe does not equal proof."},{"speaker":"one thing is obvious","startTime":2615.61,"endTime":2616.91,"body":"- If you look at the text,"},{"speaker":"one thing is obvious","startTime":2617.953,"endTime":2620.47,"body":"so example, the"},{"speaker":"one thing is obvious","startTime":2617.953,"endTime":2620.47,"body":"Bolzano-Weierstrass theorem,"},{"speaker":"one thing is obvious","startTime":2620.47,"endTime":2622.39,"body":"if you look at what they're doing,"},{"speaker":"one thing is obvious","startTime":2622.39,"endTime":2623.71,"body":"it will give you the recipe"},{"speaker":"one thing is obvious","startTime":2623.71,"endTime":2626.2,"body":"of how to construct the relevant thing,"},{"speaker":"one thing is obvious","startTime":2626.2,"endTime":2628.06,"body":"it doesn't then, at the end, go,"},{"speaker":"one thing is obvious","startTime":2628.06,"endTime":2630.16,"body":"now let me provide a certification,"},{"speaker":"one thing is obvious","startTime":2630.16,"endTime":2633.49,"body":"you're meant to understand"},{"speaker":"one thing is obvious","startTime":2630.16,"endTime":2633.49,"body":"and extract from the recipe"},{"speaker":"one thing is obvious","startTime":2633.49,"endTime":2635.11,"body":"why it's guaranteeing the properties"},{"speaker":"one thing is obvious","startTime":2635.11,"endTime":2636.88,"body":"that you end up with at the end."},{"speaker":"one thing is obvious","startTime":2636.88,"endTime":2639.28,"body":"So it's written in a recipe style,"},{"speaker":"one thing is obvious","startTime":2639.28,"endTime":2641.08,"body":"but the understanding of the recipe,"},{"speaker":"one thing is obvious","startTime":2642.22,"endTime":2645.43,"body":"the enacting bit of doing"},{"speaker":"one thing is obvious","startTime":2642.22,"endTime":2645.43,"body":"the mathematics yourself"},{"speaker":"one thing is obvious","startTime":2645.43,"endTime":2649.27,"body":"is not just stepwise obedience,"},{"speaker":"one thing is obvious","startTime":2649.27,"endTime":2650.98,"body":"but it's also the understanding"},{"speaker":"one thing is obvious","startTime":2650.98,"endTime":2652.09,"body":"of why you're doing each step."},{"speaker":"one thing is obvious","startTime":2652.09,"endTime":2653.5,"body":"So you may be right"},{"speaker":"one thing is obvious","startTime":2653.5,"endTime":2656.41,"body":"that there's a slight sense in"},{"speaker":"one thing is obvious","startTime":2653.5,"endTime":2656.41,"body":"which the analogy breaks down"},{"speaker":"one thing is obvious","startTime":2656.41,"endTime":2657.243,"body":"at the end there,"},{"speaker":"one thing is obvious","startTime":2657.243,"endTime":2661.24,"body":"but I think, essentially,"},{"speaker":"one thing is obvious","startTime":2661.24,"endTime":2663.07,"body":"you're still going through a recipe"},{"speaker":"one thing is obvious","startTime":2663.07,"endTime":2663.91,"body":"and you're understanding"},{"speaker":"one thing is obvious","startTime":2663.91,"endTime":2665.71,"body":"why you're doing each of the things."}]}