[
  {
    "start": 5.1,
    "end": 6.27,
    "text": "Hi, welcome everybody."
  },
  {
    "start": 7.09,
    "end": 14.75,
    "text": "This is a new sequence or new edition of our premises about logic podcast."
  },
  {
    "start": 15.65,
    "end": 27.37,
    "text": "so where we try to discuss the bit off background and you don't have any guest it's just us too Dennis and myself and maybe Dennis can ask start asking the question."
  },
  {
    "start": 28.43,
    "end": 36.17,
    "text": "Yeah there something I wanted to ask for quite awhile now And that's your take on dependent type theory."
  },
  {
    "start": 37.71,
    "end": 42.69,
    "text": "There is a lot of talk about it, but I'm not entirely sure about the details."
  },
  {
    "start": 43.19,
    "end": 48.35,
    "text": "so maybe let me start even more basic... Even if that's a repetition with type theory per se."
  },
  {
    "start": 49.29,
    "end": 53.05,
    "text": "So for me types are like sets But they aren't sets."
  },
  {
    "start": 53.19,
    "end": 56.11,
    "text": "there's this difference you always make"
  },
  {
    "start": 62.23,
    "end": 62.91,
    "text": "in type theory."
  },
  {
    "start": 63.33,
    "end": 70.75,
    "text": "The word set is such a nice short word that we use it as well, an entire theory."
  },
  {
    "start": 71.01,
    "end": 88.49,
    "text": "so there's lots of situations where type theoreticians called typesets and the start was Pierre-Martin Lueff himself I mean a pair introduced the type theory at two levels."
  },
  {
    "start": 88.57,
    "end": 93.97,
    "text": "he had theory of types, which we will now would call a logical framework."
  },
  {
    "start": 94.57,
    "end": 98.89,
    "text": "And within it he defined what is the set for sets versus types?"
  },
  {
    "start": 100.09,
    "end": 108.53,
    "text": "So so that's said as overloaded and more modern or recently in homotopy type theory."
  },
  {
    "start": 109.75,
    "end": 112.75,
    "text": "The word set is used to describe a type."
  },
  {
    "start": 114.23,
    "end": 129.009,
    "text": "So the word set is not so clear to identify what does it mean."
  },
  {
    "start": 129.93,
    "end": 135.17,
    "text": "But Curly, we can distinguish a set theory and tag theory..."
  },
  {
    "start": 136.71,
    "end": 137.99,
    "text": "Maybe at that tangent."
  },
  {
    "start": 138.05,
    "end": 155.41,
    "text": "for second I think this is same phenomena as that the curly brackets in Laatech their own command because I mean set theory is so interwoven or with the language of mathematics and Being a discipline at the same time."
  },
  {
    "start": 156.13,
    "end": 167.29,
    "text": "So these curly brackets are used in The syntax, let's speak of latech and similar talking about sets Simply the general concepts where that's everywhere."
  },
  {
    "start": 167.41,
    "end": 170.87,
    "text": "And then they're said as a mathematical object with our own axioms."
  },
  {
    "start": 176.65,
    "end": 182.67,
    "text": "set theory influence on latex because I think they were just running out of brackets."
  },
  {
    "start": 182.89,
    "end": 189.23,
    "text": "I mean you have round brackets, square bracket and curly brackets in ASCII yeah?"
  },
  {
    "start": 192.83,
    "end": 199.55,
    "text": "They used the round brackets for other purposes Square brackets, so they used to curly brackets."
  },
  {
    "start": 200.11,
    "end": 204.79,
    "text": "I could be wrong but as far as i know..."
  },
  {
    "start": 205.11,
    "end": 208.59,
    "text": "I don't think it's a deliberate philosophical decision that"
  },
  {
    "start": 209.33,
    "end": 209.49,
    "text": "this"
  },
  {
    "start": 209.91,
    "end": 225.23,
    "text": "symbol is used twice and once used as syntax or the mathematical symbol curly bracket you have these always to distinguish by the spec slash."
  },
  {
    "start": 225.79,
    "end": 227.33,
    "text": "But I'm staying with the type theory."
  },
  {
    "start": 227.91,
    "end": 234.33,
    "text": "So and then for me, The main difference is type tourists don't think about objects per se."
  },
  {
    "start": 234.83,
    "end": 237.31,
    "text": "they say there's an object And it has a type."
  },
  {
    "start": 238.03,
    "end": 240.15,
    "text": "most of that time It had exactly one type right?"
  },
  {
    "start": 255.67,
    "end": 261.089,
    "text": "basically using all the existing objects and put them together in a set."
  },
  {
    "start": 261.649,
    "end": 267.11,
    "text": "And then do this again, any object can be elemented as many sets you like right?"
  },
  {
    "start": 267.75,
    "end": 273.75,
    "text": "So those sets are really defined by their enumeration ,by definition or intention."
  },
  {
    "start": 276.61,
    "end": 280.13,
    "text": "there is an axiom of extensionality but it just means that."
  },
  {
    "start": 283.67,
    "end": 296.07,
    "text": "So multiplicity of elements doesn't really matter, but I would still call set theory very intentional because you really define a set by its elements."
  },
  {
    "start": 296.97,
    "end": 301.03,
    "text": "And in type theory that's different."
  },
  {
    "start": 304.55,
    "end": 309.17,
    "text": "We introduce the types and elements as at same time."
  },
  {
    "start": 309.47,
    "end": 317.91,
    "text": "so when i define a type of natural numbers, then it introduces zero element and successor."
  },
  {
    "start": 318.55,
    "end": 324.75,
    "text": "And you can be confused because there is some notation like maybe zero in suck."
  },
  {
    "start": 325.37,
    "end": 327.87,
    "text": "but really what does that mean?"
  },
  {
    "start": 327.91,
    "end": 344.53,
    "text": "There are just two ways to make an actual number either by zero or by successor to look into the actual definition."
  },
  {
    "start": 345.17,
    "end": 349.27,
    "text": "You only have an interface, what you can do with natural numbers?"
  },
  {
    "start": 349.53,
    "end": 354.11,
    "text": "I mean...you can form them and you could do recursion over them."
  },
  {
    "start": 354.15,
    "end": 357.01,
    "text": "And that's all we know about natural numbers."
  },
  {
    "start": 358.25,
    "end": 375.37,
    "text": "So if i give a different definition of natural numbers so maybe one which is more space efficient like binary number and we can turn this into a precise inductive definition."
  },
  {
    "start": 376.09,
    "end": 382.79,
    "text": "So, yeah... It's a bit- We have to be a little careful to avoid leading zeros but you can do it!"
  },
  {
    "start": 384.87,
    "end": 398.35,
    "text": "And then for you there is no way from outside to distinguish the binary natural numbers and unary natural numbers because all what you are able to do with one or other."
  },
  {
    "start": 400.07,
    "end": 406.53,
    "text": "So in this sense, types theory and here I'm talking about type theory even with before homotopy type theory."
  },
  {
    "start": 407.09,
    "end": 419.47,
    "text": "Type theory by its nature is more extensional or even intangential type theory funnily enough because you cannot talk about intangible aspects of your type."
  },
  {
    "start": 420.29,
    "end": 424.19,
    "text": "You can not talk like how it's constructed."
  },
  {
    "start": 424.93,
    "end": 428.53,
    "text": "You can only use the type in computer science."
  },
  {
    "start": 428.71,
    "end": 430.77,
    "text": "The single would be via their interface, yeah?"
  },
  {
    "start": 431.41,
    "end": 434.25,
    "text": "But I think it makes also lots of sense from a mathematical point review."
  },
  {
    "start": 435.01,
    "end": 447.65,
    "text": "so national numbers they are not... It doesn't really matter how they're defined whether you zero and success or maybe we used binary encoding over maybe at another one."
  },
  {
    "start": 448.31,
    "end": 450.87,
    "text": "what matters is what you can do with them."
  },
  {
    "start": 452.27,
    "end": 452.71,
    "text": "Yeah i see."
  },
  {
    "start": 454.21,
    "end": 459.93,
    "text": "And I mean, let's take this as our approximation to type theory for now."
  },
  {
    "start": 460.29,
    "end": 462.51,
    "text": "That is more or less sufficient for the audience."
  },
  {
    "start": 464.99,
    "end": 477.19,
    "text": "Now before all of these programs really went off there was a key moment we already mentioned per Martin Löw Of this step-to-dependent type theory."
  },
  {
    "start": 477.91,
    "end": 478.93,
    "text": "and what's innovation?"
  },
  {
    "start": 479.27,
    "end": 480.19,
    "text": "What new there?"
  },
  {
    "start": 480.57,
    "end": 482.19,
    "text": "Then"
  },
  {
    "start": 482.23,
    "end": 488.29,
    "text": "we first, personally I don't like the word dependent type theory but that's a great personal thing."
  },
  {
    "start": 489.49,
    "end": 495.77,
    "text": "I talk about types theory and one feature of types theory are dependent types okay?"
  },
  {
    "start": 496.63,
    "end": 513.37,
    "text": "But it is true this was an important idea due to Kermit and Wolf in the nineteen seventies And what was already around at the time What we now call simple type theory."
  },
  {
    "start": 514.589,
    "end": 532.19,
    "text": "So for example system T, I think which introduced by Gürtel Which was a way to turn arithmetic, the piano arithmetic or hiding arithmetic into sort of functional programming language right?"
  },
  {
    "start": 532.85,
    "end": 539.17,
    "text": "Where you have a type of natural numbers and your functions as in You can do things with it."
  },
  {
    "start": 540.07,
    "end": 552.39,
    "text": "and system T, the theory which exactly models what you can compute in arithmetic."
  },
  {
    "start": 552.63,
    "end": 559.17,
    "text": "So there's a... so this sense it is an implementation of arithmetic as a type-theory."
  },
  {
    "start": 560.31,
    "end": 573.67,
    "text": "And next was Gerard, Johnny Giras, System F where he tried to do the same or something similar for higher order logic."
  },
  {
    "start": 573.77,
    "end": 578.21,
    "text": "So system F is corresponds to a higher order of logic, so what's high order logic?"
  },
  {
    "start": 579.77,
    "end": 592.77,
    "text": "In normal predicate logic you can quantify over individuals like natural numbers but in higher-order logic also quantify over propositions and that makes it much more powerful."
  },
  {
    "start": 592.81,
    "end": 597.87,
    "text": "This was an open problem."
  },
  {
    "start": 598.27,
    "end": 604.87,
    "text": "Could one also have a syntactic definition of what is computable in higher order logic?"
  },
  {
    "start": 605.59,
    "end": 624.75,
    "text": "And Gerard provided an answer out in the seventies to say, okay here we use system F. You do System F. so yeah there's all interesting story about System U which maybe could talk another type just inconsistent."
  },
  {
    "start": 626.61,
    "end": 629.75,
    "text": "It's super important."
  },
  {
    "start": 629.79,
    "end": 631.13,
    "text": "And I mean giraffes, paradox..."
  },
  {
    "start": 633.25,
    "end": 636.05,
    "text": "Okay yeah okay let me quickly describe this."
  },
  {
    "start": 636.13,
    "end": 651.27,
    "text": "so when you do system T You can combine system t with arithmetic and this is called h a omega or arithmetic in higher types."
  },
  {
    "start": 651.35,
    "end": 659.33,
    "text": "So basically we have predicate logic And that's fine, it is completely conservative."
  },
  {
    "start": 659.43,
    "end": 660.13,
    "text": "Nothing has happened."
  },
  {
    "start": 664.23,
    "end": 668.99,
    "text": "It certainly consistent and doesn't give you anything new computations."
  },
  {
    "start": 669.71,
    "end": 678.13,
    "text": "but then Girard when he captured higher order logic wanted to do the same thing."
  },
  {
    "start": 678.19,
    "end": 689.63,
    "text": "He said okay let us take higher order logics and system F playing the same game again and he called this system you."
  },
  {
    "start": 690.49,
    "end": 700.75,
    "text": "And I think, maybe it was trying to prove consistency of his system?"
  },
  {
    "start": 701.59,
    "end": 702.79,
    "text": "He didn't manage."
  },
  {
    "start": 703.61,
    "end": 723.03,
    "text": "but then he went for opposite and was able to derive a paradox which is based on the so-called Burali Forty Paradox an ordinal of all ordinals, and he was able to encode this ordinal for all ordinances."
  },
  {
    "start": 723.09,
    "end": 726.99,
    "text": "in his view then concluding it's inconsistent."
  },
  {
    "start": 728.21,
    "end": 729.17,
    "text": "So that also... Okay!"
  },
  {
    "start": 729.23,
    "end": 732.49,
    "text": "This is important for our next part of the story."
  },
  {
    "start": 733.53,
    "end": 742.33,
    "text": "so then along came Pierre-Martin Reuves."
  },
  {
    "start": 743.89,
    "end": 749.37,
    "text": "He was interested in this sort of Currie-Howard idea."
  },
  {
    "start": 749.95,
    "end": 757.07,
    "text": "So that there is a close connection between logic and these type theories, the simple types series."
  },
  {
    "start": 758.89,
    "end": 773.41,
    "text": "The observation is that the provability corresponds to... In simple type theory it corresponds to constructing terms in simply type, in lambda calculus."
  },
  {
    "start": 774.71,
    "end": 779.61,
    "text": "But then the question is okay that's quite cool for propositional logic."
  },
  {
    "start": 780.15,
    "end": 788.37,
    "text": "but really you want to do predicate logic and true capture-predicate logic pair."
  },
  {
    "start": 788.45,
    "end": 790.89,
    "text": "Martin lives than introduced dependent types."
  },
  {
    "start": 791.01,
    "end": 795.63,
    "text": "so what does a proposition?"
  },
  {
    "start": 795.69,
    "end": 801.61,
    "text": "A proposition is just a type and the predicate is a dependent type."
  },
  {
    "start": 802.09,
    "end": 808.49,
    "text": "So, to predicate over natural number it's basically... ...a dependent type giving an actual number that returns a type."
  },
  {
    "start": 808.95,
    "end": 809.05,
    "text": "Yeah?"
  },
  {
    "start": 809.31,
    "end": 811.89,
    "text": "It was a type depending on our actual numbers!"
  },
  {
    "start": 816.01,
    "end": 816.09,
    "text": "Is"
  },
  {
    "start": 816.25,
    "end": 825.75,
    "text": "this okay to imagine as like indices I have my being a multiple of And then by itself nothing."
  },
  {
    "start": 827.43,
    "end": 831.55,
    "text": "But then I scan plug-in to being multiple of two, even."
  },
  {
    "start": 833.29,
    "end": 841.43,
    "text": "Yes so in modern types this is also again related to the notion of a function but... ...the function into universe."
  },
  {
    "start": 841.49,
    "end": 845.53,
    "text": "that's sort more involved The basic idea as dependent type."
  },
  {
    "start": 846.17,
    "end": 853.29,
    "text": "As you say it's a type which depends on value And these dependent types they show up in logic."
  },
  {
    "start": 854.05,
    "end": 855.53,
    "text": "For example let us say prime."
  },
  {
    "start": 856.13,
    "end": 857.57,
    "text": "So prime is the predicate."
  },
  {
    "start": 857.73,
    "end": 872.79,
    "text": "That means if you plug in a number like six, then prime of six is the proposition as a type and an inhabitant of this type is proof that six is a prime number which obviously it's not."
  },
  {
    "start": 872.85,
    "end": 876.35,
    "text": "there isn't such a poof Which means the time that six does not apply."
  },
  {
    "start": 877.85,
    "end": 890.85,
    "text": "so this is an idea And This is also important in computer science because we use dependent types, so for example a sequence of certain lengths."
  },
  {
    "start": 890.99,
    "end": 903.75,
    "text": "So the vector or sequence of a fixed length n it's a dependent type and you can plug-in seven which are useful."
  },
  {
    "start": 905.07,
    "end": 908.41,
    "text": "to be very specific about your data."
  },
  {
    "start": 908.75,
    "end": 919.45,
    "text": "This is, I think one of the important ideas of Peer Martin Nurews' idea for dependent types."
  },
  {
    "start": 920.35,
    "end": 937.95,
    "text": "He did much more and in his first paper he introduced a type theory which was quite brave because he introduced that type theory with all kinds."
  },
  {
    "start": 938.83,
    "end": 939.91,
    "text": "So that was... Yes,"
  },
  {
    "start": 940.05,
    "end": 940.93,
    "text": "sounds dangerous."
  },
  {
    "start": 943.19,
    "end": 945.15,
    "text": "But maybe I don't know what he thought?"
  },
  {
    "start": 945.45,
    "end": 967.49,
    "text": "He certainly knew Russell's paradox but maybe because Type-Z is more disciplined than he could get away from Russell's Paradox Because you cannot construct the set of all sets straight away because you only have types."
  },
  {
    "start": 967.69,
    "end": 974.21,
    "text": "So I don't know what he thought, but here also said look this system is quite strong."
  },
  {
    "start": 974.43,
    "end": 976.69,
    "text": "i can also encode."
  },
  {
    "start": 977.01,
    "end": 984.55,
    "text": "Girard's System F and i can System T and System F Can all be encoded in my talk theory."
  },
  {
    "start": 985.47,
    "end": 997.35,
    "text": "And then Girard say that true But actually You can also include System U In your theory And that's known to be inconsistent, right?"
  },
  {
    "start": 998.29,
    "end": 1008.17,
    "text": "So this shows the first version of Permanent Taxi was actually inconsistent."
  },
  {
    "start": 1008.85,
    "end": 1018.61,
    "text": "He then as always happens when these things... The standard reaction is to become super cautious!"
  },
  {
    "start": 1020.47,
    "end": 1025.27,
    "text": "Per Martin Lösch then rejected this whole idea of impredicativity."
  },
  {
    "start": 1026.75,
    "end": 1027.25,
    "text": "What does it mean?"
  },
  {
    "start": 1027.31,
    "end": 1036.569,
    "text": "Impredicativity is the ideal for higher order logic, that a proposition is just typed like any other type or that you can quantify over types."
  },
  {
    "start": 1038.15,
    "end": 1040.05,
    "text": "and he abandons us."
  },
  {
    "start": 1040.19,
    "end": 1058.33,
    "text": "so he introduced what's called a predicative-type theory where if we quantify all our types So this is unlike in system F, you can quantify all propositions which correspond to types and get a new type."
  },
  {
    "start": 1059.45,
    "end": 1063.63,
    "text": "That's what we call impredicative."
  },
  {
    "start": 1064.25,
    "end": 1086.79,
    "text": "And Pére-Marty Leuph after he was bitten by the inconsistency of the Russell Paradox or Jibar paradox here became very careful type theory, the predicate of theory which he stuck through with since."
  },
  {
    "start": 1088.95,
    "end": 1115.33,
    "text": "He ever looked again or even considered in predicate theories who introduced the calculus of constructions, which is basically a combination of system F and Martin Lüftag's theory."
  },
  {
    "start": 1115.41,
    "end": 1126.29,
    "text": "And it was first base for implementation of systems like rock now and coke earlier even other names."
  },
  {
    "start": 1127.31,
    "end": 1140.37,
    "text": "but this was calculus of construction attempted or attempts to combine system F and modular tag theory."
  },
  {
    "start": 1140.41,
    "end": 1140.51,
    "text": "Yeah, I"
  },
  {
    "start": 1140.55,
    "end": 1145.97,
    "text": "think there's a nice story so far."
  },
  {
    "start": 1146.41,
    "end": 1150.93,
    "text": "now i'm thinking but why is this?"
  },
  {
    "start": 1151.13,
    "end": 1155.29,
    "text": "So powerful to allow one parameter in your type theory?"
  },
  {
    "start": 1155.45,
    "end": 1158.65,
    "text": "And so this revolutionize everything."
  },
  {
    "start": 1158.95,
    "end": 1171.27,
    "text": "couldn't I encode that somehow saying for every time infinitely other types, which are just the pair of type and your parameter or something like that."
  },
  {
    "start": 1173.21,
    "end": 1178.63,
    "text": "It feels so minimal tweak but it seems to have been such a revolution."
  },
  {
    "start": 1178.69,
    "end": 1197.61,
    "text": "There is very nice talk by Connor McBride about some comics cartoon about a proletarian revolution in type theory."
  },
  {
    "start": 1198.45,
    "end": 1203.13,
    "text": "So he says, okay there is the Coulon and on the rest of the coulons are the workers."
  },
  {
    "start": 1203.21,
    "end": 1204.13,
    "text": "so she draws red."
  },
  {
    "start": 1204.21,
    "end": 1222.57,
    "text": "they just have to shovel them do all their work And he smokes, whatever."
  },
  {
    "start": 1225.37,
    "end": 1239.11,
    "text": "He's never around when there is any work to be done but sort of controls everything here and then dependent type theory opens the cool on separating them in so as a workers walk over on other side?"
  },
  {
    "start": 1240.33,
    "end": 1243.25,
    "text": "The question this like really bad as an evolution?"
  },
  {
    "start": 1243.93,
    "end": 1252.21,
    "text": "yeah actually it was positive one because become much more expressive, they can talk about a lot of things."
  },
  {
    "start": 1255.37,
    "end": 1270.97,
    "text": "Because yeah... They can distinguish and you could talk about terms in the form of theory where previously the types were strictly separated."
  },
  {
    "start": 1271.55,
    "end": 1274.41,
    "text": "now that the type are mixed And"
  },
  {
    "start": 1279.49,
    "end": 1280.77,
    "text": "can you help me with one?"
  },
  {
    "start": 1282.49,
    "end": 1288.29,
    "text": "As a philosopher, I like this revolution because it was driven by conceptual thought."
  },
  {
    "start": 1288.83,
    "end": 1293.39,
    "text": "It's not only technical tweak for technical reasons."
  },
  {
    "start": 1294.23,
    "end": 1300.85,
    "text": "as far as i know the story goes that there were based on philosophical reflections."
  },
  {
    "start": 1300.99,
    "end": 1304.41,
    "text": "Do You Know The Rough Idea?"
  },
  {
    "start": 1304.83,
    "end": 1306.67,
    "text": "What Was Thinking To Do That?"
  },
  {
    "start": 1306.75,
    "end": 1310.79,
    "text": "What Has This to do With Types As Judgments Something in that direction?"
  },
  {
    "start": 1311.55,
    "end": 1313.77,
    "text": "Yeah, I mean...I'm not sure."
  },
  {
    "start": 1314.17,
    "end": 1325.01,
    "text": "As i say the main story is that Pierre Martin-Werff wanted to extend the Kerry Howard equivalence to predicate logic."
  },
  {
    "start": 1325.21,
    "end": 1335.05,
    "text": "right this is very formal but actually if you do mathematics You're very easily introduced something which corresponds to dependent types."
  },
  {
    "start": 1335.23,
    "end": 1347.85,
    "text": "I have a sequence of the length n, so like a sequence or vector of real numbers."
  },
  {
    "start": 1349.49,
    "end": 1369.53,
    "text": "Then you maybe have a process which for every natural number gives you a sequence a function."
  },
  {
    "start": 1370.01,
    "end": 1380.53,
    "text": "So normally we think of functions as having an input and have an output, but here you have the function which has an input ,but the outputs then varies with the input ."
  },
  {
    "start": 1380.81,
    "end": 1386.27,
    "text": "You say for every natural number n I get this sequence of the length N. so what is it?"
  },
  {
    "start": 1386.37,
    "end": 1387.13,
    "text": "As a function ?"
  },
  {
    "start": 1387.55,
    "end": 1389.01,
    "text": "As an input or as a national number ?"
  },
  {
    "start": 1389.43,
    "end": 1392.71,
    "text": "But the output depends on the inputs right?"
  },
  {
    "start": 1394.71,
    "end": 1396.35,
    "text": "That's from very naive perspective."
  },
  {
    "start": 1397.83,
    "end": 1406.59,
    "text": "As soon as you do some constructions in mathematics, it's hard to avoid having some phenomenon like dependent types."
  },
  {
    "start": 1406.67,
    "end": 1417.55,
    "text": "And it turns out to be extremely powerful and extremely useful for also reflecting language talking about language in-language on."
  },
  {
    "start": 1417.63,
    "end": 1417.69,
    "text": "so"
  },
  {
    "start": 1419.57,
    "end": 1424.73,
    "text": "I think that a good primer tool dependent type theory is there?"
  },
  {
    "start": 1425.79,
    "end": 1428.89,
    "text": "Anything else we should talk about otherwise I would call it."
  },
  {
    "start": 1430.35,
    "end": 1431.89,
    "text": "Yeah, maybe a quick word."
  },
  {
    "start": 1431.97,
    "end": 1435.51,
    "text": "what i said previously is the type theory."
  },
  {
    "start": 1436.25,
    "end": 1441.73,
    "text": "okay first of all there are all sorts of variations and extensions of types theory."
  },
  {
    "start": 1442.51,
    "end": 1451.99,
    "text": "so for example they're intersection types And other variations of type theory."
  },
  {
    "start": 1454.05,
    "end": 1458.35,
    "text": "These variations, they often have a subtype relation."
  },
  {
    "start": 1460.17,
    "end": 1479.13,
    "text": "And to me this is not what I understand as type theory because now types behave like sets at certain time kind of lots of times and that's sort of useful in some applications."
  },
  {
    "start": 1479.87,
    "end": 1492.93,
    "text": "but personally when i say type theory A type theory where terms have a specific type and not just lots of types."
  },
  {
    "start": 1494.27,
    "end": 1505.39,
    "text": "We sometimes introduce notational conventions to say, oh if I write the number three then it could be natural numbers or an integer or a real number."
  },
  {
    "start": 1506.25,
    "end": 1508.67,
    "text": "but i think this is mainly in notation convention."
  },
  {
    "start": 1511.17,
    "end": 1513.11,
    "text": "Okay, depending on what you're looking for."
  },
  {
    "start": 1513.69,
    "end": 1525.13,
    "text": "I coerced into these different types but it doesn't really mean that from the foundational point of view That these types need to be formally related."
  },
  {
    "start": 1525.31,
    "end": 1531.83,
    "text": "so i would distinguish The sort of underlying theory and the presentation."
  },
  {
    "start": 1531.97,
    "end": 1536.69,
    "text": "And to me this sub typing and so On as no not so people do not agree with Me."
  },
  {
    "start": 1537.27,
    "end": 1539.33,
    "text": "they are part Of the presentation."
  },
  {
    "start": 1541.67,
    "end": 1550.19,
    "text": "I mean, you can always have a canonical injection right to relate them even without having relationships of types"
  },
  {
    "start": 1550.23,
    "end": 1550.35,
    "text": "per se"
  },
  {
    "start": 1551.21,
    "end": 1552.17,
    "text": "in your formalism."
  },
  {
    "start": 1553.35,
    "end": 1568.77,
    "text": "I think it's important also for practical use for formalism to introduce this layer of how we write things and that should be done in the formal way."
  },
  {
    "start": 1569.53,
    "end": 1575.93,
    "text": "But all I would say for me, the sort of foundational theory should be separated from the presentation."
  },
  {
    "start": 1576.03,
    "end": 1580.97,
    "text": "Otherwise it creates lots of confusions and strange phenomena"
  },
  {
    "start": 1581.51,
    "end": 1591.59,
    "text": "which is maybe a circle back to the beginning where we talked about the mix Of things as your construction of mathematics And there's mathematical objects themselves."
  },
  {
    "start": 1594.19,
    "end": 1597.65,
    "text": "Yeah that's uh A good arc For this premises."
  },
  {
    "start": 1598.45,
    "end": 1600.33,
    "text": "So, thank you for giving this primer."
  },
  {
    "start": 1600.61,
    "end": 1612.41,
    "text": "Thank you all for listening and I'm not sure how many news we have but i can probably already say that next week's episode will be a little historic."
  },
  {
    "start": 1612.49,
    "end": 1623.77,
    "text": "We had great logician and historian of logic around And we started the membership program which doesn't yet come with many perks."
  },
  {
    "start": 1626.19,
    "end": 1633.89,
    "text": "With time we are planning a few things like having proper lectures, maybe having monthly meetings or anything."
  },
  {
    "start": 1634.75,
    "end": 1636.49,
    "text": "We will figure that out already."
  },
  {
    "start": 1636.79,
    "end": 1644.89,
    "text": "huge thanks to those who support us before getting any perks from the bottom of their hearts."
  },
  {
    "start": 1644.99,
    "end": 1645.95,
    "text": "thank you for the coffees."
  },
  {
    "start": 1646.69,
    "end": 1647.75,
    "text": "Thank You for commenting!"
  },
  {
    "start": 1648.05,
    "end": 1649.61,
    "text": "We try be very active there."
  },
  {
    "start": 1650.29,
    "end": 1658.25,
    "text": "Feel free to follow up if you don't think your question was addressed correctly Then thank you all for watching and everything else."
  },
  {
    "start": 1659.15,
    "end": 1663.77,
    "text": "And then see you in a week when we talk about the history of logic!"
  },
  {
    "start": 1664.41,
    "end": 1665.51,
    "text": "Bye bye."
  }
]