[
  {
    "start": 0.0,
    "end": 5.78,
    "text": "I mean, I don't think most people's view or maybe... ...I don' t think mathematics is anything holy."
  },
  {
    "start": 5.98,
    "end": 15.34,
    "text": "All foundations are absolute but this Wittgenstein view does displace a deep misunderstanding about the whole of logic in mathematical reasoning."
  },
  {
    "start": 20.46,
    "end": 21.08,
    "text": "Hello everybody!"
  },
  {
    "start": 21.16,
    "end": 35.62,
    "text": "Welcome to our newest episode and today we have great friend of ours Jose Perez Escobar who is professor at UNED which is The Distance University Of Spain In Madrid And he's a longtime collaborator of mine."
  },
  {
    "start": 36.26,
    "end": 43.7,
    "text": "We, too with Jordy Ferrest who was also a guest in the last season we started The Young Network for Wittgensteinian Philosophy."
  },
  {
    "start": 44.0,
    "end": 53.86,
    "text": "He is doing a lot on Wittkenstein and will definitely talk about it But his also Neuroscientist and Psychologist Turned Philosopher."
  },
  {
    "start": 54.74,
    "end": 62.06,
    "text": "So I'm very happy that you accepted our invitation to be part of About Logic."
  },
  {
    "start": 63.32,
    "end": 64.16,
    "text": "Thank you for inviting me."
  },
  {
    "start": 65.12,
    "end": 70.9,
    "text": "Yeah, I'm also pleased to see you again and i think we met once at this in Rübeck."
  },
  {
    "start": 70.94,
    "end": 75.74,
    "text": "right there was a meeting on Fibonacci's garden."
  },
  {
    "start": 77.32,
    "end": 77.86,
    "text": "yeah so"
  },
  {
    "start": 78.52,
    "end": 80.8,
    "text": "couple of years ago or one year-and-a-half ago?"
  },
  {
    "start": 82.14,
    "end": 85.32,
    "text": "We made indeed the Fibonnacci Garden workshop!"
  },
  {
    "start": 85.9,
    "end": 88.48,
    "text": "It was quite an interesting meeting lots."
  },
  {
    "start": 90.64,
    "end": 95.64,
    "text": "But maybe you can tell our listeners and also remind me what are your working on in the moment?"
  },
  {
    "start": 96.46,
    "end": 102.9,
    "text": "Well, at the moment I'm working on several topics in The Philosophy of Mathematics but also generally In The Philosophy Of Science."
  },
  {
    "start": 104.2,
    "end": 111.96,
    "text": "Maybe i would say my main topic right now if i have to choose one is the applicability of mathematics To the sciences."
  },
  {
    "start": 113.14,
    "end": 122.42,
    "text": "So who'd say the philosophy for applied mathematics... ...but it has worked quite a bit on Wienerstein and a bit on the philosophy of biology."
  },
  {
    "start": 123.3,
    "end": 128.419,
    "text": "So yeah, maybe can you comment why is mathematics so useful in sciences?"
  },
  {
    "start": 129.199,
    "end": 131.16,
    "text": "Have we got any idea about this?"
  },
  {
    "start": 132.2,
    "end": 133.78,
    "text": "Well are they useful?"
  },
  {
    "start": 134.14,
    "end": 139.6,
    "text": "as I said that's the first question was used a lot indeed."
  },
  {
    "start": 139.64,
    "end": 139.8,
    "text": "uh Yeah!"
  },
  {
    "start": 139.84,
    "end": 141.76,
    "text": "They're very usefull of course actually..."
  },
  {
    "start": 142.98,
    "end": 143.98,
    "text": "Of course!"
  },
  {
    "start": 144.02,
    "end": 144.52,
    "text": "And"
  },
  {
    "start": 144.62,
    "end": 144.8,
    "text": "why?!"
  },
  {
    "start": 145.1,
    "end": 156.0,
    "text": "There's a paraphrase of what we call Wigner's Enigma, which is basically he was puzzled that mathematics are so useful in science."
  },
  {
    "start": 156.06,
    "end": 167.26,
    "text": "That maybe one should find out why but it's something like the fact."
  },
  {
    "start": 169.68,
    "end": 174.02,
    "text": "Of course, for philosophers this is not an acceptable state of affairs."
  },
  {
    "start": 175.22,
    "end": 178.1,
    "text": "So we need to get into the details."
  },
  {
    "start": 178.52,
    "end": 182.18,
    "text": "what makes mathematics so applicable and useful?"
  },
  {
    "start": 182.5,
    "end": 186.2,
    "text": "And why does it yield predictions that otherwise you wouldn't be able?"
  },
  {
    "start": 187.86,
    "end": 192.84,
    "text": "So that's actually one of my main concerns in the philosophy of Applied Mathematics."
  },
  {
    "start": 193.52,
    "end": 213.46,
    "text": "And I did work on a paper, it has been published for three years already... ...that directly engages with that question and this actually combines this topic with Wittgenstein because i use Data-Wittgensteiner Philosophy of Mathematics to explain why mathematics is so applicable to real world."
  },
  {
    "start": 214.24,
    "end": 215.74,
    "text": "Yeah, I was just thinking."
  },
  {
    "start": 216.02,
    "end": 223.14,
    "text": "Sometimes in the history we misapplied mathematics."
  },
  {
    "start": 224.04,
    "end": 233.78,
    "text": "wasn't there a theory that distances of the planets could be explained by platonic solids to know this story?"
  },
  {
    "start": 234.7,
    "end": 245.02,
    "text": "There were some calculations where all the planets are explained polyadors, but which wasn't nothing."
  },
  {
    "start": 245.06,
    "end": 248.82,
    "text": "I mean predicted something that was not compatible with reality."
  },
  {
    "start": 249.96,
    "end": 250.62,
    "text": "or do you know?"
  },
  {
    "start": 250.68,
    "end": 251.82,
    "text": "maybe other examples?"
  },
  {
    "start": 252.62,
    "end": 252.7,
    "text": "Yeah"
  },
  {
    "start": 254.4,
    "end": 263.56,
    "text": "So uh That's a good point because we used to think that mathematics is Something that works all the time."
  },
  {
    "start": 263.62,
    "end": 272.06,
    "text": "But anyone who knows A certain episodes in there in the history of science knows that mathematics can also fail in science."
  },
  {
    "start": 273.86,
    "end": 277.44,
    "text": "I mean, at the point some people would say to play devils advocate."
  },
  {
    "start": 277.62,
    "end": 287.14,
    "text": "They will say that it's... The problem is not the mathematics but it's the mathematician or the scientist who uses them."
  },
  {
    "start": 287.28,
    "end": 297.88,
    "text": "That maybe over-generalized mathematical model or misapplied in a model for instance."
  },
  {
    "start": 299.14,
    "end": 301.42,
    "text": "But this isn't the mathematics fault."
  },
  {
    "start": 301.92,
    "end": 304.02,
    "text": "I guess that what many people would say."
  },
  {
    "start": 305.54,
    "end": 306.44,
    "text": "In, in that direction?"
  },
  {
    "start": 306.76,
    "end": 309.14,
    "text": "It's not only about doing math right."
  },
  {
    "start": 309.2,
    "end": 317.3,
    "text": "you use math for theory development like physicists saying it should be simple or elegant whatever exactly means."
  },
  {
    "start": 318.38,
    "end": 338.48,
    "text": "and assuming some planets form a platonic solid is beautiful because we liked these forms but its not empirically valence simply While it could still be an interesting Hypothesis and maybe its useful to do that."
  },
  {
    "start": 338.54,
    "end": 347.78,
    "text": "Maybe It's not useful, maybe in the way of getting new hypothesis To switch to messier mathematics that describe things better."
  },
  {
    "start": 348.08,
    "end": 353.82,
    "text": "I mean In some sense Einstein's math is harder than Newton's math right?"
  },
  {
    "start": 353.9,
    "end": 355.88,
    "text": "And also more beautiful for people."
  },
  {
    "start": 356.68,
    "end": 357.08,
    "text": "So yeah,"
  },
  {
    "start": 359.78,
    "end": 369.38,
    "text": "I just thought some of these ideas were born out a platonic idea that everything has to be perfect and ideal."
  },
  {
    "start": 370.04,
    "end": 377.78,
    "text": "And the real world is sort of an imperfect reflection on this perfect world of ideas."
  },
  {
    "start": 378.54,
    "end": 387.86,
    "text": "Then it was also assumed that planets would... go in a circular orbit because that was just more perfect than what they actually do."
  },
  {
    "start": 389.06,
    "end": 393.78,
    "text": "Even epic circles and circles on circles stuff, just cause we love circles?"
  },
  {
    "start": 394.6,
    "end": 394.7,
    "text": "Yes!"
  },
  {
    "start": 395.32,
    "end": 399.58,
    "text": "And it's just about Einstein I mean isn't it... What Einstein was doing?"
  },
  {
    "start": 399.74,
    "end": 425.88,
    "text": "he had sort of an intuitive idea which was based on observations into mathematics and that became, I mean first special relativity which is based on the simple ideas at speed of light it's same for every observer."
  },
  {
    "start": 426.52,
    "end": 437.76,
    "text": "And then general relativity also based some simple idea sort gravity acceleration are indistinguishable."
  },
  {
    "start": 440.18,
    "end": 441.42,
    "text": "Can you comment on this, Rosé?"
  },
  {
    "start": 442.44,
    "end": 442.44,
    "text": "I"
  },
  {
    "start": 443.7,
    "end": 446.24,
    "text": "guess we regard to the topic."
  },
  {
    "start": 446.36,
    "end": 449.72,
    "text": "The point would be why did he choose mathematics as the language?"
  },
  {
    "start": 450.0,
    "end": 463.08,
    "text": "translate his intuition onto instead of another language and that's what states his confidence in mathematics is a language for his science or in these cases intuitions."
  },
  {
    "start": 463.86,
    "end": 467.1,
    "text": "at the point is even with idealizations."
  },
  {
    "start": 468.1,
    "end": 470.98,
    "text": "There are plenty of idealizations in physics, even with that."
  },
  {
    "start": 471.58,
    "end": 478.66,
    "text": "Mathematics remains extremely useful in physics and not only physics also other domains maybe not that much another domain."
  },
  {
    "start": 480.46,
    "end": 484.22,
    "text": "In biology mathematics has its own specific problems for instance."
  },
  {
    "start": 484.72,
    "end": 487.56,
    "text": "but despite that where is it so useful?"
  },
  {
    "start": 489.42,
    "end": 494.86,
    "text": "Why don't we use natural language to formulate theories on physics?"
  },
  {
    "start": 497.88,
    "end": 509.3,
    "text": "Well, one of the intuitions that I formulate in this work is that mathematics does not only describe reality."
  },
  {
    "start": 510.38,
    "end": 519.0,
    "text": "It's a language which is rich enough to capture structural properties of the world."
  },
  {
    "start": 519.179,
    "end": 528.3,
    "text": "In the line of the later videos on philosophy and mathematics, mathematics can also constitute rules of description."
  },
  {
    "start": 528.88,
    "end": 529.98,
    "text": "Not just descriptions."
  },
  {
    "start": 530.58,
    "end": 532.94,
    "text": "So let's say you have a mathematical model."
  },
  {
    "start": 533.96,
    "end": 540.02,
    "text": "The typical example that... A mathematical model for the apple addition And your model is two plus two is four."
  },
  {
    "start": 540.96,
    "end": 550.48,
    "text": "Then if you are in the supermarket You've got two apples On one hand there're two apples on another And then put them together And only count three."
  },
  {
    "start": 551.22,
    "end": 554.86,
    "text": "then you know that probably your model is not wrong."
  },
  {
    "start": 555.44,
    "end": 560.06,
    "text": "What's wrong, it's the measurement procedure for instance has to put a very simple example."
  },
  {
    "start": 560.86,
    "end": 566.66,
    "text": "so your model acts as reference for correct measurements."
  },
  {
    "start": 566.78,
    "end": 570.24,
    "text": "So in practical evidence Your three will not disprove your model."
  },
  {
    "start": 570.66,
    "end": 572.42,
    "text": "You're model would correct your measurements."
  },
  {
    "start": 572.56,
    "end": 576.86,
    "text": "It's a rule to describe reality instead of a description of reality."
  },
  {
    "start": 578.96,
    "end": 584.14,
    "text": "This is quite obvious in such an example, but it happens in the sciences."
  },
  {
    "start": 584.2,
    "end": 584.74,
    "text": "that's the point!"
  },
  {
    "start": 584.82,
    "end": 586.1,
    "text": "It happens in neuroscience."
  },
  {
    "start": 586.78,
    "end": 592.16,
    "text": "I have a case study on this paper showing that... ...it happens in physics and it happens you know in other domains."
  },
  {
    "start": 593.78,
    "end": 612.12,
    "text": "Let say that mathematics besides being rules of reference sometimes instead of descriptions they also enjoy a specific and special kind of credibility that other languages or other epistemic objects do not meet"
  },
  {
    "start": 612.72,
    "end": 613.08,
    "text": "for us."
  },
  {
    "start": 614.54,
    "end": 625.48,
    "text": "I know, first off all i would say describing the rule of reality is part of a description of reality but it seems to be not what you were saying right?"
  },
  {
    "start": 626.58,
    "end": 626.64,
    "text": "No!"
  },
  {
    "start": 626.84,
    "end": 639.18,
    "text": "Not exactly... about mathematics being rules of description is that they are uh, rules that we humans internalize in order to interact with the world."
  },
  {
    "start": 639.86,
    "end": 654.2,
    "text": "So if you expand a nature to include facts about human beings then yes there would be parts of nature as well but the point is that it will not be part of physics."
  },
  {
    "start": 655.18,
    "end": 656.12,
    "text": "so what happens?"
  },
  {
    "start": 656.28,
    "end": 658.74,
    "text": "A model about physics or a model about neuroscience?"
  },
  {
    "start": 659.2,
    "end": 666.16,
    "text": "And then we humans have internalized rules on how to describe things in neuroscience or thinks about physics."
  },
  {
    "start": 666.36,
    "end": 672.46,
    "text": "So these rules are expressed mathematically, In the case of models but"
  },
  {
    "start": 673.26,
    "end": 676.28,
    "text": "what they could also be expressed in initial language?"
  },
  {
    "start": 676.6,
    "end": 677.36,
    "text": "I mean yes"
  },
  {
    "start": 677.88,
    "end": 679.72,
    "text": "indeed The difference would"
  },
  {
    "start": 679.98,
    "end": 682.58,
    "text": "not a specific property of mathematics."
  },
  {
    "start": 683.9,
    "end": 685.64,
    "text": "That's a very good point, exactly."
  },
  {
    "start": 686.08,
    "end": 701.52,
    "text": "So actually letter B in senior philosophy of mathematics If I were to summarize it extremely briefly It would be about applying the philosophical investigations To mathematics."
  },
  {
    "start": 701.82,
    "end": 706.98,
    "text": "so things about ordinary language that apply to mathematics as well."
  },
  {
    "start": 709.94,
    "end": 716.4,
    "text": "you have ordinary language that can be conceived as, let's say an activity of rule following."
  },
  {
    "start": 717.54,
    "end": 726.22,
    "text": "There are certain aspects of mathematics that could be reduced to the same rule-following we find in natural science and would find a natural language."
  },
  {
    "start": 726.52,
    "end": 734.98,
    "text": "so then this aspect at least mathematics is not distinctive Of course, it is distinctive in other aspects."
  },
  {
    "start": 735.1,
    "end": 740.5,
    "text": "Otherwise natural language would be as effective in science as mathematics but its not."
  },
  {
    "start": 741.32,
    "end": 742.3,
    "text": "But they will share."
  },
  {
    "start": 742.56,
    "end": 748.0,
    "text": "according to the later Wittgenstein They both shared this root photo with an aspect."
  },
  {
    "start": 749.22,
    "end": 752.3,
    "text": "Maybe just to take deeper and one example."
  },
  {
    "start": 752.38,
    "end": 755.7,
    "text": "maybe that helps us if we look at it as a next sample."
  },
  {
    "start": 756.84,
    "end": 763.42,
    "text": "That's fine because you have this case study mentioned in neuroscience where this process also goes wrong."
  },
  {
    "start": 763.78,
    "end": 767.58,
    "text": "Maybe that's something helpful to understand the idea?"
  },
  {
    "start": 769.22,
    "end": 771.54,
    "text": "Okay, I will try my best to present it in a simple way."
  },
  {
    "start": 772.08,
    "end": 789.58,
    "text": "so let's go back to the Apple case because as most basic toy cases you have two plus true as model and then your empirical findings do not adjusts to the model measure things again, or count things again."
  },
  {
    "start": 789.72,
    "end": 793.3,
    "text": "Or thinking about the thing that went wrong for instance I lost an apple."
  },
  {
    "start": 794.22,
    "end": 807.64,
    "text": "Okay so now Let's let's imagine we have a much more complex model of Neuronal activity and i have a model That predicts the electrical activity of a neuron."
  },
  {
    "start": 809.74,
    "end": 810.7,
    "text": "What it predicts is?"
  },
  {
    "start": 810.88,
    "end": 813.54,
    "text": "The activities of a certain neuron are Gaussian."
  },
  {
    "start": 813.86,
    "end": 814.34,
    "text": "in this sense."
  },
  {
    "start": 815.38,
    "end": 829.38,
    "text": "there's a neuron has electrical activity only when I'm looking at a certain point in angular space for instance here, but it stops firing."
  },
  {
    "start": 830.0,
    "end": 833.84,
    "text": "It doesn't have an electrical activity anymore as i move away from that point."
  },
  {
    "start": 834.68,
    "end": 849.18,
    "text": "so let's see... that neuron has a preferred direction which is ninety degrees and when I look away from the direction then the activity becomes more sparse and it forms a Gaussian curve."
  },
  {
    "start": 850.3,
    "end": 864.24,
    "text": "So I have a model that includes causal information from other connections in the neural network, so on... And then it explains why this neuron fires in this way."
  },
  {
    "start": 864.88,
    "end": 867.26,
    "text": "It predicts that it will fire in this Gaussian way."
  },
  {
    "start": 867.72,
    "end": 873.86,
    "text": "But we measure the neuron We actually insert into brain wires that record electrical activity of the neurons."
  },
  {
    "start": 877.5,
    "end": 880.74,
    "text": "seems like a Gaussian curve, it's not that perfect Gaussian curve."
  },
  {
    "start": 881.34,
    "end": 899.88,
    "text": "That certain data points are higher than they should be others at lower and so on... So what we usually do in neuroscience is to smooth the data To smoothed the data is to impose the model ON THE DATA!"
  },
  {
    "start": 900.98,
    "end": 901.72,
    "text": "You have the data."
  },
  {
    "start": 901.84,
    "end": 903.02,
    "text": "its an imperfect Gaussian."
  },
  {
    "start": 903.5,
    "end": 905.86,
    "text": "Well, it should be Gaussian according to my model."
  },
  {
    "start": 906.3,
    "end": 914.58,
    "text": "So I will just smooth the curve and say well the neuron actually has a smoother signal."
  },
  {
    "start": 915.54,
    "end": 922.72,
    "text": "The deviations that i get from the gaussian is noise electrical noise for instance And there's a point to that."
  },
  {
    "start": 923.1,
    "end": 932.42,
    "text": "It's not as stupid at the same because uh There's plenty of noise in the brain, how you record electrical activity and so on."
  },
  {
    "start": 932.72,
    "end": 941.6,
    "text": "So for us philosophers we immediately see that there is epistemic circularity because the empirical data should falsify a model potentially."
  },
  {
    "start": 942.78,
    "end": 946.84,
    "text": "but when the data does not match to the model will do not always change."
  },
  {
    "start": 946.92,
    "end": 950.7,
    "text": "the model would change the data so then it adjusts to the mother."
  },
  {
    "start": 951.36,
    "end": 953.64,
    "text": "But we have to do that Because The Data Is Imperfect."
  },
  {
    "start": 955.6,
    "end": 960.54,
    "text": "There's something that happens in neuroscience and in physics, any other domains like models."
  },
  {
    "start": 960.6,
    "end": 967.28,
    "text": "they should be falsified by data actually are a reference for the data."
  },
  {
    "start": 967.92,
    "end": 969.9,
    "text": "And we adjust the database on these models."
  },
  {
    "start": 971.28,
    "end": 983.54,
    "text": "if we do that of course mathematics is going to be effective because We're taking them mathematical models as a reference to tune our data were doing it at Hock."
  },
  {
    "start": 985.16,
    "end": 991.18,
    "text": "This is fine in many cases and it's essential for science, because think about the apple example."
  },
  {
    "start": 992.58,
    "end": 1000.78,
    "text": "It helps us realize that we lost an apple... ...and then will look around at a supermarket to find the apple that we dropped."
  },
  {
    "start": 1001.64,
    "end": 1003.34,
    "text": "It is useful in science as well…."
  },
  {
    "start": 1003.42,
    "end": 1005.52,
    "text": "…because when we cleaned electrical noise."
  },
  {
    "start": 1005.9,
    "end": 1029.14,
    "text": "this works on many scenarios but can also go wrong because by smoothing data, but using mathematical models as rules of measurement instead of descriptions we can let's say... We can mess with the daytime waste that we shouldn't and then we could prevent the discovery of new features for instance."
  },
  {
    "start": 1029.2,
    "end": 1034.16,
    "text": "And this was the case in the neuroscience example I mentioned."
  },
  {
    "start": 1034.2,
    "end": 1044.42,
    "text": "on these papers there are certain neurons whose firing does not adjust to this gaussian's shape, but actually they have two preferred directions."
  },
  {
    "start": 1044.819,
    "end": 1061.08,
    "text": "So there are like two bumps in the signal and because the model specifies that is only one bump on a curve then what these cells with two bumps were thought to be artifacts of measurement?"
  },
  {
    "start": 1062.18,
    "end": 1066.1,
    "text": "That there were two neurons that we're stuck together in the process of measurement."
  },
  {
    "start": 1066.2,
    "end": 1073.38,
    "text": "So this is an artifact, so It took almost twenty years for us to realize that these neurons existed."
  },
  {
    "start": 1073.72,
    "end": 1073.88,
    "text": "Why?"
  },
  {
    "start": 1074.0,
    "end": 1076.8,
    "text": "Because who are using them all as a rule or description?"
  },
  {
    "start": 1076.94,
    "end": 1082.88,
    "text": "As I ruled off what counts as good measurements and this point it can't go wrong."
  },
  {
    "start": 1083.14,
    "end": 1105.92,
    "text": "We may take many years in realizing but All-in-all As I said, to summarize things it makes sense that if we use mathematical models as references or rules of description then mathematics is going be effective because we make it the way our scientific practice conforms in a way where we adjust other measurements."
  },
  {
    "start": 1109.58,
    "end": 1119.54,
    "text": "It's often like this that our expectations, what we think are going to see forms."
  },
  {
    "start": 1119.62,
    "end": 1123.5,
    "text": "What is the article?"
  },
  {
    "start": 1123.94,
    "end": 1137.96,
    "text": "What the cat's brain tells a cat's eye by Varela and... The point here is that expectations are very important but they can lead to hallucinations right?"
  },
  {
    "start": 1138.52,
    "end": 1142.34,
    "text": "So if we expect something and it doesn't fit with reality, We may hallucinate."
  },
  {
    "start": 1142.42,
    "end": 1144.94,
    "text": "And think that you see something which we didn't seen right?"
  },
  {
    "start": 1145.7,
    "end": 1156.52,
    "text": "It's also a problem by the way for legal proceedings because often people witnesses will report something they were expecting to see instead of what is actually seeing."
  },
  {
    "start": 1157.42,
    "end": 1167.78,
    "text": "so its not really unique aspect of mathematics I would say but general issue"
  },
  {
    "start": 1167.92,
    "end": 1171.84,
    "text": "It's not that you need a unique aspect of mathematics indeed, and your analogy is very nice."
  },
  {
    "start": 1172.22,
    "end": 1175.04,
    "text": "But it's a surprising aspect with mathematics."
  },
  {
    "start": 1176.44,
    "end": 1178.04,
    "text": "So if we think about"
  },
  {
    "start": 1178.62,
    "end": 1183.26,
    "text": "like the world as written in mathematics Is like a quote right this Galilean?"
  },
  {
    "start": 1184.0,
    "end": 1185.26,
    "text": "That's like effected."
  },
  {
    "start": 1185.3,
    "end": 1189.92,
    "text": "its clear that nature must follow Mathematical system ecities."
  },
  {
    "start": 1190.36,
    "end": 1204.98,
    "text": "so I think maybe We are preaching to the choir maybe at this moment to Torsten, but normally I think a lot of people would say that math has very objective role and there are other mistakes like bad measurements."
  },
  {
    "start": 1206.12,
    "end": 1206.66,
    "text": "But it's never."
  },
  {
    "start": 1208.04,
    "end": 1211.68,
    "text": "the math might be part of problem so-to speak."
  },
  {
    "start": 1212.72,
    "end": 1218.4,
    "text": "A good analogy with perception is that in vision for instance... The brain fills some gaps!"
  },
  {
    "start": 1219.6,
    "end": 1221.92,
    "text": "But the brains pretty much fill gaps in vision."
  },
  {
    "start": 1223.44,
    "end": 1233.36,
    "text": "Of course by filling gaps you are not, or you're projecting like your inner brain models and then to feel the absolute reality with that."
  },
  {
    "start": 1234.14,
    "end": 1237.62,
    "text": "It can go wrong of course but it works fine most of the time."
  },
  {
    "start": 1238.3,
    "end": 1243.64,
    "text": "so the confusion with mathematics would be thinking that mathematics is a representation of reality."
  },
  {
    "start": 1244.76,
    "end": 1252.66,
    "text": "uh-huh That is absolutely in the sense that I would say look my vision is absolutely perfect."
  },
  {
    "start": 1254.22,
    "end": 1255.52,
    "text": "Gaps I have no blind gaps."
  },
  {
    "start": 1256.4,
    "end": 1260.14,
    "text": "Well, I had them but i don't realize because the brain is feeling them."
  },
  {
    "start": 1261.76,
    "end": 1262.58,
    "text": "It's something similar"
  },
  {
    "start": 1263.66,
    "end": 1273.74,
    "text": "I mean all-I think and always feel a bit appalled put off by this idea that Mathematics in some sense Is there before?"
  },
  {
    "start": 1273.96,
    "end": 1275.2,
    "text": "And we are."
  },
  {
    "start": 1275.36,
    "end": 1284.0,
    "text": "I mean it was some water form of plagiarism that mathematics as they're what we see is sort the structures which are mathematical."
  },
  {
    "start": 1284.34,
    "end": 1300.32,
    "text": "And my view is that mathematics has been evolved by us as a product of cultural evolution and we have just in a way designed it, In this process to be very useful To do exactly what its good for."
  },
  {
    "start": 1301.34,
    "end": 1303.1,
    "text": "Would you agree with me?"
  },
  {
    "start": 1303.58,
    "end": 1303.86,
    "text": "Yeah"
  },
  {
    "start": 1306.04,
    "end": 1306.5,
    "text": "there's truth."
  },
  {
    "start": 1306.84,
    "end": 1311.76,
    "text": "definitely I would say it is not incompatible with the other expert on mathematics."
  },
  {
    "start": 1313.32,
    "end": 1323.82,
    "text": "It's not that our models do not evolve, they evolved all of their time and we make them more effective for their tasks."
  },
  {
    "start": 1324.48,
    "end": 1333.32,
    "text": "like a model can be fine-tuned by empirical data, not only can but should."
  },
  {
    "start": 1334.12,
    "end": 1353.94,
    "text": "This is not at odds with this other grammatical aspect of the model that regulates our expectations on reality, or that regulars our measurement procedures and interactions through to reality."
  },
  {
    "start": 1354.7,
    "end": 1361.64,
    "text": "So let's say we have a modelled an empirical situation which doesn't match."
  },
  {
    "start": 1363.72,
    "end": 1374.98,
    "text": "it depends then character of this mismatch, I may decide to change how interactive with reality or change the model."
  },
  {
    "start": 1376.24,
    "end": 1379.72,
    "text": "In some cases it's obvious that we should change our models."
  },
  {
    "start": 1380.22,
    "end": 1381.96,
    "text": "in others there is another"
  },
  {
    "start": 1382.58,
    "end": 1382.84,
    "text": "option."
  },
  {
    "start": 1383.5,
    "end": 1389.86,
    "text": "so as an other option do more experiments to find out why this mismatched took place and what would be my?"
  },
  {
    "start": 1389.92,
    "end": 1390.5,
    "text": "yeah"
  },
  {
    "start": 1390.56,
    "end": 1410.44,
    "text": "Yeah Of course That could help for instance finding out whether there was a technical problem in the experiment, it could also help tell me whether certain experimental results hold only uncertain scenarios and certain contexts but not others."
  },
  {
    "start": 1411.26,
    "end": 1417.24,
    "text": "And then this also leads to improvements in our models."
  },
  {
    "start": 1417.34,
    "end": 1426.98,
    "text": "we can say well I have performed these experiments in context A and context B and in context A I get data that is very close to my model."
  },
  {
    "start": 1427.12,
    "end": 1428.88,
    "text": "In context B, i got different data."
  },
  {
    "start": 1429.44,
    "end": 1451.84,
    "text": "so maybe this model or the scope of these models are just A but it does not apply to P. The other alternative would be well... My measurements on B were wrong And my model also applies to B. So..I will just smoothed out my data from B so that it adjusts to the model."
  },
  {
    "start": 1452.58,
    "end": 1456.56,
    "text": "And in principle, there is no a priori rule that tells me what I have to do."
  },
  {
    "start": 1456.72,
    "end": 1457.78,
    "text": "These are all pragmatics."
  },
  {
    "start": 1457.94,
    "end": 1463.94,
    "text": "and this the point whether i change the model"
  },
  {
    "start": 1464.46,
    "end": 1464.74,
    "text": "or"
  },
  {
    "start": 1465.92,
    "end": 1469.62,
    "text": "repeat my experiments or smoothed the data according to the model."
  },
  {
    "start": 1470.7,
    "end": 1472.9,
    "text": "This just the pragmatix of science."
  },
  {
    "start": 1473.14,
    "end": 1475.66,
    "text": "There's not an overarching rule for it."
  },
  {
    "start": 1476.56,
    "end": 1478.88,
    "text": "That isn't a rule of rules so-to say."
  },
  {
    "start": 1479.88,
    "end": 1488.08,
    "text": "I just wanted to say, those ideas are in philosophy of science everywhere and some form of variations."
  },
  {
    "start": 1488.3,
    "end": 1507.16,
    "text": "And like Hempel Oppenheim already had this model of scientific explanations roughly speaking modus ponens... ...and of course whenever you don't get your conclusion Your modus tollens tells that You need to be rid off one of the assumptions but it doesn't tell which ones."
  },
  {
    "start": 1507.9,
    "end": 1510.84,
    "text": "Right, so you can always pick."
  },
  {
    "start": 1511.22,
    "end": 1514.16,
    "text": "You can say okay the measurement when fine is an assumption?"
  },
  {
    "start": 1514.22,
    "end": 1516.96,
    "text": "You normally have and then you'll say oh this one was wrong."
  },
  {
    "start": 1517.84,
    "end": 1528.6,
    "text": "And um Then This is closely related to some quinion ideas of this web of leaves we have like core beliefs We are not willing to give up."
  },
  {
    "start": 1531.52,
    "end": 1538.12,
    "text": "But if we have other believes that we are happier to give out at math is normally considered a quite Core belief."
  },
  {
    "start": 1538.38,
    "end": 1540.42,
    "text": "normally you don't give up addition."
  },
  {
    "start": 1540.62,
    "end": 1544.96,
    "text": "Hmm, normally you give up That that person is trustworthy."
  },
  {
    "start": 1545.18,
    "end": 1546.26,
    "text": "who from whom?"
  },
  {
    "start": 1546.46,
    "end": 1552.0,
    "text": "I bought two times to apples and just got three in my bag after opening it at home."
  },
  {
    "start": 1554.08,
    "end": 1555.72,
    "text": "What we mean by giving a petition?"
  },
  {
    "start": 1556.34,
    "end": 1562.46,
    "text": "We may observe that speeds don't add upright So they're not giving up."
  },
  {
    "start": 1564.6,
    "end": 1564.78,
    "text": "That's"
  },
  {
    "start": 1564.88,
    "end": 1565.76,
    "text": "exactly the point."
  },
  {
    "start": 1565.86,
    "end": 1575.64,
    "text": "So even a physical rule that you might have had speeds at up is, You are more willing to give that up than giving up addition so-to speak."
  },
  {
    "start": 1576.08,
    "end": 1576.7,
    "text": "it all."
  },
  {
    "start": 1576.84,
    "end": 1579.76,
    "text": "It's like the core is less important then the boundaries."
  },
  {
    "start": 1580.1,
    "end": 1580.84,
    "text": "and what's the car?"
  },
  {
    "start": 1580.92,
    "end": 1582.32,
    "text": "Is very subjective thing."
  },
  {
    "start": 1582.56,
    "end": 1587.4,
    "text": "right if you If your hardcore Yep one two"
  },
  {
    "start": 1588.3,
    "end": 1595.26,
    "text": "Say if you have a web of beliefs that are intermingled At the point of quite is that?"
  },
  {
    "start": 1596.06,
    "end": 1601.28,
    "text": "You cannot test a specific strand off that web independently, but you test them well as I hope."
  },
  {
    "start": 1601.84,
    "end": 1602.84,
    "text": "If uh-huh."
  },
  {
    "start": 1602.88,
    "end": 1610.48,
    "text": "You have an experiment and then you have your addition And then there was sort of experiment doesn't match your addition procedures."
  },
  {
    "start": 1611.5,
    "end": 1620.04,
    "text": "instead of tuning this trend of Addition you tuned a different one which is my perception is reliable."
  },
  {
    "start": 1620.5,
    "end": 1625.44,
    "text": "Well, if you cut that one then you can preserve the addition."
  },
  {
    "start": 1625.58,
    "end": 1628.26,
    "text": "so your testing the web as a whole not just edition."
  },
  {
    "start": 1629.9,
    "end": 1633.18,
    "text": "There's also more specific form of this."
  },
  {
    "start": 1633.54,
    "end": 1637.16,
    "text": "It's different but it's Poincare's conventionalism about geometry."
  },
  {
    "start": 1638.32,
    "end": 1639.56,
    "text": "So what Poincaré?"
  },
  {
    "start": 1640.02,
    "end": 1644.06,
    "text": "he was very early Conventionalist on mathematics."
  },
  {
    "start": 1644.82,
    "end": 1653.0,
    "text": "He said our that many different models of geometry are compatible with the same empirical observations."
  },
  {
    "start": 1653.64,
    "end": 1658.5,
    "text": "And because of that, The model of geometry we stick to is a convention."
  },
  {
    "start": 1660.14,
    "end": 1667.0,
    "text": "So it's a convention for which one of those different compatible models I end up with."
  },
  {
    "start": 1668.22,
    "end": 1672.7,
    "text": "I know you're not so interested in foundations."
  },
  {
    "start": 1673.68,
    "end": 1679.38,
    "text": "I mean there are sort of inconsistent theories and they were proposed, right?"
  },
  {
    "start": 1682.04,
    "end": 1685.7,
    "text": "Obviously Frege is very famous but that happened again at the end."
  },
  {
    "start": 1686.36,
    "end": 1703.6,
    "text": "And nowadays we implement these logical theories on a computer... ...and then maybe an AI would exploit the inconsistency of our foundations to prove just following the rules that three equals four, right?"
  },
  {
    "start": 1704.34,
    "end": 1705.96,
    "text": "So what do we know now?"
  },
  {
    "start": 1706.18,
    "end": 1716.62,
    "text": "I mean you just applied rules and become to the results at our inconsistent observation is actually consistent with overalls."
  },
  {
    "start": 1719.16,
    "end": 1728.74,
    "text": "That's a great cut for this second half so-to speak to Wittgenstein because his idea of rule following it like very central part."
  },
  {
    "start": 1729.7,
    "end": 1733.46,
    "text": "You are the main research branch, if I may call it like that."
  },
  {
    "start": 1734.92,
    "end": 1736.64,
    "text": "So, is Wittgenstein helping us here?"
  },
  {
    "start": 1737.82,
    "end": 1738.84,
    "text": "Well...I don't know."
  },
  {
    "start": 1738.88,
    "end": 1744.48,
    "text": "he's helping you in foundational matters or or and reasons."
  },
  {
    "start": 1745.0,
    "end": 1754.58,
    "text": "It depends on perspective but for Wittgeinstein didn't make sense to add foundations to mathematical practices."
  },
  {
    "start": 1754.82,
    "end": 1757.86,
    "text": "so let say a rule two plus three four."
  },
  {
    "start": 1759.66,
    "end": 1764.08,
    "text": "We don't need a rule for that rule, or we do not have foundations."
  },
  {
    "start": 1764.9,
    "end": 1776.12,
    "text": "The foundation of the rules is already in practice and it's how I use them In the same way as natural language doesn´t need foundations."
  },
  {
    "start": 1776.84,
    "end": 1783.94,
    "text": "And still if there are rules followed then why should we need foundations for mathematics?"
  },
  {
    "start": 1784.58,
    "end": 1791.22,
    "text": "Foundations for mathematics would be needed if you think of mathematics as structures that can become vulnerable."
  },
  {
    "start": 1791.46,
    "end": 1795.72,
    "text": "And do know the typical Hilbert analogy, mathematics is a building."
  },
  {
    "start": 1796.8,
    "end": 1800.1,
    "text": "If it's a building and they can collapse then you need secure foundations."
  },
  {
    "start": 1800.74,
    "end": 1806.02,
    "text": "But if mathematics are not structured but rules there suddenly you don't need foundations."
  },
  {
    "start": 1806.08,
    "end": 1810.58,
    "text": "in that sense"
  },
  {
    "start": 1810.62,
    "end": 1812.36,
    "text": "I mean just my example."
  },
  {
    "start": 1814.02,
    "end": 1823.98,
    "text": "And I mean, there's a system of rules you could say which corresponds to the foundations."
  },
  {
    "start": 1824.98,
    "end": 1828.36,
    "text": "And...I would agree that foundations are not absolute."
  },
  {
    "start": 1828.78,
    "end": 1841.36,
    "text": "they're way too reflect our ways to use reasoning and an attempt be more precise To avoid having discuss each construction."
  },
  {
    "start": 1843.1,
    "end": 1849.2,
    "text": "So I don't buy Professor Kantor, or proof of the well-ordering theorem."
  },
  {
    "start": 1849.24,
    "end": 1850.18,
    "text": "What did you actually use?"
  },
  {
    "start": 1850.32,
    "end": 1858.26,
    "text": "Then we need to go back and really understand the structure for reasoning."
  },
  {
    "start": 1858.34,
    "end": 1861.38,
    "text": "so... I mean i do not agree with"
  },
  {
    "start": 1867.42,
    "end": 1870.16,
    "text": "Wittgenstein here."
  },
  {
    "start": 1873.1,
    "end": 1876.44,
    "text": "that the written rules had never really determines the practice."
  },
  {
    "start": 1877.36,
    "end": 1881.12,
    "text": "And they're very parallel sayings in The Philosophy of Math."
  },
  {
    "start": 1881.18,
    "end": 1892.2,
    "text": "like when Penn Maddy says, mathematicians work one step back from disaster she means yeah we worked with knife comprehension and then we see it doesn't work out?"
  },
  {
    "start": 1892.98,
    "end": 1900.02,
    "text": "Then try to keep as much comprehension of things."
  },
  {
    "start": 1900.44,
    "end": 1904.96,
    "text": "But it's like, so to speak a minimal reparation as little as possible."
  },
  {
    "start": 1906.36,
    "end": 1912.12,
    "text": "and there is this kind of meta rule don't abuse the contradiction."
  },
  {
    "start": 1912.78,
    "end": 1919.38,
    "text": "I mean in every day i'm still using knife comprehension...I don't care whether the sense really exists..i just comprehend."
  },
  {
    "start": 1919.72,
    "end": 1928.4,
    "text": "do comprehension as i wish And then later im thinking oh can i really that when it gets shaky or when its very weird move"
  },
  {
    "start": 1930.02,
    "end": 1932.84,
    "text": "I mean, it's not when i start to do something."
  },
  {
    "start": 1936.34,
    "end": 1962.18,
    "text": "And actually, maybe unlike at the time of Wittgenstein we can do this nowadays."
  },
  {
    "start": 1962.24,
    "end": 1966.76,
    "text": "We have computer implementations Of logical systems and I guess"
  },
  {
    "start": 1966.84,
    "end": 1968.68,
    "text": "if you really get to contradiction there?"
  },
  {
    "start": 1968.78,
    "end": 1973.08,
    "text": "We will change the rules a little bit even If it's a pragmatic thing."
  },
  {
    "start": 1973.14,
    "end": 1973.96,
    "text": "don't use AI."
  },
  {
    "start": 1974.04,
    "end": 1978.56,
    "text": "they are destroying your Code so-to speak."
  },
  {
    "start": 1978.76,
    "end": 1982.02,
    "text": "or may be its technical proper change of things."
  },
  {
    "start": 1982.4,
    "end": 1986.28,
    "text": "Maybe something related to the representation of things."
  },
  {
    "start": 1986.44,
    "end": 1987.98,
    "text": "and one final sentence."
  },
  {
    "start": 1988.02,
    "end": 1991.74,
    "text": "And then I pass over to Rossi, sorry for capturing."
  },
  {
    "start": 1991.8,
    "end": 1995.9,
    "text": "that is like when a formal practice doesn't work."
  },
  {
    "start": 1996.24,
    "end": 2001.72,
    "text": "we change the practice by saying there was never this spirit off-the-rules."
  },
  {
    "start": 2001.82,
    "end": 2003.54,
    "text": "Wittgenstein has his tennis example."
  },
  {
    "start": 2004.14,
    "end": 2016.46,
    "text": "if you start to throw your tennis ball two thousand meters in the end it takes five minutes before comes down as you hit it To your opponent, then we will come up with a rule."
  },
  {
    "start": 2016.9,
    "end": 2017.48,
    "text": "Don't do that."
  },
  {
    "start": 2017.56,
    "end": 2028.44,
    "text": "It's boring it's take too long while technically there was like nothing in the rules had against throwing things out Kilometers in the air Because it never wasn't an issue."
  },
  {
    "start": 2028.6,
    "end": 2030.66,
    "text": "so you didn't have to rule even and chess."
  },
  {
    "start": 2031.0,
    "end": 2039.24,
    "text": "The official guidebook says these rules aren't complete And if we get a problem the arbiter has the same."
  },
  {
    "start": 2040.94,
    "end": 2048.08,
    "text": "Which came up when the castling rule which is like an historic fun fact that if you promote your pawn, you get a new rook."
  },
  {
    "start": 2048.96,
    "end": 2057.8,
    "text": "So technically you can castle with that new Rook when your king is going to fields up before I think and the castle jumps over him?"
  },
  {
    "start": 2058.46,
    "end": 2060.719,
    "text": "And it was never intended to be castling."
  },
  {
    "start": 2061.179,
    "end": 2064.76,
    "text": "so we needed to rule it out later by changing rules."
  },
  {
    "start": 2064.86,
    "end": 2065.44,
    "text": "but sorry for that."
  },
  {
    "start": 2065.48,
    "end": 2069.84,
    "text": "It's just"
  },
  {
    "start": 2069.88,
    "end": 2071.239,
    "text": "really good exposition."
  },
  {
    "start": 2072.8,
    "end": 2082.06,
    "text": "What I would add there is that the like, The deepest aspect in this thing you're saying Is that for Bigenstein?"
  },
  {
    "start": 2084.34,
    "end": 2085.82,
    "text": "The formulation of a rule."
  },
  {
    "start": 2086.52,
    "end": 2091.44,
    "text": "so how our rules expressed in letters or symbols Under determines."
  },
  {
    "start": 2091.719,
    "end": 2095.94,
    "text": "How are we should be followed on how rules and follow the community of people."
  },
  {
    "start": 2096.48,
    "end": 2107.12,
    "text": "So You have a rule say I don't know, even a contradiction would work but i don't want to make such an extra example."
  },
  {
    "start": 2107.46,
    "end": 2107.92,
    "text": "or why not?"
  },
  {
    "start": 2108.3,
    "end": 2117.3,
    "text": "if i have a rule like leave the room and dont'leave the room maybe you are puzzled on how to follow that rule because it's a contradiction."
  },
  {
    "start": 2118.2,
    "end": 2127.1,
    "text": "But... Because the formulation of this rule does under-determine what do we did then can followed in different ways."
  },
  {
    "start": 2127.4,
    "end": 2134.08,
    "text": "one way will be well for this group of people, these rules should be followed in the following way."
  },
  {
    "start": 2135.24,
    "end": 2143.2,
    "text": "Leak the room mentally but not physically so stay in the room and fall asleep For instance."
  },
  {
    "start": 2143.6,
    "end": 2147.48,
    "text": "So if we apply that... This is called the rule-following paradox."
  },
  {
    "start": 2148.34,
    "end": 2165.12,
    "text": "If we say that mathematics are rules And that we follow rules when we do mathematics Then mathematics type are by definition exposed to the rule following paradox, meaning that we can have our symbols but then we can follow these symbols."
  },
  {
    "start": 2166.26,
    "end": 2172.06,
    "text": "We could use the symbols or follow mathematical derivation and do things with them in different ways."
  },
  {
    "start": 2172.98,
    "end": 2184.76,
    "text": "so... The elephant is why most people think mathematics almost always the same like multiplication addition."
  },
  {
    "start": 2185.9,
    "end": 2189.66,
    "text": "well this would not be a fact of mathematics."
  },
  {
    "start": 2189.98,
    "end": 2191.58,
    "text": "this would be a fact of the rule."
  },
  {
    "start": 2191.7,
    "end": 2201.34,
    "text": "followers like they are trained since little to follow rules in a similar way and because of that, They do the same things with the symbols."
  },
  {
    "start": 2201.9,
    "end": 2216.78,
    "text": "but it's not Because mathematics has structure That constrains you by virtue of structure To do one thing or other But It is a psychological constraint almost not a structural constraint of mathematics."
  },
  {
    "start": 2217.14,
    "end": 2227.4,
    "text": "So if you have a contradiction and cannot use it, It's not because the contradiction is flawed in mathematics or we need to fix our foundations."
  },
  {
    "start": 2227.52,
    "end": 2230.1,
    "text": "We should not have contradictions but rather beacuse."
  },
  {
    "start": 2230.2,
    "end": 2236.78,
    "text": "we haven't been given The proper training To make use Of this specific contradiction."
  },
  {
    "start": 2237.42,
    "end": 2240.26,
    "text": "And because we don't have that Training then were puzzled."
  },
  {
    "start": 2241.02,
    "end": 2243.48,
    "text": "so This would be my vegan science point"
  },
  {
    "start": 2244.98,
    "end": 2250.12,
    "text": "I don't know, it seems to me that this is sort of turning the actual process around."
  },
  {
    "start": 2250.98,
    "end": 2268.0,
    "text": "Because first we come up with maybe some ideas or some explanation for mathematical with some theorems which have an intuitive justification and then constructs a proof under challenge prepared."
  },
  {
    "start": 2269.14,
    "end": 2275.46,
    "text": "And this is a bit like how humans work."
  },
  {
    "start": 2275.56,
    "end": 2277.18,
    "text": "also, when we do moral judgments."
  },
  {
    "start": 2277.94,
    "end": 2285.46,
    "text": "We usually come to the conclusion or view sort of intuitively based on gut feelings yeah?"
  },
  {
    "start": 2286.12,
    "end": 2305.44,
    "text": "But then if you are challenged and say oh yes it's because I have this moral principle but its not that we reason from the moral principles and arrive at our judgment we post hoc explain how our judgment is constructed."
  },
  {
    "start": 2305.68,
    "end": 2307.56,
    "text": "And that's also how mathematics work, right?"
  },
  {
    "start": 2308.24,
    "end": 2322.26,
    "text": "We have got some intuitive insight or some idea of something which should hold and then giving more precise arguments why this holds."
  },
  {
    "start": 2322.5,
    "end": 2331.4,
    "text": "Here are the foundations which aren't fixed for all times but they're evolving as part play an important role that we say, okay."
  },
  {
    "start": 2331.9,
    "end": 2337.78,
    "text": "Based on these foundations which we share I can come to this conclusion and then it's accepted."
  },
  {
    "start": 2339.74,
    "end": 2347.44,
    "text": "So actually that is interesting because the later video was some extent influenced by intuitionism."
  },
  {
    "start": 2348.86,
    "end": 2354.64,
    "text": "so the scope of influence."
  },
  {
    "start": 2354.9,
    "end": 2356.26,
    "text": "a bit up-to debate."
  },
  {
    "start": 2356.56,
    "end": 2366.86,
    "text": "but here I would ask the following, so where do these intuitions come from in the mathematician?"
  },
  {
    "start": 2367.96,
    "end": 2378.1,
    "text": "So when i test whether something holds or what did I have let's say a string of symbols and want to produce The next step These intuitions."
  },
  {
    "start": 2378.58,
    "end": 2379.5,
    "text": "What are their basis?"
  },
  {
    "start": 2379.64,
    "end": 2387.64,
    "text": "well that they're foundations so-to-say because this is not the foundations In the sense of definitions on mathematics but the foundation of our intuition."
  },
  {
    "start": 2388.1,
    "end": 2392.76,
    "text": "And if we asked question I think we will get closer to the bigger biggest time."
  },
  {
    "start": 2394.86,
    "end": 2398.66,
    "text": "I mean, Brauer wasn't so interested in formal systems at all."
  },
  {
    "start": 2399.38,
    "end": 2412.7,
    "text": "and he also said that there are these ure intuitions through which we trade back an important one as of the ability of count because you can always have some next-time... So this is a time where it goes on."
  },
  {
    "start": 2412.86,
    "end": 2421.88,
    "text": "And the other Uwe intuition is geometry, that we have an intuitive understanding of geometry."
  },
  {
    "start": 2422.84,
    "end": 2428.24,
    "text": "We can use these spacing intuitions to reason and... It was bro us still"
  },
  {
    "start": 2428.44,
    "end": 2429.34,
    "text": "accepting both?"
  },
  {
    "start": 2429.48,
    "end": 2430.98,
    "text": "I just a curious question!"
  },
  {
    "start": 2431.1,
    "end": 2441.24,
    "text": "I thought space wasn't accepted yet but then this non-archedian geometries got rid about our... who instincts of space because we could be wrong there."
  },
  {
    "start": 2441.28,
    "end": 2443.78,
    "text": "and then Brauer said, but time is also enough."
  },
  {
    "start": 2444.24,
    "end": 2448.3,
    "text": "But maybe I'm confused that you are the expert on"
  },
  {
    "start": 2449.6,
    "end": 2449.66,
    "text": "this?"
  },
  {
    "start": 2450.54,
    "end": 2451.38,
    "text": "Maybe not."
  },
  {
    "start": 2451.44,
    "end": 2455.5,
    "text": "as far as philosophy is concerned... I mean a lot of biography about Brauer."
  },
  {
    "start": 2455.86,
    "end": 2456.82,
    "text": "Then"
  },
  {
    "start": 2456.88,
    "end": 2457.04,
    "text": "you'll"
  },
  {
    "start": 2457.12,
    "end": 2459.06,
    "text": "probably write a script we need to look"
  },
  {
    "start": 2459.12,
    "end": 2459.24,
    "text": "at."
  },
  {
    "start": 2459.48,
    "end": 2479.02,
    "text": "Maybe I'm mixing, but that's of my understanding and it is interesting just with respect for some interviews recently had where Steve Audi especially discussed this interaction between the geometric intuition."
  },
  {
    "start": 2479.76,
    "end": 2486.22,
    "text": "So, of course discreet intuition and how they interact."
  },
  {
    "start": 2486.36,
    "end": 2490.02,
    "text": "And lead to new ideas like this modern type theory?"
  },
  {
    "start": 2491.24,
    "end": 2497.02,
    "text": "So then once we are there... We're getting closer to understanding that."
  },
  {
    "start": 2497.96,
    "end": 2504.64,
    "text": "what is the relationship between foundations in the standard sense and the foundation of the intuitions of the mathematician?"
  },
  {
    "start": 2506.38,
    "end": 2507.38,
    "text": "Is one sufficient...?"
  },
  {
    "start": 2508.38,
    "end": 2514.3,
    "text": "If one of the two is sufficient for mathematical practice, which one are you going to drop"
  },
  {
    "start": 2514.5,
    "end": 2514.56,
    "text": "on?"
  },
  {
    "start": 2516.56,
    "end": 2518.4,
    "text": "I don't think that makes any sense."
  },
  {
    "start": 2520.64,
    "end": 2525.84,
    "text": "Our intuition is our main guidance and it's a way we work."
  },
  {
    "start": 2525.9,
    "end": 2526.66,
    "text": "We really think."
  },
  {
    "start": 2527.6,
    "end": 2539.2,
    "text": "And when we do mathematics then both the rules and foundation The precise rules are expressed in our foundations."
  },
  {
    "start": 2539.66,
    "end": 2549.24,
    "text": "I was just thinking earlier that you could say, that Russell was abusing the principle of comprehension as you said."
  },
  {
    "start": 2549.32,
    "end": 2561.44,
    "text": "so if Clyde used to rule anyway which wasn't actually intended by Kantor or by Frege and hence he did exactly this."
  },
  {
    "start": 2561.5,
    "end": 2571.78,
    "text": "what you described early on came up with different approaches how to stop this abuse, right?"
  },
  {
    "start": 2571.98,
    "end": 2579.0,
    "text": "I mean either ZFCO type theory or Martin Lerp."
  },
  {
    "start": 2579.04,
    "end": 2591.14,
    "text": "there are all attempts of comprehension and so on precise in a way that they hopefully cannot be abused but we never know."
  },
  {
    "start": 2593.36,
    "end": 2605.86,
    "text": "So i'm thinking In a way, the foundations and standards in this standard sense could be... Yeah they reflect our intuitions."
  },
  {
    "start": 2607.04,
    "end": 2610.96,
    "text": "So there is not strong demarcation between two senses of foundation."
  },
  {
    "start": 2611.5,
    "end": 2617.7,
    "text": "but yeah the foundations also guide our intuition."
  },
  {
    "start": 2617.84,
    "end": 2624.7,
    "text": "so what I wanted to get right here?"
  },
  {
    "start": 2625.62,
    "end": 2630.24,
    "text": "it seems like the foundations in this standard sense are subordinate to intuition."
  },
  {
    "start": 2630.68,
    "end": 2632.48,
    "text": "First, they come from intuition."
  },
  {
    "start": 2632.96,
    "end": 2636.16,
    "text": "second, they tune our intuitions."
  },
  {
    "start": 2636.82,
    "end": 2645.68,
    "text": "so In a way It's since like that intuitions They prime over mathematical foundation."
  },
  {
    "start": 2645.8,
    "end": 2655.64,
    "text": "So if we already agree on that then the next step, it's a more radical version that would be the Bidestinian one."
  },
  {
    "start": 2658.94,
    "end": 2669.8,
    "text": "It is our rule-following practices that prime so much over everything else... That everything else almost meaningless!"
  },
  {
    "start": 2670.34,
    "end": 2674.98,
    "text": "So why do you want to ground two plus three equals four in foundations or better?"
  },
  {
    "start": 2675.32,
    "end": 2679.96,
    "text": "One plus one equals two because this is the famous example from the Piscite Mathematica."
  },
  {
    "start": 2680.9,
    "end": 2683.4,
    "text": "Why do you want to ground that on logical foundations?"
  },
  {
    "start": 2683.46,
    "end": 2697.34,
    "text": "For instance, at a Russell style if You already have solid intuitions about one plus one equal two and how that behaves How is it going to improve your mathematical practice."
  },
  {
    "start": 2697.48,
    "end": 2699.5,
    "text": "I'm not saying that he doesn't i'm saying that."
  },
  {
    "start": 2704.6,
    "end": 2713.76,
    "text": "So it improves because there has been an always arguments mathematical arguments."
  },
  {
    "start": 2713.88,
    "end": 2731.74,
    "text": "I mean, there is this famous book by Imre Lakatos about obviously proof and refutation that certain proves are not acceptable."
  },
  {
    "start": 2733.24,
    "end": 2736.6,
    "text": "And okay he describes it correctly as a dialectic process."
  },
  {
    "start": 2738.58,
    "end": 2747.1,
    "text": "play a role in this dialectic process because, we say okay now...we established the rules."
  },
  {
    "start": 2748.92,
    "end": 2765.8,
    "text": "We can agree on and this way avoid having discussion about every construction whether you accept it or not but put down the law like we do an illegal proceedings as well."
  },
  {
    "start": 2766.02,
    "end": 2785.98,
    "text": "It's not that we each time when you have a potential misdemeanor, try to reason based on moral principles but then say okay hang in here some rules and we simplify the process by saying go follow the law or haven't followed."
  },
  {
    "start": 2790.78,
    "end": 2792.66,
    "text": "Yeah, that's a really interesting point."
  },
  {
    "start": 2792.88,
    "end": 2797.24,
    "text": "Actually I provide a rational reconstruction of Lakatos with Vivian senior philosophy."
  },
  {
    "start": 2798.14,
    "end": 2801.88,
    "text": "So I know this point and II really accepted so."
  },
  {
    "start": 2802.14,
    "end": 2809.38,
    "text": "but Then we are ascribing a function to foundations That is not very standard."
  },
  {
    "start": 2809.8,
    "end": 2816.62,
    "text": "it's like they're clarificatory they're like in a legal context, you have this rule and then it's applied to many cases."
  },
  {
    "start": 2817.14,
    "end": 2832.98,
    "text": "But the point here I would follow the Rule Following Paradox again is that clarifications on how to follow a rule are helpful but do not solve the Rule following paradox because then the Rule Following Paradox applies to the clarification itself."
  },
  {
    "start": 2833.9,
    "end": 2839.3,
    "text": "so now I know that the rule is longer, so to say when i have clarified it."
  },
  {
    "start": 2839.72,
    "end": 2842.08,
    "text": "Now how do I apply and find a problem again?"
  },
  {
    "start": 2842.4,
    "end": 2851.7,
    "text": "Of course sometimes its practical because I can control the rules in better way but doesn't fully solve this problem."
  },
  {
    "start": 2855.94,
    "end": 2867.56,
    "text": "The rules of mathematics are already well controlled compared with other rules But still they suffer from these problems as the foundational problem show."
  },
  {
    "start": 2869.9,
    "end": 2873.08,
    "text": "Let me think how I would put this in a"
  },
  {
    "start": 2874.98,
    "end": 2875.4,
    "text": "simple way?"
  },
  {
    "start": 2876.0,
    "end": 2884.72,
    "text": "As we know, whatever foundation of theory you have... ...we cannot use it to prove the consistency of these very foundations right?"
  },
  {
    "start": 2885.62,
    "end": 2887.54,
    "text": "What do you describe?"
  },
  {
    "start": 2887.58,
    "end": 2888.76,
    "text": "is the rule-following paradox?"
  },
  {
    "start": 2889.44,
    "end": 2901.88,
    "text": "is the sort of incompleteness of this rule-foring principle in a similar sense as Girdel describes incompletness, an incompletence off foundation idea."
  },
  {
    "start": 2902.14,
    "end": 2903.6,
    "text": "It's"
  },
  {
    "start": 2903.72,
    "end": 2903.9,
    "text": "always"
  },
  {
    "start": 2904.08,
    "end": 2913.74,
    "text": "self application which exactly what you're doing that if you apply foundations to questions about foundations then obviously..."
  },
  {
    "start": 2914.92,
    "end": 2921.34,
    "text": "I have thought of another example that, i think would clarify this and it's an analogy to what you said about a legal code with things."
  },
  {
    "start": 2922.18,
    "end": 2923.98,
    "text": "With stipulations on how we should do in each case?"
  },
  {
    "start": 2924.48,
    "end": 2937.22,
    "text": "Well... Legal codes actually suffer from the rule for word paradox as well because the concepts they use are open-textured like We have rules on Monopolies."
  },
  {
    "start": 2937.42,
    "end": 2945.56,
    "text": "For instance if you have a monopoly You had to split your company in half And you have that well-codified in legal codes, many countries."
  },
  {
    "start": 2946.04,
    "end": 2952.04,
    "text": "But then the problem is... In Alicato Ocean since what's a monopoly?"
  },
  {
    "start": 2952.34,
    "end": 2954.1,
    "text": "What are the definitions of monopolies?"
  },
  {
    "start": 2954.52,
    "end": 2956.44,
    "text": "and then you have definition of monopoly."
  },
  {
    "start": 2956.56,
    "end": 2963.88,
    "text": "Well company controls all say there for instance production of certain goods."
  },
  {
    "start": 2965.34,
    "end": 2969.74,
    "text": "Then we had case like Apple on the iPhone."
  },
  {
    "start": 2970.66,
    "end": 2976.72,
    "text": "Apple was sued for Monopoly over the App Store because the app store is a monopoly."
  },
  {
    "start": 2977.54,
    "end": 2990.12,
    "text": "It means that you have an iPhone, You cannot have any other App Store only Apple's App Store... ...you can not buy apps in the Apple's app stores so apple can afford to take a thirty percent cap of the price of the app."
  },
  {
    "start": 2990.74,
    "end": 2991.88,
    "text": "So it was sued from Monopoly."
  },
  {
    "start": 2991.96,
    "end": 3003.0,
    "text": "but then Apple said No, this is not a monopoly because monopoly does not... whether it's a monopoly or not that doesn't apply at the level of apps in my phone."
  },
  {
    "start": 3003.44,
    "end": 3013.16,
    "text": "It applies to the levels of phones and there are other producers on phones like Samsung, Xiaomi etc."
  },
  {
    "start": 3013.94,
    "end": 3015.9,
    "text": "so I do not hold a monopoly."
  },
  {
    "start": 3017.1,
    "end": 3022.06,
    "text": "So then the definition for monopoly in the legal code What does it apply?"
  },
  {
    "start": 3022.88,
    "end": 3026.54,
    "text": "To the apps within a phone or to the phone market?"
  },
  {
    "start": 3026.98,
    "end": 3029.34,
    "text": "and then you don't know how to follow their rule anymore."
  },
  {
    "start": 3030.72,
    "end": 3032.22,
    "text": "Then you have found that we're following paradox."
  },
  {
    "start": 3032.48,
    "end": 3053.04,
    "text": "You can do two different things That both conformed to the rule And that's the problem of the foundations, but sometimes they can be pragmatic because they clarify a rule-following But many other times They don't there just extra symbols in this train Because in principle, they're still subjected to the Rufalgring paradox."
  },
  {
    "start": 3054.36,
    "end": 3064.46,
    "text": "We know that Josey still wanted to show us something about Wittgenstein and Turing which is probably also related foundation... Something you wrote a while ago."
  },
  {
    "start": 3064.56,
    "end": 3071.86,
    "text": "Yeah, or don't you want to?"
  },
  {
    "start": 3071.94,
    "end": 3072.86,
    "text": "No it's fine."
  },
  {
    "start": 3073.34,
    "end": 3080.8,
    "text": "I had prepared this in case because i think just incase we went into the Turing versus Wittgenstein debate."
  },
  {
    "start": 3082.32,
    "end": 3089.98,
    "text": "but its also good in context as it illustrates Wittgernstein view of mathematics a bit more clearly with pictures."
  },
  {
    "start": 3090.14,
    "end": 3101.38,
    "text": "so this view that mathematics is not about structures And for this reason, we can afford to have contradictions and so on because there's nothing special about them."
  },
  {
    "start": 3101.88,
    "end": 3112.1,
    "text": "It's just something else that can be followed like any other rule and are subjected to the rule-following paradox Just like in legal codes."
  },
  {
    "start": 3112.28,
    "end": 3113.86,
    "text": "So there is nothing distinctive of them."
  },
  {
    "start": 3114.5,
    "end": 3117.9,
    "text": "That's why paradoxes are acceptable in mathematics."
  },
  {
    "start": 3117.94,
    "end": 3118.12,
    "text": "I would"
  },
  {
    "start": 3119.52,
    "end": 3130.06,
    "text": "be open to having an experiment unideal, but we wanted to try it as an experiment and are happy to get feedback."
  },
  {
    "start": 3130.12,
    "end": 3141.14,
    "text": "And might ask our guests in future To prepare such a small slides of presentation if they want But it's not obligatory and only if the feedback is positive."
  },
  {
    "start": 3141.9,
    "end": 3151.7,
    "text": "before you start let me do one tiny advertisement because Hosse I and Colin Jakob Rittbeck who was a guest here wrote about Joel Hamkins."
  },
  {
    "start": 3151.82,
    "end": 3151.86,
    "text": "Who?"
  },
  {
    "start": 3151.96,
    "end": 3164.42,
    "text": "Was also guessed Multiverse theory and we made there a Wittgensteinian take on his ideas of the multiverse because he's saying in a very nutshell, three lines."
  },
  {
    "start": 3165.02,
    "end": 3181.0,
    "text": "He wrote something We should not try to settle the continuum hypothesis Because our experience in CH & non-CH said theoretic universes So to speak, we use this Wittgensteinian notion of petrification."
  },
  {
    "start": 3181.62,
    "end": 3198.92,
    "text": "Something can become a rule that can harden to a rule which wasn't the rule all the time and so nowadays you might argue for the rule don't settle C-H because then you lose a lot of satiratic practice while you normally may have had the rule."
  },
  {
    "start": 3199.36,
    "end": 3202.04,
    "text": "try to settle mathematics statement."
  },
  {
    "start": 3202.42,
    "end": 3210.5,
    "text": "whichever Statements are mathematical, I mean the good statement might not be mathy so don't care about it but CH is."
  },
  {
    "start": 3210.96,
    "end": 3215.68,
    "text": "So care about and come up with new axioms to settle in one way or another."
  },
  {
    "start": 3218.18,
    "end": 3220.12,
    "text": "But Thorsten disagrees for that"
  },
  {
    "start": 3221.48,
    "end": 3221.74,
    "text": "idea?"
  },
  {
    "start": 3222.14,
    "end": 3223.46,
    "text": "No no i agree completely."
  },
  {
    "start": 3223.64,
    "end": 3225.18,
    "text": "just would question AC as well"
  },
  {
    "start": 3226.06,
    "end": 3227.48,
    "text": "ACO, I said CH."
  },
  {
    "start": 3228.2,
    "end": 3228.86,
    "text": "Maybe it's too hot"
  },
  {
    "start": 3228.9,
    "end": 3229.02,
    "text": "here?"
  },
  {
    "start": 3229.1,
    "end": 3229.16,
    "text": "No"
  },
  {
    "start": 3230.92,
    "end": 3230.98,
    "text": "no!"
  },
  {
    "start": 3231.98,
    "end": 3233.68,
    "text": "CH was what i wanted to call."
  },
  {
    "start": 3235.08,
    "end": 3238.74,
    "text": "The axiom of choice is not that problematic in some sense... It's"
  },
  {
    "start": 3238.78,
    "end": 3239.58,
    "text": "very problematic."
  },
  {
    "start": 3240.26,
    "end": 3241.4,
    "text": "Absolutely problematic."
  },
  {
    "start": 3243.1,
    "end": 3244.6,
    "text": "But then you mentally disagree.."
  },
  {
    "start": 3246.2,
    "end": 3249.24,
    "text": "Yeah but I mean clearly follows from ZFC and"
  },
  {
    "start": 3251.5,
    "end": 3252.36,
    "text": "doesn't follow from ZF."
  },
  {
    "start": 3252.4,
    "end": 3252.72,
    "text": "Sure does."
  },
  {
    "start": 3253.1,
    "end": 3253.6,
    "text": "That's so postive."
  },
  {
    "start": 3253.64,
    "end": 3254.18,
    "text": "What the quote is?"
  },
  {
    "start": 3254.38,
    "end": 3256.5,
    "text": "Ossification How it follows from ZFC?"
  },
  {
    "start": 3256.7,
    "end": 3257.34,
    "text": "Oh, that's fine."
  },
  {
    "start": 3257.4,
    "end": 3260.62,
    "text": "Okay then you just give a timing summary of ossification."
  },
  {
    "start": 3260.72,
    "end": 3261.04,
    "text": "thank you"
  },
  {
    "start": 3262.56,
    "end": 3262.78,
    "text": "very much."
  },
  {
    "start": 3262.82,
    "end": 3263.0,
    "text": "Yeah"
  },
  {
    "start": 3263.82,
    "end": 3268.82,
    "text": "the follow-up from ZFC is like similar to the model example."
  },
  {
    "start": 3268.88,
    "end": 3274.02,
    "text": "Like the model follows from data Of course The data made up by the mother."
  },
  {
    "start": 3275.24,
    "end": 3278.0,
    "text": "And I mean That's a thing we do in math all the time."
  },
  {
    "start": 3278.12,
    "end": 3281.46,
    "text": "oh Banach Tarski paradox We learn about it."
  },
  {
    "start": 3281.54,
    "end": 3288.16,
    "text": "It's evidence against AC And after a time, we just use it to teach AC and say look cool."
  },
  {
    "start": 3288.28,
    "end": 3295.84,
    "text": "We had said same with physics like the what was the name of the cat?"
  },
  {
    "start": 3296.94,
    "end": 3297.46,
    "text": "The cat in the"
  },
  {
    "start": 3298.42,
    "end": 3299.3,
    "text": "box or... Schrödinger's cat!"
  },
  {
    "start": 3300.44,
    "end": 3302.64,
    "text": "It wasn't argument against quantum physics."
  },
  {
    "start": 3302.7,
    "end": 3306.14,
    "text": "and now he used that as didactical material."
  },
  {
    "start": 3306.64,
    "end": 3312.12,
    "text": "but I talked way too long and Josey wanted talk about Wittgenstein and Turing for few more minutes."
  },
  {
    "start": 3312.64,
    "end": 3314.56,
    "text": "so please Let's just try the"
  },
  {
    "start": 3314.64,
    "end": 3315.24,
    "text": "experiment."
  },
  {
    "start": 3316.66,
    "end": 3345.68,
    "text": "Just briefly to illustrate Wittgenstein's view of mathematics, so in the lectures on the foundations and mathematics which is a short book about Wittgensstein's lectures in Cambridge where he taught many students that would eventually become world-class mathematicians like Turing Wittgerstein actually tried to convince them after of his own particular use of mathematics and this is how it went."
  },
  {
    "start": 3345.88,
    "end": 3349.94,
    "text": "Well, there's a part on that if I can share my screen."
  },
  {
    "start": 3350.74,
    "end": 3350.84,
    "text": "Yeah."
  },
  {
    "start": 3352.14,
    "end": 3352.34,
    "text": "Okay."
  },
  {
    "start": 3352.54,
    "end": 3353.4,
    "text": "Analogy five?"
  },
  {
    "start": 3354.98,
    "end": 3367.56,
    "text": "Oh yeah because they are many actually... This slide from a talk that i gave some time ago On the analogies that Wittgenstein used to convince Turing about his view of mathematics."
  },
  {
    "start": 3367.96,
    "end": 3380.82,
    "text": "so What Wiggenstein does with this analogy is quite interesting because this analogy doesn't represent Wiggensstein's view of mathematics and logic, it represents Turing's view."
  },
  {
    "start": 3381.56,
    "end": 3402.2,
    "text": "And when Turing isn't persuaded that paradoxes are not that much of a problem in mathematics the contradictions aren't as much of problems in mathematics Because you can just tune the pragmatics from your rule following paradoxes and contradictions in different ways."
  },
  {
    "start": 3403.42,
    "end": 3430.6,
    "text": "When Turing appeals to the structure of mathematics, to still insist on the point that contradictions are structural problems —and because something would go wrong with contradictions— Biggerstein says okay you think about mathematics in a following way like... You have a cup wheel at the left you actually have a full system of co-wheels, a structure of co wheels."
  },
  {
    "start": 3431.04,
    "end": 3439.18,
    "text": "So whatever it is that we're doing the first one determines by virtue of structure what other co wheels will do."
  },
  {
    "start": 3439.86,
    "end": 3445.16,
    "text": "so there's no pragmatics here and if something happens through this first some thing would happen to last."
  },
  {
    "start": 3445.74,
    "end": 3459.48,
    "text": "and its perfectly determined what you have to quote below, it says its movement determines the movements of the last coil in this change."
  },
  {
    "start": 3459.62,
    "end": 3477.18,
    "text": "So there's a determination relationship and because of these if you take mathematics to be that then as system of co-wheels like one you see now cannot ever work Like a contradiction In the typical way of dealing with it."
  },
  {
    "start": 3477.98,
    "end": 3478.72,
    "text": "I can never work."
  },
  {
    "start": 3480.34,
    "end": 3487.74,
    "text": "If you attempt to turn on one of the co-wheel the system itself is jammed, so it will not turn."
  },
  {
    "start": 3488.7,
    "end": 3501.74,
    "text": "So for this... For Turing who thinks of mathematics in th- In the typical structural sense, in this sense of structure a contradiction cannot work because it is jammed language."
  },
  {
    "start": 3502.36,
    "end": 3508.06,
    "text": "Like if I say leave there room and don't leave our room that will not work Because formally speaking thats a contradiction."
  },
  {
    "start": 3510.16,
    "end": 3512.06,
    "text": "That's why we need secure foundations."
  },
  {
    "start": 3512.3,
    "end": 3514.04,
    "text": "so then we're not to reach these contradictions."
  },
  {
    "start": 3514.48,
    "end": 3517.76,
    "text": "And our mathematics do NOT jam like this."
  },
  {
    "start": 3518.4,
    "end": 3525.48,
    "text": "But for bigger size Mathematics are not structures They are not a system of co-wheels, they're rules."
  },
  {
    "start": 3526.54,
    "end": 3539.18,
    "text": "Because because of being rules there's subjected to the rule following paradox and the rules can be followed in many different ways good ways bad ways economic way whatever."
  },
  {
    "start": 3539.7,
    "end": 3544.46,
    "text": "so The point is what our training Is?"
  },
  {
    "start": 3545.72,
    "end": 3554.14,
    "text": "And other facts of us as humans, like there are some rules that are easier for us to execute or remember and so on."
  },
  {
    "start": 3554.22,
    "end": 3561.1,
    "text": "So a rule following is about humans not about structures of language or structures of mathematics."
  },
  {
    "start": 3561.18,
    "end": 3572.94,
    "text": "So contradiction would work but it's not per se structurally jammed."
  },
  {
    "start": 3573.56,
    "end": 3586.1,
    "text": "And this is why for Wittgenstein you don't need secure foundations, because if you find a contradiction just needs good way of using that rule which would be well let"
  },
  {
    "start": 3590.44,
    "end": 3591.56,
    "text": "say"
  },
  {
    "start": 3591.65,
    "end": 3594.39,
    "text": "The paradox, the layer lies."
  },
  {
    "start": 3594.57,
    "end": 3596.45,
    "text": "Therefore it does not lie and so on."
  },
  {
    "start": 3596.81,
    "end": 3603.97,
    "text": "And therefore you have that five times five is three hundred thirty-five Because of the principle of explosion."
  },
  {
    "start": 3604.45,
    "end": 3611.11,
    "text": "Then the rule would be do NOT derive anything from a contradiction then your safe one from contradictions."
  },
  {
    "start": 3611.85,
    "end": 3615.37,
    "text": "Or find another way to use like in the leave the room and don't live the room."
  },
  {
    "start": 3616.35,
    "end": 3618.69,
    "text": "So for Beginstein."
  },
  {
    "start": 3621.13,
    "end": 3630.07,
    "text": "This view, this metaphysical view of mathematics and mathematics constitute structures like certain analogies."
  },
  {
    "start": 3630.15,
    "end": 3639.59,
    "text": "Like the Gogwheel analogy or Hilbert's analogy of The Building they reflect a metaphysical views that mathematics are almost physical objects."
  },
  {
    "start": 3640.87,
    "end": 3652.97,
    "text": "but Wittgenstein says it is actually confusion that mathematics should not be seen as physical object quite similar to ordinary language."
  },
  {
    "start": 3653.61,
    "end": 3670.01,
    "text": "And because of that, we can do similar things as we do in ordinary language without losing what makes mathematics special and maybe making it more economic or economical Because you don't need so much for national work."
  },
  {
    "start": 3671.17,
    "end": 3688.87,
    "text": "Paying attention to the pragmatics of mathematics structural view of mathematics and the mathematics are just An afterthought while they should be much more important as we have been saying in this interview And so on."
  },
  {
    "start": 3689.39,
    "end": 3693.19,
    "text": "So yeah, I thought it was a neat way of presenting what doings."
  },
  {
    "start": 3693.85,
    "end": 3701.13,
    "text": "most people's views even maybe Even for the intuition is sometimes art about mathematics"
  },
  {
    "start": 3705.99,
    "end": 3706.63,
    "text": "people's view."
  },
  {
    "start": 3707.39,
    "end": 3712.27,
    "text": "or maybe I don't think mathematics is anything holy, all foundations are absolute."
  },
  {
    "start": 3712.95,
    "end": 3718.65,
    "text": "But this Wittgenstein view displays a deep misunderstanding."
  },
  {
    "start": 3718.73,
    "end": 3722.91,
    "text": "what the whole of logic in mathematical reasoning... Yes!"
  },
  {
    "start": 3725.91,
    "end": 3744.65,
    "text": "This part of the mathematical practice that we try to avoid contradictions and we tried design systems which are contradiction free to be able to simplify our reasoning and play the role we have discussed previously, which foundations should play."
  },
  {
    "start": 3745.33,
    "end": 3753.37,
    "text": "And hence it's who plays this role that they say there shouldn't be contradiction free otherwise useless and turn this perfectly right."
  },
  {
    "start": 3753.95,
    "end": 3757.03,
    "text": "I mean you can disagree Which is fine?"
  },
  {
    "start": 3757.39,
    "end": 3760.91,
    "text": "Maybe one another example Is like Dirac delta function."
  },
  {
    "start": 3761.83,
    "end": 3764.55,
    "text": "That may be something could help clarifying that."
  },
  {
    "start": 3764.63,
    "end": 3766.13,
    "text": "Jose, do you want to explain"
  },
  {
    "start": 3766.21,
    "end": 3766.23,
    "text": "it?"
  },
  {
    "start": 3766.27,
    "end": 3769.27,
    "text": "Yeah we actually... but then I published a paper about that."
  },
  {
    "start": 3770.85,
    "end": 3773.35,
    "text": "so first to engage with what you just said Thornton."
  },
  {
    "start": 3775.65,
    "end": 3784.75,
    "text": "uh We would also say that having contradictions in our mathematics is inconvenient from a practical point of view because typically we don't know what to do with them."
  },
  {
    "start": 3785.03,
    "end": 3789.15,
    "text": "they confuse us They puzzle as and its not convenient to have them."
  },
  {
    "start": 3789.27,
    "end": 3790.15,
    "text": "That's another thing."
  },
  {
    "start": 3793.13,
    "end": 3798.15,
    "text": "They show that there's something wrong in the design of our logical system, which is exception to Julian's point."
  },
  {
    "start": 3798.25,
    "end": 3798.91,
    "text": "Yeah yeah!"
  },
  {
    "start": 3798.95,
    "end": 3799.25,
    "text": "That's"
  },
  {
    "start": 3799.41,
    "end": 3803.47,
    "text": "Julian's Point and that's different from what Dingerstein"
  },
  {
    "start": 3803.51,
    "end": 3803.61,
    "text": "says."
  },
  {
    "start": 3803.65,
    "end": 3807.73,
    "text": "There is a problem with the design for this foundational system."
  },
  {
    "start": 3808.29,
    "end": 3813.57,
    "text": "we should know where it is absolute, which reflects our intuition or practice."
  },
  {
    "start": 3814.51,
    "end": 3821.89,
    "text": "but as a point-of-reference to have this property of being contradiction free."
  },
  {
    "start": 3822.49,
    "end": 3824.99,
    "text": "So I think it's, yeah..."
  },
  {
    "start": 3825.43,
    "end": 3827.89,
    "text": "I mean Turing refers us to real-world implication."
  },
  {
    "start": 3827.95,
    "end": 3838.81,
    "text": "he says bridges will collapse if we have contradictory arithmetic because engineers would use that but the bridges then will be structurally unsound so to speak."
  },
  {
    "start": 3838.89,
    "end": 3839.49,
    "text": "as an example"
  },
  {
    "start": 3840.73,
    "end": 3840.81,
    "text": "Yeah!"
  },
  {
    "start": 3840.85,
    "end": 3857.19,
    "text": "But even in pure mathematics like there point is If we need to design and here were using already design the foundations so that they're contradiction-free, we are already like in a way coming back to this structural analogy."
  },
  {
    "start": 3857.31,
    "end": 3875.05,
    "text": "Like you need to decide if uh... The base of the buildings so that the building doesn't collapse while for Wittgenstein's point would be yeah maybe we'd need to steer clear of contradictions, but not because they constitute something structural in mathematics."
  },
  {
    "start": 3875.19,
    "end": 3879.85,
    "text": "Not because uh... They need to be part of a design and so on But just because they are inconvenient."
  },
  {
    "start": 3881.07,
    "end": 3885.45,
    "text": "So there's two ways of reaching a similar conclusion that from different point-of views."
  },
  {
    "start": 3886.63,
    "end": 3886.75,
    "text": "Yeah."
  },
  {
    "start": 3887.33,
    "end": 3894.39,
    "text": "So at some point Wingerstein torturing like maybe it's fine for you to avoid contradictions Because they confuse you."
  },
  {
    "start": 3894.93,
    "end": 3898.13,
    "text": "In that case I'm perfectly fine with you doing that."
  },
  {
    "start": 3899.55,
    "end": 3901.23,
    "text": "But you are doing that for a different reason."
  },
  {
    "start": 3902.05,
    "end": 3912.01,
    "text": "The first reason the Vettelbeer senior one allows us for the possibility That certain contradictions are trouble-free Because we know what to use with them."
  },
  {
    "start": 3912.43,
    "end": 3919.73,
    "text": "and this is where the delta function example comes in like the Delta function, it's."
  },
  {
    "start": 3920.87,
    "end": 3928.65,
    "text": "It was criticized by more mathematics driven physicists as being rigorous because it contains contradictions."
  },
  {
    "start": 3929.21,
    "end": 3933.75,
    "text": "And the more practical physicist said, this is not a problem."
  },
  {
    "start": 3934.03,
    "end": 3938.73,
    "text": "It's already economical as it is and perfectly under control Because we know what to do with it."
  },
  {
    "start": 3939.55,
    "end": 3941.23,
    "text": "So why invest in resources?"
  },
  {
    "start": 3941.77,
    "end": 3949.45,
    "text": "In making it rigorous... ...in the sense that it does not contain any possible contradiction when its' not going make any difference at all."
  },
  {
    "start": 3949.95,
    "end": 3953.75,
    "text": "We've worked fine on the delta function but still do these"
  },
  {
    "start": 3955.23,
    "end": 3955.49,
    "text": "days As"
  },
  {
    "start": 3955.63,
    "end": 3961.25,
    "text": "long as we are provided with the training to work with a data function in a nice way, why?"
  },
  {
    "start": 3962.51,
    "end": 3965.25,
    "text": "Work with the symbols in more messy ways."
  },
  {
    "start": 3965.29,
    "end": 3967.25,
    "text": "To have something that is different."
  },
  {
    "start": 3968.49,
    "end": 3977.13,
    "text": "because at least where there's structural anxiety coming... Because if you think mathematics is structured then it functions as an imperfect structure."
  },
  {
    "start": 3977.67,
    "end": 3984.01,
    "text": "but if mathematics aren't structured maybe we can work with"
  },
  {
    "start": 3984.39,
    "end": 3985.71,
    "text": "them in a more economical way."
  },
  {
    "start": 3985.75,
    "end": 3997.35,
    "text": "I agree that if this is a typical conflict between physicists and mathematicians, there are similar situations even when i want to construct something."
  },
  {
    "start": 3997.49,
    "end": 4006.37,
    "text": "I often make inconsistent assumptions or for example I ignore in type theory nor the universe hierarchy because it's just lots of extra work."
  },
  {
    "start": 4006.67,
    "end": 4021.87,
    "text": "obviously...I do exactly what you say but im aware And actually my friend, Ralf Mazzis he just told me last week at the types meeting."
  },
  {
    "start": 4021.93,
    "end": 4029.33,
    "text": "He says that he inadvertently used basically Russell's paradox to prove something which was wrong."
  },
  {
    "start": 4030.13,
    "end": 4038.65,
    "text": "So in the end we remain In the everyday use, we may do exactly this."
  },
  {
    "start": 4039.49,
    "end": 4044.77,
    "text": "We work maybe in an inconsistent system or don't use foundations and we wave our hands."
  },
  {
    "start": 4045.43,
    "end": 4055.43,
    "text": "but at some point when you are asked to provide more evidence then it's a point where they need to refer to foundation such as SHARE."
  },
  {
    "start": 4055.55,
    "end": 4060.85,
    "text": "The point about Foundation is also not that their absolute shared between different people."
  },
  {
    "start": 4061.33,
    "end": 4067.27,
    "text": "And then we can say okay, according to the foundations you need to refine what your doing."
  },
  {
    "start": 4068.45,
    "end": 4072.85,
    "text": "Otherwise it's unsound and possibly the grid will collapse yeah?"
  },
  {
    "start": 4074.29,
    "end": 4083.49,
    "text": "I would say that they share a notion on how to follow rules when face a contradiction."
  },
  {
    "start": 4084.43,
    "end": 4088.97,
    "text": "So if this is a paradox That's bad."
  },
  {
    "start": 4089.05,
    "end": 4103.05,
    "text": "if you like that You cannot have that in your system because most people when they find Russell's paradox They are allowed or they feel allow to derive anything from it, but that's Because let's change the focus."
  },
  {
    "start": 4103.65,
    "end": 4106.43,
    "text": "Let's not say that they share their Russell's Paradox."
  },
  {
    "start": 4106.85,
    "end": 4109.41,
    "text": "What they shared is what to do with Russell's paradox."
  },
  {
    "start": 4109.67,
    "end": 4111.15,
    "text": "I can now derive anything for me."
  },
  {
    "start": 4112.01,
    "end": 4114.569,
    "text": "well We also share the pragmatics, at least."
  },
  {
    "start": 4116.069,
    "end": 4124.069,
    "text": "So we may have Russell's paradox in a context where would... ...we do not share those pragmatiques and then Russell's Paradox is no problem anymore?"
  },
  {
    "start": 4124.97,
    "end": 4126.05,
    "text": "I don't agree."
  },
  {
    "start": 4126.45,
    "end": 4133.09,
    "text": "as i said it was exactly because you may inadvertently use this and that's not so obvious."
  },
  {
    "start": 4134.33,
    "end": 4136.27,
    "text": "Do you suggest using it?"
  },
  {
    "start": 4136.31,
    "end": 4140.61,
    "text": "You need to have interiorized these rules following procedure."
  },
  {
    "start": 4140.97,
    "end": 4143.39,
    "text": "Like, you can't use it inadvertently"
  },
  {
    "start": 4143.61,
    "end": 4144.03,
    "text": "because"
  },
  {
    "start": 4144.13,
    "end": 4148.77,
    "text": "we have been trained to use the theory of explosion...the principle of explosions."
  },
  {
    "start": 4149.33,
    "end": 4150.21,
    "text": "Yeah but look!"
  },
  {
    "start": 4150.25,
    "end": 4154.69,
    "text": "You may have a theorem which is intuitively true."
  },
  {
    "start": 4154.89,
    "end": 4156.19,
    "text": "that's not true in all cases."
  },
  {
    "start": 4156.83,
    "end": 4161.35,
    "text": "and now you prove this to be true in our case by inadvertently using Russell's paradoxes."
  },
  {
    "start": 4161.81,
    "end": 4165.85,
    "text": "And now you claim there are applicable four-cases for which it isn't and the bridge folds down."
  },
  {
    "start": 4166.63,
    "end": 4177.71,
    "text": "So at this point somebody says Your derivation here, I think you made... You appeared to a principle which i don't buy and it's not valid in our foundation."
  },
  {
    "start": 4178.27,
    "end": 4179.11,
    "text": "So please fix this!"
  },
  {
    "start": 4179.67,
    "end": 4183.67,
    "text": "And by fixing that your void is the bridge falling down."
  },
  {
    "start": 4187.13,
    "end": 4194.39,
    "text": "Let me just try some final words because we are over an hour long as I know that process on schedule."
  },
  {
    "start": 4194.43,
    "end": 4194.65,
    "text": "We can"
  },
  {
    "start": 4194.69,
    "end": 4196.49,
    "text": "maybe continue another time if its interesting."
  },
  {
    "start": 4198.93,
    "end": 4200.03,
    "text": "Let me just say something."
  },
  {
    "start": 4200.51,
    "end": 4211.89,
    "text": "Wittgenstein was always stressing that humans are rulebenders, so they're fine with cheating the rules whenever they cause something you cannot live with."
  },
  {
    "start": 4212.79,
    "end": 4218.43,
    "text": "for instance God isn't seeing what happens when I close my window"
  },
  {
    "start": 4220.01,
    "end": 4220.09,
    "text": "if"
  },
  {
    "start": 4220.39,
    "end": 4223.27,
    "text": "the rules of my religion is too strict or something like this."
  },
  {
    "start": 4223.35,
    "end": 4228.19,
    "text": "People use these points of relief and computers don't."
  },
  {
    "start": 4228.95,
    "end": 4238.45,
    "text": "So for somebody doing formal mathematics, it's much more an issue to have paradoxes than for someone not formalizing."
  },
  {
    "start": 4238.51,
    "end": 4262.79,
    "text": "I mean even living in a world where you do math without caring about foundation or knowing foundations which just most of the time goes right with your intuitions and quite formal reasoning in every layman terms but totally unrigorous reasoning formal derivation terms and it goes wrong all the time."
  },
  {
    "start": 4262.83,
    "end": 4272.99,
    "text": "If you look into the literature, You will find so many wrong results in the literature It's everybody knows that Especially the details don't matter too much."
  },
  {
    "start": 4273.09,
    "end": 4275.89,
    "text": "Maybe the theorem is right but the proof is not was wrong."
  },
  {
    "start": 4277.07,
    "end": 4278.73,
    "text": "We are on this situation all the Time."
  },
  {
    "start": 4279.73,
    "end": 4290.21,
    "text": "But So The issue Is really pragmatical And it shows And now I should be careful saying Krosé is not a rule follower."
  },
  {
    "start": 4290.27,
    "end": 4292.65,
    "text": "He's, hey... I think he lives within the law!"
  },
  {
    "start": 4295.67,
    "end": 4296.03,
    "text": "But maybe"
  },
  {
    "start": 4296.59,
    "end": 4301.91,
    "text": "it's something that shows that Austin has more rules following and more rigorous in that sense?"
  },
  {
    "start": 4302.59,
    "end": 4302.79,
    "text": "That's"
  },
  {
    "start": 4302.83,
    "end": 4303.71,
    "text": "what Germany knows about us."
  },
  {
    "start": 4305.15,
    "end": 4306.79,
    "text": "Maybe you have said."
  },
  {
    "start": 4306.83,
    "end": 4316.33,
    "text": "Yeah, I'm Turkish or Kurdish-German combining it as one of your biggest paradoxes very rule-following and very rule bandy at the same time."
  },
  {
    "start": 4317.69,
    "end": 4339.33,
    "text": "Maybe it's a very small comment that is also a small ad, It's about what you said about computers not being subjected to the same pragmatics as human rule followers And that way computers do have this tolerance too Let's say for instance a contradiction because they cannot adjust the rule follower or so on."
  },
  {
    "start": 4339.67,
    "end": 4347.81,
    "text": "We actually Dennis and I have a paper in philosophy and technology where we I say that this is actually the basis of the alignment problem, of AI."
  },
  {
    "start": 4348.63,
    "end": 4354.81,
    "text": "That humans follow rules in a certain way but then machines followed rules in different ways."
  },
  {
    "start": 4357.53,
    "end": 4367.27,
    "text": "The way that machines follow rules if they follow rules which are another thing it's more mechanical But humans... ...are not mechanically in this sense."
  },
  {
    "start": 4368.43,
    "end": 4375.69,
    "text": "So that's why we need to be careful on how with the difference between doing mathematics by human and a machine or..."
  },
  {
    "start": 4376.17,
    "end": 4381.79,
    "text": "But actually, LLMs are very flexible in their role following."
  },
  {
    "start": 4384.15,
    "end": 4387.83,
    "text": "They're quite happy to cheat even more"
  },
  {
    "start": 4387.87,
    "end": 4388.09,
    "text": "than humans."
  },
  {
    "start": 4388.13,
    "end": 4391.55,
    "text": "That is an interesting discussion you should really have there."
  },
  {
    "start": 4391.65,
    "end": 4407.27,
    "text": "I mean they will talk about these horror types computer science as a science, write everything formal mindset versus this data driven approaches that we contrasted in the paper."
  },
  {
    "start": 4408.15,
    "end": 4414.25,
    "text": "But maybe we should speak about that and another episode because I got a lot of messages that we should talk more about AI."
  },
  {
    "start": 4415.23,
    "end": 4425.59,
    "text": "so may be have special subseason with five episodes which focus on AI mostly rather than logic mostly because they intertwined anyway."
  },
  {
    "start": 4427.19,
    "end": 4430.51,
    "text": "Having said that, let me say thank you to Thorsten and Rossé."
  },
  {
    "start": 4430.81,
    "end": 4431.91,
    "text": "I think it was a lovely chat!"
  },
  {
    "start": 4433.45,
    "end": 4443.13,
    "text": "And do one final advertisement... ...and now Buy Me A Coffee is online in the real world timeline so we record this after that."
  },
  {
    "start": 4443.55,
    "end": 4450.31,
    "text": "So now i'm doing an advertisement which didn't happen during first episodes of season because back then It wasn´t on-line."
  },
  {
    "start": 4450.77,
    "end": 4454.75,
    "text": "We are happy about support or post production costs little bit money."
  },
  {
    "start": 4456.09,
    "end": 4459.73,
    "text": "If you feel especially generous, feel free to do that."
  },
  {
    "start": 4459.81,
    "end": 4468.69,
    "text": "And after a while we will think about more benefits and memberships... ...and other formats but let's not be discussed today."
  },
  {
    "start": 4469.35,
    "end": 4471.43,
    "text": "Let me just say again thank-you so much Rossi!"
  },
  {
    "start": 4471.95,
    "end": 4473.25,
    "text": "Is there any last words?"
  },
  {
    "start": 4473.29,
    "end": 4473.55,
    "text": "Thank-you for"
  },
  {
    "start": 4473.59,
    "end": 4473.79,
    "text": "me too!"
  },
  {
    "start": 4475.15,
    "end": 4475.53,
    "text": "Last word?"
  },
  {
    "start": 4475.57,
    "end": 4484.03,
    "text": "Well I'm very thankful to you two for inviting me.. ..and being able to say these crazy things at the Einstein set."
  },
  {
    "start": 4485.47,
    "end": 4486.49,
    "text": "So, yeah."
  },
  {
    "start": 4487.41,
    "end": 4492.51,
    "text": "Wittgenstein was clearly not consistent with the set of beliefs..."
  },
  {
    "start": 4492.81,
    "end": 4498.15,
    "text": "He said bit but inconsistent when he told me that Yeah and it's"
  },
  {
    "start": 4498.65,
    "end": 4498.65,
    "text": "a"
  },
  {
    "start": 4499.13,
    "end": 4500.75,
    "text": "protective"
  },
  {
    "start": 4500.79,
    "end": 4502.25,
    "text": "program Of Wittkensteinian philosophy."
  },
  {
    "start": 4502.43,
    "end": 4503.27,
    "text": "At least some think."
  },
  {
    "start": 4503.41,
    "end": 4504.47,
    "text": "other thing is garbage."
  },
  {
    "start": 4504.59,
    "end": 4507.97,
    "text": "But I think true for more or less every international endeavor."
  },
  {
    "start": 4508.81,
    "end": 4513.01,
    "text": "Think we need to take good things With us."
  },
  {
    "start": 4514.75,
    "end": 4515.85,
    "text": "But of course, that's not implied."
  },
  {
    "start": 4515.89,
    "end": 4521.47,
    "text": "We should accept everything he said is So rather than being in Stein philosophy we need to get Stinean philosophy."
  },
  {
    "start": 4522.23,
    "end": 4522.37,
    "text": "Yeah"
  },
  {
    "start": 4523.19,
    "end": 4528.51,
    "text": "Great then have a great day and subscribe comment And we will reply too."
  },
  {
    "start": 4528.65,
    "end": 4535.75,
    "text": "as many comments as possible most of the time actually all Then bye-bye."
  }
]