aboutlogic #08 | Deborah Kant – Talking with Set Theorists: Insights in Mathematical Philosophy

00:00:00: I think that's still an attractive goal.

00:00:03: Why not?

00:00:04: As a person, i want to understand something and find the truth but... ...the question is how do you really get to the truths?

00:00:16: Is there long truth?

00:00:19: Also good questions now!

00:00:20: We're in philosophy here.

00:00:28: Hello everybody welcome to our next episode of About Logic Our little podcast And I'm super happy that Devorah Kent joined us this time for many, many reasons.

00:00:38: She just published a lovely book based on her PhD study that was awarded by the DEFORMLG.

00:00:45: We are long-time collaborators so we organized an event together where Torsten and i also met.

00:00:52: So maybe this whole channel wouldn't exist without her?

00:00:55: And she does lovely stuff!

00:01:01: offer us a little time to talk about the foundations of set tier.

00:01:04: I'm

00:01:05: very happy to be here with you!

00:01:08: Great, can i very naively invite you?

00:01:11: To give us a few sentences?

00:01:13: what are you working on that?

00:01:14: maybe i should mention your now at the Freie Universität Brussels.

00:01:19: so we nearly met there but right yeah...

00:01:23: Yeah and now in Brussels because My expertise is especially on mathematical practice.

00:01:29: And Brussels has a great place to work with very nice colleagues there, and yeah actually I changed my path in mathematics because the notion of truth gets challenged quickly when you go into set theory.

00:01:49: Then i turned for PhD philosophy to understand more broadly what's happening.

00:01:56: And yeah, that's how I came to do satiotic practice.

00:02:01: and i gathered a lot of data about satanic practice... ...and thats what im still working on getting more complete picture.. ..of whats happening there.

00:02:15: That sounds super interesting!

00:02:17: Would you give us like very short recap?

00:02:20: About this huge study with satirists when we interviewed them?

00:02:27: I can do this.

00:02:28: This was a few years ago, so the starting idea is to understand the notion of naturalness in set theory.

00:02:39: because if you start thinking about axioms and independent problem in set theories that have questions which cannot be answered based on standard axiom established right now You start wondering how to maybe add more axioms if this is possible.

00:02:59: So that's an open philosophical question, I was interested in.

00:03:03: and one idea Is To Add Natural Axioms?

00:03:08: so people also on the literature like One Could Say Good And Others Suggested This Paths To Add natural axiom.

00:03:18: That what brought me to Satiric Practice because If you work out a definition or an account, what is naturalness?

00:03:28: I thought it should be consistent or reflective of what set theorists really find natural.

00:03:37: And so this developed and in the end I said okay i just do real interview study and ask them A lot of questions.

00:03:45: So different set theorist from different research areas within set theory thirty participants in the study and I asked them about how they work with new axioms, was a standard axiom.

00:04:00: With uh... The method of forcing which is very central on set theory because you prove um that A statement Is independent by building different models one In which the sentence is true And One in Which the Sentences Falls?

00:04:18: This technique Of Forcing is Enabling You to do this.

00:04:22: So, how do they work with this method?

00:04:24: And then also about... ...how they interpret the whole situation of satiotic independence.

00:04:31: Of existence and these sentences where you don't know whether they are true or false.

00:04:37: Can we now explain maybe this phenomenon of independence?

00:04:41: as a bit historic note here I think that we're going to see Dana Scott.

00:04:47: He was also involved in giving another proof of independent of the continuum hypothesis.

00:04:53: Can you say a little bit about what this is all about?

00:04:58: Yeah, maybe we should start with Kantor then in the nineteenth century because he was asking about the continuum hypothesis and so very quickly.

00:05:13: The question there is about the size of the real numbers because Kantor also proved that infinite sets.

00:05:24: And then the first infinite set, so natural numbers is the first size of infinity and then you can have other sizes.

00:05:33: and that question about continuum hypothesis where it's the size or real number situated in this hierarchy of infinities?

00:05:47: So Kanta wanted actually to prove.

00:05:50: but later with the axiomatic approach, people became more aware of which axioms you use to prove something.

00:05:58: And then they started to build models for a certain set of axiom and then found out with Goethe's results about the Constructible Universe L. and later was Cohen's result about model in which the continuum hypothesis is false.

00:06:17: so it gives another answer than L that Goethe constructed to say, okay you have a set of axioms but you can add the continuum hypothesis all.

00:06:29: The opposite proposition and two can get them consistent model.

00:06:37: so what does it tell us actually?

00:06:38: I mean is there actually one right answer for the continuum hypotheses or for me?

00:06:48: It seems Is there something like sets, the ideal set?

00:06:58: Personally I wouldn't accept this.

00:07:02: Maybe they're different conceptions of what sets are and in some continuum hypothesis holds but others doesn't.

00:07:10: So

00:07:10: that's one very possible answer.

00:07:16: I think there's good evidence in favor of that answer to just say, okay now we have these different models.

00:07:22: And this seems to show what we understand by a set is not something sufficiently determinate.

00:07:32: so I'm still developing it.

00:07:44: Yeah, so also in my PhD It was not about to find a right answer To this question But first to understand what's happening there?

00:07:55: What actually the problem.

00:07:56: So one goal was to pin down what's going wrong.

00:08:02: and yeah One interesting thing i found out that even The practitioners are very They have very different opinions on these matters.

00:08:12: for me an unexpected result to see that they don't have a consistent or consensus on this matter.

00:08:23: To me, I wanted... My impression was many people in the sets really want some.

00:08:33: so not completely no because of Goedl but as complete possible set of axioms Whereas I like Greek theories, because they apply in more situations.

00:08:49: If you have fewer axioms or fewer principles then whatever you prove is applicable to lots of different interpretations.

00:09:00: so obviously not to set the rotation, I do type theory which is in a sense much weaker.

00:09:11: But i think that's feature and not a bug you know?

00:09:13: So maybe i could say two things about this.

00:09:17: so one attractive feature of having A lot of axioms Is To go along with The idea That in mathematics You ask a question And then you can just Say okay thats true or its false.

00:09:32: So any Question you answer you Can Just Answer it.

00:09:37: You ask, you can just answer it.

00:09:39: And so a lot of people when they turn to mathematics They have this idea now that actually you Just get an answer to your questions.

00:09:48: Yes

00:09:49: and So that's attractive about having a lot Of axioms but still what you mentioned?

00:09:54: I would interpret This as the practice at The mathematical practice is specific one That Is also Used in set theory.

00:10:02: so To work with very weak axiomatic systems, also to maybe learn about the subtleties of the problem because if you have very powerful axioms that's a bit broader at the method to solve your question.

00:10:21: So it is still valuable and I think also set here C value in using weak

00:10:27: systems.".

00:10:28: Maybe to follow up on this view that Thorsten elaborated as there really one conception I mean, there is a model theoretic counterpart to that question which would be universalism versus multiverse view.

00:10:43: Whenever you want to share your personal views of course feel free but also asking about the views.

00:11:03: maybe again two things on this.

00:11:04: So in my PhD work, I was a bit reluctant to address these issues and these ontological terms because that's philosophically more difficult i think also to justify ontological statements like sets exist or there is universe.

00:11:24: but actually then when I spoke to different set theorists so at least the universe view by practitioners who were really in favor of, we have to find true new axioms.

00:11:37: They often articulated their view and ontological terms so that they really believed there is a unique universe of sets.

00:11:46: And what are you doing?

00:11:47: Is to describe this universe and to find out What's True In This Universe.

00:11:53: and then other Practitioners Who Were More In The Pluralist Cam.

00:11:58: So Who Said Actually, ZFC or ZF is an already very powerful theory and it's sufficient to work with this theory we have.

00:12:10: We don't need new axioms.

00:12:14: there are not so much formulating their view in terms of a multiverse view but just still say that if you build these models then they're what we do being more correct than the others.

00:12:34: So what is this multiverse versus universe view?

00:12:37: Can anybody enlighten me?

00:12:41: The Universe View just says that ZFC is starting point for the standard C offset theory and actually you have like you half of natural numbers in intended model, so one universe offsets We is the class of all sets.

00:13:03: And people don't know what it is right now, but the idea that one such determinate universe exists which can be described axiomatically—that's a universe view and The multiverse view most prominently Defended or proposed by Joel Hamkins Is on the contrary to say there's not just One Such Universe But actually the ontological basis of set theory is a multiverse where, or which encompasses like all the models that you can build in set theory.

00:13:45: To me it seems what we call SET I mean i think its close to what your doing as result of mathematical practice and sort of intuition, which is then turned into axioms.

00:14:01: And yeah I would say that's all.

00:14:06: there isn't anything else... Personally i find ZFC far too strong because it has some principles which I don't understand.

00:14:18: Which was ...I have a completely different intuition.

00:14:23: so it's ZFC I mean, for two reasons.

00:14:28: Obviously the axiom of choice which i don't understand and also has power sets Which are too much For me.

00:14:40: the collection of propositions isn't small but that's outside your area of interest.

00:14:50: So if we turn to practice what I really like and think is good for philosophy or philosophy of mathematics, trying to practice.

00:15:01: We can look at set-theoretic practice to understand our sets but we could also look into general mathematical practice.

00:15:14: that's maybe more closer to your viewpoint on these matters.

00:15:20: So, of course you would find many different viewpoints there.

00:15:25: But already I just considered set-theoric practice and already there i didn't find consensus agreement in like one idea of what sets are.

00:15:37: so that's yeah already interesting.

00:15:40: but of course the axiom of choice for example was not very controversial or it's just part.

00:15:48: So, you don't find insensitive practice views like we should use weaker systems or try to avoid the axiom of choice.

00:16:00: I mean in some areas people are very interested in definability and then they like to use other principles which were not consistent with choices such as axioms of determinacy.

00:16:14: but just do these things working and trying to get two proofs.

00:16:21: So then, for these reasons they avoid AC in some methods that apply but not a general very foundational

00:16:29: matter.".

00:16:30: Yeah I mean one consequence of choice is... That you lose the possibility having constructive interpretation.

00:16:45: You want the dual functions, that's what I want coming from Coupé computer side.

00:16:49: The dual function i construct are actually effective and can compute them in a fundamental way destroys this And it is very nice situation to be especially if you apply mathematics for computer science.

00:17:19: And actually, I think if you watch the practice or look at the practice there are more and more people using now systems like Rean to formalize mathematics which is based on type theory.

00:17:31: it also usually assumes choice but i-i...I thing can observe that most.

00:17:38: what would be do they don't really need?

00:17:42: it's like a habit, maybe or like a drug addiction.

00:17:48: There are certainly advantages in not using choice and that is very illegitimate.

00:17:54: I mean there one thing where you're not using choices has no advantage And this is equi-consistency right?

00:17:59: to be more of how did probably but ZF and ZFC are equiconsistent which means if one is consistent so the other yeah On that is consistency, something still worries people in this search for axioms or an old issue.

00:18:21: That's not relevant

00:18:23: anymore.".

00:18:24: It was philosophically puzzling but it was with Goethe since he said we can't prove it.

00:18:33: so you have to believe somehow and get another kind of evidence.

00:18:41: Certainly in the beginning when people were researching more axioms beyond ZFC, like very powerful axiom there was some worry about consistency.

00:18:53: but that's not really the case anymore.

00:18:58: People started to look into large cardinals.

00:19:03: so it is one kind of new axiomes you could add beyond Reinhardt Cardinals.

00:19:11: So, Keunen proved at some point that if you add a very large cardinal axiom that's inconsistent with the axiome of choice.

00:19:21: so that is close to being worried about consistency and he said okay here we have an actual inconsistency.

00:19:30: but today people are even going beyond this and then look into stronger assumptions, which shows that I mean they don't believe these are consistent immediately.

00:19:46: but there is confident enough about the lower ranks to say let's look at new concepts.

00:19:53: Maybe there some value in it?

00:19:54: Yeah

00:19:55: this not also...I always feel foundations should be rather weak because you can intuitively understand them.

00:20:07: So I'm a bit puzzled about this idea that you want something very strong, which I intuitively find hard to understand or accept.

00:20:22: Instead of having stronger foundations also have weak foundation and then... You can go in different directions right?

00:20:29: Are we necessarily still at the foundation?

00:20:32: domain.

00:20:32: Maybe that's also a question because if I just want to understand higher infinities, might look at systems for higher infinites even they aren't foundationally necessary?

00:20:45: Yeah!

00:20:46: That is good point you mentioned.

00:20:50: looking into these new axioms or very strong principles can be considered as an interesting project.

00:21:03: what you can do with it without having the goal of suggesting any of them as a foundation for mathematics.

00:21:12: You could also consider this is very inner satiatic program to find out axioms just for set theory, for example?

00:21:20: Maybe I'm prompting you more and more about your study... In the end we should read the book if you want to know all there's-to-know.

00:21:28: Can i ask very naively Fun fact, any super surprising thing that you would like to share with the audience from your studies besides the lack of consensus.

00:21:40: That we already mentioned?

00:21:41: Yeah maybe one surprising thing was...that's relating to this lack of Consensus on The Philosophical Issues Was To Find Out That If You Look Into The Naturalness Judgments or maybe more generally, so that's still in a hypothesis.

00:22:01: That is the topic of mathematical practice discussion on the extent to which people agree about such value judgments and I found there was more agreement from these judgment on what they find natural for example that usually things like this are not very natural objects.

00:22:28: So I can't give evidence for complete consensus there, but in comparison to this disagreement on the philosophical views.

00:22:37: There's much more agreement there and that led me how people agree and communicate via these naturalness of value judgment or if a problem is deep, where the result is very exciting.

00:22:57: That's from the viewpoint of mathematical practice crucial.

00:23:02: Maybe we can explain some things?

00:23:04: We already mentioned an axiom for choice which you didn't actually mention in the system hypothesis.

00:23:10: Can you expand this a bit ?

00:23:14: Yeah, so that's the second example besides the continuum hypothesis for an issue.

00:23:22: That is open.

00:23:25: You have a characterization of a Susan line which Is the mathematical definition?

00:23:31: Which is very close to the definition of real numbers.

00:23:35: There was just one small difference there So it almost the same object as a real line, a Switzerland Line.

00:23:44: But you can have a model where a Switzerland line is not the real line.

00:23:50: so that's really something different from the reals and thats what people find not-so natural to exist.

00:23:59: So when we say the real dimensions of data kind, the real are defined?

00:24:07: Yes Also, a usual definition is just to think about the power set of natural numbers.

00:24:18: And in topological terms you have the topological definition with density and completeness.

00:24:26: So if you don't want ask for concrete satirical question maybe I would like switch focus little bit but...

00:24:34: Okay!

00:24:41: coming from type theory, I find all this set theory actually a bit weird form point of mathematical practice.

00:24:51: When i say for all numbers m and n plus n equals or for natural number m plus n equal to n plus m then that's what i mean right?

00:25:01: I'm only talking about natural numbers whereas in set theory you basically see turn out to be an element of natural numbers.

00:25:13: This is rather a bit of a strange idea, right?

00:25:18: Why would I have to quantify over all sets if i only want to talk about natural

00:25:21: numbers?".

00:25:32: So you have bounded quantifiers also in set theory, and if they're bounded by the natural numbers that's

00:25:42: here.

00:25:43: A simple formula!

00:25:44: And you just talk about the natural number?

00:25:47: Yes... I mean as i contrast this with type theory there like to-I always introduce types and then talked about elements of types.

00:25:58: so I would never have to talk about all things.

00:26:00: but at least talk about two.

00:26:01: thing actually means

00:26:03: You never need unbounded quantifiers, you mean?

00:26:06: No.

00:26:08: And the other thing which puzzles me or I find a bit strange is in set theory when we introduce like this set of natural numbers and so the standard definition for Neumann every number has a set of all smaller numbers.

00:26:24: So that's particular encoding.

00:26:26: That's actually not relevant.

00:26:31: I mean, and II really means the natural numbers.

00:26:35: I shouldn't really...I mean in sets you can ask strange question like this to a subset of three which doesn't really make any sense.

00:26:44: And then we could okay on people Okay?

00:26:47: I leave you to answer it about people for me.

00:26:50: ok yeah but we don't do this.

00:26:51: But if that You're never sure.

00:26:56: I mean, people use representations sometimes to have elegant proofs and you'll never know whether a theorem is actually about numbers or whatever concept your heart has of all the particular in cooling?

00:27:11: Do see what they mean.

00:27:14: so yeah again i can just say that i think these subtleties are not something.

00:27:21: So because this is about formalization, you know?

00:27:24: About the formalization of your mathematical practices.

00:27:28: If you look into what people do in their practice... You couldn't study which formalization they always use and their proofs Because it's much too detailed!

00:27:44: Of course compared to I think that you did the formal language based at the foundations of set theory and of type theories.

00:27:55: So, they use different formal languages but that's not a very practical question anymore it is about practice.

00:28:03: That´s about the formal

00:28:06: foundation.

00:28:07: First of all I don't necessarily mean as formal right?

00:28:13: If you think in terms of types even if you do not formalize view of mathematical constructions.

00:28:23: And that's preformal, but what intuition you use?

00:28:28: I think of types.

00:28:29: and actually if we go back to Kantor in a way he introduced...I don't know whether it was the first one introduces in German Mengen right which we could say abstractly collections anyway.

00:28:46: He certainly emphasized this.

00:28:50: And then he said a set is something given by property, right?

00:28:56: So the sets and properties are sort of identified.

00:28:59: That's what Friede

00:29:00: said.

00:29:01: yeah

00:29:02: Yeah But has it come to already?

00:29:06: you've said that a set just give him prior property.

00:29:10: I think He was very open than also too having sets.

00:29:14: they were not defined by property.

00:29:16: but

00:29:16: oh i see okay so interesting.

00:29:22: And then, I mean... But if you use pure set theory.

00:29:27: Then you say property of what?

00:29:29: You see a property of sets right?

00:29:31: A set is the property in pure-set theory.

00:29:36: and If we look at this equation it's obviously It's negative that we call computer science a negative domain equation Which is inconsistent, which basically hustles for our dogs.

00:29:53: So an alternative instead of saying the counter didn't actually do this.

00:29:58: so I blame them on person to identify sets with properties and say okay let's just see what are collections or how can we form collections?

00:30:07: The tag theoretic approach...

00:30:12: I cannot say too much about comparison type theory because sufficiently into the motivations and intuitions, but I mean...I was thinking about this.

00:30:23: And I heard also like people talking about the attraction of type theory maybe very generally.

00:30:33: i just think that from your viewpoint it's maybe comprehensible.

00:30:38: there are some attractive questions to be asked in set theory?

00:30:43: Maybe their practice is different or works with very different methods that from the viewpoint of type three are not very attractive or really like weird.

00:30:55: But still, the questions set theorists are interested in much more than thinking about how we formalize questions about natural numbers.

00:31:07: so that's why they do set theory

00:31:11: formalization, but it's a different way to think.

00:31:17: No that is not

00:31:18: what

00:31:19: I

00:31:20: meant to say.

00:31:20: of course yeah

00:31:24: Yeah and actually as you said often you find an interest in parallelism and questions which occur in sets you show up in type steel or vice versa.

00:31:36: so why they are quite different?

00:31:43: Yeah, in some case there is lots of possibility to... Lots connections between the views.

00:31:54: So you don't, as a philosopher question mathematical practice.

00:32:01: You're more observing it.

00:32:02: This is a fair description

00:32:05: At this point.

00:32:06: yes I'm observing and trying under to understand how mathematicians deal with the things that I find philosophically puzzling.

00:32:16: So, first step to better understand these puzzling things about independence and set

00:32:23: theory.".

00:32:26: Maybe this is their ideal bridge too?

00:32:28: Here's another big topic i wanted to ask you about.

00:32:32: so allow me like there are two chauvinistic heads Why should they know anything?

00:32:45: Shouldn't a real philosopher just think about it and get it right, so to speak.

00:32:50: What does their opinion

00:32:52: matter?".

00:32:52: So maybe this is the philosophy first had we use that name.

00:32:56: where there are these two schools of philosophy?

00:32:58: you mentioned philosophy second... First look at things.

00:33:02: take They have some kind of priority And then You can develop something out of it.

00:33:08: Then there are philosophers who say I don't care about practice just make it right and then they should learn from me how to reorganize mathematics, so-to speak.

00:33:20: I mean you probably are in Camp II if we also published on that way where you should be in camp too but can tell us a little bit about this?

00:33:29: Have you met philosophers disregarding your results maybe even or... How is the

00:33:34: landscape?".

00:33:40: raising there with this, the philosopher has to get it right and just on himself find a solution or by himself.

00:33:52: I think that's still an attractive goal.

00:33:54: why not?

00:33:56: As a person i want to understand something and find the truth.

00:34:01: but The question is how do you really get to the truths?

00:34:07: so...

00:34:08: Is there long truth?

00:34:11: Also good question.

00:34:12: No, I mean we're in philosophy here.

00:34:17: There are questions where you hope there's just one answer and they have questions Where do know?

00:34:23: You can't give like one definite answer.

00:34:26: that's a very simple response to this.

00:34:30: but um i think in mathematics it is really difficult to disregard what going on because mathematicians as we also wrote together Dennis in our paper, they have so much expertise on their subject that philosophers lack.

00:34:51: That it's hard to really get the heart of the matter.

00:34:55: I think from an independent point of view this regards what is happening and mathematics.

00:35:10: philosophical answers that are just wrong because they don't take into account what's going on in the mathematical area.

00:35:22: And actually, you were asking about people reacting to my research and of course there a few voices which is still skeptical to give direct answers to philosophical questions and not first look into the practice, then maybe propose an answer.

00:35:49: But I always consider this a challenge to explain the relevance of research... ...and again build bridges because like sociological research i did this methodologically very different from what philosophers are used too.

00:36:06: so as a task for me to explain the relevance and method, what's going on in this kind of research.

00:36:15: What is it contributing to the philosophical discussion?

00:36:19: And then build a dialogue and see how things are working

00:36:23: out.".

00:36:24: May I maybe have less chauvinistic head and follow up that when you speak with mathematicians they focus on different thing than philosophers.

00:36:36: big challenge to really get out what you want know, so-to speak from them.

00:36:42: A toy example maybe not for satirists but a lot of mathematicians provable and truth are the same thing because normally you can prove every true statement – it's interchangeable cause.

00:36:53: satirist is aware this distinction.

00:36:56: Is there any nuanced philosophical content where you kept getting category mistake as your reply?

00:37:06: cannot answer this very generally, because also practitioners are formulating their things in different words.

00:37:15: But for example... In my interviews I tried to avoid misunderstandings of course but it couldn't be one hundred percent sure and one confusion was when i asked about the search for new axioms.

00:37:30: so i always asking the participant what do they think?

00:37:37: And one of them was answering yes, and since the incompleteness theorems we know this is very important.

00:37:44: I really like that search in it – We have to do this!

00:37:49: Then it turned out because the interview developed beforehand… …that this person had actually a pluralist view.

00:37:57: so... The answer wasn't saying you need new true axioms but just we have to investigate all these new axioms.

00:38:07: So things like this happen when you go into practice and I thought that interviews are a good method to avoid misunderstanding, like this?

00:38:15: And get the right interpretation of what they mean by what they describe about their practice.

00:38:23: yes but of course you have to find one transparent explainable way of interpreting what's happening in the practice.

00:38:33: Okay,

00:38:33: can I?

00:38:34: Yeah sure!

00:38:36: So because you said that people are looking for tools... That is a certain perception of mathematics.

00:38:47: In my view it reflects this naive platonic view that mathematical objects are basically there and they're discovered.

00:38:58: whereas i prefer to Mathematics is sort of storytelling and that's a question of truth, it not really the right question.

00:39:10: The question for what can you deliver evidence?

00:39:18: What your view on this has any impact on the way to view mathematics?

00:39:24: Truths are very good examples difficulty between doing philosophy and then mathematical practice.

00:39:34: And philosophers are also divided on this, so like Penelope Medi who has worked a lot on satiric practice she is open to the view that we should not talk about truth at all just about methods which are successful in practice.

00:39:57: That's how satiri works.

00:40:01: So also as a philosopher, one could say we don't want to talk about mathematical truth.

00:40:07: I still think then a philosopher should give a good explanation of what math is about if it's not about mathematical truths.

00:40:17: but that's feasible and the mathematicians themselves...I think in common or usual language use true is still used in the sense of we found out that's true and that's not true.

00:40:36: And things like this, so it's for someone interested in the practice.

00:40:43: It would be a bit of challenge to interpret This way of talking in different ways To avoid truth.

00:40:51: But That also thing you could do but You have to find solution on how to interpret.

00:40:59: And of course, if you ask the theorists or mathematicians maybe more generally about truth than so that's what I experienced in my interviews also.

00:41:10: They sense there is a deeper philosophical meaning um... That they don't want to touch and then..They don't wanna like bring down What actually is mathematical truths?

00:41:28: often the same thing and the usual language use.

00:41:33: Yeah, I would say provability or evidence right?

00:41:39: Something for which i have evidence that accept it.

00:41:42: so this idea of storytelling.

00:41:48: in a story you don't have tools but what is here before?

00:41:57: Conclusive

00:41:59: evidence.

00:42:02: Yes, whatever you mean by evidence we don't have to nail it down.

00:42:07: but that's the idea.

00:42:09: and in my view I always find mathematicians... I think interest in sociology.

00:42:19: there is a particular sort of idea doctrine in mathematics which is not usually questions by mathematics, it's a sort of like almost process or brainwashing from my view.

00:42:33: Because I don't find that the mainstream ideology of mathematics... ...is very good one for my philosophical point-of-view and i found always surprising that philosophers are unquestionable.

00:42:53: just follow what this mathematical ideology is.

00:42:57: Instead of not telling them, but to explain better what they mean and instead looking at it in a different community where the constructions are based on... Intuitionistic perception that mathematics is actually constructed, which I think it's very natural.

00:43:30: But this has consequences once you accept these ideas as consequences of following two mathematics?

00:43:37: I don't think the task for a philosopher to be absolutely quiet about mathematical practice and just license everything what happens there?

00:43:50: They can point to incoherences, to vaguenesses.

00:43:55: To ideas that are misleading... ...to things like this.

00:44:01: I just think it's not fruitful That the philosopher demands a lot of the mathematician to satisfy actually the philosophers goals Like having a clear picture what math is about.

00:44:24: Yeah I

00:44:39: don't think that every mathematical practitioner is really like a strong Platonist in this sense.

00:44:47: Yeah, I mean that's true.

00:44:49: Lots of them just like to ignore this question but they'll say we will do something and don't question it.

00:44:59: But... That makes maybe difficult to explain what other people are doing or why they're doing right?

00:45:07: So thats the interface!

00:45:09: Once you are introduced into this mathematical belief system then you just accept it and don't ask anymore.

00:45:17: But there is this outside world, the people who learn mathematics

00:45:22: etc.,

00:45:23: maybe they need a better way to explain what's the whole thing about?

00:45:31: It's not so much a philosopher but also naive practitioner or naive person with interesting mathematics... That's

00:45:40: certainly good here of mathematicians!

00:45:43: can explain well what their subject is about and maybe also question a bit of self-reflection there.

00:45:53: Okay,

00:45:54: so... Is there anything that you would really like to share with the audience or otherwise I will face?

00:46:03: Thank You!

00:46:03: Well it's very interesting conversation

00:46:04: thank you.

00:46:08: Anything urgent from you Thorsten?

00:46:09: Yeah,

00:46:10: thank you very much.

00:46:10: no i was good to hear hope to see again.

00:46:14: Yeah, so let me also say thank you very much for this really lovely chat and the insights into your big study.

00:46:20: And other work on The Relation of Math & Philosophy.

00:46:24: So then we have like a final moment For another kind of brainwashing That's the Brainwashing of the audience.

00:46:31: Please subscribe comment Press the Like button to whatever else You can do To support our channel.

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00:46:38: Friends.

00:46:40: Exactly

00:46:42: Great.

00:46:42: Thank you very much,

00:46:44: Dustin and Dennis.

00:46:47: It's

00:46:50: nice to be here.