aboutlogic #02 | Deniz Sarikaya – Philosophy of Math, Sociology, Set Theory & Universe vs Multiverse

00:00:00: I mean, I'm very astonished how many mathematicians take their time to talk with philosophers.

00:00:07: You are one of them, right?

00:00:15: Okay, so today we have Dennis Sarikaya.

00:00:18: Do I pronounce your name properly?

00:00:21: Good, okay.

00:00:23: He is one of the organizers of the channel and he is a philosopher of mathematics.

00:00:28: I hope I get this right.

00:00:30: Living in Hamburg, but working in Brussels.

00:00:34: Are you?

00:00:35: Yeah, am I a latest state?

00:00:36: Okay.

00:00:37: So I first met Dennis a number of years ago when he organized a summer school in the non-existing town of Bielefeld, which was quite fun, and on foundations of mathematics, and he invited me to talk about type theory.

00:00:55: But today we will focus on some proper philosophy, I hope.

00:01:00: I have some, I'm not a philosopher, I'm like a DIY philosopher, so I will ask some very naive questions.

00:01:07: But first of all here, so my first question to Dennis is, how do you become a philosopher?

00:01:14: I mean, everybody can be a philosopher, right?

00:01:16: But I mean, the standard way to become an academic philosopher is to study it.

00:01:21: And I myself, I started studying mathematics, but I was always so interested in logic and foundations and think how we can build up all these things that I started to shifting towards philosophy more and more.

00:01:37: And then my PhD was finally only in philosophy, so I stopped after the masters.

00:01:44: So the mathematics wasn't abstract enough for you?

00:01:49: Yeah, on the other side, maybe mathematicians are super cool people and I like to see how math as a field grows.

00:01:57: So how mathematicians convince each other.

00:02:00: And I mean, this is mathematically speaking, not too interesting, right?

00:02:03: Because, I mean, either you write down your theorem and you want to publish that one, but I was more interested in how these things develop.

00:02:12: And then, therefore, I switched to these weird little traditions called philosophy of mathematical practice.

00:02:18: And they are sociologists, math educators, anthropologists, and philosophers come together.

00:02:25: and really try to understand how we get to mathematical knowledge like in the real world.

00:02:29: So rather not abstract, not this over-idealized picture of you write down perfect theorems and everybody just comes and then applause and is convinced.

00:02:38: But to really look at the nitty-gritty, how there are errors in practice, like in my mathematical field in graph theory, you can open ten papers and you will find an error in at least one of them.

00:02:53: So it's right away from this over-idealized picture,

00:02:58: at

00:02:58: least so far when we don't have the automator to improve us yet, fully employed.

00:03:05: If ever.

00:03:07: Actually, yeah, I had a question later.

00:03:09: It's like, maybe I should ask now, what do you just say?

00:03:14: Should a philosopher just observe?

00:03:16: What mathematicians are doing, or maybe should they actively influence the direction of mathematical research?

00:03:24: I mean, for me personally, I think there is like in the first step with a practitioner's, so to speak, have priority.

00:03:32: You shouldn't come from the outside and say, oh, I don't care about your thoughts here, how you, what you should do.

00:03:39: And this is too much, right?

00:03:40: But ideal for a lot of, at least us.

00:03:46: I mean, this math is such a weird field.

00:03:48: It takes computers here at your computer science or any these more formal endeavors.

00:03:53: You need like years to develop intuitions about things.

00:03:57: So I think you should respect these intuitions.

00:04:00: But a good philosopher at times gives input that makes the practitioners think a little bit.

00:04:09: Like, for instance, somebody.

00:04:11: Yeah.

00:04:12: Sorry.

00:04:13: That's

00:04:14: came to my mind.

00:04:15: My impression is that mathematicians are extremely conservative.

00:04:19: And if you just sit there and observe what they're doing, isn't there a problem?

00:04:26: I mean, that they don't really want to leave their comfort zone and you just sort of stabilize everything by just observing.

00:04:35: I'm just one

00:04:36: way.

00:04:37: Observing is like the first step, right?

00:04:38: And then you can... like one lovely paper on the epistemic injustice of mathematics gives a case study where mathematicians disagreed and there might be some dynamics of power and some other problematic things.

00:04:56: And that's something that mathematicians don't really believe in at times or wasn't discussed so much.

00:05:03: So like this outside input can really help making them think about their practice from a different perspective.

00:05:10: I mean, this is more automathematical.

00:05:12: But like in a mathematical, the foundations, the whole debate about foundations is in between philosophy and math.

00:05:19: And I think a lot of fruitful stuff happened due to that.

00:05:23: But the other mathematicians, if you talk to them or foundations, they say, or go away.

00:05:29: And if you talk about philosophers, they run away.

00:05:31: So that's why you experience.

00:05:34: maybe you got a better experience.

00:05:38: I mean, I'm very astonished how many mathematicians take their time to talk with philosophers.

00:05:44: I mean, you are better experience.

00:05:47: You are one of them, right?

00:05:48: I know

00:05:49: a mathematician.

00:05:51: Yeah,

00:05:51: okay.

00:05:52: That's the book you mentioned featured quite some mathematicians.

00:05:58: They even write things.

00:05:59: I mean, we have the book here.

00:06:00: I see it in your after your background shred in this one.

00:06:06: It's full of mathematicians who are willing

00:06:09: to.

00:06:13: I don't get commissioned if you buy it.

00:06:14: So I don't need to drag the right way around.

00:06:21: I think they are open to talk about their craft.

00:06:24: And this is maybe like another motivating theme of my whole research.

00:06:29: I think we need to communicate to the public that mathematics and computer science is a very human endeavor because of you.

00:06:39: I had this experience that sometimes the philosopher gives a talk about mathematics and the mathematicians feel that they don't really understand what's going on.

00:06:51: So my impression is that they are not very positive.

00:06:57: a view on philosophy.

00:06:59: But maybe this is just, I've got the wrong input here.

00:07:05: I mean, it's maybe when a field gets over-specialized, it's like even in a mathematically, right?

00:07:11: If a topologist gives a talk, I mean, I can only listen, it's music for me.

00:07:17: Maybe the motivational example is fine, but after that, I'm lost because I lack the knowledge, the background.

00:07:26: And but I think for philosophers, it also depends whether you want to focus on philosophical questions, so to speak, that are only inner philosophically relevant.

00:07:36: Or you want to have this practice-based perspective looking at questions where practitioners should care or do care.

00:07:45: So examples could be what's the good explanations, what the role of pictures improve.

00:07:50: That's something working mathematicians also might want to think about when formalizing handwritten mathematics to the computer.

00:07:59: For instance, what happens with the pictorial proofs?

00:08:01: Is it a systematic way to translate them or not?

00:08:06: Those are philosophically relevant questions, but also relevant for practitioners.

00:08:10: And I think this is like the sweet spot where those disciplines engage.

00:08:16: Okay.

00:08:17: I actually had some more high level questions.

00:08:18: Maybe I go back to my schedule.

00:08:22: So here's something, it's a very naive question.

00:08:25: So what is actually mathematics and how is it different from other discipline?

00:08:32: I mean, my whole narrative, it's less different than you think.

00:08:36: I mean, the old narrative is math is like the prime source of knowledge.

00:08:41: Descartes needed like, when doubting physical knowledge in his meditation, he just said, oh, maybe you had a dream, you don't know physics.

00:08:50: But for mathematics, he said, oh, maybe there's an evil genius who's always corrupting your mind at the right moment to get like systematic wrong impressions about mathematics.

00:08:59: So for math, he needed the evil devil.

00:09:03: And for physics, you only need to say, oh, maybe you slept.

00:09:07: But I think this is like, that's not the case.

00:09:10: Mathematical knowledge is much more similar to knowledge from the other science.

00:09:14: The discipline is much more similar.

00:09:19: While we have this weird thing deduction that gives a lot of certainty, then around that there is a lot of trends.

00:09:28: What is an interesting question is a very sociological question.

00:09:31: There's no inner mathematical solution for that.

00:09:34: How do you convince people that something is right even if you are right?

00:09:38: It's another very different question.

00:09:41: Well, you think you're right.

00:09:42: Yeah, you never know, right?

00:09:45: And so for me, math is looking at these abstract systems.

00:09:49: and wants to prove theorems about them and develop understanding of them.

00:09:54: And some of these systems mirror something in reality and others don't.

00:09:59: And that's the field of math, roughly speaking.

00:10:03: So here's a question which I've seen a lot coming up at CORE.

00:10:07: And this is one of these social networks with questions.

00:10:12: So do you think mathematics is self-invented or discovered?

00:10:20: I know what you think.

00:10:22: I honestly, I think I don't care too much because mathematical objects could be abstract somewhere as I don't have access to them without like my human approximation and means.

00:10:36: So there might be an abstract reality.

00:10:38: I don't have access so I don't care.

00:10:41: So for me, For me, it's constructed, it's discussed, it's debated, and this is the reality I feel, and this is the important thing for me.

00:10:54: For me, at least.

00:10:55: But this is like, again, a philosophically interesting question is like this metaphysics of abstract objects, but maybe it's less interesting to me looking at the everyday work of the field, so to speak.

00:11:10: Yeah, maybe, okay, because you just said what I think, so maybe I should ask you another question which is on my mind.

00:11:22: So what do you think about constructivism?

00:11:26: I mean, there was a while ago, there was this famous Grundlagenstreit between Brauer and Hilbert, but that I think... most mathematician think Hilbert won.

00:11:36: So is it still relevant?

00:11:38: Or is it just sort of some some some strange sect of people who stuck in the past or something?

00:11:47: I mean, the first one observation, I never really understood how the Grundlang-Streit was settled.

00:11:52: So I mean, when this time when tuitionism came up as a alternative, I know there were some shifts in power, so to speak, like when Hilbert Probably, or allegedly, I'm not sure how the historical data is, but as far as I know, I threw out a lot of influential intuitionists from the top journals and things like that.

00:12:15: And I mean, he definitely chose a few chairs in Germany at least, and probably didn't took intuitionistic thinkers to the spots of power, so to speak.

00:12:25: So I see this level, but I never saw any settlement, any reason why the problems were solved that they used to have, they just started to ignore them, the problems of paradoxes, of weird definitions or whatever you want to look at.

00:12:44: So this is maybe the first observation.

00:12:46: And the second is that constructive mathematics has like a new boom due to computer science.

00:12:54: Because I mean, in the last interview we had with you, there is this meaning of constructive mathematics as you really have the witness.

00:13:04: all the times you don't make this abstract, oh, there is a witness, but I don't know which one kind of statements, but you provide them.

00:13:12: And this is like for application, huge benefit, right?

00:13:16: So it's like the kind of mathematics that is very useful for a growing

00:13:25: field in

00:13:26: the computer science.

00:13:28: But this view doesn't care about the truth in mathematics, so to speak, right?

00:13:34: Use whatever you want.

00:13:36: If you have a database full of errors, use something without the principle of explosion, because then you have a problem.

00:13:43: If you have a clean database, then maybe you can have that principle.

00:13:48: So just use whatever you want.

00:13:50: And their constructive mathematics has all the right to exist in a strong sense.

00:13:57: But do you think there's any issue here from a philosophical stance?

00:14:01: I mean, you already said you used the word truth.

00:14:05: Do you think that mathematics should be based on truth or are there any issues with this?

00:14:11: I mean, there is a growing school of pluralists, so to speak, saying, I think that mathematics is a very local phenomena, so to speak.

00:14:23: Of course, the style of Western mathematics we are doing nowadays is very good and helped a lot and no question there.

00:14:33: But I think it could have developed differently.

00:14:35: And I still believe that other kinds of mathematics are different in a very meaningful sense.

00:14:41: And it's not possible to say this one is true and this one isn't.

00:14:47: So I think truth is a too strong notion there.

00:14:50: It's like about local coherence.

00:14:52: Is this thing moving good?

00:14:54: Is it helpful?

00:14:55: Is it understandable for humans?

00:14:58: Those are like more important categories to think about it.

00:15:02: Yeah, okay.

00:15:03: But do you think I mean that mathematics are things which are constructed in our mind, right?

00:15:10: So the notion of truth seems to refer to the real world, right?

00:15:16: Can something which is just constantly your mind be true or false?

00:15:22: I mean truth.

00:15:22: there would often be like an if-then statement, right?

00:15:26: If you take this axioms and if you buy into this picture, then you get this theorem.

00:15:33: And this is where truth comes into the picture, at least for me.

00:15:38: So it's like this.

00:15:40: All of mathematics is in, if a then game, if you buy the axioms, then you get the consequence.

00:15:48: And if you start doubting the axioms, then you aren't doing mathematics, but philosophy at least for that moment.

00:15:58: But somehow, even when you use the axioms, for example, there's a question, if you think about sets, and then people ask, is the continuum hypothesis?

00:16:10: two or fours and...

00:16:14: In some models, it holds another, it fails.

00:16:18: So the notion of tools is not really a co-create, no?

00:16:22: I mean, like, model, theoretically speaking, task-y relativized truth to models, right?

00:16:28: Yeah,

00:16:28: but I mean, that's a cop-out, no?

00:16:30: I mean, I'm asking you, I mean, you talk about something, you talk about national numbers and then you could say, oh, yeah.

00:16:37: infinitely very prime numbers, this is certainly true.

00:16:40: And now we talk about sets and I ask you what's about the continuum hypothesis.

00:16:44: And is it true or not?

00:16:48: I mean, it's independent from ZFC, that's all I can say.

00:16:53: And now there starts discussions, whether we need extensions that settle CH and some people say that.

00:17:01: And this is a very justified research program.

00:17:05: And if they find an extension that really convinces people, then CH might be a set theoretical truth in the future, but then not relative to ZFC, but to ZFC plus something, maybe large cardinals, maybe forcing.

00:17:20: Whether the C is actually

00:17:25: okay.

00:17:25: I mean, there's a lovely large cardinal program beyond choice.

00:17:29: You can get so much larger cardinals.

00:17:34: Isn't this desire to sort of like reach some at least relative completeness?

00:17:40: Isn't this sort of influence by this idea that this mathematical reality really exists and we need to explore it?

00:17:49: I mean, it's a question of intended models, right?

00:17:51: There should be only one proper natural numbers.

00:17:54: Yeah.

00:17:55: But there could be different.

00:17:57: I mean, for in group theory, nobody would say, oh, let's finally solve commutativity.

00:18:02: Sure.

00:18:03: They are a billion group and they are not a billion group and it's good that way.

00:18:07: Yeah.

00:18:09: So when you think there's one universe of set, then better try to solve the continuum's hypothesis.

00:18:15: Otherwise, you are freer to say things like Hamkins, for instance, even says.

00:18:21: the truth is that we understand how the continuum's hypothesis behaves in different models of set theory.

00:18:26: So how it forces, so to speak.

00:18:30: That's all the truth of the knowledge we can have about the continuum's hypothesis that we know we can always extend to a CH model or a non-CH model.

00:18:40: So that's at least for him the most information we can have.

00:18:47: Which is early, it's sort of not really platonic in a way, but okay.

00:18:53: He calls himself a full-blooded platonist because he believes in the existence of all these models in a very strong sense.

00:18:59: They exist

00:19:00: in

00:19:00: what?

00:19:01: I mean, then we assume some sort of meta foundations which are still then not really affected by this idea of models, right?

00:19:12: I mean, you can then axiomatize the forcing relations into all kind of stuff there.

00:19:17: So, I mean, in some sense...

00:19:19: But there, somewhere, I mean...

00:19:22: Yeah, I mean, then you come to this meta theorem things.

00:19:26: It's like going on, right?

00:19:30: And some reverse mathematicians can tell you, oh, this is only addition without exponentiation or whatever, but...

00:19:38: Yeah, sure, that's another story.

00:19:43: Okay, I have got one more sort of main question here.

00:19:49: So what do you think, I think you mentioned this, but how do you think computer-assisted proof systems change the nature of mathematics?

00:19:58: So there's one thing, I think there are potentially a revolution like in proper philosophy of science sense.

00:20:07: So I think they will have an impact to practice in many dimensions.

00:20:14: I mean, you don't even need to go the full way to the stereo improve us.

00:20:19: Even like knowledge management is such a huge topic to have a infrastructure where you write down your theorem you want to prove and you get like candidates of lemmas and other theorems that might be relevant.

00:20:33: It's such a necessity because we produce more and more mathematical papers.

00:20:38: Nobody has even remotely an overview even about a field if it's a bigger field, right?

00:20:44: It's not like the very niche topic.

00:20:47: So I think we are at a point where the human mind gets problems to master.

00:20:54: research level mathematics in a bigger scale.

00:20:58: So the computer will have an input.

00:21:01: Whether it will be the sole authority of justifying proofs is a sociologically very interesting question, right?

00:21:09: Will people accept that or not?

00:21:12: But I don't really see, I think in the end they will do once the tools work sufficiently good enough.

00:21:21: So true.

00:21:22: And there's this, I think for Max Planck, who said, quantum theory will win because all the other people not using it will die out.

00:21:33: So nobody needs to be convinced, right?

00:21:35: There will be a new generation of mathematicians who grew up with these tools, who had such more free time because they got the theorems right much quicker.

00:21:46: And therefore they stayed in academia, the people with pen and pencil.

00:21:53: where to board needing so long and stopped.

00:21:58: The lead says to another question, which is really a passion level.

00:22:01: So what's the impact of AI on mathematics and mathematical research?

00:22:07: I hope it's little.

00:22:10: Because I'm a person, that's maybe also why I went to logic, so to speak, because I really want to understand the steps I'm doing.

00:22:19: I want to understand each step and why I'm doing them.

00:22:24: And this is something where these powerful tools, even Sledgehammer, which isn't AI, right?

00:22:31: But it's something I'm a little less comfortable with.

00:22:36: So I'm more happy being slow, but really understanding what I'm doing and why, than being like taking the big.

00:22:45: guns and just shoot at the theorem and kill it for some reason and I don't know even why but.

00:22:50: My AI picked the right theorem somewhere in the database.

00:22:53: I don't know which ones I can't even understand them.

00:22:56: They might even not have a human readable correspondence.

00:23:00: It's just a weird fact about algebra that no human ever asked or considered.

00:23:05: It's.

00:23:06: Ten thousand signs long about who cares.

00:23:08: It's it's useful.

00:23:10: And this is something where I'm scared that.

00:23:13: ten thousand signs long about who cares.

00:23:16: It's useful.

00:23:17: And this is something where I'm scared that you want to prove theorems and stress again that we want to understand things.

00:23:23: And then this all is maybe less tempting to go for the AI route.

00:23:29: Because

00:23:30: really understanding might be easier without AI, at least in the next fifty years.

00:23:37: And after that, I shouldn't.

00:23:39: I shouldn't say something about five years.

00:23:42: I was wrong with all my predictions so far in mathematics and in politics.

00:23:48: So I shouldn't predict anything.

00:23:50: It's always good.

00:23:52: Maybe the opposite of what you say is always true somehow.

00:23:55: It would be also a nice oracle.

00:23:59: Well, I mean, things like judge the critique.

00:24:01: Do you think they can help to explain mathematical concepts?

00:24:05: Or is this a bad

00:24:06: idea?

00:24:07: on an undergraduate level, I think.

00:24:09: So on a research level, probably not yet.

00:24:14: And again, whatever helps you, right?

00:24:18: If you have an automatic tutoring system, that's great.

00:24:21: You can see where it brings you.

00:24:23: And then you need this crucial competence to know where you need to stop.

00:24:27: Because at some points, it starts bullshitting right in there.

00:24:31: You need to get at least a feeling.

00:24:34: You don't know exactly when it starts.

00:24:35: But

00:24:37: so does your tutor, right?

00:24:38: At times.

00:24:40: That's true.

00:24:43: But the tutor has at least hopefully a model in mind.

00:24:46: I mean something instead of just sort of reflecting or parroting what they have seen.

00:24:55: Even though students tend to try to do this often.

00:25:03: This is lovely parallel, right?

00:25:09: Your undergraduate experience, or maybe at least my philosophy of bad mathematicians, it's so much harder to understand how you should talk and how you write stuff down than seeing that this thing is right.

00:25:25: Your whole linear algebra theorems are often trivial, but to write down, to mimic, so to speak, what the real mathematicians are doing is the whole task.

00:25:36: And there you can, of course, bullshit without knowing it because you just start mumbling around the right keywords.

00:25:41: Yeah.

00:25:42: Yeah.

00:25:43: So it's always interesting to present the wrong proof and then people identifying what is actually a good argument and what's not.

00:25:52: I think that's the test.

00:25:55: Good test indeed.

00:25:57: Especially if the theorem is still right and it's harder to see that something was fishy.

00:26:01: Yeah,

00:26:03: indeed.

00:26:04: I mean, it's often the case in mathematics, right?

00:26:06: That the final result may be right, but the reasoning is actually not sound, right?

00:26:12: Yeah,

00:26:15: often.

00:26:17: Especially

00:26:19: if you look at this very detailed level, then of course every proof is at some point at least having gaps, right?

00:26:26: or every most proofs that we have found in journals.

00:26:30: And those are trivial at the times.

00:26:32: You need to study five years to understand why they are trivial.

00:26:36: But sometimes the trivial steps are actually wrong.

00:26:44: My final question is to come to an end.

00:26:47: Have you got any recommendation on what is the most important text or book?

00:26:52: An amateur who is interested in philosophy of mathematics should read.

00:26:57: Maybe if you want to start to think about philosophy of mathematical practice, this branch I'm interested in, there is one standard book.

00:27:09: Let me pick it out.

00:27:15: I forgot the title.

00:27:16: It's edited by Paolo Mancusu.

00:27:20: I think we can fill it in the comments.

00:27:23: And it's a lovely collection because it brings in people from very different perspectives and they are asking at times for the first time, at times they consolidate existing debates about questions that weren't asked so often in philosophy of mathematics like explanations, formal proofs and their counterpart informal proofs.

00:27:49: Didn't said stop thinking about the old questions.

00:27:52: We don't want them anymore.

00:27:53: But it said maybe open up your mind that there are different interesting questions.

00:27:58: So I think Paulo Mancuso philosophy of mathematical practice by I think Oxford University Press should be the book.

00:28:06: And I think I can.

00:28:07: This is maybe a great, great starting place if you are philosophically interested.

00:28:13: And if you have no prior philosophical knowledge, there's a book by Stuart Shapiro thinking about numbers, which might be a funnier start and easier digestible.

00:28:22: Okay.

00:28:23: Good.

00:28:25: Thank you very much.

00:28:26: Is there anything you would like to add?

00:28:29: No, nothing I need to add.

00:28:30: Thank you for the interview.

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00:28:38: We hope that this channel grows.

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