aboutlogic: premises #01 | Is Math a Story? A Constructivist Perspective and Captain Ahab's Dilemma

00:00:00: in constructive math and bishop style analysis test is principle of God, right?

00:00:04: So

00:00:04: to clarify this, Bishop's Classical Mathematics is Intrusionistic.

00:00:08: Mathematics plus God.

00:00:09: If I ask you the question does Katnaha pass a blue shirt?

00:00:14: This question hasn't got a brilliant truth value because we are not told what color his shirt is.

00:00:22: In week where have very special guest?

00:00:25: think i can announce him by now.

00:00:32: Hello, everybody.

00:00:33: We wanted to try something slightly different.

00:00:35: as you see we do not have a guest today and we want to Try out to have a weekly Scheduling format at least so bi-weekly we still have guests And bi-weekly we wanted to tried to have this short in between discussions between Torsten & me.

00:00:52: great.

00:00:53: So for our first little experiment I wanted to ask about something that came up quite often And you use this metaphor of mathematics is like a narrative, like a story.

00:01:09: Let's dive into that little bit and would you like to elaborate how will see the issue?

00:01:15: Yeah so it on one hand its reaction to this idea platonic ideas in mathematics are somehow real.

00:01:28: I don't buy this for a number of reasons.

00:01:33: It seems to be clear that mathematics is evolving, it's not one thing but something fluid and part our cultural evolution.

00:01:47: so hence these ideas... It's related to the idea of God.

00:01:56: We may not believe in God as a person, but we live.

00:02:00: instead of replacement like there is this mathematics sort fundamental god-like principle permeating the words and I don't... it's not something that i feel right basically.

00:02:22: thoughts in there.

00:02:23: I mean first mathematics as culture, i find this hugely interesting and i think that the new AI technologies will force us to take more serious that math is not about proving stuff only or it's not like natural sciences but more like humanities.

00:02:42: then you're doing constructive maths and bishops style analysis test is principle of god right if all-knowing being things change, and then you don't have this fluidness.

00:02:58: But let me...

00:03:00: To clarify this, Bishop said classical mathematics is intrusionistic mathematics plus God, right?

00:03:06: Yeah that's

00:03:09: what he'll feel too!

00:03:10: So but let me push a little bit to understand how much development you would see.

00:03:15: Would you say there was like historical contingency that math would have taken another path, and then... Another theorem would be true.

00:03:25: or is there too

00:03:27: much?

00:03:27: I mean other theorems may be proven and considered.

00:03:33: So yeah i don't know!

00:03:35: There are certain aspects of mathematics which are sort-of unavoidable in a way like crime numbers..I'm sure aliens no crime numbers but does this means they're discovered or constructed.

00:03:49: still You could say Naya, but if these aliens also create the same concept then that's a fair point.

00:04:04: And maybe there is this language of discover and construct which isn't very clear.

00:04:10: But I would still say it's constructed or designed solutions being looked for, they are convergent but not because... They're already there.

00:04:24: But this is just an obvious solution.

00:04:28: in a way if you look at biological evolution the concept of a wing has been discovered again and again or it's constructed and discovered using same language which was designed once more right?

00:04:45: I would still say that its design process.

00:04:47: otherwise It's also very difficult to put a precise border there.

00:04:52: I mean, at one point it is so unavoidable that its discover and in some points sort of design.

00:05:01: when i write my program we had this joke with Conor McBride.

00:05:08: We gave the program program on a toilet in Addis Abibal was an edition programming.

00:05:16: And it's like, you're not discovering programs You create them and so there is this mass.

00:05:23: It's very difficult I mean to put the line at which point it sort of unavoidable.

00:05:29: hence discovery.

00:05:30: In one point There are some real choice or lots of choices in its discovered.

00:05:36: So i think To me Yes, there are aspects of mathematics which are unavoidable.

00:05:44: Which will always be designed by convergent evolution.

00:05:54: but I still would call this a construction sort of necessary construction But not something that just obviously is there in any sense.

00:06:08: You know, that's interesting.

00:06:10: And I mean you don't like to talk about truth but let's Talk a moment of our truth because there are I mean truth and affixion.

00:06:18: you have this example Of the captain in Moby Dick whether he has a red shirt or what exactly.

00:06:24: the examples

00:06:25: blue shirt blue shirt

00:06:29: and I Mean how would you spell this out?

00:06:34: In detail.

00:06:34: so you would say it's not true that he has a red shirt.

00:06:37: It is not true, but here's the blue shirt.

00:06:39: on what do you see?

00:06:39: I know i was at about.

00:06:42: if i ask you to question does cat now hop has a blue shirt?

00:06:50: and then the question.

00:06:52: this question hasn't got a brilliant truth where you are because Basically, we are not told about the color of a shirt.

00:07:12: So it's something which could be this way or that way and maybe have blue shirts here... We just don't know.

00:07:19: so you cannot assign it as classical truth value.

00:07:23: And there is typical aspect to storytelling.

00:07:30: interpretation, so a story is necessarily doesn't fix all the data and certain things are open.

00:07:41: So to me mathematics it's also kind of storytelling.

00:07:49: I mean there different kinds rules, what is acceptable.

00:08:03: For example in a criminal story something which would not be consistent with naive laws of physics isn't really acceptable or then it wouldn´t have been the criminal story but science fiction actually.

00:08:18: So there are certain rules for storytelling that differ from different genres And the genre of mathematics also has got quite some rules, how we can tell stories.

00:08:37: In particular there is a reasoning that needs to be acceptable and understandable.

00:08:48: So that's not an arbitrary, it is a particular genre of stories which are about mathematics.

00:08:59: And one thing I find where the story telling idea very clear to explain into logistic logic.

00:09:07: so we usually use what called Kripke models and I found Kripker balls as bait models.

00:09:15: they're often very well explained.

00:09:18: i think These logics are really the logic of storytelling.

00:09:25: So for example if I'm in a story and say, If they come over this mountain we will lose the war!

00:09:34: Then in the moment, we may not... it's a war story.

00:09:38: I don't know why I'm going to say no into war stories but anyway But at that time when you do know whether they or maybe other army comes over this mountain and if so then let us assume that statement is correct.

00:09:55: As soon as we know That they come over this mountain, we also know that the wall is lost.

00:10:03: And it's exactly the kipkel explanation of implication if A then B... If whenever in the future you know A and we also knew B We may not know at the moment neither but this is typical for storytelling right?

00:10:21: If an implication refers to future states And I would go on and explain disjunction.

00:10:33: So now this junction isn't modeled very well in Kripke models, but in Bait models... ...and people may know that I mean Kripker models are actually categorically related to pre-sheaf models….

00:10:48: …and Bait model is what's called sheaf model.

00:10:51: so i'm not going into the details here.. in a bait model, you have this principle basically if something is unavoidable we know it already now.

00:11:08: So all possible future-something happens.

00:11:12: so he knows that right.

00:11:14: and to continue with my war story Yeah, if you say this then and it's a true statement in the story.

00:11:32: Then that means that before the end of the story we will find out one or other right?

00:11:37: That is good storytelling principle.

00:11:41: so don't keep questions open.

00:11:42: okay maybe not universally accepted but... ...that idea!

00:11:49: And this exactly explanation of distraction and beat models.

00:11:54: You say, basically if you have a cover of the current world.

00:11:59: So which means that any part in future will go over this cover and something is true in all worlds of this cover then it's already or two is always forced as known there than its already know now right?

00:12:15: And actually these beat modes they give your complete semantic for intuitionistic logic.

00:12:21: so everything provable is two in orbit models and vice versa.

00:12:29: I mean, i don't want to put worth on your mouth but for me the important aspect of your story about stories it's the epistemic part of it.

00:12:39: so you have this epistamic element.

00:12:51: There is no reality in the story world.

00:12:55: And I mean, this also how you could think about stories right?

00:12:58: In talking about Harry Potter universe and there's a matter of fact here that would be more platonic idea of stories so to speak.

00:13:10: but again this notion of culture does a lot heavy lifting because some things can do.

00:13:16: other thing You might have implicit assumptions in your definitions that only come out later.

00:13:26: And there you can do some cheating, but not all so to speak... So there's a negotiation process there and if yeah

00:13:38: I mean okay depends how you interpret the word epistemic.

00:13:41: i usually understand it as trying to model And this also applies to sort of common sense reasoning, which I don't really want do here.

00:13:59: So for me so intergenistic reasoning is the logic.

00:14:04: here it's an idealization and its not epistemic in a sense as how we actually think It's... How we explain our thinking right?

00:14:18: is an attempt to make precise what reasoning, reflection on reasoning.

00:14:29: But for me the distinction between imprecise distinctions which are one thing you could call mathematical logic and idealisation not necessarily platonic but an idealization of how we reason We are not really addressing how everyday reasoning is working.

00:14:55: And to me at least epistemic attempts exactly this, but obviously it depends what we mean.

00:15:03: I think there's

00:15:04: a verbal dispute...I would call that bounded rationality and like an epistamic i would say you're closed under logical rules for instance And you get that immediately the moment.

00:15:16: You told me, The Piano Axioms are epistemically already know for much less theorem

00:15:21: but

00:15:23: again thats different from discipline to discipline.

00:15:26: I guess how you phrase it?

00:15:28: Yes so in a way i think we had this discussion with Graham Priest and there I wasn't completely happy also him sort of throwing into schnistic logic back with all these sort of philosophical logics which are an alternative to the classical logic.

00:15:54: And this includes what I would call epistemic logics, like for example he says we can reason and deal with contradictions because people epistemically can deal with contradiction.

00:16:07: but something which in this idealized logic, mathematical logic I don't want to have contradictions.

00:16:19: That's a basic requirement.

00:16:22: so those are for me... So if you look at what logic we're using when doing mathematics or not just mathematics but any science especially computer science And we want to be pure in a sense.

00:16:46: We don't really wanna rely on human fallacies, weaknesses, confusions every day thinking... ...we want to extract somehow what is relevant for mathematical reasoning.

00:17:07: and then As I have explained, so this classical logic seems to be unsatisfactory.

00:17:16: And there's an interesting connection here because the word constructive logic can be understood in two ways.

00:17:23: Constructive is a sense that we recognize that mathematical concepts are actually constructed and constructive but also in the sense our reasoning.

00:17:35: So these are two different meanings of the first constructive, which is somehow related.

00:17:43: I mean once i start with this understanding that logic is discovered then... ...I end up with a constructive logic where actually be more explicit about what I construct in the sense if say there exists something I can also name it.

00:18:10: We had one question where we have this question of having like an operator in the fiction and how it relates to other logical connectives?

00:18:21: Do you ever... The theory there, Like the prime example that shirt is red or isn't And then You could say In the Fiction the shirt Is Red Or It Isn't But You Cannot Say Its Red In The Fiction It's

00:18:37: not blue or whatever.

00:18:38: Okay, this was a question that was inside the story.

00:18:40: somebody could say The shirt is blue?

00:18:43: Or it's not Blue right and we would actually I'm Not so sure but let's see if he accepted then its Actually A Story in a story as second-order story because We are now talking about the mental state of the people in this story, right?

00:19:09: And there's a different level as such how it evolves.

00:19:16: So that is question what do these people think about their stories?

00:19:23: and they actually most likely want to understand their epistemic state Unless it's a story about mathematicians, right?

00:19:39: Then we would actually say if somebody says does Captain Garb have the blue shirt or not then people will maybe think I don't know.

00:19:51: Or yes!

00:19:53: Maybe he has red shirt... The status is not very clear.

00:20:02: It is, we are now reasoning about reasoners.

00:20:05: Not the story as such I would say.

00:20:09: Is there any other aspect you'd love to talk in this view of mathematics?

00:20:14: Maybe... To me it's a historic aspect here i think where mathematics also influenced by by culture, let's say here.

00:20:34: And for a long time the main application of mathematics was physics engineering and there you at least have this metaphor that is real world in which I build my machines reflected in physics and mathematics.

00:21:00: I mean, yeah it builds a machine but the laws that follow are always there given right?

00:21:08: So this suggests or a sub platonic view of mathematics.

00:21:15: But now recently we have been using computers And we construct programs and programs are sort of, or software in general is clearly constructed.

00:21:35: And so if the main application of mathematics now moves a bit from this real world applications to these software applications just changes.

00:21:49: how we do mathematics that developments as sort of like, a bit like software whereas previously we may have thought about them taking place in this perfect new world.

00:22:06: And so I think that leads to paradigm change.

00:22:12: There's at least the sociological fact.

00:22:15: there is more intuitionistic and constructive math computer science departments than physics department.

00:22:23: here might be a counter example.

00:22:27: It's everywhere, not necessarily the real department well at home too great.

00:22:34: and so I see we still have a lot of things to talk about.

00:22:37: i like this math as culture topic.

00:22:40: at some point i would love to talk And

00:22:50: we could also fix my ignorance about philosophy and maybe you can tell us a bit more what Wittgenstein and Hegel, stuff like this.

00:23:03: Sure!

00:23:05: We can switch next time if you think of any questions to ask me in the episodes.

00:23:12: but for now let's... see us all in a week where we have very, very special guests.

00:23:18: I think i can announce them by now.

00:23:21: it's Dana Scott touring Prize awardee so did the lot for philosophy of philosophy of satirical set theory um computability theories are one of the greats and you all can support contributing at Buy Me a Coffee and we look forward to your comments.

00:23:47: If you have wishes for this in-between filler episodes, if you need some background information on the topic... ...if we forgot to reply to a comment feel free to do that here so that we have content into weeks for another episode without a guest!

00:24:05: Can I also add something before you wave off?

00:24:09: So one thing I wanted to say about Dana Scott, who I met many times when i still was a student and young postdoc.

00:24:19: And they always liked to talk with him but also like the fact that he asked stupid questions right?

00:24:28: He was particularly cool about this...I still remember..I don't want to give any names.

00:24:34: somebody well known category citation, giving a presentation about something.

00:24:40: And Dana was sitting there and after a while he got up and said I don't understand the word.

00:24:45: what you're talking...I mean i felt the same but they didn't dare to say this yet!

00:24:51: Yeah

00:24:54: so whenever people do not understand the words we are talking about put it in the comments.

00:25:05: Okay, great.

00:25:06: Thanks for the discussion

00:25:08: and thank you for running so short this time.

00:25:12: everybody for tuning in.

00:25:15: I have

00:25:18: one but still

00:25:20: don't.

00:25:21: you're starving?

00:25:22: You are running out of coffee!

00:25:24: Exactly we had two episodes before.

00:25:27: The budget is over.

00:25:28: no then see y'all looking forward

00:25:32: bye-bye.