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  • home A triple theory.

  • We've talked a little bit a bit in the past.

  • What?

  • How does it affect people?

  • Kind of day today is a kind of a way of thinking about this for somebody who's just, you know, day today, making computer programs.

  • Yeah, um, I think that's certainly a long term thing.

  • Ah, what a change is is the foundation's off theoretical foundations on which computer science is based.

  • Um, that using Ah, this homer took type dealy and union violence principle.

  • Ah wei ableto have a morse off structural approach to the mathematical foundations of computer science, which which affect you If you do a former if you start.

  • If you build big library off 35 programs, then we want to be able to re use them not to them again and again.

  • And to be able to re use them, we need to be able to ignore implementation.

  • It is height implementation.

  • It is.

  • And this universe, since principal tells us yes, we can do this.

  • We can just ignore the implementation details and nothing will go wrong.

  • So it's it's not me.

  • Obviously it's not something which ah so will change the vote tomorrow, but in the long run, and I think it has quite a profound impact.

  • One survey The understand computer science in mathematics they have, they tend to ignore this issue.

  • The issue which computer scents are very committed sincerely, very avail off that you want to be able to ignore implementation details.

  • You have a module you've implemented in a certain way, and now you decide to change it.

  • Now the question is, does the who'll mountain collapse and everything which depends on this has to be religion?

  • Or are you able to plug in a different model, which does the same thing, but maybe better?

  • Yeah, so that's That's a question.

  • Computer scientists are very avai off.

  • Mathematicians actually have the same problem.

  • Is this something like because it's using too much memory?

  • It breaks something else or?

  • Or is it?

  • Is it more fundamental?

  • It's more fundamental.

  • The question is, you you have a different model, which is implemented in a different way, and the Christians offices, the programs which used to smile.

  • You ever be able to seize this difference?

  • Maybe somebody some stupid programmer exploited that this was implemented in a certain way, and now you change it.

  • And the whole thing stopped working because there was this undocumented feature which this podium I used.

  • So there's a lech off hiding here.

  • You want to hide implementation details so it becomes a black box that you put exactly?

  • Yeah.

  • So you want to use black boxes?

  • That's computer Scientists are very aware off this.

  • No mathematicians.

  • For them, it's all the important.

  • They may use one mathematical object, and then they use another one, which is, in some sense equivalent see its eyes.

  • Um, offic When mathematicians do this, they get away with it because maybe they do it on a on a blackboard or something else.

  • Yeah, we know everything is stable.

  • Another's everything that you have done for this object.

  • Also, for this one here.

  • Ah, but if you really want to do this precisely, it could be that he actually exploited some details off savanah object in the other.

  • So using the more self discipline, language, language of type theory, they cannot seat anymore.

  • And this union villains principal actually tense.

  • Um, yeah.

  • If you follow this discipline, then everything will be fine.

  • You can actually replace one mathematical object by another one, which is a cool and and nothing will go wrong or you're theorems will still go.

  • So you Is that why the vault's Kate was so hot on using computers to prove the mathematics?

  • I think it's Is it related?

  • It's related.

  • I think that what really happened is that he understood that if you talk about structures living into stepped in way, you need to use something like type theory.

  • I mean, he he just checked You send me any way, had some image conversation 11 years ago, uh, about this, and he started us to explore this idea and learn about type theory and and and stuff because it was related to his research.

  • But then he also became excited about the tools which already existed.

  • It's all related, but it's a different angle.

  • I mean, both fit very well together because if you have a language like time steely, that's very suitable for being used in a proof assistant.

  • And indeed, many poof assistance based on time save Volsky possibly search in Homo to PC.

  • And it was just a mathematical subject state.

  • It's very abstract geometry, and let me just explain a little bit if you think about geometric objects like let's see a ball and the bicycle tube, then innovate ball.

  • It's the same execute because we can solve if you use played or you can just turns a ball into Cuba and a bicycle to work.

  • It's the same as a mark, you know, because you can transform the bicycle tube into a market.

  • Plato Boy's bicycle troops and spheres or boards are not the same because the two piss got a hole in the middle.

  • That changes that this new self continues transformation from from from a ball into a bicycle, too.

  • Now what Homer took me serious about to say, like, How can we actually see for that?

  • The bicycle tube into sphere, the ball?

  • That's, you know it's the same thing.

  • Yeah, and think about paths.

  • So let's say single.

  • The ball could be the Earth and he started the North full revolt around, and we end back up.

  • The North Pole's away from the past from the North Pole, the North Pole.

  • Now we can also continuously transform path.

  • We're shifting them a little bit, little bit, little bit, and so any two paths on this here can be turned from the North Pole to the north but can be turned into the other.

  • The vague produces is your first self turned one, passed into what's called the empty party office.

  • Just stay at the North Pole, but you can continuously transform this part.

  • You make it shorter and shorter and shorter and intestine just That's the North Pole, so any to path from the north coast of North Pole equivalent in the sense that, as on the bicycle trip, you can walk around the tube between different ways this way or that way.

  • And this path cannot be transformed into the empty path.

  • Because you cannot go through the middle of the bicycle tube, you have to stay on the surface.

  • So this is the difference from who want Toby Point of view between a bicycle tube and the ball.

  • How is this related toe tap?

  • Siri.

  • If if you think about, we have two objects.

  • One question is what is the type of the qualities or evidence of equality between any two objects?

  • And ah, what's his idea was to say that's actually the same A surpassed.

  • So he thought the types, unlike geometric objects, and he called it the element off the types of the points on the surface.

  • And the quality is in a path from one point to another.

  • And when we study equality, that question is, are qualities obvious?

  • Trivial?

  • So, for example, ah, they say, at most one path between any two elements like we saw in the sphere is only one path.

  • So, for example, if he asked, a sleek would flee, there's only one way to stated three is equal to three.

  • But if you think about more complicated structures like types themselves, then when are two times equal?

  • Uh, this was answered by what's qihoo silky two types of equal.

  • If there's a want of encores, bonds between the elements, that's the universe since principal.

  • But there may be more than one wanted.

  • One corresponds.

  • And this corresponds to the situation on the on the bicycle tube where we have more than one path, uh, former point to a point.

  • So he was able to to answer this question about nature off qualities by making this connection with this study off very upset study of June meant ago Number five has done some things on Klein objects, so they called Klein objects with Crypt Cliff stole these just different kinds of geometric shapes that would be classified.

  • Yes, we can.

  • We can study them.

  • Ozel, from this point forward a theory and look att pass On the surface, Brodsky realized, is that ah, is this off of a He used in home atop a theory can be used to talk about any mathematical objects, so they're actually there.

  • There's a problem in mathematics that certain things work well on this offset level.

  • But if you want to work about structures, more abstract, geometric, abstract mathematical concepts, then you need sort of a different type of mathematics.

  • And he realized that using ideas for a moment of your theory, you could make up.

  • So it's more abstract.

  • They off mathematics.

  • So his understanding was, I think, that he realized that set theory, which mathematicians usually use, is no fittings, a bill, but somehow that ideas from home, a trophy theory should give rise to a different ve to do mathematics, and that then let him to type theory and connections to computer science.

  • We had a video a couple of years ago with Professor Browse feed about non Euclidean geometry and this idea of whether we can prove whether parallel lines ever meet the innate curvature of the Earth so long as you taking sightings and going journals is what votes keys done?

  • Kind of just taking this a step further Or is it something different?

  • Yeah, this desolate to compare this because initially when people thought I mean including German tree just thought off the geometry off the plane on that it.

  • But then when we're looking at different types of Jim between, non Euclidean geometries extended the geometry off office fear on DSO Ah, so what?

  • What?

  • What's key in the way?

  • We can compare this because mathematicians usually and we think about sets which are very self simple structures.

  • But it turns out that if you want to do a more abstract mathematics, if you want to be able to build these towers of obstruction in an efficient very, it's good to get away from this idea off flat structure like a set, and think of times which have a more complicated structure like that extra tools and so on.

  • So in this way it's a it's a generalization, like before we have a clean geometry, which is only the jumped off the plane, and it was generalized to include other geometries.

  • And in an analogy, very you're thinking only about sense and first structures.

  • So what are the votes he proposed is, too, to generalize this here's exist idea from home atop the theory and think about more interesting basic types of which include structures and more interesting qualities.

  • And so, from my kind of, extremely, Lehman's mathematics always think.

  • And when you mentioned sets off, then diagrams on again.

  • Obviously they are depicted on the plane.

  • So this is like having more dimensions.

  • Yeah, yes, simplifying Know.

  • In a sense, it's it's just you think of off the points and how they're connected.

  • So if you think about equality, Czar Path and the plane, there's only one connection between any two points.

  • So they're a bit like what we call a set.

  • But But if he if you think about like this is a bicycle tube, we have got a more interesting structure.

  • We have these points, which elements, but there can be equal in different ways, right?

home A triple theory.

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ホモトピー型理論の議論 - コンピュータマニア (Homotopy Type Theory Discussed - Computerphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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