Name: Homotopy Type Theory Discussed - Computerphile
Uploaded: 2020-04-04T10:05:13.000Z
Duration: 13 min 31 s
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We've talked a little bit a bit in the past.

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.

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

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

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?

Mathematicians actually have the same problem.

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

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.

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

That's computer Scientists are very aware off this.

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.

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

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.

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 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 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.

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?

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.

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.

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Homotopy Type Theory Discussed - Computerphile

discipline

equivalent

realize

vague