Computer Science ∩ Mathematics (Type Theory) - Computerphile 

The guy Gérard Huet behind Coq also wrote some data structure he called The Zipper. Apparently he said in conference during a demonstration: "Now I'm gonna open the Zipper and show you my Coq". Source: someone at INRIA who knew Huet told me.

I'm starting realize that just because I come here, it doesn't mean I'm smart.

Drink a shot every time he says "ja".

Mathematicians should wear robes and pointy hats and be interviewed in front of some copper plated bubbling apparatuses (apparati?). Then replace "theorem" by "curse", objects by "demons", and demonstrations by "invocation". "First evoke a local Riemann-integrable demon. the Fundamental Curse of Analysis allows you make sure you can restore that demon when you observe its growth." Cedric Vilani is the headmaster of the Lyon School of Wizardry and professor of Implicit Demons Taming.

Didn't know Thor was into maths

"A function which doesn't function shouldn't be called a function" Real-life James Bond Villain - 2017

As a programmer, this makes so much sense. I was in a weird situation where i learned algebra first, and then learned programming (and algorithms and data structures), and then more maths, and i couldn't help but think how i could represent the concepts i learned in computer code, but the notation used in maths was set theory, which was slightly confusing some times. Then i noted to a lecturer that a definition he wrote of something could be expressed as a short recursive function, and turns out he knew (among other things) haskell and he agreed that the recursive notation was correct, but pointed out that you can solve some such functions which may be noted recursively using transformations that allow you to do it as one computation instead of a series of computations (iterative), which can dramatically impact the time they take to run, and he had done work on that. Using maths to speed up some heavy logic functions by transforming them. Anyways, my big take-away is i'm fairly decent at maths, but i'm a bad calculator, and i strongly prefer computer science notation over set theory notation. If i had been taught maths through programming the concepts, i would have done MUCH better, and it would also have allowed running the functions and (automatically through code) visualizing the output to gain a better "feel" for it.

The jist of the curry-howard correspondence is: In a well-typed programming language (such as Coq, Agda, Idris etc.) Types correspond to theorems Programs correspond to proofs. You write the type, then you prove it by writing a body to that function.

Mathematical Poofs! I love this guy

I think this would have been more clear if we had examples of some useful types that propositions and their proofs could take.

So it seems our galactic president Beeblebrox is into type theory. BTW, great video! Love this more theoretical approach to Computerphile!

"coq could upset english speakers" LOL and they say Germans dont have a sense of humor

More vids on Haskell and functional programming would be great!

Need a proof? Bring on the COQ.

I'm pretty sure the only people who understood what he was saying already understood everything beforehand.

That was really fascinating. I have heard of type theory but didn't know much more that it resembles types in programming. He contrasts it to set theory which helps a lot and makes sense. What we seem to have here, as he is saying, a potential, qualitative (of course partial) transformation of a field, mathematics, because new tools, computers, bring new paradigms. Fascinating, thank you so much. I would be happy to see some more of this, and maybe some examples from this style of math.

Proof theorems ja. samsing wrong wis se pruuf.

RU-vid is great at suggesting videos. I am starting college and covering set theory, and I realized that there are multiple encodings for natural numbers and that implementation details were being exposed. And then I find this video which mentions the same thing. Amazing.

I think the guy behind this saw us complain about the 404 video and is now giving us some meatier content. I'm happy. :)

Thorsten is a great explainer and talks about some really subjects - liked this video! You guys should also track down Kevin Buzzard. He's a great speaker, interesting guy, and he's recently been working on proof formalisation in a language called Lean. He and students have been writing up a whole bunch of definitions/theorems/proofs, and is planning to do his whole department's curriculum.

I like listening to Dr Thor here But I'm gonna be honest, I got lost about 5 minutes in :(

I recently signed up for CS in school, and math class is doing set notation. A youtube title has never been so relatable.

This guy takes no.3 spot after the Klien bottle and Japanese guy.

I was surprised that he managed to talked about set, type and homotopy theory, abstract mathematics and new foundations of mathematics without mentioning category theory.

Man, this dude is so intelligent. He just pulls all this theory history out of his head; its awesome.

great video hope to see more abstract discussions like this for more computerphile videos! or at least a numberphile video on this but more from the other guys' perspective.

Excellent video on a relatively new field of study with a real future!

For those who don't know, "coq" means "rooster" in french.

This is exciting. Just finished Group Theory and this seems like something I can learn from a text book now.

I know I'm late but props to the one(s) who made the subtitles. Very creative with using the relevant symbols.

Underneath Type Theory there is a more foundational mathematical approach called Category Theory, which seems to unify all areas of mathematics. Unsuprisingly, Types form a Category. Just as there is a connection between Homotopy Theory and Type Theory, there is a connection between eg. Set Theory and Topology Theory via Category Theory and the theorems of one theory correspond to theorems of the other. I suggest you look into that.

This is awesome, my mind is blown and I feel much better

Please do video on backpropagation algorithm in neural networks

Ok, the comments have spoken: "Dr. Thor" it is Edit: very interesting! Thanks for doing these!

I think of mathematical proofs as a mechanism for problem-solving and gaining understanding while producing useful outcomes. Consider machine learning vs. algorithms. Machine learning can develop a function based on a training set that we can use to predict results from other inputs. That's great, but it doesn't necessarily lead to expanding our understanding of the underlying principles, things that we can use to help solve other more general problems.

It will be interesting to see if a type theory is developed in which univalence is a theorem rather than an axiom

this is really awesome and interesting, I will check out set and type theories!

This guy looks like the scientist from Independence Day!!!

6:43 or as Mr. Numberphile would call it: The Parker Function.

I super enjoyed this video

Eyyy funny stuff, I just used Coq a month ago for researching a class project. I was writing a paper on using recurrent neural nets to automatically write proofs in a Coq-like language. I was not successful - turns out it's pretty hard to do, lol

Really informative and 'enlightening'. My thanks.

gotta love the category theory diagrams on the back

Could you talk about the Haskell type system?

I like to think that I know quite a lot about both mathematics and computer science, but this goes way over my head

can you make videos explaining type theory and how math ideas can be proven with it?

Top 10 programming languages to learn in 2020: 1. Type theory 2. Type theory 3. (wait for it) Type theory ...

COQ magic :D

I've always struggled with gaining intuition in abstract mathematical theories because of the set theoretic foundations, (set theoretic definition of topology, for instance)

could we have a second video perhaps going more into detail of type theory maybe xsome examples of its uses. pls

Wow. Brady's really bringing in the big bucks. He brought in the scientist from Independence Day.

We are happy to welcome you mathematicians into the programming world, but there is one thing you need to learn fast and constantly repeat to yourselves if necessary: Single character identifiers are not acceptable! Use full words and clear and consistent naming conventions. Learn it, live it, love it.

More videos of this guy please. I have no idea what he's talking about but... more videos please.

Surprising as this may seem, Mathematicians seldom bother with sets in practice. In fact they already use mathematical constructions pretty much exactly like how it is described with types. For example the natural numbers N, defined as a set together with a preferred element 0 and a function s:N ->N that gives the successor (programmers would call it next) which is "universal" with this property (in the precise technical sense below), is enough to define N unique up to _unique_ isomorphism (= bijection), and that is perfectly good enough to work with it and ignore any implementation details and it nicely avoids asking silly questions like whether 2 \in 3 because they are not preserved under these isomorphisms. Here a set of natural numbers (N, 0 , s) is _universal_ if it has the following property: for any set X, element x \in X and any function T:X->X, there is a unique function f: N->X such that f(0) = x and f(s(n)) = Tf(n) (from which it easily flows: f(n) = T^n(x)). Of course one now has to prove that such a set exists (surprise surprise, it does).

ja?....ja?....ja?

I actually once used 2 (element) 3 in a Discrete Maths homework where we couldn't use

Thanks! Very interesting.

Set theory has the concept of universal set on which a choice function to extract a particular set can be applied.... Like moving from from everything towards being specific. Does universal set include everything that might be constructed?

Formal methods are getting more and more important as systems grow. For things like blockchain processing one approach to proving correctness is Hoare's Communicating Sequential Processes. Then there's Predicate Transformer Semantics etc.. It all gets very complicated.

For me, it always been that the sign ∩ was turned 90 degrees anticlockwise in this statement. :)

such a great thumbnail

So this may be way off, but is SFINAE (in C++ templates) for type deduction basically using the same relationship to encode logical statements?

This makes me feel guilty about not having read TAPL for two weeks...

10:49 he is onto something imo Aesthetics and genres and psychology and visual mediums paired with endless user data

ja

I just watched this video and to my understanding using HOTT you would be able to show that in fact two different proofs have the same outcome and thus are the same. So you would be able to first create a proof and then verify that in fact it is an improvement (more elegant, easier, whatever) to an already existing older proof... ? Or, the other way around, you would be able to show that a proof is really different from another proof...?

Could this be used to assist with the verification of the proof to the ABC conjecture?

If it is unable to distinguish between different representations of numbers, is it impossible to do boolean algebra? Is it possible to calculate the result of 42 AND 69 for instance without representing the numbers in binary?

More of this!

Computers along with there parts be it Monitors, Screens, Keyboards, Mouses, Remotes, Modems, Scanners, Printers, Floppy Discs, SD Cards, CPU's, and other devices can go hand in hand whether they are desktops, laptops, I pads, kindles, smart phones, cell phones, palm pilots, and other electronic devices and can serve well if they do not over heat or run out of storage space.

I was watching with subtitle, and i saw he was gonna have to pronounce Coq, and couldnt wait to hear that.

I have been studying formal mathematics for my Computer Science thesis for a little while now. And looking back on it, I think Set Theory has its priority backwards. Set Theory tries to take two Sets of elements, and tries to evaluate whether a function definition exists as a pathway between these two sets. Type Theory takes a set and a function definition, and uses it to generate elements of another set, which can be used as proofs for the other set. How I understand it is that: Set Theory is like a funky subset of Type Theory because Set Theory deals with concrete definitions generated from abstract ideas, whereas Type Theory works the other way around. In a way, this is more useful and powerful than Set Theory because you can easily create various definitions of a set of elements that would otherwise be hard to do using Set Theory. It's not impossible. It's just hard...

This video is really abstract to me compared to the other. In some videos the most basic concepts (to me) are explained but this one is really hard to follow, so much stuff is considered to be known and there doesn't seem to be much concrete stuff to relate to. It would have helped if it had been a bit more scripted, may be with examples or something. It's quite hard to get into Dr Thorsten Altenkirch's head for this topic!

Awesome!

I'm a Swedish mathematician as well.

Nice explantions but I doubt many will understand, I learned Measure Theory and thus can understand the general outline but those unfamiliar will be lost.

A function that does not function should not be called a "function". Makes sense on so many levels!

What about whitehead and russell? I guess they came up with such an idea way earlier... And the problem with a constructibility is that very important theorems couldn't be proven f.e. that every vectorspace has a basis or if you like non-standard analysis or even normal analysis you would loose many theorems that are quite crucial.

Can anybody relate this with typescript, babel and how types(shapes) are identified and are considered to be interpreted into another types. Like replacing a type instance with another equivalent instance which do the work.

How would you check the infinitude of number of primes via computing??

Dr Thorsten Altenkirch At 13:50, you talk about the Univalence Principal, where it is stated that if two things are equivalent, then they are equal. I have interpreted this as "If two things are logically equivalent, then they have the same properties. I shall show by example that this is not always the case: From the premises "forall x (Sx implies Px)", it can be proved that "Exists x (Sx & Px) iff Exists x(Sx) Using this example, I shall prove that if two logical expressions are equivalent, they don't necessarily have the same properties. In this example the property they don't have in common will be truth value. Take both Sx and Px to be "x=x". Substitution gives (1): "forall x (x=x implies x=x)" can prove "Exists x (x=x and x=x) iff Exists x (x=x)" (1) However, by the principal of transposition, it will also be true that (2): "forall x (not (x=x) implies (not x=x))" can prove "Exists x (not x=x) and (not x=x)) iff Exists x (not x=x)" (2) Example (1) has a truth value of true, however that of (2) is false. (x=x is always true, likewise (not x=x) is always false.) Thus if two statements are logically equivalent, they do not always have the same properties. Using transposition again, this means that if two statements have the same properties, they are not always the same statement.

The biggest problem will be when dealing with infinities, other than memory constraints some of the concepts become quite abstract.

isn't Formal Programming a way of proofing mathematical theorems?

Moooooore from this guy please :)

I did not find the other part of the video on numberphile maybe someone can pass me the link

I will die a happy man when the producers of James Bond finally hire this man to be their evil genius.

this guy is so cool!

So would Type Theory make it possible to compile abstract math into executable code? I ask because that is basically what I dream about.

6:43 "Dysfunctional function". LOL

6:43 Parker Function

TIL a function that does not function is not a function.

can someone explain what this math on the whiteboard behind him is about?

We are talking about a type of logic that has to be necessarily less expressive than First Order Logic (that is used for mathematical proofs) otherwise computers cannot help that much.

That was interesting and difficult to keep up with.

11:32 "It's true that 2 is an element of 3, doesn't really make any sense, right?" I mean, isn't 3 basically == | | | And 2 == | | so there indeed is a *| |* inside 3. So 2 is indeed an element of the set called 3

2 as an element of 3 doesn't seem silly to me.

this guy is my spirit animal

"That is getting abstract."

This should be on numberphile

coq is a big thick piece of software, also veiny