All the Streicher Haters in the comment section are getting a dislike from me 😡

lil bro weis garnicht was er macht , hazim ist gut in valo

There is a mathematically very important distinction between A and B are isomorphic and A and B are _uniquely_ isomorphic. In mathematical practice, one can only safely identify two structures, if the isomorphism between A and B is unique. The non uniqueness of isomorphisms is in fact easily detectable: the group of self isomorphisms (automorphisms) of A is non trivial. This is already apparent for basically the simplest mathematical object: the Booleans. As a finite set the booleans is just two element set and it is manifestly has a automorphism T <-> F. However as a Boolean algebra, it no longer has an automorphism and it is unique, and it is completely harmless to change its representation (in fact this actually done in practice: in most computer hardware a boolean is represented by F <-> 0 , T <-> 1 but in some others it is represented by F <-> 0 T <-> ~0 = 0b111111...111 = -1).

Thanks. Why no subtitle though.

In a past life, Thorsten was on the front line converting people from Roman Numerals to Arabic numerals 😂. 🎉 a hero and scholar.

Understanding with precision the samenesses and the differences (identities and deltas/differentials) between things was the motive for me finding my way to homotopy type theory. Stating the differences and samenesses (importantly the differences) on the *tools* of stating samenesses and differences is profoundly useful and helps people to become ambivalent to the tools that suit their needs best!

Real =/= Physical!

This is the new mathematics of foundations. Not new foundations.

Mathematicians are getting further away from the foundations of mathematics, not closer. Does an infinity groupoid explain the difference between cardinals and ordinals?

I can't believe this guy's teaching at a university

Sorry this is unwatchable. I stoped watching after you keep using the term token to describe the term term. Term is the right term for term for god sake. Dont get me started about witness

"Are you familiar with lambda calculus?" 🤣

What’s the motivation for using an Intuitionistic PL for the meta-logic presented here?

This is guy a fin genius... and this is the first time I've commented such a thing!

01:55 Set theory ; 02:45 Type theory ; 03:38 What's the difference? ; 13:15 Intensional vs. extensional ; 16:22 What is a function? ;

Absolutely amazing video!

i ran into this in the real world. say that you use a service that signs statements that it believes about you with a MAC; literally hashing strings: A=H("fname:thorsten"), etc. A is evidence that the string was hashed by our authority. (A(B ~ C)) = AB ~ AC. Where hashes are literally multiplied, and "~" means that they differ by some constant. Say that K is a secret key that we want to calculate to open a logical padlock, by storing values. (K - (AB ~ AC)) = K-AB ~ K-AC ... means that we can calculate K by adding AB to (K-AB), or adding AC to (K-AC). To handle not logic, we would need negated expressions (which are true), such as !D. That works fine because it has an actual witness value. AB and AC are "witnesses". But you get stuck when you want proof that a statement E is "missing". The certificate that a user has would be a list of statements "and" together, like: A B D E G H !M !N. We have evidence for these. But if we are asked to prove that X is MISSING from the certificate, such as X=H("DrugConviction2020"), we can easily walk the certificate to verify that X is not in the list. But we can't provide a specific witness value that calculates the key to unlock the padlock. ie: AB = 2394. AC =2300. K = 1300. case AB stores -1094 in the padlock, AC stores -1000. User must be able to supply the value of AB or AC to recover the key. This is very much like needing a witness. Because the key K must be calculated the same, no matter who the user is; We have a similar issue with general not(p) of propositions; in that we can't produce a consistent witness; unless the padlock brings the witness value with it. The intention is that the padlocks are totally public; and should not leak the values of any attributes, though they may leak what they are looking for. This is a reasonable definition of boolean logic, and it even admits a reasonable interpretation for values between 0 and 1: def and(a,b): return a*b def not(a): return 1-a def or(a,b): return not(and(not(a),not(b))) But, they lack witness information. I am definitely handling and/or/not, but the inputs are not just zero and one. With general circuits, it does not seem possible to make the user certificate provide all the witnesses required to handle NOT cases. The padlock may need to provide (ie: leak!) a witness value, to tell what to look for in the certificate. And there doesn't seem to be a way to enforce that the code honors a missing derogatory attribute from its certificate. It's not clear how to create a witness value for something proven missing from a certificate. I tried all kinds of crazy stuff. One thing was having user and padlock encode their beliefs into points on a polynomial (which amounts to polynomial neural nets for questioning witnesses), where they agree on what hashes to x, but disagree on what the y values are; and use the difference at a particular point between user (x,y) and padlock (x,y) (ie: H("DrugTest2020Pass")) answers in the polynomial. This is the first time that all this abstract gobbledygook I read in "Type Theory And Programming" made sense. Even when witnesses required to handle Not are not secrets, you can only walk the certificate (ie: an environment) to prove that the statement is "missing"; which is different from being explicitly asserted as false (giving an explicit witness value).

A Master Piece Lecture !!!!!!!!!!!!!!!!!

A Master Piece !!!!!!!!!!!!!!

I would never have considered mixing computer modern with comic sans, but it sort of works

Absolutely amazing video!

1:32:00 "There is not mathematical definition what stupid means." :D

This talk was from time to time hard to understand, but it is such mind expanding that is thing that you must watch. Requiesciat in pace Voevodsky, you were a great mathematician.

Very interesting and enjoyable talk. Also: ja.

Good God. Just TELL him, "No, it is not formal logic but it can still help you by pretending to be formal logic."

"Internal / External" is far more comprehensible than "propositional" or "logical". For example, I have no idea what a "proposition" is supposed to mean as a function of math or semantics. I have a very clear idea of how "internal" is relatable to "external". That has a dualistic notion (structural), whereas "proposition" is a circular, abstract term, and "logic" only concerns consistency, not meaning. The reason so many people hate math, is because mathematicians are horrible at conveying meaning and are very poor thinkers overall in the modern day. Math is a LANGUAGE. Nothing more. It is not some devine idea. Logic is a language too. It is a language with the same intuitions and assumptions of any other language. He is trying to explain the MEANING of ideas. That is what a philosopher should try to do.

I find the idea that "A function is a set of pairs" interesting. I must have learned enough computer science before set theory to not think of it that way, except (ironically?) for finite functions like &&, ||, !=, etc.. The way I think of a && b is as its logic table "00->0;01->0;10->0;11->1", which can be manipulated as a whole to show certain logic identities. For any other function, though, I think of it more as the way to *generate* the table (or graph), than as a lookup. Even for functions which I don't know how to calculate, I think of it like a blackbox, which hypothetically spits out values when you give it input.

In set theory, we can say that ℕ ⊂ ℤ ⊂ ℝ, which encapsulates the idea that 3 :: ℕ behaves the same as 3 :: ℤ, so in some sense they are the same (and in set theory they literally are). It does seem important to be able to say that a `double` can represent the same things that an `int` can, and more.

a wonderful overview!!!!

I got lot at 17:45. Not having been introduced to dependent types, this "path induction" device is quite impenetrable.

The slides are off-sync. There are two version of the slides.

Very Nice

Hey Urs, ich versteh kein Wort- kannste mir das nochmal erklären?

I have no idea what he's talking about.

1:10:40 sounded like Musk

Excellent talk. He covers a lot of subtle issues that deeply illustrate the difference between Set theory and Type theory.

Thorsten seems to interpret "extensional" and "intensional" as some complicated, hard to explain concepts. But at least in philosophy, they were never meant to be complicated, see for example: philosophy.stackexchange.com/questions/16164/what-is-the-difference-between-intensional-and-extensional-logic. What worries me even more is that Thorsten feels that the name intentional type theory is paradoxical, and that he feels a paper by Per Martin-Löf explaining some closely related concepts would be based on some confusion. Maybe Thorsten just failed to get that people mean something utterly trivial when they say "extensional"...

Pity the community of mathematicians having to deal with this man who thinks SOOOOOOOOOOOOOO outside the box. His project will probably have to wait. Sad he left us so soon.