
Homotopy Type Theory and Higher Inductive Types - tw1010
http://www.science4all.org/article/homotopy-type-theory/
======
sigsergv
This articly is quite hard to read and follow. I have some background in math
but can't follow the logic in the article because author is constantly jumping
from one topic to anonther. Artificial “questions” from imaginary layman are
not helping, instead they confuse and dilute attention.

~~~
SolarNet
The questions are not the problem. They are a variation of the Socratic method
and are super useful - when used correctly - for checking your understanding,
and following where the author is going to go next.

The problem is that the text is full of grammar, spelling, and punctuation
errors. There are missing words, confusingly run on sentences, etc. It's an
unnatural read because it hasn't had an editor.

------
auggierose
I really wish somebody could explain the Univalence axiom to me. Without
assuming that I know category theory or algebraic topology. You know, just
like you can explain the axioms of set theory pretty easily. I mean, if it is
a foundation, there should be an easy way to explain it. And not in metaphors,
please, but in a precise way.

~~~
Koshkin
I doubt that it can be explained in less than fifteen pages.

[https://math.uchicago.edu/~may/REU2015/REUPapers/Macor.pdf](https://math.uchicago.edu/~may/REU2015/REUPapers/Macor.pdf)

~~~
auggierose
Well, it is a start. Fourteen pages anyone? Once we are down to five pages
I'll give it a(nother) shot.

~~~
agumonkey
you should read it, it's an easy one, at least it's how I feel after reading 3
pages (usually I stop long before that)

~~~
dvt
It's not bad, but man the way the author throws around multivariate (e.g. non-
curried) functions all over the place is sooo confusing. I remember reading a
λ-calculus book that would do that and it's really not fun to try and parse
(see the bottom of pg. 7).

But I will say it's very cool to see how HoTT derives all the basic tools
you'd use in propositional calculus (A: A -> A, :(A -> (B -> C)) -> ((A->B) ->
(A->C)), and so on).

------
mathgenius
Computerphile did some videos on voevodsky and homotopy type theory:
[https://www.youtube.com/watch?v=v5a5BYZwRx8](https://www.youtube.com/watch?v=v5a5BYZwRx8)

(watch those eyes! I like how they bug out when he gets really excited.)

------
mathgenius
> I mean, when you actually use integers, do you really picture them as
> equivalence classes of the Cartesian square of natural numbers?

Yes, accountants really do think of integers this way: these are assets and
liabilities.

------
amenghra
[https://www.cl.cam.ac.uk/~sd601/thesis.pdf](https://www.cl.cam.ac.uk/~sd601/thesis.pdf)
might provide useful background reading for those who aren’t familiar with
type theories.

------
mbid
_In 2013, a consortium of the greatest mathematicians published a massive
volume which reboots our conception of mathematics._

Ok, here's a rant by a frustrated student who's spent way too much time
drinking the cool-aid. None of the authors, except for Voevodsky, is widely
known among professional mathematicians. And Voevodsky is not known because of
his work on HoTT. In fact, most mathematicians regard homotopy type theory as
some obscure stuff logicians and computer scientists do, with no relevance to
their research. And they're most certainly right.

"greatest mathematicians". Nobody would call a mathematician "great" for
coming up with a definition. (No, Grothendieck is not an example, because he
actually put his definitions to use and solved hard problems.) Yet there are
no interesting theorems proved via the univalence axiom which weren't known
before. Thus, the "greatest" mathematical achievement of the HoTT community
seems to have been coming up with the univalence axiom. Everybody can come up
with random axioms. Thus, I'd say their greatest accomplishment is not
actually mathematical but social, because they managed to market this
"breakthrough" the way it has been.

The whole process of how type theory research seems to work is completely
absurd. Type theorists first create a syntax and later come up with a
semantics for it. In other words, they first think about _how_ they want to
say something before they even know _what_ to say. For example, the semantics
of dependent Per Martin-Löf type theory were developed only 10-15 years after
Martin-Löf came up with his syntax. A few years ago, somebody finally
published a proof that the semantics of extensional Martin-Löf type theory is
equivalent to locally cartesian closed categories (lccc's). It's not hard to
understand what an lccc is, and state their defining operations/axioms. lcccs
pop up everywhere. The concept of dependent types, on the other hand, has not
existed in mathematics for thousands of years, seemingly without causing any
harm. Nobody gained any interesting new insights by understanding dependent
types, to my knowledge. Why, exactly, is this awkward axiomatization of lcccs
interesting?

And the same happens for intensional type theory with the univalence axiom.
It's been blurted out for years that this syntax's semantics should be
equivalent to infinity lcccs. Yet there is still no proof, no evidence that
HoTT is the easiest axiomatization or the easiest to work with, and a proof,
if it exists, will likely be extremely convoluted because it has to deal with
all the peculiarities of a syntax that was created without having infinity
categories in mind. A language that makes it hard to understand what it means
has failed.

If you don't believe me (and there is now reason why you should), please read
this thread [1]. Lurie, _the_ expert on infinity categories (well, there are
others, but he's written the canonical tome), debates with type theorists
about the usefulness of HoTT. I believe it ends with Lurie not seeing any
usefulness and calling it a day.

[1]
[https://mathematicswithoutapologies.wordpress.com/2015/05/13...](https://mathematicswithoutapologies.wordpress.com/2015/05/13/univalent-
foundations-no-comment)

~~~
naasking
> The whole process of how type theory research seems to work is completely
> absurd. Type theorists first create a syntax and later come up with a
> semantics for it. In other words, they first think about how they want to
> say something before they even know what to say.

Well no, they know sort of what they have to say. Type theory is syntax-
directed because this is what ensures it's mechanically automated and doesn't
require any intelligence to verify.

> The concept of dependent types, on the other hand, has not existed in
> mathematics for thousands of years, seemingly without causing any harm.

That's what some thought about Russell's basic theory of types before the
inconsistencies of set theory were properly acknowledged. And now type systems
are in every major programming language that drive trillion dollar industries.

> Nobody gained any interesting new insights by understanding dependent types,
> to my knowledge.

What kind of insights? There are plenty of great industry and computer science
insights, even if there have been no or few abstract mathematical insights
yet. Also, there's a growing push to use automated theorem proving tools like
Coq in mathematics because of the growing complexity of this field.

Finally, your argument that a difficult proof hasn't been found in the handful
of years since HoTT began, therefore HoTT is probably invalid or useless, is
frankly asinine.

~~~
mbid
Can't reply to everything here. I don't know to which extent Russel's type
system influenced modern-day type systems in programming languages.

 _Well no, they know sort of what they have to say._

Yes, I agree. But I'd expect more from people who literally want to make
formal, exact proofs viable.

 _Type theory is syntax-directed because this is what ensures it 's
mechanically automated and doesn't require any intelligence to verify._

Validity of terms in extensional dependent type theory is not decidable, but
that doesn't stop people from studying this system. They could just as well
study free locally cartesian closed category, but I guess it would be too
obviously trivial.

 _Also, there 's a growing push to use automated theorem proving tools like
Coq in mathematics because of the growing complexity of this field._

Err... There are _very_ few mathematicians who want to spend significant
effort on this. And one of my points is that there is no reason to believe
that a system based on intensional dependent type theory (e.g. Coq) is the
most suitable one, except that it already exists.

 _Finally, your argument that a difficult proof hasn 't been found in the
handful of years since HoTT began, therefore HoTT is probably invalid or
useless, is frankly asinine._

Why thank you. HoTT has existed for 10 years now, Martin-Löf type theory for
40, and experts in the fields about to be revolutionized are very reserved. If
you think that not expecting a revolution here is asinine, I don't think we
can come to an agreement.

~~~
naasking
> There are very few mathematicians who want to spend significant effort on
> this.

Because the tools aren't quite there yet.

> And one of my points is that there is no reason to believe that a system
> based on intensional dependent type theory (e.g. Coq) is the most suitable
> one, except that it already exists.

No one has claimed otherwise.

> If you think that not expecting a revolution here is asinine

That's not what I said. You effectively claimed that the fact that a difficult
proof had not yet been found, despite many believing the equivalence, somehow
entails that it won't be found. That's the asinine argument. I'm sure you're
very well aware of proofs that evaded mathematicians for _millenia_.

------
danidiaz
Introductory talk: "A Functional Programmer's Guide to Homotopy Type Theory"
[https://www.youtube.com/watch?v=caSOTjr1z18](https://www.youtube.com/watch?v=caSOTjr1z18)

