

Univalent Foundations: New Foundations of Mathematics [video] - colinprince
https://video.ias.edu/node/6365

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pavpanchekha
Here's a bit of perspective. By "foundations" for mathematics, we mean a
system of _inference rules_ and _axioms_ for mathematics. A very limited but
correct view of mathematics is that it is a game, where the inference rules
are the available "moves", and the axioms are the "starting position". Given
these two things, we can work out all the theorems we can possibly prove (this
is because of Gödel's completeness theorem—not his incompleteness theorem,
that's a different thing).

Now, at this point a lot of programming-oriented people get confused, because
they think, "how can you ever change the foundations? How can you do any math
at all until you've completely formally defined the foundations of the
subject?" This is a philosophical question, but in truth foundations of math
aren't as hugely important as you might think. Here's an analogy: on both
Windows and Linux you can write an email client. Now, the process will not be
identical: some things that are easy on Windows are harder on Linux, and vice
versa. But at the end of the day, the result on either platform will
recognizable be an email client, and your major challenges will probably be
email challenges, not platform challenges. Most mathematicians do not work at
the foundations—they work at foundations for their specific field (a ring
theorist cares about the properties of rings, not often the properties of
sets, for example). You can analogize this to using a framework, like GTK for
example.

On the other hand, some subjects are poorly served by certain foundations, and
some of the newer subjects in mathematics are running into that problem. The
most pressing examples are category theory and homotopy theory, both of which
are based on a study of _operations_ , not _objects_ , in a technical way that
is poorly served by the current consensus foundation for mathematics. This
would be like really wanting to write mobile apps—Windows would not be a good
platform for it, and Android would be. As a PL theorist and kind-of-
mathematician, I think the work going on in the Univalent Foundations project
to be really interesting and extremely beautiful. I think it will have a
profound impact on our understanding of mathematical foundations and higher-
order logic, and I'm excited to see what comes of this effort. But I don't
think it will really impact mathematics all that much—the impact of
categorical thinking is the really important change, and that is helped, but
not much, by new foundations.

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daturkel
This is a great and insightful comment. May I just clarify, you say you are a
"PL theorist"—is that programming language theorist? What kind of work does
that entail?

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silentvoice
This is a very interesting project and it has actually set me on an
interesting path.

In scientific computation one blends mathematics and computation. The math is
highly advanced, and it will tell you important things about your resulting
code such as what kind of stability properties you should expect, and how
"robust" the code is (deliberately ambiguous term, but in most cases it means:
does your code work equally well no matter what parameters you throw at it?).

In the realm of physics simulations, these math questions turn out to be
amazingly delicate and failing to answer them before running your code on a
very large problem very often leads to unexpected (and wrong) results, if the
code actually ever successfully finishes executing.

There is therefore a tremendous amount of literature where these questions are
answered, all containing a meticulous amount of careful detail as well as deep
insights coming from all the relevant branches: linear algebra, physics,
functional analysis.

The problem however is that as we gain more understanding and knowledge the
math hasn't gotten any easier. Sometimes the arguments are so sophisticated
that it really leads me to doubt whether the result can be trusted at all.

This leads into something which I know very little:

A better understanding of foundations really is called for here, and a
foundation which lends itself better to computation than what is currently
available. Computers need to start being able to verify proofs better. Proof
assistants exist today, but their use is currently limited to computer
scientists because they rely on deep results in that field relating type
theory to mathematics - it represents a tremendous barrier to practicing
mathematicians.

I hope in the future that "reproducibility" in my field means not only
submitting your code so that other people can reproduce your performance
claims, but submitting proof documents to be independently verified on someone
else's computer. Whenever I read the small bits about univalent foundations it
sounds like they are adding magic sauce to make this possible, and so I follow
it closely (but don't understand it all yet).

The interesting path I spoke of earlier: I'm now learning type theory and
functional programming just so that I can make sense of this branch of
computer science that I originally didn't know existed.

~~~
chas
You might be interested in Edward Kmett's automatic differentiation[0]
package, ad[1]. It is one of my favorite non-trivial blendings of computer
science techniques and more traditional mathematics. In specific, it uses
Haskell's type system to prevent subtle numerical errors which other automatic
differentiation libraries require the user to prevent by being very careful.
The resulting code is much more trustworthy.

[0]
[http://en.wikipedia.org/wiki/Automatic_differentiation](http://en.wikipedia.org/wiki/Automatic_differentiation)
[1] [https://github.com/ekmett/ad/](https://github.com/ekmett/ad/)

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baoyu
Vladimir Voevodsky received Fields medal in 2002, so please note that it's not
usual amateurish try to create new mathematics while knowing nothing about the
old one.

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dang
Please remember to append "[video]" to the title when it's a video.

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DonGateley
Just when I found myself starting to lean in he started skipping slides. Most
frustrating.

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diego898
Does anyone have another link? For some reason its dead.

