
Set Theory and Algebra in CS: Introduction to Mathematical Modeling (2013) [pdf] - lainon
https://pdfs.semanticscholar.org/d106/6b6de601c1d7d5af25af3f7091bc7ad3ad51.pdf
======
mlevental
call me a mathematical hipster (new frameworks etc.) but i could never get
into axiomatic set theory. algebra is already boring by virtue of being
abstract but then make it even more abstract by dropping all models! at least
in algebra you have applications like galois groups, lie algebras, crypto,
etc. yes i know that complexity theory is basically in the domain of set
theory (complexity hierarchy) but even that was so abstract. so does anyone
know any really cool applications of this stuff (just to be clear i don't mean
technology i mean concrete theorems).

~~~
pron
Formal systems (like formal axiomatic set theory) are more in the domain of
logic than mathematics (although there are things like model theory that lie
at the intersection of the two). From their very early conception, they were
designed to serve two roles: 1. Serve as completely precise languages, where
no ambiguity is possible, and 2. allow for _mechanical_ reasoning. As role 1
is often in service of role 2, I think it's fair to generalize and say that
the role of formal systems is mechanical reasoning.

If mechanical reasoning is not something that interests you, you have little
use for these pedantic systems. Mechanical reasoning has two current uses: 1.
mechanical verification of mathematical proofs and assistance in such proofs
-- this application is very rarely used in practice, and is commonly the
domain of a few enthusiasts who hold high hopes for improvements that would
make it more mainstream. 2. Mechanical verification of software, including
static analysis, model checkers, type systems and other software-relevant
proof systems.

~~~
antidesitter
Axiomatic set theory isn't just for "pedantry" and mechanical reasoning,
though that's an important application. It's also like a telescope array that
allows us to see ever further into Cantor's paradise and the universe
(multiverse?) of mathematics. The upper levels of transfinite set theory are
especially breathtaking and somewhat scary [1]. Hugh Woodin spoke a little bit
about this in a presentation on the Ultimate L conjecture:

"There's the other possibility: set theory is completely irrelevant to
physics... That's almost more interesting because if this conjecture is true,
we've argued there's truth here for set theory, and it's a truth that's not
about the physical universe. To me that's almost more remarkable, that there
could be some conception of truth that transcends truth that we see around
us."

The presentation is called _The Continuum Hypothesis and the search for
Mathematical Infinity_ [2].

[1]
[http://cantorsattic.info/Cantor's_Attic](http://cantorsattic.info/Cantor's_Attic)

[2]
[https://www.youtube.com/watch?v=nVF4N1Ix5WI](https://www.youtube.com/watch?v=nVF4N1Ix5WI).

------
user68858788
I learned this years ago and forgot everything. I'm curious - does anyone
actually use linear algebra in their day to day, and if so, what's your job
title?

~~~
theoh
Isn't it very important for machine learning?

------
ganzuul
It is a source of endless wonder and bemusement to me that we did not choose
geometry as our basis for logic and proof over the equality symbol. Many
brilliant minds are barred from entry into the field of mathematics because of
this error.

