
Foundations of Mathematics (2015) [pdf] - lainon
http://www.mathetal.net/data/book1.pdf
======
ocfnash
A substantial portion of this text appears to be concerned with traditional
Zermelo–Fraenkel set theory (and its extensions).

I have gradually come to believe that ZF theory has received a
disproportionate amount of attention on account of the fact that it serves as
the "official" foundations of mathematics, but that it is not an especially
beautiful, or useful, structure.

I believe that ZF theory is an interesting object, worthy of mathematical
study, but _not_ the best candidate for the foundations of mathematics! I am
very happy to see that this year's Chauvenet Prize [1] was won by Tom
Leinster's "Rethinking set theory", in which he highlights that Lawvere set
theory looks like a much better candidate. I cannot do better than recommend
you look at Leinster's superb article.

[1] MAA, Chauvenet Prize, [https://www.maa.org/programs-and-
communities/member-communit...](https://www.maa.org/programs-and-
communities/member-communities/maa-awards/writing-awards/chauvenet-prizes)

[2] Leinster, T., "Rethinking set theory",
[https://arxiv.org/abs/1212.6543](https://arxiv.org/abs/1212.6543)

~~~
nextos
Tangential question. Is there a mathematics curriculum out there that focuses
on constructive foundations? Or is it possible to assemble one?

I feel this would be much more adequate for then following up with pure CS
studies on topics such as [1-6].

Classical math bootcamps, like Math 55, are typically algebra plus analysis
and they don't even focus too much on classical foundations.

[1] [https://www.elsevier.com/books/lectures-on-the-curry-
howard-...](https://www.elsevier.com/books/lectures-on-the-curry-howard-
isomorphism/sorensen/978-0-444-52077-7)

[2]
[https://softwarefoundations.cis.upenn.edu/](https://softwarefoundations.cis.upenn.edu/)

[3]
[https://www.cis.upenn.edu/~bcpierce/tapl/](https://www.cis.upenn.edu/~bcpierce/tapl/)

[4]
[https://www.springer.com/gb/book/9783540654100](https://www.springer.com/gb/book/9783540654100)

[5] [http://www.concrete-semantics.org/](http://www.concrete-semantics.org/)

[6] [http://adam.chlipala.net/frap/](http://adam.chlipala.net/frap/)

~~~
valw
You might be interested in the works of Gérard Huet!
[http://gallium.inria.fr/~huet/PUBLIC/FSCD16.pdf](http://gallium.inria.fr/~huet/PUBLIC/FSCD16.pdf)

~~~
nextos
Thanks. I know about Gérard Huet, but not about this particular publication.

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visualstudio
Not a mathematician. If the goal is simplicity why can't you use Peano's
axioms?

~~~
Retra
The Peano axioms don't do very much, they just help model the natural numbers.
You still need something to model the logic and provide foundations more
complex subjects like analysis.

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techno_modus
It would be interesting to learn about some new (unified) theory that can
formalize the notion of infinity. Apparently, this can hardly be done on the
basis of the classical (Cartesian) view of space which is essentially a box
with points (elements).

~~~
Mugwort
Then John H. Conway's Surreal Numbers might be of interest to you. Donald
Knuth wrote a book about the subject and it's actually in the form of a story
with dialogue, not your typical math book. I'll attempt a summary. It is
possible to build all number systems at once including all integers,
rationals, algebraic, irrational, infinitesimal, hyper-reals and transfinite
all at once using a single procedure which resembles Dedekind cuts. From that
foundation one can recreate many of the results of non-standard analysis.
Conway discovered his Surreal Numbers while investigating Combinatorial Game
Theory. He wasn't even trying to create a new foundation for mathematics, he
was just playing at his games which many of his colleagues looked down upon as
useless. Apparently playing games and having fun can occasionally result in
extraordinary discoveries.

Conway's left and right "cuts" look like sets but then again I'm not really
sure what they are. Is this alien mathematics accidentally discovered on Earth
by perhaps the most intelligent Earthling who ever lived. I don't know. All I
can say is I took a detour from my usual math studies to look into Surreal
numbers and it was an interesting trip. I'm quite conservative when it comes
to math. I don't go around preaching Surreal numbers are the way to do math
but it is a curiosity none-the-less worth at least the small price of reading
Knuth's short book about it from cover to cover.

~~~
voxl
If you listen to Conway talk about the Surreals you'll note that he considered
it a game (in the game theory sense). He was interested in games of a certain
kind and what properties they had, a long the way he stumbled upon this game
that we now call the surreal numbers.

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cellular
I like Godel: "this sentence is unprovably true".

