
Constructive mathematics and computer programming (1979) [pdf] - mathetic
http://www.cs.cornell.edu/courses/cs6180/2017fa/notes/week4/lecture8/Martin-Lof-ConstructiveMathematicsAndComputerProgramming
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AnthonBerg
Per Martin-Löf writes so beautifully.

Here’s more: [https://github.com/michaelt/martin-
lof](https://github.com/michaelt/martin-lof)

Here’s another one - this one is wonderful: [https://archive-
pml.github.io/martin-lof/pdfs/Bibliopolis-Bo...](https://archive-
pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf)

As far as I understand the theory, then dependently typed languages such as
Idris have their roots in his work on intuitionistic type theory:
[https://en.wikipedia.org/wiki/Intuitionistic_type_theory](https://en.wikipedia.org/wiki/Intuitionistic_type_theory)

Intuitionism in math is beautiful imo:
[https://en.wikipedia.org/wiki/Intuitionism](https://en.wikipedia.org/wiki/Intuitionism)

~~~
jhedwards
Out of curiosity as someone less mathematically inclined but very interested
in mathematics: what do you do with the equations in books like this?

Do you look at them and already intuitively know what they mean so you don't
have to dig deeper? Do you slowly go over each element individually and then
grasp the whole thing afterwards? Or are you so used to them that you can read
them more or less the way you read English? I really want go deeper into this
stuff but when I look at these books I feel like I would need years of
specialized training to understand them.

~~~
intuitionist
Written mathematics is quite informationally dense and, in general, reads much
more slowly than typical English prose. Sometimes you do have to go over an
equation symbol-by-symbol; sometimes you have to convince yourself of the
truth of a sentence with no symbols at all.

In this particular case, though, you're in luck, as Martin-Löf gave a less
technical and more philosophical series of lectures on the same subject, which
are a much easier read: [http://archive-pml.github.io/martin-
lof/pdfs/Meanings-of-the...](http://archive-pml.github.io/martin-
lof/pdfs/Meanings-of-the-Logical-Constants-1983.pdf)

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ex3xu
As a layman I'm curious about the state of constructivist mathematics and
intuitionism since the tragic passing of Vladimir Voevodsky in 2017. I recall
reading about the Coq proof assistant, homotopy type theory, and univalent
foundations with some interest, but I haven't been keeping up with any new
developments -- is it still an active field of research?

I'm also curious if someone can weigh in on the fact that this paper by
Martin-Löf finishes with a proof of the axiom of choice -- isn't that a bit of
a controversial axiom? Does all of constructivist mathematics still depend on
the axiom of choice to this day?

~~~
mauricioc
One exciting new development is "Cubical Type Theory: a constructive
interpretation of the univalence axiom" [0], a constructive version of
homotopy type theory.

The "axiom of choice" in this context is not really an axiom, just a provable
theorem whose proof in this setting is given by Martin-Löf. The reason, I
believe, is that constructive mathematics has a different meaning for the
exists (∃) quantifier. In a constructive context, existence is "stronger"
since you have to specify a witness of existence; this makes the hypothesis of
the "axiom" of choice stronger, allowing for a proof.

Here's a stab at a very informal version of this. Suppose you have a set
composed of "slices". The "axiom" of choice says that "IF for every slice (x
in A) there exists a good element of that slice (y in B_x, and 'good' is
defined by satisfying property C), THEN there exists a function which takes a
slice and returns a good object of that slice." But the very definition of the
(constructive) word "exists" in the hypothesis means you already have to
specify, as input, a way of finding one good object of each slice. With this,
finding a function that provides a good object of each slice is a tautology.

[0] [https://arxiv.org/abs/1611.02108](https://arxiv.org/abs/1611.02108)

~~~
ex3xu
Thank you, this helps my understanding.

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r00tanon
Very interesting. Amazing how much we owe, without realizing, to the early
pioneers and designers of the first computers, who often prognosticated issues
we deal with today.

I recently have been researching Leslie Lamport's TLA+ project. Definitely in
line with the ideas in this paper.
[https://lamport.azurewebsites.net/tla/tla.html](https://lamport.azurewebsites.net/tla/tla.html)

~~~
auggierose
Nah, no! Leslie Lamport is quite far away from constructive mathematics. You
will for example see his explanation of logic in the TLA+ video lectures is
entirely classical.

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codedrome
Thanks, I'll copy this to my tablet so I can carry it around with me.

Looking forward to being impressed (hopefully) that it is still relevant
today.

~~~
AnthonBerg
That paper is still the future, it looks like to me.

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ukj
Few years ago I joined some metaphysical dots between "construction",
"constructivism", "creation" and "creationism" in my head, which is what led
me on to explore 100 years of history/theory behind computational
trinitarianism, intuitionistic logic, Curry-Howard isomorphism etc.

This conceptual understanding of Monist metaphysics was the end of my militant
atheism.

The expression of knowledge is the creation of knowledge.

It pleases me to see that Per Martin-Löf joined the same dots in his paper "A
path from logic to metaphysics".

