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.
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?
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.
Constructivist mathematics does not depend on the axiom of choice.
That said, when you only admit a universe of mathematical objects that admit of a finite construction, it is not unreasonable that you can have a computer program that, at least in principle, will enumerate all possible constructions and proofs of statements about them. Given that, you can create a construction which is literally for each set Y, the primary x in Y is the first x which in that enumeration a statement of the form "x is in Y" is proved.
This construction give a choice function of the kind that the Axiom of Choice says exists. However that said, the bizarre consequences of the Axiom of Choice that you find in classical set theory can't be proven for other reasons.
For me (intuitively, conceptually and abstractly), the AC is just a particular case of Hilbert's Entscheidungsproblem.
It doesn't matter what the AC states. All that matters is that you interpret it as a proposition. And then it's sufficient to assume that some algorithm can be constructed, or some Oracle can be consulted to determine the truth-value of the proposition.
Like, for example. I can construct you an Oracle which evaluates the "Law" of Non-Contradiction as true.
I think you have it backwards? From a mathematical perspective, the constructivist framework is defined as "provable without the axiom of choice." Constructivists are very concerned with avoiding the axiom of choice.
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.
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.
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".
AnthonBerg|6 years ago
Here’s more: https://github.com/michaelt/martin-lof
Here’s another one - this one is wonderful: https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Bo...
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
Intuitionism in math is beautiful imo: https://en.wikipedia.org/wiki/Intuitionism
jhedwards|6 years ago
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.
ex3xu|6 years ago
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|6 years ago
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
btilly|6 years ago
That said, when you only admit a universe of mathematical objects that admit of a finite construction, it is not unreasonable that you can have a computer program that, at least in principle, will enumerate all possible constructions and proofs of statements about them. Given that, you can create a construction which is literally for each set Y, the primary x in Y is the first x which in that enumeration a statement of the form "x is in Y" is proved.
This construction give a choice function of the kind that the Axiom of Choice says exists. However that said, the bizarre consequences of the Axiom of Choice that you find in classical set theory can't be proven for other reasons.
hackermailman|6 years ago
There's also the OPLSS lectures if you're interested: https://www.cs.uoregon.edu/research/summerschool/summer16/cu...
intuitionist|6 years ago
ukj|6 years ago
It doesn't matter what the AC states. All that matters is that you interpret it as a proposition. And then it's sufficient to assume that some algorithm can be constructed, or some Oracle can be consulted to determine the truth-value of the proposition.
Like, for example. I can construct you an Oracle which evaluates the "Law" of Non-Contradiction as true.
https://repl.it/repls/GiftedHumiliatingProcess
This is the fundamental (only?) difference between mathematics and programming. Mathematics assumes pure functions.
garmaine|6 years ago
r00tanon|6 years ago
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
auggierose|6 years ago
codedrome|6 years ago
Looking forward to being impressed (hopefully) that it is still relevant today.
AnthonBerg|6 years ago
ukj|6 years ago
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".