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> It was to provide a small, parsimonious foundation for all of mathematics with a minimal number of "obvious" commitments, to give us confidence that the mathe

> It was to provide a small, parsimonious foundation for all of mathematics with a minimal number of "obvious" commitments, to give us confidence that the mathematics we're doing is consistent I would argue that type theory does a better job at this than set theory. With set theory, you need to believe in two separate things: (1) the language of first-order logic (or some other logic) with its inference rules, (2) the set theory axioms. With type theory, there is only the language of lambda terms. And the rules for type theory are straightforward and intuitive for programmers, e.g., you can only call a function on an argument if the function's domain matches the type of the argument. Contrast that with set theory, where you have highly counterintuitive and seemingly arbitrary axioms like the axiom of separation.

spekcular 4y ago
I suppose to some extent this is a matter of taste. I'll just say that, in my experience, people are typically very comfortable with, e.g., logical connectives and the primitive notion of a set of objects from grade school mathematics education. So this framework is "natural" and readily believed. Further, in ZFC, the only basic notation is that of a set. In something like the calculus of constructions, there are five fundamental notions (if I remember correctly). From the standpoint of ontological parsimony, that's a win for ZFC. Axiom of separation just says we can make subsets of things - I think this is not so hard to swallow. I'm curious what you find counterintuitive about it.
- ogogmad 4y ago
  Nobody's comparing the aesthetics of sets to types. That's not the point... People want mathematical proofs to automatically determine programs, or to be more than just proofs somehow. They want to exploit the capabilities of constructive logic. The fact that arbitrary sets can intersect each other makes extracting computational meaning from set theory proofs harder. The fact that types can be like sets but don't have to be is also why they're interesting: The notion has more flexibility.
c-cube 4y ago
I'm not sure you can call the type rules for CiC "intuitive for programmers". They're quite powerful and go further than "type of argument matches expected type". Compared to CoC, first order logic is a model of simplicity, and I think it's reasonable to argue that adding the inductive types to get to CiC is as complex as adding a handful of axioms to Fol to obtain ZFC. And don't forget to pick a flavor of universe polymorphism or cumulativity to make it usable. That's not exactly simple. I think there's a good case for CiC or HoTT being nice and usable for mathematicians. I don't think they're simple or more appealing to programmers. A kernel for metamath is the simplest, and it has more independent implementations than any other system.

spekcular 4y ago

I suppose to some extent this is a matter of taste. I'll just say that, in my experience, people are typically very comfortable with, e.g., logical connectives and the primitive notion of a set of objects from grade school mathematics education. So this framework is "natural" and readily believed. Further, in ZFC, the only basic notation is that of a set. In something like the calculus of constructions, there are five fundamental notions (if I remember correctly). From the standpoint of ontological parsimony, that's a win for ZFC. Axiom of separation just says we can make subsets of things - I think this is not so hard to swallow. I'm curious what you find counterintuitive about it.

ogogmad 4y ago
Nobody's comparing the aesthetics of sets to types. That's not the point... People want mathematical proofs to automatically determine programs, or to be more than just proofs somehow. They want to exploit the capabilities of constructive logic. The fact that arbitrary sets can intersect each other makes extracting computational meaning from set theory proofs harder. The fact that types can be like sets but don't have to be is also why they're interesting: The notion has more flexibility.

c-cube 4y ago

I'm not sure you can call the type rules for CiC "intuitive for programmers". They're quite powerful and go further than "type of argument matches expected type". Compared to CoC, first order logic is a model of simplicity, and I think it's reasonable to argue that adding the inductive types to get to CiC is as complex as adding a handful of axioms to Fol to obtain ZFC. And don't forget to pick a flavor of universe polymorphism or cumulativity to make it usable. That's not exactly simple. I think there's a good case for CiC or HoTT being nice and usable for mathematicians. I don't think they're simple or more appealing to programmers. A kernel for metamath is the simplest, and it has more independent implementations than any other system.