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user 1176
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
11
votes
How exactly are realizability and the Curry-Howard correspondence related?
I am sure more than one exact correspondence can be made, but here's at least one that is technically precise. We shall employ categorical logic.
Executive summary: realizability is the interpretatio …
- 48.8k
37
votes
What are some ways to stay engaged with the mathematical community from outside academia?
If you like computers, you could consider getting into formalized mathematics, which is mathematics done completely formally and verified by computer programs, known as proof assistants. Formalized ma …
233
votes
Accepted
What makes dependent type theory more suitable than set theory for proof assistants?
I apologize for writing a lengthy answer, but I get the feeling the discussions about foundations for formalized mathematics are often hindered by lack of information.
I have used proof assistants for …
- 48.8k
24
votes
Old books you would like to have reprinted with high-quality typesetting
Just for fun, Principia mathematica.
76
votes
On proof-verification using Coq
Coq is a proof assistant, and not the only one. Other popular ones are Agda, Isabelle and the related HOL light. They all use type theory as a mathematical foundation (as opposed to first-order logic …
- 48.8k
10
votes
Accepted
Progress towards a computational interpretation of the univalence axiom?
Cubical type theory is a variant of type theory which has all the usual (and some unusal) computational properties, and the Univalence Axiom is a theorem of cubical type theory. As was already pointed …
- 48.8k
1
vote
Formal Definition of Finite Conditions
It might be the case that the definition of finite elements from domain theory is what you are looking for. It makes precise the idea that an element of a poset carries finite amount of information.
…
- 48.8k
39
votes
Accepted
What is some current research going on in foundations about?
It is quite difficult to answer this question comprehensively. It's a bit like asking "so what's been going on in analysis lately?" It is probably best if logicians who work in various areas each answ …
24
votes
Does the "propositions-as-types" paradigm match mathematical practice?
There are many aspects to the question "does a logical formalism reflect mathematical practice?" I will focus just on a very simple but important detail that every mathematician is familiar with.
In …
- 48.8k
26
votes
What are some important but still unsolved problems in mathematical logic?
The modern logic (and foundational mathematics in general) of the 20th century gave us many important things: Russell's type theory, Zermelo-Fraenkel's set theory, meta-theorems about first order logi …
6
votes
type theory that does not treat the terms of $\mathrm{Prop}$ as types
Before I actually answer the question asked, let me try to explain one way of thinking about proofs as elements of propositions. It is not the only way, but it should appeal to a mathematician with a …
- 48.8k
4
votes
classical typed higher order logic natural deduction
Russell & Whiteheads theory is perhaps a bit on the heavy side, but here are some references to support Andreas Blass' comment:
An early formulations of classical higher-order logic was given by Alo …
- 48.8k
23
votes
What technical and/or theoretical challenges are involved in automatically extracting proofs...
I am going to answer the question as if you asked about massive formalization of proofs, not automatic extraction of formal proofs from existing informal proofs written in books, because that's a fair …
10
votes
Salvaging Leibnizian formalism?
You seem to think that synthetic differential geometry only handles squarenil infinitesimals, but this is not so. The generalized Kock-Lawvere axiom allows us to work with infinitesimals of any order, …
- 48.8k
30
votes
What does a theoretical mathematician do?
There are several things that mathematicians do:
teachers of mathematics teach math, and you surely know some of those,
applied mathematicians use their knowledge of mathematics to help engineers, p …