Mathematics and Computation

A blog about mathematics for computers

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Posts in the year 2021

Proof assistant communities have grown quite a bit lately. They have active Zulip chats: Lean, Coq, Agda, Isabelle. These are good for discussions, but less so for knowledge accumulation and organization, and are not indexed by the search engines.

I have therefore created a proposal for a new “Proof assistants” StackExchange site. I believe that such a site would complement very well various Zulips dedicated to specific proof assistants. If you favor the idea, please support it by visiting the proposal and

To pass the first stage, we need 60 followers and 40 questions with at least 10 votes to proceed to the next stage.

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Here are the slides of my talk "The dawn of formalized mathematics" from the 8th European Congress of Mathematics, which is taking place online and in Protorož, Slovenia, from June 20 to 26, 2021:

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A while ago, my former student Chuangjie Xu and I computed an integer using a sheaf topos. For that purpose,

  1. we developed our mathematics constructively,
  2. we formalized our mathematics in Martin-Löf type theory, in Agda notation,
  3. we pressed a button, and
  4. after a few seconds we saw the integer we expected in front of us.

Well, it was a few seconds for the computer in steps (3)-(4), but three years for us in steps (1)-(2).

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This is joint work with Marc Bezem, Thierry Coquand, Peter Dybjer.

We use the Burali-Forti argument to show that, in homotopy type theory and univalent foundations, the embedding $$ \mathcal{U} \to \mathcal{U}^+$$ of a universe $\mathcal{U}$ into its successor $\mathcal{U}^+$ is not an equivalence. We also establish this for the types of sets, magmas, monoids and groups. The arguments in this post are also written in Agda.

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It is my pleasure to have the opportunity to speak at the University of Wisconsin Logic seminar. The hosts are graciously keeping the seminar open for everyone. I will speak about a favorite topic of mine, synthetic computability.

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