martin-loef

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Editorial note

Really, Martin-Loef type theory is the mathematics foundation, and the Calculus of Constructions is the computer science foundation.

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Quick take

Really, Martin-Loef type theory is the mathematics foundation, and the Calculus of Constructions is the computer science foundation.

Example sentences

Really, Martin-Loef type theory is the mathematics foundation, and the Calculus of Constructions is the computer science foundation.

There are also foundations of mathematics that are not using FOL (Martin-Loef type theory and successors such as HoTT).

No, HoTT is not based on the Calculus of Constructions, but instead on intensional Martin-Loef type theory with identity typed.

The novelty of HoTT, and only extension over intensional Martin-Loef type-theory, is the univalence axiom.

In other words, Agda, the proof calculus for Martin-Loef type theory.

But anyway, what happened is that Martin-Loef's Intuitionist Type Theory has always been hard to work with, because you have to pull all kinds of funny definitional tricks with equality.

Consider Type Theory by Per Martin-Loef.

Yes, Vladimir Voevodsky says HoTT is based on Martin-Loef type theory, but he discovered HoTT by learning the CoC used in Coq, and is implementing his Univalent Foundations library of proofs in Coq.

Quote examples

Per your comment on "use classes for data modeling" I would like to provide you with two historical facts: it took more than 25 years to get Simula (and OO) types right [1] and it took not more than 8 years for Per Martin-Loef to have intuitionistic type theory right [2].