About Logic interview with Giovanni Sambin

You can read notes on Martin-Löf's lecture given in Padua in 1980 here: Giovanni Sambin's notes on Per Martin-Löf's Intuitionistic Type Theory.

My comment:

40:32 Thorsten "You only need the notion of a type... Then I can explain a proposition,..." I remember when you did that a few weeks ago, and you explained it, but did anybody actually understand your explanation? I know I didn't! 🤣

30:38 On intensional and extensional theories and "running them in parallel". I wonder how far you could get by modelling functions with bijections between pairs, i.e. permutations on graphs. I did something like this once in the iterative construction I described in this post: Brady Haran interview with Terence Tao. It produced some very interesting constructions full of amazing symmetries. See also Category Theory May Actually be a Disease and Emma Tye on Martin-Löf Type Theory. What I did was construct something like a closure operator on an abstract algebra of natural transformations which automatically preserved congruences in the process. Unfortunately on February 15th, 2010 I lost all my work leading up to that and I still feel a bit sick when I think about it. 

See https://www.math.unipd.it/~sambin/ and his most recent book Positive Topology A New Practice in Constructive Mathematics:

In dynamic constructivism, mathematics is seen as the result of a dynamic process of interaction between the construction of mathematical entities, by abstraction and by idealization, and their selection according to their efficiency in applications to reality and in the organisation of mathematics itself.

The crucial benefit of this vision is its independence from dogmas and external authorities. A practical consequence is full respect for the diverse areas of mathematics - mainly computation, spatial intuition, deduction, and abstract axiomatic method - without reducing one to another. As a second consequence, a dynamic interaction between different 'epistemological levels' is always active and present, in the development of mathematics in practice, the study of its foundations and its formalisation in a computer language.

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