Re: [HoTT] Re: Why do we need judgmental equality?

[Thorsten Altenkirch <Thorsten.Altenkirch@nottingham.ac.uk>]

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You are right that you don't need to explain elements in terms of objects. That doesn't necessarily make it a bad idea.

Not necessarily but I think it is a bad idea. Why do I need to understand objects to understand elements? Why do I have to capture all possible constructions if I just want to talk about some specific ones?

This is something that irritates me about set theory: if I want to say for all natural numbers …, I am actually saying for all sets which turn out to be natural numbers… At least conceptually this is rubbish. Who thinks like this?

Yes and in computer science we think about types sorting untyped programs. But why? Why do I need to think about untyped programs to understand typed ones. Untyped programs are just weird. Do you understand the untyped lambda calculus? It’s syntax is simple but its meaning is a headfuck. It took Dana Scott a while to come up with a mathematical semantics.

I think we should put our mouth where our heart is. Typed objects are easier to understand, they make sense by themselves so lets refer to them. Yes we implement them based in intyped things, trees, strings, bit sequences but we don’t need low level concepts to understand high level concepts!! Yes we need them to implement them but this is another issue.

Thorsten
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