Re: [HoTT] A small observation on cumulativity and the failure of initiality

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

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On 16 Oct 2017, at 00:49, Andrej Bauer <andrej...@andrej.com<mailto:andrej...@andrej.com>> wrote:
The theorem that beta-reduction preserves matching brackets is  mixing
levels of abstraction in bad way. As we are happy to reason about terms as
trees knowing that we can parse strings into trees, we should think about
intrinsic terms (what Thomas calls "derivations") knowing that by type
checking and scope checking we can convert trees into derivations.
Assigning meaning to purely syntactic entities is a confusing levels of
abstractions in the same way as the matching brackets theorem.

I do not understand these remarks. Did I say somewhere that we should
worry about matching brackets in beta-reductions, or that we should do
semantics at the wrong level of abstraction?

Sorry if this was not very clear. I was trying to make an analogy. Some novice programmers seem to think that strings are the only data structure. From this point of view we may define beta reduction as an operation on strings and then prove that it preserves matching brackets.

I hope we all agree that this makes little sense. Instead beta reduction should be defined on trees or even better alpha-equivalence classes of trees. Analogously we should think of derivations and not preterms when studying the semantics of Type Theory.




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