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

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

On 14/10/2017, 10:37, "homotopyt...@googlegroups.com on behalf of
Andrej Bauer" <homotopyt...@googlegroups.com on behalf of
andrej...@andrej.com> wrote:

>> In our recent work on intrinsic
>> syntax of Type Theory we present a typed syntax that never refers to
>> preterms. In this framework theories with coercions cannot be expressed
>> and the syntax is the initial algebra of the corresponding notion of
>> algebras.
>
>This is quite cool, but I'd just like to point out that there are
>levels of abstraction, some of which cannot be ignored.
>
>At a very banal level, we need to know how to convert concrete syntax
>into abstract syntax.
>
>At a slightly less banal level, we have to admit that preterms
>(expressed as abstract syntax) are unavoidable. An important aspect of
>type theory revolves around making sure that the thing written down
>makes sense. The "thing written down" is a preterm (something that
>survived parsing but has yet to be type checked).

Indeed, but it isn't even preterms but strings we really see (or bits?).
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.

>
>There is also the question of breaking the vicious circle: we express
>type theory inside type theory, but where did the original type theory
>come from? I think it is quite reasonable to take the possition that
>type theory is primitive, but it is also equally reasonable to take
>the position that one should analyse how type theory arises from a
>more concrete setup, for instance, there is some value in knowing that
>the syntactic models based on concrete syntax are initial for type
>theory.

The original type theory is in our mind and we may use english or any
other natural language to express it and then we want to be more precise
and use the type theory in our minds to explain the type theory we
implement, which then in turn can be implemented. No vicious circle.

Thorsten


>
>With kind regards,
>
>Andrej
>
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