I just had a chance to catch up and watch the video whichprompted this disucussion last night. I'm a huge fan of the ideaof trying to understand and make precise the notion of "highercategory with families" and what that might say about the syntaxof type theory. So my question is probably mostly directed atThorsten, but I am curious to hear other people's responses aswell.
I guess my question is pretty simple: why should we insist,as Thorsten seems to, that the "intrinsic" syntax of typetheory form a set?
I can, I think, anticipate the first response: well, we want tohave a type-checking algorithm. And, in light of the conversionrule, this typically means that we will have to reduce typeexpressions to a normal form and check if they are equal. Hence,if we don't have decidability of the syntax, we cannot havedecidability of typechecking. Am I right that this is the principlemotivation for having decidable syntax?
But, then, this seems to be a statement about *extrinsic* syntax.
If,as Thorsten advocates, we somehow manage to produce a highlystructured, internal description of the syntax of type theory,then typechecking for this syntax is, by definition, unnecessary! After all,the whole point is that in such a setup, only the well-typed termseven make sense (i.e., are typeable in the meta-language).
So from an internal perspective, I cannot think of any reason toinsist on decidability. And consequently, insisting that aninternal higher category with families be univalent does not seemin any way strange to me.
But maybe there is some other objection that I'm not seeing?
On Thursday, 31 May 2018 20:02:51 UTC+1, Alexander Kurz wrote:
> On 31 May 2018, at 12:05, Michael Shulman <shu...@sandiego.edu> wrote:
>
> It sounds like Thorsten and are both starting to repeat ourselves, so
> we should probably spare the patience of everyone else on the list
> pretty soon. I'll just make my own hopefully-final point by saying
> that if "properties of the typed algebraic syntax" can imply that the
> untyped stuff we write on the page has a *unique* typed denotation,
> independent of a particular typechecking algorithm, as mentioned in my
> last email, then I'll (probably) be satisfied.
I am interested in this question of translating the untyped stuff we write on the page into type theory.
To give a concrete example of what I am thinking of as untyped, but nevertheless conceptual and structural mathematics, I would point at Tom Leinster’s elegant description of the solution to the problem of Buffon’s needle, see the first paragraphs of the section “What is category theory?” at https://golem.ph.utexas.edu/category/2010/03/a_perspective_on_higher_catego.html
I call this argument type free because I see no obvious or canonical way to make the types precise enough in order to implement the proof in, say, Agda. Even if this can be done, it is still important that mathematicians can discuss this argument first without having to make the types precise. So there will always be mathematics outside of type theory.
--I don't understand why you call this argument untyped. Do you feel that a formalization in set theory, which is untyped, would be easier than a formalization in type theory? How is untypedness helping with this argument?
Martin
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