[Agda] Why dependent type theory?

[James McKinna]

Martin, on Fri, 06 Mar 2020, you wrote:

> In other words, choose your proof assistant as a function of what  
> you want to talk about *and* how you want to talk about it. Martin
>
> On 06/03/2020 21:05, Martin Escardo wrote:
>> The troubling aspect of proof assistants is that they not only  
>> implement proof checking (and definition checking, construction  
>> checking etc.) but that also that each of them proposes a new  
>> foundation of mathematics.
>>
>> Which is sometimes not precisely specified, as it is the case of  
>> e.g. Agda. (Which is why I, as an Agda user, I confine myself to a  
>> well-understood subset of Agda corresponding to a (particular)  
>> well-understood type theory.
>>
>> For mathematically minded users of proof assistants, like myself,  
>> this is a problem. We are not interested in formal proofs per se.  
>> We are interested in what we are talking about, with rigorously  
>> stated assumptions about our universe of discourse.

Martin,

I largely agree with you (at least in terms of my practice as an Agda  
user; users of other proof assistants should look to their own  
consciences regarding well-known and less-well-known sources of  
potential inconsistency in (implementations of) their favourite  
foundations), but I had a mind a separation even at the level you  
allude to:

-- that mathematicians are not/need not be bound by the  
restrictions/stipulations of a given foundational system; rather that  
they develop appropriate language/meta-language for their own eventual  
mathematical needs, and that such processes are very much historically  
bound, and subject to the dynamics of paradigm change in terms of the  
(greater) explanatory power of the paradigm within which they work;  
the example of Grothendieck developing a raft of categorical  
techniques in order to be able to successfully carry out his research  
programme in algebraic geometry being only one of the most  
familiar/famous/notorious; Kevin's and others' use of lean in the  
formalisation of perfectoid spaces suggest another such case, where  
the tool(s), and in particular their expressivity wrt the concepts  
being studied, made them more immediately ready-at-hand than any mere  
reduction to set theory (even if that were a possibility-in-principle  
with ZFC/TG implementations in Isabelle, Mizar or Egal);

-- that concern for (consistency of such) foundations has,  
historically at least, typically lagged behind the mathematical  
developments; though against that, one might say that the Grothendieck  
school were precisely concerned with developing such foundations,  
hand-in-hand with the dazzling mathematics they carried out therein  
(an example of a much longer gap between the mathematical development,  
and putting it on satisfactory foundations might lie in the history of  
distributions from Heaviside to Schwartz; or the foundations of  
geometry after the discovery of non-Euclidean geometries...); the  
contemporary frenzy of activity on a number of fronts in  
higher-dimensional (categorical) algebra via homotopy type theory  
suggests a similar interplay between the development of  
'mathematics-of' and 'foundations-for'.

The Kolmogorov paper (as well as, for example, Lawvere's insistence on  
(a) 'logic' as somehow a conceptual secondary notion to the  
categorical structure which supports it, and indeed Brouwer's  
conception of logic as part of mathematics (and not the other way  
round, which perhaps seems strange to those who see the field of proof  
assistants as somehow the triumph of the logicist/formalist  
programme)) suggests that the logic of mathematical 'problems' (and  
their solutions) emerges from the domain of such problems. (What K  
points out as a "remarkable fact" is that such logic turns out to be  
(Heyting's formalisation of) Brouwer's intuitionistic logic, and such  
congruence would repeat itself 40 years later with the internal logic  
of toposes. )

It almost seems (to me at least), that the development and use of  
proof assistants has refocused our attention on the possibility that  
the 'Grundlagenstreit' was not a single historical moment, with a  
right/wrong outcome, but part of the dynamics of mathematical  
development: to each age its mathematics, and to each such  
mathematics, not only its freedom, but also its appropriate  
foundations. But that's not to say that we can, or should, be blas'e  
about such things. I'm sorry if my earlier post suggested otherwise.

James.

-- 
The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336.