From: Nathan Carter <nathancarter5@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: [HoTT] my first 3 questions about HoTT
Date: Thu, 20 Jun 2019 09:16:35 -0700 (PDT) [thread overview]
Message-ID: <d62ccb9e-10d7-4884-bb09-aa1cce32bcb2@googlegroups.com> (raw)
[-- Attachment #1.1: Type: text/plain, Size: 3184 bytes --]
Hello, HoTT community.
I've learned a bit about HoTT in bits of spare time over the past few
years, and have come up with some questions I can't answer on my own. It
was suggested that I ask them on this list. I will start with a few small
questions, and if anyone in the community here has time to answer them,
then I'll continue with others as needed. Thank you in advance for any
help you can give.
(Where I'm coming from: I'm a mathematician; my dissertation was on
intermediate logics, but I haven't focused on logic much for the past 15
years, instead doing mathematical software and some applied mathematics. I
have a passion for clear exposition, so as I learn about HoTT, I process it
by writing detailed notes to myself, explaining it as clearly as I can.
When I can't explain something clearly, I flag it as a question. I'm
bringing those questions here.)
Here are three to start.
1. Very early in the HoTT book, it talks about the difference between
types and sets, and says that HoTT encourages us to see sets as spaces.
Yet in a set of lecture videos Robert Harper did that I watched on YouTube
(which also seem to have disappeared, so I cannot link to them here), he
said that Extensional Type Theory takes Intuitionistic Type Theory in a
different direction than HoTT does, formalizing the idea that types are
sets. Why does the HoTT book not mention this possibility? Why does ETT
not seem to get as much press as HoTT?
2. When that same text introduces judgmental equality, it claims that it
is a decidable relation. It does not seem to prove this, and so I
suspected that perhaps the evidence was in Appendix A, where things are
done more formally (twice, even). The first of these two formalisms places
some restrictions on how one can introduce new judgmental equalities, which
seem sufficient to guarantee its decidability, but at no point is an
algorithm for deciding it given. Is the algorithm simply to apply the only
applicable rule over and over to reduce each side, and then compare for
exact syntactic equality?
3. One of my favorite things about HoTT as a foundation for mathematics
actually comes just from DTT: Once you've formalized pi types, you can
define all of logic and (lots of) mathematics. But then the hierarchy of
type universes seem to require that we understand the natural numbers,
which is way more complicated than just pi types, and thus highly
disappointing to have to bring in at a foundational level. Am I right to
be disappointed about that or am I missing something?
Thanks in advance for any help you have time and interest to provide!
Nathan Carter
--
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/d62ccb9e-10d7-4884-bb09-aa1cce32bcb2%40googlegroups.com.
For more options, visit https://groups.google.com/d/optout.
[-- Attachment #1.2: Type: text/html, Size: 3666 bytes --]
next reply other threads:[~2019-06-20 16:16 UTC|newest]
Thread overview: 7+ messages / expand[flat|nested] mbox.gz Atom feed top
2019-06-20 16:16 Nathan Carter [this message]
2019-06-20 16:37 ` Cory Knapp
2019-06-20 16:39 ` Thorsten Altenkirch
2019-06-20 16:56 ` Michael Shulman
2019-06-20 23:11 ` Nathan Carter
2019-06-21 1:04 ` Michael Shulman
2019-06-20 16:48 ` Ali Caglayan
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=d62ccb9e-10d7-4884-bb09-aa1cce32bcb2@googlegroups.com \
--to=nathancarter5@gmail.com \
--cc=HomotopyTypeTheory@googlegroups.com \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).