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) 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 indexThread overview:7+ messages / expand[flat|nested] mbox.gz Atom feed top2019-06-20 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

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