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First, I'm not a set theorist. Second, one of the major motivations for introducing type theory in mathematics was to assist with computer-aided formalization e

First, I'm not a set theorist. Second, one of the major motivations for introducing type theory in mathematics was to assist with computer-aided formalization efforts, which has had some success and has a promising future (e.g. the stuff the Lean community has been doing lately). This is great! I'm not complaining at all. But I do think trumpeting HoTT as a new, improved foundation of mathematics evinces a misunderstanding of what people who work on foundations actually want out of a foundation of mathematics; or even more broadly, the mistaken belief that finding a new foundation of mathematics is an interesting project in the 21st century. Lurie's comment here explains a bit more: https://mathematicswithoutapologies.wordpress.com/2015/05/13/univalent-foundations-no-comment/comment-page-1/#comment-299 https://mathematicswithoutapologies.wordpress.com/2015/05/13... (And clearly, Lurie is not some ignorant traditionalist. There's a substantive disagreement that can't be reduced to people not wanting to learn "your stuff.")

Ericson2314 5y ago
> evinces a misunderstanding of what people who work on foundations actually want out of a foundation of mathematics This is still gatekeeping who gets to use the word "foundations". I will go read https://mathematicswithoutapologies.wordpress.com/2015/05/13 https://mathematicswithoutapologies.wordpress.com/2015/05/13 but my hopes are not high for a sudden breakthrough.
- spekcular 5y ago
  I'm sorry, but this is not gatekeeping; that term has an established meaning and mathematical history, and corresponds to a productive vein of research going back to Gödel (at least). If you'd like to argue that the study of foundations should be expanded to include the things that HoTT people are interested in, that's one thing. But can we at least agree it does not have a ton of relevance to the questions people have traditionally considered in foundational work? (This is almost by definition if we set it up so that it is mutually interpretable with ZFC.)
  - throwaway81523 5y ago
    > going back to Gödel (at least) See, that is the basic misunderstanding here. Foundations now is a wider topic area than it was in Gödel's day. It's like saying that automobiles run on internal combustion engines in a productive vein of manufacturing going back to Henry Ford (at least). Since those days though, there is this guy Elon who you might like to meet...
    - spekcular 5y ago
      I absolutely agree that it's a wider, more sophisticated topic today. This doesn't mean that its main concerns have fundamentally changed. The Maddy article linked in a comment above is extremely clear explanation of this.
      - throwaway81523 5y ago
        I don't see the Maddy link, but anyway try for something written in the current century.
        spekcular 5y ago
        Here, from three years ago: http://www.socsci.uci.edu/~pjmaddy/bio/What%20do%20we%20want%20-%20final http://www.socsci.uci.edu/~pjmaddy/bio/What%20do%20we%20want...
        throwaway81523 5y ago
        Thanks, that looks interesting and I'll try to read it. But it appears to mostly be about philosophy, and it doesn't say much about what (mathematical) logicians are doing today.
        Ericson2314 5y ago
        I did start to read it last night, and finished today. My understanding is that regular MLTT already can do all the bold things Set theory does, and HFTT (and other developments) aims to make "Proof Checker" better realized (no more "setoid hell"). Ultimately what us CS people care about is a unity of theory and practice. What drew us to computers int the first place was trying to make explicit and tangible ideas that previously were only laboriously communicated from one generation of big brains to the next in the ivory tower. In that regards, I would say don't even look HoTT or fancy theory developments. Look at Lean, written by a non-type-theorist, and mathlib, written by a non-constructivist. Lean and Mathilb are plainly plowing a head in ways that no set theory based tool has ever achieved, and covering all the set theory bold points in the process. Perhaps a new more elegant type theory will replace Lean soon, but honestly I am not even worried about that. The important thing is making math much more accessible without loss of rigor with these tools. If better theories can follow rather than lead that methological revolution, that's fine with me, As to Category Theory and essential guidance, I have a simple test :). As mathlib (or its sucesssors) matures, let us see how many proofs are refactored to use category theory! I don't want to poo-poo the deeper connections between type theory and cateogory theory, but merely using lots of category theory "library code" in a type theoretic proof assistant not itself designed with explicit reference to category theory is good enough a success story for me. In conclusion: - Risk Assessment: depends how many "large cardinal" type things we want (unlike the 1970s category theorists I personally don't really care) - Regular type theory: *check* - HoTT: *check* - Set theory: *check*, bonus points for more exotic large cardinals - Generous Arena, Shared Standard, Meta- mathematical Corral - Regular type theory: *check* - Set theory: *check* - HoTT: WIP because culture shock of weird equality - Proof Checker: - Set theory: *FAIL* - Regular type theory: *partial support*, full if we discard all constructive side goals (Lean) - HoTT: *check*! (Very recent cubical results - Essential guidance: - Category theory does best, but can use as "library" rather than "language". good enough for now.
Twisol 5y ago
I think Mike Shulman's side of the discussion in that thread is also very worth reading, and it's somewhat unfair to (perhaps unintentionally) characterize Lurie as the representative of mathematicians trying to talk sense into the homotopy type theorists. They have a very balanced discussion where each respectfully tries to understand where the other is coming from. I'm desperately trying to find a lovely philosophical writeup on the different purposes to which a "foundation" is put, and why different foundationalists want those foundations for distinct purposes. I can't seem to hunt it down (Google has become steadily less helpful over time). But the thrust is, I don't think anybody is claiming that HoTT ought to replace ZFC for reverse mathematics in particular; it's "just" a capturing of some pervasive intuitions for concepts at the ground floor of the formalism itself. It takes a perspective that makes grappling with some concepts easier to approach. If you're already a deft hand with a different (very well-established) framework, as Lurie exhibits throughout, maybe it's worth it to keep using the entirely viable tools you have!
- spekcular 5y ago
  You write: "I don't think anybody is claiming that HoTT ought to replace ZFC for reverse mathematics in particular; it's "just" a capturing of some pervasive intuitions for concepts at the ground floor of the formalism itself. It takes a perspective that makes grappling with some concepts easier to approach. If you're already a deft hand with a different (very well-established) framework, maybe it's worth it to keep using the entirely viable tools you have!" My claim is that "capturing [...] some pervasive intuitions" is not what ZFC is trying to do, or what people in foundations are interested in. Lurie puts the point well when he says a foundation "mean[s] a small number of concepts and assumptions from which the rest of mathematics can be derived." The purposes here are first, to figure out what exactly one must assume to get the rest of mainstream mathematics, and second, to provide a convenient base for doing metamathematics (since the fewer elementary operations/objects you have to reason with, the easier it is to write proofs). Capturing intuitions is completely orthogonal to this endeavor; no one is claiming that set-theoretic translations of everyday mathematical arguments capture the intuitive content of those arguments, because no one one intended ZFC to ever do that in the first place. Similarly, no one is claiming that handing someone a Turing machine that performs quicksort is a great way to capture the intuition behind quicksort, or explain the algorithm in an understandable way. Turing machines exist as a special-purpose construct to probe the limits of computation, not to reproduce or communicate high-level algoritmhic reasoning. ZFC is like a Turing machine for (meta)math.

Ericson2314 5y ago

> evinces a misunderstanding of what people who work on foundations actually want out of a foundation of mathematics This is still gatekeeping who gets to use the word "foundations". I will go read https://mathematicswithoutapologies.wordpress.com/2015/05/13 https://mathematicswithoutapologies.wordpress.com/2015/05/13 but my hopes are not high for a sudden breakthrough.

spekcular 5y ago
I'm sorry, but this is not gatekeeping; that term has an established meaning and mathematical history, and corresponds to a productive vein of research going back to Gödel (at least). If you'd like to argue that the study of foundations should be expanded to include the things that HoTT people are interested in, that's one thing. But can we at least agree it does not have a ton of relevance to the questions people have traditionally considered in foundational work? (This is almost by definition if we set it up so that it is mutually interpretable with ZFC.)
- throwaway81523 5y ago
  > going back to Gödel (at least) See, that is the basic misunderstanding here. Foundations now is a wider topic area than it was in Gödel's day. It's like saying that automobiles run on internal combustion engines in a productive vein of manufacturing going back to Henry Ford (at least). Since those days though, there is this guy Elon who you might like to meet...
  - spekcular 5y ago
    I absolutely agree that it's a wider, more sophisticated topic today. This doesn't mean that its main concerns have fundamentally changed. The Maddy article linked in a comment above is extremely clear explanation of this.
    - throwaway81523 5y ago
      I don't see the Maddy link, but anyway try for something written in the current century.
      - spekcular 5y ago
        Here, from three years ago: http://www.socsci.uci.edu/~pjmaddy/bio/What%20do%20we%20want%20-%20final http://www.socsci.uci.edu/~pjmaddy/bio/What%20do%20we%20want...
        throwaway81523 5y ago
        Thanks, that looks interesting and I'll try to read it. But it appears to mostly be about philosophy, and it doesn't say much about what (mathematical) logicians are doing today.
        Ericson2314 5y ago
        I did start to read it last night, and finished today. My understanding is that regular MLTT already can do all the bold things Set theory does, and HFTT (and other developments) aims to make "Proof Checker" better realized (no more "setoid hell"). Ultimately what us CS people care about is a unity of theory and practice. What drew us to computers int the first place was trying to make explicit and tangible ideas that previously were only laboriously communicated from one generation of big brains to the next in the ivory tower. In that regards, I would say don't even look HoTT or fancy theory developments. Look at Lean, written by a non-type-theorist, and mathlib, written by a non-constructivist. Lean and Mathilb are plainly plowing a head in ways that no set theory based tool has ever achieved, and covering all the set theory bold points in the process. Perhaps a new more elegant type theory will replace Lean soon, but honestly I am not even worried about that. The important thing is making math much more accessible without loss of rigor with these tools. If better theories can follow rather than lead that methological revolution, that's fine with me, As to Category Theory and essential guidance, I have a simple test :). As mathlib (or its sucesssors) matures, let us see how many proofs are refactored to use category theory! I don't want to poo-poo the deeper connections between type theory and cateogory theory, but merely using lots of category theory "library code" in a type theoretic proof assistant not itself designed with explicit reference to category theory is good enough a success story for me. In conclusion: - Risk Assessment: depends how many "large cardinal" type things we want (unlike the 1970s category theorists I personally don't really care) - Regular type theory: *check* - HoTT: *check* - Set theory: *check*, bonus points for more exotic large cardinals - Generous Arena, Shared Standard, Meta- mathematical Corral - Regular type theory: *check* - Set theory: *check* - HoTT: WIP because culture shock of weird equality - Proof Checker: - Set theory: *FAIL* - Regular type theory: *partial support*, full if we discard all constructive side goals (Lean) - HoTT: *check*! (Very recent cubical results - Essential guidance: - Category theory does best, but can use as "library" rather than "language". good enough for now.

Twisol 5y ago

I think Mike Shulman's side of the discussion in that thread is also very worth reading, and it's somewhat unfair to (perhaps unintentionally) characterize Lurie as the representative of mathematicians trying to talk sense into the homotopy type theorists. They have a very balanced discussion where each respectfully tries to understand where the other is coming from. I'm desperately trying to find a lovely philosophical writeup on the different purposes to which a "foundation" is put, and why different foundationalists want those foundations for distinct purposes. I can't seem to hunt it down (Google has become steadily less helpful over time). But the thrust is, I don't think anybody is claiming that HoTT ought to replace ZFC for reverse mathematics in particular; it's "just" a capturing of some pervasive intuitions for concepts at the ground floor of the formalism itself. It takes a perspective that makes grappling with some concepts easier to approach. If you're already a deft hand with a different (very well-established) framework, as Lurie exhibits throughout, maybe it's worth it to keep using the entirely viable tools you have!

spekcular 5y ago
You write: "I don't think anybody is claiming that HoTT ought to replace ZFC for reverse mathematics in particular; it's "just" a capturing of some pervasive intuitions for concepts at the ground floor of the formalism itself. It takes a perspective that makes grappling with some concepts easier to approach. If you're already a deft hand with a different (very well-established) framework, maybe it's worth it to keep using the entirely viable tools you have!" My claim is that "capturing [...] some pervasive intuitions" is not what ZFC is trying to do, or what people in foundations are interested in. Lurie puts the point well when he says a foundation "mean[s] a small number of concepts and assumptions from which the rest of mathematics can be derived." The purposes here are first, to figure out what exactly one must assume to get the rest of mainstream mathematics, and second, to provide a convenient base for doing metamathematics (since the fewer elementary operations/objects you have to reason with, the easier it is to write proofs). Capturing intuitions is completely orthogonal to this endeavor; no one is claiming that set-theoretic translations of everyday mathematical arguments capture the intuitive content of those arguments, because no one one intended ZFC to ever do that in the first place. Similarly, no one is claiming that handing someone a Turing machine that performs quicksort is a great way to capture the intuition behind quicksort, or explain the algorithm in an understandable way. Turing machines exist as a special-purpose construct to probe the limits of computation, not to reproduce or communicate high-level algoritmhic reasoning. ZFC is like a Turing machine for (meta)math.