4 ms·

In what sense is type theory "easier to work with"? For what purpose? ZFC exists largely because it's a useful framework for proving meta-mathematical theorems.

In what sense is type theory "easier to work with"? For what purpose? ZFC exists largely because it's a useful framework for proving meta-mathematical theorems. In this sense, it is extremely convenient to work with. Also, I'm not sure what "all of this stuff" refers to here re: your comment about inaccessible cardinals. You don't need large cardinals to do 99.99% of modern mathematics (essentially, everything but set/proof theory). ZFC is good enough (and you can usually get away with something weaker).

Ericson2314 5y ago
Frankly we are sick of set theorist traditionalists complaining about type theory, category theory, and other new things without actually bothering to understand them at all. After multiple decades, it's clear you all are never going to bother to learn our stuff nearly as well as we bothered to learn your stuff. This "debate" is thus completely unproductive without any good agreed-upon foundation to argue form. (How ironic!)
- spekcular 5y ago
  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

Ericson2314 5y ago

Frankly we are sick of set theorist traditionalists complaining about type theory, category theory, and other new things without actually bothering to understand them at all. After multiple decades, it's clear you all are never going to bother to learn our stuff nearly as well as we bothered to learn your stuff. This "debate" is thus completely unproductive without any good agreed-upon foundation to argue form. (How ironic!)

spekcular 5y ago
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