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A substantial portion of this text appears to be concerned with traditional Zermelo–Fraenkel set theory (and its extensions). I have gradually come to believe

A substantial portion of this text appears to be concerned with traditional Zermelo–Fraenkel set theory (and its extensions). I have gradually come to believe that ZF theory has received a disproportionate amount of attention on account of the fact that it serves as the "official" foundations of mathematics, but that it is not an especially beautiful, or useful, structure. I believe that ZF theory is an interesting object, worthy of mathematical study, but _not_ the best candidate for the foundations of mathematics! I am very happy to see that this year's Chauvenet Prize [1] was won by Tom Leinster's "Rethinking set theory", in which he highlights that Lawvere set theory looks like a much better candidate. I cannot do better than recommend you look at Leinster's superb article. [1] MAA, Chauvenet Prize, https://www.maa.org/programs-and-communities/member-communities/maa-awards/writing-awards/chauvenet-prizes https://www.maa.org/programs-and-communities/member-communit... [2] Leinster, T., "Rethinking set theory", https://arxiv.org/abs/1212.6543 https://arxiv.org/abs/1212.6543

ernst_klim 7y ago
Speaking of the foundations: is intuitionism still alive? How is constructive math doing nowadays? Do mathematicians embrace or reject it? Did the recent HoTT change their attitude towards constructive math?
- ocfnash 7y ago
  I'm not at all qualified to answer such a broad question but I'll share my amateur opinions for what they're worth. I believe people are gradually coming round to the point of view that constructive mathematics (including intuitionism) is a useful generalization of classical mathematics, rather than a handicap. Proof assistants, while still fairly niche, continue to get more attention and the fact that constructive mathematics tends to give "proofs that compute" helps. For what it's worth, I waffled on about this a little a few months ago right here: https://news.ycombinator.com/item?id=18404914 https://news.ycombinator.com/item?id=18404914 In my opinion HoTT is the most interesting subject in this area but we have yet to figure out exactly what its role is, and the extent to which it should serve in foundations. However the fact that it is a deep theory with significant value independent of its foundational role must surely help the constructive point of view. I think it's fair to say "Book HoTT" is not fully constructive since univalence is an axiom but this seems to have been resolved with the introduction of cubical type theory. I also suspect the majority of mathematicians still spend very little time thinking about foundations.
- maxiepoo 7y ago
  Constructive mathematics is alive and well for 2 reasons. 1. The type theorists/homotopy theorists that are exploring homotopy/cubical type theory as a foundational approach. 2. The objective fact that in general the internal language of a topos (in general) does not validate the law of excluded middle or the axiom of choice. This means that if you prove a result while using constructive logic you can apply it to a broader class of models than classical logic. This viewpoint of logic makes the whole "argument" between constructive and classical logic seem very silly: it's like arguing whether or not it is "true" that groups are abelian. Some are and some aren't! Intuisionism as in Brouwerian intuitionism with choice sequences etc that disprove LEM is still a niche subject, especially foundationally but is still being explored, with the help of understanding of sheaf models that allow it to be studied from a classical perspective.
- Gondolin 7y ago
  Oh yes, it is very much alive. For instance any topos (with a nno) is a model of intuitionist set theory. And since topos are a very natural type of categories (a topos= acategory which has finite limits and a power object), intuitionist logic is important in category theory. Two examples: in algebraic geometry, mathematicians use schemes (introduced by Grothendieck), which have allowed to make tremendous progress. But from the point of view of intuitionist logic, a scheme is just a ring, and a quasi-coherent sheave is just a module over this ring (in more details they are ring objects and module objects in the big Zariski topos). In synthetic differential geometry, one use a certain intuitionist topos to formalize intuitive aspects of reasoning with infinitesimal quantities (without relying on delta and epsilons). Note that non standard analysis also allows to work with infinitemismal quantities (by using ultraproducts), but has the same power has standard analysis (and in particular is classical). Synthetic differential geometry is intuitionnist, so is less powerful, but on the other hands its theorems are automatically valid for more models (complex models, formal smooth schemes...). In SDG, your reals R are a field with an (actually several) element epsilon !=0 (the infinitesimals) and such that epsilon^2=0 (that's why this only work in intuitionist logic, in classical logic R being a field would imply that epsilon=0). In this model every function f: R->R is differentiable, and we have f(x+epsilon)=f(x)+f'(x) epsilon. In proof theory (like coq) one work with Martin Lof type theory, which is the type theory of topoi (and so is intuitionist). A recent development is to work with homotopy type theory instead, the type theory of \infty-topos. This is more powerful (and can be used as fundations for all of maths), and solve real problems, like correctly defining equality (equality is actually a tricky concept, for instance it may not be the case in the prover's theory that "for any two functions f and g such that forall x, f(x)=g(x); then f=g", but this is obviously a feature we want.) On the other hand it is quite a bit trickier to define (we need (infty,1)-categories).
  - Gondolin 7y ago
    On the other hand, constructive set theory is still rather a niche part of mathematics (this may change with the recent development of HOTT). It is hard not to use the LEM (and I am not good at it, I am not an expert on this subject. Expert say it gets easier over time to get used to not use LEM). For instance, in the intuitionist reals, from "x >= 0 and x != 0" you cannot conclude that "x != 0". To prove x>0 you would need to exhibit a natural number n such that x>=1/n (so in other words you need a constructive proof that x>0). But this is not always possible, you have models of R where there is an x != 0 which is smaller than all the 1/n. There is however one nice feature of intuitionist theory: every geometric sentence which is provable in classical theory is actually true in intuitionist theory. So if you have a geometric sentence (https://ncatlab.org/nlab/show/geometric+theory https://ncatlab.org/nlab/show/geometric+theory) you can use LEM in your proofs, the proof will still be valid intuitionally!
pfortuny 7y ago
Thanks for the links!
sidusknight 7y ago
What's your background in math?
- ocfnash 7y ago
  I am a geometer by training.
nextos 7y ago
Tangential question. Is there a mathematics curriculum out there that focuses on constructive foundations? Or is it possible to assemble one? I feel this would be much more adequate for then following up with pure CS studies on topics such as [1-6]. Classical math bootcamps, like Math 55, are typically algebra plus analysis and they don't even focus too much on classical foundations. [1] https://www.elsevier.com/books/lectures-on-the-curry-howard-isomorphism/sorensen/978-0-444-52077-7 https://www.elsevier.com/books/lectures-on-the-curry-howard-... [2] https://softwarefoundations.cis.upenn.edu/ https://softwarefoundations.cis.upenn.edu/ [3] https://www.cis.upenn.edu/~bcpierce/tapl/ https://www.cis.upenn.edu/~bcpierce/tapl/ [4] https://www.springer.com/gb/book/9783540654100 https://www.springer.com/gb/book/9783540654100 [5] http://www.concrete-semantics.org/ http://www.concrete-semantics.org/ [6] http://adam.chlipala.net/frap/ http://adam.chlipala.net/frap/
- hackermailman 7y ago
  Besides these lecture notes http://www2.masfak.ni.ac.rs/cmfp2013/Nis%20lecture%20170113.pdf http://www2.masfak.ni.ac.rs/cmfp2013/Nis%20lecture%20170113.... I haven't found a curriculum. This is one of the goals of Robert Harper et. all involved in type theory research https://existentialtype.wordpress.com/2018/01/15/popl-2018-tutorial/ https://existentialtype.wordpress.com/2018/01/15/popl-2018-t... There is a guy however with a rational foundations of Math curriculum, doing algebraic calculus and rational Trig that I find easy to work with (ie: building a library) https://www.youtube.com/user/njwildberger/playlists https://www.youtube.com/user/njwildberger/playlists
- throwawaymath 7y ago
  I'm not familiar with any curriculum focusing on constructivist foundations. Out of curiosity, why do you feel constructivist foundations would be more complementary to theoretical computer science?
  - logicchains 7y ago
    Constructive maths is generally computable maths, which means less worry about whether whatever you just did is computable or not.
triska 7y ago
Thank you for the links! As stated in Tom Leinster's paper that you link to, both ZFC and Lawvere set theory are first-order theories: "In logical terminology, both axiomatizations [i.e., ZFC and Lawvere set theory] are simply first-order theories." This means that with both theories, we cannot categorically define important concepts such as the natural numbers, due to the compactness theorem and its consequence, the upward Löwenheim-Skolem theorem that hold in first-order logic: https://en.wikipedia.org/wiki/Categorical_theory https://en.wikipedia.org/wiki/Categorical_theory https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_... Also, the downward Löwenheim-Skolem theorem leads to Skolem's paradox, because ZFC proves (as we expect from a set theory) Cantor's theorem, and at the same time we know that if ZFC is consistent, then it has a countable model: https://en.wikipedia.org/wiki/Skolem%27s_paradox https://en.wikipedia.org/wiki/Skolem%27s_paradox So, even the rather basic notion of countability is relative. As long as you stay within first-order predicate logic, a theory of sets will run into such issues. From what I see, such issues are often "resolved" when working with ZFC by stating explicitly that a very specific (i.e., the "intended") model is meant, with the caveat that, from a logical perspective, we do not even know whether the theory is consistent and hence whether such a model in fact exists. Thus, even though ostensibly working "in" ZFC, a meta-logical statement about the intended models seems often necessary that cannot be formulated in first-order logic because with first-order predicate logic, we cannot control the cardinality of infinite models. Do other considered set theories offer any improvements over ZFC in these specific respects?
- ocfnash 7y ago
  Your remarks bring me close to the border of my limited knowledge of model theory so I must be careful with my response. I believe the answer to your question: > Do other considered set theories offer any improvements over ZFC in these specific respects? is "no", and that you essentially answer this yourself with the following paraphrasing of Löwenheim–Skolem: > As long as you stay within first-order predicate logic, a theory of sets will run into such issues. So if we opt Lawvere set theory I think we just do the same thing as if we had opted for ZCF, i.e., just assume we have a model (perhaps provided by some higher order logical system). However in my opinion, we do have a theory with a much more intuitive set of axioms.
throwawaymath 7y ago
I'm inclined to agree with you, with a few caveats. I do think ZFC isn't the superlative candidate for foundations. That being said, most mathematicians don't really think about the foundations of mathematics often (if ever). They use ZFC's set theory because it's a succinct, versatile and useful notation which is frankly very easy to learn. In that sense there's a compelling argument that a revised set theory might be called for, in the sense that it could be a better fit for the way most mathematicians use set theory in practice. But since Leinster's set theory is axiomatically equivalent to ZFC without regularity, there's also a compelling argument that it's not worth it unless it's significantly more expedient or easy to learn.

ernst_klim 7y ago

Speaking of the foundations: is intuitionism still alive? How is constructive math doing nowadays? Do mathematicians embrace or reject it? Did the recent HoTT change their attitude towards constructive math?

ocfnash 7y ago
I'm not at all qualified to answer such a broad question but I'll share my amateur opinions for what they're worth. I believe people are gradually coming round to the point of view that constructive mathematics (including intuitionism) is a useful generalization of classical mathematics, rather than a handicap. Proof assistants, while still fairly niche, continue to get more attention and the fact that constructive mathematics tends to give "proofs that compute" helps. For what it's worth, I waffled on about this a little a few months ago right here: https://news.ycombinator.com/item?id=18404914 https://news.ycombinator.com/item?id=18404914 In my opinion HoTT is the most interesting subject in this area but we have yet to figure out exactly what its role is, and the extent to which it should serve in foundations. However the fact that it is a deep theory with significant value independent of its foundational role must surely help the constructive point of view. I think it's fair to say "Book HoTT" is not fully constructive since univalence is an axiom but this seems to have been resolved with the introduction of cubical type theory. I also suspect the majority of mathematicians still spend very little time thinking about foundations.
maxiepoo 7y ago
Constructive mathematics is alive and well for 2 reasons. 1. The type theorists/homotopy theorists that are exploring homotopy/cubical type theory as a foundational approach. 2. The objective fact that in general the internal language of a topos (in general) does not validate the law of excluded middle or the axiom of choice. This means that if you prove a result while using constructive logic you can apply it to a broader class of models than classical logic. This viewpoint of logic makes the whole "argument" between constructive and classical logic seem very silly: it's like arguing whether or not it is "true" that groups are abelian. Some are and some aren't! Intuisionism as in Brouwerian intuitionism with choice sequences etc that disprove LEM is still a niche subject, especially foundationally but is still being explored, with the help of understanding of sheaf models that allow it to be studied from a classical perspective.
Gondolin 7y ago
Oh yes, it is very much alive. For instance any topos (with a nno) is a model of intuitionist set theory. And since topos are a very natural type of categories (a topos= acategory which has finite limits and a power object), intuitionist logic is important in category theory. Two examples: in algebraic geometry, mathematicians use schemes (introduced by Grothendieck), which have allowed to make tremendous progress. But from the point of view of intuitionist logic, a scheme is just a ring, and a quasi-coherent sheave is just a module over this ring (in more details they are ring objects and module objects in the big Zariski topos). In synthetic differential geometry, one use a certain intuitionist topos to formalize intuitive aspects of reasoning with infinitesimal quantities (without relying on delta and epsilons). Note that non standard analysis also allows to work with infinitemismal quantities (by using ultraproducts), but has the same power has standard analysis (and in particular is classical). Synthetic differential geometry is intuitionnist, so is less powerful, but on the other hands its theorems are automatically valid for more models (complex models, formal smooth schemes...). In SDG, your reals R are a field with an (actually several) element epsilon !=0 (the infinitesimals) and such that epsilon^2=0 (that's why this only work in intuitionist logic, in classical logic R being a field would imply that epsilon=0). In this model every function f: R->R is differentiable, and we have f(x+epsilon)=f(x)+f'(x) epsilon. In proof theory (like coq) one work with Martin Lof type theory, which is the type theory of topoi (and so is intuitionist). A recent development is to work with homotopy type theory instead, the type theory of \infty-topos. This is more powerful (and can be used as fundations for all of maths), and solve real problems, like correctly defining equality (equality is actually a tricky concept, for instance it may not be the case in the prover's theory that "for any two functions f and g such that forall x, f(x)=g(x); then f=g", but this is obviously a feature we want.) On the other hand it is quite a bit trickier to define (we need (infty,1)-categories).
- Gondolin 7y ago
  On the other hand, constructive set theory is still rather a niche part of mathematics (this may change with the recent development of HOTT). It is hard not to use the LEM (and I am not good at it, I am not an expert on this subject. Expert say it gets easier over time to get used to not use LEM). For instance, in the intuitionist reals, from "x >= 0 and x != 0" you cannot conclude that "x != 0". To prove x>0 you would need to exhibit a natural number n such that x>=1/n (so in other words you need a constructive proof that x>0). But this is not always possible, you have models of R where there is an x != 0 which is smaller than all the 1/n. There is however one nice feature of intuitionist theory: every geometric sentence which is provable in classical theory is actually true in intuitionist theory. So if you have a geometric sentence (https://ncatlab.org/nlab/show/geometric+theory https://ncatlab.org/nlab/show/geometric+theory) you can use LEM in your proofs, the proof will still be valid intuitionally!

pfortuny 7y ago

Thanks for the links!

sidusknight 7y ago

What's your background in math?

ocfnash 7y ago
I am a geometer by training.

nextos 7y ago

Tangential question. Is there a mathematics curriculum out there that focuses on constructive foundations? Or is it possible to assemble one? I feel this would be much more adequate for then following up with pure CS studies on topics such as [1-6]. Classical math bootcamps, like Math 55, are typically algebra plus analysis and they don't even focus too much on classical foundations. [1] https://www.elsevier.com/books/lectures-on-the-curry-howard-isomorphism/sorensen/978-0-444-52077-7 https://www.elsevier.com/books/lectures-on-the-curry-howard-... [2] https://softwarefoundations.cis.upenn.edu/ https://softwarefoundations.cis.upenn.edu/ [3] https://www.cis.upenn.edu/~bcpierce/tapl/ https://www.cis.upenn.edu/~bcpierce/tapl/ [4] https://www.springer.com/gb/book/9783540654100 https://www.springer.com/gb/book/9783540654100 [5] http://www.concrete-semantics.org/ http://www.concrete-semantics.org/ [6] http://adam.chlipala.net/frap/ http://adam.chlipala.net/frap/

hackermailman 7y ago
Besides these lecture notes http://www2.masfak.ni.ac.rs/cmfp2013/Nis%20lecture%20170113.pdf http://www2.masfak.ni.ac.rs/cmfp2013/Nis%20lecture%20170113.... I haven't found a curriculum. This is one of the goals of Robert Harper et. all involved in type theory research https://existentialtype.wordpress.com/2018/01/15/popl-2018-tutorial/ https://existentialtype.wordpress.com/2018/01/15/popl-2018-t... There is a guy however with a rational foundations of Math curriculum, doing algebraic calculus and rational Trig that I find easy to work with (ie: building a library) https://www.youtube.com/user/njwildberger/playlists https://www.youtube.com/user/njwildberger/playlists
throwawaymath 7y ago
I'm not familiar with any curriculum focusing on constructivist foundations. Out of curiosity, why do you feel constructivist foundations would be more complementary to theoretical computer science?
- logicchains 7y ago
  Constructive maths is generally computable maths, which means less worry about whether whatever you just did is computable or not.

triska 7y ago

Thank you for the links! As stated in Tom Leinster's paper that you link to, both ZFC and Lawvere set theory are first-order theories: "In logical terminology, both axiomatizations [i.e., ZFC and Lawvere set theory] are simply first-order theories." This means that with both theories, we cannot categorically define important concepts such as the natural numbers, due to the compactness theorem and its consequence, the upward Löwenheim-Skolem theorem that hold in first-order logic: https://en.wikipedia.org/wiki/Categorical_theory https://en.wikipedia.org/wiki/Categorical_theory https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_... Also, the downward Löwenheim-Skolem theorem leads to Skolem's paradox, because ZFC proves (as we expect from a set theory) Cantor's theorem, and at the same time we know that if ZFC is consistent, then it has a countable model: https://en.wikipedia.org/wiki/Skolem%27s_paradox https://en.wikipedia.org/wiki/Skolem%27s_paradox So, even the rather basic notion of countability is relative. As long as you stay within first-order predicate logic, a theory of sets will run into such issues. From what I see, such issues are often "resolved" when working with ZFC by stating explicitly that a very specific (i.e., the "intended") model is meant, with the caveat that, from a logical perspective, we do not even know whether the theory is consistent and hence whether such a model in fact exists. Thus, even though ostensibly working "in" ZFC, a meta-logical statement about the intended models seems often necessary that cannot be formulated in first-order logic because with first-order predicate logic, we cannot control the cardinality of infinite models. Do other considered set theories offer any improvements over ZFC in these specific respects?

ocfnash 7y ago
Your remarks bring me close to the border of my limited knowledge of model theory so I must be careful with my response. I believe the answer to your question: > Do other considered set theories offer any improvements over ZFC in these specific respects? is "no", and that you essentially answer this yourself with the following paraphrasing of Löwenheim–Skolem: > As long as you stay within first-order predicate logic, a theory of sets will run into such issues. So if we opt Lawvere set theory I think we just do the same thing as if we had opted for ZCF, i.e., just assume we have a model (perhaps provided by some higher order logical system). However in my opinion, we do have a theory with a much more intuitive set of axioms.

throwawaymath 7y ago

I'm inclined to agree with you, with a few caveats. I do think ZFC isn't the superlative candidate for foundations. That being said, most mathematicians don't really think about the foundations of mathematics often (if ever). They use ZFC's set theory because it's a succinct, versatile and useful notation which is frankly very easy to learn. In that sense there's a compelling argument that a revised set theory might be called for, in the sense that it could be a better fit for the way most mathematicians use set theory in practice. But since Leinster's set theory is axiomatically equivalent to ZFC without regularity, there's also a compelling argument that it's not worth it unless it's significantly more expedient or easy to learn.