Prospects of teaching/learning elementary math with computed-checked type theory











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I've read as much as I can understand about type theory and homotopy type theory (HoTT) and it seems like these are very promising directions for re-foundationalizing mathematics in a way where complicated math proofs can be programmed and verified mechanistically. Unfortunately, but not surprisingly, all of this activity going on with type theory/HoTT is at a graduate-level and above.



Would it be useful as an exercise to try to formalize my understanding of relatively elementary subjects such as calculus, linear algebra, geometry, etc in a proof assistant such as Agda and just ignore the more advanced aspects of HoTT? Or is HoTT only useful for super advanced math










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    It might be useful for you. I doubt that it would be useful for "teaching/learning elementary math".
    – Ethan Bolker
    Nov 25 at 19:48










  • In several ways, HoTT is more like normal math than Agda is (e.g. function extensionality holds). Developing any traditional math should only be easier in HoTT than in CIC, say.
    – Derek Elkins
    Nov 25 at 20:30










  • Incidentally, there's nothing particular about type theory that makes machine-checked proofs possible. There are set-theory-based proof assistants such as Mizar. Also ones that are type-theory-based, but on non-dependent type theories, such as Isabelle/HOL. That said, formalizing any non-trivial chunk of mathematics ("elementary" or otherwise) is a challenging task in any of these proof assistants. Formalizing calculus would not be the task of a weekend but of many months, and that's assuming you're already familiar with the system you are using.
    – Derek Elkins
    Nov 25 at 20:37












  • "teaching/learning elementary mathematics" has several centuries of success and resources behind it. Given that, though, you are welcome to try alternate approaches by yourself. Be prepared for a lot of extra very difficult work.
    – Somos
    Nov 25 at 20:41










  • @DerekElkins Surely formalizing all of calculus would be a massive undertaking, but one could formalize the basics in a synthetic way right? If I wanted to formalize an entire field of math from a few foundational axioms, that would of course take a large team of people working a long time. However, in a traditional programming language I can import and use a function that is a black box to me. Can I do the same with a proof assistant? Just add axioms and develop them as proved theorems later, drastically improving the rate of progress?
    – Brandon Brown
    Nov 25 at 20:58

















up vote
0
down vote

favorite












I've read as much as I can understand about type theory and homotopy type theory (HoTT) and it seems like these are very promising directions for re-foundationalizing mathematics in a way where complicated math proofs can be programmed and verified mechanistically. Unfortunately, but not surprisingly, all of this activity going on with type theory/HoTT is at a graduate-level and above.



Would it be useful as an exercise to try to formalize my understanding of relatively elementary subjects such as calculus, linear algebra, geometry, etc in a proof assistant such as Agda and just ignore the more advanced aspects of HoTT? Or is HoTT only useful for super advanced math










share|cite|improve this question


















  • 1




    It might be useful for you. I doubt that it would be useful for "teaching/learning elementary math".
    – Ethan Bolker
    Nov 25 at 19:48










  • In several ways, HoTT is more like normal math than Agda is (e.g. function extensionality holds). Developing any traditional math should only be easier in HoTT than in CIC, say.
    – Derek Elkins
    Nov 25 at 20:30










  • Incidentally, there's nothing particular about type theory that makes machine-checked proofs possible. There are set-theory-based proof assistants such as Mizar. Also ones that are type-theory-based, but on non-dependent type theories, such as Isabelle/HOL. That said, formalizing any non-trivial chunk of mathematics ("elementary" or otherwise) is a challenging task in any of these proof assistants. Formalizing calculus would not be the task of a weekend but of many months, and that's assuming you're already familiar with the system you are using.
    – Derek Elkins
    Nov 25 at 20:37












  • "teaching/learning elementary mathematics" has several centuries of success and resources behind it. Given that, though, you are welcome to try alternate approaches by yourself. Be prepared for a lot of extra very difficult work.
    – Somos
    Nov 25 at 20:41










  • @DerekElkins Surely formalizing all of calculus would be a massive undertaking, but one could formalize the basics in a synthetic way right? If I wanted to formalize an entire field of math from a few foundational axioms, that would of course take a large team of people working a long time. However, in a traditional programming language I can import and use a function that is a black box to me. Can I do the same with a proof assistant? Just add axioms and develop them as proved theorems later, drastically improving the rate of progress?
    – Brandon Brown
    Nov 25 at 20:58















up vote
0
down vote

favorite









up vote
0
down vote

favorite











I've read as much as I can understand about type theory and homotopy type theory (HoTT) and it seems like these are very promising directions for re-foundationalizing mathematics in a way where complicated math proofs can be programmed and verified mechanistically. Unfortunately, but not surprisingly, all of this activity going on with type theory/HoTT is at a graduate-level and above.



Would it be useful as an exercise to try to formalize my understanding of relatively elementary subjects such as calculus, linear algebra, geometry, etc in a proof assistant such as Agda and just ignore the more advanced aspects of HoTT? Or is HoTT only useful for super advanced math










share|cite|improve this question













I've read as much as I can understand about type theory and homotopy type theory (HoTT) and it seems like these are very promising directions for re-foundationalizing mathematics in a way where complicated math proofs can be programmed and verified mechanistically. Unfortunately, but not surprisingly, all of this activity going on with type theory/HoTT is at a graduate-level and above.



Would it be useful as an exercise to try to formalize my understanding of relatively elementary subjects such as calculus, linear algebra, geometry, etc in a proof assistant such as Agda and just ignore the more advanced aspects of HoTT? Or is HoTT only useful for super advanced math







proof-verification foundations type-theory homotopy-type-theory univalent-foundations






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asked Nov 25 at 19:46









Brandon Brown

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  • 1




    It might be useful for you. I doubt that it would be useful for "teaching/learning elementary math".
    – Ethan Bolker
    Nov 25 at 19:48










  • In several ways, HoTT is more like normal math than Agda is (e.g. function extensionality holds). Developing any traditional math should only be easier in HoTT than in CIC, say.
    – Derek Elkins
    Nov 25 at 20:30










  • Incidentally, there's nothing particular about type theory that makes machine-checked proofs possible. There are set-theory-based proof assistants such as Mizar. Also ones that are type-theory-based, but on non-dependent type theories, such as Isabelle/HOL. That said, formalizing any non-trivial chunk of mathematics ("elementary" or otherwise) is a challenging task in any of these proof assistants. Formalizing calculus would not be the task of a weekend but of many months, and that's assuming you're already familiar with the system you are using.
    – Derek Elkins
    Nov 25 at 20:37












  • "teaching/learning elementary mathematics" has several centuries of success and resources behind it. Given that, though, you are welcome to try alternate approaches by yourself. Be prepared for a lot of extra very difficult work.
    – Somos
    Nov 25 at 20:41










  • @DerekElkins Surely formalizing all of calculus would be a massive undertaking, but one could formalize the basics in a synthetic way right? If I wanted to formalize an entire field of math from a few foundational axioms, that would of course take a large team of people working a long time. However, in a traditional programming language I can import and use a function that is a black box to me. Can I do the same with a proof assistant? Just add axioms and develop them as proved theorems later, drastically improving the rate of progress?
    – Brandon Brown
    Nov 25 at 20:58
















  • 1




    It might be useful for you. I doubt that it would be useful for "teaching/learning elementary math".
    – Ethan Bolker
    Nov 25 at 19:48










  • In several ways, HoTT is more like normal math than Agda is (e.g. function extensionality holds). Developing any traditional math should only be easier in HoTT than in CIC, say.
    – Derek Elkins
    Nov 25 at 20:30










  • Incidentally, there's nothing particular about type theory that makes machine-checked proofs possible. There are set-theory-based proof assistants such as Mizar. Also ones that are type-theory-based, but on non-dependent type theories, such as Isabelle/HOL. That said, formalizing any non-trivial chunk of mathematics ("elementary" or otherwise) is a challenging task in any of these proof assistants. Formalizing calculus would not be the task of a weekend but of many months, and that's assuming you're already familiar with the system you are using.
    – Derek Elkins
    Nov 25 at 20:37












  • "teaching/learning elementary mathematics" has several centuries of success and resources behind it. Given that, though, you are welcome to try alternate approaches by yourself. Be prepared for a lot of extra very difficult work.
    – Somos
    Nov 25 at 20:41










  • @DerekElkins Surely formalizing all of calculus would be a massive undertaking, but one could formalize the basics in a synthetic way right? If I wanted to formalize an entire field of math from a few foundational axioms, that would of course take a large team of people working a long time. However, in a traditional programming language I can import and use a function that is a black box to me. Can I do the same with a proof assistant? Just add axioms and develop them as proved theorems later, drastically improving the rate of progress?
    – Brandon Brown
    Nov 25 at 20:58










1




1




It might be useful for you. I doubt that it would be useful for "teaching/learning elementary math".
– Ethan Bolker
Nov 25 at 19:48




It might be useful for you. I doubt that it would be useful for "teaching/learning elementary math".
– Ethan Bolker
Nov 25 at 19:48












In several ways, HoTT is more like normal math than Agda is (e.g. function extensionality holds). Developing any traditional math should only be easier in HoTT than in CIC, say.
– Derek Elkins
Nov 25 at 20:30




In several ways, HoTT is more like normal math than Agda is (e.g. function extensionality holds). Developing any traditional math should only be easier in HoTT than in CIC, say.
– Derek Elkins
Nov 25 at 20:30












Incidentally, there's nothing particular about type theory that makes machine-checked proofs possible. There are set-theory-based proof assistants such as Mizar. Also ones that are type-theory-based, but on non-dependent type theories, such as Isabelle/HOL. That said, formalizing any non-trivial chunk of mathematics ("elementary" or otherwise) is a challenging task in any of these proof assistants. Formalizing calculus would not be the task of a weekend but of many months, and that's assuming you're already familiar with the system you are using.
– Derek Elkins
Nov 25 at 20:37






Incidentally, there's nothing particular about type theory that makes machine-checked proofs possible. There are set-theory-based proof assistants such as Mizar. Also ones that are type-theory-based, but on non-dependent type theories, such as Isabelle/HOL. That said, formalizing any non-trivial chunk of mathematics ("elementary" or otherwise) is a challenging task in any of these proof assistants. Formalizing calculus would not be the task of a weekend but of many months, and that's assuming you're already familiar with the system you are using.
– Derek Elkins
Nov 25 at 20:37














"teaching/learning elementary mathematics" has several centuries of success and resources behind it. Given that, though, you are welcome to try alternate approaches by yourself. Be prepared for a lot of extra very difficult work.
– Somos
Nov 25 at 20:41




"teaching/learning elementary mathematics" has several centuries of success and resources behind it. Given that, though, you are welcome to try alternate approaches by yourself. Be prepared for a lot of extra very difficult work.
– Somos
Nov 25 at 20:41












@DerekElkins Surely formalizing all of calculus would be a massive undertaking, but one could formalize the basics in a synthetic way right? If I wanted to formalize an entire field of math from a few foundational axioms, that would of course take a large team of people working a long time. However, in a traditional programming language I can import and use a function that is a black box to me. Can I do the same with a proof assistant? Just add axioms and develop them as proved theorems later, drastically improving the rate of progress?
– Brandon Brown
Nov 25 at 20:58






@DerekElkins Surely formalizing all of calculus would be a massive undertaking, but one could formalize the basics in a synthetic way right? If I wanted to formalize an entire field of math from a few foundational axioms, that would of course take a large team of people working a long time. However, in a traditional programming language I can import and use a function that is a black box to me. Can I do the same with a proof assistant? Just add axioms and develop them as proved theorems later, drastically improving the rate of progress?
– Brandon Brown
Nov 25 at 20:58

















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