Prospects of teaching/learning elementary math with computed-checked type theory
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
asked Nov 25 at 19:46
Brandon Brown
1987
|
show 3 more comments
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
asked Nov 25 at 19:46
Brandon Brown
1987
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
1
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
|
show 3 more comments
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
asked Nov 25 at 19:46
Brandon Brown
1987
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
proof-verification foundations type-theory homotopy-type-theory univalent-foundations
asked Nov 25 at 19:46
Brandon Brown
1987
asked Nov 25 at 19:46
Brandon Brown
1987
asked Nov 25 at 19:46
Brandon Brown
1987
asked Nov 25 at 19:46
Brandon Brown
1987
asked Nov 25 at 19:46
Brandon Brown
1987
1987
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
1
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
|
show 3 more comments
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
1
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
1
1
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
|
show 3 more comments
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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
1
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