Re: Homotopy Type Theory
On 11 Jan 2014, at 22:41, Alberto G. Corona wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, This is non sense. Computer programs have born in math. but mathematical structures and both are essentially the same thing: The computable is purely mathematical since birth (excepting Babbage, but even Babbage discovered it was mathematical at the end of his life, arguably, from a work due to Jacques Lafitte). But the mathematical, classically conceived, is *much* larger than the computable. N^N is not enumerable. the computable restriction of N^N is enumerable. both are paths from premises to conclussion in a space with topological properties That does not make them identical. And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. You might study the book by Szabo, on the category approach on the algebra of proofs. But proofs and computations are not equivalent concept at all. There is a Church's thesis for computability, not for provability and definability which are machines or theories dependent. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. That is very interesting, and category provides nice model for constructive subpart of the computable, like typed lambda calculus. But category becomes very hard on the complete algebra of computation. the partial nature of the fiunctions involved makes hard to even compute a co-product. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures They are since Church thesis. That is all what computability or recursion theory is all about. The rest is semantics of languages, more useful in computing theory than in computability theory, which is born, I insist, before we implement physical computer. The computer have been disocvered by mathematicians, in mathematics, indeed, in arithmetic. Those notions are born mathematical. Only later, some physicists have tried to get, without any success, a notion of physical computation. The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ Guiseppe Longo wrote also nice book on that subject. It is a vast field, but Gödel made proof into arithmetical objects well before, as the notion of computations will follow soon after (if not before if we take Post's unpublished anticipation into account). Bruno -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Homotopy Type Theory
On 12 Jan 2014, at 02:41, Alberto G. Corona wrote: But the proofs where not studied before as mathematical structures. Godel and any mathematician did profs, but proofs where meta-mathematical, in the sense that they were not mathematical objects, No, that is not true at all, and meaningless. Gödel did arithmetize meta-arithmetic. His whole proofs is based on this. although they could be formalized in a language. And then translated in math, even arithmetic. The same happened with the notion of equality and equivalence etc That are defined in a fuzzy or ad-hoc way. HOTT study how equal are two things depending on the path from the one to the other, that is , what topology has the proof of equality between the two. That is interesting work, but it is a restriction on some typed or constructive approach. It does not make things more mathematical, as it was elementary arithmetic from the start, as Gödel and the sequel have proven. Note that, computation can be seen as a particular case of proof, and proof can be seen as a particular case of computations, but those concept are quite different and obeys to quite different mathematics. That happens often. You can see a function as particular case of a relation (functional relation), and you can see a relation as a particular case of a function (by the characteristic function), but relation and function are not the same notion. Any way, both computation and proof are mathematical object in computer science and mathematical logic, since the start. Bruno 2014/1/11, LizR lizj...@gmail.com: That sounds like (some of) what Bruno talks about. The computer programme known as the UD (and its trace) are in maths. (And didn't Godel make proofs paths of maths?) On 12 January 2014 10:41, Alberto G. Corona agocor...@gmail.com wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, but mathematical structures and both are essentially the same thing: both are paths from premises to conclussion in a space with topological properties And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com . Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything- l...@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Homotopy Type Theory
Phisical computation was discovered by nature 4000 Million years BT (Before Turing) . And even before. 2014/1/12, Bruno Marchal marc...@ulb.ac.be: On 11 Jan 2014, at 22:41, Alberto G. Corona wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, This is non sense. Computer programs have born in math. but mathematical structures and both are essentially the same thing: The computable is purely mathematical since birth (excepting Babbage, but even Babbage discovered it was mathematical at the end of his life, arguably, from a work due to Jacques Lafitte). But the mathematical, classically conceived, is *much* larger than the computable. N^N is not enumerable. the computable restriction of N^N is enumerable. both are paths from premises to conclussion in a space with topological properties That does not make them identical. And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. You might study the book by Szabo, on the category approach on the algebra of proofs. But proofs and computations are not equivalent concept at all. There is a Church's thesis for computability, not for provability and definability which are machines or theories dependent. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. That is very interesting, and category provides nice model for constructive subpart of the computable, like typed lambda calculus. But category becomes very hard on the complete algebra of computation. the partial nature of the fiunctions involved makes hard to even compute a co-product. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures They are since Church thesis. That is all what computability or recursion theory is all about. The rest is semantics of languages, more useful in computing theory than in computability theory, which is born, I insist, before we implement physical computer. The computer have been disocvered by mathematicians, in mathematics, indeed, in arithmetic. Those notions are born mathematical. Only later, some physicists have tried to get, without any success, a notion of physical computation. The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ Guiseppe Longo wrote also nice book on that subject. It is a vast field, but Gödel made proof into arithmetical objects well before, as the notion of computations will follow soon after (if not before if we take Post's unpublished anticipation into account). Bruno -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Homotopy Type Theory
Physical computation was discovered by nature 4000 Million years BT (Before Turing) . And even before. 2014/1/12, Bruno Marchal marc...@ulb.ac.be: On 11 Jan 2014, at 22:41, Alberto G. Corona wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, This is non sense. Computer programs have born in math. but mathematical structures and both are essentially the same thing: The computable is purely mathematical since birth (excepting Babbage, but even Babbage discovered it was mathematical at the end of his life, arguably, from a work due to Jacques Lafitte). But the mathematical, classically conceived, is *much* larger than the computable. N^N is not enumerable. the computable restriction of N^N is enumerable. both are paths from premises to conclussion in a space with topological properties That does not make them identical. And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. You might study the book by Szabo, on the category approach on the algebra of proofs. But proofs and computations are not equivalent concept at all. There is a Church's thesis for computability, not for provability and definability which are machines or theories dependent. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. That is very interesting, and category provides nice model for constructive subpart of the computable, like typed lambda calculus. But category becomes very hard on the complete algebra of computation. the partial nature of the fiunctions involved makes hard to even compute a co-product. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures They are since Church thesis. That is all what computability or recursion theory is all about. The rest is semantics of languages, more useful in computing theory than in computability theory, which is born, I insist, before we implement physical computer. The computer have been disocvered by mathematicians, in mathematics, indeed, in arithmetic. Those notions are born mathematical. Only later, some physicists have tried to get, without any success, a notion of physical computation. The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ Guiseppe Longo wrote also nice book on that subject. It is a vast field, but Gödel made proof into arithmetical objects well before, as the notion of computations will follow soon after (if not before if we take Post's unpublished anticipation into account). Bruno -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Homotopy Type Theory
2014/1/12, Bruno Marchal marc...@ulb.ac.be: On 12 Jan 2014, at 02:41, Alberto G. Corona wrote: But the proofs where not studied before as mathematical structures. Godel and any mathematician did profs, but proofs where meta-mathematical, in the sense that they were not mathematical objects, No, that is not true at all, and meaningless. Gödel did arithmetize meta-arithmetic. His whole proofs is based on this. That is not the same than study proofs themselves inside math than reduce them to arithmetic or set theory. Moreover, what Godel did was called meta-mathematics AFAIK although they could be formalized in a language. And then translated in math, even arithmetic. The same happened with the notion of equality and equivalence etc That are defined in a fuzzy or ad-hoc way. HOTT study how equal are two things depending on the path from the one to the other, that is , what topology has the proof of equality between the two. That is interesting work, but it is a restriction on some typed or constructive approach. It does not make things more mathematical, as it was elementary arithmetic from the start, as Gödel and the sequel have proven. Note that, computation can be seen as a particular case of proof, and proof can be seen as a particular case of computations, but those concept are quite different and obeys to quite different mathematics. That happens often. You can see a function as particular case of a relation (functional relation), and you can see a relation as a particular case of a function (by the characteristic function), but relation and function are not the same notion. Any way, both computation and proof are mathematical object in computer science and mathematical logic, since the start. Bruno 2014/1/11, LizR lizj...@gmail.com: That sounds like (some of) what Bruno talks about. The computer programme known as the UD (and its trace) are in maths. (And didn't Godel make proofs paths of maths?) On 12 January 2014 10:41, Alberto G. Corona agocor...@gmail.com wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, but mathematical structures and both are essentially the same thing: both are paths from premises to conclussion in a space with topological properties And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com . Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything- l...@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from
Re: Homotopy Type Theory
On 12 Jan 2014, at 11:28, Alberto G. Corona wrote: Phisical computation was discovered by nature 4000 Million years BT (Before Turing) . And even before. Show me the publication. Come one, with argument like that I could answer that mathematical computation has been discovered already out of time and space, and indeed the physical computation (a notion you might try to define, btw) have develop from them, which is the computationalist point I try to explain. Bruno 2014/1/12, Bruno Marchal marc...@ulb.ac.be: On 11 Jan 2014, at 22:41, Alberto G. Corona wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, This is non sense. Computer programs have born in math. but mathematical structures and both are essentially the same thing: The computable is purely mathematical since birth (excepting Babbage, but even Babbage discovered it was mathematical at the end of his life, arguably, from a work due to Jacques Lafitte). But the mathematical, classically conceived, is *much* larger than the computable. N^N is not enumerable. the computable restriction of N^N is enumerable. both are paths from premises to conclussion in a space with topological properties That does not make them identical. And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. You might study the book by Szabo, on the category approach on the algebra of proofs. But proofs and computations are not equivalent concept at all. There is a Church's thesis for computability, not for provability and definability which are machines or theories dependent. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. That is very interesting, and category provides nice model for constructive subpart of the computable, like typed lambda calculus. But category becomes very hard on the complete algebra of computation. the partial nature of the fiunctions involved makes hard to even compute a co-product. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures They are since Church thesis. That is all what computability or recursion theory is all about. The rest is semantics of languages, more useful in computing theory than in computability theory, which is born, I insist, before we implement physical computer. The computer have been disocvered by mathematicians, in mathematics, indeed, in arithmetic. Those notions are born mathematical. Only later, some physicists have tried to get, without any success, a notion of physical computation. The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ Guiseppe Longo wrote also nice book on that subject. It is a vast field, but Gödel made proof into arithmetical objects well before, as the notion of computations will follow soon after (if not before if we take Post's unpublished anticipation into account). Bruno -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com . Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything- l...@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post
Re: Homotopy Type Theory
On 12 Jan 2014, at 11:36, Alberto G. Corona wrote: 2014/1/12, Bruno Marchal marc...@ulb.ac.be: On 12 Jan 2014, at 02:41, Alberto G. Corona wrote: But the proofs where not studied before as mathematical structures. Godel and any mathematician did profs, but proofs where meta-mathematical, in the sense that they were not mathematical objects, No, that is not true at all, and meaningless. Gödel did arithmetize meta-arithmetic. His whole proofs is based on this. That is not the same than study proofs themselves inside math than reduce them to arithmetic or set theory. Moreover, what Godel did was called meta-mathematics AFAIK The point is that computation have been discovered in arithmetic, and were initially defined in arithmetic or principia mathematica, and then in other mathematical theories (like combinatirs, or lamda calculus). And meta-mathematics is just the name Kleene gived to the study of the mathematical notions of formal proofs, theories, models, and computations. It is the same as mathematical logic. Bruno although they could be formalized in a language. And then translated in math, even arithmetic. The same happened with the notion of equality and equivalence etc That are defined in a fuzzy or ad-hoc way. HOTT study how equal are two things depending on the path from the one to the other, that is , what topology has the proof of equality between the two. That is interesting work, but it is a restriction on some typed or constructive approach. It does not make things more mathematical, as it was elementary arithmetic from the start, as Gödel and the sequel have proven. Note that, computation can be seen as a particular case of proof, and proof can be seen as a particular case of computations, but those concept are quite different and obeys to quite different mathematics. That happens often. You can see a function as particular case of a relation (functional relation), and you can see a relation as a particular case of a function (by the characteristic function), but relation and function are not the same notion. Any way, both computation and proof are mathematical object in computer science and mathematical logic, since the start. Bruno 2014/1/11, LizR lizj...@gmail.com: That sounds like (some of) what Bruno talks about. The computer programme known as the UD (and its trace) are in maths. (And didn't Godel make proofs paths of maths?) On 12 January 2014 10:41, Alberto G. Corona agocor...@gmail.com wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, but mathematical structures and both are essentially the same thing: both are paths from premises to conclussion in a space with topological properties And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com . Visit this group at http://groups.google.com/group/everything- list. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything- l...@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com . Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails
Re: Homotopy Type Theory
On 1/12/2014 1:57 AM, Bruno Marchal wrote: You might study the book by Szabo, on the category approach on the algebra of proofs. But proofs and computations are not equivalent concept at all. There is a Church's thesis for computability, not for provability and definability which are machines or theories dependent. But isn't every computation an instantiation of a proof, relative to the computer. Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Homotopy Type Theory
By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, but mathematical structures and both are essentially the same thing: both are paths from premises to conclussion in a space with topological properties And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Homotopy Type Theory
That sounds like (some of) what Bruno talks about. The computer programme known as the UD (and its trace) are in maths. (And didn't Godel make proofs paths of maths?) On 12 January 2014 10:41, Alberto G. Corona agocor...@gmail.com wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, but mathematical structures and both are essentially the same thing: both are paths from premises to conclussion in a space with topological properties And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
Re: Homotopy Type Theory
But the proofs where not studied before as mathematical structures. Godel and any mathematician did profs, but proofs where meta-mathematical, in the sense that they were not mathematical objects, although they could be formalized in a language. The same happened with the notion of equality and equivalence etc That are defined in a fuzzy or ad-hoc way. HOTT study how equal are two things depending on the path from the one to the other, that is , what topology has the proof of equality between the two. 2014/1/11, LizR lizj...@gmail.com: That sounds like (some of) what Bruno talks about. The computer programme known as the UD (and its trace) are in maths. (And didn't Godel make proofs paths of maths?) On 12 January 2014 10:41, Alberto G. Corona agocor...@gmail.com wrote: By the way, what about if you find a mathematical theory that show that: computer programs and matematical proofs are no longer something out of math, but mathematical structures and both are essentially the same thing: both are paths from premises to conclussion in a space with topological properties And the theory stablish topological relations between these paths so that proofs and computer algorithms are classified according with these relations. That is homotopy type theory. http://homotopytypetheory.org/ I´m starting to learn something about it, It is based on type theory, category theory and topology. The book introduction is nice (HOTT link at the bottom of the page). It seems to be a foundation of computer science and math that unify both at a higher level, since proofs and programs become legitimate mathematical structures The book: http://homotopytypetheory.org/2013/06/20/the-hott-book/ -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- Alberto. -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
