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 computerprogrammeknown as the UD (and its trace) are "in maths". (And didn't Godelmakeproofs 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 showthat:computer programs and matematical proofs are no longer somethingoutof math, but mathematical structures and both are essentially thesamething: 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 typetheory,category theory and topology. The book introduction is nice (HOTTlinkat 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 GoogleGroups"Everything List" group.To unsubscribe from this group and stop receiving emails from it,send anemail 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 GoogleGroups"Everything List" group.To unsubscribe from this group and stop receiving emails from it,send anemail 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 GoogleGroups "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.