On 12 Jan 2014, at 11:36, Alberto G. Corona wrote:

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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 aredefined in a fuzzy or ad-hoc way. HOTT study how equal are twothingsdepending 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. Bruno2014/1/11, 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 samething: both are paths from premises to conclussion in a spacewithtopological propertiesAnd the theory stablish topological relations between thesepaths sothat 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 linkat the bottom of the page). It seems to be a foundation ofcomputerscience and math that unify both at a higher level, since proofsandprograms 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 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.

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