On 11 Jan 2014, at 22:41, Alberto G. Corona wrote:

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By the way, what about if you find a mathematical theory that showthat: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

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