# Re: Homotopy Type Theory

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On 12 Jan 2014, at 02:41, Alberto G. Corona wrote:```
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```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,
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No, that is not true at all, and meaningless. Gödel did arithmetize meta-arithmetic. His whole proofs is based on this.
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```although they could be formalized in a language.
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And then translated in math, even arithmetic.

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```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.
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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.
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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.
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Any way, both computation and proof are mathematical object in computer science and mathematical logic, since the start.
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Bruno

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2014/1/11, LizR <lizj...@gmail.com>:
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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?)

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On 12 January 2014 10:41, Alberto G. Corona <agocor...@gmail.com> wrote:
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By the way, what about if you find a mathematical theory that show that:
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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/

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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.

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--
Alberto.

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