On 12 Jan 2014, at 11:36, Alberto G. Corona wrote:
2014/1/12, Bruno Marchal <[email protected]>:
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 <[email protected]>:
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 <[email protected]>
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/
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