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Toward a Foundation of Mathematics based on A New Kind of Science

POSTED BY: José Manuel Rodríguez Caballero
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In the video you quoted, Mikhael Gromov said nothing about Algebras containing relations that are not computations.

The statement:

All traditional descriptions of mathematics, in my view, are greatly faulty

includes the description of Algebras as a particular case.
POSTED BY: José Manuel Rodríguez Caballero

I believe Dr. Gromov is referring to popular historic literature, such as Dantzig's book "Number".
POSTED BY: Richard Frost

Interesting. From my perspective it is a valid relation in the English algebra and is neither computable nor uncomputable but something else. Now it is true that you can measure the relation (e.g., for validity) but this is very different from whether or not it is a computation.

So at least from my perspective, Mathematics contains more than just computation and I'd suggest rewording your definition. No big deal though.
POSTED BY: Richard Frost
POSTED BY: José Manuel Rodríguez Caballero
POSTED BY: Richard Frost
POSTED BY: José Manuel Rodríguez Caballero
POSTED BY: Richard Frost

Let algebra E = the English language. From your perspective, what if anything is being computed in the relation "I am going to the store"?
POSTED BY: Richard Frost
POSTED BY: José Manuel Rodríguez Caballero

Define algebra A to have at least two types of subgroups: those related to computation and those which are not. Would you consider A to be part of Mathematics?
POSTED BY: Richard Frost

Sure, the subgroups of A that are not related to computation are an abstraction of the computational notion of subgroup. My claim "mathematics is an abstraction of computation" does not mean that "mathematics is just computation". Mathematics is obtained from computation by ignoring some features. In this case, you are ignoring a feature known as computability.
POSTED BY: José Manuel Rodríguez Caballero
POSTED BY: Richard Frost
POSTED BY: José Manuel Rodríguez Caballero

José, Your discussion of quotient here is interesting. But in consideration of "Mathematics is just the abstraction in a computation", I am reminded of the theorem (often assigned homework problem) in Modern Algebra 4/5:

Every language is an algebra.

This exposes your premise as too narrow. I believe a better definition is:

"Mathematics is the language of measurement" -- Steven Krashen
POSTED BY: Richard Frost
POSTED BY: José Manuel Rodríguez Caballero
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