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

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15 Replies

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

The definition

"Mathematics is the language of measurement" -- Steven Krashen

involves a computation, since the act of measuring is a (concurrent) computation: the transmission of information from one system to another. To overcome the computational foundations of mathematics, it is important to develop a foundation of mathematics that does not involve computation of any kind. Then computation should be derived from such a hypothetical system without using computation in this derivation process. I do not know such an approach; maybe another person knows.

A little history: Four decades ago in the U.S. it was necessary to justify computation as a science in order to receive NSF funding for what we now refer to as computational science. The need to do so has long since vanished.

Now in consideration of your definition of Mathematics, if one considers the following as true:

1. Every language is an algebra
2. There are several languages with subgroups unrelated to computation

then we receive the Lemma:

L. Not every algebra is entirely for computation

and consequently

"Mathematics is just the abstraction in a computation" is False.

By considering the premises:

  1. Every language is an algebra
  2. There are several languages with subgroups unrelated to computation

to derive the conclusion

L. Not every algebra is entirely for computation

you have shown an example of the claim

Mathematics is just the result of abstraction in a computation

because an algebra that is unrelated to computation is an abstraction from an algebra that is related to computation, indeed, you have the computational concept of algebra, and you make abstraction by ignoring the property of being computational.

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?

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.

Let algebra E = the English language. From your perspective, what if anything is being computed in the relation "I am going to the store"?

According to the above-mentioned framework, the sentence "I am going to the store" is an expression, not a mathematical object. To transform it into a mathematical object, it is necessary to make some abstraction. For example, to say "I am going to x", where x is an arbitrary place. Now, the mathematical object "I am going to x" can be represented by the following set of expressions:

"I am going to the store", "I am going to the zoo", "I am going to the mountain," etc.

Any transformation of the mathematical object "I am going to x" should transform all its instantiation. For example, the transformation from "I am going to x" to "x is my destiny" is defined by the family of transformations of expressions (computations)

from "I am going to the store" to "the store is my destiny." from "I am going to the zoo" to "the zoo is my destiny." from "I am going to the mountain" to "the mountain is my destiny." etc.

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.

Your conclusion

Mathematics contains more than just computation

which is justified from the perspective that you are talking about, seems to be equivalent to my conclusion that the whole point of mathematics is that it contains less than computation. In other words, the non-computational mathematics in your approach is the analog of computations from which the property of computability was ignored during a process of abstraction. Considering numbers, if you say that some real numbers are non-computable, I may equivalently say that these non-computable real numbers are obtained from the computable real numbers by ignoring the property of computability during a process of abstraction. Hence, non-computable real numbers are less than computational real numbers from a conceptual point of view: they are real numbers minus the property of computability. This conceptual subtraction is what makes mathematics so powerful: it allows shortcuts in irreducible computation.

In your undergraduate mathematics studies, did you take the course series in group theory, rings and fields, and then algebras? It is from that viewpoint that I state there are algebras which contain relations that are neither computable nor uncomputable, but instead fall in other categories. As an example, I cited human languages. In fact, beyond those courses there is the discipline of abstract linguistics which you might find related to your quest to define Mathematics.

In your undergraduate mathematics studies, did you take the course series in group theory, rings and fields, and then algebras? It is from that viewpoint that I state there are algebras [...]

I totally agree, you derived the right consequence of that viewpoint. Nevertheless, according to M. Gromov, that viewpoint is wrong (I do not claim to have the right viewpoint neither):

All traditional descriptions of mathematics, in my view, are greatly faulty, and this is the reason why there is no model of mathematics realized in computers, why there is no model of understanding languages, because we have an absolute wrong perception traditionally build in development of numerical mathematics in logic distorted our perception of ourselves, of mathematics and of languages. It is as distorted as the Sun orbiting the Earth, [...] it is the wrong language, the wrong description.

M. Gromov, reference here.

In the video you quoted, Mikhael Gromov said nothing about Algebras containing relations that are not computations.

In your statements on the matter, you conflate non-computation with uncomputable. But the latter term refers to questions that can't be answered. A transitive relation such as the sentence I provided is neither a question nor a computation.

Your statements regarding the need to restate "I am going to the store" as "The store is my destination" reveal that you don't fully grasp what an algebra is, nor why every language is an algebra. Instead it seems you have misunderstood the statement to mean that a language can be used to make mathematical statements - a very different idea.

There are many different concepts concerning the definition of Mathematics. Here is one often assumed by children whose arithmetic instruction is solely by rote:

enter image description here

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.

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

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