Petar Maymounkov

A math philosophy question

Wed, Jun 25, 2014

Dear All,

I have a philosophical math question which I wanted to ask here, if I may. Here goes:

Math is a system of knowledge that is unambiguously stored within an “accounting” language which underlies all work ever written on the subject, regardless of its form (symbolic, English, hybrid, etc). The uniqueness of this underlying formal language is manifested in the consensus amongst scientists when they go about their daily communications.

That language is—by choice or necessity or taste, but certainly by agreement—acyclic: It begins with axioms (of Set Theory) and grows with operations (of Formal Logic).

Acyclic graphs (which are the formal system here) are not the only ones we know. Systems of truth can be circular—as if suspended in air without axioms—and their sophistication can be accomplished by refinement as opposed to growth. This is what we call CSP problems in Computer Science.

Why is that we study formal systems with no beginning or end (CSP problems in combinatorial optimization), but we use a language to describe our findings (Formal Logic) which is an acyclic infinite system?

This question rings familiar when you put it in the context of two painful paradoxes in math: the search for correlation-cause-effect resolution and the question of existence of one-way functions.

Could it be that our choice to use an axiomatic language to talk about math is creating singularities: one at the place when information comes into math (the act of observation) and at the other end of the math universe (the term for the “unknown” in math as language—randomness).

Could the resolution of both paradoxes simply be to rethink the language of mathematical expression so that those two open questions are collapsed into one and converted into a theorem—thereby going from two conundrums to one result, or (-1) + (-1) = +1.

Here’s where this problem emerges in practice:

I find it odd that experimental sciences (when they are correct) report: “The observations show a correlation between … and we are going to use some side theory to disambiguate between cause-and-effect.” What side theory?

The very side theory they are appealing to is Math and Math resolves their meaningless and inane (due to cognitive bias and short historic memory) question (of “Is correlation cause or effect?”) into another meaningless one, which is the collection of paradoxes intrinsic in Math—the provider of the answer—as an axiomatic language.

But if Math wasn’t serving to the asking experimentalist and rather aiming to build a language that explains everything as seen and observed and testified to by its users (people), in a consistent impartial and non-arguable manner, it would have to offer a new tool to record and reason about correlation without breaking it into cause and effect. No?

Thank you so much,
Petar

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