The History of Discovery of 21 Century Math Method

The Problem with Axiomatic Method and Principle of Simplicity

[TODO]  

Following the principle of infinite development as contained in the Christian concept of children of God (to which I believe to pertain myself), I found that the main principle of my thinking which was axiomatic method is a wrong path leading away from God to void infinite decrease of perfection, I believed in existence of an other better method which can be the next stage of development of thinking.

I mean that if we limit ourselves only to simple axioms systems, then generalizing axioms system following axiomatic method, we necessarily will go by circles due a limited number of simple axioms. Alternatively we go to the more and more complex systems that is to the direct opposite to the stated purpose of axiomatic method that is writing math in simple axioms systems. So axiomatic method is a dead end.

These thoughts were called when I (probably) rediscovered real numbers giving them an AGT-based (see below) definition and realized that I'm going to make a big circle possibly returning back to old pre axiomatic method math that is to study of real numbers.

I set before myself to find a method which does not leads to going by circles being limited by the principle of simplicity. I'm not yet sure whether 21MM is such a method but anyway it is a step forward in math.    

Algebraic General Topology

I found a way to write certain general topology statements as equations and called this new field of math Algebraic General Topology™. It is yet unpublished.

I generalized and wrote algebraically some of general topology structures but later I thought that it is of limited usefulness for practical tasks as we had no general way to rewrite a problem from math analysis in the form of AGT™ equations where it is simply solvable, nor a way to extend AGT further except of using intuition.

So I started to think about an algorithm which would transform math analysis theorems into AGT equations and more generally to transform traditional math into quantifiers-free math. 21MM provides us with an algorithm to free ourselves from quantifiers. That is I searched for a way back from old less general theories to new theories. (BTW, it would be a kind of formalization of History of Mathematics, a self-teaching algorithm.)

Historically 21MM was constructed to be a formal system where it is possible to write certain basic definitions of Algebraic General Topology.

Before invention of 21MM I developed AGT in axiomatic method which appeared to be inappropriate for this new theory as in axiomatic method it appears hard (or impossible) to write the most general cases of AGT axioms and the systems of axioms appear to be somehow antinatural and hard to catch in mind for some people. (I thought about these general cases long time and was unable to fix such a system in axiomatic method.) So I'm going to re-develop AGT in 21MM possibly extending 21MM by the way as needed.

Extension-Definition Method, an Old Version

Older Versions of "21 Century Math Method" article

Here is the history of important changes of this document, for the History:

Some Fragments of Old Versions

Older Generalization Definition Attempt

Here is my older former (wronger) attempt to define generalization:

Let γ is a proven expression. Then γ ∈ ((α→β)(γ))(...). If δ ∈ parts(γ), then δ ∈ β(...) ⇒ δ ∈ α(...). So δ ∈ α(...) ⇔ δ ∈ β(...). That is T ∩ α(...) = T ∩ β(...).

A generalization is a pair of expressions (α, β) such that all the following conditions hold for any (valid) generalization:

• α ∈ β(...);

• (α -all→ β)(γ) matches no already written axiom or definitinal construct if γ is an already written axiom or definitinal construct which matches α.

After a generalization for every proven expression γ the expression δ = (α -all→ β)(γ) is also written.

References