Tuesday, March 1, 2011

Mathematics Axiomatic Frontier and How Outer Worlds Puncture it to Generate Numbers

Mathematics Axiomatic Frontier and How Outer Worlds Puncture it to Generate Numbers.

Any number within Mathematical World (bounded by ZFC Axiomatic Frontier, including Axiom of Choice :-) is seen within Math World as generated directly from the Fundamental Axioms. But, applied mathematicians and other people using math within their disciplines (Civil Engineers, Traders, Physicist, as shown in this illustration) have to keep tab what they have just counted, i.e. what a particular number is associated with, and where it comes from. And that, the language, the particular discipline's language and logic, as it is illustrated, is outside mathematics, and that fact should be signified to students when introducing them to math. There has to be a clear distinction between a particular discipline discourse, discipline's specific logic, language of explanation, definitions, and pure math, math that can be traced, essentially, to the fundamental ZFC axioms.

Each thin arrow that goes from "Outer" world, world where the math is "applied", can generate a number, inside Math world, can generate a relationship, formula, even differential and integral equation, right there. When the calculation is done, within Math, the result will be returned via thick arrow to the discipline where the "request" for calculation came from.

The drawing shows one more property. Math is not "applied" to the fields, but, it's rather used as a direction of thinking in order to solve some particular problem. The world "applied" probably came in use because the mathematical constructs, and math development in general, can be done without any of the Outer World "bubbles", disciplines (Economics, Physics, Civil Engineering, Electrical Engineering), and then, when there is a need, that math results can be matched with some requests from the Outer World. While certain domains of math development have been motivated by physical processes, that math can be developed completely from mathematical axioms, without analyzing any physical process.

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