## Tuesday, April 24, 2012

### Mathematics and Real World Applications Links. World #1 and World #2 Approach

In mathematics, it is all about numbers, sets of numbers, and the sequences of operations on them.

Axioms how to link real world scenarios to mathematical axioms and theorems, have to be defined. The closer you are to the point of complete axiomatization of the real world domains you want to apply math at, the better. Mathematics will reward you with meaningful results. Quantitative aspects of real world axioms, theorems and laws are usually theorems within mathematics.

How that can help you to learn mathematics and the domains where the mathematics is apply to, whatever that means? The answers is that you have to separate the two systems of axioms. The real world system, let's call it World #1 and the pure mathematical system (with its own axioms!), let's call it World #2. Note that there are World #1 axioms (ideally), World #2 axioms and axioms how to link the premises, and eventually theorems, from these two worlds.

If you quantify objects and their relationships in the World #1, the quantities, and their relations, you obtained enter mathematics as initial or boundary conditions, or simply as numbers, set of numbers, pairs of numbers, as the starting points or starting premises. Note that these numerical starting points can be obtained inside math as well, without any extraneous motivation, i.e. within the realm of pure math only, sometimes directly from fundamental axioms.

The interpretation of numerical results is, apparently, up to you. It is you who will keep track of what is counted and why. The logic why you would do certain mathematical operations on the specified numbers, if coming from World #1, has to be firm and, ideally, has to be derived from an firm axiomatic system. Of course, math doesn't care if you axiomatized your real world domain or not. Mathematics will, without asking any questions, follow your instructions for calculations, and give you back results. Interpretation and usage of those results will depend on the correctness of your real world domain assumptions, accuracy of its logic, and  completeness and correctness of the real world domain axioms.

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