Thursday, May 3, 2012

Axioms and Theorems in Relation to the Mathematical Models of Real World Processes

A mathematical model of a real world process is a set of numerical, quantitative premises driven and postulated by that real world environment, by its rules and by its logical systems extraneous to mathematics. Yet, these premises can be also derived directly from mathematical axioms. Moreover, while the premises are motivated by the real world processes and scenarios, the proofs of theorems, theorems built on these premises, are done and can be done only within the world of pure mathematics, using pure mathematical terms, concepts, axioms, and already proven theorems.

When illustrating to students applied math, it should be shown which premises are introduced from, and by the field, of mathematics application, and, as the second step, how these premises can also be defined from the inside of pure mathematics, without any influence of, or reference to the real world process or environment. Then, it has to be shown that the proofs of the theorems that use those premises are completely within mathematics, i.e. no real world concepts are part of the proof.

Successful assumptions will give predictable consequences. Axiomatizing that set of assumptions should ensure no contradictions in consequences. Usually, we are after a certain type, a particular set of consequences. We either know them, or investigate them, or we want to achieve them. Hence dynamics in our world of assumptions.

Theoretical, pure logic doesn't care what are your actual assumptions. It just assume that something is true or false and go form there. Sure, results in that domain are very valuable. But, we are after the particular things and statements we assume or want to know if they are true or false. Not in general, but in particular domain. Any scientific field  can be an example. Logic cannot tell us what are we going to chose and then assume its truth value. Usually it is the set of consequences we are after that will motivate the selection of initial assumptions. Then logic will help during the tests if there are any contradictions. If you are interested in specific consequences, in particular effects, investigate what causes those effects. When you have enough information about causes, make every attempt to axiomatize them. And, again, as mentioned, axiomatizing that set of causes should ensure no contradictions in consequences.

Axiomatizing the set of causes should ensure no contradictions in effects (consequences). When tackling the topic of applied mathematics, it should be explained how the mathematical proofs contain no concepts or objects from the real world areas to which mathematics is applied to. That very explanation will shed light on the realtionship between mathematical axioms, theorems and the logical structures in the field of mathematical application (physics, engineering, chemistry, physiology, economics, trading, finance, commerce...).

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