Tuesday, March 29, 2011

One Insight into Mathematics, Axioms, Logic, and Their Relations to Other Disciplines

Mathematics is, in the sense of methodology, like any other scientific discipline, including law, economy, psychology, biology, physics, chemistry. Or, more precisely, all other disciplines should be very similar to mathematics, if they are to discover new truths and solutions. I will demonstrate what are the two major similarities and one major difference. What differs mathematics from ALL of these disciplines is that mathematics deals exclusively with numbers, counts. It is, sometimes, hard to imagine that mathematics is independent discipline, given how much we, as students, and later in career, are fed (and fed up!) with numerous examples, starting with apples, pears, meters, acceleration, force, atomic mass, light wavelength etc.. Perhaps surprisingly to many of us, mathematics is an INDEPENDENT discipline and can be developed completely outside any example, i.e. examples (physical processes, decision generated numbers, measures) are not required for development and research in mathematics. Again, mathematics deals with counts, numbers exclusively, and that's it.

Now, similarities.

First major similarity, and the reason why mathematics is called one of the most precise sciences, is that it uses strong logic methodology. But note, this logic is used as a tool of thinking to solve problems in math, and develop mathematics. Logic is a separate discipline that can, and should, UNIVERSALLY be applied to any other scientific discipline.

Second major similarity is that mathematicians succeeded to define initial truths in mathematics, truths to start with, starting postulates or AXIOMS. But note, while axiom might be considered a mathematical term, it's meaning can be applied to ANY other scientific discipline or any other creative direction of thinking. Axioms are everywhere, we use it every day, but we do not call them axioms. Usually, these are initial assumptions in our directional thinking to solve a problem or to come up with a creative answer for something. The same truths are present in law, biology, physics, of course, the way they are discovered are different for each discipline. But, this is how it should be done. Define initial assumptions, make sure they are correct and start develop the system you are interested in.

Now, note how mathematics used LOGIC and AXIOMATIC approach to deal with counts! It is not that counts triggered development of logic and axioms. It's vice versa. Logic and axioms were there before, and are used to develop and enhance mathematics and are and should be used to enhance and develop other scientific and other creative human disciplines.

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