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

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

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.

[ mathematics, real life math applications, learning math, teaching math, math applications, real world math applications ]

The Concept of a Mathematical Function

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

The concept of a mathematical function should not be first introduced as a formula, but as an arbitrary (ordered) pairs of numbers. Pupils are conditioned to think of a function as a continuous line or a firmula. Later there are issues with statistics when function is shown to be function a set of distinct dots, representing ordered pairs of numbers, i.e. there is no formula at all. Moreover, in probability and statistics numbers appear to be showing at random! 

The rule how you pair one number with another can be a formula, but also can be a completely random event. Math function is, first and foremost about pairing two numbers (or more in multivariable functions). Students should be aware that they can pair random chosen numbers, they do not need to calculate second number from first. The rule can be linked input or output, but that restricts the function in the way that you have to know input to get the other paired number, the output. Because, function can have a pairing rule "pick first number, then, ask another person to pick another number without looking at the first number, then pair two numbers". Rule is one thing. Paired numbers are another. I want to emphasize that function need not to be defined in a restrictive way by using words "inputs" and “outputs", which is more related to computer science. You do not have to know input to get output, in a function. Both elements of the ordered pair of an function can be completely random and independent from each other. Function is first and foremost a pair of ordered numbers. My examples show why the \"input\" \"output\" definition is restrictive and possibly misleading. In my view, the word pair, or more precisely definition "ordered pair" best describes the function. Then we can use word map, association of two numbers etc. Input and output really leads someone to think that there need to be formula or some dependency between output and input. But, it is not so. It can be, but that's too restrictive for function definition. As in my example, a function can be "pick an output that in no way depends on input". Or, pick one number, then cover it (hide it) then ask another person to pick another number. Pair these two numbers. Here, output in no way depends on input, yet this is a function.

 

[ math function, function, concept of a function, concept of function, mathematics, map, mapping, teaching math ]