Friday, January 20, 2012

The Concept of a Mathematical Function

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 ]

Sunday, January 8, 2012

Axioms and Propositions

Axiom. Theorem. Starting point. Proposition. Premise. Assertion. Proof. Argument.
Before any argument, containing, say, two propositions, axioms for each premise should be stated first, so we know where the propositions are coming from and why we assume they are true.

It's not enough to say you based your decision on logic. Logic, but based on which set of axioms? Axioms of principles, values, feelings, or physics laws? Or, logic that uses axioms and premises on a hybrid axiomatic system, perhaps a combination of two or more mentioned? Perfect logical reasoning with wrong assumption is useless. That kind of logical reasoning is almost always worse than using intuition.



[ mathematics, math, applied mathematics, applied math, logic, mathematical logic, inventions, innovations,  ]

Friday, January 6, 2012

An Example How to Learn Probability and Statistics Using World 1 and World 2 Approach

This is an example, actually guidelines, how to learn mathematics, specifically material in the book "Business Statistics" by D. Downing and J. Clark.



Please read my previous posts about separating mathematical world, World # 2 from non mathematical axioms and logic, called World #1. Here World # 1 is motivation to develop probability and statistics material. World # 2 is pure mathematics.

It is sufficient to recognize premises in World # 2 motivated by World # 1. Note that mathematics has well established set of axioms, and that these premises can be developed without any mentioning of real world examples related to the statistical analysis. Again, please read my previous post or my book.

Hence, it is sufficient, and necessary, to learn these premises. Note that they do not require proof, or more precisely, many of them follow direct from basic math axioms. Then, learn real world explanations that can motivate their selection and introduction. Clearly separating these two worlds you will be able to firmly understand mathematical treatment of business statistics. At any point you should be able to define the premises and show the separation boundary between pure mathematics and business field (hint: they even use different vocabulary). To help you further, no business term can ever be used to prove any mathematical theorem mentioned in this book no matter how business situation "motivates" mathematical concept introduction.

It is interesting that math students are taught how real examples motivate math new concepts and new directions of math development, but then it's not emphasized how no real world object or concept can be used in any mathematical proof. 

A Guide to Interdisciplinary Innovative Thinking

Application of one scientific field (or any system that has a logical structure) in the other, usually means that, when both systems are axiomatized, the link via logical connectives between the two fields' postulates, hence creating new premises in the new, hybrid system, will mean that postulates in one system (field!) will dictate selection of premises in the other. Some particular combination of these can lead to an invention.

Sometimes, the key to an invention is a selection of two (or more) systems and linking them together. Linking primarily means links via logical connectives, i.e. selecting premises, forming theorems.Sometimes, we already know the fields (systems!) but we need to find winning connection between the two (or more) of them. Where intuition fits in? It fits in selecting appropriate fields and selecting correct and useful links between them. Don't forget, an axiom is not provable within the system it defines, i.e. within the system it is developed from them. Choosing right premises and choosing to search for axioms is usually inspired by the linkage to the world outside the one that we look to find the axioms for.

Here are some examples. Each field, or system, is assumed to have its own set of axioms, postulates, and theorems, whatever that means in that system. As you will see, the system does not have to be mathematics. Note the selection and links.


Music -> Emotions.

Instrument -> Music -> Emotions

Physics -> Mathematics -> Human Language

Engineering Design Requirements -> Physics -> Mathematics -> Human Language.

Emotions, morals -> Paints, canvas -> Painting

Electric Power Systems -> Economic dispatch

For readers' exercise, try to define axioms in each of the systems and illustrate how the postulates, theorems in one system dictates premises in the other.

The power of a good question is that it can point to the areas of knowledge you need to familiarize yourself with. It can also initiate effective knowledge filtering and selection of the right facts that will be connected in a new, original way, to answer your question.

You probably got your engineering degree for knowing how to solve differential equations, not how to select useful and innovative initial and boundary conditions.

A mathematician and an artist. An accomplished NASA and IBM statistician and scientist talks about his sculptures.

Tuesday, January 3, 2012

A conception of an idea - axioms and brain

An idea is conceived in your mind. But that's the different question than axioms in mathematics. How an idea came into an existence at the first place is a question for biochemistry, energy paths, oxygen driven, in our brain's biochemical processes. But, when we talk mathematics, we use our oxygen driven conceptual mechanism to limit our ideas that can be generated from math axioms only. Note that first axioms must be conceived, then thoughts from axioms. They are all "puff" generated from energetic processes in our brain.

We can think, that's apparently given. How the idea is created in our head, or, even worse what is it, is not a part of mathematical study. We can say that an idea is a state of our mind, molecular, energetic, a dynamic state of biochemical processes, that keeps the idea present in our brains, purely on an energetic level.

We can conceive an idea or a thought, that can be called a postulate, and then use logical thinking to derive theorems from the postulate or axioms. You have to be sure that your next mathematical thought is originated in axioms and that it can be derived, proved by them.

Thinking freely, without axiomatic boundaries, is also an attractive scenario. Free train of thoughts can give initial and starting conditions, initial premises in, most likely, any axiomatic system.