The meaning of mathematics

Mathematics has many definition, suspended from the point if view of everyone. But generally, mathematics specificity to cover how to know description, type, characteristic about something.
We were learn mathematics from school until now in university. But, mathematics in school and in university is different.
In school, we are learn the basic of mathematics and how mathematics generally. But, in university we are learn pure mathematics and applied mathematics.

As we know, characteristic of mathematics are abstract. It means that the object of mathematics is abstract. We can’t see or touch this object. But, we can think about it.
I’m sure we are think, how to learn the object abstract?
There are 2 ways to learn it
1.abstraction
Abstraction isn’t same with abstract. It’s different. Abstraction has mean as drawing about something. Example, from all the characteristic of something we just choose one of them. Mr. marsigit was give us good example. It’s number 5. Number 5 is outside of our mind. But, it has many characteristic like colour, size, price, material of number 5. From all of characteristic we just choose one to thinking or discussion it. it’s call abstraction.
2.idealization
Idealization can we say as how someone see an object and it’s real which has relation with it.
Example, in this world nothing is taper. The edge of Needle are consist of atom’s track which shape is oval. So it’s impossible to us for thinking that needle is taper.
So the conclusion, what it call idealization example taper is truly taper and straight is truly straighty whitout squiggly.

The object of Mathematics:
• Definition
• Axiom
• Lemma
• Theorems
• Pattern
• Formula
• rule
Characteristic of mathematics are logic and consistent.
According mr.kayatesi from Melbourne university mathematics consist of 2:
1. Conjecture
It used to think, predict, previse to solve problem.
2. Confience
It used to communicate the result to other people, so the other can use this result.

According professor Shigeo Katagiri from japan, what it call mathematics are mathematical thinking.
According him, mathematical thinking are consist of 3 aspect.
This is the copying version from mr.marsigit posting from the version of shigeo katagiri(2004).
I. Mathematical Attitudes
1. Attempting to grasp one’s own problems or objectives or substance clearly, by oneself
(1) Attempting to have questions
(2) Attempting to maintain a problem consciousness
(3) Attempting to discover mathematical problems in phenomena
2. Attempting to take logical actions
(1) Attempting to take actions that match the objectives
(2) Attempting to establish a perspective
(3) Attempting to think based on the data that can be used, previously learned items, and assumptions
3. Attempting to express matters clearly and succinctly
(1) Attempting to record and communicate problems and results clearly and succinctly
(2) Attempting to sort and organize objects when expressing them
4. Attempting to seek better things
(1) Attempting to raise thinking from the concrete level to the abstract level
(2) Attempting to evaluate thinking both objectively and subjectively, and to refine thinking
(3) Attempting to economize thought and effort
II. Mathematical Thinking Related to Mathematical Methods
1. Inductive thinking
2. Analogical thinking
3. Deductive thinking
4. Integrative thinking (including expansive thinking)
5. Developmental thinking
6. Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)
7. Thinking that simplifies
8. Thinking that generalizes
8. Thinking that specializes
9. Thinking that symbolize
10. Thinking that express with numbers, quantifies, and figures

I want to give an example from syllogism (conclusion from premise).
1st Premise: If the mathematics teacher didn’t come then all of the student are happy.
2nd Premise: if the situation of the class not noisy then some student not happy
3rd Premise: the mathematics teacher didn’t come
The conclusion: the situation of the class noisy.
III. Mathematical Thinking Related to Mathematical Contents
1. Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of sets)
2. Focusing on constituent elements (units) and their sizes and relationships (Idea of units)
3. Attempting to think based on the fundamental principles of expressions (Idea of expression)
4. Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)
5. Attempting to formalize operation methods (Idea of algorithm)
6. Attempting to grasp the big picture of objects and operations, and using the result of this understanding (Idea of approximation)
7. Focusing on basic rules and properties (Idea of fundamental properties)
8. Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)
9. Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of formulas)

One thing that you must remember, power of mathematics is our critical thinking.