what is mathematics?

<The Detail of This Chapter.pdf>

the first event: 1, 2, 3,...

then, we know that counting means +1

the first principle: no more, nor less.

addition, subtraction, multiplication, division

subtraction lead to 0 and negatives

division lead to fraction

the closeness of operation=>the set of number: natural number, integral number, rational number.

<from the demand of closeness to completeness, we introduce the notion of infinity.>

complete induction

how to denote number? base and power

how to solve algebraic equation?

the complete of order=>real number set

the second principle: completeness

any two elements of set R, a>b, or b>a,

【the base of analysis】

【the power of set】

but, when we need describe something like a point on a plane, or a solution of a special equation, such a set is not enough.

complex number: if x2=-1, whether x > or < any real number?

【complex number】

vector: is (3, 6) > or < (9, 2)?

the completeness of field: algebraic equation

【algebraic field】

the completeness of linear space: the structure of linear space

【linear space】

so, when we create mathematics from NATURE, we get algebra at first!

in succession, we need add more detail into the algebraic "coordinatisation" to realize or represent our understanding about NATURE.  

the third principle: sign variable


the second event: to describe the causalities and the relationships that come from NATURE.

algebraic function, transcend function based on real or complex set or any other algebraic set.


realization of function

the forth principle: geometry - the logic from NATURE

algebra is the primary method that we grasp NATURE mathematically. and then, because geometry is one of the basic gates that we enter NATURE, we usually get useful notions from geometry as a part of mathematics.