Poincaré还是从球面入手

## 仿佛琢空而现

“任何一种科学，如果我们不是把它理解为一种表达力量和控制能力的工具，而只是当作我们人类世代所推进的知识探险，那她不是别的，就只是那个可以名之为和谐的东西。从一个时代到另一个时代，这个和谐或广或窄，或丰或乏，历经一代又一代，一个世纪又一个世纪，依次所显现出来的对于各个主题的精妙映照，仿佛就是琢空而现。”
--译自收割与播种(Reapings and Sowings)（"And every science, when we understand it not as an instrument of power and domination but as an adventure in knowledge pursued by our species across the ages, is nothing but this harmony, more or less vast, more or less rich from one epoch to another, which unfurls over the course of generations and centuries, by the delicate counterpoint of all the themes appearing in turn, as if summoned from the void."）

## 数学是什么？

what is mathematics?

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The Detail of This Chapter.pdf

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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.

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,