Category Archives: mathematical logic

1.1.2 Logic and Pure Mathematics

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Pure Mathematics is a system of (nontrivial) tautologies. Roughly speaking, a tautology[2] is an analytic statement.

For example, consider this mathematics statement:

2 + 2 = 4

You do not have to do any kind of real world experiments in order to verify the statement. As long as you know the meanings of the symbols “2”, “+”, “=”, and “4”, you know that the statement is correct, and always. Of course, it says nothing about the physical world.

In pure mathematics, since you cannot and do not have to say anything about the real physical world, you can do[3] anything you like. Just like what you do when designing the rules of chess.[4] You can do anythings as long as they are

consistent and

interesting.[5]

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Is a tautology just a nonsense?

Maybe, maybe not. It depends on context:

When you present an analytic statement as an analytic statement, it is not a nonsense.

When you present an analytic statement as a synthetic statement, it is a nonsense.

[2] 重言句, 恆真式
[3] define
[4] or when programming a software
[5] i.e. useful

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Imagine the following scenes.

1.1.3 Scene One

A primary school student wrote 2 + 2 = 5 in his homework. His mathematics teacher told him that 2 + 2 = 5 was incorrect, “Two plus two should equal Four.” In such a context, the statement is, although analytic, not a nonsense.

1.1.4 Scene Two

After 30 years of research, a physicist declared his research result, “Two plus Two equals Four!!!” In such a context, the statement is, although true, a nonsense.

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2008.05.12 Monday \copyright CHK^2

1.1.1 Analytic and Synthetic

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Mathematics is about statements.

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To know what mathematics itself is, we have to realize that there are two kinds of statements: analytic statements and synthetic statements.

For an analytic statement, there is no information about the objective world. Whether an analytic statement is true or not depends on only the meanings of the component words. No real world experience is needed.

For a synthetic statement, there is some information about the objective world. Whether a synthetic statement is true or not depends on not only the meaning of the component words, but also the objective facts of the world.

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For example, consider this statement:

I have passed the exam or I have not.

It is an analytic statement … because you do not have to check my examination result to verify the statement. As long as you know the meanings of “or” and “not”, you know that the statement is always true. But it says nothing about the world.

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Consider another statement:

I have passed the exam.

It is a synthetic statement … because you have to check my examination result to verify the statement. Even if you know the meanings of “or” and “not”, you do not know whether the statement is true or not. But the statement says something about the world.

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2008.05.08 Thursday \copyright CHK^2

Yin and Yang

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Principles

Everything can be described as either yin or yang.

1. Yin and yang are opposites.

Everything has its opposite—although this is never absolute, only comparative. No one thing is completely yin or completely yang. Each contains the seed of its opposite. For example, cold can turn into hot; “what goes up must come down”.

2. Yin and yang are interdependent.

One cannot exist without the other. For example, day cannot exist without night.

3. Yin and yang can be further subdivided into yin and yang.

Any yin or yang aspect can be further subdivided into yin and yang. For example, temperature can be seen as either hot or cold. However, hot can be further divided into warm or burning; cold into cool or icy.

4. Yin and yang consume and support each other.

Yin and yang are usually held in balance—as one increases, the other decreases. However, imbalances can occur. There are four possible imbalances: Excess yin, excess yang, yin deficiency, yang deficiency.

5. Yin and yang can transform into one another.

At a particular stage, yin can transform into yang and vice versa. For example, night changes into day; warmth cools; life changes to death.

6. Part of Yin is in Yang and part of Yang is in Yin.

The dots in each serve as a reminder that there are always traces of one in the other. For example, humans will always be both good and evil, never completely one or the other.

— Wikipedia

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2007.10.27 Saturday CHK2

Contents

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Contents
Preface 緣起

1 Additional Mathematics
1.1 General Mathematics
1.1.1 Analytic and Synthetic
1.1.2 Logic and Pure Mathematics
1.1.3 Scene One
1.1.4 Scene Two
1.1.5 Contrast
1.1.6 What is Mathematics

1.2 Additional Additional Mathematics
1.2.1 Deduction and Induction
1.2.2 Mathematical Induction
1.2.3 數學歸納法
1.2.4 數學感應法

1.13 Newton’s Binomial Theorem
1.13.1 \displaystyle{(1 + x)^{-1}}

1.13.2 \displaystyle{(1 + x)^{\frac{1}{2}}}

1.16 Exercises

2 Applied Mathematics
3 Storyline
4 Master (On career planning)
5 Writing
6 Doctor (On studying skills)
A Storyarc 

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July 21, 2007 (c) CHK2