機會率哲學 2.2

Posted on November 6, 2012 by rpflee

The problem of induction 1.2 | 西瓜 6.2

這段改編自 2010 年 4 月 3 日的對話。

第二個「解答」，來自英國哲學家 A. J. Ayer。那不是一個正式的「解答」，而是指出「歸納法是理性的」這句說話，根本是重言句。又或者說，「歸納法為何符合理性」這個問題，根本是多餘的，問來也沒有意思。根據正常人對「理性」和「歸納法」這兩個字眼的用法，「理性」已經包含了「歸納法」。正如「亞洲人是人」是重言句，因為「亞洲人」的意思，已包括了「人」。

換句話說，「歸納法」這個概念，一早已經裝嵌於「理性」之中。「理性」的其中一個必要元素是，懂得使用「歸納法」。例如，有一個嬰兒，手指不慎接觸到蠟燭的火光，感覺到痛，他就立刻縮手，以免再受傷。自此以後，他對蠟燭的火光，都存有戒心，不敢再接觸。那樣，我們會覺得那個嬰兒，是一個「理性的嬰兒」。「理性」的地方在於，縱使不自覺，他也運用了「歸納法」：「上次我手指碰到火光時，感覺到痛。下次碰到時，很可能都會那樣，所以最好避之則吉。」

相反，如果有另一個嬰兒，太早有哲學思考的話，他就可能會質疑「歸納法」：「雖然我上次被火灼傷，但那並不代表，我下次都會被火灼傷，所以，我可以再把手指，放於火光之中，再試一試。」那樣，我們會覺得那個嬰兒，是一個「瘋狂的嬰兒」。

這個答法的問題在於，「歸納法」和「理性」的關係，並不如「亞洲人」和「人」的關係那麼明顯。「亞洲人」這個概念，很明顯包括了「人」的元素。我們不會追問，究竟「人」這個概念，是如何嵌入「亞洲人」之中？但是，我們卻可以繼續追問，究竟「歸納法」這個概念，是如何嵌入「理性」之中呢？

— Me@2012.11.05

機會率哲學 2.1

Posted on November 5, 2012 by rpflee

The problem of induction 1.1 | 西瓜 6.1

這段改編自 2010 年 4 月 3 日的對話。

大學二年級時，我們曾經旁聽李教授的「哲學分析」科。其中一課是講「歸納法問題」。「歸納法」的意思是，過往重複發生過很多次的事件，我們估計將來都會發生。

但是，「歸納法」是沒有「必然性」的。例如，中學甲於過去十年，每年公開試英文科的合格率，都達九成以上。但是，你最多只能「預測」，而不能百分百「保證」，來年都是那樣。

又例如，根據你的電視機例子，這個牌子這個型號，來自同一條生產線的電視機，即使之前十萬部的壽命，都超過三年，售貨員最多只能「預測」，而不能百分百「保證」，你買的那一部，都是那樣。

過去會發生的事情，即使已重複發生了很多次，也不代表，將來會發生。那就引發了，你所追問的「歸納法問題」。「歸納法問題」的意思是，既然我們運用「歸納法」所得的結論，是沒有保障的，為何我們要接受「歸納法」呢？又或者說，既然我們運用「歸納法」所預測的將來，不一定是正確的，「歸納法」還符合「理性」嗎？

其中一個答法是，人類利用「歸納法」，過往無論是在日常生活、科學研究，還是科技發展，都取得了鉅大的成功，所以「歸納法」是可信可用的。

但是，這個說法正正是利用了「歸納法」本身，去辯護「歸納法」，循環論證也。

— Me@2012.11.05

Existence, 4

Posted on October 23, 2012 by rpflee

Why does the universe exist? 2

The sentence “the universe does not exist” is meaningless. However, its limited version “there are nothing” or “the universe has nothing” may be meaningful.

If the universe is finite in space, in principle, you can search all over the space to confirm that there are really nothing. So it seems that the sentence “the universe has nothing” does not violate the confirmation principle. However, there are three problems.

First, spacetime is also a “thing”, provided that the definition of the word “thing” is not limited to “object” or “matter”. Second, “spacetime” has no meaning if there are no matter and no energy. Moreover, you, as an observer, is also a “thing”.

After all, “the universe has nothing” is meaningless, in the sense that it violates the confirmation principle.

— Me@2012.10.15

Why does the universe exist? 1.2

Posted on October 18, 2012 by rpflee

Existence, 3.2

Verification principle, 3

The sentence “the universe does not exist” has no meanings, because it violates the confirmation principle. When we say that “dogs do not exist in this room“, we can search all over this room to prove the non-existence of dogs. However, the definition of the word “universe” is “everything”. So the universe has no “outside”. The universe is not contained within a bigger system. So when we say that “the universe does not exist“, we cannot search all over some bigger environment to prove the non-existence of the universe, even in principle.

The sentence “the universe exists” has the same problem. It also violates the confirmation principle. When we say that “a dog exists in this room“, as long as we can find a dog within the room, we prove the existence of the dog. However, the definition of the word “universe” is “everything”. So the universe has no “outside”. The universe is not contained within a bigger system. So when we say that “the universe exists“, we cannot “find” the universe, even in principle.

Whatever we find, such as a dog, a room, a house, a city, etc., is only part of the universe. “Part of the universe exists” does not imply “the universe exists“. For example, I have part of one million dollars, such as 500 thousand dollars, doesn’t mean that I have one million dollars.

— Me@2012.10.15

— Me@2012.10.18

西瓜 5

Posted on September 28, 2012 by rpflee

[physical geometry]

In so far as the statements of geometry speak about reality, they are not certain;

[mathematical geometry]

and in so far as they are certain, they do not speak about reality.

— Albert Einstein

Analytic statements are about the languages.

Synthetic statements are about the world.

Choosing the best language describing the world is itself a synthetic problem.

— Me@2012-03-24 12:02:44 AM

“Is logic empirical?” is not a valid question, because it does not specify the meaning of “logic”.

“Is logic empirical?” is due to the confusion of two different concepts.

If you have no such confusion, the answer to the question is trivial.

As systems of analytical statements, the different theories of logic are not empirical.

But choosing the best among the logic systems to describe the real world is itself empirical.

— Me@2012-09-23 05:10:23 PM

連繫智力 2.2

Posted on March 26, 2012 by rpflee

無足夠資料 7.2 | 西瓜 4

無知（matters of facts）綜合句

要去除無知，就要博覽群書，看破紅塵，以獲取充足的思考材料。

愚蠢（relations of ideas）重言句

要刪減愚蠢，就要連繫意念，融會貫通，以建構高速的思考網絡。

— Me@2012.03.26

連繫智力

Posted on March 20, 2012 by rpflee

軟硬智力 3 | 西瓜 3 | 程式員頭腦 13 | Amazing Gags 1.2

這段改編自 2010 年 3 月 20 日的對話。

（安：如果「比喻」是那麼重要，可不可以這樣說：只要觀察一個人，是否能夠善於利用比喻，就可以估計到他的思考水平？）

不一定，因為那有一個技術上的困難。如果一個人不是從事教學工作，他就近乎沒有需要，透過運用比喻，去解釋複雜的意念。那樣，你就沒有理據，由他利用比喻的頻率，去判斷他智力的高低。

「Intelligence is the ability to recognize connections. – Carolus Slovinec」

智力的一個核心元素是，連繫意念的能力。提出比喻，是這種能力的一個典型示範，因為所謂「比喻」，就是察覺到，原本貌似互不相干的兩個意念，背後原來有關係。利用那個關係，或者相似之處，把那兩個意念相連起來，就有助人增強理解之效。

如果一個「比喻」，格外奇特有趣，加上能夠給予聽眾，一點魔幻感覺的話，就為之一個「笑話」。讀書的致勝之道，在於將千般知識和萬種意念，融會貫通。所以，無論是「比喻」還是「笑話」，都對教學有很大的幫助。這種「認清不同意念之間關係」的本領，構成了智力的主要部分。

但是，意念的關係，並不一定是比喻。例如「1 + 1 = 2」，道出了「一」、「二」、「加」和「等如」這四個概念的關係。但是，它並不是一個「比喻」。換句話說，即使你少用比喻，也不代表思考水平低，因為智力的化身，除了「善用比喻」外，還有其他。

— Me@2012.03.20

Verification principle, 2

Posted on October 21, 2011 by rpflee

Verification- and falsification-principles

The statements “statements are meaningless unless they can be empirically verified” and “statements are meaningless unless they can be empirically falsified” are both claimed to be self-refuting on the basis that they can neither be empirically verified nor falsified.

— Wikipedia on Self-refuting idea

Verification principle: That meaningful statements should be analytic, verifiable or falsifiable

Falsifiability: The possibility that an assertion may be disproved

— Wikipedia on Verification theory

Strong verification principle (aka verification principle) may or may not be self-refuting, depending on whether your regard it as an analytic statement or not. It can be regarded as an analytic statement (aka tautological statement) in a sense that verification principle defines what “meaningful” means, distinguishing meaningful statement from meaningless one. It is related to the definitions of “analytic statement” and “synthetic statement”.

Weak verification principle (aka confirmation principle) is not self-refuting.

Falsification principle is not self-refuting. Falsification principle is about science statements. Itself is not a science statement. Instead, it is part of the definition of “science statements” (aka synthetic statements). So it should not be applied to itself.

— Me@2011.10.21

Make a difference, 2

Posted on September 15, 2011 by rpflee

Verification principle

Verificationism is the view that a statement or question is only legitimate if there is some way to determine whether the statement is true or false, or what the answer to the question is.

Verification principle: That meaningful statements should be analytic, verifiable or falsifiable.

— Wikipedia on Verificationism

檢證原則

一句句子要有意思，你要講得出，至少在原則上，它在什麼情況下為之真、在什麼情況下為之假。換句話說，你要講得出，至少在原則上，如何證明或者否證它。否則，那句句子就沒有意義。

檢證原則 –> 印證原則

但是，有時即使一些句子明明是有意義的，在原則上，也沒有可能百分百證明，它們是正確的。

例如，科學理論句子的特性是，只要有一個妥當執行的實驗，和它的預測不相符，就足以否證它。相反，無論有多少次實驗的結果，和該個科學理論的預測吻合，你也不能保證，下一次的實驗結果，仍然會是那樣。換句話說，無論你做多少次實驗，也不能百分百證明，那句科學理論句子是正確的。

所以，我們放寬了「檢證原則」的要求，把它改編為「印證原則」。一句句子即使不能通過「檢證原則」，如何能夠通過「印證原則」的話，我們仍然可以視之為有意義。

印證原則

一句句子即使不能通過「檢證原則」，如果你可以講得出，至少在原則上，如何提高它的可信度，我們仍然可以視之為有意義。

— Me@2011.09.15

西瓜 2

Posted on August 12, 2010 by rpflee

這段改篇自 2010 年 3 月 18 日的對話。

（CN：為什麼在「可見光譜」中（visible light spectrum），紅色光的波長是最長的？）

你彷彿是在問我：「為什麼爺爺是爸爸的爸爸？」

那是因為「爺爺」這個詞語，是「爸爸的爸爸」的簡稱。同理，「紅色光的波長是最長的」的原因是，我們把「可見光」中，最長波長的光，簡稱為「紅色」。

— Me@2010.08.12

西瓜

Posted on February 15, 2010 by rpflee

有些句子有意思，有些句子沒有意思。而有意思的句子之中，可再分成兩類：analytic propositions（重言句/恆真式）和 synthetic propositions（綜合句）。

重言句只是概念之間的關係（relations of ideas）。例如：

1. 冰箱內有西瓜或者沒有西瓜。

2. 我爺爺是我爸爸的爸爸。

重言句的好處是它絕對準確。不好處是它沒有任何信息內容，對世界沒有任何描述。

綜合句是對事實的陳述（matters of facts）。例如：

1. 冰箱內有西瓜。

2. 愛因斯坦是我爸爸的爸爸。

綜合句的不好處是有可能錯。好處是它有信息內容，對世界有描述。

邏輯學和純數學是重言句系統。

物理學和其他科學是綜合句系統。

— Me@2010.02.15

Contents Chapter 1

Posted on May 24, 2008 by rpflee

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 Constrast
1.1.6 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

2 Applied Mathematics
3 Storyline
4 Master
5 Writing
6 Doctor
7 Painting

A Storyarc

2008.05.24 Saturday $\copyright CHK^2$

1.1.6 What is Mathematics?

Posted on May 20, 2008 by rpflee

As long as you can realize the difference between analytic statements and synthetic statements, you can know what pure mathematics is.

Pure Mathematics is a system of useful analytic statements.[10]

Pure Mathematics is a system of useful tautologies, whether obvious or unobvious.

In Physics, every generation of physicists have to update the previous theories. For example, Einstein’s theory of gravity has updated Newton’s, explaining what Newton’s gravitation could not explain. But for Pure Maths, although every generation of mathematicians also create new mathematics, the new theories do not and cannot contradict the old ones. For example, “1+1=2” is always true, even in thousands of years later.[11]

1.1.7 Why maths is always true but physics is not?[12]

Pure Maths is a system of analytics statements. Analytic statements say nothing about the world. When you say nothing, you cannot be wrong.

[10] In Philosophy of Mathematics, this is called the Formalist’s theory of Mathematics. There is a bug in the formalist’s system. It is about the status of the axiom of infinity. For reference, see Bertrand Russell’s Introduction to Mathematical Philosophy.

[11] Mathematics is eternal, as it is timeless, or outside time.

[12] Mr. Lee

2008.05.20 Tuesday $\copyright CHK^2$

1.1.5 Contrast

Posted on May 17, 2008 by rpflee

Table 1.1: Contrasting

Analytic

Synthetic

Logic
Pure Mathematics

Applied Mathematics
Physics

Relations of ideas

Matter of fact[6]

Deduction

Induction

Say nothing

Say something[7]

Always correct

Maybe wrong

Theory

Experiment

Software

Hardware

Computer Science

Computer Engineering

Mathematical Geometry

Physical Geometry

[6] David Hume
[7] about the physical world

2008.05.17 Saturday $\copyright CHK^2$

1.1.2 Logic and Pure Mathematics

Posted on May 12, 2008 by rpflee

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]

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

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.

2008.05.12 Monday $\copyright CHK^2$

1.1.1 Analytic and Synthetic

Posted on May 8, 2008 by rpflee

Mathematics is about statements.

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.

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.

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.

2008.05.08 Thursday $\copyright CHK^2$

Chapter 1 Additional Mathematics

Posted on May 4, 2008 by rpflee

1.1 General Mathematics

Biology is about living things.
Chemistry is about transformations of matter.
Physics is about matter and energy.

Mathematics is about what?

What is Mathematics?

2008.05.04 Sunday $\copyright CHK^2$

Posted on July 21, 2007 by rpflee

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

July 21, 2007 (c) CHK2

Physics Town

continues

Category Archives: analytic-synthetic | 重言經驗之別