Category Archives: mathematical logic

Make a difference, 2

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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

2011.09.15 Thursday (c) All rights reserved by ACHK

From Heisenberg to Godel

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My student Mike Stay did computer science before he came to UCR. When he was applying, he mentioned a result he helped prove, which relates Godel’s theorem to the Heisenberg uncertainty principle:

2) C. S. Calude and M. A. Stay, From Heisenberg to Godel via Chaitin, International Journal of Theoretical Physics, 44 (2005), 1053-1065. …

Now, this particular combination of topics is classic crackpot fodder. People think “Gee, uncertainty sounds like incompleteness, they’re both limitations on knowledge – they must be related!” and go off the deep end. So I got pretty suspicious until I read his paper and saw it was CORRECT… at which point I definitely wanted him around! The connection they establish is not as precise as I’d like, but it’s solid math.

— This Week’s Finds in Mathematical Physics (Week 230)

— John Baez

2011.05.18 Wednesday ACHK

A Fraction of Algebra

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As a mathematician there is a story I hear a lot. It tends to come up whenever I tell someone what I do for the first time, and they admit that they don’t really like, or aren’t very good at, mathematics. In almost every case, if I bother to ask (and these days I usually do), I find that the person, once upon a time, was good at and liked mathematics, but somewhere along the way they had a bad teacher, or struck a subject they couldn’t grasp at first, and fell a bit behind. From that point on their experiences of mathematics is a tale of woe: because mathematics piles layer upon layer, if you fall behind then you find yourself in a never ending game of catch-up, chasing a horizon that you never seem to reach; that can be very dispiriting and depressing.

— The Narrow Road, Zen and the Art of Mathematics

2011.05.12 Thursday ACHK

Autological

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1. An adjective is autological if and only if it describes itself. For example “short” is autological, since the word “short” is short. “English,” “unhyphenated” and “pentasyllabic” are also autological.
  
2. An adjective is heterological if it does not describe itself. Hence “long” is a heterological word, as are “abbreviated” and “monosyllabic.”

  
Is “autological” autological?

It can be chosen consistently to be either:

    * if we say that “autological” is autological, and then ask if it applies to itself, then yes, it does, and thus is autological;
    * if we say that “autological” is not autological, and then ask if it applies to itself, then no, it does not, and thus is not autological.

This is the opposite of the situation for heterological: while “heterological” logically cannot be autological or heterological, “autological” can be either.

In logical terms, the situation for “autological” is:

    “autological” is autological if and only if “autological” is autological
    A if and only if A, a tautology

while the situation for “heterological” is:

    “heterological” is heterological if and only if “heterological” is autological
    A if and only if not A, a contradiction.

— Wikipedia on Grelling–Nelson paradox

2011.02.24 Thursday ACHK

Metamagical Themas

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There are three articles centered on the Lisp programming language, where Hofstadter first details the language itself, and then shows how it relates to Godel’s incompleteness theorem.

The title is an example of wordplay: it is an anagram of Mathematical Games, the title of Martin Gardner’s column that Hofstadter’s column succeeded in Scientific American.

— Wikipedia on Metamagical Themas

2011.02.23 Wednesday ACHK

樹 | 根, 2

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超世 3

樹要往高往上往光輝處生，根要往下往深往黑暗處長。

— 改篇自尼采

A man does not show his greatness by being at one extremity, but rather by touching both at once.

— Blaise Pascal

越能入世的人越能出世；越能出世的人越能入世。

— Me@2011.02.08

2011.02.13 Sunday (c) All rights reserved by ACHK

天堂力量

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這段改篇自 2010 年 4 月 8 日的對話。

我年輕時有極嚴重的情緒病。「唯有經過地獄的磨難，才有創造天堂的力量。」（改篇自尼采）那我有什麼「創造天堂的力量」呢？

我可以幫人解決心理問題。很多時，當有學生對我訴說讀書的困難時，我都發現他所謂的「讀書困難」，其實是情緒疾病和心理問題。而那些麻煩往往是我當年遇過的，所以我可以詳細描述解決方法：

「你的問題其實是這個。你可以用這個方法。第一步是這樣。第二步是那樣… 有八成機會會成功。不行的話，你可以用這個後備計劃 …」

— Me@2010.12.01

What does not kill me, makes me stronger.

— Friedrich Nietzsche

2010.12.01 Wednesday (c) All rights reserved by ACHK

互化法

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這段改篇自 2010 年 4 月 8 日的對話。

人生，就是一場考不完的試。

以我現在的心理年齡，我再不會有特別的考試壓力，因為對我來說，每一日的生活，就好像考試一樣。

試想想，如果在某一天你原本打算做七件事，而到頭來只做到三件的話，你會睡不著覺。又例如，如果你在某次演講中詞不達意，你會在演講後鬱鬱不歡，無法釋懷，直至下一次演講成功為止。

我以前做中學教師時，每日的生活就好像考幾次口試一樣。平均每二十課課堂，我覺得只有一課是表達得好，沒有遺憾的。其餘的十九課，都是失敗收場，從而令我之後神志失常。那也沒有辦法。人生就是那樣。

將「日常生活」化成「考試」，就等如將「考試」化成「日常生活」。如果你將日常生活化到好像考試那麼緊張的話，你的考試就會好像日常生活那麼不緊張。

— Me@2010.11.25

You do not have to fear the exams, for every day is an exam.

— Me@2009.11.04

2010.11.25 Thursday (c) All rights reserved by ACHK

心理火警演習

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這段改篇自 2010 年 4 月 8 日的對話。

我時常叫你在平日「按年份、計時間、計分數」做 past paper（歷屆試題）。其中一個主要目的，是要預演考試時候的心理狀態。

除非你以前有案例，曾經試過在考試時，因為自己「恐懼」而導致做不到任何題目，否則，你毋須為自己的「恐懼」而擔憂。

考試時你可以「恐懼」。但是，不要「恐懼」自己的「恐懼」。知不知個微妙之處？

（CN：我知。）

如果你考試時介意自己的「恐懼」，而企圖迫自己「不要恐懼」的話，你會更加「恐懼」。相反，如果你知道「考試時會恐懼」是人之常情，而接受自己的「恐懼」的話，你反而不會過份「恐懼」。你的「恐懼」會在一個可控制的範圍內，不會對你的考試表現，有明顯的實質影響。

有很多時候，「解決問題」的最好方法，未必是「直接解決」，而是「把問題 transcend 掉」，令到原本的問題不再重要。換句話說，有很多時候，「解決問題」的最好方法，是令到自己毋須再解決那個「問題」。

現在的重點，並不是研究如何令到自己「在考試期間不要恐懼」，而是令到自己不再需要「沒有恐懼」。換言之，你要 transcend 了「恐懼」，令到自己無論在有沒有「恐懼」的情況下，都可以達到最佳成績。

所以，你要在平日時常「按年份、計時間、計分數」做 past paper，以作大量的考試心理演習。

— Me@2010.10.30

2010.10.30 Saturday (c) All rights reserved by ACHK

西瓜 2

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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

2010.08.12 Thursday (c) All rights reserved by ACHK