Category Archives: mathematical logic

Alexander Grothendieck

Posted on by

In other words, the philosophy was this: If a phenomenon seems hard to explain, it’s because you haven’t fully understood how general it is. Once you figure out how general it is, the explanation will stare you in the face.

It is, I believe, just plain impossible, without trying to teach you a lot of mathematics, to convey the extremes to which Grothendieck carried this philosophy, or the magnificence of its success. Of course it might still have had its limits.

— The Generalist

— Steve Landsburg

— November 17, 2014

2014.11.25 Tuesday ACHK

Alfred Tarski

Posted on by

Alfred Tarski diagnosed the paradox as arising only in languages that are “semantically closed”, by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the “object language”, while the referring sentence is considered to be a part of a “meta-language” with respect to the object language. It is legitimate for sentences in “languages” higher on the semantic hierarchy to refer to sentences lower in the “language” hierarchy, but not the other way around. This prevents a system from becoming self-referential.

— Wikipedia on Liar paradox

2014.11.21 Friday ACHK

語言科目

Posted on by

這段改編自 2010 年 8 月 11 日的對話。

.

（TK: 通常，最勁（厲害）的人，都是讀數學出身的。彷彿只要數學勁，就幾乎什麼行業，都可以做得到。）

無錯。你知不知道，為什麼數學和中、英文一樣，都歸入「主科」？

漫畫化地講，因為三科也是「語言科目」。

中文，是中國人的語言；

英文，是地球人的語言；

而數學，則是宇宙人的語言。

「宇宙人語言」的意思是，在宇宙中的任何一個角落，一加一都是等於二。無論你移居去哪一個星球，你也不用擔心，你已學的數學知識，不再適用。數學可以說是，應用範圍取廣的知識。

中、英、數三科也是「語言科目」的意思是，它們不只是一般的知識，而且還直接是，其他科目的工具。其他知識中的大部分，也要用中、英、數來表達。

微觀而言，中、英、數的好壞，會直接影響到其他科的成績高低。宏觀來說，缺少其中一項語言的話，你將來的主修，或者專業的選項，就會大大減少。

而更嚴重的是，你的數學成績，會直接反映了，你智力的高低。假設你的數學一流，你很難可以說服到人，其實你的智力奇低。

— Me@2014.10.29

.

.

2014.10.29 Wednesday (c) All rights reserved by ACHK

Functional Paradox

Posted on by

paradox ~ mixing a level with its meta-level

There is no mixing-level problem for the equation

f(x) = x,

because it just means that two variables have the same value. In other words, the value of a function of x is equal to the value of x.

However, for the equation

f = x,

there is a mixing-level problem, resulting no meanings; because

the equation means that a function (aka a formula of a number) is equal to a variable (aka a number).

A formula of a number is a structure. It cannot be a number.

— Me@2013-07-16 10:45 AM

2014.10.13 Monday (c) All rights reserved by ACHK

I am a Strange Loop, 4.3

Posted on by

記憶奇異圈 1.3 | Godel, Escher, Bach, 2.3

這段改編自 2010 年 4 月 10 日的對話。

（安：他那兩本書，也和「哥德爾定理」有關？）

有關，但它只是例子，而不是主旨。Douglas 想帶出的是，「自我」意識，其實來自一些「Strange Loop」（奇異圈）。

「奇異圈」的意思是，一些有「自我指涉」的系統。例如，「這句話是假的」這句子就有「自我指涉」的成份。那樣，你就可以把它視為，有一個「奇異圈」。

「奇異」之處在於是，如果句子「這句話是假的」是真的，它就是假的。但是，如果「這句話是假的」是假的，它就是真的。

「自我指涉」的程度越高，「自我意識」就會越強烈。（留意，「自我指涉」中的「自我」，和「自我意識」中的「自我」，意思不同。）例如，（我估計，）狗的「自我意識」比老鼠強，是因為狗的「反思」能力比老鼠高。同理，人的「自我意識」比狗強，是因為人的「反思」能力比狗高。

…

Douglas 的成名作是《Godel, Escher, Bach》，就是透過眾多「奇異圈」的例子，企圖帶出「自我」和「意識」的來源。但是，由於書中的例子，橫跨了太多課題，例如，邏輯、數學、物理、繪畫、音樂、文字等，而又橫跨得十分精采，導致近乎沒有人，知道該書的主旨是什麼。

所以，作者於大概三十年後，寫了《I am a Strange Loop》（「我」是一個奇異圈）。為免再令讀者誤會，作者把該書的主旨，直接用作書名。

雖然，在我讀了那兩本書之後，仍然覺得，對於「『自我』和『意識』從何而來」這問題，作者最終也沒有給予，一個滿意的答案，但是，我不會怪作者，因為過程之中，我獲取了各門學問中，大量的靈感。

情況就有如，尋寶圖中所指的地點，最終發現沒有任何寶藏。但是，在尋寶的路途上，遇到的寶藏，比尋寶圖中的預定目標還要多。

— Me@2014.09.29

2014.09.29 Monday (c) All rights reserved by ACHK

記憶奇異圈

Posted on by

I am a Strange Loop, 4.2 | Godel, Escher, Bach, 2.2 | Copy Me 11

這段改編自 2010 年 4 月 10 日的對話。

如果不是該書的精湛展示，我相信可能要，用多十年的思考和研究，才能領悟到那些道理。

（安：他那兩本書，也和「哥德爾定理」有關？）

有關，但它只是例子，而不是主旨。Douglas 想帶出的是，「自我」意識，其實來自一些「Strange Loop」（奇異圈）。

「奇異圈」的意思是，一些有「自我指涉」的系統。例如，「這句話是假的」這句子就有「自我指涉」的成份。那樣，你就可以把它視為，有一個「奇異圈」。

「奇異」之處在於是，如果句子「這句話是假的」是真的，它就是假的。但是，如果「這句話是假的」是假的，它就是真的。

「自我指涉」的程度越高，「自我意識」就會越強烈。（留意，「自我指涉」中的「自我」，和「自我意識」中的「自我」，意思不同。）例如，（我估計，）狗的「自我意識」比老鼠強，是因為狗的「反思」能力比老鼠高。同理，人的「自我意識」比狗強，是因為人的「反思」能力比狗高。

（安：其實你用「記憶」會不會較容易理解呢？

例如，你可以說：「人的『意識』比狗強，是因為人的『記憶』，比狗的『記憶』，較詳細和較多元化。」）

都可以，因為，雖然「自我指涉」和「記憶」意思不同，但十分相關。例如，如果有一隻昆蟲是，近乎完成沒有記憶力的，例如牠的任何記憶，都只能維持到三秒以內，那樣，在每一刻，牠也只能以當時的本能即時反應來行事。在這情況下，我們把那隻昆蟲視為，沒有「意識」的。

（安：你又怎麼知道，昆蟲是沒有記憶的呢？）

我是說「如果」。你不喜歡的話，用「機器」作例子，可能會好一點。

假設在十年後（2024 年），工程師製造了，一些有意識的機械人。那樣，市面上就會有兩種機械人 —— 「有意識」和「無意識」的。

有一天，有朋友帶了，他新買的機械人見你，要你估計一下，那機械人有沒有意識。

如果那隻機械人是沒有記憶的 —— 在每一刻，它也只能以當時的本能即時反應，來行事的話，你就可以肯定，它是沒有意識的。例如，它的功用就只是，每當偵測到主人，身邊的空氣過熱時，就立刻開風扇。

如果那隻機械人是有記憶的，那樣，它就有可能有意識。

「記憶」，是「意識」的先決條件，但不是充份條件。有「意識」的生物或機器，就一定有「記憶」；但是，有「記憶」的東西，亦未必有「意識」。

— Me@2014.09.27

2014.09.27 Saturday (c) All rights reserved by ACHK

I am a Strange Loop, 4

Posted on by

Godel, Escher, Bach, 2

這段改編自 2010 年 4 月 10 日的對話。

如果你接受到，「自我」其實是「軟件」，你就會明白，莫講話「教學」，即是只是「對話」，也是（部分）「自我」（互相）「複製」的過程。所以，如果你大大受過一位神級導師的影響，在很大程度上，你就是他。他的思想已經複製到你的腦中。

我有很多這類想法，都是來自《I am a Strange Loop》（「我」是一個奇異圈）這本書。

（安：那本書的內容是關於什麼呢？）

「自我」。

《I am a Strange Loop》的作者是 Douglas R. Hofstadter。而 Douglas 的成名作是《Godel, Escher, Bach》。

我從《I am a Strange Loop》中，吸收很深厚的教學功力。我當時的感覺是相當震撼的。這書竟然可以將，那麼高深的概念，例如「哥德爾不完備定理」、「自我來源」、「多重自我」、「自我程式」等，逐步舖排，表達到連初學者的我，也能明白。而它的舖排，往往是橫跨幾個章。如果不是作者對那幾門知識，有極深刻的瞭解，他並不可能作到，那樣宏觀的佈局。

雖然當年的我，程度不低，但是，那時的我，並沒有那幾門學問的詳細背景知識。例如，在那之前，我只知道「數理邏輯」這個學問中，有幾條重要的定理，都叫做「哥德爾定理」。除了名字以外，我對「哥德爾定理」的理解近乎是零。但是，經過《I am a Strange Loop》的介紹，我就了解到「哥德爾定理」的核心思想是什麼。

如果不是該書的精湛展示，我相信可能要用多十年的思考和研究，才能領悟到那些道理。

（安：他那兩本書，也和「哥德爾定理」有關？）

…

— Me@2014.09.20

2014.09.23 Tuesday (c) All rights reserved by ACHK

Physical information

Posted on by

Information itself may be loosely defined as “that which can distinguish one thing from another”. The information embodied by a thing can thus be said to be the identity of the particular thing itself, that is, all of its properties, all that makes it distinct from other (real or potential) things. It is a complete description of the thing, but in a sense that is divorced from any particular language.

— Wikipedia on Physical information

2014.09.12 Friday ACHK

Extreme physical information

Posted on by

In a theory developed by B. Roy Frieden, “physical information” is defined as the loss of Fisher information that is incurred during the observation of a physical effect. Thus, if the effect has an intrinsic information level J but is observed at information level I, the physical information is defined to be the difference I − J. This defines an information Lagrangian. Frieden’s principle of extreme physical information or EPI states that extremalizing I − J by varying the system probability amplitudes gives the correct amplitudes for most or even all physical theories. The EPI principle was recently proven. It follows from a system of mathematical axioms of L. Hardy defining all known physics.

— Wikipedia on Physical information

2014.08.31 Sunday ACHK

Integrating factor

Posted on by

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.

— Wikipedia on Integrating factor

2014.08.26 Tuesday ACHK

Entropy (arrow of time)

Posted on by

Why did the universe have such low entropy in the past, resulting in the distinction between past and future and the second law of thermodynamics? Why are CP violations observed in certain weak force decays, but not elsewhere? Are CP violations somehow a product of the Second Law of Thermodynamics, or are they a separate arrow of time? Are there exceptions to the principle of causality? Is there a single possible past? Is the present moment physically distinct from the past and future or is it merely an emergent property of consciousness? Why does time have a direction?

— Wikipedia on List of unsolved problems in physics

2014.08.17 Sunday ACHK

Self-replication

Posted on by

Von Neumann’s crucial insight is that part of the replicator has a double use; being both an active component of the construction mechanism, and being the target of a passive copying process. This part is played by the tape of instructions in Von Neumann’s combination of universal constructor plus instruction tape.

The combination of a universal constructor and a tape of instructions would i) allow self-replication, and also ii) guarantee that the open-ended complexity growth observed in biological organisms was possible. The image below illustrates this possibility.

This insight is all the more remarkable because it preceded the discovery of the structure of the DNA molecule by Watson and Crick, though it followed the Avery-MacLeod-McCarty experiment which identified DNA as the molecular carrier of genetic information in living organisms. The DNA molecule is processed by separate mechanisms that carry out its instructions and copy the DNA for insertion for the newly constructed cell. The ability to achieve open-ended evolution lies in the fact that, just as in nature, errors (mutations) in the copying of the genetic tape can lead to viable variants of the automaton, which can then evolve via natural selection.

— Wikipedia on Von Neumann universal constructor

2014.08.11 Monday ACHK

Change

Posted on by

By the definition of “time”, anything in time must be changing. Anything not in time, such as the dead, cannot change.

If you do not keep changing for better (improve, grow), you are effectively dead.

— Me@2011.10.10

Although this paragraph is not 100% accurate, its spirit is useful.

— Me@2014.08.09

2014.08.09 Saturday (c) All rights reserved by ACHK

For all, 7

Posted on by

Whatever you (physically or mentally) encircle, it is part of the universe, not the universe.

If you claim that what you have encircled is the whole universe, I will ask,

“How can you know?” 

If you define the word “universe” as “everything“, how can you know that there are no other parts unknown to you?

— Me@2014-07-27 02:50:20 PM

2014.07.27 Sunday (c) All rights reserved by ACHK