Category Archives: mathematical logic

Eternal return

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Copy Me, 8 | Identical particles 5

Eternal return (also known as “eternal recurrence”) is a concept which posits that the universe has been recurring, and will continue to recur, in a self-similar form an infinite number of times across infinite time or space. It is a purely physical concept, involving no supernatural reincarnation, but the return of beings in the same bodies.

— Wikipedia on Friedrich Nietzsche

The identity of indiscernibles is an ontological principle which states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa; to suppose two things indiscernible is to suppose the same thing under two names. It states that no two distinct things (such as snowflakes) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science.

— Wikipedia on Identity of indiscernibles

The difference that makes no difference makes no difference.

—

Eternal recurrence is not a useful concept.

If two periods of time are identical in all details, they are actually the same period, not two periods. If two periods of time are not identical in all details, the second period is not an “eternal return” of the first period.

— Me@2013-06-04 01:30:16 AM

2013.06.11 Tuesday (c) All rights reserved by ACHK

Clifford algebra

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More precisely, Clifford algebras may be thought of as quantizations (cf. quantization (physics), Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra.

— Wikipedia on Clifford algebra

2013.06.04 Tuesday ACHK

Consciousness | 自我 | 意識

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There is no direct self-interaction.

An observation or measurement is an interaction between the observer and the observed, involving two objects. So there is logically impossible to have direct self-observation.

— Me@2013-05-29 12:09:46 AM

You cannot see yourself directly from your own point of view. Instead, you can only see your mirror, photo, and video images. In other words, you can only indirectly see yourself.

— Me@2013-01-20 01:08:34 AM

— Me@2013-05-31 10:45:49 PM

2013.05.31 Friday (c) All rights reserved by ACHK

Cumulative concept of time, 14

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cause

~ necessary condition

~ part of

Event A is a cause of event B

= Event A is a necessary condition of event B

= Event A is part of event B

We can remember the past but not the future because the past is part of the future; the whole contains its parts, but not vice versa. 

— Me@2011.08.21

2013.05.19 Sunday (c) All rights reserved by ACHK

Terence Tao 1

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A powerful way to prove a mathematical result (e.g. an identity of the form A=B) is to introduce a new object or concept (say C) and connect it in two different ways to the original problem. For instance, if one can show that A=C and one can also show that C=B, then one can deduce that A=B. More generally, one can introduce n new objects or concepts, and establish at least n+1 non-trivial connections between these objects and each other, or to the original problem; for instance, if one introduces two new objects C,D and three connections, two of which A=C, D=B are to the original problem, and one of which C=D is between the newly introduced objects, then one has again established A=B.

A typical example of this is the use of complex analysis methods to solve a real analysis problem, as per Hadamard’s famous dictum “The shortest path between two truths in the real domain passes through the complex domain”. For instance, suppose one wants to compute some real integral A and show that it equals some value B. To do this, one can introduce two new concepts (the complex contour integral C, and the notion of a residue of a pole D); write the real integral as a contour integral (establishing a result of the form A=C), invoke the residue theorem (which is a result of the form C=D), and then compute the residues (a result of the form D=B), to obtain the final desired result A=B.

— Terence Tao

2013.05.11 Saturday ACHK

Pains

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Pains occupy a distinct and vital place in the philosophy of mind for several reasons. One is that pains seem to collapse the appearance/reality distinction. If an object appears to you to be red it might not be so in reality, but if you seem to yourself to be in pain you must be so: there can be no case here of seeming at all. At the same time, one cannot feel another person’s pain, but only infer it from their behavior and their reports of it.

— Wikipedia on Private language argument

2013.03.31 Sunday ACHK

The Metagame

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I’ve been bored at work for many reasons at many different times, but three things stand out as real killers:

1. working on the same project with the same people for years and years,
  
…

These are the symptoms of a problem, not the cause, and I think most jobs will have elements of them. But surprisingly it turns out that – for programmers at least – boredom is a choice. Recently, I chose not to be bored. I chose to think one abstraction level higher. I chose to play the metagame.

…

— Work Is Fascinating: The Metagame

— Mark O’Connor

2013.03.28 Thursday ACHK

Gödel’s speedup theorem

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I only showed that the shortest proof of P(n) using Peano arithmetic is insanely long. I did provide a short proof of P(n). But I did this assuming Peano arithmetic is consistent!

So I didn’t give a short proof of P(n) using Peano arithmetic. I gave a short proof using Peano arithmetic plus the assumption that Peano arithmetic is consistent!

So, if we add to Peano arithmetic an extra axiom saying ‘Peano arithmetic is consistent’, infinitely many theorems get vastly shorter proofs!

This is often called Gödel’s speedup theorem.

— Insanely Long Proofs

— John Baez

2013.03.18 Monday ACHK

微積分 6.6

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無限年 3.6

這段改編自 2010 年 4 月 3 日的對話。

（安：但是， (\delta) 是什麼呢？

你還未賦予 \delta 意義。亦即是話，你對「無限小」的定義，尚未完成？）

無錯。我還需要講清楚，\delta 究竟是什麼。

不過，每一題極限（limit）題目的 \delta 都會不同。\delta 並沒有通用的定義，而是要經過一點運算才知道。例如，以這一題極限題目而言，\delta 剛好等於 (\epsilon)。

( \lim_{x \to 3} \frac{x^2-9}{x-3} ) = 6

意思是，如果你要求數式 和 6 的距離，小於 (\epsilon)，無論 \epsilon 有多麼小，你都一定可以達成，只要你設定 x 和 3 的偏差，小於 (\epsilon)。例如，如果你要求數式和 6 的距離，小於 0.001（\epsilon），只要你設定 x 和 3 的偏差小於 0.001（\delta = \epsilon），就可以達成。

（安：還有，你說那些後期數學家，就是用了這一套避開了「無限小」這個詞彙的語言，來描述牛頓和萊布尼茲，在「微積分初版」中，原本想帶出的意念。

你是否暗示了，其實「微積分初版」中的結果是正確的，雖然運算步驟含糊其詞？）

可以那樣說。「微積分初版」的運算結果大致正確；對於日常用家而言，可信可用。現在中學的「微積分」課程，也是「微積分初版」。

（安：那為什麼還要嚴格定義「無限小」？那是否庸人自擾？）

因為「微積分初版」的運算結果，只是「大致正確」，並非「完全正確」。在高深一點的理論或應用中，「微積分初版」會完全瓦解。

還有，「『微積分初版』的運算結果大致正確」，是事後孔明。「微積分初版」並不知道自己，原來「大致正確」。那是「微積分再版」對它的評語。

一日「無限小」這個邏輯漏洞尚未修補，一日也不知道，「微積分初版」在什麼情況下可以用，什麼情況下不可以。

— Me@2013.03.11

2013.03.11 Monday (c) All rights reserved by ACHK

微積分 6.5

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無限年 3.5

這段改編自 2010 年 4 月 3 日的對話。

當 x 接近 3 時，(x+3) 很明顯會接近 6。所以，結論是，

( \lim_{x \to 3} \frac{x^2-9}{x-3} ) = 6

如果用粗疏的語言，我們會說，當 x 非常接近 3 時， 就會非常接近 6。

如果用「微積分初版」的語言，我們會說，當 x 和 3 的距離是「無限小」時， 和 6 的距離，都會是「無限小」。

如果準確一點的語言，我們會說，當 x 足夠接近 3 時， 就會足夠接近 6；又或者說，無論你要數式的數值，多麼接近 6 都可以，只要 x 足夠接近 3。

如果用後期數學家，所創製的嚴格語言，我要會說，

0\ \exists \ \delta > 0 : \forall x\ (0 < |x – 3 | < \delta \ \Rightarrow \ \left| \frac{x^2-9}{x-3} – 6 \right|

\forall \epsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |x – 3| < \delta \ \Rightarrow \ \left| \frac{x^2-9}{x-3} – 6 \right| < \epsilon)

意思是，如果你要求數式   和 6 的距離，小於 (\epsilon)，無論 \epsilon 有多麼小，你都一定可以達成，只要你設定 x 和 3 的偏差，小於 (\delta)。換句話說，這裡定義了，何謂「足夠接近」。

那些後期數學家，就是用了這套「(ε, δ)-definition of limit」(epsilon-delta definition of limit) 的語言，來描述牛頓和萊布尼茲，在「微積分初版」中，原本想帶出的意念，而又避開了「無限小」這個詞彙。

（安：但是， (\delta) 是什麼呢？

你還未賦予 \delta 意義。亦即是話，你對「無限小」的定義，尚未完成？）

— Me@2013.03.09

2013.03.09 Saturday (c) All rights reserved by ACHK