Author Archives: rpflee

Generalizing a problem

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One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you generalize something then you are trying to prove a stronger statement). I know that I’ve run into this phenomenon many times, and sometimes it has been extremely striking just how much simpler the generalized problem is.

edited Sep 26 2010 at 8:34
gowers

Great question. Maybe the phenomenon is less surprising if one thinks that there are ∞ ways to generalize a question, but just a few of them make some progress possible. I think it is reasonable to say that successful generalizations must embed, consciously or not, a very deep understanding of the problem at hand. They operate through the same mechanism at work in good abstraction, by helping you forget insignificant details and focus on the heart of the matter.

answered Sep 26 2010 at 10:27
Piero D’Ancona

— Generalizing a problem to make it easier

— MathOverflow

A general case has less information (details) than a special case. 

— Me@2012.03.10

2012.03.13 Tuesday (c) All rights reserved by ACHK

明不明白, 2.1

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這段改編自 2010 年 3 月 20 日的對話。

我以前提過，有一本書叫做「I am a Strange Loop」。書內研究的其中一個要點是，怎樣才為之「明白」？

作者發現，在很大程度上，「明白」就是「比喻」。

（安：什麼意思？）

你試想想，我們學習新東西時，在什麼情況下，才有「明白」的感覺呢？

第一種情況是，把新東西表達成舊東西。例如，「3 乘以 2」為什麼會等如「6」呢？

那是因為「乘」的意思是，把同一個數，加很多次：

3 x 2 = 3 + 3 = 6

第二種情況是，把新東西反覆背誦和運用，令到自己對它熟練到，成為一個習慣為止。那樣，即使沒有任何實質的理解，你也會有「明白」的幻覺。例如，小時候你背誦了「乘數表」，所以你覺得自己明白，為何「九八七十二」。

9 x 8 = 72

但是，你之所以「明白」，並不是因為，你曾經把「9 x 8」化成加數，真正如實地運算「9 + 9 + 9 + 9 + 9 + 9 + 9 + 9」。

第三種情況是，把新東西類比成熟悉的事物。例如，如果你教一個小孩「物質是由粒子組成的」，他可能會一頭霧水。要他「明白」的話，你可以試試這樣說：

物質，是由一些超微小的彈珠（波子）組成的。那些超微小的彈珠，叫做「粒子」。

（安：第三種情況，可以看成第一種的一個特例，因為它都是把未知的東西，翻譯成已知的事物。）

可以這樣說。但是，第一種中的翻譯，是「解釋」；而第三種的，是「比喻」。或者我這樣分比較好：

第一種「明白」，是通過「解釋」而得來的。

第二種是通過「熟習」。

第三種是用「比喻」。

我現在想集中討論的，是第三種情況。

— Me@2012.03.13

2012.03.13 Tuesday (c) All rights reserved by ACHK

Holographic principle, 6

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During this period, Tom Banks, Willy Fischler, Stephen Shenker and Leonard Susskind formulated matrix theory, a full holographic description of M-theory using IIA D0 branes. This was the first definition of string theory that was fully non-perturbative and a concrete mathematical realization of the holographic principle. It is an example of a gauge-gravity duality and is now understood to be a special case of the AdS/CFT correspondence.

— Wikipedia on String theory

2012.03.12 Monday ACHK

功夫傻瓜

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中學實驗報告 2 | 間時間表 3.2

這段改編自 2010 年 5 月 26 日的對話。

無論是你過份懶惰，還是你的老師過份勤力，都足以令你欠下功課鉅債。更加不幸的是，很多時，這兩種情況會同時發生。所以，深明「歸還功課」之道，至關重要。歸還功課的有效方法，可參考我「間時間表 2」和「間時間表 3」兩篇文章。

但是，如果考試臨近，就千萬不要期望，自己可以有足夠時間，去「追回」以前所有欠交的功課。

你要記住，做功課的最終目的，並不是「完成功課」。做功課的最終目的，是透過做功課，去奪取最多的學術知識和考試分數，從而升上大學。所以，真正重要的，並不是你完成了多少份功課，而是你入不入到大學。

即使你以前在校內，間中有欠交功課，如果公開試的成績上佳，一定會有大學取錄你。相反，即使你在校內，一份功課也沒有欠，如果公開試的成績不足，大概也沒有大學會取錄你。勤奮做功課，而成績仍然差，就足以證明，你是一個傻瓜。

不做傻瓜的話，你就要懂得權衡輕重，先做成本效益最高的東西。眾多功課之中，令你的考試分數提升得最多的，就是第一要務。如果時間不足，你就千萬不要企圖，去完成「所有」功課的「所有」部分。你真正必須「完成」的，是第一要務。其他功課，無論你有多麼掛念，都應暫時完全拋諸腦後，直到完成第一要務為止。完成了第一要務後，才開始第二要務，如此類推。

— Me@2012.03.12

2012.03.12 Monday (c) All rights reserved by ACHK

Douglas Hofstadter

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Douglas Richard Hofstadter (born February 15, 1945) is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics. He is best known for his book Godel, Escher, Bach: an Eternal Golden Braid, first published in 1979, for which he was awarded the Pulitzer Prize for general non-fiction and the National Book Award for Science.

Hofstadter has said that he feels “uncomfortable with the nerd culture that centers on computers”. He admits that “a large fraction [of his audience] seems to be those who are fascinated by technology”, but when it was suggested that his work “has inspired many students to begin careers in computing and artificial intelligence” he replied that he was pleased about that, but that he himself has “no interest in computers.”

— Wikipedia on Douglas Hofstadter

2012.03.10 Saturday ACHK

Bitcoin 2

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That being said, there is a mechanism in place to combat the obvious consequences. Extreme deflation would render most currencies highly impractical: if a single Canadian dollar could suddenly buy the holder a car, how would one go about buying bread or candy? Even pennies would fetch more than a person could carry. Bitcoin, however, offers a simple and stylish solution: infinite divisibility. Bitcoins can be divided up and trade into as small of pieces as one wants, so no matter how valuable Bitcoins become, one can trade them in practical quantities.

In fact, infinite divisibility should allow Bitcoins to function in cases of extreme wallet loss. Even if, in the far future, so many people have lost their wallets that only a single Bitcoin, or a fraction of one, remains, Bitcoin should continue to function just fine. No one can claim to be sure what is going to happen, but deflation may prove to present a smaller threat than many expect.

— Won’t loss of wallets and the finite amount of Bitcoins create excessive deflation, destroying Bitcoin?

— Bitcoin Wiki on FAQ

2012.03.09 Friday ACHK

功夫種子 2

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種子論起點 15

這段改編自 2010 年 3 月 20 日的對話。

小時候所背的詩詞歌賦、所聽的人生道理 和 所學的無用知識，即使當時體會不到，也不會是浪費時間。年紀大了，歷練多了，就自然會明白，自然會受用。

— Me@2012.03.09

2012.03.09 Friday (c) All rights reserved by ACHK

Wheeler–DeWitt equation

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Well, in some sense, the Wheeler-DeWitt equation is nothing else than Einstein’s equations that you encountered elsewhere – potentially simplified and reduced in a minisuperspace formulation.

But if something like the Wheeler-DeWitt equation is the only equation that defines what can happen in a quantum gravity system, your task actually becomes harder, not easier. You must artificially reparameterize your physical states according to a new observable (e.g. the total volume of the Universe at some “moment”) that will play the role of your time (recall that there’s no preferred choice of coordinates in general relativity) – and with this new “gauge-fixed” choice of your time, you will find out that the Wheeler-DeWitt “H=0” equation becomes non-trivial.

— Why and how energy is not conserved in cosmology

— Lubos Motl

Bryce DeWitt first published this equation in 1967 under the name “Einstein–Schrodinger equation”; it was later renamed the “Wheeler–DeWitt equation”.

…

Solutions to the Wheeler-DeWitt equation have therefore been interpreted as the Universal wave function.

This property is known as timelessness [disambiguation needed]. The reemergence of time requires the tools of decoherence and clock operators.

— Wikipedia on Wheeler–DeWitt equation

Timelessness is a property of the Wheeler-deWitt equation in canonical quantum gravity.

— Wikipedia on Timeless

2012.03.08 Thursday ACHK

Worry

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You cannot choose not to worry. You can only choose what to worry. So choose wisely.

— Me@2012.03.07

2012.03.08 Thursday (c) All rights reserved by ACHK

Posted in OCD

逃避發問

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有東西不明白，就應該盡量立刻發問。千萬不要有一個逃避的心態：

在學校有東西聽不明白，就打算回家才慢慢看教科書；在家裡有東西看不明白，就打算回校才問老師。

結果，被這個循環的邏輯結構鎖死了。不明白的東西，永遠也不明白。

— Me@2012.03.08

2012.03.08 Thursday (c) All rights reserved by ACHK

Bitcoin

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Money Supply

While the number of bitcoins in existence will never exceed 21 million, the money supply of bitcoins can exceed 21 million due to Fractional-reserve Banking.

Deflation

Because the monetary base of Bitcoins cannot be expanded, the currency would be subject to severe deflation if it becomes widely used. Keynesian economists argue that deflation is bad for an economy because it incentivises individuals and businesses to save money rather than invest in businesses and create jobs.

The Austrian school of thought counters this criticism, claiming that as deflation occurs in all stages of production, entrepreneurs who invest benefit from it. As a result, profit ratios tend to stay the same and only their magnitudes change.

In other words, in a deflationary environment, goods and services decrease in price, but at the same time the cost for the production of these goods and services tend to decrease proportionally, effectively not affecting profits. Price deflation encourages an increase in hoarding – hence savings – which in turn tends to lower interest rates and increase the incentive for entrepreneurs to invest in projects of longer term.

— Bitcoin Wiki on Controlled Currency Supply

2012.03.07 Wednesday ACHK

反轉意圖

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種子論起點 14.3

這段改編自 2010 年 3 月 20 日的對話。

「種子論」的其中一個可能名字，是「paradoxical intention」（反轉意圖）。

（安：什麼意思？）

例如，當我失眠時，如果刻意迫令自己入睡，反而會更加不能入睡。相反，如果我任由自己繼續睡不著，反而會有機會成功入睡。

入睡的意圖，會增添自己的壓力，令自己緊張起來，導致適得其反。放棄入睡的意圖，會去除自己的壓力，令自己放鬆下來，入睡的機會自然會大增。

（安：那只是「放棄意圖」，而不是「反轉意圖」。）

我有「反轉意圖」的版本。

當你失眠時，身體維持住躺在床上的狀態。但是，你要張開眼睛，心裡迫令自己，千萬不要入睡。那樣，你就有很大機會，在不知不覺間睡著。

但是，「反轉意圖」所描述的，只是「種子論」的一個小部分。

— Me@2012.03.07

2012.03.07 Wednesday (c) All rights reserved by ACHK

Time horizon

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Bezos: It does fit into my view. Our first shareholder letter, in 1997, was entitled, “It’s all about the long term.” If everything you do needs to work on a three-year time horizon, then you’re competing against a lot of people. But if you’re willing to invest on a seven-year time horizon, you’re now competing against a fraction of those people, because very few companies are willing to do that. Just by lengthening the time horizon, you can engage in endeavors that you could never otherwise pursue. At Amazon we like things to work in five to seven years. We’re willing to plant seeds, let them grow—and we’re very stubborn. We say we’re stubborn on vision and flexible on details.

In some cases, things are inevitable. The hard part is that you don’t know how long it might take, but you know it will happen if you’re patient enough. Ebooks had to happen. Infrastructure web services had to happen. So you can do these things with conviction if you are long-term-oriented and patient.

— Jeff Bezos Owns the Web in More Ways Than You Think

— By Steven Levy
   
— Wired December 2011

2012.03.06 Tuesday ACHK