The Top Idea in Your Mind

事業愛情觀 4

I realized recently that what one thinks about in the shower in the morning is more important than I’d thought. I knew it was a good time to have ideas. Now I’d go further: now I’d say it’s hard to do a really good job on anything you don’t think about in the shower.

Everyone who’s worked on difficult problems is probably familiar with the phenomenon of working hard to figure something out, failing, and then suddenly seeing the answer a bit later while doing something else. There’s a kind of thinking you do without trying to. I’m increasingly convinced this type of thinking is not merely helpful in solving hard problems, but necessary. The tricky part is, you can only control it indirectly.

I think most people have one top idea in their mind at any given time. That’s the idea their thoughts will drift toward when they’re allowed to drift freely. And this idea will thus tend to get all the benefit of that type of thinking, while others are starved of it. Which means it’s a disaster to let the wrong idea become the top one in your mind.

What made this clear to me was having an idea I didn’t want as the top one in my mind for two long stretches.

I’d noticed startups got way less done when they started raising money, but it was not till we ourselves raised money that I understood why. The problem is not the actual time it takes to meet with investors. The problem is that once you start raising money, raising money becomes the top idea in your mind. That becomes what you think about when you take a shower in the morning. And that means other questions aren’t.

I’d hated raising money when I was running Viaweb, but I’d forgotten why I hated it so much. When we raised money for Y Combinator, I remembered. Money matters are particularly likely to become the top idea in your mind. The reason is that they have to be. It’s hard to get money. It’s not the sort of thing that happens by default. It’s not going to happen unless you let it become the thing you think about in the shower. And then you’ll make little progress on anything else you’d rather be working on.

— The Top Idea in Your Mind

— July 2010

— Paul Graham

2013.03.07 Thursday ACHK

Power 2

The devil is in the details.

The angel is also in the details.

— Me@2013-03-06 02:09:48 AM

Whoever gets the details gets the power.

— Me@2013-03-06 02:09:48 AM

2013.03.07 Thursday (c) All rights reserved by ACHK

微積分 6.4

無限年 3.4 | 0/0 2

這段改編自 2010 年 4 月 3 日的對話。

(安:你的意思是,牛頓和萊布尼茲發明「微積分」之初,雖然必須使用「無限小」這個概念,但卻沒有賦予它,一個嚴格的定義。而這個「微積分」的漏洞,是後人幫他們修補的。)

無錯。那些數學後人,用了「(ε, δ)-definition of limit」(“epsilon-delta definition of limit”),來定義「無限小」。

(安:那樣,「無限小」的嚴格定義是什麼?)

例如,數式

\frac{x^2-9}{x-3}

在 x = 3 時,並沒有數值,因為那會導致分母變成零。分母等於零的分數,沒有任何數學意義。但是,我們卻可以研究,

\lim_{x \to 3} \frac{x^2-9}{x-3}

等於什麼。換句話說,雖然 x = 3 並不合法,但是,我們仍然可以追問,「x 非常接近 3」時,這題數式會得到什麼數值。

正式的運算方法是這樣的:

\lim_{x \to 3} \frac{x^2-9}{x-3}

= lim_{x \to 3} \frac{(x+3)(x-3)}{x-3}

然後,我們約了分子和分母的(x-3):

= lim_{x \to 3} (x+3)

= 6

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

— Me@2013.03.07

2013.03.07 Thursday (c) All rights reserved by ACHK