If mathematics is the perception of this realm of pure ideas, then doing mathematics requires our brains to somehow reach beyond the physical world.

如果数学是对纯理念王国的感知的话，那么研究数学就需要让自己的大脑以某种方式抵达现实世界之外的某地。

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When something so fundamental to life as food is priced beyond the reach of the poor, we know that something in our world has gone terribly wrong.

对生命至关重要的食品的价格上涨到穷人难以问津的地步，我们知道大事不妙，情况实在过于离谱了。

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These are the things I want for you-to grow up in a world with no limits on your dreams and no achievements beyond your reach.

这些是我想要让你们得到的东西：在一个梦想不受限制、无事不能成就的世界中长大。

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For digital CEOs used to the susceptibility of problems to the power of algorithms, the analog world of politics is a domain frustratingly beyond their reach.

WSJ: Sandberg Wins Admirers, But What Will Come Next?

We see the same fierce hope in those who defy censorship and reach beyond an electronic curtain to connect with people around the world.

WHITEHOUSE: Presidential Proclamation -- Captive Nations Week, 2012

Amazon helps more than two million small businesses and individuals expand their reach beyond their hometowns to 150 million customers around the world.

NPR: Big Technology A Threat To Small Businesses?