Although most of us are fascinated with astrophysics — the science of what naturally forms all throughout the entirety of the cosmos — very few of us actually embark down the path of becoming an astrophysicist. To get there, one must not only master a large suite of mathematical tools, but also learn how to obtain solutions when faced with difficult problems. Sometimes, we run up against fundamental limitations:
尽管大多数人对天体物理学充满好奇——这是研究宇宙中自然形成的一切的科学——但是很少有人真正走上成为天体物理学家的道路。要达到这个目标,不仅需要掌握大量的数学工具,还要懂得在面对困难问题时如何找到解决方法。有时候,我们会遇到一些根本性的限制:
As physicists, we don’t get to control or dictate the laws of nature; we can only approximate reality as closely as possible in our attempts to describe nature.
作为物理学家,我们无法掌控或改变自然规律,只能尽可能准确地近似描述自然。
Still, there is a particular set of skills that one must develop in order to be competent as an astrophysicist. While most fret about the difficulty of the math involved in solving such problems — from algebra to calculus to differential equations to even more advanced topics — it turns out that math skills can be practiced and gained just like any other skill: painting, swimming, or playing the piano. If you want to get good at painting, you must paint; if you want to get good at swimming, you must swim; and if you want to get good at solving math problems, you simply practice solving them. But along the way to becoming an astrophysicist, especially once you’ve acquired those necessary skills, a series of deeper insights emerges. Here are eight key lessons that apply near-universally to lifelong learners everywhere.
成为一名合格的天体物理学家,需要掌握一系列特定的技能。很多人都会担心解决这类问题所需的数学难度,比如从代数到微积分,再到微分方程,甚至更高级的数学。但事实上,数学技能和其他技能一样,是可以通过练习获得的。就像要画得好,就需要不断画画;要游得好,就需要不断游泳;同样,要解决数学问题能力强,就需要不断练习解题。在成为天体物理学家的过程中,尤其当你掌握了这些必要技能后,会有一些更深刻的领悟出现。以下是 8 个对终身学习者来说几乎普遍适用的关键体会。
1. The most critical step in problem-solving is setting up the problem correctly....
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