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by inverse_pi
2771 days ago
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There's a long version and a short version. The long version is you have to learn to write mathematics by yourself. Start with an intro course and start deriving theorems by yourself. Do not look at the proofs. At this stage, details are very important and can't be overlooked. You need to be your own critic and keep asking why and how to every single detail and step until you can convince yourself that you would be able to naturally come up with the theorem and proof. Continue doing this to higher level courses. This is how I learned Math since middle school all the way throughout graduate school. The short version is you have to ask the right questions. Naturally for every theorem or equation, there are 3 big questions: 1) What does the theorem/equation say? What's the intuition behind it? 2) Why is it true? 3) How does one come up with it? One must ask these questions in the exact order. To understand what the equation really means, you should break it down further to smaller components. What is this variable? What does it represent? What is the intuition behind what it represents? What's the implication when the variable increases, decreases, etc? Do that for every single component in the equation/theorem. One should fully understand the intuition and clearly describe all quantities before trying to look at the equation/theorem as a whole. To understand why an equation/theorem is true you need to build up a repertoire of theorems related to the quantities of interest. The bigger your repertoire, the easier you can prove or disprove something. The more advanced way is to build up intuition around the quantities of interest then come up with intuitive hypotheses. The hypotheses are often easier to prove/disprove. The process repeats. |
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edit: patience, self-forgiveness, and a willingness to accept frustration are important traits. You might spend a whole week banging your head against the wall, feeling like you're making no progress, and then one day everything falls beautifully into place. That doesn't mean you did something correctly on the final day - it means you did everything correctly for the whole week before. Don't view a difficult and unrewarding day as wasted time. You're building something very difficult and that takes a bewildering amount of time.