Surely some of the scientists working today will make equally groundbreaking or insightful discoveries or develop innovative theories and thus can fairly be labeled “genius” or as having the same level of smarts? I think I have developed a stronger aptitude for language than for math and due to suffering from depression in high school and middle school I didn’t push myself nearly as much as I could and lost much of my motivation.
However, I don’t see why it’s not possible for me to develop mathematical abilities as strong as my linguistic abilities or even pursue a career in astronomy (which I love) or physics or even pure mathematics.
See also Eric Schechter’s “Common errors in undergraduate mathematics“. Hi, Not to be rude, but a translation of Descartes that captures the original poetry of his phrase better might be: Each truth I discovered was a rule that then served to discover other truths.
I also have a post on problem solving strategies in real analysis. Thanks for your advice on Solving mathematical problems. [Corrected, thanks – T.] Dear Professor Tao, here are two articles on the benefits of clever note-taking for math problem solving: PS_R_A_with a strong emphasis on math competitions and Hi dear Professor Tao, I am very interested in elementary geometry and higher dimension Euclidean geometry, could you please upload chapter 4 in your problem book (I see it is about geometry), thank you very much.
It’s worth persisting in trying to understand how to do these problems, and not just for the immediate goal of getting a good grade; if you have a difficulty with the homework which is not resolved, it is likely to cause you further difficulties later in the course, or in subsequent courses.
I find that “playing” with a problem, even after you have solved it, is very helpful for understanding the underlying mechanism of the solution better.When you think of having fun, math may not be the first thing that comes to mind.But there are some math problems that people just can’t seem to get enough of. This seemingly simple problem that has the internet stumped: Do you think you know the answer? In the 1980s, 90 percent of people who tried to solve this problem got it correct.There are of course several other problem-solving books, such as Polya’s classic “How to solve it“, which I myself learnt from while competing at the Mathematics Olympiads.Solving homework problems is an essential component of learning a mathematical subject – it shows that you can “walk the walk” and not just “talk the talk”, and in particular identifies any specific weaknesses you have with the material.And I know you say similar things on your career advice blog, and I know it’s important to be realistic and plan for graduate school and beyond I just really don’t like how people put this label of genius or prodigy on certain people to (in my opinion) make them seem able to achieve things that most other people cannot–even the levels of Einstein or Mozart.And I don’t think that’s arrogant or unrealistic.I wanted to get your honest opinion. Hey Leif, This book might be useful in pursuing the answer for your question: Disclaimer: I just read the summary and reviews of that book. I’m, at the moment, too busy with studying Maths stuff. Tao, I am a high school student, I loved math got good grades in my middle school years.I hope you are interested in elementary geometry, too, nice to meet you here! Hi Prof Tao, As an undergraduate student I often face the problem of deciding how many textbooks problems I should do before moving on, for example, Is ten questions per chapter of Rudin’s Principles of Math Analysis adequate?The more problems I do on a specific topic the slower it takes to reach graduate level mathematics. Tao: I hava translated this essay into chinese, I’m sorry I couldn’t translated it well enough, as my ability in english is as poor as mathematics.But I find math hard and i often make many mistakes now.In fact ,i think i can work out many problems while doing my homework . Dear Professor Tao, I am a fifteen year old student currently in high school. Should I study some analysis or is it group theory that you recommend? Well, after calculus, one usually studies multivariable calculus. While trying to solve problems from my text books (like Stein’s Complex Analysis ), I notice that very often I cannot solve the hardest problems from them.