Personal life and work extremely benefit from independent thinking. Good mathematics research may only be done through independent thinking, through new ways of thinking. Most institutions and teachers train the opposite of independent thinking. Indeed, by definition, you can’t induce an independent thought. I usually found it quite hard to ask questions in research-level talks that don’t involve my research area. And I don’t think my lack of understanding of the other areas was the main reason. I realize that good questions in that context are good research directions. In startups, good ideas come from noticing something important that you lack. Then, we see that behind a good idea there is a good question.

How do we ask independent questions? Not being satisfed makes you see what the world lacks. The best literature works may come from frustrated childhood experiences, because that is what the work expresses. The writer of novels, then, may be advised to experience greatly and feel fully. But what about the scientist? It is clear that hard work is required, this enables you to be intimately familiar with the objects you are concerned about, and the possibilities of what can be built. Great scientist (eg, Talagrand and Mezard) often say that they work with simple examples that they can understand fully. Simple is useful because it is general. But simple is a corollary of trying to understand something fully. We are tempted to move towards complicated objects, where we end up going in circles. On the other hand, it is clear that hard work is not enough. China, for instance, has a very hard-working population but is underrepresented in world-class academic achievement. An interesting contrast is the massive amount of olympic medals China has, which usually does not require independent questions.

But if I am intimately familiar with the objects, how could I not see what I can do with them? It is often said that if we do envision all the uses of an object, or impliciations of a statement, we wouldn’t even be able to move. Then, independent questions come at the cost of efficiency. But efficiency is overrated, and usually hides fear to the unknown, being rejected, or defying authority. This means that trying to look like the smartest in the room or to impress my mentor will never lead to independent thinking. It cannot be independent because it is based in the opinion of other person. In fact, you have to be prepared for people disliking you and thinking you are dumb.

The people that ask independent questions are usually also weird. What is being weird? Being weird is being non-standard for apparently no reason [1]. An innate disregard for customs

I had the illusion that in this highly-developed society, we are doomed to accept some things as true. The reasoning goes like “there’s so much knowledge out there that I have to believe the doctor, they studied for that”. Well yes and no. It is surprising how much of common knowledge we can rederive from scratch (this is something descartes also said.) This is not localized to math: why do I do this gym exercise, why do I behave this way around this group of people. besides there is a massive difference between discovering something and validating its truth.

[1] More generally, if we have an expectation of a situation to turn out certain way and it turns out in other way without apparent reason, we may say “this is weird.” Note the lack of reason matters: it stops being ‘weird’ upon learning why the situation turned the other way. Is the weird person weird for themselves? doing action without reason is interesting