Nonstandard
I used to think everything was invented and discovered. Or everything worth it, at least. And that's an unexciting place to be, but apparently a common one.
While I was a student at Polytechnique, there was a talk by an old student who was working in the technology industry, around Silicon Valley or similar. He said he used to think that everything had been invented too, but that after graduating and starting to work “in the real world” he discovered otherwise. He saw how every day someone wakes up with a revolutionary idea!
Exciting! But I didn't hold my hopes high. I thought that by the time I would graduate, everything worth it would already have been found. Guess what? Not at all.
Related to this original idea that everything worth it is already known, I used to think that the standard teachings were the best. That what is taught at school and at the university are, with possibly minor adjustments, the best ideas that are out there to make sense of the world.
I started to suspect that it may not be the case when I learnt about Nonstandard Analysis. You learn at school how to do derivatives and integrals using an epsilon-delta kind of game to define different kind of limits, learn to ban infinitesimals and to contort your thoughts to write with some rigor (big words!) things that seem way easier to grasp in an informal way. But such is life, it seemed, and that's the best you can do. Except, it is not. Nonstandard Analysis is an approach to calculus that, in my view, makes much more sense, is close to our intuitions about infinitesimals and more straightforward. And it has been invented for many decades now, so being too new is not exactly an excuse.
A glitch in the standard curriculum? Soon after, I also discovered that Bayesian Statistics, which I had heard of before but always in a dismissing manner, beats the crap out of standard statistics. The insights and the techniques that come from the Bayesian viewpoint are way more powerful than what one sees in a standard course in statistics.
Those are fundamental fields, that so many people have thought about and that have wide applications. How can the standard be so grotesquely sub-optimal? I was shocked! Even more shocking was to realize that those are not the only fields. Not by a long shot.
Some of the other subjects I've learnt about and convinced myself that the standard ways are way inferior to them are: the Condorcet methods for voting (which show how lacking our current voting systems are, and for no good reason), Geometric Algebra instead of the standard matrix algebra, and the causal “do-calculus” statistics.
Somehow related, other important subjects were much less in the spotlight than they deserved (fortunately for many it may be less so nowadays than when I first met them), for example Loop Quantum Gravity in physics, or Universal Basic Income in society.
How many more examples are there? Why are they not better known? We are not talking about a limit to our knowledge that in the future is going by pushed by some people that will figure things out (though that too will happen). It's simpler. It's about getting those non-standard ways to do things, which already exist and are better than the standard, to be known and become the standard.
Rather than being the exception, I now suspect that for most standard ways of doing things there are already much better ways, already discovered and waiting to get attention. The world is much more exciting than living in a world where everything worth it has already been figured out — not only there are new ideas coming in all the time, but we can listen to nonstandard alternatives to gain an insight that most people are still lacking.