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CCC comments on Welcome to Less Wrong! (7th thread, December 2014) - Less Wrong
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While yes, that can make life easier, it also means that if the reason why you can't follow the proof is because you're misunderstanding the finding in question, then you're not applying any error checking and anything that you do that depends on that misunderstanding is going to potentially be incorrect. So, if you're going into any field where mathematics is important, it can also make life significantly harder.
It's hard to put in words what I mean. There a certain ability to think in abstract concepts that you need in math. Wanting things to feel like you "understand" can be the wrong mode of engaging complex math.
That doesn't mean that understanding math isn't useful but it's abstract understanding and trying to seek a feeling of common sense can hold people back.
I... think I learnt math in a very different way to you. If I didn't feel that I understood something, I went back until I felt that I did.
I do not understand the difference between an "abstract understanding" and a "feeling of common sense". Is a feeling of common sense not a subtype of an abstract understanding (in the same way that a "square" is a subtype of a "rectangle")?
On the contrary, failing to feel common sense is usually a sign that you don't really understand what's going on. Your understanding of an abstract concept is only as good as that of your best example. The abstract method in mathematics is just a way of taking features common to several examples and formulating a theory that can be applied in many cases. With that said, it is a useful skill in math to be able to play the game and proceed formally.
There's an anecdote about a famous math professor who had to teach a class. The first time, the students didn't understand. A year later, he taught it again. Learning from experience, he made it simpler. The students still didn't understand. When he taught it a third time, he made it simple enough that even he finally understood it.
I will concede that in practice it can be expedient to trust the experts with the complications and use ready-made formulas.
This doesn't seem to be true for anything that's normally analyzed statistically: the stock market, for example, or large-scale meteorology.