Is it true that if you understand the content well enough, you won't need to know the solution (proofs)

Question

I am currently self-studying an intermediate level martingales textbook that does not have solutions. I read somewhere that if you understand the content well enough, you won't need to verify, as you will just know. This makes logical sense, but if you're learning the content for the first time, how will you know if your initial ideas can be extended to the theorems in the textbook? If you're not sure if your solution is correct, do you just keep working on it until you're 100% sure you are correct?

Answer 1

When you say solutions, do you mean proofs, or solutions to exercises? Lack of proof is rarely a good thing for studying basic to intermediate stuff - usually there are a few results requiring novel ideas, that the reader cannot be expected to generate themselves. And while knowing the results but not the proofs in a certain field can be useful for quickly gaining some tools that you can use later, it is a bad idea for foundational theories.

Lack of solutions to exercises, however, is common, and useful: It deters you from looking at a solution too early. Exercises vary a lot in difficulty, and importance, across math books, so sometimes looking up solutions online may be a help.

In both cases, I would agree that you often know your argument is correct, sometimes to a degree that you don't even have to fill out all the details. But, as you seem to be describing, if you have an argument that you think is correct, but you are not sure, I would encourage you to fill out all the details of the argument and check them. That's the strength of mathematics - intuition is good and important, but when in doubt, even the largest statements can be reduced to a sequence of very precise, direct steps that can be checked almost mechanically.