For some tasks, this is absolutely true. We can identify e.g. a dog in picture in 100ms or so, and nobody is capable of explaining how. We can understand and speak languages, without the slightest idea of how it works. Maths is of course not comparable to these cognitive functions, but people with high levels of expertise do have a lot of their knowledge "automated", and not open to introspection.

You can also understand the very simple basic essence of something, but get lost in the complexity when scaling up.

Binary is very simple, but scaled up: look what we've created with software.

When it comes to explanation: pulling from rote memory, requires someone to attempt to hold all the short-term details in mind.

There are biological limitations to how well we can do this, but we can also exercise our brains to improve this ability.

But when something is deeply learned, in long-term memory, the effort of recall is much less than rote memory of short-term details. Our context window is limited, fills up, and we must recover. When you're remembering long-term details, context seems easier to swap in and out (sorry to sound like an LLM, but they do simulate thinking).

Whether or not someone is a master of any given domain of knowledge comes from demonstration. Maybe that is teaching the essence of a subject in a way that demonstrates you can visualize and move around the subject with ease. Or maybe you can create something very useful, or tasteful.

We accept that you have spent time in this area and probably can revral truth to us. You are credible.

If you can't demonstrate mastery through teaching, exchanging ideas to bring me closer to your level: them other forms of credentials are sought: like how well they code, or how useful their products become.

But life isn't about usefulness and will just lead to unhappiness. Just be the best version of yourself you can be. Life is too much to understand all at once.

Expert mathematicians may not be able to explain how they find a proof of a difficult mathematical statement, but once a correct proof is found, it can be, with sufficient work, be formalized (most mathematicians don't do this part). This formal proof can be mechanicaly checked, without any creativity, step-by-step according to the axioms and inference rules of a mathematical logic.

Up to the limits of the Goedels incompletness theorem.

Dependenting on the used notation the formal proof can be very long. For example, Principia Mathematica took about 300 pages to prove that 1 + 1 = 2.

https://commonplacefacts.com/2022/07/27/principia-mathematic...