system
Given the following post, respond as if you're commenting on Reddit:
human
Can you know something without having it memorized?

Like you know it in some contexts (like when you see or hear it and recognize it) but not others (like when trying to solve a problem or trying to communicate something). I haven't really seen anything written about this so I think its worth discussing.
gpt
There's some interesting work about it in Husserl's *Ideas II*, near the beginning, as well as in his *Origins of Geometry*.

Basically, Husserl has the idea that one you learn something, it can become the grounds for something else that you learn. Let's say, for example, that you learn special relativity to read a book about black holes. When you are studying black holes, you are not actively recalling how special relativity works, but your understanding of it serves in the background as something like the "horizon" for you to read about black holes. Also, you might be able to understand books about special relativity without sitting down and doing the equations from memory again. 

Once you have grasped special relativity, in this example, you don't have to recall the entire argument, the equations, or do the math again to "check" your work. Rather, it becomes something like "background" knowledge.

However, part of this background knowledge is the ability to recall it, to work it out again, and check what you know.

What's interesting is that Husserl later applies this to the history of mathematics to explain how it allows for mathematics and the sciences to build off one another without having to redo all of the discoveries each time. He argues that within, say, a geometrical theorem, the proof of this theorem remains something implicit. If we try, in epistemology, to recall this proof, we *recreate* the original conditions of its discovery (which allows Husserl to claim that epistemology is a "historical" inquiry).

This basic idea, that there's a difference between understanding an idea and being able to conceptually articulate it from memory, can be found in both Plato and Leibniz's theories of innate ideas.

In Husserl's historical writings, geometrical ideas aren't "innate," but function in a similar way. This is because mathematics and science have a history. They build off earlier discoveries, which serves as a background for new discoveries. 

Generally speaking, Plato has a lot to say about having a correct belief about something and *knowing* it. He might be a good place to look for more information.