Yes - and writing is a way to extend your memory as well. You can work more complex ideas with yourself, using the paper as storage for what you have already thought through.
Can you imagine doing some even simple Calc-type proofs without paper? Not possible. Many things are like that.
You give up maybe better memory in your head in exchange for being able to make progress that your couldn’t have made without the tool of writing.
Humans are ultimately tool-using animals more or less... makes sense that we use tools that are valuable.