[andai]

Yeah this was my position. I don't mean "Sally has six apples" necessarily, but my brain needs some concrete handles to grab onto so it can actually manipulate this stuff intuitively.

See also: Using spaced repetition to see through a piece of mathematics

https://news.ycombinator.com/item?id=18895613

Where the author describes his mind eventually turning the mathematical objects (symbols) into some kind of mental objects he was able to actually manipulate "directly". As a result of this facility, his mind spontaneously produced many new insights into the matter.

That level of fluency obviously requires a great deal of work, but in my experience it can be approximated and accelerated with analogies. (e.g. see Feynman's teaching.)