feeling mathematically inarticulate

There's a term from Victorian England called a monoconvolute.

It's a derogatory term and brought up after the study of phrenology became passé and neurology became its successor. The idea was that the convolutions (wiggles) on the surface of the cortex provide "space" for storing information and memory, and if you had only one convolution on your brain, then you must be a very stupid person.

At any rate, the existence of such a word demonstrates some importance about having a complicated brain for higher cognitive skills and whatnot.

But I believe that the opposite extreme can be troublesome, because I suspect it in myself. Either my brain is too muddled and twisted, or my thoughts require more discipline: something in me is muddled, at least.

In fact, this reminds me of an instance from high school: I was a sophomore taking the usual English class and my teacher was a fine editor but a harsh grader, and used red fine-tipped pens by the box.

There was one essay assignment we had to write one week .. but so that you know, we had essays every week. Monday was our deadline and Wednesday she returned them to us, mires of scarlet and scrawl atop once-cleanly typed white pages. But this one assignment everyone had jaws dropped, grades slashed,

.. in the same way that I grade Calc II quizzes, come to think of it ..

and there was general despondence and "gnashing of teeth," as one friend of mine would say.

I could swear that my essay looked the worst. It might as well been dipped in the red ink, but the strange thing was, there was no grade on it: not on the first page, the second, nor the third. I was able to read the last of the scrawls, which I interpreted as either:

Keep these ideas. Rewrite this.
or
Kaput the ideas. Reunite us.

Deciding to use common sense, I rewrote my essay and brought it back on Friday. Being a curious child, I couldn't help but ask my teacher, "Was it really that bad? Was it really ungrade-able?"

"It wasn't finished, and the language was awkward," she replied, "I wanted to see what your completed thoughts were."

I nodded, handed her the rewrite, and that next Wednesday I received two essays back. My rewrite looked only speckled with red this time, and later I learned that nobody had ever been given a rewrite option from her before. I don't know if she ever did it again.

Suffice to say that English is my native language, but mathematics isn't. It seems like I never quite phrase things properly, and every time I discuss some matter with a fellow maths person, we would discuss the same object or concept, but their viewpoint is much grander and more precise than mine.

When I studied literature and history, I never felt like a plagiarist, but now that I study mathematics, I feel like I steal all the good stuff from other people.

Take my meeting with my advisor today: it was the third session we've sat down (well, figuratively) to hash out this lemma of Milnor's, concerning existence of diffeomorphisms of topological n-spheres when given a smooth homotopy condition (more precisely, a differential isotopy) on maps of (n-1)-spheres.

I still feel as if it shouldn't have taken that long. Had I remembered to tend more to geometric issues, then this could have been put to rest sooner. Unfortunately I still think too formulaically and not very intuitively, and honestly, I use rather terrible notation.

Had I not drawn enough diagrams, my advisor might have thought I spoke nonsense .. which I probably did. Happily and fortunately, he's a forgiving man.

Perhaps there's no real conclusion in all of this .. only that today is an "mathematically inarticulate" day for me. \: