On asking and answering questions.

Today my mathematical studies move slowly. I just feel less mentally capable than on most days (on which I blame an exceedingly late Friday night). The best I can do is remember those lessons after meeting with my advisor.

I walked in two weeks ago to ask about an exercise from his book, and two meetings later we've discussed enough facts and made enough observations to solve it. (Rather, having written the book, he referred to useful facts and I did my best to think quickly and understand the logic and line of thought. The remainder of the time involved my struggling to explain something and his patient waiting.)

Looking back through these notes, I'm amazed. It doesn't appear easy to write down this question in its present form, because it is so condensed. It's done to the point where it can be stated with simple ideas, and without a hint to the level of sophistication needed to solve it.

There is a good deal to remember .. the toil from honest lessons and facts that I wouldn't have thought to check .. and today I've written them down. It gives the day and night purpose at least, even if I have no strong mental capacity for new and creative things.

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At the very least this experience agrees with the popular belief that it is far easier to ask a question than to answer one. One can even reduce this difference to how language works: take subsets of words that everyone knows, say

{who, what, where, when}
{run, make, eat, sleep, sit}
{sky, wind, dirt, water, apple, food}

One selection of words gives: "Who eats (an) apple?"

It's a very simple question, and to answer it, we search the catalog of our memory as to what creatures like apples. It takes some thought, but the process is mechanical and a computer can do the same.

Another selection is: "What makes (the) sky?"

Does this mean the colors, how many, and how they are mixed? Does this refer to physical chemistry and the interactions of gases? Or is this a philosophical question, concerning the origins of the earth, and ourselves?

Sometimes it is a matter of interpreting what a question means, and it is a separate process to understand what can be used to answer it. The second primitive question is not so primitive: parsing it, it's not very clear what is involved in the answer or in this form, if it can be answered.

Every so often I wonder how we know anything at all.
It's a confusing world, and maths are confusing too.