Hewitt response

Arbitrary.

"I describe something as arbitrary if someone could only

come to know it to be true by being informed of it by some

external means - whether by a teacher, a book, the internet,

etc."

Necessary.

"They are parts of the mathematics curriculum which

are not social conventions but rather are properties which

can be worked out from what someone already knows."

I agree with Hewitt when he says "[i]nviting students to 'think about it' is appropriate for

what is necessary, but not for what is arbitrary." There is indeed, no point in asking a student to ponder on what the word is for a three-sided regular polygon. 

Hewitt says "[s]uch division of the mathematics curriculum into arbitrary and necessary is based upon the philosophical roots of the notions of 'contingent' and 'necessary'." He also makes a distinction between "awareness" and "memory," the former associated with "necessary" and the latter with "arbitrary." Later on, he says that "what is necessary comes as a consequence of certain accepted givens," which contrasts somewhat with his earlier hinting that "the necessary" is the objective heart of mathematics. Or maybe he shares my view that the objective is a consequence of accepted truths, mostly subconscious, and the distinction between awareness and memory is not clear-cut - but I somewhat doubt this. I don't see the distinction between contingent and necessary as clear-cut either - sure, we can say that it is necessary that the definition of a Lie algebra forces their classification into the simple and exceptional Lie algebras - but without the contingency of high-energy physics, powered by uranium and with applications in war and industry - would anyone care about Lie algebras enough to classify them?

Despite my philosophical rambling I think Hewitt's article brings some interesting points to mind that may affect my teaching. There is no point for me to ask students to ponder the arbitrary, and as much as possible I want to let them discover the necessary for themselves. When it comes to Hewitt's division of math into the arbitrary and necessary - I was already thinking about this implicitly, after guidance from my SA to let the class figure out more things on their own, without giving away the answer so soon - and this is part of my frustration with academia: its tendency to give names and symbols to concepts that many people already understand, and then one person can claim this knowledge as their own when really they were just the first to put it to paper, not the first to have the idea. As someone who tends to learn things first implicitly, and then later (if ever) explicitly, I find the act of watching an experienced teacher much more helpful to my teaching practice than any article.