I vote for "axiom" as the term which, as much as any other, crystallises this question. For the ancient Greeks, "axiom" meant a proposition for which nobody could think of a proof, but the truth of which had to be accepted on trust, or the discourse could never get started. In our time, all of the sciences are much more developed, and we tend to think of "axiom" as meaning any proposition which can be "taken as read". For example, if I want to propose an extension to the theory of Special Relativity, I will probably take the Theory of Special Relativity itself as an axiom, without rehashing all of the arguments in its favour.

A further (whimsical) thought: if we understand "terms" to include symbols, what is the etymology of "+" and "x"? Have you noticed that "+" refers to a singular addition (as in, 2+3=5) but, by turning "+" through 45 degrees, it transforms to an iterative addition (as in 2x3=2+2+2=6)?

too snippy. — scherz0

Too snippy? You've been spamming this lame question all over the Internet. Consider yourself snipped.

Please don't take this wrong.

What're you getting out of asking these questions to us as well as stackexchange?

I don't know what replies the question may have garnered in other forums, but I was hoping for something a bit more interesting than my own in this one... my Dictionary of Philosophy doesn't list "snippy" (must be out-of-date)... the question is actually deeply interesting, and cuts across many lines of philosophic thought, from Euclid to J L Austin. Surely somebody out there has something positive to contribute?

I was an American high school algebra math teacher for five years. One of the problems with the way we teach math in the West is we teach formulas for students to memorize. The alternative is how to arrive at the formulas, and why they are needed.

I had a day early during the school year in which I would take my spare tire into class and lean it against the wall. People would come in wondering what it was for, and I would make up some lame teacher joke like, "In case I need a quick getaway."

I would then take the kids out to a back field in groups of 3 with meter sticks, as well as the tire under my arm. I would place the tire down and then tell the kids, "Today we're going to measure the length of this field!" After students would grown, they would start to use the meter sticks to attempt to measure the field. After about a minute or two, I would shout, "Wait, wait, wait! This seems pretty hard right. Is there a better way we could measure this field?" Then I would slowly look at the tire.

We look at the tire, and I would ask them questions about the distance of one rotation on the tire: Circumference. Ok, how do we measure it? Some kids would try to get the radius, but there's no center area of the tire. Diameter. We would mark the side of the tire afterwards, a volunteer would roll it while another counted, then we would go back to the classroom to find out how long the field was.

Never had a student forget circumference after that. Math makes sense when you show how the formula is formed, and why its useful to them in particular. So yes, etymology added into the classroom could possibly help substantially. I would add that's just one part: the history and practical application are paramount for real understanding.

The problem of course is that math classes are jam packed with formulas and theorems. A careful explanation of how these form takes more time then a quick, "Memorize this formula and practice 30 math problems in your homework until you have it down." We need real educators cutting down exactly what we teach in math, focusing only on the most important aspects that would apply to practical life, as well as further possible academic exploration.