> Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out

> Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet. From one viewpoint, math deals with pure logic. It's something of a poor viewpoint from my perspective: Newton couldn't back up his calculus with logic, Euler and Riemann made numerous unfounded assumptions when looking at the zeta function, Heaviside built a telegraph across the Atlantic despite lacking a proper logical foundation. My personal viewpoint is that mathematics is a series of shortcuts for understanding and manipulating a wide variety of phenomena, and mathematical research is the development of further shortcuts. Often logic comes in, but it's usually after you get the result. Nothing wrong with logic (I sure do like computers, for example!), it's just not the panacea for anything mathematical. > But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students. You're absolutely right about this, and I was specifically thinking of two very bad high school teachers when I wrote my post. The fact is that I know very few HS math teachers, and so my opinion is quite clouded by these two.