4.1

1. Once again I am burdened with the question: when are we just manipulating symbols and when are we actually manipulating ideas (obviously the latter is probably the answer and necessary for any rigorous study)? This chapter has challenged my conceptions of what x is. I guess even in my old way of thinking, it's just an element with certain properties. This actually did clear up my question of why we would say that polynomials in a ring had coefficients in that ring since it seemed stupid to have some polynomials in Z but use elements of R in place of x.

2. Talking about the degree of functions, I am wondering: how do you prove something that is obvious? The interesting thing is that obvious is subjective. I have been bored in class when a seemingly obvious concept has been proven. I have been confused in class when some method of proof is substituted with hand-waving because the result is obvious. Experience breeds intuition. We're all being taught to think in a certain way, to think with certain biases and conventions. But I think the world is too complex, and the subject matter far too advanced, for someone with real intuition to build it up from scratch.