I have always had a tenuous relationship with the concept of angular momentum, but recently my concerns resurfaced when I did my studies on Kepler, and in particular his “equal areas law” and Newton’s elegant geometric proof. I love the fact that a simple geometric argument, seemingly totally divorced from the physical situation, can provide an explanation for why the line from the Sun to a planet sweeps out equal areas in equal time as the planet orbits the Sun, solely under the influence of the gravitational force between them. However, modern physics books invariably cite the conservation of angular momentum as the “explanation.” I indicated before in my “Kepler’s Laws and Newton’s Laws” essay that this “explanation” irritated me. In this essay I go into detail about my reservations concerning this line of argument. See Angular Momentum.
This post is the first on a meditation on the nature of mathematics as I see it. I have been thinking about this for some time, and my thoughts were again stimulated by a March 2014 article I read in Slate by Brian Palmer that attempted a popularized explanation of the mathematical concepts associated with Zeno’s Paradox. It was a laudable effort that I applaud. So it is a bit churlish of me to critique it, but I felt its misconceptions got at the heart of some fundamental ideas about mathematics that I wanted to clarify.
The key idea exemplified in this article is the role “making it up” plays in math. That is, the general impression seems to be that math is dealing with things as they actually are if we can just be brought to see it. Whereas the idea that mathematicians make things up or define things is given little credence. For example, 0 x 2 “is” 0 doesn’t make any sense if you arrive at multiplication inductively from the intuitive idea of its being repeated addition. That is, 2 x 0 = 0 + 0 = 0 makes sense, but 0 x 2 = 0 does not. So mathematicians just say let’s define 0 x 2 = 0. If we do, it will be consistent with the other rules we have abstracted from the repeated addition idea, such as the commutative and distributive rules – that is, nothing breaks. (Try defining 0 x 2 to be any other number than 0 and see what breaks.) To put it another way, the reason we want to have 0 x 2 = 0 is for a different reason than we originally thought was meant by multiplication. We have extended the original idea into new territory. A similar thing happens with the advent of negative numbers. This is a very sophisticated idea and a challenge to present at an elementary stage.
In Part I, I will first present the article, heavily annotated with my critique. Then in Part II I will try to explain in more depth the admittedly philosophical concepts I am trying to get at. See Meditation on “Is” in Mathematics I – Zeno’s Paradox.