Category Archives: Math Commentary

Newton and the Declaration of Independence

One of the books that has stuck with me over the years is Carl Becker’s The Declaration of Independence (1922, reprint 1942), not only for its incredibly clear and beautiful writing but also for its emphasis on the impact of the revolution most prominently caused by Isaac Newton, which was later subsumed under the term Scientific Revolution covering the entire 17th century. A consequence of this remarkable period was the so-called Enlightenment that followed in the 18th century and became the soil from which our nation’s founding ideas and documents sprang. Both these centuries have been further optimistically called the Age of Reason.

Our current times, awash in lies, corruption, and such terms as “alternative facts”, have been characterized as an assault on the rationalism and Enlightenment that shaped our founding. Any revisiting of these origins would seem to be a valuable endeavor to see if they still have validity. What makes Becker’s essay particularly relevant to me is the current pervasiveness of the mathematical view of reality that was launched by Newton some 300 years ago. Becker shows how this new way of thinking spread far beyond the bounds of mathematics and engendered a new “natural rights” philosophy that formed the foundation for the Declaration of Independence. Essentially the idea was that if the behavior of the natural world was based on (mathematical) laws, then so must the behavior of man be based on natural laws.

See Newton and Declaration of Independence

(Update 10/31/2019) Steven Strogatz Confirmation. Continue reading

The Essence of Mathematics

It is a bit presumptuous to think I could reduce the universe of mathematics to some succinct essence, but ever since I first saw a column in Martin Gardner’s Scientific American Mathematical Games in 1967, I thought his example illustrated the essential feature of mathematics, or at least one of its principal attributes. And he posed it in a way that would be accessible to anyone. I especially wanted to credit Martin Gardner, since the idea resurfaced recently, uncredited, in some attractive videos by Katie Steckles and James Grime. (This reminds me of the Borges idea that “eighty years of oblivion are perhaps equal to novelty”.) See the Essence of Mathematics.

Restructuring the US Senate

This subject admittedly has only a tenuous relationship to mathematics (via arithmetic), but perhaps it can join more mathematically challenging political topics like voting and gerrymandering. In any case, I was stimulated to consider the idea of reapportioning the US Senate by the % US population of each state by an 8 December 2018 article in the Atlantic by former Congressman John Dingell, who advocated abolishing the Senate. I thought this a bit too Draconian and considered the percent population idea as a better compromise. It turned out I was not alone in having this (obvious) thought: I just came across a more extensive 2 January 2019 Atlantic article by Eric Orts that concurs with my idea about reapportionment of the Senate, discusses the legal ramifications in more detail, and echoes the benefits I mentioned as well as others. See Restructuring the US Senate.

Angular Momentum

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.

Meditation on “Is” in Mathematics II – Mathematical Reality

This post continues a meditation on the nature of mathematics begun in Part I. It involves the perennial question about whether mathematics is invented or discovered, and consequently evokes questions about mathematical reality. This subject is probably of little interest to most people, and even most mathematicians. But the extremely heavy involvement of mathematics in the descriptions of quantum mechanics, and the even more mathematically abstruse excursions into ideas such as string theory in an effort to wed quantum mechanics to general relativity, force us to confront the central place mathematics has in “explaining” our physical reality. Of course, this essay has no definitive answers, and leaves the situation as a mystery.  See Meditation on “Is” in Mathematics II – Mathematical Reality.

Meditation on “Is” in Mathematics I – Zeno’s Paradox

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.