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.
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.
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 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.
I have long been fascinated by Newton’s proof of Kepler’s Equal Areas Law and wanted to write about it. Of course, others have as well, but I wanted to emphasize an aspect of the proof that supported my philosophy of mathematics.
Before I get to Newton, however, I wanted to discuss how Kepler himself justified this law, since his approach has a number of fascinating historical aspects to it. I have previously discussed Kepler’s ellipse and in the process of doing that research, I came across a number of articles about how Kepler arrived at his equal areas law. One notable result is that even though now we call the idea that a planet orbits the Sun in an elliptical path with the Sun at one focus, Kepler’s First Law, and the idea that the line from the Sun to the planet sweeps out equal areas in equal times, Kepler’s Second Law, Kepler actually discovered these laws in reverse order. See Kepler’s Equal Areas Law
Years ago (1963) I got the paperback The Calculus:A Genetic Approach, by Otto Toeplitz, which presented the basic ideas of the differential and integral calculus from a historical point of view. One thing Toeplitz did at the end of his book that I had not seen in other texts was to show the equivalence of Kepler’s Laws and Newton’s Law of Gravity. (Since 1963 David Bressoud has developed this theme in his excellent 1991 text.) I thought I would try to emulate Toeplitz’s approach with more modern notation (vectors) and arguments in hopes of extracting the essential ideas from the clutter.
A by-product of this effort was to reveal strongly the different paths that physics and mathematics follow in understanding physical reality. The mystery is that the mathematics ends up describing the physics so well. I will return to this theme a number of times in other posts. See Kepler’s Laws and Newton’s Laws.
This 2011 article gives some thoughts I had after reading Michael Dirda’s review in the Washington Post of Larrie D. Ferreiro’s Measure of the Earth. The book described the 1735 Geodesic Mission, whose purpose was to resolve the question of the shape of the earth, that is, whether it was a sphere, or like an egg with the poles further from the center than the equator, or like an oblate spheroid with the equator further from the center than the poles, as Newton averred due to centrifugal force. In the review Dirda said, “A team, sympathetic to Newton’s view, would travel to what is now Ecuador and measure the exact length of a degree of latitude near the equator. This would then be compared with the same measurement taken in France. If the latter was larger, Newton was right.” I wondered at first if Dirda got it right. It turned out my confusion stemmed from a mistaken definition of a degree of latitude. See Degree of Latitude.