For a number of years I have collected excerpts that portray mathematical ideas in a literary or philosophical setting. I had occasion to read a few of these on the last day of some math classes I was teaching, since there was no point in introducing a new subject before the final exam.
I thought it might be interesting to present some of these excerpts now. They roughly fall into three categories: logic, infinities (Zeno’s Paradoxes, infinite regress), and permutations.
The September 2019 Special Issue of Scientific American is a must read. Unfortunately it is behind a paywall, so you should purchase a copy at a store or digitally online. All the articles are fascinating and relevant, and address basic questions of epistemology—how do we know what we know? The first section, “Truth”, is the most pertinent to my thinking, as it covers three subjects I have been pondering for years.
Physical Reality. The first article in the section is “Virtually Reality: How close can physics bring us to a truly fundamental understanding of the world?” by George Musser. I have addressed this issue of physical reality in my article Angular Momentum, with an emphasis on the role of mathematics. Musser cites the difficulties of trying to understand quantum mechanics after almost one hundred years or the failure to marry quantum mechanics with Einstein’s theory of gravitation as possible indications that there might be limits to our human endeavor to comprehend physical reality. This frustration is not new:
Over the generations, physicists have oscillated between self-assurance and skepticism, periodically giving up on ever finding the deep structure of nature and downgrading physics to the search for scraps of useful knowledge. Pressed by his contemporaries to explain how gravity works, Isaac Newton responded: “I frame no hypotheses.”
This essay introduces a topic I have been thinking about for a number of years. It also may allow me to connect the math impulse to a wider range of thoughts than just those based on math or even science.
It all begins with the perennial question of “why” that drives our curiosity about the nature of things and how various situations came about, such as our physical universe, our biology, the origin of life, or historical events. The explanations are usually couched in terms of causal links: such and such happened because some other thing happened. In the physical sciences we think the causal links follow certain physical, chemical, or biological laws that we provisionally hypothesize. In the historical realm we think there are still causes, such as the physical environment (geography, climate, weather, etc.) or the imprint of individuals. But the historical chains of events are often disrupted by chance and coincidences, and some supposed links degenerate into imagined connections or associations.
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.
Recently I viewed a startling video by Matt Parker about the Tupper Self-Referential Formula. It is a formula that visually represents itself when graphed at a specific location in the (x, y) plane. I found it difficult to fathom, so I looked it up on Wikipedia and Google. After reading different explanations, I finally think I have the idea. So thought I would add my version to the mix. See Tupper Self-Referential Formula.