Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host. And now, on to the math.
Recently I read an interesting book, “A Fuller Explanation”, that attempts to explain the mathematical philosophy and ideas of Buckminster Fuller, written by a former student and colleague of his named Amy Edmondson. Fuller is probably best known as the architect who popularized geodesic domes, those sphere-like structures built of a large number of connected triangles, which evenly distribute forces to make the dome self-supporting. He had a unique approach to mathematics, defining a system of math, physics, and philosophy called “synergetics”, which coined a lot of new terms for what was essentially the geometry of 3-dimensional solids. It’s a very difficult read, which is what led to the need for Edmonson’s book.
One of the basic insights guiding Fuller was the distrust of the casual use of infinities and infinitesimals throughout standard mathematics. He would not accept the definition of infinitely small points or infinitely thin lines, as these cannot exist in the physical world. He was also bothered by the infinite digits of pi, which are needed to understand a perfect sphere. Pi isn’t equal to 3.14, or 3.142, or 3.14159, but keeps needing more digits forever. If a soap bubble is spherical, how does nature know when to stop expanding the digits? Of course, in real life, we know a soap bubble isn’t truly spherical, as it’s made of a finite number of connected atoms. Or as Fuller would term it, it’s a “mesh of energy events interrelated by a network of tiny vectors.” But can you really do mathematics without accepting the idea of infinity?
With a bit of googling, I was surprised to find that Fuller was not alone in his distrust of infinity: there’s a school of mathematics, known as “finitism”, that does not accept any reasoning that involves the existence of infinite quantities. At first, you might think this rules out many types of mathematics you learned in school. Don’t we know that the infinite series 1/2 + 1/4 + 1/8 … sums up to 1? And don’t we measure the areas of curves in calculus by adding up infinite numbers of infinitesimals?
Actually, we don’t depend on infinity for these basic concepts as much as you might think. If you look at the precise definitions used in many areas of modern mathematics, we are talking about limits— the convenient description of these things as infinites is really a shortcut. For example, what are we really saying when we claim the infinite series 1/2+1/4+1/8+… adds up to 1? What we are saying is that if you want to get a value close to 1 within any given margin, I can tell you a number of terms to add that will get you there. If you want to get within 25% of 1, add the first two terms, 1/2+1/4. If you want to get within 15%, you need to add the three terms 1/2+1/4+1/8. And so on. (This is essentially the famous “epsilon-delta” proof method you may remember from high school.). Thus we are never really adding infinite sums, the ‘1’ is just a marker indicating a value we can get arbitrarily close to by adding enough terms.
You might object that this doesn’t cover certain other critical uses of infinity: for example, how would we handle Zeno’s paradoxes? You may recall that one of Zeno’s classic paradoxes says that to cross a street, we must first go 1/2 way across, then 1/4 of the distance, and so on, adding to an infinite number of tasks to do. Since we can’t do an infinite number of tasks in real life, motion is impossible. The traditional way to resolve it is by saying that this infinite series 1/2+1/4+… adds up to 1. But if a finitist says we can’t do an infinite number of things, are we stuck? Actually no— since the finitist also denies infinitesimal quantities, there is some lower limit to how much we can subdivide the distance. After a certain amount of dividing by 2, we reach some minimum allowable distance. This is not as outlandish as it seems, since physics does give us the Planck Length, about 10^-35 meters, which some interpret as the pixel size of the universe. If we have to stop dividing by 2 at some point, Zeno’s issue goes away, as we are now adding a finite (but very large) number of tiny steps which take a correspondingly tiny time each to complete. Calculating distances using the sum of an infinite series again becomes just a limit-based approximation, not a true use of infinity.
Thus, you can perform most practical real-world mathematics without a true dependence on infinity. One place where finitists do diverge from the rest of mathematics is in reasoning about infinity itself, or the various types of infinities. You may recall, for example, Cantor’s ‘diagonal’ argument which proves that the infinity of real numbers is greater than the infinity of integers, leading to the idea of a hierarchy of infinities. A finitist would consider this argument pointless, having no applicability to real life, even if it does logically follow from Cantor’s premises.
In Fuller’s case, this refusal to accept infinities had some positive results. By focusing his attention on viewing spheres as a mesh of finite elements and balanced force vectors, this probably set him on the path of understanding geodesic domes, which became his major architectural accomplishment. As Edmondson describes it, Fuller’s alternate ways of looking at standard areas of mathematics enabled him and his followers to circumvent previous rigid assumptions and open “rusty mental gates that block discovery”. Fuller’s views were also full of other odd mathematical quirks, such as the idea that we should be “triangling” instead of “squaring” numbers when wanting to look at higher dimensions; maybe we will discuss those in a future episode.
And this has been your math mutation for today.