The last post asked, “Is mathematics fiction or nonfiction?” Here’s a recap of the discussion so far:
- One definition for math is “the science of patterns.”
- Science and technology depend on math. Disciplines like chemistry, physics, and computer science are made possible by the quantitative study of real-world patterns. This makes it seem like math is nonfiction.
- Some mathematical scenarios, like nine-dimensional space, do not correspond to the real world in any obvious way. Since it is possible to study patterns that do not exist, it would seem that math is fiction.
If you went to a typical American high school, you almost certainly learned Euclidean geometry. This geometry was formalized* by the ancient Greek mathematician Euclid. The principles of Euclid’s system seem straightforward enough. Any two points can be connected by a straight line. All right angles are equal. Given a line and a point not on that line, there is exactly one line that can be drawn through the point that is parallel to the original line.
*This was important for a lot of reasons and will hopefully be discussed in a future post.
Screenshot from here.
Our real-world experience tells us that making things bigger won’t change their shape. A triangle with sides of 3 feet, 4 feet and 5 feet should be the same shape (and have the same angles) as a triangle with sides 30, 40 and 50 feet. Fortunately, Euclidean geometry meshes with our idea of reality. It also does a fantastic job describing things like tables, beach balls, and skyscrapers. There is just one problem: apparently, the universe doesn’t work that way.
Space is curved. You may have heard this before—the idea became famous when Einstein developed his theory of relativity. Einstein did not use Euclidean geometry in his model. He used a curved Riemannian geometry. Riemannian geometry is just as mathematical as Euclidean geometry is, but it clashes with concrete intuition. Small changes in its basic structure shove curved geometry outside the realm of experience. But curved geometry, not Euclidean geometry, fits relativity. In other words, physics says that curved geometry describes our universe, and Euclidean geometry does not. Now, which of these ideas is a fiction? Is Euclidean geometry “nonfiction” because it is true to human experience? Is Riemannian geometry “nonfiction” because it is used in physics? Are both nonfiction, or maybe neither?
In the end, it depends what you mean by “fiction” and “nonfiction.” Math is not information we gather about the natural world. You can’t pick up a piece of math or look at it under a microscope. No scientific experiment can prove a theorem. No historical record contains math that “happened.” Every bit of math in the history of civilization occurred in someone’s mind. If “nonfiction” is defined as “a description of something physical,” then math is fiction. However, math gives us insight into the natural world. We use math to organize the information we gather with our senses. Math can be used to explain and predict physical patterns. It makes the difference between drinking “coffee, then coffee, then coffee” and drinking “three cups of coffee.” It’s what takes you from “This coin sometimes comes up heads” to “What are the odds it will happen?” If nonfiction includes “that which humans use to understand reality,”then math is nonfiction.
So, that is an answer. It depends on how you define nonfiction. But the debate is far from over. For one thing, these posts have not discussed the controversial idea of “mathematical truth.” The tremendous enjoyment of seeing the order and harmony in math is often accompanied by a sense that math is “real” in some way: that it is something humans perceive, not imagine. This adds another facet to the fiction vs. nonfiction debate. Maybe it will get its own post later on.
Thank you for reading!
There were many, many people whose ideas went into the formulation of this series. I tried to present the points smoothly, without pointing out whose idea was whose. But to everyone who discussed this with me: thanks for your input!