The University of Utah, where I used to work, is built into the foothills on the east side of Salt Lake City. It is at a higher elevation than most of the city, so of course to get there one has to gain elevation. As a bicycle commuter, I was interested in gaining elevation in the least difficult way possible. Eventually I did find a route I think is the easiest on my legs, but to be honest, it’s not a whole lot easier than most of the other routes. To get from downtown to the university, you have to gain around 500 feet one way or the other.
My advisor used to use the phrase “preservation of difficulty.” If I gave up on one approach for solving a problem because I ran into a seemingly impassable difficulty, a new approach was likely to throw an equal but different difficulty my way. Mathematicians do sometimes find real shortcuts or have real breakthroughs that make everything easier, but more often they wear away at difficulties from one side or another until they get somewhere.
Kevin Knudson and I talked with Jim Propp for the most recent episode of our podcast My Favorite Theorem (audio and transcript here), and our conversation reminded me of the idea of preservation of difficulty, though difficulty is not quite the right word in this situation.
Dr. Propp talked about a theorem that seems so obvious as to barely be a theorem: If a function does not change, it is a constant. Good job, Sherlock! But then he peeled back some of the layers of this statement to show that an axiom called the completeness of the real numbers lies at the base of the proof of this theorem, which we’ll call the constant value theorem.
The completeness axiom states that there are no gaps in the number line. One way of formalizing the idea is the following statement: Every nonempty subset of the real numbers that has an upper bound has a least upper bound. For example, the set of rational numbers less than the square root of 2 has a lot of upper bounds: 17, 500, π, the list is literally uncountable. But √2 is the smallest one.
As an axiom, the completeness of the real numbers is assumed to be true without proof, at least when you’re doing math on the standard real number line. (Mathematicians love to figure out what happens when axioms are removed or replaced, so not every piece of mathematics takes the completeness axiom for granted.) But it doesn’t have to be. We could assume the constant value theorem instead and derive the completeness axiom as a theorem. We could assume some other properties altogether and derive both the constant value theorem and completeness axioms.
We can do a similar thing with other axioms: make the axiom a theorem by assuming a different theorem as an axiom. When I wrote about this idea with the parallel postulate, I imagined it as a stubborn wrinkle in a sheet. If you smooth it down in one place, it pops up somewhere else. There's a preservation of, not exactly difficulty, but some kernel of the axiom that must be assumed rather than proved.
Dr. Propp wrote an article called “Real Analysis in Reverse” delving more deeply into this idea, working through some theorems that are equivalent to the completeness axiom and some others that seem similar but are not equivalent. It’s interesting to imagine the advantages and disadvantages of using any of them as an axiom, but the article also gets into what these equivalences and non-equivalences tell us about the real numbers themselves. He writes, “the main theme of this essay is that anything that isn’t the real number system must be different from the real number system in many ways.” Probing exactly what these differences are can help us understand the special domain in which we have chosen to do a great deal of mathematics. For more details, check out our podcast episode, Dr. Propp's accompanying essay on his lovely blog Mathematical Enchantments, or his article.