Burn Math Class: And Reinvent Mathematics for Yourself (2016)

2. The Infinite Power of the Infinite Magnifying Glass

2.5. A Brief Sermon on Rigor

Give me the fruitful error any time, full of seeds, bursting with its own corrections. You can keep your sterile truth for yourself.

—Vilfredo Pareto

In this chapter, we discussed (and made use of) the idea of infinitely small numbers, an idea that some consider to be mathematically taboo. Before the end of the chapter, I want to make a general point about the place of rigor and certainty in mathematics. Sometimes in mathematics, you find yourself making an odd argument like the ones we made above, and you’re not sure if it’s “right,” or if it might lead to contradictions somewhere down the road. That’s okay! It is not a sin to simply forge ahead, and attempt to make sense of what you’ve done later. Much of the mathematics that is written in modern textbooks was discovered in just this way, and “cleaned up” later, often long after the original pioneers were dead. If someone else prefers the job of formalizing ideas and cleaning things up, that’s perfectly fine too! As they say: To each own god I hate English pronouns.

If you ever run into a macho mathematician who thinks that this approach renders our discussion undeserving of the name “real mathematics,” kindly ask them to remember the name Leonhard Euler, or for that matter, virtually any other mathematician before the days of Bourbakism. Leonhard Euler was one of the greatest mathematicians of all time, and one day he wrote down this bizarre expression:

The thing on the left is a sum of infinitely many positive terms. It’s essentially

1 + 2 + 4 + 8 + 16 + · · · (forever)

How could one of the best mathematicians of all time have thought this was equal to −1?! Well, it turns out he was following a surprisingly reasonable chain of argument. He wasn’t crazy. When you see the argument for the first time, it’s easy to become convinced that he was right! The point of this is to say that the ultra-cautious “how do I know if I’m right?” feeling we all develop in mathematics classes is a feeling that we should, at least partially, learn to abandon. When we’re inventing mathematics (or anything) for ourselves, we don’t know if we’re right. No one ever does. We might be uncertain of a given argument, and only later think of a different argument that arrives at the same results. This later argument might appear more convincing to us, and to the extent that we can find independent ways of getting to the same conclusion, we can become more convinced that the conclusion makes sense. But we can never be unquestionably sure that what we’re doing makes sense.¹¹

I’ll resist the temptation to mention Gödel’s second incompleteness theorem here, but I won’t resist the temptation to mention apophasis.^ω
ω. I’ll also resist the temptation to explain the joke(s?) in the above footnote, but I won’t resist the temptation to mention that the recursive footnote is a much less popular literary device than it deserves to be.

I understand the desire for rigor, I really do. For several years I planned to go into mathematical logic, focusing especially on the foundations of mathematics. There are astonishingly few modern mathematicians who focus primarily on foundations. As the logician Stephen Simpson has said in his phenomenal textbook Subsystems of Second Order Arithmetic, “Regrettably, foundations of mathematics is now out of fashion.” In spite of its unpopularity, however, I was always drawn to the field. During that time, my obsession with rigor was stronger and more totalizing than that of most. Only later did I realize how much this mindset was killing my mathematical creativity. When I began to read some of the most famous papers by the field’s early pioneers — Kurt Gödel, Alonzo Church, Alan Turing, Stephen Kleene, and modern-day giants like Harvey Friedman and Stephen Simpson — I found that the field’s greatest minds reasoned and spoke in a surprisingly informal way about the formal languages and formal theories they studied. Their proofs were entirely rigorous by the standards of mathematics, but none of them appeared to confine themselves to that level of rigor when thinking about their field. This is not a shortcoming, but a virtue. The old saying among physicists appears to be true: too much rigor can, and does, lead to rigor mortis.