The Eternal Question of Rigor

Just how rigorous should a math textbook, tutorial, or other kind of reference be? How much can be taken without proof, how many steps can be skipped, how precise must the definitions be? How paranoid about "rigor" must you be yourself? All the smart kids took those "rigorous" math classes, so you should, too, right? The answers lie in understanding the concept of rigor properly and then determining what it is you're really after.

The dilemma is best illustrated by a seemingly odd instance of redundancy. In the United States, calculus is often the most advanced course taught in high schools. In contrast, real analysis is a college-level course that covers much of the same material but in a much more proofy, theoretical way. Worse yet, there graduate classes in real analysis beyond the undergraduate ones. Do mathematicians just come up with ever more pedantic phrasing for the same concepts? Or does "rigor" just translate into ever more difficult integrals, less informative lectures and textbooks, and an exponentially thickening miasma of condescension that holds that only simpletons (certainly not you) would be unable to understand everything.

The dictionary defines rigorous as "extremely thorough, exhaustive, or accurate," but this can cause aspiring mathematicians to end up with rigor choice paralysis. If we're to do calculus, we first need to understand limits. But to understand limits, we need to understand numbers. After all, how do we really know that something lies between some epsilons and deltas? But then we learn that the real numbers, which answer that first question, are actually constructed from the rationales, which in turn are constructed from the integers. And where do those come from? (Kronecker's divine revelation is distinctly unsatisfying.) Maybe we'll get into number theory? Or maybe Church numerals look promising (maybe they really are from God?). And of course we can't just have naive set theory, we need to be running something like ZFC because this is Serious Business. But why not ZF, or some other system? (NBG sounds hipsterish.) And what really are the laws of logic that underpin this whole operation, allowing us to make logical steps? Oh, and we totally missed formal algebra on our way down, so we're probably just boned six ways to Sunday.

At this rate, it's impossible for any study of math to be truly "rigorous" without descending into the crushing depths of the bottom of the mathematical bedrock. The way out of this quagmire is to clearly and precisely enumerate all of the mathematical dependencies that are being assumed, and then work from there. To the infuriation of mathematics students everywhere, this is rarely done well.

Take the proof that the square root of 2 is irrational. This problem is often introduced as the first problem in a course on real analysis, often without any context, and students are demanded to simply "get it" by the instructor's (or book's) sheer will to power once the proof has been presented. Proofs of the irrationality of the square roots of 3 and then all primes are left as homework, and class is over. This is a total failure of communication, to say nothing of pedagogy. Instead, the course of study should explain that the students may use any properties of the integers they like, along with the definitions (and strictly the definitions) of rational and irrational numbers, and that anything else concerning rationals or reals is off-limits. This is a clean, exact line in the sand. Any confusion can be resolved by pointing out which side of the line things lie on. Proving the properties of the rationals by breaking them down into two integers, applying the known integer properties, and then reassembling the rationals is immensely helpful in clarifying how this line works. The alternative is eternal anguish in the pit of "why are you allowed to use that knowledge but I can't use this knowledge?"

The requirement for clear lines between regions of math under study implies a second requirement: that the scope and goals of the educational endeavor to be well defined. The longer a resource spends on one topic, the less it spends on another. A book dedicated to solving all manner of differential equations but that does not cover the definitions and proofs of the theorems from analysis that underpin them is no less rigorous than a book dedicated to analysis that does not cover the more general metric-free topology that underpins its own popsicle stand. To wit, Baby Rudin makes this exact move, yet nobody would say it is anything short of pitilessly rigorous. And remember, kids, unless you took those weird "logic" classes that for some reason are located in the philosophy department, you don't really know what it means for something to "logically follow" from something else anyway.

So don't ask how rigorous a given math resource is. Ask whether its area of study is cleanly and precisely defined, and if it is, then ask yourself if it's what you want to learn about. If you're bothered by not taking some alternatively "more rigorous" approach, just remember that that approach is hardly as "rigorous" as the one yet beyond it. (Just remember those metaphysical logicians waiting for you to visit their dingy offices in the basement - and the fact that they're not employed to solve fluid simulations.)