mathematical ideas I understood long after encountering
Not every mathematical result is intuitive, but that's not necessarily the problem of the result. More often, it's about how it's presented, especially in primary and secondary education. I've described two examples from my childhood where there was a long wait between my first encounter with an idea and learning about a simple concept that made it "click".
There's nothing about these concepts that I wouldn't have understood at the time, but no one told me about them. I eventually found out, which is how I'm able to write this, and though I might like to have known about them earlier, I suppose it's never too late to understand something.
Factorials and Empty Products
I don't remember the first time I found out about factorials, but it must have been before I was 11, because I remember talking about it then. There was a problem, something like "what's the largest number you can make out of four 4's". It was open ended, more of a kind of "why are manhole covers round" question than a calculation. You can probably tell that I answered that you could get an arbitrarily large number by doing "4!!!...".^{[1: multifactorials]}
Anyway, I say this not because I want to brag about how smart I was as an 11 year old, but because it was only many years later, when I was 18, that I could say I actually understood why 0! = 1. Often, when this came up in school, the teacher would refer to the recursive definition for the function, and perhaps say that it's useful for things that we calculate using factorials, like permutations.
This never really made sense to me, until I was watching one of Knuth's Christmas lectures and he mentioned offhand that the product of a list with no elements was 1. I thought about this for a while, and realised that the reason 0! = 1 is because 1 is the multiplicative identity. In the same way that the sum of an empty list is 0, the product of an empty list is 1.
I thought about this in two ways. The first was something like: given a list (a, b, c, ...), and 1, the multiplicative identity, ×(a, b, c, ...) = ×(1, a, b, c, ...) = ×(1, 1, a, b, c, ...) and so on. I thought of this like "a sum can have +0 anywhere you want without changing the value, so a product can have ×1 anywhere without changing the value". Hence, after removing (a, b, c, ...) from the list, you're left with as many 1's as you like.
The other way is probably better. To eliminate 3 from ×(1, 2, 3) = 6, you divide by 3. To eliminate 2 from ×(1, 2) = 2 you divide by 2. To eliminate 1 from ×(1) = 1 you divide by 1. Which leaves you with ×() = 1, which is 0!.
The thing about this is that the recursive definition isn't really useful pedagogically, where 0! = 1 is framed as an exception to a rule, rather than the emergent pattern that it is.
- Apparently the notation n!!... is used for multifactorials, where n!! = ×(n, n-2, n-4, ...), n!!! = ×(n, n-3, n-6, ...), &c. I don't really care though.^{[back]}
Proof by Infinite Descent
If you show that a proposition implies something known to be incorrect, you show that the statement is false. This is called proof by contradiction. Infinite descent is a type of proof by contradiction — infinite descent can't happen. If you show that a statement leads to infinite descent, you show that the statement is false.
As for what infinite descent actually is, it means that if something is true for a given number, it's also true for a smaller number (and therefore true for an even smaller number than that one, and so on). Since there are only a finite amount of smaller (integer) numbers —you eventually reach 0— this infinite descent cannot occur.
So, having said this, I'll reproduce a proof that sqrt(2) is irrational.
- Assume that there exists some rational number in lowest terms m/n = sqrt(2).
- Then m²/n² = 2, and m² = 2n²,
- so m² is divisible by 2, and since 2 is prime, m is also divisible by 2.
- Let a = m/2 and substitute 2a = m into the equation.
- So we have (2a)² = 2n²,
- 4a² = 2n²,
- 2a² = n²,
- so n² is divisible by 2, and since 2 is prime, n is also divisible by 2.
- Since m and n are both divisible by 2, this means we have a contradiction, since m/n can't be in lowest terms. Therefore, no such m/n exists.
This is a correct proof, but the violated assumption can feel like a technicality. A proof phrased in terms of infinite descent is much clearer. The conclusion would be something like this:
- If there did exist some m/n = sqrt(2), there would necessarily be a smaller m and smaller n such that m/n = sqrt(2). Since m and n are integers, this is impossible. Therefore, no such m/n exists.
Once again, it was a long time (several years) between when I first saw this proof, and felt that it was unsatisfying because the violated assumption seemed so unimportant, and when I came across the concept of infinite descent watching an interview with Terrence Tao. Later, I realised that this proof could be phrased as an infinite descent proof, after which point it seemed a lot more meaningful and less annoyingly subtle.
In fact, the context he mentioned it in was that he didn't know about this technique during a maths competition where it was required for the solution to a problem. That probably means something.