I think if I interpret the explanations correct, a really simple way to put it is like:

Some math nerds were looking at some really hard formulas, and they thought, “Man, if THIS math worked like this OTHER kind of math that looks a lot like it, things would be a lot easier.” But if they tried to logically prove the two things should work alike, there was no way to form an argument that they should.

Then some smart-aleck said, “You know what, what if I do it anyway?” and it turned out *it worked*. There’s still not a great logical framework for explaining why. But there’s also no proof *it doesn’t* despite many people trying.

So it’s basically a big math shortcut, like noticing that if you add up all the digits of a number, and that sum is a multiple of 3, then that number is a multiple of 3. Try it. 12345’s digits sum to 15, and 15 is divisible by 3. Ta da, the number is divisible by 3.

That’s kind of crazy, because our number system wasn’t designed to MAKE that true. And there’s probably not some actual rule that defines WHY it happens. It’s just some kind of weird coincidence that is ALWAYS true. (Maybe there is a proof, and I’m unaware.)

My feeling is Umbral Calculus is a lot like that. It’s a weird trick that lets mathematicians treat one kind of complicated equation as if it were a different kind. Sometimes that makes using the equation for what they want simpler. If you are very strict about logic and proofs it’s *not supposed to work*. But it does.

## Latest Answers