Tried reading wikipedia, watched some videos introducing it. Still have no idea what it’s about.
In: Mathematics
In layman’s terms, the “original” theory of “umbral calculus” (called that because it seemed to involve shadowy operations, but produced results verifiable with cumbersome but provable methods) was a seemingly paradoxical relationship between subscript index numbers (e.g. “*Bₙ*(x)”) in one type of equation, and superscript exponent numbers in related equations (e.g. y^n + y^k-n or the like). This seemed *very weird* to the mathematicians who were studying this, because normally there is either no special relationship at all between index numbers and exponents, *or* the relationship is a trivial one, that comes from just flat defining something to be that way. Early on, this did not seem to be what was happening. It seemed like an almost magical connection that didn’t need to be defined.
Now (or, rather, 50ish years ago), we can see that there *is* a specific operation happening here which makes the “magic” happen. It’s called a “linear functional,” which (for this specific application) is a way of mapping exponential things to indexed numbers so that the map is easier to work with, but one that preserves all of the important structure and behavior of the original exponents. This general *idea,* of converting to a different perspective where a problem becomes (in some sense) “easy”, solving it, and then converting back to the original form, is extremely common in math and physics. I recently (in another ELI5) mentioned the “method of images,” a technique used for solving differential equations, e.g. “what does the electric field around this conductor look like with these charges near it?”, by effectively pretending that there are extra “mirror” charges on the other side, which let you exploit symmetry arguments and the like. Or, to use a different example, the proof of Fermat’s last theorem came from proving something seemingly unrelated (an equivalence relationship between equations like a^n + b^n = c^n and a different branch of mathematics), and then showing that a then-recently proven conjecture in that other area of math necessarily meant that Fermat’s last theorem had to be true.
Latest Answers