Be back in a bit for this.
(The excerpted words below, keeping my place, are from a wondrous post at https://betterexplained.com/ )
When Richard Feynman was asked what information he would leave to humanity to rebuild in the event all knowledge was lost, he said that information would be "All Things Are Made of Atoms." When Cornelius Lanczos was asked what mathematical equation he would leave to humanity, above all others, he said "The Fourier Transform." And Lanczos was probably the very best person to ask this question, perhaps above all others.
"From Smoothie to Recipe
"A math transformation is a change of perspective. We change our notion of quantity from "single items" (lines in the sand, tally system) to "groups of 10" (decimal) depending on what we're counting.
"Tally it up.
"The Fourier Transform changes our perspective from consumer to producer, turning What do I have? into How was it made?
"In other words: given a smoothie, let's find the recipe.
"Why? Well, recipes are great descriptions of drinks. You wouldn't share a drop-by-drop analysis, you'd say "I had an orange/banana smoothie". A recipe is more easily categorized, compared, and modified than the object itself.
"So... given a smoothie, how do we find the recipe?"
Life carries and obeys a basic recipe committed to memory by evolution that most of us "understand" well enough to follow from birth without knowing anything about protons and electrons, quarks, or even proteins of which our bodies (our living clay tablets :) are composed. We just know we are hungry, and what to do to try to fix that. Later, we may want to copy the recipe, and pass it on, but it won't feel to us like passing on a memorized ancient family secret, like following directions, even though it will be.
Important counsel from Kalid Azad, the creator of this Fourier post and his wonderful website Better Explained.
"Get a map, not directions.
"Memorization isn’t understanding:
"you follow the recipe, apply the formula, and get from A to B without
knowing why. Directions “work”, but what about wrong turns? A new
destination? Helping a friend who’s lost at point C, not A? This site is
about sharing maps, the intuitions that get you from any point to any
other point. We’ll leave the raw details for the encyclopedias."