Question

Allegories of mathematical results in real life?

Answer 1

I guess it's a more philosophical point, but the Yoneda lemma can be interpreted as "if you know how some thing X interacts with every thing (of the same kind), then you know X completely".

For example if you have, say some specific physical body X, then if you know, for every physical body how it interacts if you throw it at X, then you know everything about X.

I think that example encapsulates the idea of how the Yoneda lemma is mostly applied: just throw everything at the object you want to understand, then if you know what happens there, you understand the object.

Answer 2

Not sure if this is exactly what you're talking about, but I've always been super fascinated with the idea of emergent behavior. I don't know if I'd call it a mathematical result, but it pops up in statistics, computer science, and physics in a whole bunch of different ways.

One of my favorite examples is Conways Game of Life. Here you have just 4 simple rules describing the behavior of individual cells, and yet what emerges is a system capable of producing both chaotic randomness and dynamic but stable patterns. The rules seem to create multi cell objects that behave in interesting ways, even though the rules are only defining the most basic relationships between cells.

And that's how the real world feels sometimes. There's chaos everywhere, but there's also stability. We can describe objects that have unique rules, but when you get down to it they're all made of the same building blocks that follow a simpler set of rules (or so we hope). Life itself seems to just be some sort of emergent behavior of chemistry.

Answer 3

Maybe a bit of a stretch but in algebraic number theory, to understand all the abelian extensions of a field is equivalent to understanding the internal multiplicative structure of it's ideals.

To me it's always felt like some cheesy saying "To understand what is around you, you must first understand what is within you"

Answer 4

'Euclid's first common notion is this: "Things which are equal to the same thing are equal to each other." That's a rule of mathematical reasoning. It's true because it works; has done and always will do. In his book, Euclid says this is "self-evident." You see, there it is, even in that two-thousand year old book of mechanical law: it is a self-evident truth of things which are equal to the same thing, are equal to each other. We begin with equality. That's the origin, isn't it? That balance -- that's fairness, that's justice.'

Answer 5

I think what you’re looking for is something like “Shortest distance between two points is a straight line” could I guess be translated to some sentiment about “just going for it” but I don’t think you’re going to get what you need from this subreddit.

Answer 6

The intuitionist mathematician would not accept the law of excluded middle, analogous to statements in real life being neither true nor false. Although the more philosophical ideas in math are not the results but more what are the axioms we take for granted. Do we take AoC for granted because we believe we can make an infinitude of arbitrary decisions? Do infinite things really exist in reality? Is induction valid in real life?