What are the most mindblowing things in mathematics?

cll7793@lemmy.world to No Stupid Questions@lemmy.world – 617 points –

What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?

Here are some I would like to share:

  • Gödel's incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
  • Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)

The Busy Beaver function

Now this is the mind blowing one. What is the largest non-infinite number you know? Graham's Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.

  • The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don't even know if you can compute the function to get the value even with an infinitely powerful PC.
  • In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
  • Σ(1) = 1
  • Σ(4) = 13
  • Σ(6) > 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10 (10s are stacked on each other)
  • Σ(17) > Graham's Number
  • Σ(27) If you can compute this function the Goldbach conjecture is false.
  • Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

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I'm not sure Earth would be a correct analogy for spherical geometry. Correct me if I'm wrong, but spherical geometry is when the actual space curvature is a sphere, which is different from just living on a sphere.

CodeParade made a spherical/hyperbolic geometry game, and here's one of his devlogs explaining spherical curvature: https://www.youtube.com/watch?v=yY9GAyJtuJ0

Oh yeah I'm subscribed to codeparade! I know it's not a perfect analogue but since it's such a large scale, our perspective makes it look flat. So at long distances it feels like you're moving in a straight line when you're really not.

A sphere is a perfect model of spherical geometry. It's just a 2-dimensional one, the spherical equivalent of a plane we might stand on as opposed to the space we live in. A sphere is locally flat (locally Euclidean/plane-like) but intrinsically curved, and indeed can have triangles with 3 right angles (with endpoints on a pole and the equator.)

Ah, gotcha! Spherical and hyberbolic geometries always mess with my mind a bit. Thanks for the explanation!