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) > 101010101010101010101010101010 (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:

  • _pheel_@lemmy.world
    1·
    3 years ago

    There are more ways to arrange a deck of 52 cards than there are atoms on Earth.

    52 Factorial

    I feel this one is quite well known, but it’s still pretty cool.

  • Artisian@lemmy.worldEnglish
    1·
    3 years ago

    For the uninitiated, the monty Hall problem is a good one.

    Start with 3 closed doors, and an announcer who knows what’s behind each. The announcer says that behind 2 of the doors is a goat, and behind the third door is a car student debt relief, but doesn’t tell you which door leads to which. They then let you pick a door, and you will get what’s behind the door. Before you open it, they open a different door than your choice and reveal a goat. Then the announcer says you are allowed to change your choice.

    So should you switch?

    The answer turns out to be yes. 2/3rds of the time you are better off switching. But even famous mathematicians didn’t believe it at first.

  • Gogo Sempai@programming.dev
    1·
    3 years ago

    Goldbach’s Conjecture: Every even natural number > 2 is a sum of 2 prime numbers. Eg: 8=5+3, 20=13+7.

    https://en.m.wikipedia.org/wiki/Goldbach’s_conjecture

    Such a simple construct right? Notice the word “conjecture”. The above has been verified till 4x10^18 numbers BUT no one has been able to prove it mathematically till date! It’s one of the best known unsolved problems in mathematics.

  • parrottail@sh.itjust.worksEnglish
    1·
    3 years ago

    Godel’s incompleteness theorem is actually even more subtle and mind-blowing than how you describe it. It states that in any mathematical system, there are truths in that system that cannot be proven using just the mathematical rules of that system. It requires adding additional rules to that system to prove those truths. And when you do that, there are new things that are true that cannot be proven using the expanded rules of that mathematical system.

    "It’s true, we just can’t prove it’.

  • Rhynoplaz@lemmy.world
    0·
    3 years ago

    I heard that Pythagoras killed a man on a fishing trip because he solved a problem first.

    That’s a pretty wild math tale!