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:

  • CHINESEBOTTROLL
    14•1 year ago

    Maybe a bit advanced for this crowd, but there is a correspondence between logic and type theory (like in programming languages). Roughly we have

    Proposition ≈ Type

    Proof of a prop ≈ member of a Type

    Implication ≈ function type

    and ≈ Cartesian product

    or ≈ disjoint union

    true ≈ type with one element

    false ≈ empty type

    Once you understand it, its actually really simple and “obvious”, but the fact that this exists is really really surprising imo.

    https://en.m.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

    You can also add topology into the mix:

    https://en.m.wikipedia.org/wiki/Homotopy_type_theory

    • @febra
      2•1 year ago

      One could say that Homotopy type theory is really HoTT right now (pun intended.) I’ve never actually used it in connection with topology though.

      Also, overall higher order logics are really cool if you’re a programmer and love the abstract :)