this post was submitted on 03 Aug 2023
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What are the most mindblowing things in mathematics? (lemmy.world)
submitted 1 year ago* (last edited 1 year ago) by [email protected] to c/[email protected]
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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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[–] gogosempai 102 points 1 year ago* (last edited 1 year ago) (7 children)

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.

[–] [email protected] 2 points 1 year ago (5 children)

How can you prove something in math when numbers are infinite? That number you gave if it works up to there we can call it proven no? I'm not sure I understand

[–] gogosempai 5 points 1 year ago

As you said, we have infinite numbers so the fact that something works till 4x10^18 doesn't prove that it will work for all numbers. It will take only one counterexample to disprove this conjecture, even if it is found at 10^100. Because then we wouldn't be able to say that "all" even numbers > 2 are a sum of 2 prime numbers.

So mathematicians strive for general proofs. You start with something like: Let n be any even number > 2. Now using the known axioms of mathematics, you need to prove that for every n, there always exists two prime numbers p,q such that n=p+q.

Would recommend watching the following short and simple video on the Pythagoras theorem, it'd make it perfectly clear how proofs work in mathematics. You know the theorem right? For any right angled triangle, the square of the hypotenuse is equal to the sum of squares of both the sides. Now we can verify this for billions of different right angled triangles but it wouldn't make it a theorem. It is a theorem because we have proved it mathematically for the general case using other known axioms of mathematics.

https://youtu.be/YompsDlEdtc

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