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]

236 comments fedilink hide all child comments

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:

(page 3) 50 comments

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[–] [email protected] 7 points 1 year ago

One thing that definitely feels like "magic" is Monstrous Moonshine (https://en.wikipedia.org/wiki/Monstrous_moonshine) and stuff related to the j-invariant e.g. the fact that exp(pi*sqrt(163)) is so close to an integer (https://en.wikipedia.org/wiki/Heegner_number#Almost_integers_and_Ramanujan.27s_constant). I hardly understand it at all but it seems mind-blowing to me, almost in a suspicious way.

[–] [email protected] 7 points 1 year ago* (last edited 1 year ago) (2 children)

Saving this thread! I love math, even if I'm not great at it.

Something I learned recently is that there are as many real numbers between 0 and 1 as there are between 0 and 2, because you can always match a number from between 0 and 1 with a number between 0 and 2. Someone please correct me if I mixed this up somehow.

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[–] [email protected] 7 points 1 year ago* (last edited 1 year ago) (1 children)

Integrals. I can have an area function, integrate it, and then have a volume.

And if you look at it from the Rieman sum angle, you are pretty much adding up an infinite amount of tiny volumes (the area * width of slice) to get the full volume.

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[–] [email protected] 6 points 1 year ago

An arithmetic miracle:

Let's define a sequence. We will start with 1 and 1.

To get the next number, square the last, add 1, and divide by the second to last. a(n+1) = ( a(n)^2 +1 )/ a(n-1) So the fourth number is (2*2+1)/1 =5, while the next is (25+1)/2 = 13. The sequence is thus:

1, 1, 2, 5, 13, 34, ...

If you keep computing (the numbers get large) you'll see that every time we get an integer. But every step involves a division! Usually dividing things gives fractions.

This last is called the somos sequence, and it shows up in fairly deep algebra.

[–] [email protected] 6 points 1 year ago

Szemeredis regularity lemma is really cool. Basically if you desire a certain structure in your graph, you just have to make it really really (really) big and then you're sure to find it. Or in other words you can find a really regular graph up to any positive error percentage as long as you make it really really (really really) big.

[–] [email protected] 6 points 1 year ago (2 children)

The infinite sum of all the natural numbers 1+2+3+... is a divergent series. But it can also be shown to be equivalent to -1/12. This result is actually used in quantum field theory.

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[–] [email protected] 4 points 1 year ago

This one isn't terribly mind-blowing, though it does have some really cool uses. I always remember it because of its name: Witch of Agnesi

[–] [email protected] 4 points 1 year ago

Small nit: you don't compute sigma, you prove a value for a given input. Sigma here is uncomputable.

was not aware of the machines halting only iff conjectUres are true, tho. Thats a flat out amazing construction.

[–] [email protected] 3 points 1 year ago

The birthday paradox

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