What are the most mindblowing fact in mathematics?

Submitted ⁨⁨1⁩ ⁨year⁩ ago⁩ by ⁨cll7793@lemmy.world⁩ to ⁨[deleted]⁩

What concepts or facts do you know from math that is mindblowing, 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 mindblowing 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
Σ(17) > Graham’s Number
Σ(27) If you can compute this function the Goldbach’s conjecture is false.
Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

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Comments

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jabberati@social.anoxinon.de ⁨1⁩ ⁨year⁩ ago
@cll7793 A useful, well known fact in statistics:
If you take the average of indepedent variables that come from the same distribution, you suddenly end up with a normal distribution: https://en.m.wikipedia.org/wiki/Central_limit_theorem
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GooseFinger@lemmy.world ⁨1⁩ ⁨year⁩ ago
The Banach - Tarski Theorm is up there. Basically, a solid ball can be broken down into infinitely many points and rotated in such a way that that a copy of the original ball is produced. Duplication is mathematically sound! But physically impossible.

en.m.wikipedia.org/wiki/Banach–Tarski_paradox

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- sneezycat@sopuli.xyz ⁨1⁩ ⁨year⁩ ago
  
  Duplication is mathematically sound!
  
  Only if you accept the axiom of choice :P
  
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HexesofVexes@lemmy.world ⁨1⁩ ⁨year⁩ ago
Non-Euclidean geometry.

A triangle with three right angles (spherical).

A triangle whose sides are all infinite, whose angles are zero, and whose area is finite (hyperbolic).

I discovered this world 16 years ago - I’m still exploring the rabbit hole.

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- julianh@lemm.ee ⁨1⁩ ⁨year⁩ ago
  Spherical geometry isn’t even that weird because we experience it on earth at large scales.
  
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  - Boogeyman4325@reddthat.com ⁨1⁩ ⁨year⁩ ago
    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
    
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Varyk@sh.itjust.works ⁨1⁩ ⁨year⁩ ago
Great thread. I’m just reading and watching stuff this afternoon now

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- cll7793@lemmy.world ⁨1⁩ ⁨year⁩ ago
  Glad you are enjoying it! Please feel free to share any other channels like Computerphile, Numberphile, Mutual Information, & Veritasium. It’s difficult to find gems nowadays on YT.
  
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  - jantin@lemmy.world ⁨1⁩ ⁨year⁩ ago
    I like Stand-up Maths, usually starts with a real-life situation and escalates into proper math, sometimes also programming shows up.
    
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  - Blyfh@lemmy.world ⁨1⁩ ⁨year⁩ ago
    Well, if you ever want to find some more mathtubers, just browse the #SoMe2 or #SoMe3 hshtags on YouTube. It led me to Morphocular. Another great channel is Another Roof. VSauce also has some good videos on math topics, if you like him.
    
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  - bird@aussie.zone ⁨1⁩ ⁨year⁩ ago
    Late to the party but one channel I came across was Combo Class, who should get more love! I adore his setup and his passion for weird maths stuff.
    
    Absolutely strange in the most endearing way.
    
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  - SandLight@lemmy.world ⁨1⁩ ⁨year⁩ ago
    As mentioned by @Blyfh, 3Blue 1Brown is great.
    
    I also came across this channel recently that does more Long Lecture videos on math for video game programmers. I found the the one on splines to be quite enlightening (as I had recently been doing a lot of spline stuff and struggling with third party packages).
    
    www.youtube.com/@Acegikmo
    
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zenharbinger@lemmy.world ⁨1⁩ ⁨year⁩ ago
There are more infinite real numbers between 0 and 1 than whole numbers.

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cia@lemm.ee ⁨1⁩ ⁨year⁩ ago
The Julia and Mandelbrot sets always gets me. That such a complex structure could arise from such simple rules. Here’s a brilliant explanation I found years back: www.karlsims.com/julia.html

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SamSpudd@lemmy.lukeog.com ⁨1⁩ ⁨year⁩ ago
As someone who took maths in university for two years, this has successfully given me PTSD, well done Lemmy.

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nx@kbin.ectolab.net ⁨1⁩ ⁨year⁩ ago
The 196,883-dimensional monster number (808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
≈ 8×10^53) is fascinating and mind-boggling. It's about symmetry groups.

There is a good YouTube video explaining it here: https://www.youtube.com/watch?v=mH0oCDa74tE

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CHINESEBOTTROLL@lemm.ee ⁨1⁩ ⁨year⁩ ago
Maybe a bit advanced for this crowd, but there is a three-way correspondence between logic, type theory (like in programming languages), and topology. 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.

You can also add topology into the mix:

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- febra@lemmy.world ⁨1⁩ ⁨year⁩ ago
  HoTT is really HOT right now. Pun intended. Yeah, type theory is great. I’ve never actually used it in connection with topology though
  
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problematicPanther@lemmy.world ⁨1⁩ ⁨year⁩ ago
The Monty hall problem makes me irrationally angry.

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- zenharbinger@lemmy.world ⁨1⁩ ⁨year⁩ ago
  I found the easiest way to think about it as if there are 10 doors, you choose 1, then 8 other doors are opened. Do you stay with your first choice, or the other remaining door? Or scale up to 100. Then you really see the advantage of swapping doors. You have a higher probability of choosing the last remaining door than of having correctly choosen the correct door the first time.
  
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  - dirtbiker509@lemm.ee ⁨1⁩ ⁨year⁩ ago
    Holy shit this finally got it to click in my head.
    
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  - _g_be@lemmy.world ⁨1⁩ ⁨year⁩ ago
    I find the easiest way to understand Monty Hall is to think of it in a meta way: Situation A - A person picks one of three doors, 1 n 3 chance of success. Situation B - A person picks one of two doors, 1 in 2 chance of success. If you were an observer of these two situations (not the person choosing doors) and you were gonna bet on which situation will more often succeed, clearly the second choice.
    
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AlmightySnoo@lemmy.world ⁨1⁩ ⁨year⁩ ago
The fact that complex numbers give you a better approximation of the derivative than classical finite difference at almost no extra cost:

rac{1}{arepsilon}{athrm{Im}}eft[ f(x+iarepsilon) ight] = f’(x) + athcal{O}(arepsilon^2)

Higher-order derivatives can also be obtained using hypercomplex numbers.

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- efrique@lemm.ee ⁨1⁩ ⁨year⁩ ago
  Well, that’s something I hadn’t seen before.
  
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- MBM@lemmings.world ⁨1⁩ ⁨year⁩ ago
  This is the first one that’s new to me! Really interesting, I guess the big problem is that you need to actually be able to evaluate the analytic extension.
  
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that_leaflet@lemmy.world ⁨1⁩ ⁨year⁩ ago
Rieman sums and 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 area slices to get the volume.

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- Reliant1087@lemmy.world ⁨1⁩ ⁨year⁩ ago
  Geometric interpretation of integration is really fun, it’s the analytic interpretation that most people (and I) find harder to understand.
  
  If you work in numerically solving integrals using computers, you realise that it’s all just adding tiny areas.
  
  I finally understand what divergent integrals are intuitively when I encountered one while trying to do a calculation on a computer.
  
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dbaner@lemmy.world ⁨1⁩ ⁨year⁩ ago
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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- MuThyme@lemmy.world ⁨1⁩ ⁨year⁩ ago
  I don’t suppose you know the exact application in QFT? I assume it’s used for some renormalization scheme?
  
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theodewere@kbin.social ⁨1⁩ ⁨year⁩ ago
Incompleteness is great.. internal consistency is incompatible with universality.. goes hand in hand with Relativity.. they both are trying to lift us toward higher dimensional understanding..

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mathemachristian@lemm.ee ⁨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.

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keenanpepper@sopuli.xyz ⁨1⁩ ⁨year⁩ ago
One thing that definitely feels like “magic” is Monstrous Moonshine (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 (en.wikipedia.org/wiki/Heegner_number#Almost_integ…). I hardly understand it at all but it seems mind-blowing to me, almost in a suspicious way.

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Artisian@lemmy.world ⁨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.

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TheGiantKorean@lemmy.world ⁨1⁩ ⁨year⁩ ago
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 betweeb 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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- Badland9085@lemm.ee [bot] ⁨1⁩ ⁨year⁩ ago
  You are correct. This notion of “size” of sets is called “cardinality”. For two sets to have the same “size” is to have the same cardinality.
  
  The set of natural numbers (whole, counting numbers, starting from either 0 or 1, depending on which field you’re in) and the integers have the same cardinality. They also have the same cardinality as the rational numbers, numbers that can be written as a fraction of integers. However, none of these have the same cardinality as the reals, and the way to prove that is through Cantor’s well-known Diagonal Argument.
  
  Another interesting thing that makes integers and rationals different, despite them having the same cardinality, is that the rationals are “dense” in the reals. What “rationals are dense in the reals” means is that if you take any two real numbers, you can always find a rational number between them. This is, however, not true for integers. Pretty fascinating, since this shows that the intuitive notion of “relative size” actually captures the idea of, in this case, distance, aka a metric. Cardinality is thus defined to remove that notion.
  
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  - TheGiantKorean@lemmy.world ⁨1⁩ ⁨year⁩ ago
    Fantastic explanation. Thank you!
    
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Snipe_AT@lemmy.atay.dev ⁨1⁩ ⁨year⁩ ago
I love this thread! It shall be saved.

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swirle13@lemmy.world ⁨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

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uniqueid198x@lemmy.dbzer0.com ⁨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.

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MonkderZweite@feddit.ch ⁨1⁩ ⁨year⁩ ago
Told ya. That’s why i don’t like math: the syntax is awkward.

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donut4ever@lemmy.world ⁨1⁩ ⁨year⁩ ago
Infinity

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- Pons_Aelius@kbin.social ⁨1⁩ ⁨year⁩ ago
  Which one?
  
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  - donut4ever@lemmy.world ⁨1⁩ ⁨year⁩ ago
    Exactly
    
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Tolstoshev@lemmy.world ⁨1⁩ ⁨year⁩ ago
The birthday paradox

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