this post was submitted on 03 Aug 2023

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No Stupid Questions

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516

What are the most mindblowing things in mathematics? (lemmy.world)

submitted 1 year ago* (last edited 1 year ago) by cll7793@lemmy.world to c/nostupidquestions@lemmy.world

172 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 2) 50 comments

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[–] nx@kbin.ectolab.net 10 points 1 year ago* (last edited 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

[–] BitSound@lemmy.world 9 points 1 year ago* (last edited 1 year ago) (1 children)

Not so much a fact, but I've always liked the prime spirals: https://en.wikipedia.org/wiki/Ulam_spiral

Also, not as impressive as the busy beaver, but Knuth's up-arrow notation is cool: https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

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[–] that_leaflet@lemmy.world 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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[–] AlmightySnoo@lemmy.world 7 points 1 year ago* (last edited 1 year ago) (4 children)

The fact that complex numbers allow you to get a much more accurate approximation of the derivative than classical finite difference at almost no extra cost under suitable conditions while also suffering way less from roundoff errors when implemented in finite precision:

$\frac{1}{\varepsilon}\,{\mathrm{Im}}\left[ f(x+i\,\varepsilon) \right] = f'(x) + \mathcal{O}(\varepsilon^2)$

(x and epsilon are real numbers and f is assumed to be an analytic extension of some C^2 real function)

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

Another related and similarly beautiful result is Cauchy's integral formula which allows you to compute derivatives via integration.

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[–] SamSpudd@lemmy.lukeog.com 7 points 1 year ago

As someone who took maths in university for two years, this has successfully given me PTSD, well done Lemmy.

[–] problematicPanther@lemmy.world 7 points 1 year ago (9 children)

The Monty hall problem makes me irrationally angry.

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[–] cia@lemm.ee 7 points 1 year ago* (last edited 1 year ago)

The Julia and Mandelbrot sets always get me. That such a complex structure could arise from such simple rules. Here's a brilliant explanation I found years back: https://www.karlsims.com/julia.html

[–] mathemachristian@lemm.ee 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.

[–] Artisian@lemmy.world 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.

[–] dbaner@lemmy.world 5 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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[–] TheGiantKorean@lemmy.world 5 points 1 year ago* (last edited 1 year ago) (1 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.

[–] Badland9085@lemm.ee 6 points 1 year ago (1 children)

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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[–] theodewere@kbin.social 5 points 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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