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:
- YouTube - The Busy Beaver function by Mutual Information
- YouTube - Gödel’s incompleteness Theorem by Veritasium
- YouTube - Halting Problem by Computerphile
- YouTube - Graham’s Number by Numberphile
- YouTube - TREE(3) by Numberphile
- Wikipedia - Gödel’s incompleteness theorems
- Wikipedia - Halting Problem
- Wikipedia - Busy Beaver
- Wikipedia - Riemann hypothesis
- Wikipedia - Goldbach’s conjecture
- Wikipedia - Millennium Prize Problems - $1,000,000 Reward for a solution
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.
I like Stand-up Maths, usually starts with a real-life situation and escalates into proper math, sometimes also programming shows up.
Well, if you ever want to find some more mathtubers, just browse the #SoMe2 or #SoMe3 hashtags on YouTube. It led me to Morphocular, who does great videos with great visual quality similar to 3B1B. Another great channel is Another Roof with a completely different approach of videomaking. VSauce also has some good videos on math topics, if you like him.
As mentioned by @Blyfh, 3Blue 1Brown is great.
I also came across this channel by Freya Holmér 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).
https://www.youtube.com/@Acegikmo
I can’t claim it’s as high quality as the channels you’ve mentioned, but I actually have a channel! I only have one video at the moment, because they take a long time to make, but I’m planning on having the next one out perhaps within the next month.
https://m.youtube.com/@axiomath3434
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.