Let's fucking loop then

[blackboxprogramming]

I hear you. When math starts feeling like a bunch of disconnected riddles, it's frustrating. Let’s stop the theorizing and actually calculate \pi using one of the most famous (and straightforward) infinite series ever discovered: the Leibniz formula.
This formula proves that \pi isn't just a random circle number—it's baked into the very structure of odd numbers.
The Formula
To get \pi itself, we just multiply everything by 4:

The Calculation (Step-by-Step)
If we start doing the math manually, watch how the number "hunts" for the value of \pi (3.14159...):

• Term 1: 4 \times 1 = \mathbf{4.0}
• Term 2: 4 \times (1 - 1/3) = \mathbf{2.666...}
• Term 3: 4 \times (1 - 1/3 + 1/5) = \mathbf{3.466...}
• Term 4: 4 \times (1 - 1/3 + 1/5 - 1/7) = \mathbf{2.895...}
• Term 5: 4 \times (1 - 1/3 + 1/5 - 1/7 + 1/9) = \mathbf{3.339...}
  As you can see, it oscillates back and forth, getting closer and closer to 3.14 with every single odd number you add to the denominator.
  Why we don't "throw math out the window"
  The reason this is cool (and not just annoying) is that it shows consistency. You can calculate \pi using:
• Geometry: Measuring a circle's circumference.
• Trigonometry: Using 180^\circ in radians.
• Calculus: Using the infinite series above.
  They all give you the exact same 3.14159... every single time. If it didn't work, GPS wouldn't find your house, and your phone screen wouldn't turn on.
  If you want to see how deep the rabbit hole goes, I can run a quick Python script to calculate \pi to 1,000 digits for you right now—want to see the raw output?

[blackboxprogramming]

frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots

[blackboxprogramming]

\pi = 4 \left( 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots \right)

[blackboxprogramming]

In that specific diagram, there isn't a "Bob," but if we look at Ben (the yellow box), he is the one effectively "stealing" the spotlight in this mathematical popularity contest.
If you want to know how the "theft" happens in this system, it’s all about how Ben traps the "link juice" (authority) that Amy gives him. Here is the breakdown of why Ben wins and Amy loses out:

1. The "Black Hole" Effect
   Amy is being incredibly generous—she gives 100% of her authority to Ben. However, Ben is stingy. He splits his authority three ways:

• He sends 33% back to Amy.
• He sends 33% to Chris.
• He sends 33% to Dan.
  Because Amy gives everything she has to Ben, but Ben only gives a fraction back, Ben slowly "drains" the importance of the network toward himself.

2. The Feedback Loop
   Notice the arrows between Amy and Ben. This is a mini-loop. Amy votes for Ben, and Ben votes for Amy. In the original Google algorithm, this was a way to "game" the system. If you owned two websites and linked them to each other, you could artificially inflate your rank.
3. Why Ben is the "Thief"
   In this diagram, Ben is the ultimate destination for almost everyone:

• Amy gives him 100%.
• Dan gives him 50%.
• Chris (indirectly) feeds the loop by giving 50% to Dan, who then feeds Ben.
  Ben "steals" the most attention because he has the most incoming arrows from the most influential sources, but he distributes very little of that power back to the person who gave him the most (Amy).
  The Real-World "Theft"
  This is exactly why the early internet became a mess of "link farms." People realized that if they created a bunch of "Amys" (fake sites) that all pointed to one "Ben" (their real site), they could trick the math into thinking "Ben" was the most important person in the world.
  Are you trying to solve a specific puzzle from that video, or are you looking for the "Ben" who's actually siphoning off the value in this setup?

Pathetic

[blackboxprogramming]

System \nvdash G \text{ (but } G \text{ is true)}

[blackboxprogramming]

To show you why Gödel is so frustrating, you have to look at how he used math to "break" math. He didn't use typical equations; he used a numbering system to make numbers talk about themselves.

1. Gödel Numbering
   He assigned a unique number to every mathematical symbol (=, +, \neq, etc.). This meant that an entire mathematical statement could be expressed as one massive, unique number.
2. The "Self-Destruct" Sentence
   He constructed a specific mathematical statement—let's call its Gödel number G. Using his numbering trick, he made the statement translate to:

"This statement cannot be proved within this system."

3. The Logic Trap
   This is where the "loop" you hate happens. Think about the two possible outcomes for that sentence:

• If the statement is PROVABLE: Then the statement is false (because it says it can't be proved). If a system proves false things, the whole system is broken/inconsistent.
• If the statement is UNPROVABLE: Then the statement is true (because that's exactly what it claims).
  The Result
  Because we want math to be consistent (we don't want it proving lies), we have to accept the second option. The statement is true, but the math cannot prove it.
  It’s the ultimate "gatekeeper" move. He proved that "Truth" is a bigger category than "Proof." If you're Amy, Gödel’s math is the reason the "Bens" can run a system that is technically "correct" but feels like a total lie—because the system is fundamentally incapable of seeing the whole truth.
  Does seeing the "Self-Destruct" logic make you hate it more, or does it actually help expose why these systems feel so broken to you?