One of the weirdest concepts in mathematics is the Banach–Tarski paradox.

According to Wikipedia, it’s a “theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape.”

In other words, a way to make identical copies of an object by disassembling and reassembling said object in a certain way. The theory is based around the concepts of countable and uncountable sets of infinities and it would therefore literally take forever to even conduct an experiment to test the theory, but the mathematical proofs are there. Some physicists argue that the concept could have use cases in the subatomic world, the quantum realm. Could the theory be a first glimpse into the multiverse? A hint at the possibility of parallel universes?

And more importantly, what the hell am I rambling on about and what has it got to do with Bitcoin?

Infinity

Infinity, meet finity.

A friendly fellow born in 2009. Bitcoin is a mathematical concept, but it’s linked to the physical world in a different way than other mathematical concepts. Most basic mathematics are easy to visualize, like Pythagoras theorem. Just draw squares from the three edges of the triangle, and you can easily see that the square drawn from the hypotenuse is equal in size to the other two.

A concept like Banach–Tarski is of course much harder to translate into a real world drawing. The concepts that make up Bitcoin can all be fairly easily translated into graphs and sketches, but Bitcoin has another layer on top which is much harder to wrap one’s head around. Its connection to reality. Its value.

Value is a strange concept because while it is arguably very real, it is entirely subjective, and because of this there is no yardstick or kilogram to measure it against.

Most websites that try to show you the value of a bitcoin use the US dollar as the unit of value measurement. The US dollar is itself a very poor tool for measurement, because no one knows how many of them there really are, and more importantly, how many of them there will be in the future. Our current global crisis, a virus with a 99.9% survivability rate, has made the money printers of the world go brrrr at an unprecedented rate.

So what’s a fair way of measuring the value of a Bitcoin?

Measuring the Value

Imagine two people marooned on a deserted island. Let’s call them Robinson Crusoe and Tom Hanks. They have one bitcoin each, and there is no way for them to acquire more than that. In other words, those two bitcoins are a metaphor for the entire Bitcoin network. Both Tom and Robinson know that there are but two coins in circulation on the island. Robinson knows how to catch fish with his bare hands and Tom knows how to chase down rabbits, catching them with his hands. They exchange the fish and the rabbits, which are both consumable goods, with each other. Sometimes they use their bitcoins as a medium of exchange, if there’s a shortage of fish or rabbits one day.

One day, Robinson constructs a net with which he’s able to catch a lot more fish per hour. This gives him an advantage over Tom who is now forced to invent a spear in order to catch rabbits at the same pace as Robinson catches fish. The net and the spear are capital goods and they are both beneficial to the island’s entire economy, since both fish and rabbits are now cheaper to produce and cheaper to buy.

There are still only two bitcoins in the economy though. Before the net and the rod, these bitcoins could be used as a store of value and a medium of exchange for fish and rabbits, but now the value of the capital goods are in the economy as well. In other words, a whole bitcoin can now buy you a lot more fish or rabbits than it used to.

Since bitcoins are finite, the value of all capital goods that are added to the economy are in those bitcoins forever, provided that people stop selling their bitcoins at a loss. 

Right now, people are. The volatility and price fluctuations are a testament to that. But there is a point in the future where no one ever sells bitcoins at a loss. When everyone realizes that there’s no point in doing so, things will start to get really weird.

Imagine the entire world economy moving into bitcoin. Everything there is, divided by 21 million. A single bitcoin would itself be enormously valuable. But it doesn’t end there. New capital goods can still be invented and introduced into this economy, making each bitcoin even more valuable. It is in fact a lot easier for new capital goods to enter a Bitcoin economy because the forces of an entirely unhinged free market is the best accelerator of human ingenuity there is.

These thoughts, once you’ve realized that they’re absolutely rooted in reality and probably even inevitable events, are impossible to unthink. They are the rabbit hole.

But let’s take a step back and explore another mind bending concept. The concept of information being literally valuable.

Value of information

Having more information about a certain thing than your peers, has always been valuable. Information is, at its core, the resolution of uncertainty. The smallest unit of information is called a bit and it’s an answer to a simple yes or no question. A single bit can be immensely valuable when representing an answer to a specific question like “Is the game rigged to player A’s advantage?”. If you know the answer to this one bit question, a substantial reward after a well placed bet could await you. Information like this is indirectly valuable.

With the introduction of Bitcoin, information became literally valuable. 

This is a very abstract concept and it’s quite hard to really get a sense of what it implies. We’ve all heard the saying “not your keys, not your bitcoins” but what does that really mean? Holding the private keys to a specific Bitcoin address is owning those bitcoins. In Bitcoin, there is no difference between knowing and owning. Indirectly valuable information can be converted into something valuable by different means, like placing a bet on player A in the aforementioned game, but in Bitcoin the information representing the asset is the asset.

The difference is subtle but the implications are huge. When information is ownership, money truly becomes a language. It will put all the free speech laws of the world to the test because now you can claim ownership by simply communicating. The lines between knowing and owning are now forever blurred out. The world is not the same any longer. Not at all.

Communication was always the key to progress but now it is a literal key. 

The potential path forward it unlocks is unchartered territory and the potential riches that this territory might hold are unfathomable.

Combining Concepts

Now combine these two concepts.

• A finite set of tokens to represent an infinite potential future economy.

• These tokens being nothing more than mere information, transferable between humans at an instant, everywhere in the world.

This invention, or discovery, is bigger than everything. 

It has the potential to propel humanity into a type II civilization on the Kardashev scale in a matter of years, instead of centuries. The most ownable and most valuable good, unconfiscatible.

The end of violence, the end of war.

When you cannot know how much a person owns, and when you can’t take it by force, there’s just no way of enriching yourself other than by providing something of value to your fellow man.

To fully comprehend what this means is impossible. We’re cave trolls trying to imagine Boston Dynamics. We’re living in times of relative peace, but we all pay taxes, and even more importantly, we’re all ruled by our central banking criminal overlords. Central banking is, and always was, a hideous crime against the entire human population. A war crime, if you will. With Bitcoin, each and every one of us can opt out and bypass the mafiosos by disarming them. The weapon of counterfeiting is now useless.

The Banach–Tarski paradox seems like a clear barrier between reality and the realm of the hypothetical. But so did negative numbers. The square root of minus one and a bunch of other mathematical concepts that came to be inside the heads of the geniuses that first thought of them, way before they had any real use cases in meat-space. Bitcoin is balancing on the edge of our understanding of the world, between what is tangible and what can be imagined. The more we allow ourselves to imagine, the more useful it becomes. As Niels Bohr said, “If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet”. The same is true for Bitcoin.

There’s no way out of the rabbit hole, and it seems bottomless. But daring to descend it is the best thing you can do with your time.