27.03.2024

*Can something be both finite and infinite at the same time? Bitcoin forces us to rethink our notions of scarcity, value, and the boundless potential of a finite creation with seemingly limitless possibilities.*

**I.**

In the 5th century BC, the ancient Greek philosopher Zeno of Elea used a paradox to explore the concept of infinity. Known as "Achilles and the Tortoise," the paradox involves a race between the legendary Greek hero Achilles and a slow tortoise. The tortoise is given a head start to make the race more challenging.

According to common sense and our understanding of motion and speed, Achilles should quickly overtake the tortoise. However, Zeno argues that Achilles will never actually reach the tortoise. Zeno's reasoning is based on the idea that Achilles must first cover half the distance between them before he reaches the tortoise's starting point. But before he can cover half the distance, he must first cover a quarter of the distance, and so on. This process can be divided infinitely, requiring Achilles to take infinite steps.

This counterintuitive notion challenges our common sense and practical reasoning. It suggests that the nature of infinity can defy our expectations and lead to unexpected conclusions. Once we grasp the concept of infinity, we can no longer ignore its implications for our understanding of the world.

**II.**

Pythagoras and his followers, the Pythagoreans, did not deal well with infinity. They focused on integers and ratios (rational numbers) as the building blocks of the universe. Their "All is number" philosophy meant that everything could be understood through ratios of integers. This philosophy led to music theory and geometry discoveries, including the famous Pythagoras' theorem.

Nevertheless, their world was shaken by the discovery of incommensurable lengths, now known as irrational numbers. These numbers cannot be expressed as ratios of two integers.

A prime example is the square root of 2.

For the Pythagoreans, the existence of irrational numbers like the square root of 2 was an absurdity. It contradicted their core belief that whole number ratios could represent everything. According to some possibly legendary accounts, a Pythagorean named Hippasus discovered incommensurability and was ostracized or even killed for challenging the Pythagorean doctrine.

**III.**

Bernhard Riemann offered an alternative, more refined approach to apprehending the concept of infinity. The 19th-century German mathematician shaped infinity into a sphere. The Riemann sphere provides a powerful tool for dealing with infinity in more complex analysis. Imagine a plane where all numbers reside, stretched and wrapped around a sphere. A single point at the top of the sphere represents infinity.

It’s like a lighthouse beam sweeping across a flat complex plane. It might seem like the beam disappears the further it goes. But if you imagine the beam hitting the Riemann sphere, it simply converges on the single point at the top, representing infinity.

By adding this point at infinity, the Riemann sphere allows us to treat functions that might blow up or become undefined at infinity on the flat plane as well-behaved functions on the sphere. Operations like division by zero, which are undefined on the complex plane, can now be assigned specific interpretations at this "infinite point" on the Riemann sphere.

**IV.**

The concept of infinity has preoccupied mathematicians and philosophers for millennia. From Pythagoras's reverence for whole numbers to Zeno's paradoxes and the elegance of the Riemann sphere, notions of endlessness and limits have continually pushed the boundaries of our understanding. Surprisingly, these abstract ideas resonate within the very heart of bitcoin, a digital currency seemingly built on the most definite of foundations – a finite supply.

**Bitcoin's Core Finiteness.**

At its core, bitcoin embodies a fundamental finiteness. The protocol irrevocably dictates a maximum of 21 million bitcoins that will ever be mined. Bitcoin mining is the process of verifying transactions on the bitcoin network and creating new bitcoins. Miners use specialized computers to solve complex mathematical problems, and the first miner to solve a problem is rewarded with a certain number of bitcoins. The reward is currently 6.25 BTC per block, halved every 210,000 blocks (roughly every four years). In addition to the block reward, miners also receive transaction fees from the transactions they include in their blocks.

This absolute scarcity starkly contrasts traditional currencies, where central banks hold the power to create more money, leading to potential inflation and devaluation. With its hard cap, bitcoin presents itself as an infinitely inflation-resistant asset, a concept appealing to those seeking a store of value insulated from the arbitrariness of monetary policy.

Yet, counterintuitively, within this rigid finitude lies the potential for a different kind of infinity. Each bitcoin is divisible into 100 million units, known as satoshis. This echoes the idea of the Riemann sphere, where a seemingly infinite plane is mapped onto a finite surface. While there's an absolute limit to the number of bitcoins, the divisibility into these near-infinitesimal units allows for a near-endless range of transactional possibilities.

**Bitcoin's Mining Will Endure Infinitely.**

The bitcoin mining mechanism further plays off the notion of infinity – with a twist. bitcoin's difficulty recalibrates, ensuring a steady pace of block creation. Miners compete to solve complex puzzles, the reward for which halves periodically.

Even if the block reward were to drop to zero, mining would still be profitable for the right mining power at the right difficulty adjustment due to transaction fees. Transaction fees are paid by users who want their transactions to be processed. The higher the fee, the faster the transaction will be processed.

This system recalls Zeno's paradox of Achilles and the tortoise; similarly, the mining reward will approach zero but never truly disappear, as bitcoin will always be transferred and there will always be someone to mine, as the difficulty - and therefore the power necessary to mine the next block - is always adjusted to make the average block time 10 minutes. The process of mining stretches out infinitely within the protocol's prescribed rules.

The mining landscape will likely change significantly in the future. Renewable energy sources, such as solar and wind, are becoming increasingly affordable, making it more feasible to mine bitcoins using renewable energy. Additionally, the Lightning Network and other Layer 2 solutions could reduce the number of transactions that need to be processed on the bitcoin blockchain, reducing the demand for mining power. At the same time, phenomena like ordinals create high demand for direct on-chain transactions that are welcome to be (very) expensive, as they make the inscription more expensive. Already today, transaction fees add up somewhere between 0.2 and 0.4 BTC per block (see https://mempool.space/).

We don't need to wait 120 years until the last mined bitcoin in 2140: Already within the next 12 years (three halvings) transaction fees will surpass the creation of new bitcoin per block for the first time. So the big test for bitcoin's mining mechanism will come sooner than expected and once it's passed, it will continue infinitely.

**The paradoxical nature of bitcoin's value.**

The perception of bitcoin's value also dances with infinity. While the currency itself is quantifiably finite, its price in fiat currencies is anything but. The volatile swings in Bitcoin's price, from mere cents to tens of thousands of dollars, suggest the possibility of infinite value growth (or, conversely, an infinite collapse toward zero). This lack of a fixed price ceiling creates a sense of open-endedness that mirrors the boundless nature of infinity.

The true promise of bitcoin may lie in the infinite potential of its decentralized structure. A global network with no central authority, resistant to any single point of failure, hints at a system that could endure and evolve infinitely. Its open-source nature facilitates a boundless realm of applications, from simple payments to complex smart contracts, opening the door to a new economic paradigm of infinite possibilities.

Bitcoin, therefore, is a curious juxtaposition. Its finite supply offers a mathematically verifiable scarcity, appealing to those yearning for a safe haven asset in a world of inflationary currencies. Simultaneously, its micro-divisibility, volatile price, approaching total supply, and the expansive potential of its decentralized ledger all flirt with various shades of infinity and applications we cannot even imagine yet.

Bitcoin embodies the quest for sound money and value, a pursuit as old as civilization, but manifested in a digital asset that encapsulates both the tangible and the intangible, the definite and the limitless.