The Number Zero and Bitcoin (Part 2)
Bitcoin is an invention as impactful as its distant digital ancestor: the number zero.
Functions of Zero
Zero’s first function is as a placeholder in our numeric system: for instance, notice the “0” in the number “1,104” in the equation above, which indicates the absence of value in the tens place. Without zero acting as a symbol of absence at this order of magnitude in “1,104,” the number could not be represented unambiguously (without zero, is it “1,104” or “114”?). Lacking zero detracted from a numeral system’s capacity to maintain constancy of meaning as it scales. Inclusion of zero enables other digits to take on new meaning according to their position relative to it. In this way, zero lets us perform calculation with less effort—whether it’s pen strokes in a ledger, finger presses on a calculator, or mental gymnastics. Zero is a symbol for emptiness, which can be a highly useful quality—as Lao Tzu said:
“We shape clay into a pot, but it is the emptiness inside that holds whatever we want.”
More philosophically, zero is emblematic of the void, as Aczel describes it:
“…the void is everywhere and it moves around; it can stand for one truth when you write a number a certain way — no tens, for example — and another kind of truth in another case, say when you have no thousands in a number!”
Drawing analogies to the functions of money: zero is the “store of value” on which higher order of magnitude numerals can scale; this is the reason we always prefer to see another zero at the end of our bank account or Bitcoin balance. In the same way a sound economic store of value leads to increased savings, which undergirds investment and productivity growth, so too does a sound mathematical placeholder of value give us a numeral system capable of containing more meaning in less space, and supporting calculations in less time: both of which also foster productivity growth. Just as money is the medium through which capital is continuously cycled into places of optimal economic employment, zero gives other digits the ability to cycle—to be used again and again with different meanings for different purposes.
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Zero’s second function is as a number in its own right: it is the midpoint between any positive number and its negative counterpart (like +2 and -2). Before the concept of zero, negative numbers were not used, as there was no conception of “nothing” as a number, much less “less than nothing.” Brahmagupta inverted the positive number line to create negative numbers and placed zero at the center, thus rounding out the numeral system we use today. Although negative numbers were written about in earlier times, like the Han Dynasty in China (206 BCE to 220 BCE), their use wasn’t formalized before Brahmagupta, since they required the concept of zero to be properly defined and aligned. In a visual sense, negative numbers are a reflection of positive numbers cast across zero:
Interestingly, negative numbers were originally used to signify debts—well before the invention of double-entry accounting, which opted for debits and credits (partly to avoid the use of negative numbers). In this way, zero is the “medium of exchange” between the positive and negative domains of numbers—it is only possible to pass into, or out of, either territory by way of zero. By going below zero and conceptualizing negative numbers, many new and unusual (yet extremely useful) mathematical constructs come into being including imaginary numbers, complex numbers, fractals, and advanced astrophysical equations. In the same way the economic medium of exchange, money, leads to the acceleration of trade and innovation, so too does the mathematical medium of exchange, zero, lead to enhanced informational exchange, and its associated development of civilizational advances:
Zero’s third function is as a facilitator for fractions or ratios. For instance, the ancient Egyptians, whose numeral system lacked a zero, had an extremely cumbersome way of handling fractions: instead of thinking of 3/4 as a ratio of three to four (as we do today), they saw it as the sum of 1/2 and 1/4. The vast majority of Egyptian fractions were written as a sum of numbers as 1/n, where n is the counting number—these were called unit fractions. Without zero, long chains of unit fractions were necessary to handle larger and more complicated ratios (many of us remember the pain of converting fractions from our school days). With zero, we can easily convert fractions to decimal form (like 1/2 to 0.5), which obsoletes the need for complicated conversions when dealing with fractions. This is the “unit of account” function of zero. Prices expressed in money are just exchange ratios converted into a money-denominated price decimal: instead of saying “this house costs eleven cars” we say, “this house costs $440,000,” which is equal to the price of eleven $40,000 cars. Money gives us the ability to better handle exchange ratios in the same way zero gives us the ability to better handle numeric ratios.
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Numbers are the ultimate level of objective abstraction: for example, the number 3 stands for the idea of “threeness” — a quality that can be ascribed to anything in the universe that comes in treble form. Equally, 9 stands for the quality of “nineness” shared by anything that is composed of nine parts. Numerals and math greatly enhanced interpersonal exchange of knowledge (which can be embodied in goods or services), as people can communicate about almost anything in the common language of numeracy. Money, then, is just the mathematized measure of capital available in the marketplace: it is the least common denominator among all economic goods and is necessarily the most liquid asset with the least mutable supply. It is used as a measuring system for the constantly shifting valuations of capital (this is why gold became money—it is the monetary metal with a supply that is most difficult to change). Ratios of money to capital (aka prices) are among the most important in the world, and ratios are a foundational element of being:
“In the beginning, there was the ratio, and the ratio was with God, and the ratio was God.” — John 1:1*
*(A more “rational” translation of Jesus’s beloved disciple John: the Greek word for ratio was λόγος (logos), which is also the term for word.)
An ability to more efficiently handle ratios directly contributed to mankind’s later development of rationality, a logic-based way of thinking at the root of major social movements such as the Renaissance, the Reformation, and the Enlightenment. To truly grasp the strange logic of zero, we must start with its point of origin—the philosophy from which it was born.
Philosophy of Zero
“In the earliest age of the gods, existence was born from non-existence.”—The Rig Veda
Zero arose from the bizarre logic of the ancient East. Interestingly, the Buddha himself was a known mathematician — in early books about him, like the Lalita Vistara, he is said to be excellent in numeracy (a skill he uses to woo a certain princess). In Buddhism, the logical character of the phenomenological world is more complex than true or false:
“Anything is either true,
Or not true,
Or both true and not true,
Or neither true nor not true.
This is the Lord Buddha’s teaching.”
This is the Tetralemma (or the four corners of the Catuṣkoṭi): the key to understanding the seeming strangeness of this ancient Eastern logic is the concept of Shunya, a Hindi word meaning zero: it is derived from the Buddhist philosophical concept of Śūnyatā (or Shunyata). The ultimate goal of meditation is the attainment of enlightenment, or an ideal state of nirvana, which is equivalent to emptying oneself entirely of thought, desire, and worldly attachment. Achievement of this absolute emptiness is the state of being in Shunyata: a philosophical concept closely related to the void—as the Buddhist writer Thich Nhat Hanh describes it:
“The first door of liberation is emptiness, Shunyata
Emptiness always means empty of something
Emptiness is the Middle Way between existent and nonexistent
Reality goes beyond notions of being and nonbeing
True emptiness is called “wondrous being,” because it goes beyond existence and nonexistence
The concentration on Emptiness is a way of staying in touch with life as it is, but it has to be practiced and not just talked about.”
Or, as a Buddhist monk of ancient Wats temple in Southeast Asia described the meditative experience of the void:
“When we meditate, we count. We close our eyes and are aware only of where we are at in the moment, and nothing else. We count breathing in, 1; and we count breathing out, 2; and we go on this way. When we stop counting, that is the void, the number zero, the emptiness.”
A direct experience of emptiness is achievable through meditation. In a true meditative state, the Shunyata and the number zero are one and the same. Emptiness is the conduit between existence and nonexistence, in the same way zero is the door from positive to negative numbers: each being a perfect reflection of the other. Zero arose in the ancient East as the epitome of this deeply philosophical and experiential concept of absolute emptiness. Empirically, today we now know that meditation benefits the brain in many ways. It seems too, that its contribution to the discovery of zero helped forge an idea that would forever benefit mankind’s collective intelligence — a sort of software upgrade to our global hive-mind.
Despite being discovered in a spiritual state, zero is a profoundly practical concept: perhaps it is best understood as a fusion of philosophy and pragmatism. By traversing across zero into the territory of negative numbers, we encounter the imaginary numbers, which have a base unit of the square root of -1, denoted by the letter i. The number i is paradoxical: consider the equation ±x² + 1 = 0; the only possible answers are positive square root of -1 (i) and negative square root of -1 (-i or i³). Ascending into a higher dimension, the equation ±x³ + 1 = 0 yields the possible answers of +1 or -1. These answers continue to alternate between the real and imaginary domains as their underlying formulae exponentiate higher. Visualizing them in the real and imaginary domains, we find a rotational axis centered on zero with orientations reminiscent of the tetralemma: one true (1), one not true (i), one both true and not true (-1 or i²), and one neither true nor not true (-i or i³):
Going through the gateway of zero into the realms of negative and imaginary numbers provides a more continuous form of logic when compared to the discrete either-or logic, commonly accredited to Aristotle and his followers. This framework is less “black and white” than the binary Aristotelean logic system, which was based on true or false, and provides many gradations of logicality; a more accurate map to the many “shades of grey” we find in nature. Continuous logic is insinuated throughout the world: for instance, someone may say “she wasn’t unattractive,” meaning that her appeal was ambivalent, somewhere between attractive and unattractive. This perspective is often more realistic than a binary assessment of attractive or not attractive.
Importantly, zero gave us the concept of infinity: which was notably absent from the minds of ancient Greek logicians. The rotations around zero through the real and imaginary number axes can be mathematically scaled up into a three-dimensional model called the Riemann Sphere. In this structure, zero and infinity are geometric reflections of one another and can transpose themselves in a flash of mathematical permutation. Always at the opposite pole of this three-dimensional, mathematical interpretation of the tetralemma, we find zero’s twin—infinity:
The twin polarities of zero and infinity are akin to yin and yang — as Charles Seife, author of Zero: Biography of a Dangerous Idea, describes them:
“Zero and infinity always looked suspiciously alike. Multiply zero by anything and you get zero. Multiply infinity by anything and you get infinity. Dividing a number by zero yields infinity; dividing a number by infinity yields zero. Adding zero to a number leaves it unchanged. Adding a number to infinity leaves infinity unchanged.”
In Eastern philosophy, the kinship of zero and infinity made sense: only in a state of absolute nothingness can possibility become infinite. Buddhist logic insists that everything is endlessly intertwined: a vast causal network in which all is inexorably interlinked, such that no single thing can truly be considered independent — as having its own isolated, non-interdependent essence. In this view, interrelation is the sole source of substantiation. Fundamental to their teachings, this truth is what Buddhists call dependent co-origination, meaning that all things depend on one another. The only exception to this truth is nirvana: liberation from the endless cycles of reincarnation. In Buddhism, the only pathway to nirvana is through pure emptiness:
Some ancient Buddhist texts state: “the truly absolute and the truly free must be nothingness.” In this sense, the invention of zero was special; it can be considered the discovery of absolute nothingness, a latent quality of reality that was not previously presupposed in philosophy or systems of knowledge like mathematics. Its discovery would prove to be an emancipating force for mankind, in that zero is foundational to the mathematized, software-enabled reality of convenience we inhabit today.
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Zero was liberation discovered deep in meditation, a remnant of truth found in close proximity to nirvana — a place where one encounters universal, unbounded, and infinite awareness: God’s kingdom within us. To buddhists, zero was a whisper from the universe, from dharma, from God (words always fail us in the domain of divinity). Paradoxically, zero would ultimately shatter the institution which built its power structure by monopolizing access to God. In finding footing in the void, mankind uncovered the deepest, soundest substrate on which to build modern society: zero would prove to be a critical piece of infrastructure that led to the interconnection of the world via telecommunications, which ushered in the gold standard and the digital age (Bitcoin’s two key inceptors) many years later.
Blazing a path forward: the twin conceptions of zero and infinity would ignite the Renaissance, the Reformation, and the Enlightenment — all movements that mitigated the power of The Catholic Church as the dominant institution in the world and paved the way for the industrialized nation-state.
Thank you for reading The Number Zero and Bitcoin (Part 2).
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I like how you are serializing your previous works into bite sized pieces. Would love to see this done with the other previous works as well in between new writings.