Jorge Luis Borges wrote a short story in 1983, “Tigres azules,” Blue Tigers. It captures the secret nightmare of every logician: a professor from Scotland is in India and discovers a pile of blue stones that defy arithmetic. He picks up a handful, and they multiply. He divides them, and they vanish. They are the incomprehensible and impossible reality, and they are pure, terrifying flux. They lack the one quality required for our universe to be countable: the ability to remain the same, to be identical with themselves over time.
We tend to think of mathematics as a discovery of the world’s deep structure. But reading David and Ricardo Nirenberg’s book “Uncountable, I am reminded that mathematics is less a discovery than a defense mechanism. It is a fortress humans built to keep the “blue tigers” of chaos at bay.
Western Civilization is constructed on a philosophical foundation built by Parmenides, Plato, and Aristotle. Parmenides insisted that “Being is One” and that change is an illusion. To survive the terrifying flux of reality, we decided to treat the world as if it were made of unchanging, countable units. We chose the apathic (that which cannot suffer or change) over the pathic (that which suffers and flows). The Nirenbergs argue that Western thought—from Plato to Augustine, from Descartes to modern physics – essentially signed a pact with Parmenides. The world gets treated as if it were made of unchanging pebbles, simply because we couldn’t bear the alternative. We built our science on an identity principle: things are what they are, and if we separate them (analysis) and put them back together (synthesis), they remain unchanged.
This desire for the “Countable One,” ultimate oneness, didn’t stop at math; it went straight to heaven. Monotheism adopted Greek anxiety and invented God as the ultimate signifier on unity. If the world is messy and changing, then God, as Creator, must be the source of ultimate unity, the indivisible, the unchanging origin that also creates self-identity. “I am who I am.” (Exodus 3:14)
These were not philosophical decisions; they were unconscious adaptations to make the universe livable, and in the process of carrying this out, humans developed not only religion, but also different types of numbers that make reality countable. What do we learn from mathematics about the unity of numbers? To say the least, it is very complicated, and the journey is truly amazing.
Natural Numbers: The Invention of Sameness
The first great leap was the natural number (1, 2, 3…). To count “one sheep, two sheep,” our ancestors had to perform a radical mental act: they had to ignore the reality that no two sheep are identical. One is sick, one is fat, one is nursing. To count them, you must strip them of their uniqueness and reduce them to a unit.
Then came the Integers, the inclusion of negative numbers, that allows us to calculate debt and loss. This is the creation of the apathic object. An integer like “3” is safe because it has no history, no hunger, and no death. It is the same today as it was in Babylon. By projecting this grid of integers onto the world, we gained the power to track loss, create debt, collect taxes, to trade and to build. Negative numbers track non-existent things, but the “such-ness” of reality evaporates, negative objects appear, and we exchanged the unique for the universal.
Zero: Creation ex Nihilo
If the integer was a mental leap, Zero was a mental abyss. For the Greeks, “nothing” could not exist; the void was a logical scandal. Romans had no zero in their number system, because you cannot count what isn’t there. But what happens if you subtract a number from itself? You get zero, and it took Indian mathematicians (and later Islamic scholars) to realize that “nothing” is a something—a placeholder, a pivot point, and a little more than an empty space in a decimal system. It took the rise of banking and debt to finally make the Western world accept that you could indeed have less than nothing.
Modern set theory zero gives us a new spin: it uses the “empty Set (∅) as the foundation of the number system. (Numbers are understood as sets.) The empty set is the void from which all other numbers are born. You start with the representation of nothing (∅), then you include this representation into another set to make one: {(∅)}, then you reinsert the empty set again to get to Two ( [{(∅), {(∅)}] ), and so on. It is a terrific magic trick that creates a universe of meaning out of a pocket of emptiness.
The Real Numbers: Covering the Void
For the Pythagoreans, the universe was a harmony of rational numbers, expressed in fractions. They believed that if you looked closely enough, all reality could be measured by the ratio of two integers. Think of the Parthenon in Athens, a magnificent expression of mathematical beauty in stone.
Traumatizing discoveries came swiftly. To name just one: The diagonal of a square cannot be measured by its sides. In our terms, the Pythagoreans collided with the square root of two, √2, which is an irrational number, a number that cannot be expressed as the ratio of two integers. To us, this is nothing special. To them, it was an unnamable, irrational flaw in the structure of reality, a fundamental and terrifying break between geometry and arithmetic. And similarly: How do you calculate the relationship between a circle’s edge (circumference) and its width (diameter)? The same goes for the golden ratio, and many other geometrical relations. The discovery of irrational numbers meant that the smooth line of geometry was full of holes. A new type of number was needed that combines the rational and the irrational numbers on a continuous number line, and the real numbers, a countable infinite set, an continuous one-dimensional number line, came into existence.
In the 19th century, mathematicians like Richard Dedekind invented a way to represent real numbers. He realized that even if a number like √2 doesn’t exist in the rational world, the gap it leaves behind does. He defined the real number as the gap itself. This was a massive shift: we moved from counting discrete things to constructing a continuous reality. We poured a new, infinite type of number into the cracks of the universe, allowing for calculus, physics, and the smooth perception of time and change.
Imaginary and Complex Numbers: The Lateral Step
The adaptations continued. When mathematicians encountered the square root of negative one (√-1), which is a physical impossibility and violates a basic rule, they didn’t stop. (Multiplying a number with itself should always give a positive value: -1 x -1 = 1). The Imaginary Number i (√-1) was used to add a virtual space to the number line, and turn it into the plane of complex numbers.
This required a profound shift in thinking: numbers stopped being just about quantity (how much?) and became transformations (which way?). By combining real and imaginary numbers into Complex Numbers, we gained the ability to calculate rotation and oscillation. We stepped out of the one-dimensional line of “more or less” and into a lateral reality. Without this “impossible” number, we would have no understanding of electricity, no quantum mechanics, and no modern world.
Transcendental Numbers: The Problem of Pi
The story does not end here. Many irrational numbers behave erratically, like Pi: the decimal digits of π appear to be randomly distributed, there is no pattern, and no proof either way. Transcendental Numbers, like Pi or e, (Euler’s number for natural growth) cannot be the solution to any algebraic equation using integers. They are not just irrational; they are, in a sense, unreachable by the standard tools of algebra.
The “problem of Pi”—squaring the circle—obsessed the ancients because it represented the clash between the straight line (the countable, the rational) and the curve (the infinite, the transcendental). Pi is infinite and non-repeating; it refuses to settle into a pattern. It reminds us that despite all our sophisticated concepts of real and rational numbers, the universe remains fundamentally wild, even in a simple circle. There are not many straight lines in the natural world, we find mostly curves, growth, randomness, and decay.
Continuum, Cardinality, and Transcendence
Irrational and transcendental numbers share certain fundamental properties: both belong to the reals, both resist expression as simple fractions, and both extend infinitely in non-repeating decimal expansions. Yet transcendental numbers represent a far more exclusive—and paradoxically far more populous—category. While every transcendental number is necessarily irrational, the converse fails to hold.
The irrational numbers themselves divide into two classes. The algebraic irrationals, such as √2, satisfy polynomial equations with integer coefficients (x² − 2 = 0, for instance). The transcendental numbers—π and e among them—satisfy no such equation. Transcendence thus imposes a stricter condition than irrationality, and proving it demands considerably more work.
The distinction becomes even more striking when we consider cardinality (how many objects are in the set.) The algebraic numbers, which encompass both the rationals and the algebraic irrationals, form a countable set no larger than the natural numbers. The transcendental numbers, by contrast, are uncountable and possess the same cardinality as the entire continuum of reals. In precise mathematical terms, this means that almost all real numbers are transcendental—the algebraic ones, despite their familiarity, constitute a vanishingly rare exception.
Serious reflection will conclude that the countable world is just a tiny island in a sea of uncountable transcendence.
The Philosophical Cost (and Benefit)
So, where does the “deeper philosophical insight” lie if we accept this sweeping view of number theory?
We are just beginning to explore the relationship between numbers and reality, Numbers are concepts fabricated by human minds that confront the mysteries of the real, they are names for phenomena we encounter at the boundary of thought. To say it in another way: numbers are maps, not the territory.
The Nirenbergs argue that our civilization is in trouble because we have confused the map for the territory. We see a house and express its value in the purchase price. We look at a student and see a GPA (a number). We look at a patient and see a diagnostic code (a category). We treat the fluid, changing, “uncountable” reality of human beings through socio-economic facts (age, gender, race, etc). We translate our world into stable integers, and this is the way we measure progress. And, so the argument goes, the more we do this, the more we get it wrong – we forget that our countable world is almost nothing when compared to the vastness of the uncountable continuum.
We should acknowledge this error – but the insight is also that we should not stop counting. We must start somewhere: Our ancient ancestors left the ocean for the land, and today we need to remind ourselves to remain amphibious as well.
We need to be able to live on the dry land of the “countable”—where we build spaceships, dose medications, and code software. But we also need to be able to swim in the river of the “Uncountable”—the pathic world of emotion, relationship, and the feeling of the passing moment and its existential weight.
The danger isn’t the number itself; the danger is the totalitarianism of the number. It’s thinking that only what can be counted counts. So, perhaps the ultimate adaptation is not just inventing new numbers, but developing the wisdom to know when to use them. That, I think, is the true adaptation: knowing that while we need the pebbles to survive, we live in a river that flows into the ocean.