Are Numbers Real? And What Does It Matter?
What are numbers? Are they real? Do they exist outside the mind of humans?
As denizens of the digital age, we encounter them daily, embedded in our transactions, our measurements, and algorithms. Yet, beneath the surface of this numerical landscape lies a complex and philosophical realm that has intrigued thinkers for thousands of years.
When we take a break from our Thanksgiving dinners and all those loving family members and walk outside into the winter night beneath the stars, are we seeing a mathematical universe where numbers are part of the structure of all things? When we listen to Bach, are we hearing numbers in the music?
Beyond theories of what grounds numbers, what does it matter if they are “real” or “exist?”
In the kaleidoscope of mathematical abstraction, numbers are the fundamental building blocks. Yet, defining them proves surprisingly elusive. Philosophers ask whether a number is a mere symbol or possesses a more intrinsic quality.
To grasp the essence of numbers, we can take the historical route and journey back to the dawn of human thought. Early civilizations grappled with the concept of quantity, using rudimentary symbols to represent sheep, grain, or the stars overhead. The Babylonians etched cuneiform numerals into clay tablets, laying the groundwork for numerical language.
According to Pythagoras, numbers were musical notes, composing a celestial symphony. Plato envisioned numbers as archetypes, existing in a realm of ideal forms. This intellectual freedom in the domain of numerical abstraction birthed the concept of pure mathematics—a pursuit of truth that transcended the empirical. Mathematics was so important to Plato that he took a ten-year break from philosophy to study it. Above the door to his school was written: "Let no one unversed in geometry enter here."
As the Renaissance dawned, the mathematician Leonardo Fibonacci introduced Europe to the numerical sequence that bears his name. Suddenly, numbers weren't just about counting; they were a rhythmic dance of ratios and proportions. The stage was set for the arithmetic revolution that would propel humanity into the scientific enlightenment. Newton discovered binomial theorem and calculus, which would ground physics for several hundred years. This was an age when the universe seemed to run like a beautifully precise clock.
However, numbers took an unexpected turn with the advent of calculus. Infinity and infinitesimals emerged as mathematical frontiers, challenging our very understanding of limits and continuity. Imaginary numbers, once deemed fanciful, found a home in the equations of quantum mechanics, revealing the interplay between abstraction and reality.
As we step into the quantum realm, the nature of numbers takes on an even more surreal quality. Quantum states are described not by definite values, but by probabilities—a numerical puzzle where uncertainty becomes the norm. Mathematical formalism becomes a portal to the mysterious. And what is the mystery?
In the early 20th century, British philosophers Bertrand Russell and Alfred Whitehead attempted to formally ground numbers in a comprehensive theory in their treatise, Principia Mathematica. The book took ten years to write. But several years in, they came up against what’s now called Russell’s paradox, which threatened the entire project. Indeed, when Russell told the German mathematician and logician, Frege, who was engaged in a similar attempt to ground numbers, of the paradox, the latter went into a deep depression which ended in his death.
Russell’s paradox: Consider the set of all sets that do not contain themselves. Let's call this set R. Now, the question is: Does R contain itself? If R contains itself, then it shouldn't be in the set of sets that don't contain themselves. But if R doesn't contain itself, then it should be in the set of sets that don't contain themselves. This creates a logical contradiction, leading to the paradox.
Later in the 21st century, numbers have transcended their abstract origins. In the realm of computers, they pulse through the circuits of artificial intelligence, decoding the genome, and predicting financial markets. The once-abstract notion of numbers has become the lifeblood of a technologically driven society. However, the perennial question casts its shadow: Do numbers truly exist, or are they merely figments of our abstract imagination? Are numbers real? And if they are, what else outside the world of matter might exist? How about properties, or values like truth and beauty?
Going back to history, Greek philosophers also grappled with the metaphysical question about numbers. For Plato, numbers were not mere symbols but ethereal entities existing in a world of ideal forms. The shadows cast upon the walls of his allegorical cave were not just illusions; they were the very fabric of reality. Numbers transcended the mundane. They pointed to a realm beyond matter.
Contrary to his mentor, Aristotle grounded numbers in the material world. Numbers, for him, were not otherworldly entities but rather reflections of the relationships inherent in physical objects.
The introduction of the infinite and imaginary numbers added another layer to the existential drama. Rooted in the square root of negative one, these numbers seemed to defy the very essence of reality. Were they fantastical imaginings or essential categories in the question of what exists? In the quantum realm, particles flipped between states, and numerical probabilities replace deterministic certainties. The very fabric of the cosmos seems woven with uncertainty, prompting physicists to question the nature of numerical existence.
Today, in a society where numbers dictate financial markets, steer technological advancements and underlie the difference between the computer and the human brain, the line between mathematical abstraction and concrete reality blurs. Modern techies and scientists mostly question metaphysics and any realm outside matter. They question the reality of numbers.
I recently encountered a terrific argument for the existence of numbers by the philosopher Jonathan Schaffer. He offers the following proof.
1. There are prime numbers.
2. Therefore, there are numbers.
As Schaffer says, 1 is a mathematical truism. And 2 immediately follows by a standard adjective-drop reference. When you drop the adjective, the same proposition holds true. For example, “There are red roses, therefore there are roses.”
Schaffer anticipates three objections to his proof. First, someone may say that it is only according to the fiction of numbers that there are prime numbers. Schaffer argues that 1 is obviously true despite the argument that it must be paraphrased with the conditional.
Second, one might argue that the meaning of “are” changes between 1 and 2 from some kind of internal meaning to an external meaning. But Schaffer replies that this is an argument against the adjective-drop inference itself. Those who propose this objection could not automatically infer that “there are roses” from “there are red roses” for fear of a meaning shift.
Third, Schaffer anticipates the objection that all quantification is ontologically neutral and thus accepts 2 while denying that numbers exist. The rebuttal is that the neutralist seems committed to the unfathomable conjunction: "Numbers do not exist, and there are numbers.”
A belief that numbers exist is a challenge to the materialist who believes that matter is all there is. Numbers are the gateway to the metaphysical, a pathway both important and fundamental to the ideas of Plato and Aristotle.
This recurring question, "Do numbers exist?” continues to undermine the foundations of our intellectual edifice. The answer, like the digits themselves, may be infinite in its complexity, an eternal mystery confounding what is real.