Are Numbers Real?

1. Numbers: Human Inventions Mirroring Reality

Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible?

Numbers as constructs. Numbers, seemingly fundamental, are human inventions. They are valuable tools for understanding the world, but are they real in the sense of existing independently of human thought? This question probes the very nature of science and our ability to perceive factual existence in the universe.

Mathematics' effectiveness. The remarkable effectiveness of mathematics in representing reality is a central theme. Numbers, initially tied to tangible objects, have evolved into abstract entities that can predict and explain phenomena across diverse fields. This raises the question of why these human inventions are so adept at mirroring the universe's behavior.

Math vs. Reality. The book explores whether mathematics is at the heart of the universe or simply a powerful tool for understanding it. It questions whether scientists sometimes confuse math and reality, and whether modern science places too much emphasis on mathematics.

2. From Counting Goats to Abstract Digits

Some very basic arithmetic seems to be preprogrammed.

Tally marks. The journey begins with the invention of numbers, starting from simple tallies to keep track of possessions. Early humans likely counted without numbers, using fingers or notches to represent quantities. This system evolved as societies settled and trade emerged, creating a need for more sophisticated record-keeping.

Digits. The development of numbers involved assigning words and symbols to specific quantities, such as "hand" to represent five. These early numbers were closely tied to tangible objects, like goats or bags of grain. The concept of universality, where a number could represent any set of objects, was a crucial leap in abstraction.

Babylonian base-60. The Babylonians developed an advanced base-60 number system, which was a positional system where the position of a number in a row of numbers shows whether “1” is just “1” or “60” or “3600”. This system, written in cuneiform, was used for accounting, trade, and astronomical calculations. The Babylonians dedicated vast quantities of their clay tablets that they used as a cuneiform writing material to numbers.

3. Pythagoras's Theorem: All is Number

As far as the Pythagoreans were concerned, many of the most important patterns revolved around whole numbers.

Pythagorean cult. The Pythagoreans, an ancient Greek cult, believed that numbers were the underlying structure of reality. They assigned characteristics to numbers, linking them to concepts like mind, opinion, and wholeness. This elevated numbers beyond mere counting tools, imbuing them with a deeper significance.

Patterns in music. The Pythagoreans also recognized numerical patterns in music, such as the harmonic ratios produced by different string lengths. This discovery suggested that numbers could be used to derive scientific laws, marking a shift from qualitative to quantitative understanding of nature.

Irrational numbers. The discovery of irrational numbers, like the square root of 2, challenged the Pythagorean worldview. These numbers could not be expressed as ratios of whole numbers, threatening the idea that the universe was built on a foundation of integers. This discovery was so disturbing that legend tells us that the cultist Pythagoreans would be prepared to kill to preserve intact their obsession with the significance of numbers.

4. Euclid's Elegant, Imperfect World

The mathematics of Euclid operates in a little world of its own, a model for the way math has operated ever since.

Euclid's Elements. Euclid's Elements was a foundational textbook that defined geometry for over 2,000 years. It presented a system of mathematical proofs based on a set of axioms and definitions, creating a self-contained world of logic and deduction. Euclid's geometry operates in a little world of its own, a model for the way math has operated ever since.

Limitations of Euclid. Euclid's geometry assumes a flat surface, which is not true for the curved surfaces found in the real world. This limitation highlights the difference between the perfect shapes of mathematical proofs and the imperfect objects of reality.

Plato's Cave. Plato's metaphor of the cave illustrates the relationship between mathematics and reality. Mathematical structures exist in a realm of perfection, while the shapes and objects we experience are mere shadows of these pure originals. Scientific models are mathematical shadows of the natural world, simpler than the complex systems they represent.

5. Archimedes: Practical Math and Infinite Sand

From his notebooks it seems clear that Newton gradually assembled his thoughts on fluxions over a couple of decades, while he would not publish full details until a considerable time after he had developed it.

Archimedes's engineering feats. Archimedes, a Greek mathematician, applied mathematics to practical engineering problems, designing engines of war and machines for everyday use. This demonstrated the power of math as a tool for understanding and manipulating the physical world.

The Sand-Reckoner. In The Sand-Reckoner, Archimedes devised a new number system to calculate the number of grains of sand needed to fill the universe. This demonstrated the potential of mathematics to go beyond mundane thinking and explore the vastness of the cosmos.

Astronomy. Archimedes got a long rod with a movable disc attached to it. Waiting until just after sunrise, when he believed the Sun’s light was safe to look at (it isn’t, but at least it is weakened by the extra air it has to pass through), he pointed the rod at the Sun and slid the disc up and down until it just covered the Sun’s surface. From the angle the disc made to the eye, he worked out that the Sun’s disc covered between 1/164th and 1/200th of a right angle, then he multiplied this up for the whole orbit and rounded up to get to his approximation of 1/1,000th of the distance around the orbit.

6. Zero: The Revolutionary Placeholder

The less anything is, the less we know it: how invisible, how unintelligible a thing, then, is this nothing!

The absence of zero. The Greeks and Romans struggled with numbers because they lacked a way to represent nothing. This absence hindered their ability to perform complex calculations and develop advanced mathematical systems.

Zero as a placeholder. The Babylonians used a diagonal marker to indicate an empty column in their number system, but they never fully embraced zero as a number. It wasn't until Indian mathematicians developed a fully functional zero that numbers could take on their true power.

Fibonacci. Fibonacci introduced the Arabic numerals and the concept of zero to the West, revolutionizing mathematics and enabling more complex calculations. Zero's dual role as a placeholder and a number transformed arithmetic and paved the way for algebra.

7. Roger Bacon: Math's Indispensable Role in Science

He who is ignorant of mathematics cannot know the other sciences and the things of this world.

Bacon's vision. Roger Bacon, a thirteenth-century philosopher, emphasized the importance of mathematics for understanding science and the world. He believed that mathematics was the key to unlocking the secrets of nature.

Importance of math. Bacon dedicated a whole section of his Opus Majus to mathematics, though he used the term to cover everything from calendar reform to astrology. This last may seem unnerving to those brought up to consider astrology totally unscientific, but Bacon was a person of his time.

Grandfather effect. Bacon set a broad direction for the future, emphasizing the essential nature of mathematics, but did not actually come up with a viable means to head in that direction. Some time later, the father figures take the first significant step that is in direct line of inheritance to the finished product.

8. Imaginary Numbers: Unveiling Hidden Dimensions

As subtle as it is useless.

The square root of -1. Imaginary numbers, defined as the square root of a negative number, initially seemed to have no connection to reality. However, they proved surprisingly versatile in modeling two-dimensional phenomena, such as waves.

Gauss's number plane. Carl Friedrich Gauss realized that imaginary numbers made it easy to extend the number line to turn it into a two-dimensional number plane. This allowed mathematicians to perform algebra on complex numbers, which combined real and imaginary components.

Applications. Imaginary numbers have become essential tools in physics and engineering, particularly in calculations involving electricity and quantum mechanics. They provide a way to solve real-world problems by venturing into the abstract world of mathematics.

9. Newton and Laplace: The Clockwork Universe

Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it—an intelligence sufficiently vast to submit these data to analysis—it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.

Newton's mechanical universe. Isaac Newton constructed a mechanical picture of the universe, where mathematics could be used to predict the behavior of objects and forces. His laws of motion and gravity provided a framework for understanding the cosmos as a vast clockwork mechanism.

Laplace's determinism. Pierre-Simon Laplace extended Newton's vision, suggesting that with perfect knowledge of the universe's initial conditions, it would be possible to predict the future with certainty. This deterministic view emphasized the power of mathematics to unlock the secrets of the universe.

Calculus. Newton's method of fluxions, later refined into calculus, provided a powerful tool for understanding change and motion. Calculus enabled scientists to model complex phenomena and make accurate predictions about the behavior of the natural world.

10. The Intriguing Dance of Chance and Statistics

There are three kinds of lies: lies, damned lies, and statistics.

Statistics and probability. Statistics, initially used for gathering data on states, evolved into a tool for predicting future outcomes. The combination of statistics and probability allowed mathematicians to quantify chance and uncertainty.

Insurance industry. The insurance industry emerged from the application of probability to human life, using statistics to assess risk and calculate premiums. This demonstrated the practical power of mathematics in managing uncertainty.

Limitations. Statistics can be misused or misinterpreted, leading to flawed conclusions. The quote, "There are three kinds of lies: lies, damned lies, and statistics," reflects the skepticism surrounding the use of numbers to manipulate or distort reality.

11. Maxwell's Equations: Math Takes the Lead

He who is ignorant of mathematics cannot know the other sciences and the things of this world.

Maxwell's electromagnetism. James Clerk Maxwell's work on electromagnetism marked a turning point in the relationship between mathematics and physics. His equations, derived from pure mathematical reasoning, described the behavior of electricity and magnetism.

Abstract equations. Maxwell's equations stood apart from any consideration of physical reality, representing a new level of abstraction in scientific theory. This shift made it more difficult for non-experts to understand the underlying principles, increasing reliance on the mathematical framework.

Electromagnetic waves. Maxwell's equations predicted the existence of electromagnetic waves, which were later confirmed by experiment. This success demonstrated the power of mathematics to lead the way in scientific discovery.

12. Cantor and Gödel: Exploring the Edges of Infinity

I protest against the use of an infinite quantity as an actual entity; this is never allowed in mathematics.

Cantor's set theory. Georg Cantor revolutionized mathematics by developing set theory, a framework for understanding the nature of numbers and infinity. He showed that there are different sizes of infinity, challenging traditional notions of the infinite.

Gödel's incompleteness theorem. Kurt Gödel's incompleteness theorem demonstrated that any mathematical system, above a certain level of complexity, must contain problems that cannot be solved within that system. This highlighted the inherent limitations of mathematics.

Axiom of choice. The axiom of choice in set theory introduces an element of arbitrariness, raising questions about the foundations of mathematics and its relationship to reality. These discoveries reveal the complex and sometimes paradoxical nature of mathematical truth.

Last updated: April 10, 2025