Less Than Nothing? The Centuries-Long Argument About Negative Numbers

Think about numbers. They start simply enough: 1, 2, 3… for counting things. Then maybe fractions, for parts of things. Zero comes along to represent nothing. But what about numbers less than nothing? What about negative numbers, like -5 or -10?

Today, we see negative numbers everywhere. Thermometers show temperatures below zero. Bank accounts can be overdrawn (definitely less than nothing!). We learn rules in school for adding, subtracting, multiplying, and dividing them. They seem like a normal part of the number family.

But would you believe that for hundreds, even thousands of years, negative numbers were viewed with deep suspicion, confusion, and outright hostility by some of the world’s greatest thinkers? They were called “absurd,” “fictitious,” or “false.” How could anything truly be less than nothing?

The journey of negative numbers from mathematical outcasts to everyday tools is a fascinating story. It reveals how our ideas about numbers are tied to what we can physically see or count, and how accepting abstract concepts can be a real struggle, even for geniuses. It’s a tale filled with paradoxes, philosophical debates, and the slow, reluctant acceptance of a truly useful idea. Understanding this history might even give you a new perspective on why these numbers sometimes feel tricky, and how they connect to broader ideas of balance and opposition.

Where Did Negatives Come From? The Equation Problem

Numbers first arose from practical needs: counting sheep, measuring land, trading goods. In these activities, having less than nothing doesn’t make much sense. The smallest amount you can have is zero.

So, where did the strange idea of negative numbers first pop up? It emerged from solving equations – mathematical puzzles where you need to find an unknown value.

Consider a simple problem:

“I am 7 years old and my sister is 2. In how many years will I be exactly twice as old as my sister?”

If we call the unknown number of years x, this translates to finding x such that 7 + x = 2 * (2 + x). The answer here is x = 3 (in 3 years, you’ll be 10 and she’ll be 5). That’s straightforward.

But what if the starting ages were different?

“I am 18 years old and my brother is 11. When will I be exactly twice as old as my brother?”

This translates to finding x such that 18 + x = 2 * (11 + x). If you work through the steps to find x, the answer comes out to be x = -4.

What does -4 years mean? It means the condition happened 4 years in the past, when you were 14 and your brother was 7.

Ancient problem-solvers in Egypt and Mesopotamia could handle the first type of problem. But when their methods led to a situation requiring something like subtraction that resulted in less than zero, they usually just stopped, concluding the problem was impossible or had no sensible answer in that context.

Greek Philosophy vs. Negative Numbers

The ancient Greeks, who laid so much of the foundation for Western mathematics and philosophy, completely ignored negative numbers. Why? It stemmed from their core understanding of what numbers were.

For Greek thinkers, numbers represented magnitudes – lengths of lines, areas of shapes, volumes of objects. These were properties of things in the real world. You could have a line segment of length 5, but a “length of negative 5” was meaningless. You could have an area of 10 square units, but not “-10 square units.” Since numbers were tied to these physical, measurable quantities, the idea of a quantity less than nothing was logically impossible within their framework.

Even Diophantus, a later Greek mathematician (around 250 A.D.) who wrote a whole book about solving equations (Arithmetica), only ever considered positive fractional answers. If solving an equation led him to a negative result, he dismissed it as “absurd.” For example, if solving a problem led to needing 4x + 20 = 4, he simply stated this was absurd because the 20 was already more than the 4. Adding something positive (4x) couldn’t possibly result in 4.

Glimmers in the East: India and China

Interestingly, evidence suggests that mathematicians in India and China were more open to the idea of negative quantities much earlier than Europeans were.

• India: As early as the 7th century A.D., Indian mathematicians like Brahmagupta were working with negative numbers. They sometimes thought of them in practical terms, like debts versus possessions (positive numbers). Brahmagupta stated rules for adding, subtracting, multiplying, and even dividing positive and negative numbers. He knew, for instance, that a negative times a negative yields a positive. Later Indian mathematicians, like Bhaskara II (around 1100 A.D.), continued this tradition. When solving quadratic equations, Bhaskara correctly found both positive and negative roots. Yet, even he expressed caution, sometimes noting that the negative answer shouldn’t be taken “because of its inapplicability. For people have no clear understanding in the case of a negative quantity.” The concept was useful for calculation, but its meaning was still unsettling.
• China: Ancient Chinese mathematical texts also show evidence of working with negative quantities, perhaps using different colored counting rods to represent positive and negative amounts during calculations.

These Eastern traditions, however, didn’t directly influence early European mathematics in this specific area. European scholars primarily inherited the Greek aversion to negatives.

Europe’s Reluctance: “False” and “Fictitious” Numbers

When algebra began to flourish in Europe during the Renaissance (starting around the 13th-14th centuries, based on translations of Arab works like Al-Khwarizmi’s), the Greek discomfort with negatives persisted.

Al-Khwarizmi, writing in the 9th century, didn’t use negative numbers in his algebra. When solving quadratic equations, he recognized that sometimes there could be two answers, but only considered cases where both answers were positive. His methods often involved thinking about geometric areas and lengths, where negative values made no sense.

European mathematicians who learned algebra from these sources inherited this limitation. For centuries, right through the 16th and 17th centuries – the time of the Scientific Revolution! – many leading figures rejected negative solutions to equations.

• They called them “false roots” or “fictitious roots,” distinct from the “true” (positive) roots.
• Girolamo Cardano (mid-1500s), who published the solution to the cubic equation, acknowledged negative roots might exist but focused his attention elsewhere.
• Michael Stifel, a German mathematician around the same time, called negative numbers “absurd.”
• Even René Descartes (early 1600s), who revolutionized geometry with coordinates, called negative solutions “false” roots. His coordinate system, while groundbreaking, initially focused primarily on positive lengths and distances. The idea of negative coordinates and axes took time to fully develop.

Why were they so resistant, even when negative numbers seemed to pop out of their own calculations? The core problem remained philosophical: what is a number less than nothing? How can it exist?

Knowing the Rules vs. Understanding the Meaning

One fascinating aspect of this history is that many mathematicians who rejected negative numbers conceptually still knew the rules for calculating with them.

They understood, for instance, the “laws of signs” needed when multiplying expressions involving subtractions (like (x – a)(x – b)). They knew that the equivalent of “negative times negative is positive” and “negative times positive is negative” worked within these calculations, as long as the final answer came out positive.

But they didn’t see these rules as applying to independent entities called “negative numbers.” They saw them as rules about the operation of subtraction. The idea of -5 as a standalone concept, a number existing on its own, was the difficult part.

Paradoxes and Puzzles: Where Do Negatives Fit?

As mathematicians began (hesitantly) to allow negative numbers into their work in the 17th century, new problems arose. Where did these strange numbers fit in relation to the others? The results were sometimes baffling.

• Antoine Arnauld (1612-1694) argued that if -1 is less than 1, then the proportion -1 : 1 = 1 : -1 (smaller is to larger as larger is to smaller) seemed absurd.
• John Wallis (around 1655) argued that since dividing by zero gives infinity, dividing by a negative number (which he considered somehow “less than nothing”) must result in something larger than infinity! He concluded that negative numbers must actually be greater than infinity – a truly bizarre idea.

Even Isaac Newton, in his 1707 algebra textbook, offered a less-than-helpful definition: “Quantities are either Affirmative, or greater than nothing, or Negative, or less than nothing.” This circular statement, carrying Newton’s huge authority, didn’t exactly clarify the conceptual problem. How can a quantity be less than nothing?

These paradoxes show that even brilliant minds struggled deeply with the meaning of negative numbers, long after the rules for using them were known.

Gradual Acceptance: Usefulness Wins Out

So, how did negative numbers finally win their place? There wasn’t a single moment of revelation. Instead, it was a slow, gradual process driven mainly by their undeniable usefulness.

• Algebraic Completeness: Accepting negative numbers made the theory of equations much neater. Quadratic equations could always have two roots (though sometimes they might be equal or complex). Patterns emerged more clearly.
• Coordinate Geometry: As Descartes’s coordinate system evolved to include negative axes, it provided a clear visual representation. Negative numbers weren’t just abstract concepts; they were locations on the other side of zero, perfectly mirroring the positive side.
• Physics and Science: As mathematics was applied more widely to physics (like Newton’s work on motion and gravity), negative numbers became useful for representing direction (e.g., velocity in the opposite direction), deficits, or states below a baseline.

By the middle of the 18th century, mathematicians like Leonhard Euler were using negative numbers extensively, even if their philosophical explanations still felt a bit shaky. Euler famously used the analogy of debt:

“Since negative numbers may be considered as debts, because positive numbers represent real possessions, we may say that negative numbers are less than nothing. Thus, when a man has nothing of his own, and owes 50 crowns, it is certain that he has 50 crowns less than nothing…”

This practical analogy helped, but even Euler sometimes stumbled when explaining why the rules (like negative times negative equals positive) worked, falling back on formal arguments rather than intuitive ones.

The Last Doubters:

Skepticism lingered, especially in Britain. As late as 1831, the logician Augustus De Morgan wrote that negative quantities like -b were just as “imaginary” or “inconceivable” in terms of real meaning as square roots of negatives like √-a.

Abstraction Sets Negatives Free

What finally cemented the place of negative numbers wasn’t a perfect physical analogy, but a shift in mathematics itself towards abstraction.

In the 19th century, mathematicians like Gauss, Galois, Abel, and others began focusing less on what numbers represented in the real world and more on the structure and rules governing how numbers behaved in relation to each other within a system.

In this more abstract view, negative numbers weren’t problematic concepts needing justification; they were simply the necessary additive opposites of positive numbers. They were the elements needed to make subtraction always possible. They completed the number system, giving it a beautiful symmetry around zero. Their existence was defined by their role within the mathematical structure, not by whether they corresponded to a pile of apples.

Ironically, this move away from needing a concrete physical meaning paved the way for negative numbers to be confidently applied to concrete situations – temperatures, elevations, finances, physics – wherever the concept of “opposite direction” or “below a baseline” was relevant.

Lessons from Less Than Nothing

The long, difficult journey of negative numbers holds valuable lessons:

1. Intuition vs. Logic: Our everyday intuition, based on counting physical objects, can sometimes be a barrier to accepting more abstract mathematical ideas. Negative numbers felt “wrong” for centuries because they didn’t fit the simple counting model.
2. Usefulness Drives Acceptance: Often, mathematical concepts gain acceptance not because their philosophical meaning is perfectly clear, but because they prove incredibly useful for solving problems or simplifying theories.
3. Knowing vs. Understanding: It’s possible to know the rules for manipulating symbols (like negative numbers) long before fully understanding their conceptual basis. True understanding often takes time and perspective.
4. Math Anxiety is Understandable! If geniuses like Descartes, Wallis, and Newton struggled with the concept of negative numbers, it’s okay if they sometimes feel confusing to students today! Their history shows these aren’t “easy” ideas.

Negative numbers remind us that mathematics is a human endeavor, constantly evolving. Ideas that seem strange or impossible in one era can become fundamental tools in another, often pushed forward by their sheer utility, even while the philosophical dust takes centuries to settle.

Balance, Opposition, and the Number Line: A Numerological View

The story of negative numbers offers rich territory for numerological reflection.

Duality and Polarity: The acceptance of negative numbers fundamentally changed the number line from a ray starting at zero (or one) and going infinitely in one direction, to a line extending infinitely in two opposite directions from zero. This introduces the core concept of polarity or duality – positive/negative, forward/backward, possession/debt. Many spiritual traditions explore the interplay of opposites. Negative numbers give mathematical form to this fundamental duality, centered around the balancing point of zero.

Zero as the Fulcrum: As negatives gained acceptance, zero’s role shifted from just “nothing” to being the crucial center point, the fulcrum balancing the positive and negative realms. This aligns with zero’s numerological meaning as the void, the source, but also the point of equilibrium before duality emerges (with 1 and -1 being the first steps away from the center).

Shadow Aspects: Negative numbers could be seen as the “shadow” side of positive numbers. They were resisted, seen as “false” or “absurd.” In psychology and spirituality, integrating the shadow – acknowledging and working with aspects of ourselves or reality we find difficult or negative – is crucial for wholeness. The struggle to integrate negative numbers into mathematics mirrors this process. Accepting them made the number system more complete and powerful.

Beyond the Physical: Like zero, negative numbers challenged the Greek view of numbers tied only to physical magnitudes. Accepting them required a step into abstraction, recognizing numbers could represent concepts like direction or debt, not just physical quantity. Numerology operates in this abstract, symbolic realm, suggesting numbers represent energies or archetypes, not just counts. The acceptance of negatives was a step towards this more symbolic understanding within mathematics itself.

Master Number 11 and Intuition: Could the eventual acceptance be linked to higher vibrations? The 18th century, when acceptance grew (Euler, etc.), saw major shifts in thought (the Enlightenment). Perhaps the intuitive leaps needed to fully embrace negatives align with the energy of Master Number 11 (intuition, illumination, bridging higher and lower understanding). While specific dates for individual insights are hard to pin down, the collective shift might reflect this energy.

The centuries-long argument about negative numbers shows how deeply our concept of “reality” influences our acceptance of mathematical ideas. It took a shift towards abstraction, driven by the undeniable usefulness of these “absurd” quantities, for them to finally find their essential place on the number line, bringing balance and completeness to our understanding of numbers.