Pythagoras: More Than Just a Triangle

Pythagoras of Samos, born around 570 BCE, was far more than the mathematician associated with the famous theorem bearing his name. He was a philosopher, mystic, and founder of a religious and philosophical school that viewed mathematics as the fundamental principle underlying all reality. The Pythagorean Brotherhood, as his followers were known, combined mathematical inquiry with spiritual beliefs in a way that would be unimaginable in modern academia.

The Pythagorean School was established in Croton, a Greek colony in southern Italy, around 530 BCE. This wasn’t merely an academic institution but a secretive society with strict rules, rituals, and beliefs. Members took vows of silence regarding the school’s teachings, and mathematical discoveries were attributed to Pythagoras himself rather than individual contributors. The school functioned as both a philosophical community and a political organization, with significant influence in the local government of Croton.

The central belief of the Pythagorean Brotherhood was that “all is number.” They believed that numbers were not just tools for calculation but the very essence of reality. This philosophy arose from their observations of mathematical relationships in music, astronomy, and geometry. They discovered that musical harmony could be expressed through numerical ratios, that the movements of celestial bodies followed mathematical patterns, and that geometric shapes embodied numerical perfection.

The Pythagoreans lived according to strict rules that reflected their mathematical worldview. They believed in the transmigration of souls (reincarnation) and practiced vegetarianism, partly because they believed beans contained souls. They had peculiar taboos, such as refusing to eat beans, not to pick up what had fallen, and not to stir the fire with iron. These practices, while seemingly irrational to modern observers, were connected to their broader philosophical system.

The organization was hierarchical, with two classes of members: the mathematikoi (learners) and the akousmatikoi (listeners). The mathematikoi lived communally, owned no personal possessions, and were taught the inner workings of Pythagorean philosophy and mathematics. The akousmatikoi lived outside the community, had personal possessions, and were only taught the outer doctrines. This structure reflected the Pythagorean belief in the importance of mathematical knowledge as a path to spiritual enlightenment.

Pythagorean mathematics was fundamentally different from modern mathematics. They focused primarily on the properties of whole numbers and their ratios, viewing these as the building blocks of reality. They associated numbers with geometric shapes (triangular numbers, square numbers, etc.) and believed that the universe’s harmony could be understood through numerical relationships. This approach led to important mathematical discoveries but also created problems when they encountered mathematical concepts that didn’t fit their worldview.

The Pythagorean Brotherhood made significant contributions to mathematics beyond the famous theorem. They discovered perfect numbers (numbers equal to the sum of their proper divisors, like 6 = 1+2+3), amicable numbers (pairs where each is the sum of the other’s divisors), and various properties of numbers that would later become fundamental in number theory. They also made advances in geometry, astronomy, and music theory, always seeking to express these subjects in numerical terms.

The political influence of the Pythagorean School eventually led to its downfall. Around 500 BCE, the democratic party in Croton attacked the Pythagorean meeting place, killing many members. Pythagoras himself may have died in this persecution, though accounts vary. The school’s secretive nature and political power had made it enemies, and its destruction marked the end of the original Pythagorean Brotherhood.

However, Pythagorean ideas survived and influenced later philosophers and mathematicians. Plato was deeply influenced by Pythagorean thought, incorporating mathematical mysticism into his philosophy. The belief that mathematics underlies reality became a cornerstone of Western philosophy and science, influencing thinkers from Kepler to Einstein. The Pythagorean emphasis on mathematical relationships in nature helped lay the groundwork for the scientific revolution.

The Pythagorean School represents a unique fusion of mathematics, mysticism, and communal living that has no parallel in modern academia. While their specific beliefs may seem strange to us, their core insight—that mathematics reveals fundamental truths about reality—has proven remarkably prescient. Their approach to mathematics as a spiritual and intellectual discipline, rather than merely a practical tool, influenced the development of mathematical thinking for centuries.

The legacy of the Pythagorean Brotherhood extends beyond mathematics to encompass an entire approach to understanding the world. Their belief in the mathematical nature of reality, their emphasis on rigorous reasoning, and their view of mathematics as a path to truth all contributed to the development of Western mathematical and scientific thought. While we may not share their mystical beliefs, we can appreciate their profound influence on the intellectual tradition that led to modern mathematics and science.

The Pythagorean Theorem, stating that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (a² + b² = c²), is one of the most recognizable mathematical results in history. However, the story behind this theorem is more complex than commonly understood, involving ancient civilizations, competing claims of discovery, and the transmission of mathematical knowledge across cultures.

While the theorem bears Pythagoras’s name, evidence suggests that knowledge of this relationship predates him by centuries. Babylonian tablets from around 1800 BCE, more than 1,000 years before Pythagoras, contain lists of Pythagorean triples (sets of three integers that satisfy the equation a² + b² = c², such as 3, 4, 5). The Egyptians used the 3-4-5 triangle to create right angles in construction, indicating practical knowledge of the relationship. These examples suggest that the theorem was discovered independently by multiple ancient civilizations.

What may be original to Pythagoras or his school is not the theorem itself but the first rigorous proof that it holds for all right triangles, not just specific examples. The Babylonians and Egyptians had empirical knowledge of the relationship, but the Pythagoreans may have been the first to prove it deductively. This distinction between empirical observation and mathematical proof marks an important development in the history of mathematics.

Euclid’s proof of the Pythagorean Theorem in his “Elements” (around 300 BCE) is one of the most famous proofs in mathematics. Euclid’s approach uses geometric methods, showing that the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides. The proof involves constructing squares on each side of the triangle and demonstrating that the areas are related as the theorem states.

There are hundreds of different proofs of the Pythagorean Theorem, each offering a unique perspective on why the relationship holds. Some are algebraic, using symbolic manipulation to show that the equation is satisfied. Others are geometric, using visual constructions to demonstrate the relationship between areas. Still others are trigonometric, using properties of right triangles and trigonometric functions.

One particularly elegant proof involves rearranging four identical right triangles within a square. By calculating the area of the large square in two different ways—first as the square of the hypotenuse, then as the sum of the areas of the four triangles plus the square of the difference of the two sides—algebraic manipulation leads directly to the Pythagorean relationship.

The theorem’s importance extends far beyond geometry. In trigonometry, it leads to the fundamental identity sin²θ + cos²θ = 1. In coordinate geometry, it provides the distance formula between two points in a plane. In physics, it’s essential for calculating resultant vectors and understanding the relationship between space and time in special relativity. The theorem’s applications are so widespread that it appears in virtually every branch of mathematics and science.

The converse of the Pythagorean Theorem is also important: if the sides of a triangle satisfy a² + b² = c², then the triangle is a right triangle. This result provides a method for determining whether a triangle is right-angled without measuring its angles, which is useful in both theoretical and practical contexts.

Pythagorean triples, sets of three positive integers that satisfy the equation a² + b² = c², have fascinated mathematicians for millennia. The simplest example is (3, 4, 5), but there are infinitely many such triples. The general formula for generating Pythagorean triples was known to the ancient Greeks and involves two parameters that can be used to generate all primitive triples (those with no common factors).

The theorem also connects to deeper mathematical concepts. In number theory, the question of when a² + b² = c² has integer solutions leads to the study of Diophantine equations. In algebra, the theorem is related to the concept of norms in vector spaces. In higher dimensions, the theorem generalizes to the distance formula in n-dimensional space.

Fermat’s Last Theorem, one of the most famous problems in mathematics, asks whether equations of the form aⁿ + bⁿ = cⁿ have integer solutions for n > 2. This problem, which remained unsolved for over 350 years, can be seen as a generalization of the Pythagorean Theorem. The connection between these problems illustrates how a simple geometric relationship can lead to deep mathematical questions.

The cultural impact of the Pythagorean Theorem extends beyond mathematics. It has become a symbol of mathematical reasoning and deductive proof, representing the power of human intellect to discover universal truths. The theorem appears in art, literature, and popular culture as a shorthand for mathematical sophistication. Its enduring fame reflects not just its mathematical importance but also its role as an entry point into the world of mathematical thinking.

One of the most significant events in the history of mathematics was the discovery of irrational numbers, which fundamentally challenged the Pythagorean worldview and led to what historians call the first crisis in the foundations of mathematics. This discovery, traditionally attributed to Hippasus of Metapontum (a member of the Pythagorean Brotherhood), revealed that not all lengths could be expressed as ratios of whole numbers, undermining the Pythagorean belief that “all is number.”

The discovery likely arose from investigations of the Pythagorean Theorem itself. When applying the theorem to a right triangle with both legs of length 1, the hypotenuse has length √2. The Pythagoreans knew that 1² + 1² = 2, so the hypotenuse squared equals 2. However, they were unable to find two whole numbers whose ratio equals √2. This failure was initially perplexing but eventually led to a profound mathematical discovery.

The proof that √2 is irrational (cannot be expressed as a ratio of integers) is a masterpiece of ancient mathematical reasoning. The proof uses the method of reductio ad absurdum (proof by contradiction). Assume that √2 = a/b, where a and b are integers with no common factors. Then 2 = a²/b², so 2b² = a². This means a² is even, so a must be even (since the square of an odd number is odd). Let a = 2c. Then 2b² = (2c)² = 4c², so b² = 2c². This means b² is even, so b is even. But if both a and b are even, they have a common factor of 2, contradicting our assumption. Therefore, √2 cannot be expressed as a ratio of integers.

This proof was devastating to the Pythagorean worldview because it demonstrated that geometric reality contained quantities that could not be expressed within their numerical system. The Pythagoreans believed that all relationships in nature could be described using ratios of whole numbers, but √2 showed that this was not the case. The existence of incommensurable quantities (those without a common measure) challenged their fundamental assumptions about the mathematical nature of reality.

The discovery had profound implications for Greek mathematics. It showed that arithmetic (the study of numbers) was insufficient for geometry (the study of shapes), leading to a greater emphasis on geometric methods in Greek mathematical practice. Greek mathematicians developed sophisticated geometric techniques for dealing with incommensurable quantities, but they remained wary of treating them as numbers in the same sense as integers and fractions.

The secrecy surrounding this discovery reflects its threatening nature to Pythagorean beliefs. According to legend, Hippasus was drowned at sea for revealing the existence of irrational numbers, though this story is likely apocryphal. The Pythagoreans may have tried to keep the discovery secret because it undermined their central doctrine, but the mathematical truth could not be suppressed indefinitely.

The crisis of irrationals led to important developments in mathematical thinking. It forced mathematicians to distinguish between different types of mathematical objects and to develop more sophisticated concepts of number. The Greeks eventually accepted irrational quantities as legitimate mathematical entities, though they continued to treat them geometrically rather than arithmetically for centuries.

Eudoxus of Cnidus (around 400 BCE) developed a theory of proportions that could handle irrational quantities without treating them as numbers. His approach, described in Euclid’s “Elements,” allowed Greek mathematicians to work with incommensurable ratios while avoiding the need to represent them numerically. This theory was a major achievement in ancient mathematics and influenced mathematical thinking for over two millennia.

The full resolution of the crisis of irrationals required the development of the real number system, which wouldn’t occur until the 19th century. The ancient Greeks’ geometric approach to irrationals was ingenious but limited, as it didn’t provide the computational tools needed for more advanced mathematics. The eventual acceptance of irrational numbers as legitimate mathematical objects was essential for the development of calculus, analysis, and modern mathematics.

The discovery also had philosophical implications beyond mathematics. It showed that sensory intuition could be misleading and that rigorous reasoning was necessary to understand mathematical reality. This lesson influenced Greek philosophy and contributed to the development of the axiomatic method, in which mathematical truths are derived from explicitly stated assumptions through logical deduction.

The crisis of irrationals demonstrates how mathematical discoveries can challenge fundamental assumptions and lead to conceptual revolutions. The Pythagorean belief that all reality could be reduced to ratios of whole numbers was elegant and appealing, but it was ultimately incorrect. The willingness to abandon cherished beliefs in favor of mathematical truth is a hallmark of mathematical thinking and has led to many of the field’s most important advances.

Modern mathematics has fully integrated irrational numbers into the number system, treating them as equal partners with rational numbers in the real number system. The development of rigorous theories of limits and infinite processes has resolved the ancient paradoxes surrounding irrationals and provided powerful tools for mathematical analysis. The crisis that once threatened the Pythagorean worldview has become a cornerstone of modern mathematical understanding.