Pythagoras: Education, Mathematical Achievements, and Enduring Legacy
Pythagoras, a Greek mathematician and philosopher who lived around 570-495 BC, is a legendary figure renowned for his contributions to mathematics and his transformation of our understanding of the world through numbers and shapes. Born on the island of Samos, Greece, Pythagoras is believed to have spent time in Egypt and Babylon, where he was exposed to advanced mathematical and astronomical knowledge. Although there are no direct written records from Pythagoras, his teachings and philosophies were widely disseminated through his followers who formed the Pythagorean school.
Early Life and Education
Pythagoras was born in Samos around 570 BC. During Pythagoras's formative years, Samos was a thriving cultural hub known for its feats of advanced architectural engineering, including the building of the Tunnel of Eupalinos, and for its riotous festival culture and was a major center of trade in the Aegean where traders brought goods from the Near East.
The dates of his life cannot be fixed exactly, but assuming the approximate correctness of the statement of Aristoxenus that he left Samos to escape the tyranny of Polycrates at the age of forty, we may put his birth round about 570 BC, or a few years earlier. Some writers call him a native Samian, a Tyrrhenian from Lemnos, or a Phliasian from Peloponnesus. Due to this obscurity, some modern scholars deem it safer to accept "that Pythagoras and his father were pure-blooded Greeks, of undiluted Samian stock".
Pythagoras likely went to Egypt and Babylon as a young man. He traveled widely, to Egypt and Babylonia and perhaps even Persia, to learn how math was used in those ancient cultures. He emigrated to southern Italy about 532 BCE, apparently to escape Samos’s tyrannical rule, and established his ethico-political academy at Croton (now Crotone, Italy). Because of anti-Pythagorean feeling in Croton, he fled that city in 510 BCE for Metapontum (now Metaponto, Italy) where he died.
The Pythagorean School
Pythagoras and his followers developed the idea that numbers have deep meanings and important relationships with the universe, believing that all aspects of life could be explained through numbers and harmonious ratios. Around the age of 40 or so, Pythagoras headed west to southern Italy, settling in the Greek colony town of Croton. There he initiated a new phase of ancient science, mixing religion, music and math in a cult devoted to living in harmony with nature.
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The so-called Pythagoreans applied themselves to mathematics, and were the first to develop this science; and through studying it they came to believe that its principles are the principles of everything. Pythagoras believed that, at its root, reality was made from numbers. Numbers, he taught his followers, are the seeds from which all reality grows. Specifically, Pythagoras identified the root of reality in what he called the tetractys, consisting of the first four integers: 1, 2, 3 and 4. Added together, those numbers equal 10.
Following Croton's decisive victory over Sybaris in around 510 BC, Pythagoras's followers came into conflict with supporters of democracy, and their meeting houses were burned.
Mathematical Contributions
Although Pythagoras is most famous today for his alleged mathematical discoveries, classical historians dispute whether he himself ever actually made any significant contributions to the field. Many mathematical and scientific discoveries were attributed to Pythagoras, including his famous theorem, as well as discoveries in the fields of music, astronomy, and medicine.
Since at least the first century BC, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that "in a right-angled triangle the square of the hypotenuse is equal [to the sum of] the squares of the two other sides"-that is. The Pythagorean theorem reveals that in every right triangle, the relationship between the lengths of the triangle’s sides can be analyzed in a very interesting way. Although Pythagoras is often identified with this theorem, similar concepts actually existed earlier in Mesopotamian and Indian cultures.
It difficult to distinguish Pythagoras’s teachings from those of his disciples. Pythagoras himself likely wrote no books, and Pythagoreans invariably supported their doctrines by indiscriminately citing their master’s authority. Pythagoras, however, is generally credited with the theory of the functional significance of numbers in the objective world and in music. Other discoveries often attributed to him (the incommensurability of the side and diagonal of a square, for example, and the Pythagorean theorem for right triangles) were probably developed only later by the Pythagorean school.
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The Music of the Spheres
For Pythagoras, the historian of Greek philosophy W.K.C. Guthrie wrote, "all things are numbers". Of course, if your goal is to live in harmony with the universe, you need to know something about the universe. Pythagoras believed that, at its root, reality was made from numbers. That sounds crazy to modern minds taught that matter is made of atoms and molecules. But in ancient times, nobody really knew anything about what reality is. Every major philosopher had a favorite idea for what sort of substance served as reality’s foundation. Thales, for instance, thought everything derived from water. His student Anaximander rebelled, arguing that reality at its root consisted of some infinite, featureless material called the apeiron, or the unlimited. Anaximenes, Anaximander’s successor, rejected the apeiron in favor of air - everything could be explained by air either rarefying or solidifying. Pythagoras chose numbers.
Imagine plucking a taut string of fixed length, producing a musical note. If you pluck a string half as long, you get another note, separated from the first note by an octave. If the strings are plucked simultaneously, the two notes are harmonious. In other words, a 2:1 ratio of string lengths produces a pleasing sound. In a similar way, other harmonious musical intervals called the fourth and the fifth represent string length ratios of 4:3 and 3:2. Pythagoras realized that these harmony-producing ratios all involved the numbers 1, 2, 3 and 4.
Pythagoras himself may not have developed that theory fully, but his later followers produced a vision of the universe consisting of heavenly bodies revolving around a “central fire.” That fire was NOT the sun, which was just one of the other heavenly bodies. The Pythagoreans surmised that the motions of the heavenly bodies generated pleasant music. As Aristotle later explained it, those bodies move rapidly and therefore they must make sound, because anything moving quickly on Earth makes sound (think arrows whizzing through the air). Proper ratios of the planets’ speeds (which depended on their distances from the central fire) guaranteed that the sounds would be harmonious. So in a sense, the Pythagoreans believed that the universe itself could be regarded as a gigantic musical instrument.
But this picture had a problem. The Pythagoreans knew of only eight heavenly bodies: Earth, moon, sun, Mercury, Venus, Mars, Jupiter and Saturn. A ninth, outermost sphere transported the fixed stars. But to be perfect, the cosmos needed a 10th body. So the Pythagoreans proposed the existence of another planet, a “counter-Earth,” orbiting the central fire inside the orbit of the Earth. Nobody could see that planet because it was always on the other side of the fire. This theory was wrong, of course. But it nevertheless foreshadowed the modern use of mathematics to predict unseen phenomena. In the 19th century, for instance, James Clerk Maxwell used equations to predict the existence of radio waves. In the 20th century, Paul Dirac used math to predict the existence of antimatter.
Pythagoras's Philosophical Influence
Pythagoras’s legacy lies not only in his geometric theorem but also in his profound philosophical influence. His thoughts on the relationship between numbers, music, and harmony impacted many great thinkers in the future, including Plato and Aristotle.
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Pythagoras influenced Plato whose dialogues (especially Timaeus) exhibit Pythagorean ideas. Aristotle states that the philosophy of Plato was heavily dependent on the teachings of the Pythagoreans. Cicero repeats this statement, remarking that Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned all things Pythagorean").
A major revival of his teachings occurred in the first century BC among Middle Platonists, coinciding with the rise of Neopythagoreanism. Pythagoras continued to be regarded as a great philosopher throughout the Middle Ages and Pythagoreanism had an influence on scientists such as Nicolaus Copernicus, Johannes Kepler, and Isaac Newton.
Albert Einstein believed that a scientist may also be "a Platonist or a Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research." The English philosopher Alfred North Whitehead argued that "In a sense, Plato and Pythagoras stand nearer to modern physical science than does Aristotle.
Pythagoras in Modern Education
The influence of Pythagoras can still be seen today in mathematics education. The theorem that bears his name is taught in schools around the world and continues to be an important part of the curriculum.
Pythagoras's broader impact
Using math for understanding nature was unknown before Pythagoras. It was his idea. Previously math had been a tool for scribes or surveyors or cooks. “Pythagoras freed mathematics from these practical applications,” the Dutch mathematician B.L. van der Waerden wrote in his classic history of ancient math.
As for the music of the spheres, one issue remained. If the heavens made harmonious sounds, why didn’t anybody hear them? Aristotle rejected that explanation, just as he rejected the idea of a “counter-Earth” as well as the whole notion that everything was made from numbers. And yet, the importance of numbers in science, first expressed by Pythagoras, ultimately proved to be much more resilient than most of Aristotle’s ideas.
During the Middle Ages, Pythagoras was revered as the founder of mathematics and music, two of the Seven Liberal Arts. He appears in numerous medieval depictions, in illuminated manuscripts and in the relief sculptures on the portal of the Cathedral of Chartres.
In his preface to his book On the Revolution of the Heavenly Spheres (1543), Nicolaus Copernicus cites various Pythagoreans as the most important influences on the development of his heliocentric model of the universe, deliberately omitting mention of Aristarchus of Samos, a non-Pythagorean astronomer who had developed a fully heliocentric model in the fourth century BC, in effort to portray his model as fundamentally Pythagorean. Johannes Kepler considered himself to be a Pythagorean. He believed in the Pythagorean doctrine of musica universalis and it was his search for the mathematical equations behind this doctrine that led to his discovery of the laws of planetary motion.
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