Pythagoras was a demigod who went around performing miracles. He talked to the animals and they listened to him.1 Once, he convinced a bear to stop harassing the townspeople and the bear gave its word that it would. He also was renowned for having a “golden thigh.”
These are just some of the legends that surround this historical figure.2 In addition to all of that, he did not invent the Pythagorean theorem. Consequently, when we deal with Pythagoras, we are dealing with an enigmatic figure who is partly mythical and partly real. Like Socrates, he did not leave any writings, but also like Socrates, his followers attributed their ideas to him.3 Pythagoras and his followers lived in a highly secretive community.
Similar to the the other Presocratics, we know almost nothing about his life.4 We do know that he was born around 570 BC and died around 495 BC.5 Like the Milesian philosophers that we’ve studied – Thales, Anaximander, and Anaximenes – Pythagoras was an Ionian by birth. He was born on the island of Samos, off the coast of what is today’s western Turkey. Around the age of 40, he moved to Italy, apparently because he did not get along with the local tyrant at Samos. Local tyrants can be difficult to get along with. He settled in at a Greek town called Kroton (modern Croton) on the east coast of southern Italy. Just like Asia Minor, southern Italy and Sicily had been populated with Greek colonies.
The School at Croton, Italy
The school that he founded at Croton, an Italian school of philosophy, was partly philosophical and partly religious.6 In this way, it reminds me of a medieval European monastery. The Pythagoreans had some strange practices. Pythagoras prohibited his followers from eating beans since he said that they contained the souls of the dead. He believed in the transmigration of souls and reincarnation, beliefs he possibly picked up while traveling to India.
We are all familiar with Pythagoras from studying the Pythagorean theorem in school where we learned the relationship between the sides of a right triangle as a2 + b2 = c2 Even though Pythagoras’s name is attached to the theorem, the equation was widely known over 1000 years before he was born.7 He probably came up with a proof for the formula and that is most likely how his name became attached to it.
Regardless of his relationship to the theorem, Pythagoras is considered the first mathematician.8 People had been doing mathematics for centuries, but Pythagoras started the first school dedicated to discovering mathematical theorems and principles. Prior to the Pythagoreans, mathematics mainly had been done for practical purposes such as commerce and construction.
Numbers as the Basis of Everything
If Thales was the first philosophical philosopher, then Pythagoras was the first mathematical philosopher since he was the first to combine philosophy with mathematics.9 Aristotle said the following in his Metaphysics:
“…the Pythagoreans, in their interest in mathematics, were the first (early philosophers) to bring in numbers and stated that the principles of mathematics were the basic principles of all things.”10
Contrasted to the three Milesian philosophers that I previously discussed, Pythagoras chose numbers rather than water, air, or infinity as the arche of the universe. The arche of the universe is that fundamental principle from which everything originates. (See Post 26 for a further discussion of the term “arche.”)
Even though the Greek philosophers were all so unique and had different approaches to things, they had one thing in common – they all sensed an underlying unity to the diversity of the universe. They realized that the cosmos was characterized by change as well as constancy, having both static and dynamic aspects. There were universal principles that worked themselves out through the particulars. For Pythagoras, the universals were numbers. They were abstract and unchanging. That is what gave constancy to a changing world and why he said that numbers created the cosmos and that everything was made of numbers. In his view, numbers were, in fact, divine.
According to Iamblichus, a Neoplatonist who lived from the third to the fourth century AD, Pythagoras’s journey to discover the arche of the universe started when he walked by a blacksmith shop and heard the sounds of various hammers hitting metal.11 He realized that certain combinations of hammers produced sounds pleasing to the ear, where others did not. According to the legend, he went into the shop and after observing for a time, surmised that the differing weights of the hammers in combination produces either consonance or dissonance.
Numbers and the Harmony of the Universe
He noticed that consonance was produced by hammers that had a certain weight proportionate to one another. He then started to ponder why this was so – why proportion was so important. I find it interesting that the Presocratic philosophers, given to abstract thinking, were so concrete and earthy in their approach to things.
After his eureka moment in the blacksmith shop, he started experimenting with strings and found that strings of differing lengths produced different notes, and that combinations of strings of various lengths in certain proportion to one another produced harmonies pleasing to the ear, while other proportions did not. For example, if the notes of two strings, one string half the length of the other, were played simultaneously, this combination would make what is called a perfect octave.12 The ratio of this harmonious proportion is 2:1. If the ratio of the string lengths is 3:2, then we have a perfect fifth and if the ratio is 4:3, we have a perfect fourth and so on. As such, Pythagoras discovered that mathematical proportions govern the relationship between notes.
It was a major achievement to discover that the pitch of a note was determined by the rate of the string’s vibration and that it depended upon its length. This discovery started the prominence in Pythagoreanism of the relationship between mathematics and music.
Pythagoras realized that the three fundamental harmonies were generated by the first four numbers – 1, 2, 3, and 4. Adding these together made 10, which Pythagoras stated had a mystical significance since it was the basis of counting. The Pythagoreans liked to make shapes out of numbers, and they represented the number 10 via the following shape, which they called the tetractys:
The tetractys had such religious significance for the Pythagoreans that they would swear oaths on that image.13 Boethius, a philosopher who lived from the fifth to the sixth centuries AD, expanded upon these harmonies and created his own diagram to include the harmonies involving the numbers 6, 8, 9, and 12:14
This is the same diagram that appears in the above painting by Raphael. See if you can find it. This illustrates how important these discoveries were. Raphael was demonstrating a continuity of knowledge that spanned a millennium, from Pythagoras’s day until his own.
Numbers as the Link between the Physical and Metaphysical
Pythagoras illustrated that numbers were the link between the physical and metaphysical realm and music was just the beginning. The harmonious relationship produced by two or more objects did not have to be limited to music. The Pythagoreans extended this principle to the heavenly bodies and all of nature including human beings. They surmised that the distances between various planets in proportion to one another produced a celestial harmony of sorts. This was termed “celestial music.”15 It was not an audible sound, although some Pythagoreans said that the celestial bodies emitted hums that harmonized with one another. The “music” laid in the relationship of the harmonious proportions between them. Boethius would later term this as music universalis to contrast it from the music heard from musical instruments or singing.
The Pythagoreans also applied these principles to the human body. They thought of the soul as a kind of harmony. They said that if our bodies and our souls are in correct proportion, then we will be in harmony and thus be healthy. When our bodies and souls are unharmonious, then we get sick and even die. They also said that the musical harmonies had a nourishing effect upon the soul and thus the body. This belief laid the foundation for the idea of dualism that would later developed by Plato and end up having a long philosophical shelf life.
Boethius termed the harmony of body and soul as “human music” or musica humana.16 So to summarize, harmonies found in audible music are also found in the celestial bodies as celestial “music” and in humans as human “music.” Indeed they are found throughout all of nature.
Goodness is found when all of creation is in harmony with itself. This is why the Greeks put such an emphasis on harmony and proportion.
The Pythagoreans extended this concept of human music to ethics, morality, and government. A good government is one that governs harmoniously with itself and society. A person will live a harmonious life if he or she lives within ethical means and avoids the extremes. As you can see, the practical applications of harmonious proportion are endless. This is why it continued to be developed by Plato, and why his followers after him, in the first century B.C., combined the ideas of Plato and Pythagoras and developed them even further.17
The Legacy of Pythagoras
Plato was attracted to the ideas of Pythagoras because he, too was an abstract thinker and a lover of beauty, whereas Aristotle, being a more concrete thinker and not as concerned with beauty, was more dismissive of Pythagoras.18 Finally, these ideas reached their full flowering with the Neoplatonists beginning with Plotinus in the third century BC and including people like Iamblichus, mentioned above.
Pythagoras’s ideas would go on to affect not only philosophy and mathematics, but art and architecture, especially in the Middle Ages. Much of Gothic art and architecture was based on this idea of harmonious proportion.19 The Roman architect Vitruvius (85 BC to 15 AD) was heavily influenced by these ideas as well as the 16th century Italian architect Andrea Palladio. Through Palladio, Pythagoras’s legacy reached even to the United States, being responsible for the architectural beauty of Monticello, Harvard Hall, and the Capitol Building.
Speaking of Harvard Hall, one of the buildings below was designed with the principle of harmonious proportion and one wasn’t. Can you guess which is which and do you think that harmonious proportion has anything to do with beauty? The building on top is a modern office building and the one on the bottom is Harvard Hall.
Pythagoras’s journey really started with a search for beauty and led him to the divine. The Greeks recognized not only beauty in music and art, but in the cosmos and nature as well. They all came up with various explanations for beauty, where Pythagoras happened to find the mathematical basis for it. For Pythagoras and many other Greek philosophers, the beauty of the cosmos reflected not only a harmonious order but divinity itself. That is why they were also a community that worshipped.
It seems strange to us that for Pythagoras and his followers, there was no incongruity with being a philosopher or scientist and one who worships God. It seems odd because these things were separated from one another in the Enlightenment. As a result of that, we moderns are living in an unnatural state of existence. We see the universe as meaningless and explain our love of beauty in purely evolutionary terms.
To use Pythagoras’s ideas, we are living in dissonance rather than consonance with the order of the universe as created by God. As a result, we have become alienated, not only from our environment, but from each other and our own selves. We, as modern people, feel the tension of this alienation as loneliness and despair. This dissonance is also reflected in our art and architecture. By rejecting the principles of harmonious proportion that have guided artists for hundreds of years, we moderns have produced some of the ugliest works of art and architecture ever seen.
Our hope is found in returning to the rich treasures of Western civilization that we have abandoned. This not only includes the Christian faith, but the rich heritage of philosophy, art, music, literature, and science. These disciplines need to be integrated with one another into a unified whole as they were in the Middle Ages rather than remaining compartmentalized and disjointed. There was a time when artists and architects would consult with philosophers and theologians in order to improve their craft. We are now a one note performance- we look at everything through the cold lens of technology.
This does not mean that we should attempt to recreate the Middle Ages. We do not want to commit the fallacy of nostalgic thinking. But it means that we return to rebuilding the edifice of human civilization upon the foundation that we have inherited from our pre-Enlightenment forebears. When we do that, we will find that eventually all roads lead to Jesus Christ. What begins with the search for beauty should end with the worship of Him.
Note the following quote from Kitty Ferguson’s book, Pythagoras, His Lives and the Legacy of a Rational Universe:
“When the Pythagoreans, with their discovery of the mathematical ratios underlying musical harmony, caught a glimpse of the deep, mysterious patterned structure of nature, the conviction became overwhelming that in numbers lay power, even possibly the power that had created the universe. Numbers were like the key to vast knowledge – the sort of knowledge that would raise one’s soul to a higher level of immortality, where it would rejoin the divine.”20
Finally, consider the following question:
Why do you think we find it so odd to combine religion, especially mysticism, with science? Please leave your comment below. Thank you!
- Adamson, Peter, Lecture 4, “The Man with the Golden Thigh: Pythagoras,” History of Philosophy without any Gaps, King’s College, London, Dec. 27, 2010, https://historyofphilosophy.net/pythagoras
- Huffman, Carl, “Pythagoras”, The Stanford Encyclopedia of Philosophy (Winter 2018 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/win2018/entries/pythagoras
- Maor E. (1987) The Discovery of Irrational Numbers. In: To Infinity and Beyond. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-5394-5_8
- Britannica, The Editors of Encyclopaedia. “Pythagoras”. Encyclopedia Britannica, 19 Feb. 2021, https://www.britannica.com/biography/Pythagoras
- Cornford, F. M. “Mysticism and Science in the Pythagorean Tradition.” The Classical Quarterly, vol. 16, no. 3/4, 1922, pp. 137–150. JSTOR, www.jstor.org/stable/636499
- Britannica, The Editors of Encyclopaedia. “Pythagorean theorem”. Encyclopedia Britannica, 26 May. 2020, https://www.britannica.com/science/Pythagorean-theorem
- Veljan, Darko. “The 2500-Year-Old Pythagorean Theorem.” Mathematics Magazine, vol. 73, no. 4, 2000, pp. 259–272.
- Brann, Eva, The Logos of Heraclitus, p. 29,Paul Dry Books, Philadelphia, 2011
- Aristotle, The Metaphysics, [Alpha 5], p. 19, Translated by Hugh Lawson-Tancred, Penguin Books, New York, 2004
- Anderson, Gene H. “Pythagoras and the Origin of Music Theory.” Indiana Theory Review, vol. 6, no. 3, 1983, pp. 35–61.
- Clayton, David, The Way of Beauty, p. 106, Angelico Press, Kettering, Ohio, 2015
- Cornford, F. M. “Mysticism and Science in the Pythagorean Tradition (Continued).” The Classical Quarterly, vol. 17, no. 1, 1923, pp. 1–12.
- Clayton, David, The Way of Beauty, p. 148-151
- Leighton R. Scott. “Pythagorean Proportion and Music of the Spheres in Richard II.” Albion: A Quarterly Journal Concerned with British Studies, vol. 10, no. 2, 1978, pp. 104–117. JSTOR; Clayton, David, The Way of Beauty, p. 148-149
- Clayton, David, The Way of Beauty, p. 110-1112, 151-153
- Lippman, Edward A. “Hellenic Conceptions of Harmony.” Journal of the American Musicological Society, vol. 16, no. 1, 1963, pp. 3–35. JSTOR
- Gress, Dr. Carrie, “A Survey of the Philosophy of the Good, the True, and the Beautiful,” Lectures 7-9 on Aristotle and Beauty, Master of Sacred Arts, Pontifex University, https://www.pontifex.university/page/show/217276
- Clayton, David, The Way of Beauty, pp. 140-142, Angelico Press, Kettering, Ohio, 2015
- Ferguson, Kitty, Pythagoras, His Lives and the Legacy of a Rational Universe, p. 67, Walker Publishing Company, New York, 2008
Bibliography and Sources
Clayton, David, The Way of Beauty, Angelico Press, Kettering, Ohio, 2015
Also, check out David Clayton’s blog post on Pythagoras:
Aristotle, On the Soul, translated by Fred D. Miller, Jr., Oxford University Press, Oxford World Classics, Oxford, England, 2018
Aristotle, The Metaphysics. Translated by Hugh Lawson-Tancred, Penguin Books, New York, 2004
Aristotle, Physics, David Bostock, author, translated by Robin Waterfield, 1st ed., Oxford University Press, Oxford, England, 2008
Brann, Eva, The Logos of Heraclitus, Paul Dry Books, Philadelphia, 2011
Copleston, Frederick, A History of Philosophy, Book 1, Image Press, Cicero, N.Y., 1981
Ferguson, Kitty, Pythagoras, His Lives and the Legacy of a Rational Universe, Walker Publishing Company, New York, 2008
Grayling, A.C., The History of Philosophy, Penguin Press, New York, 2019
Hollis, Christopher, The Noble Castle, Longmans, Green and Co., London, New York, Toronto, 1941
Taylor, Thomas, Iamblichus’s Life of Pythagoras, Inner Traditions International, Rochester, Vermont, 1986
Waterfield, Robin, The First Philosophers: The Presocratics and Sophists, 1st ed., Oxford University Press, Oxford, 2009
Adamson, Peter, Lecture 4, “The Man with the Golden Thigh: Pythagoras,” History of Philosophy without any Gaps, King’s College, London, Dec. 27, 2010, https://historyofphilosophy.net/pythagoras