Plato and Archytas
Agree to disagree. Sometimes it is this agreement that holds an unexpected friendship together. Such was the case between two great philosophers, Plato and Archytas.
Plato was born around 428 BCE in ancient Greece. He was the child of an aristocrat. Yet, his father claimed to have been a descendant of Codrus, the semi-mythical king of Athens, who was a descendant of Poseidon. Plato was instructed in grammar, music, and gymnastics and might have been a wrestler. He was a dedicated student of philosophy and a devoted student of Socrates. The Socratic Method enthused Plato’s logic and exposition, which contributed to Euclid’s axiomatic-deductive proofs.[i] Plato wanted to motivate the Athenians to understand the value of justice by defending Socrates as the “gadfly” of the state.
After Socrates’s death, Plato left Athens and traveled to southern Italy and Egypt. During his travels, he sought out the acquaintances of the Pythagoreans. Utilizing Socrates’s methodology of persistent questioning, he found mathematical inspiration in the Pythagorean science of numbers. In 387 BCE, around the time that Athens began to enter into a new era of intellectualism, Plato returned to Athens. Back in Athens, Plato started a school where he taught other great thinkers and mathematicians, including Aristotle and Eudoxus of Cnidus. Plato’s mathematical teachings focused on proofs that used precise descriptions, accurate and coherent assumptions, and logic in the evidence. Plato’s logical process of proofs was groundbreaking to geometry.
Archytas was also born around 428 BCE. There are no specific years for both Archytas and Plato. So it is debated as to who was the older philosopher in this friendship. Archytas lived for about 80 years and built a legacy of respect as a Pythagorean philosopher, philanthropist, public official, educator, mathematician, and general. His notoriety as a general was so well received that he was elected seven consecutive years in a row. His teacher, Philolaus, taught him Pythagoras’s belief that mathematics is the path to know all things. Hence, Archytas was a Pythagorean, and his mathematical theories were foundational to projective geometry and number theory. Around 335 BCE Eudemus of Rhodes, one of the world’s first science historians, wrote that Archytas “enriched the science with original theorems and gave it a sound arrangement.”[ii] Archytas was a polymath who studied and was fascinated with many subjects. He gathered many of his studies under one belief, which was that mathematics consisted of four branches: arithmetic, astronomy, geometry, and music. By the middle ages, these four branches came to be known as the quadrivium. The quadrivium were four of the seven liberal arts taught after the trivium, which were grammar, logic, and rhetoric. Unlike the mathematicians before him, who were mathematical theorists, Archytas used mechanics to describe geometry. He was one of our first mechanical engineers.
Around 388 BCE, Plato had left Athens after the death of his dear teacher and friend, Socrates. From Athens, Plato traveled to southern Italy and then to the island of Sicily. It was his first time there. Since Plato was developing an interest in mathematics, he was intent on meeting the mathematician Archytas. By the time Plato reached Sicily, he had the opportunity to meet Archytas. Thus, they began a working relationship, which became a friendship, which saved Plato’s life. But, more on this later.
Plato did not fully agree with Archytas’s mathematical methods and distanced himself from the school of Pythagoras. Nevertheless, Archytas’s math had a considerable influence on Plato. In Plato’s Book VII of Republic, which tells the Allegory of the Cave, Plato writes that “as the eyes are framed for astronomy so the ears are framed, for the movements of harmony; and these are in some sort kindred sciences, as the Pythagoreans affirm and we admit.”[iii]
Around the time that the two had met, a math puzzle buzzed around philosophical circles that involved doubling a cube. In trying to solve this math puzzle, Hippocrates of Chios, a much older contemporary of Plato and Archytas, simplified the duplication of the cube by finding two mean proportionals in continued proportion.[iv] As a result, he found that if
\frac{a}{x}=\frac{x}{y}=\frac{y}{b}, \text {then}\frac{a^3}{x^3}=\frac{a}{b}Therefore, if
a=1 \space \text{and} \space b=2, \text {then} \space x=\sqrt[3]{2}After this discovery, the Delian problem became a problem of finding these two mean proportionals geometrically.
According to the ancient historian Eratosthenes, while Plato was in Sicily, he had proposed this math puzzle of Doubling the Cube to the citizens of Delia. That why the Doubling of the Cube problem is often referred to as a Delian Problem. For Plato, as a philosopher, his goal was to introduce the Delians to geometry and provide them a unique and different way of looking at mathematics.
Plato’s new friend, Archytas, was also contemplating this same puzzle. However, Archytas had a different approach. Archytas utilized theories of proportions and empirical observations. Thus, instead of thinking through the problem, Archytas constructed the problem and the solution using solids. Archytas’s technique was groundbreaking because all other methods up to this point were applied using only two-dimensional geometry. Using three-dimensional geometry, Archytas intersected a quarter of a cone, a cylinder, and a section of a torus. It was brilliant and provided a new way of understanding mathematics.
The following Website provides a phenomenal tutorial on how to recreate Archytas’s model of his solution in AutoCAD. http://www.takayaiwamoto.com/Greek_Math/Delian/Archytas_Delian.html
Archytas applied a mechanical approach to his mathematics, which did not bode well with Plato. According to Eratosthenes, Plato reproached Archytas for resolving the problem of duplication of volume by using tangible solids, calling it “irrational.” Eratosthenes states, “In proceeding in this way, did not one lose irredeemably the best of geometry, by a regression to a level of the senses, which prevents one from creating and even perceiving the eternal and incorporeal images among which God is eternally god.”[v]
In other words, Plato believed that mathematics must be imagined and perceived. In contrast, Archytas believed that understanding mathematics sometimes requires a mechanical process. Nevertheless, they agreed to disagree and remained friends.
While traveling in South Italy, he traveled to Syracuse. There he met a philosopher named Dion, who was the brother-in-law of the despot Dionysius I. Dion and Plato became close friends. As a result, Plato found himself involved in a family dispute.
Around 390 BCE, Archytas asked Plato to serve as a teacher and philosopher to Dionysius I. Up until this point, Plato often entertained the idea of being a political leader. Plato believed that kings could be made into philosophers, that philosophers could be made into kings, and that there could be an ideal philosopher-king.
Dionysius and his family were a rather violent and paranoid group. Dionysius was so paranoid that someone would murder him. He forced his visitors, even his son, to strip down to show that they were unarmed completely. Dionysius then made them change into different clothing. Dionysius was so violent that he even killed a man who dreamt of killing him.
Upon arriving, Plato was shocked by the Syracusian lifestyle. Plato, years later, wrote, “And when I came I was in no wise pleased at all with “the blissful life,” as it is there termed, replete as it is with Italian and Syracusan banquetings; for thus one’s existence is spent in gorging food twice a day and never sleeping alone at night, and all the practices which accompany this mode of living.”[vi]
Plato informed Dionysius that no person can become wise if he is focused on gluttonous behavior. This advice did not go over well with the elder Dionysius. As a result, Dionysius became angry with the philosopher and sold Plato into slavery. Luckily for Plato, a friend and philosopher, Anniceris, paid for Plato’s freedom and sent him back to Athens. It was a narrow escape for Plato. Back in Athens, Plato succeeded as a philosopher. During this time, he founded the Academy, where he produced his works Republic and The Symposium.
While Plato was in Athens, Dionysius had died either by alcohol poisoning or possibly by a sleeping potion given to him by his son Dionysius II. At the urging of Dion and possibly Archytas, Plato agreed to work with the family. This time, it was Dionysius II that Plato tried to turn into a philosopher. So, at the age of 60, he traveled again to Syracuse, stating, “I ultimately inclined to the view that if we were ever to attempt to realize our theories concerning laws and government, now is the time to undertake it.”[vii]
Plato began to make an impression on Dionysius II. However, Dionysius II was surrounded by sycophants who were concerned with Plato’s persuasive methods. As a result, they started to inform Dionysius II that Dion was using Plato to convince the younger Dionysius to step down from power. The lackeys were successful. After four months of trying to work with Dionysius II, Plato was unable to make him a philosopher. Instead, the younger Dionysius charged Dion with tyranny and exiled him from Syracuse. Plato’s work was done, and he returned to Athens.
Then several years later, Archytas reached out to Plato, urging him to go back to Syracuse to try to work with Dionysius II again. Though Plato was seventy years old, he held steadfast to his philosophical beliefs and went to Syracuse. However, though his efforts were partially successful, Dionysius II could never prove himself a true philosopher. He plagiarized other philosophers. Additionally, he misunderstood Plato and regurgitated what Plato had taught him without any thoughts or revelations of his own. According to Plato, Dionysius’s efforts were simply a means to gain more power. During their teachings, the younger Dionysius tried to use Plato as a tool to strip Dion of his property and earnings. This manipulation did not bode well for Plato, and he would not go along with Dionysius’s plot. As a result, Dionysius II banned Plato from the palace and sent him to the garden’s lodge. Dionysius II instructed his guard to watch over Plato and not let him leave Syracuse.
While trapped in the servants’ quarters, too much time had passed to Plato to escape. All of the trading ships departed Syracuse. Plato had no way off the island. While living in the servants’ quarters, he communicated with mercenaries from Athens paid to fight Dionysius’s war. Some of these mercenaries informed him that the people he was living among were plotting his murder. Plato, worried about his demise, arranged to have a messenger send a letter to Archytas in Tarentum. Luckily, the message got to Archytas. Archytas, realizing the danger his friend was in, arranged for a ship to go to Syracuse and rescue Plato from Dionysius II’s grips. Plato returned safely back to Athens, where he lived to the age of 80.
Even though they agreed to disagree, and even though they had differing views on mathematics, Plato had a deep respect for Archytas. Some scholars theorize that in Plato’s work, Republic, the rulers he refers to are Archytas and Dionysius. In a sense, Republic is his way of honoring his friend Archytas. In Republic, Plato, without naming names, questions how society creates rulers who are tyrants, much like Dionysius, and rulers who are conscientious philosophers, like Archytas. Possibly, Plato even considered Archytas as the ideal philosopher-king, which for Archytas is the highest compliment this profound philosopher could endow. And such is the relationship between two peers, two reliable friends, and two successful men who could truly count on each other.
[i]. Jyl Gentzler, Method in Ancient Philosophy (Oxford: Clarendon, 2007), 362.
[ii]. David Eugene Smith, History of Mathematics, Vol. I (New York: Dover Publications, 1958), 85.
[iii]. Plato and G. M. Grube, Plato’s Republic (Indianapolis: Hackett Publishing, 1974), 530(d), 182.
[iv]. Thomas Heath, A History of Greek Mathematics, Vol. I (Oxford: Clarendon Press, 1921), 200.
[v]. Nicholas Nicastro, Circumference: Eratosthenes and the Ancient Quest to Measure the Globe (New York: St. Martin’s Press, 2008), 95.
[vi]. Plato, “Plato, Letters, Letter 7, Page 326,” Perseus Digital Library, accessed March 9, 2021, https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0164%3Aletter%3D7%3Apage%3D326.
[vii]. Plato , “Plato, Letters, Letter 7, Page 326,” Perseus Digital Library, accessed March 9, 2021, http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0164:letter=7