Cherry Hagen Pie

π
Paideia

John 3:13
"And no man hath ascended up to heaven, but he that came down from heaven, even the Son of man which is in heaven."

San Marco - S.Giorgio Maggiore - Internal Cloister
San Giorgio Maggiore, Venice
Paideia

In ancient Greek, the word paideia (παιδεία) means "education" or "instruction." Paideia was the process of educating humans into their true form, the real and genuine human nature.^[1]

Since self-government was important to the Greeks, paideia, combined with ethos (habits), made a man good and made him capable as a citizen or a king.^[2] This education was not about learning a trade or an art—which the Greeks called banausos, and which were considered mechanical tasks unworthy of a learned citizen—but was about training for liberty (freedom) and nobility (the beautiful). Paideia is the cultural heritage that is continued through the generations.

The term paideia is probably best known to modern English-speakers through its use in the word encyclopedia, which is a combination of the Greek terms enkyklios, or "complete system/circle", and paideia.

Origins and foundations

The Greeks considered Paideia to be carried out by the aristocratic class, who were said to have intellectualized their culture and their ideas; the culture and the youth are then "moulded" to the ideal. Starting in archaic times, love played an important part in this process,^[4] as adult aristocrats in most cities were encouraged to fall in love with the youths they mentored. The aristocratic ideal is the Kalos Kagathos, "The Beautiful and the Good." This idea is similar to that of the medieval knights, their culture, and the English concept of the gentleman.

Greek paideia is the idea of perfection, of excellence. The Greek mentality was "to always be pre-eminent"; Homer records this charge of King Peleus to his son Achilles. This idea is called arete. "Arete was the central ideal of all Greek culture."^[5]

In The Iliad, Homer portrays the excellence of the physicality and courage of the Greeks and Trojans. In The Odyssey, Homer accentuates the excellence of the mind or wit also necessary for winning. Arete is a concomitant of what it meant to be a hero and a necessary component in warfare in order to succeed. It is the ability to "make his hands keep his head against enemies, monsters, and dangers of all kinds, and to come out victorious."^[6]

This mentality can also be seen in the Greeks' tendency to reproduce and copy only the literature that was deemed the "best"; the Olympic games were also products of this mentality. Moreover, this carried over into literature itself, with competitions in poetry, tragedy, and comedy. "Arete" was infused in everything the Greeks did.

The Greeks described themselves as "Lovers of Beauty," and they were very much attuned to aesthetics. They saw this in nature and in a particular proportion, the Golden Ratio (roughly 1.618) and its recurrence in many things. They also referred to the need for balance as the Golden mean (philosophy) (choosing the middle and not either extreme). Beauty was not in the superficialities of color, light, or shade, but in the essence of being—which is structure, line, and proportion.

The Greeks sought this out in all aspects of human endeavor and experience. The Golden Mean is the cultural expression of this principle throughout the Greek paidea: architecture, art, politics, and human psychology.

In modern discourse, the German-American classicist Werner Jaeger, in his influential magnum opus Paideia (3 vols. from 1934; see below), uses the concept of paideia to trace the development of Greek thought and education from Homer to Demosthenes. The concept of paideia was also used by Mortimer Adler in his criticism of contemporary Western educational systems, and Lawrence A. Cremin in his histories of American education.

Sayings and proverbs that defined Paideia

"'Know thyself' and 'Nothing in Excess,' which were on everyone's lips."^[7] Words inscribed on the temple at Delphi.
Hard is the Good.

Pi

Pi or π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159 in the usual decimal notation (see the table for its representation in some other bases). π is one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve π.^[1]

π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.

The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter "περίμετρος", first by William Jones in 1707, and popularized by Leonhard Euler in 1737.^[2] The constant is occasionally also referred to as the circular constant, Archimedes' constant (not to be confused with an Archimedes number), or Ludolph's number (from a German mathematician whose efforts to calculate more of its digits became famous).

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:^[3]

Irrationality and transcendence

Main article: Proof that π is irrational

Being an irrational number, π cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert.^[3] In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known.^[7][8] A somewhat earlier similar proof is by Mary Cartwright.^[9]

Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root.^[10] An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.^[11] This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity; many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.