PythagoreansPythagoreans

Why Pythagoreans?Why Pythagoreans?

We are about to begin a unit which will We are about to begin a unit which will focus heavily on the focus heavily on the Pythagorean Pythagorean TheoremTheorem—a theorem about right —a theorem about right triangles and how to find the measures of triangles and how to find the measures of the sides of these triangles.the sides of these triangles.

Pythagoreans have an interesting Pythagoreans have an interesting history…history…

PythagorasPythagoras

Historically, Historically, PythagorasPythagoras means much more means much more that the familiar theorem about right triangles. that the familiar theorem about right triangles.

The philosophy of Pythagoras and his school The philosophy of Pythagoras and his school has impacted the very fiber of mathematics and has impacted the very fiber of mathematics and physics, even the western tradition of liberal physics, even the western tradition of liberal education no matter what the discipline. education no matter what the discipline.

Pythagorean  philosophy was the prime source Pythagorean  philosophy was the prime source of inspiration for Plato and Aristotle; the of inspiration for Plato and Aristotle; the influence of these philosophers is without influence of these philosophers is without question and is immeasurable. question and is immeasurable.

Pythagoras HimselfPythagoras Himself

Little is known of his life. Little is known of his life. Pythagoras (580-500 BC) Pythagoras (580-500 BC)

was born on the western was born on the western coast of what is now Turkey. coast of what is now Turkey. His town was always in His town was always in constant conflict.constant conflict.

He was reportedly the son of He was reportedly the son of a substantial citizen, a substantial citizen, Mnesarchos. Mnesarchos.

Pythagoras’ EducationPythagoras’ Education

He met Thales, likely as a young man, who He met Thales, likely as a young man, who recommended he travel to Egypt. It seems recommended he travel to Egypt. It seems certain that he gained much of his knowledge certain that he gained much of his knowledge from the Egyptians, as had Thales before him. from the Egyptians, as had Thales before him.

Probably because of continual conflicts and Probably because of continual conflicts and strife in his homeland, Pythagoras settled in strife in his homeland, Pythagoras settled in Croton, on the eastern coast of Italy, a place of Croton, on the eastern coast of Italy, a place of relative peace and safety. relative peace and safety.

This is where Pythagoras began his society. This is where Pythagoras began his society.

Pythagoras and the Pythagoras and the PythagoreansPythagoreans

The school of Pythagoras was every bit as The school of Pythagoras was every bit as much a religion as a school of mathematics. much a religion as a school of mathematics. For example, here are some of the rules: For example, here are some of the rules: To abstain from beans. To abstain from beans. Not to pick up what has fallen. Not to pick up what has fallen. Not to touch a white chicken. Not to touch a white chicken. Not to stir the fire with iron. Not to stir the fire with iron. Do not look in a mirror beside a light. Do not look in a mirror beside a light.

Vegetarianism was strictly practiced probably Vegetarianism was strictly practiced probably because Pythagoras preached the because Pythagoras preached the transmigration of souls. transmigration of souls.

Pythagoras and the Pythagoras and the PythagoreansPythagoreans

The school of Pythagoras represents the The school of Pythagoras represents the mystic traditionmystic tradition in contrast with the scientific!! in contrast with the scientific!!

Indeed, Pythagoras regarded himself as a Indeed, Pythagoras regarded himself as a mystic and even mystic and even semi-divinesemi-divine. Said Pythagoras . Said Pythagoras

"There are men, gods, and men like "There are men, gods, and men like Pythagoras." Pythagoras."

It is likely that Pythagoras  was a charasmatic. It is likely that Pythagoras  was a charasmatic.

Pythagoras and the Pythagoras and the PythagoreansPythagoreans

Life in the Pythagorean society was more-or-Life in the Pythagorean society was more-or-less less egalitarianegalitarian. .

The Pythagorean school regarded men and The Pythagorean school regarded men and women equally. women equally.

They enjoyed a common way of life. They enjoyed a common way of life. Property was communal. Property was communal. Even mathematical discoveries were Even mathematical discoveries were

communal and by association attributed to communal and by association attributed to Pythagoras himself -- even from the grave. Pythagoras himself -- even from the grave. Hence, exactly what Pythagoras discovered Hence, exactly what Pythagoras discovered personally is difficult to ascertain. personally is difficult to ascertain.

Pythagorean PhilosophyPythagorean Philosophy

The basis of the Pythagorean philosophy is simply The basis of the Pythagorean philosophy is simply stated: stated:

"There are three kinds of men and three sorts of "There are three kinds of men and three sorts of people that attend the Olympic Games. people that attend the Olympic Games. 1.1. The lowest class is made up of those who come to The lowest class is made up of those who come to

buy and sell, buy and sell, 2.2. the next above them are those who compete. the next above them are those who compete. 3.3. Best of all, however, are those who come simply to Best of all, however, are those who come simply to

look on. look on. The greatest purification of all is, therefore, The greatest purification of all is, therefore,

disinterested science, and it is the man who disinterested science, and it is the man who devotes himself to that, the true philosopher, who devotes himself to that, the true philosopher, who has most effectually released himself from the has most effectually released himself from the 'wheel of birth.'"'wheel of birth.'"

Pythagorean PhilosophyPythagorean Philosophy

The dictum of the Pythagorean school was The dictum of the Pythagorean school was All is number.All is number. What this meant was that all things of the universe had a What this meant was that all things of the universe had a

numerical attribute that uniquely described them. For example, numerical attribute that uniquely described them. For example, The number The number oneone : the number of reason. : the number of reason. The number The number twotwo: the first even or female number, the number : the first even or female number, the number

of opinion. of opinion. The number The number threethree: the first true male number, the number of : the first true male number, the number of

harmony. harmony. The number The number fourfour: the number of justice or retribution. : the number of justice or retribution. The number The number fivefive: marriage. : marriage. The number The number sixsix: creation : creation The number The number tenten: the : the tetractystetractys, the number of the universe. , the number of the universe.

Pythagorean PhilosophyPythagorean Philosophy

One point:One point: generator of dimensions. generator of dimensions. Two pointsTwo points: generator of a line of dimension one : generator of a line of dimension one Three pointsThree points: generator of a triangle of dimension : generator of a triangle of dimension

two two Four pointsFour points: generator of a tetrahedron, of : generator of a tetrahedron, of

dimension three. dimension three.

The sum of these is ten and represents all The sum of these is ten and represents all dimensions. Note the abstraction of concept. dimensions. Note the abstraction of concept. This is a distance from ``fingers and toes".This is a distance from ``fingers and toes".

Findings Credited to Findings Credited to PythagorasPythagoras

Classification of numbers.Classification of numbers. The distinction between The distinction between even and odd. (Originally, two was not considered even and odd. (Originally, two was not considered even)even)

PrimePrime or or incompositeincomposite numbers and numbers and secondarysecondary or or composite composite numbers numbers

They proposed that there were an infinite number of They proposed that there were an infinite number of primes.primes.

Figurate Numbers.Figurate Numbers. Numbers geometrically Numbers geometrically constructed had a particular importance to the constructed had a particular importance to the Pythagoreans. (Traingular, Square, Hexagonal, etc)Pythagoreans. (Traingular, Square, Hexagonal, etc)

Of course, the Pythagorean Theorem is Of course, the Pythagorean Theorem is creditedcredited to to Pythagoras.Pythagoras.

Effects of Effects of PythagoreanismPythagoreanism

The combination of mathematics and The combination of mathematics and theology began with Pythagoras.theology began with Pythagoras. It characterized the religious philosophy in Greece, It characterized the religious philosophy in Greece,

in the Middle ages, and down through Kant. in the Middle ages, and down through Kant. In Plato, Aquinas, Descartes, Spinoza and Kant In Plato, Aquinas, Descartes, Spinoza and Kant

there is a blending of religion and reason, of moral there is a blending of religion and reason, of moral aspiration with logical admiration of what is timeless. aspiration with logical admiration of what is timeless.

Platonism was essentially Pythagoreanism. Platonism was essentially Pythagoreanism. The whole concept of an eternal world revealed The whole concept of an eternal world revealed to intellect but not to the senses can be to intellect but not to the senses can be attributed from the teachings of Pythagoras. attributed from the teachings of Pythagoras.

Pythagoras Dies but Pythagoras Dies but Tradition Continues…Tradition Continues…

The Pythagorean School gained considerable influence The Pythagorean School gained considerable influence and became politically active -- on the side of the and became politically active -- on the side of the aristocrasy. Probably because of this, after a time the aristocrasy. Probably because of this, after a time the citizens turned against him and his followers, burning citizens turned against him and his followers, burning his house. Forced out, he moved to another city in his house. Forced out, he moved to another city in Southern Italy. Here he died at the ages of eighty. His Southern Italy. Here he died at the ages of eighty. His school lived on, alternating between decline and re-school lived on, alternating between decline and re-emergence, for several hundred years. emergence, for several hundred years.

Tradition holds that Pythagoras left no written works, Tradition holds that Pythagoras left no written works, but that his ideas were carried on by his eager but that his ideas were carried on by his eager disciples. disciples.