Pythagoras and the Pythagoreans

Historically, Pythagoras means much more that the familiar theorem about right triangles. The philosophy of Pythagoras and his school has impacted the very fiber of mathematics and physics, even the western tradition of liberal education no matter what the discipline.

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

Pythagoras and the Pythagoreans

Little is known of his life. Pythagoras (fl 580-500, BC) was born in Samos on the western coast of what is now Turkey. He was reportedly the son of a substantial citizen, Mnesarchos. There he lived for many years under the rule of the tyrant Polycrates, who had a tendency to switch alliances in times of conflict — which were frequent.

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

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

Even so, just as he arrived, Croton lost a war to neighboring city Locri, but soon thereafter defeated utterly the luxurious city of Sybaris.

This is where Pythagoras began his society.

The Pythagorean School

tex2html_wrap_inline492 The school of Pythagoras was every bit as much a religion as a school of mathematics. For example, here are some of the rules:

  • To abstain from beans.
  • Not to pick up what has fallen.
  • Not to touch a white cock.
  • Not to stir the fire with iron.
  • tex2html_wrap_inline494
  • Do not look in a mirror beside a light.

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

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

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

“There are men, gods, and men like Pythagoras.”

It is likely that Pythagoras  was a charasmatic.

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

  • The Pythagorean school regarded men and women equally.
  • They enjoyed a common way of life.
  • Property was communal.
  • Even mathematical discoveries were communal and by association attributed to Pythagoras himself — even from the grave. Hence, exactly what Pythagoras discovered personally is difficult to ascertain.

The Pythagorean Philosophy

The basis of the Pythagorean philosophy is simply stated:

“There are three kinds of men and three sorts of people that attend the Olympic Games. The lowest class is made up of those who come to buy and sell, the next above them are those who compete. Best of all, however, are those who come simply to look on. The greatest purification of all is, therefore, disinterested science, and it is the man who devotes himself to that, the true philosopher, who has most effectually released himself from the ‘wheel of birth.'”

The message of this passage is radically in conflict with modern values. We need only consider sports and politics.

tex2html_wrap_inline492 Is not reverence these days is bestowed only on the “super-stars”?

tex2html_wrap_inline492 Are not there ubiquitous demands for accountability!!

The Pythagorean Philosophy

The gentleman, of this passage, has had a long run with this philosophy, because he was associated with the Greek genius, because the “virtue of comtemplation” acquired theological endorsement, and because the ideal of disinterested truth dignified the academic life.

The Pythagorean Philosophy ála Bertrand Russell

From Bertrand Russell, we have

“It is to this gentleman that we owe pure mathematics. The comtemplative ideal — since it led to pure mathematics — was the source of a useful activity. This increased it’s prestige and gave it a success in theology, in ethics, and in philosophy.”

Mathematics, so honored, became the model for other sciences. Thought became superior to the senses; intuition became superior to observation.

The combination of mathematics and theology began with Pythagoras. It characterized the religious philosophy in Greece, in the Middle ages, and down through Kant. In Plato, Aquinas, Descartes, Spinoza and Kant there is a blending of religion and reason, of moral aspiration with logical admiration of what is timeless.

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

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

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

What is known of the Pythagorean school is from a book written by the Pythagorean, Philolaus of Tarentum. From this book Plato learned the philosophy of Pythagoras.

The dictum of the Pythagorean school was All is number.

What this meant was that all things of the universe had a numerical attribute that uniquely described them. For example,

  • The number one : the number of reason.
  • The number two: the first even or female number, the number of opinion.
  • The number three: the first true male number, the number of harmony.
  • The number four: the number of justice or retribution.
  • The number five: marriage.
  • The number six: creation
  • tex2html_wrap_inline494
  • The number ten: the tetractys, the number of the universe.

Pythagorean Mathematics

One point: generator of dimensions.

Two points: generator of a line of dimension one

Three points: generator of a triangle of dimension two

Four points: generator of a tetrahedron, of dimension three.

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

Pythagorean Mathematics

Classification of numbers. The distinction between even and odd numbers certainly dates to Pythagoras. From Philolaus, we learn that

“…number is of two special kinds, odd and even, with a third, even-odd, arising from a mixture of the two; and of each kind there are many forms.”

And these, even and odd, correspond to the usual definitions. But even-odd means a product of an even and odd number. Note: orginally the number 2 was not considered even.

Prime or incomposite numbers and secondary or composite numbers are defined in Philolaus:

  • A prime number is rectilinear, meaning that it can only be set out in one dimension. The number 2 was not originally regarded as a prime number, or even as a number at all.
  • A composite number is that which is measured by some number. (Euclid)
  • Two numbers are prime to one another or composite to one another if their greatest common divisorgif is one or greater than one, respectively.

Proposition. There are an infinite number of primes.

Proof. (Euclid) Suppose that there exist only finitely many primes tex2html_wrap_inline502 . Let tex2html_wrap_inline504 . The integer N-1, being a product of primes, has a prime divisor tex2html_wrap_inline508 in common with N; so, tex2html_wrap_inline508 divides N– (N-1) =1, which is absurd!

The search for primes goes on. Eratsothenes (276 B.C. – 197 B.C.), who worked in Alexandria, devised a seive for determining primes. This seive is based on a simple concept:

Lay off all the numbers, then mark of all the multiples of 2, then 3, then 5, and so on. A prime is determined when a number is not marked out. So, 3 is uncovered after the multiples of two are marked out; 5 is uncovered after the multiples of two and three are marked out.

The Primal Challenge

The search for large primes goes on: Below is a list of the largest found to date. For a great deal of information on primes, including the numbers below, check out the The Primes Home Page  A special method, the Lucas-Lehmer test has been derived to check primality.

 2^859433-1 (258,716 digits); Slowinski and Gage, 1994
 2^756839-1 (227832 digits); Slowinski and Gage, 1992
 391581*2^216193-1 (65087 digits); Noll and others, 1989
 2^216091-1 (65050 digits); Slowinski, 1985
 3*2^157169+1 (47314 digits); Jeffrey Young, 1995 
 9*2^149143+1 (44898 digits); Jeffrey Young, 1995
 9*2^147073+1 (44275 digits); Jeffrey Young, 1995
 9*2^145247+1 (43725 digits); Jeffrey Young, 1995
 2^132049-1 (39751 digits); Slowinski, 1983
 9*2^127003+1 (38233 digits); Jeffrey Young, 1995

Subdivisions of even numbers are reported by Nicomachus (a neo-Pythagorean, ~100 A.D.).

tex2html_wrap_inline492  even-even — tex2html_wrap_inline518

tex2html_wrap_inline492  even-odd — 2(2m+1)

tex2html_wrap_inline492  odd-even — tex2html_wrap_inline526

Similar subdivisions of odd numbers are:

tex2html_wrap_inline492  prime and incomposite — ordinary primes excluding 2,

tex2html_wrap_inline492  secondary and composite — ordinary composite with prime factors only,

tex2html_wrap_inline492  relatively prime — two composite numbers but prime and incomposite to another number, e.g. 9 and 25.

Actually the third category is wholy subsumed by the second.

Also ascribed to the Pythagorean s is the study of perfect and amicable and deficient numbers.

A number n is perfect if the sum of its divisors is itself: Examples: ( 6, 28, 496, 8128, …) In Euclid, we find the proposition: If tex2html_wrap_inline534 is prime, then tex2html_wrap_inline536 is perfect. (Try , p= 2, 3, 5, 7 to get the numbers above.)

The pair of numbers a and b are called Amicable If the divisors of a sum to b and if the divisors of b sum to a. Example: 220 and 284.

In addition, the number a was classified as abundant or deficient according as their divisors summed greater or less than a, respectively.

Example: 12-divisors are: 6,4,3,2,1– tex2html_wrap_inline554 . So, 12 is abundant.

Example: All primes are deficient.

Mersenne Primes

Just the fact of finding perfect numbers using the previous propositions has spawned a cottage industry of determining those numbers p for which tex2html_wrap_inline534 is prime. Such primes are called Mersenne (1588-1648) primes after the friar of the 17 tex2html_wrap_inline560 century So far 33 have been found, though it is unknown if there is another between the 32nd and 33rd. It’s not known if there are an infinity.



Mersenne Primes

number    p                        year  by
 1-5   2,3,5,7,13         in or before the middle ages
 6-7   17,19              1588  Cataldi
 8     31                 1750  Euler
 9     61                 1883  Pervouchine
10     89                 1911  Powers
11     107                1914  Powers
12     127                1876  Lucas
13-14  521,607            1952  Robinson
15-17  1279,2203,2281     1952  Lehmer
18     3217               1957  Riesel
19-20  4253,4423          1961  Hurwitz & Selfridge
21-23  9689,9941,11213    1963  Gillies
24     19937              1971  Tuckerman
25     21701              1978  Noll & Nickel
26     23209              1979  Noll
27     44497              1979  Slowinski  & Nelson
28     86243              1982  Slowinski
29     110503             1988  Colquitt  & Welsh jr.
30     132049             1983  Slowinski
31     216091             1985  Slowinski
32?    756839             1992  Slowinski & Gage
33?    859433             1994  Slowinski and Gage,

Figurate Numbers. Numbers geometrically constructed had a particular importance to the Pythagorean s.

Triangular numbers. These numbers are 1, 3, 6, 10, … . The general form is the familiar



Square numbers These numbers are clearly the squares of the integers 1, 4, 9, 16, and so on. Represented by a square of dots, they prove(?) the well known formula



The gnomon is basically an architect’s template that marks off “similar” shapes.

Note the gnomon has been placed so that at each step, the next odd number of dots is placed.

Figurate Numbers of any kind can be calculated.




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