Sci 122 Telecourse Study Guide Program 6

©1998 RCBrill. All rights reserved

Pythagorean Mysticism

Program 6
Lesson 1.6

Coming Up



Number Game



Six Hundred Years

Pre Socratic Philosophy

Pythagorean Mysticism


Text References

Spielberg & Anderson 21-25

Booth & Bloom 7-11

Coming Up

Before we're done with this program we will have seen how the early ancient Greek philosophers replaced superstition with mysticism as they began to recognize numerical patterns in nature. Unable to explain these pattern they developed a numerological mysticism which became rationalized into a paradigm of perfection in the heavens.

In this and the next two programs we will trace the development of the geocentric paradigm over six hundred years culminating in the Ptolemaic system, which dominated Western thought for ffteen hundred years thereafter.


1.1. Name and briefly describe the four schools of pre Socratic Greek philosopy.

1.2. What is meant by "Pythagorean Mysticism?"

1.3. Give an example of Pythagorean harmony.

1.4. Define symmetry and discuss the relationship between truth, simplicity, perfection and symmetry.

1.5. What is the Pythagorean theorem?

1.6. According to Pythagoras, what is the perfect symmetrical figure?

1.7. Describe and illustrate the Pythagorean universe.


1. Understand the bases of ancient Greek philosophy.

2. Describe the characteristics of the Greek world view regarding nature.

3. Be able to write about the role of the supernatural in the Greek world view

4. Distinguish between purpose and principle

5. Understand the concept of numerical mysticism.

6. Be able to write about the significance of numerical patterns and geometic shapes in nature.

7. Understand the relationship between symmetry and perfection.

8. Describe the structure of the Pythagorean universe.

1. Number Game

Pick any number between one and ten, any number at all.

Multiply it by nine.

Add 1 to the first digit, you get your number back.

Subtract the second digit from 10, Again you get your number back, right.?

The sum of the two digits in the number is nine, the number you multiplied by. Nine is ten minus one.

Is there magic in the number nine because this works for any number between one and ten? Try it, it works. Is it magic, or is it just a property of numbers?

2. Introduction

This is the first of three programs dealing with the ancient Greeks, who contributed so much to our current world view, and to our science. The Greeks gave us much, and not just the Olympics, which is definitely physical, but not in the realm of our physical science course. They were the first civilization of record who placed a high value on thinking for the sake of thinking. Because of that, they considered many questions about the nature of life, the universe, and everything. Although we now disagree with much of what they believed, their thoughts are an important parrt of our heritage and the natural history of science.

The system of logic which the Greeks formalized is with us today in many forms, permeating every aspect of our lives, both in physical science and in the world at large. Logic,and mathematics, especially geometry, are among their greatest gifts to us,

Over a six hundred year period beginning around six hunded B.C., Greek intellectuals pondered and philosophized about everything imaginable. Many of our modern ideas about matter and the physical universe originated with the Greeks in one form or another. In this program we will consider the Pre Socratic period, especially the philosopher and mathematician known as Pythagoras and the cult of followers he attracted. In the next two programs we wlll consider the classical philosophers of the golden age and Greece after the death of Alexander.

The Greek influence on European thought was especially strong because of the stagnation which occured after the fall of the Roman empire left Europe in chaos linguistically, sociall, politically, and economically. During the dark ages that followed, the works of the great Greek thinkers was mostly unavailable in Europe, but limited amounts were studied and reworked in n monestaries where they were treated as sacred texts.

When the works of the Greek masters were rediscovered in the twlefth century, it precipitated a renaissance of learning, which established the geocentric paradigm with Aristotle as the authority on everything and which ultimately led to the scientific revolution six centuries later.

The question we must answer in the next three programs is: If the Greeks were such good thinkers, how come they got it all wrong and in doing so confused the Christian world for two thousand years?

3. Fundamentals of Greek Science

Anyway, we cannot explain cultural preferences except to note that the habits, beliefs and practices of one culture almost always seem strange at first from the outside.

It is all too easy to look back on the Greeks and wonder how they could have believed some of the things they believed, but we must remember, these were not stupid people, They were just like us, locked into a paradigm over which they had little control, just like us.

Like their predecessors in the Mediterranean, the Greeks were interested in astronomy, agriculture, and navigation. They differed from other Mediterranean cultures in their approach to nature. Rather than speculate on the nature of the heavens in mythological terms, they preferred a rational approach.


3.1. World view is consistent with general philosophy

Anyway, we cannot explain cultural preferences except to note that the habits, beliefs and practices of one culture almost always seem strange at first from the outside.

It is all too easy to look back on the Greeks and wonder how they could have believed some of the things they believed, but we must remember, these were not stupid people, They were just like us, locked into a paradigm over which they had little control, just like us.

Like their predecessors in the Mediterranean, the Greeks were interested in astronomy, agriculture, and navigation. They differed from other Mediterranean cultures in their approach to nature. Rather than speculate on the nature of the heavens in mythological terms, they preferred a rational approach.


3.2. Physical theories arise out of prior metaphysical considerations

Because of the interconnectedness of the physical, moral, and political worlds it was a necessity that physical theories arise out of prior metaphysical considerations. This means that metaphysics drove physics, the study of the material world.

Does this begin to make sense? The material world is intimately linked with being, the structure of the universe, and the nature of knowledge. We will see this linkage develop as we make our way through the Greek philosophy.

3.2.1. metaphysics

That branch of philosophy which considers first principles. Generally includes three branches. Ontology is the study of the nature of being, cosmology is the study the nature of the origin and structure of the universe, epistemology is the study of the nature of knowledge. ontology: study of being cosmology: study of origin and structure of the universe epistemology: study of knowledge

3.3. Knowledge of natural world is called natural philosophy

Knowledge of the natural world was called natural philosophy, which included all aspects of the natural world, including it's overlap with the political, esthetic, and moral world.

It is a nice scheme, to have everything tied somehow to everything else. The problem is that it is awfully hard to to it without encountering contradictions somewhere along the line. As we will see, this got even the greatest thinkers in a lot of logic trouble, kind of like painting yourself into an intellectual corner.

3.4. Preference for understanding "purpose" rather than principle

Built in to the Greek philosophy was a preference for understanding purpose rather than principle. It was more important to understand why something happened rather that to understand how it happened. We cannot explain a cultural preference for the idea that everything must have a purpose, but is it consistent with the idea of destiny.

3.5. Mathematical relationships exist between observable quantities

The importance of mathematics in the Greek world view cannot be overstated. As we saw in the last program, the Greeks were the first recorded culture to value the concept of mathematics in its general forms. Relationships between numbers came to be seen as mystical, largely through the efforts of Pythagoras and his followers, the topic of this lesson.

3.6. Mechanistic, geometric universe

he cosmology of the Greeks was mechanistic and geometric. That means that the natural philosophy sought answers not in terms of Gods and heavenly entities, but rather in terms of material objects moving in some sort of connected system. The heavenly objects, stars, planets, sun, and moon were viewed as material objects, although unlike any matter here on earth, but matter of some kind.

3.7. Separation of science and religion

With the Greeks we see the separation of science and religion, the former concerning itself with mechanical principles such as the motions of the heavens, the later accounting for divine purpose and destiny. The Greek gods were really rather indifferent to man, except as a nuisance or a curiosity. In Greek mythology, the Gods essentially lived in Olympos having parties and betraying one another, and occasionaly interferring in human afairs, producing demigods, half human and half god. In this scheme the gods did not control the heavenly motions although they were responsible for such earthly things as weather, earthquakes, volcanic eruptions and the like.

3.7.1. science: mechanical principles

3.7.2. religion: divine purpose

3.7.3. no supernatural causes, no astrology

3.8. Many systems were considered

The Greek philosophers considered many different alternative systems when establishing their world view. There were lively debates about geocentrism vs. heliocentrism, the nature of matter, the existence of atoms, elemental principles and the nature of motion.

3.8.1. lacked observational tools to confirm or deny

One of the major shortcomings of Greek science was the lack of observation. The Greeks generally did not trust the senses, believing that logical discourse could ascertain the truth from first principles. Unfortunately, conclusions derived logically are no better than the starting premises. Many of the conclusions reached by the great Greek philosophers, Aristotle especially, might have been disproven with a heavier reliance on observation, as in the case of freefall.

3.8.2. paradigm prevented seeing holes in model

Like the rest of us, their paradigm of unity and reliance on logic derived from metaphysical principles prevented them from seeing the holes in their theories, many of which seem obvious to us today.

3.8.3. lack of evidence used to confirm theory of heavens

One important difference between our modern view of reality and that of the ancient Greeks was their use of the lack of evidence to confirm a theory. This is like the old elephant repellant joke.

What! You must have heard that one. A guy is at the doctor's office in the waiting room. Occasionally he takes a small bottle out of his pocket, pours a small amount of liquid into the palm of his hand and flings it about the room as he yells something incomprehensible very loudly. After several episodes of this the receptionist says to the guy, "Excuse me, sir but is everything all right." The guy replies "Sure, I'm just keeping the elephants away," to which the receptionist replies, "But there aren't any elephants around here." The guy looks up at her and says, "See, it works."

Specifically it was the failure to see the crecscent of Venus and the parallax of stars which contributed to the rejection of the heliocentric system. Had these two phenomena been visible to the naked eye it is likely that the heliocentric theory would have developed two thousand years ago. As it was the world had to await the invention of the telescope in order to see these things. We will cover these in greater detail later in the course.

3.8.4. misunderstanding of motion allowed minimal contradictions

More than any other factor, it was the misunderstanding of motion that kept the Greeks from a better understanding of natural philosophy. Today we take for granted our understanding of motion because we have a paradigm inherited from the seventeenth century. When we study the details of motion in a later program we will see that there are some aspects of motion which are not intuitively obvious at first. Galileo, who is responsible for our modern understanding of motion from his work in the early seventeenth century, recogized the weakness in Aristotle's understanding of motion, and undertook studies of motion specifically to discredit Aristotle after viewing the crescent of Venus through the telescope and for the first time seeing proof of heliocentrism.

Anyway, we cannot explain cultural preferences except to note that the habits, beliefs and practices of one culture almost always seem strange at first from the outside.

It is all too easy to look back on the Greeks and wonder how they could have believed some of the things they believed, but we must remember, these were not stupid people, They were just like us, locked into a paradigm over which they had little control, just like us.

Like their predecessors in the Mediterranean, the Greeks were interested in astronomy, agriculture, and navigation. They differed from other Mediterranean cultures in their approach to nature. Rather than speculate on the nature of the heavens in mythological terms, they preferred a rational approach.

4. Six Hundred Years of Thought

The Greek civiliation flourished for approximately six hundred years. That's about three times longer than the United States has existed as a country. Just as life and ideas in our own country have changed over the centuries, so did the Greeks. It would be incorrect to think of the Greek culture as stagnant and unchanging. It would also be incorrect to think that the famous philosophers of which we learn today were the only ones who contributed to the world view.

This is not a philosophy course, but Greek philosophy had a n extremely strong influence on the development of physical science. Sol we want to focus on the main ideas of only a few of these great thinkers without giving the impression that we are covering Greek philosophy in any detai. I would encourage you to look at some of the reference materials in the study guide bibliography to get a much clearer picture of this philosophical heritage. We will consider several of the major philosophers and the ideas they contributed , while concentrating on only five names.

The names we will become familiar with are: Pythagoras, Socrates, Plato, Aristotle, and Ptolemy.

In Greek philosophy and science, there are three major periods to consider. Today we will consider the pre Socratic period. In program seven we will look at the great triad of Golden Age philosophers, Socrates, Plato, and Aristotle. In program eight we will look at the Hellenistic Greek culture which shifted to Alexandria, in Eggypt, after the death of Alexander the Great who had united Greece and destroyed the independent city states.

Thales (625-545 B.C.)

Anaxamander (611-547 B.C.)

Pythagoras (582-500 B.C.)

Socrates (470-399 BC)

Plato (427-347 B.C.)

Euxodos (400 B.C.)

Aristotle (384-322 B.C.)

Aristarchus (250 B.C.)

Eratosthenes (235 - 195 B.C.)

Euclid (330? - 275 B.C.)

Hipparchus (140 B.C.)

Ptolemy (A.D.100 -170 )

5. Pre Socratic Philosophy

Western philosophy and science traces its origins to the pre-Socratics, those who came before Socrates. (600 to 400 BC). They are generally credited with being the first to separate thinking and reflection about the world and reality from religion and mythology and to use logical and rational means to consider ultimate questions. The ideas of pre-Socraticphilosophers fall roughly into four major groups or schools: the MILESIAN SCHOOL, the PYTHAGOREANS, the ELEATIC SCHOOL, and the SOPHISTS.

We are not sure of the exact teachings of individuals of the four groups because many of the original documents have been lost. Much of what we do know of this period comes from historical accounts of later writers in the Greek era. Nonetheless, it is important to get a glimpse of the overall philosophies of these various schools of thought, because they wind up being incorporated, one way or the other, into Aristotle's system of the world which served as a cosmological model for the next two thousand years.

5.1. Four Schools of Thought

5.1.1. Milesian

The Milesians attempted to distinguish between the appearance and the underlying reality of the physical world. They tried to discover the essential "stuff" of which all matter is composed. Two of the earliest Milesian philosophers were Thales and Anaxamander. Thales (625-545 B.C.)

Link to The Internet Encyclopedia of Philosophy

THALES OF MILETUS taught that the most basic substance was water; but his successors isolated other substances as fundamental matter, first earth, then air, then fire.

Thales is generally credited with removing the gods from nature by stating that the heavenly objects are solid, material objects, rather that gods. It is here that we see the beginnings of the preference to consider the natural world and the supernatural separately. Thales taught that nature is impersonal, that natural events happen naturally, without regard for human affairs. This was around the same time that philosophers in other parts of the world began to consider the same ideas. The gods, in Thales view, were reserved for concern with the spiritual welfare of man rather than the workings of the heavens. Removed Gods from nature heavenly objects are solid, material objects, not gods natural causes: nature is impersonal around the same time as Hebrew, Persian, Zoroastrians, Buddha did the same thing Gods reserved for concern with spiritual welfare of man Anaxamander (611-547 B.C.)

Anaxamander, another Milesian, was the first of record to recognize that the heavens revolve around Polaris, and also argued that fire was a fundamental constituent of matter, along with earth, air, and water. heavens revolve around Polaris added fourth element (fire) to constituents of Earth

5.1.2. Pythagorean

The Pythagoreans, who we will study in detail later in this program, are named for the school's founder, PYTHAGORAS OF SAMOS (6th century). They were mystified by the nature of and relationship between numbers, shapes and human affairs. They taught that the fundamental nature of all things was to be found in the basic limiting quality of number, specifically the counting numbers. They also developed many theorems of arithmetic and geometry which advanced the study of mathematics.

5.1.3. Eleatic

The Eleatics proposed two separate solutions to important problems which the Milesians had encountered. The Milesians had been unable to account for the nature and possibility of change in the basic elements they had proposed as fundamental to reality. One proposal, advanced by HERACLITUS, was that all things ultimately become one and the same, and that the basic quality of reality is change. Opposing this was the view that the most basic statement that could be made about anything was that it must either be or not be. So being was the essential quality of which all things partake. Change vs. being as the ultimate reality. What a choice!

Philosophers responded to the Eleatic position with various arguments. EMPEDOCLES taught that all things have their roots in the four elements of fire, air, earth, and water, which are fused or divided by the forces of Love and Strife. ANAXAGORAS proposed Mind as the ordering force in a mechanistic universe. The atomists LEUCIPPUS and DEMOCRITUS held that nothing exists but "atoms and void" and that the atoms constantly rearrange themselves in accordance with mechanistic laws. Heraclitus all things ultimately become one and the same basic quality of reality is change. opposed the view that the most basic quality was being Empedocles four elements are basic to reality Anaxagoras mind is the ordering force in a mechanistic universe Leucippus and Democritus nothing exists but atoms and void atoms constantly rearrange themselves in accoradnce with mechanical laws

5.1.4. Sophist

The Sophists were not really philosophers. They were more masters of rhetorical argument. They were intensely skeptical about everything and proud of their ability to argue any side of any dispute. Socrates considered their philosophy to be amoral and countered with his moral philosophy. masters of rhetorical arugment intensely skeptical argued all sides of a dispute amoral philosophy rejected by Socrates

6. Pythagorean Mysticism

The views and logical arguments that these ancient philosophers argued are incorporated in one form or another in our modern world view. Although they were unable to decide for sure which view was correct, we have learned that parts of all of it is corect. Their ideas wll continue to pop up throughout the course.

Now we turn our attention to the Pythagoreans. Of all the pre Socratic philosophers, Pythagoras and his followers had the greatest influence on the newly developing cosmology. The most significant legacies of the Pythagoreans was the idea of the perfection of the circle and the spherically concentric universe, concepts that would remain in the paradigm for centuries.

6.1. Pythagoras (582-500 B.C.)

"Seek truth and beauty together; you will never find them apart"

Pythagoras of Samos

Little is known of Pythagoras personally. This is not surprising since he lived 2500 years ago and was the founder of a secret cult who tried to keep their ideas and even their existence secret. His work is known largely through Plato, who combined the Phythagorean ideas with this own one hundred years later.

6.1.1. often gets credit for Pythagorean discoveries

6.1.2. difficult to separate his from the cult's

6.2. Mystic Cult

Pythagoras fomed a mystic cult which was devoted to mathematical speculation and religious contemplation. Men and women were admitted on equat terms, an unusual attitude for those times. As part of the inititation, every cult member had to surrender ownership of all posessions, including ideas, to the cult. All property and ideas were held in common as a commune. Because of the mystical nature of the cult, all mathematical discoveries were kept secret from outsiders.

6.2.1. devoted to mathematical speculation and religious contemplation

6.2.2. men and women admitted on equal terms

6.2.3. all property and ideas held in common

6.2.4. mathematical discoveries kept secret from outsiders

6.3. Numbers And Geometry

In the Pythagorean scheme, numbers and geometry provided a conceptual model of the universe.

6.3.1. a conceptual model of the universe

6.3.2. quantities and shapes determine the forms of natural objects

Pythagoras saw that natural objects mimicked geometric shapes and could often be described by numbers. relationship between geometry (shape) and arithmetic (quantity)

6.3.3. numbers = counting numbers = integers

By numbers, he specifically meant the counting numbers, or the integers (without the zero which is a relatively modern invention). It is important to keep in mind that our modern numbers are an invention of the Arabic world nearly one thousand years after Pythagoras. The Greeks used a number system based on letter of the alphabet, making calculations and pattern recognition extremely more difficult than we find it today with our decimal number system. It is not too hard to see why pattern might have seemed magical.

According to the story, Pythagoras' fascination with these patterns began when he noticed, as a fairly young man, that pleasant musical tones are generated by pipes or chimes whose lengths are in small whole number relationships. Here we see the ratios of the notes in a major scale. Pythagoras thought that there was mystical significance in this relationship between number and harmony. It does seem kind of magic doesn't it?

Another of the mystical relationships which fascinated Pythagoras and his followers was the existence of certain triplets of numbers which related to the side of a right triangle. We have already seen how these Pythagorean triplets were used by the Babylonians and Egyptians to build square buildings.

It was the Pythagoreans who discovered in these triplets the general relationship that we call the Pythagorean theorem today. The relationship is a simple one: The square of the hypoteneuse is equal to the sum of the squares of the other two sides. I want to stress the important difference between simply recognizing that some numbers will yield a right triangle and recognizing a general relationship between certain numbers and the triangular shape.

In modern times we rely heavily on this Pythagorean theorm in our analysis of forces and other physical quantities. We will see more of this relationship as we progress through the course. pleasant musical tones have integral relationships

Try it! Use a magic marker to mark levels of water on water glasses or plastic water bottles. It will take some thought and a little measurement to make the fractions but you can estimate them. Then arrante the bottles and play a scale. Does it work? Pythagorean theorem The square of the hypoteneuse equals the sum of the squares of the other two sides.

6.3.4. irrational numbers can't be expressed as integers

One outgrowth of the Pythagorean theorem was the realization that there are certain numbers, called irrational numbers, which cannot be expressed as the ratio of two integers. Fractions such as three-fourths or thirteen thirty seconds fit nicely into the integers. Although the fractions are not integers, and may even have no end when expressed as decimals (numbers like 1/3 for example), they are still composed of integers.

Other number, such as pi (the ratio of the circumference to diameter of a circle, or the square root of 2 are irrational. There are no integers which can be divided one into the other to generate these numbers. Sure we can use 22/7 as an approximation to pi, but it is only an approximation. In modern decimal notation after the second decimal place 22/7 no longer matches pi.

The existence of these irrational numbers bothered the Pythagoreans, in fact they were downriight irritated by them, so they attempted to hide their existence by suppressing all speculation about them.

Consider the square root of 2 for example. In a right triange with sides of 1, how long is the hypoteneuse? For those of you who have not had the pleasure of studying geometry, the hypoteneuse is the side of a right triangle which is opposite the right angle. numbers such as pi and the square root of 2 irritated Pythagoreans, so they suppressed knowledge

6.4. Numerology & Mysticism

Based on these numerical pattern arose the belief among the Pythagoreans that numbers and shapes represent something fundamental about the nature of the universe. Once again we cannot find any rational explanation for the belief, but it is not much different that other superstitions. It takes little introspection to realize that even today, most people believe that our destiny is linked to something other than our own choices. It is more comforting to think that there is something rather than random chance which influences our lives.

6.4.1. numbers and shapes influence natural and human affairs

Gradually the belief grew that numbers and shape play a role in destiny and so influence both natureal and human affairs. Numerology was born with the concept of lucky and unluck numbers, magic numbers, and even the idea that the numbers representing the letters in one's name can provide information about the destiny of that individual.

Note that this is very much like the astrology of the Babylonians, except it is not based on the locations of the stars, and is a good deal more abstract. lucky and unlucky numbers magic numbers add up numbers in name

6.4.2. triangular and square numbers

Another of the Pythagorean mysteries of note was the relationshp between the triangular and square numbers. The triangular numbers are those which represent a triangular array of objects , like the ten pins of bowling, or the fifteen balls used in a rack in pool. The square numbers are those which arise from building an array which contains equal numbers of rows and columns.

It is the relationship between the triangluar and square numbers, not the numbers themselves, in which the Pythagoreans saw the mystery. We see that each square number can be represented as the sum of two successive triangular numbers. Geometrically speaking, each square array can be built by adding the the next highest triangular array to it.

Why does this relationship exist? It is the nature of the numbers and shapes. Is it magic? I dunno. What do you think? If not magic, it is certainly an interesting property of numbers. Whether or not it has meaning or significance is another question.

At this point I must raise the question: Does every pattern have meaning? Here we find a pattern which is not just created by the brain, unlike the patterns in clouds and tea leaves that we examined in program 3. Here it is the signiificance of the pattern, and not the pattern itself which is in the mind of the beholder.

6.5. Symmetry & Perfection

Speaking of patterns and order, what about symmetry? What is symmetry anyway?

In the visual arts we speak of symmetry as an esthetic quality of balance. In a more general sense, symmetry is a type of ordering which involves repetition, or more precisely, symmetery refers to something which remains unchanged after some action is taken or some operation is performed.

See the demonstration of the symmetry of the polygons in the video program.

Geometric shapes such as the regular polygons were among those studied by the early Pythagoreans. They figured out, for example, formulas to calculate the areas and perimeters of these and other figures. They also thought there was a certain purity of form in these simplest of shapes.

6.5.1. Symmetry refers to something which is unchanged after an action

6.5.2. Rotational and mirror symmetry are the simplest

The simplest forms of geometric symmetry are the symmetry of rotation and reflection. Certain shapes appear the same when wiewed upside down, like the double arrow, or in a mirror, like the left and right hands. Try it. Your right hand in a mirror looks like a left hand.

Question to Ponder: If mirrors reverse things then why don't we appear upside down when we look in a mirror?

6.5.3. Regular polygons show high degrees of symmetry

These regular polygons are highly symmetrical. They all show various degrees of rotational symmetry. They also have mirror symmetry, which means that one half is a mirror image of the other half.

If the triangle is rotated one third of a turn, the it looks the same as it did bofore. The square can be rotated four time, a quarter turn each time to come back to the original position. Each quarter turn leaves it looking as if no rotation has occurred.

Mirror symmetry means you can replace one half of the figure with the mirror image of the other half and it will look the same.

Note that the symmetry of the hexagon and octagon is much higher than the other three, meaning that there are more different kinds of symmetry. triangle, square, pentagon, hexagon, octagon, etc.

6.5.4. Food for thought: symmetry

I'ts time for another food for thought. We've drawn the mirror planes on the triangle and the square for you. Can you find them on the other three figures? How does the symmetry of the triangle compare with that of the hexagon. How do the square and the octagon compare?


6.5.5. Symmetry and perfection

What connection is there between symmetry and perfection? From the point of view of esthetics, there is a certain elegance to a simple, symmetrical figure. Think of it as beauty without complexity.

Do you see the connection? The more beautiful something iswith the less complexity, the more elegant. The ideal, perfect work, if it could exist, would be that which is the most beautiful, while expressing basic truth, in the simplest, most symmetrical way possible.

You may not agree with this description, but it is the basis for the Greek concept of perfection whether it be in art or in nature. It is not just a Greek idea. Many of the great philosophies of the world have similar visions of perfection. beauty, truth, perfection, simplicity are related concepts

6.5.6. The Perfect Circle

Now we come to the crux of the matter of symmetry. What is the most perfect geometric shape? What shape has the most symmetry with the most simplicity?

The circle is the most symmetrical shape with the greatest simplicity, so it must be the most perfect shape.

The circle is not altered by rotation, it has and infinite number of spoke like diameters which are all planes of mirror symmetry. It is like a polygon with an infinite number of sides.


By extension, the sphere is the most perfect three dimensional shape, since it can be generated by rotating a circle around a diameter. the circle is not altered by rotation the circle is the most symmetrical shape the circle has both elegance and simplicity the circle has infinite number of diameters of mirror planes the circle is like a polygon with an infinite number of sides by induction the sphere is the most perfect three dimensional shape

6.6. The Pythagorean Universe

All of which now brings us to the Pythagorean cosmology. Remembering that we must not try to explain cultural preferences by applying our own cultural criteria, we now can see the next step in the Pythagorean thought process which might go something like this.

The ideal universe would be perfect because it is ideal. The stars and everything else goes around earth as if they were on a spherical bubble. So the universe outside of earth appears to be spherical and therefore is probably perfect, except for here on earth, where we know for sure that things are not perfect.

So, how do you visualize the structure of a universe which is spherical and apparently flawless, except for earth.

Here we get only brief glimpses of truth, beauty and perfection, in the form of simple geometric shapes which nature mimics, and which have what might be best described as magical properties. We also briefly see truth and beauty in the mathematical relationships of musical harmony, harmonies which bring out our better qualities, the perfection in us (did you ever try to be angry while listening to harmonious music?) That imperfection resides inside a bubble which appears to be spherical, against which the planets, the moon and the sun move in circular paths through the sky?

Whew? So how would you imagine this universe is structured?

Simple, geometric, mechanical, symmetrical, circular, and perfect except at the center!

The universe of the Pythagoreans was spherical, and consisted of three concentric regions. The inner part held the corruption of man and other imperfections of the earthly realm. This was Uranos. It comprised the earth and everything up to the moon.

The middle sphere, Cosmos was the sphere of the moveable heavens. In this region the seven planets (Mercury, Venus, Mars, Jupiter, Saturn, Sun, Moon) moved in their heavenly ways.

The outer sphere was Olympos, the home of the Gods. I think I mentioned before that some philosophers spoke of the stars as if they were windows to the lights of heaven.

Uranos, Cosmos, Olympos. Stars, wanders, earth. How simple, how elegant, how symmetrical.

How incorrect!

This three tiered structure will become the model of the universe which gets modified and incorporated into Aristotle's cosmology, and which reemerges sixteen hundred years later in the Scholasic cosmology of St. Thomas and the Church in Dante's "Divine Comedy".

We will soon see how engrained this idea of circular perfection became. So much so that by the time of the astronomer Ptolemy it no longer mattered why the heavens were circular, they just were. There was no other way to even consider the heavenly motions. This preference persisted until the middle of the sixteenth century.

Am I stressing this point enough? Does that give you a clue that it is important?

7. Summary

It is interesting to see how an idea which began as one of mystcial esthetics became locked into a paradigm, which we will call the circular paradigm.

The ancient Greeks "discovered the mind" in the sense that they were the first on record to "think for the sake of thought".

The four schools of PreSocratic Greece had an extremely important influence on the development of scientific ideas right up until the present. Almost all natural philosophers from the fifth century B.C. forward repeated or refined one of the ideas of these early thinkers.

The mysticism of the Pythagoreans led to the idea that the heavens were perfect and therefore must be represented by circles.

The perfection of the circle and the necessity for perfect circular motion in the heavens was very hard to overcome.

It was Plato, a hundred years after Pythagoras who really established the circular paradigm. Based on Plato's Question, which we'll study the next time, the need for uniform circular motion of all heavenly bodies dominated astronomy until Kepler broke out of in the the sixteenth century A.D., more than two thousand years after the Pythagoreans expressed a preference for a perfect circular universe.