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.
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.
188.8.131.52. ontology: study of being
184.108.40.206. cosmology: study of origin and structure of the universe
220.127.116.11. epistemology: study of knowledge
3.7.1. science: mechanical principles
3.7.2. religion: divine purpose
3.7.3. no supernatural causes, no astrology
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.
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.
18.104.22.168. 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.
22.214.171.124.1. Removed Gods from nature
126.96.36.199.1.1. heavenly objects are solid, material objects, not gods
188.8.131.52.1.2. natural causes: nature is impersonal
184.108.40.206.1.3. around the same time as Hebrew, Persian, Zoroastrians, Buddha did the same thing
220.127.116.11.2. Gods reserved for concern with spiritual welfare of man
18.104.22.168. 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.
22.214.171.124.1. heavens revolve around Polaris
126.96.36.199.2. added fourth element (fire) to constituents of Earth
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.
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.
188.8.131.52.1. all things ultimately become one and the same
184.108.40.206.2. basic quality of reality is change.
220.127.116.11.3. opposed the view that the most basic quality was being
18.104.22.168.1. four elements are basic to reality
22.214.171.124.1. mind is the ordering force in a mechanistic universe
126.96.36.199. Leucippus and Democritus
188.8.131.52.1. nothing exists but atoms and void
184.108.40.206.2. atoms constantly rearrange themselves in accoradnce with mechanical laws
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.
220.127.116.11. masters of rhetorical arugment
18.104.22.168. intensely skeptical
22.214.171.124. argued all sides of a dispute
126.96.36.199. amoral philosophy rejected by Socrates
6.1. Pythagoras (582-500 B.C.)
"Seek truth and beauty together; you will never find them apart"
Pythagoras of Samos
6.1.1. often gets credit for Pythagorean discoveries
6.1.2. difficult to separate his from the cult's
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
6.3.1. 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.
188.8.131.52. 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.
184.108.40.206. 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?
220.127.116.11. Pythagorean theorem
18.104.22.168.1. 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.
22.214.171.124. numbers such as pi and the square root of 2
126.96.36.199. irritated Pythagoreans, so they suppressed knowledge
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.
188.8.131.52. lucky and unlucky numbers
184.108.40.206. magic numbers
220.127.116.11. 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.
6.5.1. 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.
18.104.22.168. 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.
22.214.171.124. 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.
126.96.36.199. the circle is not altered by rotation
188.8.131.52. the circle is the most symmetrical shape
184.108.40.206. the circle has both elegance and simplicity
220.127.116.11. the circle has infinite number of diameters of mirror planes
18.104.22.168. the circle is like a polygon with an infinite number of sides
22.214.171.124. by induction the sphere is the most perfect three dimensional shape