PSYC85H3 Chapter Notes - Chapter 2: Pythagoreanism, Pythagorean Theorem, Gnomon
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C85: History of Psychology
Chapter 2: Touchstones - The Origins of Psychological Thought
Pythagoras has often been characterized as having had a peerless influence on the course of western thought.
He is often described as having miraculous powers and was even referred to as the incarnation of Apollo, the Greek god
most associated with otherworldly wisdom.
He was thought to have studied in Egypt and to have been initiated into the mathematical and religious secrets of the
It is the myth of Pythagoras that has had such a great influence on the way many people in the West think about the world.
One of the central features of the Pythagorean myth is that he founded a semi-secret society in Italy
•The members were supposed to have been able to attune themselves to the harmony that ordered the
•harmony is central to his thought
•It is closely linked to the notion that the structure of mathematics is the structure of reality.
It is said that Pythagoras was astounded by his discovery of mathematical relationships in everyday experiences such as
listening to music.
•He felt he had an insight into the unity of the cosmos.
•Believed that the phenomena of our experience are united mathematically.
Pythagoreans were convinced of the virtue of unity and all that has concept implies.
•Unity implies wholeness and oneness, a simplicity that is perfect.
there are surprising similarities in ancient world views, ranging from the Pythagorean cosmology to the yin/yang
philosophy of Chinese Taoism
Cosmology- the study of the structure and development of the universe
Many see the universe as initially a unity that becomes differentiated into pairs of opposites
•These opposites are then reunited, or harmonized to generate the various forms of life we witness.
In contemporary psychology, the process of ontogeny (individual development) is sometimes understood in a similar way.
Several features of the Pythagorean world view that are elaborated in a distinctive way are:
a) the nature of opposites
b) the importance of number in regulating phenomena
The Pythagorean Opposites
believed that some pairs of opposites were particularly useful for describing our experience.
The most important is limit vs unlimited.
•This distinction is regarded as crucial to the process of creation.
•Opposites arise out of an original unity.
•The integration of opposites produces the limited.
•We do not experience things that are unlimited; everything we experience has a limit.
•The union of the unlimited and limited produces the world we experience.
i.e. as I shovel the snow, it may seem never endless and limitless.
However, as Johann Kepler observed if you carefully examine particles of snow under a magnifying glass, you will see
that all snowflakes have ‘six corners’, each snow crystal demonstrates a mathematical structure. In the same way in our
everyday experiences, there is to an extent some sort of limit.
Central to Pythagorean doctrine is the concept of ‘proportion’:
•When opposites are mixed in the right proportion, there is a harmonious outcome, a union of opposites
that achieves the unity and integrity we value.
•Union of opposites- 1. A harmonious outcome that achieves balance between opposing forces. 2. The
balancing of opposing internal tendencies; this is a person’s overriding goal (Jung).
•The psyche (soul), seeks precisely such a harmony
•The soul can resonate to the mythic music of the spheres that surrounds us always
•We seldom hear the ‘music’ because we are like ‘bronze smiths who have become so accustomed to the
noise of their forge that they grow oblivious of it.’
•We overlook the beauty that surrounds us, not having reached the state where we can mimic that which
would make us most happy.
•A Pythagorean viewpoint is that the things we overlook are those harmonies developing out of the
interplay of opposites.
While limit and unlimited are one of the most important of the Pythagorean opposites, there are others such as good vs evil,
light vs dark, unity vs disunity....
•The first member is positive, the second is negative.
(figure 2.3 of the book, page 21)
gnomon refers to a carpenters square, but figueratively it means the addition or
subtraction of one figure of the same shape.
We begin with unity (one pebble), and as we add successive gnomons, odd numbers
of pebbles i.e. 3,5,7...are enclosed, the figure is always a square.
The successive gnomons enclose even nmbers, and the figure tends to no definite shape, but rather becomes increasingly
The basic contrast between the two figures is that the first has a definite limit, and thus the properties of odd, square and
unity, while the second is unlimited, being also plural, oblong and even.
These examples show that the Pythagorean opposites were intended to refer to specific occurences. Moreover, they denote
mathematically precise relationships, illustrating the Pythagorean belief that everything that happens can be described
The Pythagoreans viewed numbers as underlying all phenomena; numbers are responsible for uniting the opposites in a
•Number was a property for everything.
•the famous theorem of Pythagoras is a demonstration of unvariant proportion; the square on the
hypotenuse of a right angled triangled traingle ALWAYS equaly the sum of the squares on the other two sides.
Pythagoras, Plato, and the Problem of the Irrational
(figure 2.5, page 23)
the pythagorean theorem itself led to some difficulty later on:
•for example, consider a right angle triangle with sides of 1 unit each, the sum of
squares the sides 1^+1^ equals 2.
•This means tha the square on the hypotenuse must equal 2, and that the length of the hypotenuse is the
square root of 2, which is a decimal...this is not expressed as a single number, it is an irrational number.
•Thus, the pythagoreans came upon the irrational as an unavoidable aspect of reality
The problem of the irrational became a preoccupation for the Greek mathematicians.
Many attempts were made to solve the problem of the irrational; this was done for the golden section- a famous proportion
in the history of Western thought.
The golden section can be obtained by dividing a line into two segments such that the smaller is to the larger as the larger is
to the whole line.
Chapter 2: touchstones - the origins of psychological thought. Pythagoras has often been characterized as having had a peerless influence on the course of western thought. He is often described as having miraculous powers and was even referred to as the incarnation of apollo, the greek god most associated with otherworldly wisdom. He was thought to have studied in egypt and to have been initiated into the mathematical and religious secrets of the priesthood there. It is the myth of pythagoras that has had such a great influence on the way many people in the west think about the world. One of the central features of the pythagorean myth is that he founded a semi-secret society in italy. The members were supposed to have been able to attune themselves to the harmony that ordered the universe. harmony is central to his thought. It is closely linked to the notion that the structure of mathematics is the structure of reality.