Thales-Pythagoras
Thales
Thales, our first western philosopher, scientist, and mathematician, lived in Greece around the end of the sixth century B.C.E (2600 years ago). He is viewed as the first person to use reason instead of religion to explain natural phenomena, and the first to believe in an underlying, natural, unity within all things. Though Thales left no written record of his achievements, Pythagoras, Plato, and Aristotle all referenced and noted him as an important and essential teacher.
Abstract and Concrete
Thales used correspondence, or the property of similar triangles, to measure the distance of a ship at sea. He measured the shadow it cast on the sand and then interpolated its distance from the shore. Though the triangles are different in size (concrete), they are identical in concept (or abstract relationship). The Egyptians used this method to measure the height of their pyramids. However, they hadn't used it to generalize, abstract, or deduce, solutions to other, larger, natural phenomena (eclipses, for example). Thales showed understanding the relationship within one triangle would accurately predict the underlying relationship in all triangles, totally independent of time (speed) or space (size). Thales proved abstract deduction is the most practical device for understanding anything concrete (form predicts, and produces, substance).
Power of Deduction - Demonstration of a Circle
Thales used unified, rational, repetitive cycles (circles) to predict a solar eclipse. He was the first to do this using observation and deduction only, without referencing an outside god, goddess, or religious system. Predicting a solar eclipse using mathematics and deduction alone, with no telescope, no pictures of the universe, and no accepted body of 'scientific knowledge,' was a phenomenal achievement 2600 years ago (perhaps more phenomenal than our scientific achievements today). Other people may have predicted solar eclipses before Thales, more than 2600 years ago, but we have, or know of, no written record today.
Pythagoras
Pythagoras, 50 years Thales' junior, is best known for the Pythagorean Theorem. This theorem shows the hypotenuse of a right triangle squared is equal to the sum of the length of each of the sides squared. The two squares formed by each side of a right triangle are exactly equal in area to the square formed by the hypotenuse. Or, in other words, two squares must always produce, or are eternally, and infinitely, tied to, the same third square.
Irrational and Rational
By discovering the relationship among three squares (three fours), Pythagoras also, quite by accident, discovered the irrational number. An irrational number cannot be represented by the ratio of two whole (rational) numbers. This was probably Pythagoras' most important achievement. Pythagoras' discovery tied the rational and the irrational together. One cannot, and does not, occur without the other. Pythagoras' discovery of the irrational number also demonstrates solving one problem will (must) always create another.
Symbolic Opposites
Pythagoras' work with sound (harmonics), dividing strings into different lengths (1/2's, 1/3's, 1/4's, etc), created a system of notes and scales which is the basis of our musical system today. An important part of Pythagoras' work with sound was his discovery of the octave, an equal but different tone pitch that is created by dividing a string in two. The octave demonstrates a circle, or a set of equal, and unequal, opposite pairs, repeating and expanding, and contracting, into infinity.
Circles, Triangles, Squares in Everything
Pythagoras viewed all entities as a set of corresponding opposites (pairs, or relationship of two). The opposites he chose were: odd, even; limited, unlimited; one, many; right, left; male, female; still, moving; straight, crooked; light, dark; good, bad; square, oblong. Pythagoras also believed numbers have either a triangular or square 'feeling' quality (relationship of three and/or four). Twos, threes, and fours, or circles, triangles, and squares, from one point of view, are all circles. The parts, no matter how they are organized, always comprise, or result in, and therefore must come from, the same whole.
The Idealist's Unity
Like Thales, Pythagoras was interested in the power of abstraction and generalization, for he, too, was looking for a grand scheme or an underlying unified theory of things. Thales was our first (recorded) realist, and Pythagoras our first (recorded) idealist. Both used circles, triangles and squares, 2600 years ago, to understand and explain the larger ideas underlying reality. Most certainly other men had thought these thoughts before (and since) but we have no written record.
The Circle
Using symbols to abstract and/or deduce reality proves everything's (most basically) a circle (one form is equivalent to every other). The square and triangle are specific versions of the more general circle, and, therefore, all forms are equivalent, in some sense, or in some dimension, to the circle. Thales and Pythagoras, two of our earliest 'thinkers,' our first philosopher/scientists, were demonstrating, without necessarily articulating, Conservation of the Circle.
Conservation of the Circle is the basis for reality.
How to think in a circle...