In March 2017, I organized a philosophical activity at Shanghai, the topic is about ‘philosophy of science’ by WeiFu An(安维复) from East China Normal University, first time I heard of ‘All is number’ by Pythagorean School:
“All is number” is the motto of the Pythagorean School. This school was founded by the Greek mathematician and philosopher Pythagoras (ca. 580–500 B.C.) Pythagoras was born in Samos, a Greek island near Turkey. He is said to have traveled to Egypt and Babylon. Then he settled in Croton, in southern Italy, where he established his school.
The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.”
— Aristotle, Metaphysics 1–5, c. 350 BC
Many portions of Euclid’s Elements is contributed by Pythagorean School:
Pythagoras (c. 570–495 BC) was probably the source for most of books I and II, books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.(from wikipedia) The propositions contained in Books 7–9 are generally attributed to the school of Pythagoras.(form book note)
At ancient Greek era, there are no Arabic numerals, so in number Theory parts, Euclid use two right angle to express 180 degree, use straight-line combine and truncate to express plus and subtract, use straight-line ration change to express multiple, this method is quite surprise to me, also because the limit of numeral expression, we regard Euclid as father of Geometry but Number theory, though have much content about number theory in his book
Since I knew the important of Mathematics from that philosophical activity, I marked several books include Euclid’s Elements, which create the fundamental of Mathematic axiom system, influenced two thousand years and billions of people, direct prosper Renaissance, Euler, Newton, Non-Euclidean geometry and so on
Euclid’s Elements consists of thirteen books:
- Book 1 outlines the fundamental propositions of plane geometry, including the three cases in which triangles are congruent, various theorems involving parallel lines, the theorem regarding the sum of the angles in a triangle, and the Pythagorean theorem.
- Book 2 is commonly said to deal with “geometric algebra”, since most of the theorems contained within it have simple algebraic interpretations.
- Book 3 investigates circles and their properties, and includes theorems on tangents and inscribed angles.
- Book 4 is concerned with regular polygons inscribed in, and circumscribed around, circles.
- Book 5 develops the arithmetic theory of proportion.
- Book 6 applies the theory of proportion to plane geometry, and contains theorems on similar figures.
- Book 7 deals with elementary number theory: e.g., prime numbers, greatest common denominators, etc.
- Book 8 is concerned with geometric series.
- Book 9 contains various applications of results in the previous two books, and includes theorems on the infinitude of prime numbers, as well as the sum of a geometric series.
- Book 10 attempts to classify incommensurable (i.e., irrational) magnitudes using the so-called “method of exhaustion”, an ancient precursor to integration.
- Book 11 deals with the fundamental propositions of three-dimensional geometry.
- Book 12 calculates the relative volumes of cones, pyramids, cylinders, and spheres using the method of exhaustion.
- Book 13 investigates the five so-called Platonic solids.
