On Numbers from the Perspective of the Pythagorean School of Mysticism
The purpose of this article is not to discuss the influence of Daoist thought, nor whether Pythagoras himself qualifies as Greek philosophy, which is not what we intend to debate. Rather, we aim to discuss an intuitive feeling or realization of the Dao, exploring these essential thoughts of humanity or early-stage understandings, which are beneficial for comprehending or recognizing the concept of the Dao. Should there be any errors in the text, we welcome criticism and correction.
First, remember these quotes:
“The common people use it daily but do not know it.”
“I do not know its name; I forcibly call it Dao.”
“In learning, one gains day by day; in Dao, one loses day by day. Losing and losing again until reaching non-being.”
“Dao is empty, yet its use may never be exhausted. Profound and deep, it seems to be the ancestor of all things.”
“There is something formless and complete that existed before heaven and earth. Silent and empty, it stands alone and does not change; it circulates everywhere and does not wear out. It can be regarded as the mother of the world.”
Let us first discuss why we are talking about the Pythagorean school. First, because it is a mystical school, a way of observing and understanding the world. In the historical records of Freemasonry, Pythagoras is claimed to be the source of the order, which is worth noting.
In legends, people claim to have seen Pythagoras in different places at the same time. From a Daoist perspective, this seems to have reached the realm of the manifestation of the Yang spirit. However, legends are just legends, and the truth or falsehood within them is not what this article needs to concern itself with.
They had a core idea: all things are numbers. That is, everything in the world is a manifestation of numbers. In the thought of the I Ching (Book of Changes), this has a similar meaning to the 'number' in the concepts of image, principle, and number. However, the initial consciousness of this school defined 'number' narrowly, referring only to integers. For decimals, they could be expressed in the form of fractions.
The most intuitive way to understand the world is that everything has a number. For example, one apple, two apples; this is the manifestation of integers in reality. Inside one apple, how many seeds are there, or how much flesh makes up the apple, these can all be quantified. For instance, from a modern perspective, we can consider that the flesh of an apple is composed of molecules, and the number of molecules can be determined. Even if there is a loss of aroma, assuming a sealed system where all material content is fixed, the number of molecules can be counted. This is something that can be expressed using integers.
What if the units of molecules are different? They can be subdivided further. What if atoms are different? They can be refined into quarks. But no matter how far down we divide, it always returns to a unified unit for integer quantity expression. The ultimate unit is unnamed. In the Pythagorean school, it is believed that this most fundamental expression, consistently, is an expression of numbers. 'All things are numbers' means that everything can be quantified and expressed using integer addition, subtraction, multiplication, and division. For example, 1.2 can be expressed as one and one-fifth; this is the concept of integer expression. (Of course, in the primitive Pythagorean school period, there were no concepts of atoms or quarks; mentioning these is for modern people to facilitate understanding of this mode of thinking.)
These are measurements of the tangible. So how are measurements of the intangible done? For example, music. In this school, a relatively significant discovery was finding some rhythmic relationships.
One day, Pythagoras passed by a blacksmith's shop. The harmonious sound produced by the blacksmith striking iron inspired him. He determined the mathematical relationships of various tones by comparing the different sounds emitted by hammers of different weights. Later, Pythagoras continued experiments on strings, finding the relationships of octaves, fifths, and fourths. Thus, Pythagoras concluded: the harmonious relationship of music is a relationship of numbers.
In terms of intangible music, this school proposed that even intangible things are manifestations of numbers. They went a step further to believe that God manages all things in the world using numbers. A characteristic here is that this school did not care about the flow of energy or the process of material manifestation; they simply believed that all things are numbers, so numbers could express everything about all things.
Because of this, the school was keen on studying the relationships between mathematics and numbers, and then using these relationships for research into magic or divination. For example, amicable numbers: if two numbers a and b, the sum of all factors of a except itself equals b, and the sum of all factors of b except itself equals a, then a and b are called a pair of amicable numbers.
Some people might get a headache seeing such expressions. In fact, this is saying that there are two numbers; if you exclude themselves and sum the factors obtained, they are equal, then they can be called amicable numbers. Amicable numbers express a state where, although they look different on the surface, their internal qualitative effects are the same.
If we analyze the human soul, for example, 220. 220 itself represents the person's explicit consciousness, outward personality traits, or ways of dealing with the world. The decomposition of 220 is 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220. Since 220 represents oneself, we exclude 220, and the remaining numbers constitute the internal soul structure of this 220. The number calculated comprehensively from this internal soul structure is: 1+2+4+5+10+11+20+22+44+55+110=284. Similarly, if 284 is decomposed, excluding 284, and the remaining numbers are added, we get 220. Such numbers are mutually amicable.
According to mystical说法, the projection of the soul of 220 can gather to become 284, and the soul projection of 284 can gather to become 220; they possess perfect mutual affinity.
So what is the application in magic? In the Pythagorean period, everything regarding this was very obscure. We can only look at the usage methods of later people and find that one application is expressing 220 and 284 in a certain way, and then applying them to two people, utilizing the natural affinity relationship of numbers to enable the two to generate affinity.
The Pythagorean school's research on amicable numbers is very meaningful. Generally, Yin and Yang attract each other, while Yang and Yang, Yin and Yin are basically repulsive (like repels like). However, the concept of Yin and Yang combining is based on relativity. In reality, it is sometimes difficult to explain why substances that are both Yang have deep mutual assistance or undergo harmonization phenomena. For example, soldiers on the same battlefield who risk life and death for each other, establishing deep affection, or homosexual phenomena, where both are Yang or both are Yin, how can harmonization occur? Borrowing the analysis of amicable numbers makes it easy to explain the reason from another perspective. This approach, which does not seek affinity between Yang and Yang or Yin and Yin from the angle of contradiction and opposition, has positive reference significance.
In reality, perfect affinity is almost impossible, so one can pursue partial affinity and semi-affinity, and then adjust it mathematically. This involves more in-depth mathematical matters.
The theory and practice of the Pythagorean school's 'all is number' was originally quite successful. However, later a member of the school discovered that if you take a square with a side length of 1, what should be the length of its diagonal? After calculation, it was found that this is an infinite non-repeating decimal, which cannot be expressed precisely using integers.
This caused a huge shock to the theory that all things are numbers (mainly integers), because a number that could not be expressed appeared. The disciples of the Pythagorean school were panicked. To keep it secret, they even killed the person who discovered the problem and threw him into the sea to drown.
In fact, the Pythagorean school was not that fragile. The fragility at the time was due to only recognizing rational numbers. Infinite non-repeating decimals were simply an incomprehensible existence. Why incomprehensible? Today, anyone who understands mathematics knows irrational numbers and considers them common sense, no longer strange. But at that time, it was a very serious problem.
Why is it serious? Actually, this is an ancient problem that has influenced quantum mechanics later on, specifically the measurement problem.
In reality, existing physical objects, such as a 1 cm square cookie in your hand, we are very confident that measuring its length and width with a ruler is no problem. If the ruler is precise enough, we can measure its length and width precisely to millimeters, micrometers, or even smaller units (ignoring things like molecular dispersion at the edges of the cookie).
For such a cookie, we can hold it in our hands, perceive it at any time, and even take a bite. It is clearly right in front of our eyes. But if you are told that there is a data point on it that we can never measure, namely its diagonal length.
No matter how precise the ruler you use, you will find that this measurement is endless. You will never get its true length. More precise measurement just means adding a few more digits after the decimal point. Meanwhile, what is more depressing is that this infinite non-repeating decimal cannot be expressed using integers at all. Unable to be expressed by integers means you cannot make a ruler that can precisely express its length, nor can you make a direct measuring tool that can precisely express it.
The thing right in front of you is actually unquantifiable. The problem of this diagonal later triggered the First Mathematical Crisis, which was ultimately resolved by adding the concept of irrational numbers, i.e., adding the concept of the square root, using the expression √2 to indirectly express that existing number. Actually, this was the first compromise with reality.
If we switch to philosophical thinking, we will find that there are some things in this world that can be perceived, calculated, and reasoned, but cannot be directly observed or measured.
Speaking of this, can you roughly feel it? Something exists, but this thing cannot be directly perceived or measured. However, we still indirectly recognize its existence through other measurements. If we were to use the Dao De Jing to describe a problem similar to √2, we could say: 'I do not know its name; I forcibly call it √2.'
The appearance of irrational numbers actually introduced a way of expressing numbers using expressions. Rational numbers and irrational numbers are collectively called real numbers. Corresponding to real numbers are imaginary numbers, which take a step further on the basis of irrational numbers. Because what is inside the square root are positive integers, and the square of both positive and negative numbers is a positive integer. For example, the square of either positive 1 or negative 1 is positive 1. In a sense, this is asymmetric. Where is its corresponding opposite? Later, imaginary numbers were discovered, i.e., numbers whose squares can be negative.
Imaginary numbers are called 'imaginary' because they cannot be found in reality at all. For example, a cookie cannot have an imaginary length. But in the world of mathematics, they truly exist. If we further consider coordinate systems, they magically connect with circles, i.e., the beautiful Euler's identity.
It can be seen that the development of numbers first evolved from directly measurable numbers, to objectively existing but indirectly expressible numbers, and then to numbers that seem to no longer exist objectively but can be used as indirect calculation tools. That is to say, the development of numbers is moving towards things that are difficult to describe, striving to include concepts of the real world more broadly.
This school had some other strange cognitions that seemed absurd at the time.
For example, they believed that space was filled with ether, and ether would emit sound when in motion. If we understand 'sound' here as广义的噪音 (broad-spectrum noise), we would surprisingly find that they actually predicted the discovery of cosmic noise.
For example, they believed the Earth was round. In their cosmic model, it consisted of ten celestial bodies, three of which were the Sun, Moon, and Earth, combined with the five planets, one anti-Earth, and a central fire ball that remained stationary in the center. Although this model deviated significantly from the real world, it was after all a celestial model.
After the Pythagorean school gradually withdrew from history, some of its doctrines were continued by Plato and passed down through Aristotle. However, although Aristotle had great achievements and contributions, he distorted the original concepts of Pythagoras. As a result, some more important secret arts, as the Pythagorean school dispersed, turned into the inheritance of some secret sects.