Journey to Enlightenment: The Pythagorean School (Part Two)
The Pythagorean doctrine that all is number long held a dominant place in mathematics; its perfect union of theory and practice offered humanity a new vantage from which to know the world. Yet this seemingly unbreakable system once met an unprecedented challenge.
Although abstract, imaginary numbers are not beyond grasp. In fact they have wide applications in daily life. In electrical engineering, for example, the phase difference between voltage and current can be expressed with complex numbers, and the imaginary part supplies the phase information. In quantum mechanics the wave function is often expressed with complex numbers, and the imaginary part plays a key role.
This school held some distinctive views that to people of the time seemed absurd. They firmly believed that a substance called "aether" pervaded space, and that when aether moved it emitted a sound. If we try to understand this sound in a broad sense, we find to our surprise that the school had in effect predicted what modern science calls cosmic noise.
In addition they insisted that the earth is round, and built a distinctive cosmic model. In this model the whole cosmos was composed of ten heavenly bodies: sun, moon, and earth; the five planets; a counter-earth; and a stationary ball of fire at the center. Though this model differs markedly from the real world, it was at least an attempt to understand the cosmos as a system of heavenly bodies.
As time passed the Pythagorean school gradually exited the stage of history. Yet part of their teaching was inherited by Plato and passed on to Aristotle. Although Aristotle achieved enormous scholarly success and made great contributions, he distorted Pythagoras' original ideas. Therefore some of the more important esoteric teachings, with the decline of the Pythagorean school, passed instead into certain secret sects, where they were preserved.
A scholar of the Pythagorean school, while studying the relation between the diagonal of a square and its side, struck a startling fact. Asked what the diagonal of a square with side 1 ought to be, he found the value to be an infinite non-repeating decimal. This meant it could not be expressed exactly as an integer, nor by any ordinary rational number.
For the doctrine that all things are number, this discovery was a massive blow. In Pythagorean theory all things were composed of integers, and every relation and law could be revealed and described in integers. Yet this newly found irrational number overturned the idea completely — like a chasm that could not be filled, lying across the theoretical body of numerism.
The news caused an uproar within the school. Disciples were thrown into panic; they could not accept this subversive conclusion. To defend the school's authority and faith, some even took extreme measures, secretly executing the scholar who had found the problem and casting him into the sea to drown.
Yet the course of history cannot be stopped. Although the Pythagorean school once tried to cover up the fact, the existence of irrational numbers was eventually recognized and accepted by more and more people. As time went on, people came gradually to see the value and significance of irrational numbers, and they became an indispensable part of mathematics.
Yet here there actually lies an ancient problem that, one may say, was never resolved at the root — a problem that continued to influence quantum mechanics long afterward.
In our real world a physical object is like that square centimeter of biscuit in your hand — within reach. With a ruler we are confident we can measure its length and width precisely, whether in millimeters, micrometers, or even smaller units (setting aside for the moment subtle details such as molecules escaping from the biscuit's edge).
Yet this biscuit — an object we can at any moment feel, see, and bite — hides a mystery we can never touch: the length of its diagonal.
However fine the ruler, the process of measuring the diagonal never ends. The true length is a fog; no matter how we strive, the digits after the decimal point are an unending game. More frustrating still, this non-repeating decimal simply cannot be expressed as a definite number.
This means we cannot make a ruler precise enough to express its length, nor design an instrument that measures it exactly. Faced with this biscuit, we have met the predicament of quantification — a predicament that even provoked the first crisis in the history of mathematics. In the end the introduction of the irrational number, and especially the expression √2, became our symbol of bowing to reality, indirectly conveying that value we cannot reach.
If we examine this philosophically, we realize that many things in the world can be sensed, calculated, and reasoned about, yet cannot be directly observed or measured. They are like invisible dimensions, challenging the bounds of our knowledge and stirring deep reflection on the true face of the world.
At this point, do you feel something? It is like an invisible existence — not directly perceivable, not precisely measurable — yet we still come to know it through other means, in an indirect way. If we used the language of the Daodejing to describe an existence such as the square root of two, it would be: "I do not know its name; I style it 'square root of two.'"
The appearance of irrational numbers in fact opened a new mode of number: number expressed by an expression. Rational and irrational numbers together are called the real numbers; corresponding to the real numbers there are also the imaginary numbers. Imaginary numbers go a step beyond irrationals, because what stands under the radical is always a positive integer, and the square of either a positive or a negative number is a positive integer. The square of +1 and the square of −1 are both +1 — in a sense this is asymmetrical. Where, then, is its opposite? Imaginary numbers were later discovered: numbers whose squares can be negative.
Imaginary numbers, this seemingly mysterious mathematical concept, conceal profound wisdom within. Why call them "imaginary"? The name comes from their elusive nature in the real world. In daily life the length, weight, and time of objects are all substantial and can be measured with concrete numbers. Imaginary numbers break this convention: no corresponding entity can be found in real life. The length of a biscuit, for example, is a real number, and we can measure it precisely. Imaginary numbers are utterly helpless in describing the length of a biscuit, because they do not exist in reality as a concrete number.
Yet in the mathematical world imaginary numbers are an indispensable part. Through deep study mathematicians discovered that imaginary numbers have unique advantages in solving certain complex problems — in certain physics problems and in circuit analysis, for example, they provide convenient mathematical models that let problems be solved smoothly.
Moreover, imaginary numbers are inseparable from complex numbers; together they form the coordinate system of the complex plane. In that system imaginary and real numbers work together to form a complete number line. We therefore cannot simply dismiss imaginary numbers as a useless mathematical concept. On the contrary, they play an important role in mathematics and engineering. By understanding their properties and applications more deeply, we can better grasp the nature of mathematics and use imaginary numbers to solve practical problems.
Imaginary numbers also have important applications in signal processing, control systems, and other fields. In signal processing they can be used to analyze the frequency components of a signal, helping us better understand its characteristics. In control systems they describe system stability and help engineers design more reliable and effective controllers.
More wondrous still, imaginary numbers form a marvelous connection with the circle in geometry. Euler's identity is a landmark in the history of mathematics: it binds trigonometric functions, imaginary numbers, and the circle tightly together. Through Euler's identity we can convert between imaginary numbers and geometric figures, presenting the beauty of mathematics in visual form. The discovery of this identity not only opened new fields of mathematical research but also gave us a deeper understanding of imaginary numbers.
As we can see, the development of number is a process of continuous exploration and expansion. It grew from numbers that can be measured directly, to numbers that exist objectively but must be expressed indirectly, and on to numbers that seem not to exist at all — yet can serve as tools of indirect calculation.
In other words, the development of the concept of "number" itself has moved steadily toward what is harder and harder to describe, because only in this way can it encompass the real world more broadly.