General Preface

The great virtue of Heaven and Earth is generation; the great treasure of the sage is his position. Since the high antiquity when Fuxi gazed upward to observe the celestial and looked downward to examine the terrestrial, drawing trigrams and establishing images, the ancestors of Chinese civilization have used the methods of "image and number" (xiangshu) to penetrate the bond between Heaven and humanity. The Way thereof is vast and profound; its arts have branched and multiplied endlessly. As for what later ages call "Bazi" (the Eight Characters) and "Qimen Dunjia" (Mysterious Gates and Hidden Shield), though they differ in form and diverge in technique, both are rooted in the great system of Heavenly Stems, Earthly Branches, Yin-Yang, and the Five Phases. Their headwaters may be traced to the pre-Qin era, indeed to the very dawn when sage-kings created calendars and charted the movements of the heavens.

In the present age, those who debate the arts of calculation often quarrel: between Bazi and Qimen Dunjia, which carries the greater volume of information$1 This question may appear merely technical, yet it touches upon the deep structure of mathematics and the fundamental principles of metaphysics. For "information volume" is not a concept invented by moderns. The Xici Zhuan (Appended Statements, the Great Commentary of the Yijing) declares: "The Yi as a book is vast and complete: it contains the Way of Heaven, the Way of Humanity, and the Way of Earth. It combines the Three Powers and doubles them, hence the number six." This notion of "vast and complete" (guangda xibei) is the ultimate expression of information-carrying capacity. The sages established hexagrams and observed images with the aim of "encompassing the Way of Heaven and Earth" (milun tiandi zhi dao) -- how immense, then, is the information contained in the hexagram images! The arts of Stems-and-Branches and Dunjia alike were fashioned from this root of "encompassing"; the magnitude of their information content is determined by their respective mathematical architectures and layers of symbolic meaning.

The purpose of this essay is to trace matters back to their origins, proceeding from pre-Qin texts and the heritage of high antiquity, with mathematical structure as the warp and metaphysical principle as the weft, to conduct a systematic comparative study of the information volume of Bazi and Qimen Dunjia, and finally to attempt a unified conclusion combining mathematics and metaphysics. All citations herein are drawn from pre-Qin and early Han texts, striving for well-grounded evidence and well-supported reasoning.

What is "information volume"$2 This is not a concept created solely by moderns. The Xici says: "Writing cannot exhaust speech; speech cannot exhaust meaning." It further says: "The sages established images to exhaust meaning, set up hexagrams to exhaust the true and the false, and appended statements to exhaust their words." Between speech and meaning there is an unbridgeable gap of "not exhausting," and thus "images" and "hexagrams" are needed to fill it -- this is the classical expression of information encoding and information capacity. How much "meaning" a divinatory system can carry, how much of "the true and the false" it can express, how many "incipient moments" of change it can encompass -- therein lies its information volume.

We must inquire:

By what power can Stems and Branches record the cycles of Heaven and Earth$3 By what power can Dunjia unfold the stratagems of war$4 Whence arises this "power"$5 And how shall we measure its magnitude$6

This is the central thesis of the present essay.