Introduction
This is just a fun little attempt at interpreting the Dao De Jing mathematically. In other words, can I make sense of the Dao De Jing by metaphorically equating it to concepts in math? Only one way to find out!
Chapter 1
道可道 非常道
The Dao that can be spoken of is not the ever-constant Dao.
Here, I interpret the Dao as the natural constant $e$. The natural constant is itself transcendental, meaning that it cannot be expressed as the root of any non-zero polynomial with rational coefficients; in other words, “the natural constant that can be ‘spoken of’ is not the true constant.” Furthermore, it plays a vital role in mathematics and physics, frequently appearing in the most fundamental equations across many fields.
$$道 = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n$$
I likewise include here the notion of the exponential family of distributions as very natural sets of probability distributions for modeling the world, in which the natural constant plays a central role. (This will come up again later in the text.)
$$f(x \mid \theta) = h(x) 道^\left( \eta(\theta) T(x) – A(\theta) \right)$$
where $x$ is observed data, $\theta$ is the parameter of the distribution, $h(x)$ is the base measure (a function of $x$ independent of $\theta$), $\eta(\theta)$ is the natural parameter (a function of $\theta$), $T(x)$ is the sufficient statistic, $A(\theta)$ is the log-partition function (ensuring normalization).
名可名 非常名
The name that can be named is not the ever-constant name.
I interpret this as the aphorism “all models are wrong.” We could perhaps represent this as the following:
$$\lim_{n \to \infty} 名_n = 常名 + \epsilon$$
where $名 = f(\theta)$ is some “name” or applied function approximating $常名$, the true function $f^{\ast}(\theta)$ or the impossibly “ever-constant name,” and $\epsilon$ is the error.
無名 天地之始
Nameless, it is the beginning of heaven and earth.
I interpret this both as an extension of the “all models are wrong” aphorism, but also as the property of the exponential function to equal 1 with 0 as the input, from which all other natural numbers can be constructed.
$$道^0 = 1$$
有名 萬物之母
Named, it is the Mother of ten thousand things.
Similarly, the exponential function, or “function of the Dao” $道^x$, returns itself with input 1, implicating that the Dao is the base of the natural logarithm.
$$道^1 = 道$$
This function likewise equals its own derivative, meaning that its rate of change is itself, the function of the Dao.
$$\frac{d}{dx} 道^x = 道^x$$
This function naturally models processes involving continuous growth or decay and forms the basis of the exponential family of distributions.
故常無欲 以觀其妙
Therefore, always desireless, you observe its subtle mysteries.
I interpret this as some set of observations $x$ generated via an unseen parameter $\theta$, a 常名 or “ever-constant name,” that shapes the world. We define a function $f$ as the sampling distribution of $x$, so that $f(x \mid \theta)$ is the probability of $x$ when the underlying parameter is $\theta$. We can call this function 妙 or the “subtle mystery.”
$$\theta \mapsto f(x \mid \theta) = 妙$$
In other words, the “subtle mystery” is a likelihood function.
常有欲 以觀其徼
Always desiring, you observe its manifestations.
We can then define a separate function $g$ of the parameter $\theta$, representing the bounds of our current knowledge about the parameter, i.e., the 名 or “name that can be named.” We can call this function 徼 or the “manifestation.”
$$\theta \mapsto g(\theta) = 徼$$
In other words, the “manifestation” is a prior distribution.
此兩者 同出而異名 同謂之玄
These two arise from the same source but have different names; together they are called the deep mystery.
Finally, we combine them, 妙 or the “subtle mystery” as a likelihood function and 徼 or the “manifestation” as a prior distribution, to derive Bayes’ theorem. Together, they form a posterior distribution, 玄 or the “deep mystery.”
$$\theta \mapsto f(\theta \mid x) = \frac{f(x \mid \theta) g(\theta)}{\int_{\Theta} f(x \mid \vartheta) g(\vartheta) d\vartheta} = 玄$$
$$\text{i.e., } 玄 \propto 妙 \times 徼$$
玄之又玄 衆妙之門
Deep mysteries upon deep mysteries, this is the gateway to all that is subtle and wonderful.
Accordingly, we can use posterior probabilities to update our priors as we gather more data, refining our understanding of the world and the hidden parameters governing it.
$$f(\theta \mid x_1, x_2, \dots, x_t) \propto f(x_t \mid \theta) f(\theta \mid x_1, x_2, \dots, x_{t-1})$$
$$\text{i.e., } 玄_t \propto 妙_t \times 玄_{t-1}$$
Chapter 2
天下皆知美之為美 斯惡已
皆知善之為善 斯不善已
故有無相生 難易相成 長短相較
高下相傾 音聲相和 前後相隨
In the world, all know beauty as beauty, and thus there is ugliness.
All know kindness as kindness, and thus there is unkindness.
Therefore, being and non-being create each other:
Difficult and easy complete each other, long and short contrast each other,
High and low lean on each other, sound and syllable harmonize with each other,
Before and after follow each other.
I interpret this as simply the idea of complementation in set theory. If you define a subset $A$ of a universal set $U$, then the complement of $A$ naturally forms another subset of $U$.
$$A^C = \{ x \in U | x \notin A \}$$
$$A = \{ x \in U | x \notin A^C \}$$
In other words, when you define $A$, you are implicitly partitioning U into two parts: elements in $A$ and elements not in $A$. Once we designate $A$, we inevitably create $A^C$, e.g., naming something as “beauty” automatically induces “ugliness,” etc. This is an inevitability of categorization in making sense of the world, that is, in attempting to name the “ever-constant name” which cannot be named. This can again be interpreted as connected to the “all models are wrong” aphorism.
是以聖人處無為之事 行不言之教
萬物作焉而不辭 生而不有
為而不恃 功成而弗居
夫唯弗居 是以不去
Therefore, the Sage lives without striving and teaches without words.
The ten thousand things arise and none are neglected. Creating without possessing;
Acting without expectation for results; accomplishing tasks without dwelling on achievement.
Only when there is no dwelling, do the results last.
I interpret this as the Sage abiding in the complement of the universal set, that is, the empty set, denoting the concept of Wu Wei (無為).
$$U^C = \emptyset$$
The Sage does not dogmatically take sides in any subdivided region of $U$, identifying with none of the “partial distinctions.” In other words, because the Sage does not partition $U$, they can embrace $U$ holistically without partiality, remaining open to every schema of conceptualization and recognizing their usefulness, the empty set itself being a subset of every set.
$$\emptyset \subseteq A$$
To extend this interpretation, the Sage recognizes that all models follow from a set of assumptions and are therefore wrong, in that, real world phenomena never actually follow the exact assumptions, and all conceptualizations of the world leading to complementation are, in fact, not real. In other words, seeking a “correct” or “perfect” model is not the answer, nor is the seeking of an answer a valid objective; rather, the Sage learns and refines their knowledge in a neverending journey undertaken solely for learning’s sake.