An important principle in the philosophy of dialectics is the transformation of quantity into quality.

That is, a system which undergoes a significant enough “quantitative change” which eventually experience a “qualitative change”, with the appearance of a discrete or discontinuous state change even though the actual process was continuous.

The classic example is that of state changes in water: at a low temperature, water takes the form of solid ice. Raise the temperature a bit and it melts into water. Raise it further and eventually it becomes a gas. Each of these phases – solid, water, gas – have distinct “qualitative” properties even though the passage between them is a continuous one.

The other is the transition from one mode of production to another, say from capitalism to feudalism. Merchants, who existed as a class under feudalism, over centuries acquired enough power in society that the logic of capital accumulation was directing most of production, and then it was capitalism.

How is this possible? It is never really explained in any account I’ve seen. Here is an explanation. In the dialectical-materialist view, reality is composed of processes constantly in flux rather than static things.

Moreover, these processes are all embedded in a unitary reality. As discussed before on this blog, I think a good model is to think of a “material process” as a stochastic process $\{ X_t \}_{t \in \mathbb{R}}$, with state set $X$, equipped with a map $x \colon U \to X$ where $U$ is the state set of a stochastic process representing the universe, $\{ U_t \}_{t \in \mathbb{R}}$.

The map $x$ captures the fact that a “material process” is one which is playing out in “the universe”, and thus in some relation to all other processes.

Now, it is important to note that the “material processes” $(\{X_t\}, x)$ do not exist in the same way as the universal process. In reality, there is only the single process of the entire universe. We carve out pieces of it to model, for a time, as “material processes” $(\{ X_t \}, x)$. These are mental devices, models for us to understand the universe which we cannot grasp directly due to its complexity.

We project (literally, in a mathematical sense) the world onto them, to allow us to understand some aspect of it.

Because these processes are projections of the world and not the world itself, they are imperfect. At any moment in time, they correspond to a greater or lesser degree to the world. Now we are in a position to understand the “transformation of quantity into quality”.

Given a material processes $\mathcal{X} = (\{ X_t \}, x)$, we can define a function $e_\mathcal{X} \colon \mathbb{R} \to [0, 1]$ which measures the degree to which $\mathcal{X}$ faithfully models reality at a moment $t_0$. Intuitively it will measure the divergence between the stochastic processes $\{ X_t \}$ and $\{ x(U_t) \}$ near the time $t_0$, but I won’t give a formal definition here.

Now suppose you have two material processes $\mathcal{X} = (\{ X_t \}, x)$ and $\mathcal{Y} = (\{ Y_t \}, y)$. Think, $\mathcal{X}$ is the “solid model” for the particles of a given quantity of water, in which each particle assumes a fixed position over time, and $\mathcal{Y}$ is the “liquid model”, in which the particles change position according to the laws of fluid dynamics.

Now say at time $0$ we start the universe at $-10$ C and start turning up the temperature on an ice cube. Say it hits $0$ degrees at time $1$. From time $0$, to close to time $1$, $e_\mathcal{X}$ is close to $1$, indicating that the ice is solid, in that its constituent particles’ positions are not moving over time. Near time $1$, it rapidly decreases towards $0$. Conversely with $\mathcal{Y}$ which starts off near $0$, and rapidly climbs to a number close to $1$.

Around $0$ the graphs of the two functions intersect, which is the moment of phase transition or “transformation of quantity into quality.” This is in reality, not a physical event¹, but a mental one, since the two models (“solid” vs “liquid”) exist only in the mind of the observer trying to make sense of the unitary process of reality.