In today’s post, I want to explore an observation about how we make distinctions and what this reveals about the structure of our thinking. I am inspired by the ideas in Spencer-Brown’s “Laws of Form” and broader themes in cybernetics about how observers construct meaning.

The starting point is simple. When we make a distinction, we create a boundary that separates what is inside from what is outside. Spencer-Brown formalized this with his notation of the Mark, showing how any act of indication simultaneously creates both the indicated and the non-indicated. This is shown below:

As we look closer, things get more interesting.

The Basic Operation of Distinction-Making:

When I make one distinction to mark “A,” I create two states. There is A (the marked state) and not-A (the unmarked state). This seems straightforward enough. We can depict this as below:

(A) not-A

Spencer-Brown showed that this basic operation has interesting algebraic properties. The unmarked state is not simply absence or void. It is the enabling condition that gives the marked state its meaning. Without the background of the unmarked, the mark itself would be meaningless.

This relationship between marked and unmarked is fundamental to how meaning emerges. The marked state exists only in relation to what it excludes.

We can take this further. Consider what happens when we make multiple distinctions. If I distinguish both A and B within the same unmarked space, Spencer-Brown’s notation shows this as ((A)(B)).

This actually creates three categories, not four. There is A, there is B, and there is everything else that is neither A nor B. We can represent this as ((A)(B))X, where X represents the remainder of the unmarked space.

In Spencer-Brown’s system, A and B are mutually exclusive by the nature of how the distinctions are made. They are separate marks within the same unmarked background, not overlapping regions as in classical set theory.

This gives us the pattern that n distinctions create n+1 categories. Three distinctions would create four categories, four distinctions would create five, and so on.

The Persistent Unmarked State:

What interests me most is how something remains unmarked regardless of how many distinctions we make. No matter how extensively we mark up our space with categories and boundaries, there is always an unmarked background that enables those markings to have meaning.

This unmarked background is not just everything else we have not thought of yet. It is the condition that makes thinking and categorizing possible in the first place. When we argue about categories like hot versus cold, we often treat these as exhaustive alternatives, often as dichotomies. But there is always the unmarked space that contains the ideas of moderate temperatures, context-dependent judgments, and the framework of assumptions that makes temperature distinctions seem natural and meaningful.

Connection to Self-Reference Problems:

This observation about the persistent unmarked state connects to well-known problems in formal systems, though the connection is analogical rather than mathematically precise.

Russell discovered that attempts to create completely comprehensive sets run into contradictions when they try to include themselves. The set of all sets that do not contain themselves creates a paradox when we ask whether it contains itself. Gödel showed that formal systems strong enough to express arithmetic cannot prove their own consistency without appealing to principles outside the system.

These results point to a general pattern. Complete self-inclusion appears to be impossible. There is always something outside the system that the system requires but cannot fully capture within its own terms.

The unmarked state in Spencer-Brown’s system suggests a similar limitation. The observer making distinctions cannot fully mark their own position as observer. There is always something unmarked that enables the marking process itself.

Implications for How We Think:

This has practical implications for how we approach knowledge and categories. It suggests epistemic humility. If our categorical frameworks always rest on unmarked assumptions and background conditions, then we should hold our categories lightly. They are tools for navigating experience, not mirrors of an independent reality.

In addition, it points toward the value of examining our own distinction-making processes. When we notice ourselves categorizing something, we can ask what remains unmarked in that process. What assumptions are we making? What alternatives are we not seeing?

And it also suggests why different observers can legitimately make different distinctions. The unmarked background that enables distinctions varies with the observer’s purposes, biological capabilities, and cultural context. The distinctions we make depend entirely on the purpose(s) of the observer. Different observers make different distinctions. This viewpoint supports the idea of pluralism.

Final Words:

Spencer-Brown’s insight about the marked and unmarked states reveals something fundamental about the structure of thought itself. Every act of indication creates both what it marks and what it leaves unmarked. The unmarked is not simply absence but the enabling condition for meaning.

This leads to both epistemic humility and intellectual pluralism. Different ways of making distinctions reveal different aspects of complex situations. No single framework captures everything. The wisdom lies in working skillfully with multiple perspectives while recognizing what each obscures.

Most importantly, the unmarked space always exceeds our attempts to mark it completely. As Heinz von Foerster observed, “Objectivity is a subject’s delusion that observing can be done without him.” The observer making distinctions cannot fully step outside their own process of observation.

This is not a limitation to overcome but a fundamental feature of how minds engage with complexity. “The environment as we perceive it is our invention,” von Foerster also noted, pointing to the active role we play in constructing the realities we inhabit.

Understanding this process of distinction-making is essential for navigating complexity with wisdom. Think about how this affects the popular frameworks with neat triads, 2×2 matrices, etc. that promise to carve up the world into manageable categories. Every one of these frameworks commits the same fundamental error. They erase the observer who created the distinctions and ignore the vast unmarked space of assumptions, context, and excluded possibilities that makes their tidy categories seem meaningful.

The unmarked state reminds us that thinking is always an ongoing process within contexts we can never fully transcend. This recognition opens us to continued learning and the possibility of seeing familiar situations in new ways.

Stay Curious and Always Keep on Learning.

If you found value in this exploration of thinking and categories, check out my latest book on the Toyota Production System, Connecting the Dots…

The soft copy is available here. And the hard copy is available here.