Impossible But Lawful: 3 When Systems Over-Achieve

In the previous post, we saw that Escher’s impossibilities emerge from perfect local lawfulness. Each element obeys its rules, yet the whole cannot exist as a coherent world. Here, we generalise this pattern beyond art. Some systems fail not through lack, but through excess: they succeed too well locally, and in doing so, foreclose global coherence.

Over-Achievement as Structural Failure

Escher demonstrates a fundamental principle: maximal local compliance can produce global impossibility. In other words:

• Underachievement can cause obvious failure: a misdrawn step, an inconsistent rule, a broken transition.

• Overachievement can also cause failure: when every part obeys its rules flawlessly, the system can become rigid, self-referential, or unintegrable.

Escher’s staircases and multiple gravitational planes are perfect examples. Each local transition is lawful, yet they collectively form an impossible architecture. This is failure by excess, not error.

Real-World Analogues

1. Bureaucracy

   • Regulations can be internally consistent and detailed.

   • Yet a bureaucracy that maximizes compliance and precision at every node often collapses in practice: forms contradict each other, processes loop indefinitely, and no single path allows coherent action.

   • Local efficiency produces global gridlock.

2. Metrics and Data Systems

   • Each indicator or algorithm may be rigorously designed and internally valid.

   • Yet when metrics proliferate and are enforced without regard to integration, the system produces perverse or meaningless outcomes: everyone follows the rules, but the system as a whole stops functioning in a way anyone can inhabit or make sense of.

3. Language and Recursion

   • Sentences and clauses may obey strict grammatical rules.

   • Recursive embedding, however, can produce text that is locally valid but globally incomprehensible.

   • Lawful structures can render the whole unreadable.

The common thread is clear: perfect local success can block global inhabitation. Systems can “over-achieve” in the very dimension that would normally support coherence.

Escher as Paradigm, Not Exception

What distinguishes these examples is structural logic, not aesthetic form. Escher is paradigmatic because he encodes over-achievement in a perceptible, visual form:

• Local rules are transparent and graspable.

• Global impossibility is immediately apprehensible, even if we cannot inhabit it.

In bureaucracy, computation, or language, the same principle operates invisibly, often with consequences that are harder to detect but no less real.

The Ontology of Excess

From a relational perspective:

• Local lawfulness = the adherence of individual nodes and transitions to their rules.

• Global integrability/worldhood = the composition of nodes into an inhabitable whole.

• Over-achievement = the condition in which local lawfulness is so strict or pervasive that global integrability is blocked.

Escher shows this in miniature: a stairway that climbs forever while returning to its starting point, a gravity system that works everywhere yet nowhere simultaneously. These are lawful failures—failures that obey all rules.

Conceptual Caution

We might be tempted to analogise this to Gödel incompleteness, and the temptation is understandable: both involve local validity and global limitation. But caution is warranted. Escherian over-achievement is structural and perceptual, not formal-mathematical. The lesson is philosophical and systemic, not a theorem to be invoked lightly.

Conclusion

Some failures of worldhood arise not from deficiency, but from excess precision and maximal local adherence. Escher demonstrates this elegantly: a world that succeeds everywhere locally yet fails entirely globally.

In the next post, we pivot: having seen the mechanics of lawful failure, we will ask what distinguishes inhabited worlds from uninhabitable ones. By contrasting Escher with fictional worlds—Wonderland, Gormenghast, Prospero’s Books—we will begin to articulate the conditions that make a world liveable.