Paradoxes Reconsidered

From Cat to Constraint

Jonathan Maram

June 2025

Part 2.5 in the Recursive Observation Series

I. Orientation and Recap

As we move past the greatest hits of quantum paradox, some questions still linger: What does it mean for possibilities to “collapse”? Where does probability come from? And does it matter whether we’re dealing with quantum riddles or game-show puzzles?

In this essay, we take one last tour through the borderlands of paradox, testing our constraint-first logic against a few notorious edge cases. The aim is not just to resolve lingering doubts, but to clarify how structure, probability, and “collapse” all reduce, whether in physics or in life, to nothing more (and nothing less) than the logic of constraint.

Reminder:
As defined in Essay 0, a system is the set of all constraints in play for a given perspective. A structure is whatever satisfies all those constraints at once, often in an abstract “constraint space,” not just in physical geometry. Throughout this essay, “collapse,” “probability,” and “interaction” will be treated strictly as consequences of which constraints are present, not as processes, events, or acts of belief.

II. Schrödinger’s Cat: Constraint, Not Collapse

Few paradoxes have captured the public imagination like Schrödinger’s cat. We’re told that until someone opens the box, the cat is both alive and dead, a superposition “waiting” for observation to decide its fate.

But under the principle of constraint, the cat’s fate is never in limbo. The only real question is: Which constraints define the system under consideration?

• Inside the box, everything that matters: the radioactive decay, the Geiger counter, the cat, the poison, forms a fully entangled web. For any observer within the box, the structure is defined by all the local constraints. There is no superposition, only fact: the cat is either alive or dead, because every possible interaction inside has already imposed its constraint.

• Outside the box, the observer lacks those constraints. Their set of viable structures includes both “cat alive” and “cat dead” outcomes, not because reality is split, but because their constraint set hasn’t yet narrowed the field.

There is no mystical process, no “wavefunction collapse.” The solution set changes precisely when new constraints (such as opening the box) enter your system. The shift is instantaneous, structural, and entirely local to the active constraints, never global, never magical.

The Constraint Connection:
A “superposition” is not a ghostly blend of worlds, but simply the set of all structures compatible with the current constraints. When new information enters, the solution set sharpens. There is no metaphysics, only constraint logic.

III. Monty Hall: Probability, Knowledge, and Constraint

Let’s turn to a puzzle from game shows rather than physics: the infamous Monty Hall problem. “Should you switch doors after Monty reveals a goat?” The answer feels counterintuitive, and for many, the sense of “collapse” is as real as anything in quantum mechanics.

But here’s the key: For Monty, there is no probability. He knows exactly where the car is. The “system,” from his perspective, is completely determined, a single structure fits all his constraints.

For the contestant, though, only some constraints are present. Probability, in this sense, is not a property of the world, but a measure of how unconstrained one’s perspective remains.

• At first pick: The contestant has no information, just three doors, equal chance; the solution set is wide.

• Monty opens a goat door: This is not a “process” or event in the world. It is a new constraint added to the contestant’s system, instantly reshaping the set of viable possibilities.

It feels like probability “collapses”, from three options to two, and then to certainty when the final choice is made. But structurally, nothing has changed in the world itself; only the set of constraints in play for each perspective has shifted.

The Constraint Connection:
Probability is not about belief or ignorance; it is simply the count of structures left after all known constraints are applied. Collapse, whether in Monty Hall or quantum mechanics, is just the local update as new constraints sharpen the solution set.

Note:
Some schools (Bayesianism, QBism) treat probability as “degree of belief.” In our structural view, belief is not required, probability is just what remains viable given the present constraints. This applies even when there is no agent or observer at all.

IV. Dangerous Musings: Paradox Sampler

Not every puzzle that haunts science and philosophy needs a full page to unravel. Sometimes, it is enough to see that the same logic dissolves them all. Here are a few scenarios, each famous in its way, each stripped to its structural core:

If a Tree Falls in the Forest…

Does it make a sound? Is there even a tree?
In our framework, “tree,” “sound,” and “falling” only exist as structures if their constraints are present in the system under consideration. If no system encodes “sound” (no ears, microphones, or pressure sensors), then “sound” is not a realized structure, though pressure waves may still fit the remaining constraints. Similarly, “tree” is just a structure: it persists wherever the relevant constraints define it, and nowhere else. Existence is always local to constraints.

Elitzur–Vaidman Bomb Tester (Interaction-Free Measurement)

Can you detect a bomb so sensitive that a single photon would set it off, without triggering it?
Quantum mechanics says yes, and the method seems paradoxical: you can infer the presence of a “live” bomb (a detector) even when no photon interacts with it. In our terms, the presence of the bomb changes the constraints of the system, altering the set of viable structures (i.e., interference patterns). “Interaction-free” is a misnomer: it’s a shift in what fits the constraints, not a process or a signal.

Quantum Zeno Effect (“A Watched Pot Never Boils”)

Does observing a system constantly prevent it from changing?
If you measure a quantum system repeatedly and rapidly, the probability of it changing state drops toward zero. Structurally, each measurement introduces new constraints, successively shrinking the set of viable structures until only the “unchanged” outcomes remain. The “freezing” is not mystical, but a cumulative effect of constraint accumulation.

Common Theme:

All these paradoxes, classical or quantum, whimsical or technical, trade on a single confusion:
They treat “event,” “object,” or “interaction” as things, rather than as solutions to a set of constraints.
When the constraints are made explicit, the paradox evaporates:

• What exists is whatever fits the constraints present in the system under consideration.

• “Collapse,” “interaction,” and even “existence” are not universal events, but structural shifts, local, perspective-dependent, and governed by constraint.

The Constraint Connection:
Every paradox is a question about structure: What is realized, and under what constraints? The only real magic is forgetting to count the rules.

V. Reflections on Paradox and Perspective

Many paradoxes, e.g. Schrödinger’s cat, Monty Hall, quantum “collapse”, arise not from any mystery in the world, but from shifting perspectives on what constraints are present.

• In quantum mechanics, “collapse” is not a physical jump, but a narrowing of what’s possible when new constraints (measurements) enter.

• In Monty Hall, the “collapse” is simply a change in the contestant’s viable solution set when new information arrives.

There is no difference in kind between these “collapses”, both are updates to the set of structures compatible with the current constraints. Probability measures the slack left in the system; collapse is what happens when slack runs out.

The Constraint Connection:
Every so-called paradox in probability or physics is, at its core, a question of which constraints have been counted, and from whose perspective.

VI. Interaction and the Meaning of Structure

What does it mean to “interact”, or, for that matter, to be a “particle”?

For decades, physics has wrestled with wave/particle duality, as if the world must decide which mask to wear. But, apologies to Bohr, one needn’t find a paradox here, or a deep problem to solve. “Matter,” from our perspective, has always been one underlying thing, with different manifestations depending on the constraints in play.

In the logic of constraint, the question is never whether something is a “wave” or a “particle,” but: What structures are possible given all the current constraints?

• In some settings, the solution set supports wave-like behaviors, i.e. interference, spreading, entanglement.

• In others, the constraints admit only “particle-like” outcomes, e.g. localized impacts, discrete counts, and definite events.

Neither “wave” nor “particle” is fundamental. They are two faces of the same deeper reality: a structure that satisfies all present constraints. Which face you see depends on the questions you ask, or the constraints you impose.

The Constraint Connection:
Wave/particle duality isn’t a riddle; it’s a reminder that structure is always local to the constraints of the system. There is no third thing hiding behind the curtain, only the intersection of rules.

To interact, then, is simply to alter the web of constraints, reshaping what structures can exist. “Particles” are not indivisible bits of stuff; they are persistent solutions to a rich mesh of constraints. The world is not built out of objects, but out of whatever fits.

VII. Foreshadowing ε₂: From Filtering to Anticipation

As we close the loop on paradoxes, the next climb up the epsilon ladder beckons. When structures begin to layer, when they can remember prior constraints, or anticipate future ones, we step into new territory: the birth of anticipation, modeling, and recursive memory.

Gentle warning:
Even words like “anticipate” and “remember” will be recast, not as mental acts, but as features of structures shaped by recursive constraint.

The journey continues: from filtering, to memory, to the first glimmer of law and expectation, not by process or belief, but by the relentless narrowing of what can persist.

See all essays in the Recursive Observation Series