Where Physics Stops and Made-Up Ideas Start

How We Tell Things Apart

Every experiment comes down to a simple act: this, not that. Physics is built on telling things apart in a finite way. But the math physics uses — the real numbers — includes things no finite step could ever tell apart. This paper says the edge of real physics matches exactly the computable reals: numbers a step-by-step method can get as close as you want to. The paper builds on earlier work about laws of form, quantum versions of those laws, and ideas about what machines can physically do.

Two Basic Moves

The paper starts with two simple moves from Spencer-Brown's work. The mark says "I pick this, not that." The enclosure draws a line: inside versus outside. Two rules cut these forms down. Calling says doing the same mark twice is just doing it once. Crossing says a mark that goes into a boundary and comes back out cancels to nothing. A later extension gives these moves physical meaning. The mark becomes a measurement outcome. The enclosure becomes the physical step linking states. Calling becomes two same measurements giving one result. Crossing becomes a distinction you can make and unmake by finite steps — so it is not a new physical thing.

Building Numbers Step by Step

Numbers are not just given to us. We build them from the two basic moves through finite steps. One mark gives the number one. Repeating marks gives the counting numbers. Grouping marks by position gives decimal writing. Putting groups inside boundaries gives fractions. Countable sequences of marks and boundaries give the computable reals. But one more step goes too far. The Monna-map projection throws a discrete branching tree onto a smooth continuous space. This step loses information. It makes things — the non-computable reals — that have no matching piece in the tree. They are artifacts of the projection, not things we can build.

Where the Edge Lies

Four different views meet at the same edge. Only computable reals can be approximated to any needed closeness by finite steps. Only computable reals have a finite description that generates them. Finite physical systems cannot hold infinite information. Only computable reals can be reached from the two basic moves through finite uses of the two cutting rules.

The paper offers three claims. First, all physical theories that made testable predictions used only computable real amounts. No physical theory ever needed a non-computable number to give a prediction we could tell apart by measurement. This one is established. Second, the computable and non-computable edge matches the physics and made-up ideas edge. Any formal system using non-computable amounts cannot be tested — so it is not physics by definition. This one is speculative. Third, what cannot be told apart by any finite physical step cannot be physically real. This one is the author's own guess.

Other work backs this up. One researcher proved formally that given a black box making bits of a non-computable number, no step-by-step method can tell it from a computable stand-in at any finite measurement closeness. This is not a technology limit. It is a logic limit. Another researcher argued that physics must focus on step-by-step ways to get numbers we can check by experiment, and that non-constructive proofs in quantum theory are empty for physics. A third gave the formal bridge between computable numbers and physical models.

What Cognitive Fiction Means

The paper defines cognitive fiction: a formal system built from the two basic moves, with no inner contradictions, but unable to be cut down to nothing by any finite sequence of the two rules. Three kinds of systems fall here. Non-computable reals: expressible and consistent, but no finite method can make them, and no finite measurement can tell them from computable stand-ins. Non-measurable sets: expressible in standard set theory, consistent with it, but needing a special axiom with no constructive way to show them. Unfalsifiable formal systems: any theory with no measurement-distinguishable predictions but inner coherence. What is not cognitive fiction: computable reals, which can be cut down by the machine that makes them, and systems with inner contradictions.

How Our Senses Shape What We See

The paper extends earlier work on the Monna-map. It argues that the usual way we measure size in physics — where errors add up in a straight line — is an artifact of human senses. Our senses evolved in roughly three-dimensional space, picking this way of measuring as the natural one. We built math around it, and physics inherited that structure. But the underlying space of physical distinctions branches in a tree shape where errors do not add up. The continuous real line is the shadow of a fundamentally discrete, branching tree. This claim is speculative but testable: if quantum error correction works better with tree-shaped encoding than with straight-line encoding, the idea gains support.

The History of What We Can Tell Apart

The history of number systems tracks what we learned to tell apart. Tallies came first: mark once, twice, and so on. Then ratios: put marks inside boundaries and compare. Then limits: countable sequences of marks and boundaries. Each step was a new way to distinguish. Then came the continuum: not a new way to distinguish, but a projection that makes things we cannot tell apart by any finite step. It was a new way to obscure, not to see.

What This Does and Does Not Show

The paper ends by saying physics is the fixed point of this loop: the set of forms from the two basic moves through finite uses of the two rules. This fixed point is exactly the computable reals. Everything past this closure — non-computable reals, non-measurable sets, untestable systems, creative stories — shares one structure: inner consistency without outer checkability. They are forms that cannot be cut down to nothing by any finite sequence of the two basic laws.

The paper does not show that mathematics is unreal. It says mathematics holds two classes: things reachable by finite steps from basic acts of telling apart, and things not so reachable. It does not prove that non-computable numbers can never enter physics. It says the burden is on those who claim they must: show a measurement that needs a non-computable number to describe its result at finite closeness. Until then, we may treat non-computable things as artifacts of the projection — the shadow a discrete tree casts on a continuous screen. The argument uses math to draw the edge of math. The paper says this self-reference is not a flaw. It is the structure of the thing itself.

Words used here

computable real

a number a step-by-step method can get as close to as you want

cognitive fiction

a consistent formal system that cannot be checked by any finite physical step

mark

the basic act of saying "this, not that"

enclosure

a boundary that separates inside from outside

Learn more in plain language

• Alan TuringSimple English Wikipedia
• Set theorySimple English Wikipedia
• Quantum error correctionWikipedia (English, more technical)
• Model theoryWikipedia (English, more technical)
• Laws of FormWikipedia (English, more technical)
• Computable numberWikipedia (English, more technical)