Absolute Quantitative Numbers
and Variable Numbers

I. The Origin of the Problem

Modern mathematics is built upon rigorous axiomatic systems. Since Kolmogorov, probability theory — as a branch of measure theory — has achieved impeccable formal completeness. But does formal completeness equate to epistemological validity? Does a mathematically self-consistent system necessarily describe the physical world effectively?

The classic “Two Children Problem” in probability theory offers an entry point: given that a family has two children and at least one is a boy, what is the probability that both are boys? The standard answer is 1/3. Yet the sexes of two children are physically independent. Change the method of acquiring the condition — “a randomly selected child turns out to be a boy” — and the answer becomes 1/2. Is it 1/3 or 1/2? No physical anchor can adjudicate; the answer drifts with the conditions.

Pursuing this question leads to a deeper proposition: How should we determine the validity of mathematical concepts?

II. The Core Dichotomy

This dichotomy divides the entire trajectory of mathematical development into two paths: the Path of Discovery — revealing the quantitative structures embedded in the universe; and the Path of Invention — performing deductions within human-constructed condition spaces and symbol systems. The products of the Path of Invention are not “wrong” — they simply lack the standing to claim they describe physical reality. They may be useful tools or ingenious intellectual constructions, but their conclusions do not possess the status of physical truth.

III. Key Examples of Absolute Quantitative Numbers

3.1 Pi (π) — A Natural Topological Relation

Pi is the purest representative among Absolute Quantitative Numbers. It requires no human construction whatsoever — as long as space exists in the universe, π exists. It is the inevitable product of continuous rotational symmetry in space, an irreducible natural topological relation. Even if the symbols 0 through 9 were completely scrambled, the physical quantity corresponding to π could still be realigned and expressed. Because π is not a symbol — it is the metric structure of space itself.

π appears in the normal distribution of probability theory, in the Schrödinger equation of quantum mechanics, and in the distribution of primes in number theory. A number that repeatedly emerges across entirely unrelated fields is the most powerful evidence of its physical-topological irreducibility.

3.2 √2 — A Derived Geometric Relation

The essential difference between √2 and π is this: √2 requires first constructing a right angle, then defining two sides of equal length, and then computing the hypotenuse. It depends on a human-constructed geometric condition — the right isosceles triangle. Although right angles exist in nature, √2 is a derived relation that can be reduced to “the diagonal relationship between equal-length measurements in two orthogonal directions.” It can be further decomposed into more fundamental geometric operations. π cannot be decomposed; √2 can. This is the boundary between L1 and L3.

3.3 The Speed of Light, Planck’s Constant, and Absolute Zero

The speed of light c is the absolute limit of causal propagation in spacetime — it is not composed of other quantities. Planck’s constant h is the smallest quantum of action — it cannot be further divided. Absolute zero 0K is the absolute lower bound of thermodynamic entropy, the ultimate ground state of molecular motion. These quantities share a common characteristic: remove any one of them, and the structure of physical laws ceases to exist. They are not parameters of physical laws — they are the preconditions upon which physical laws depend.

IV. Variable Numbers and Layered Validity

4.1 The “Two Children Problem” — Zero Anchoring

Is it 1/3 or 1/2? No physical anchor can adjudicate. The answer is not converging toward an objective value — it is drifting between different human-constructed premises. This is zero anchoring — no physical operation can lock the result.

4.2 The Weak Anchoring of Probability Theory

Probability theory is not entirely invalid. Predictions of radioactive decay half-lives and quantum mechanical measurement probabilities — these results are strongly anchored, repeatedly verifiable through physical experiments. The probabilistic values relied upon by insurance actuarial science and epidemiological models have no single physical anchor, but under the framework of the Law of Large Numbers they produce repeatable practical value — this is weak anchoring, where the “anchor” is statistical convergence rather than a single physical operation.

4.3 Temperature Scales — Variable Shells Around an Absolute Core

Absolute zero is an absolute quantity (L1); the Celsius zero point is an arbitrarily defined marker (variable). The same physical concept can contain both an absolute quantitative core and a variable shell. The key is identifying which layer is anchored and which is human-defined. The physical world contains absolute quantities; humans use symbols to approximate them. The approximation process introduces error and ambiguity, but the absolute quantity itself does not move.

V. The Physical-Topological Anchoring Depth Model

5.1 Layering Methodology: Physical-Topological Irreducibility

The stratification criterion of the four-layer model is not the traditional physics classification of “dimensioned vs. dimensionless,” nor the number-theoretic classification of “rational vs. irrational,” nor the philosophical classification of “realism vs. anti-realism.” It is an entirely new standard: Physical-Topological Irreducibility — can the mathematical quantity be reduced to more fundamental physical operations or relations?

The speed of light c is irreducible — it simply is the limit of the causal structure of spacetime. π is irreducible — it simply is the rotational symmetry of space. But √2 is reducible — it requires first constructing a right angle and then calculating the diagonal, depending on a prerequisite geometric operation. This criterion arranges all mathematical quantities along a continuous spectrum from “absolutely irreducible” to “entirely dependent on human-defined conditions.”

5.2 The Four-Layer Stratification

5.3 Dynamic Boundaries Between Layers

The boundaries between the four layers are not static. History has witnessed multiple “layer migrations”: the imaginary unit i was once regarded as a purely human invention (variable), until quantum mechanics proved that it anchors the physical structure of wave functions, pulling it from the “Path of Invention” into the “Path of Discovery.” Non-Euclidean geometry was once considered a purely mathematical construction (L3), until general relativity proved that Riemannian geometry anchors the physical reality of curved spacetime, elevating it to an L1-level topological tool.

This layering model contains an implicit dynamic prediction: Constants classified today as L3 or L4 may be “promoted” in the future if deeper physical anchors are discovered; conversely, certain L1 constants currently deemed irreducible may be “demoted” to L3 if a deeper theory derives them. This endows the entire framework with falsifiability.

VI. Falsifiability Analysis

A good theory must be falsifiable. The following is a layer-by-layer analysis of the falsification conditions of this model in terms of both logical alignment and physical alignment.

6.1 Layer L1: Falsification Conditions for Each Constant

6.2 Layer L2: Bidirectional Falsification

Each constant in L2 has two directions of falsification: Upward — if a future theory derives the value of α from first principles, it would be “promoted” from L2 to L3 (a derivable quantity), thereby falsifying its L2 status; Outward — if α is found to take different values in different regions of the universe (astronomical observations have already hinted at this possibility), then even its status as a “constant” would require reassessment. Both directions are falsifiable.

6.3 Layer L3: Falsification Through Upward Promotion

L3’s direction of falsification is upward. If √2 or the golden ratio φ is discovered to anchor an irreducible physical-topological structure (currently unknown), they should be promoted to L1 or L2. The definition of L3 itself presupposes this fluidity.

6.4 Layer L4: Built-In Falsifiability

The very existence of L4 is a statement awaiting falsification — “these parameters are currently irreducible.” Every major breakthrough in physics reduces the number of L4 parameters. The Standard Model expanded from 19 parameters to 26 (after the discovery of neutrino masses), demonstrating that L4 can also expand in the reverse direction.

6.5 Falsifiability of the Framework Itself

The criterion of “irreducibility” itself evolves with theoretical development — Planck’s constant h is considered in some frontier theories to be derivable from more fundamental natural units. If the standard of “irreducibility” is historical rather than absolute, then the boundary between L1 and L3 is dynamic. But this is precisely the advantage — a framework that self-corrects as physics advances is more honest, and more scientific, than one that claims to be eternally immutable.

VII. Three Principles of Physical Anchorism

VIII. Comparison with Existing Philosophies of Mathematics

8.1 Against Mill’s Empiricism

Mill held that all mathematics derives from empirical induction. Physical Anchorism disagrees. π is not a product of induction — it is an intrinsic property of geometric space, a discovery, not an induction. Physical Anchorism acknowledges the existence of a priori objective structures in mathematics, but insists that the validity of these structures must be verifiable through physical operations. Mill cannot explain why π appears in fields unrelated to circles; Physical Anchorism’s explanation is that π anchors not “the circle” but the topological structure of space itself.

8.2 Against Quine’s Indispensability Argument

Quine argued that mathematical entities “exist” because scientific theories cannot do without them. Physical Anchorism does not ask whether mathematical entities “exist” — it asks whether mathematical conclusions are “anchored.” Probability theory is indispensable to insurance actuarial science, but the 1/3 answer to the Two Children Problem does not thereby acquire the status of physical truth. Indispensability is not the same as anchoring.

8.3 Against Lakatos’s Quasi-Empiricism

Lakatos focused on how mathematical knowledge advances through conjectures and refutations. Physical Anchorism addresses a more fundamental question: how to stratify the ontological status of mathematical conclusions. Lakatos’s conjecture-and-refutation model cannot distinguish between “refuted by physical experiment” and “overturned by condition re-specification” — the former is an anchor test, the latter is scope drift. This is precisely the core problem that Physical Anchorism aims to solve.

IX. Corollaries and Applications

9.1 Wrong Anchor, Total Collapse

Ptolemy’s geocentric model — an elaborate mathematical system built on the wrong anchor. Epicycles, deferents, eccentrics: mathematically self-consistent, roughly matching observations. But the entire system was a sandcastle. Copernicus re-anchored to the Sun; Kepler corrected with elliptical orbits; everything became simple and clear. The mathematics didn’t change. The anchor changed. Everything changed.

9.2 Absolute Quantitative Mathematics as the Steel Skeleton of Science

GPS depends on the speed of light. Semiconductor chips depend on Planck’s constant. Space travel depends on the gravitational constant. Nuclear energy depends on E=mc². These absolute quantities are the steel skeleton of the edifice of science: remove any single one, and the entire structure collapses immediately.

9.3 The Two Divergent Paths of Mathematical Development

The dispute between Euclidean and non-Euclidean geometry — which better anchors real physical space. General relativity proved that Riemannian geometry won. The imaginary unit i went from “pure invention” to “physically anchored” through quantum mechanics. Today’s “purely computational mathematics” may well discover its physical anchor in the future, crossing from the Path of Invention to the Path of Discovery.

X. Conclusion

Physical Anchorism provides a complete and falsifiable philosophical framework for mathematics: the core dichotomy divides mathematics into Absolute Quantitative and Variable categories; Physical-Topological Irreducibility provides a single stratification criterion; the four-layer depth model (L1 Irreducible → L2 Underivable → L3 Derivable → L4 Pending Determination) offers an operational classification tool; the Three Principles provide a philosophical manifesto; and the layer-by-layer falsifiability analysis ensures the framework’s scientific quality.

This framework self-corrects as physics advances — layer boundaries are dynamic, and constants can be promoted or demoted. A theory that acknowledges its own capacity for revision is closer to truth than one that claims eternal immutability.

True science is built upon anchors that do not move.

References