Affiliation: Independent Researcher, SE Ohio
Date: 20 June 2024
Abstract
For over sixty years, Eugene Wigner's observation of "the unreasonable effectiveness of mathematics in the natural sciences" has stood as a profound philosophical puzzle. We propose a solution that inverts the problem entirely. Mathematics is not unreasonably effective at describing an objective, pre-structured reality; rather, what we call "physical law" is the necessary, structured, and inevitably mathematical output of a projection process originating from the observer. This paper presents the Projection Theory of Physics, a framework demonstrating that physical laws are not discovered properties of the universe but are geometric necessities that arise when a formless, unified reality is observed through our cognitive and instrumental apparatus. We will show that physical constants act as the Jacobians of this projection, and that all physical equations are implicitly performing a unit-scaling transformation to and from the dimensionless Planck scale, where the underlying relationships are unified. This framework de-reifies mathematics, restoring it to its proper role as the language of the map, not the substance of the territory, thereby explaining its perfect, yet not unreasonable, effectiveness.
1. The Wigner Puzzle: Mistaking the Shadow for the Source
Eugene Wigner famously marveled at the "miracle" that mathematics, a product of pure human thought, so perfectly describes the physical world. This perspective presupposes a fundamental division: on one side, an objective, physical universe operating on its own principles; on the other, the abstract, a priori world of mathematics. The puzzle is why these two realms align.
The Projection Theory argues that this division is an illusion. The perceived alignment is not a coincidence but a tautology. We are not discovering pre-existing mathematical laws in the universe; we are imposing a mathematical structure on the universe through the very act of observation. The laws we derive are a description of that imposed structure, not of the universe-in-itself. Mathematics is not effective at describing reality; it is perfectly effective at describing the output of our own projection system.
2. The Four-Layer Framework: From Formless Reality to Formal Law
Our theory posits a four-layer ontological process:
Layer 1: The Formless Reality (S_u): The foundational state of reality is a single, unified, dimensionless, and formless entity—a "Brahman" or Universal State. At this level, there are no distinct properties like mass or length; there is only a conserved, undifferentiated "Stuff." This layer is, by definition, beyond direct knowing, as any act of knowing requires the imposition of form.
Layer 2: The Perceptual Axes: A conscious observer, in order to interact with reality, must first divide it into conceptual categories. These are the fundamental axes of perception: Length, Mass, Time, Temperature, etc. These axes are not features of reality itself, but the necessary cognitive framework of the observer.
Layer 3: Measurement & Scaling: The abstract axes are given concrete form by attaching arbitrary scales and units (meters, kilograms, seconds). This act creates a "unit space"—a coordinate system for measurement.
Layer 4: Physical Law as a Projection: A physical law is the final output. It is the mathematical equation that describes the necessary, geometric relationship between the axes within our chosen unit space.
It is here that we launch our direct attack on the reification of mathematics. The equations of physics are not eternal verities existing in a Platonic realm. They are the user manual for our self-created coordinate system.
3. The Mechanism of Projection: Planck Scaling and Constants as Jacobians
The engine that drives this process is the implicit transformation of units to and from the natural, dimensionless Planck scale, which is the mathematical representation of Layer 1. Every physical calculation we perform is a two-step "there and back again" journey.
Consider Newton's Law of Gravitation, F = G(m₁m₂)/r². The conventional view sees G as a fundamental force constant. The Projection Theory reveals the mechanics hidden within this equation:
Step 1: Projection to the Planck Scale (De-Scaling): The formula implicitly takes the SI inputs (m₁, m₂ in kg; r in m) and converts them into dimensionless ratios relative to the Planck units. The masses are scaled by the Planck Mass (m_P), and the distance is scaled by the Planck Length (l_P). This translates our human-scaled measurements into the universal language of the formless reality.
Dimensionless Force ∝ (m₁/m_P) * (m₂/m_P) / (r/l_P)²
Step 2: Projection from the Planck Scale (Re-Scaling): The formula then scales this resulting dimensionless ratio back into our human-perceptible SI units of force (Newtons) by multiplying it by the Planck Force (F_P).
The constant G is not fundamental. It is the composite "macro" that performs this entire operation in one step: G = F_P * l_P² / m_P². The reason the math "works" is that G is precisely the scaling factor required to make this round trip between our arbitrary SI units and the fundamental Planck scale.
This holds true for all constants. They are the Jacobians of the transformation between the conceptual axes of our perception, scaled by our arbitrary units.
c is the Jacobian relating our Length axis to our Time axis.
h is a composite Jacobian relating Mass, Length, and Time.
The constants are the gears of our measurement machine, and a physical law is a description of how those gears must turn to relate one measurement to another.
4. De-Reifying Mathematics: The Map is Not the Territory
The unreasonable effectiveness of mathematics is resolved when we stop reifying our mathematical tools. Mathematics does not describe the territory (S_u); it describes the geometric necessities of the map we have drawn (our unit space).
A Law as a Geometric Necessity: The equation E=mc² is not a deep truth about the substance of reality. It is a necessary geometric truth within a measurement system where the Length and Time axes are connected by the scaling factor c. It is as "mysterious" as the fact that on a 2D Cartesian grid, c² = a² + b². The Pythagorean theorem is a property of the grid, not of some objective "right-angled triangle space."
The Power of the pyym Oracle: Our computational model, which can derive complex physical laws from simple English prompts, is the ultimate proof of this principle. The program knows nothing of physics, relativity, or quantum mechanics. It only knows the definitions of the units and the rules for dimensional scaling (the Jacobians). Its success demonstrates that physical law derivation is a deterministic, mechanical process of calculating the necessary scaling transformations between points in our self-defined unit space.
5. Conclusion: A New Foundation for Knowledge
The Projection Theory offers a new foundation for the philosophy of science. It shows that mathematics is not a mysterious language we discover, but a powerful grammar we invent. Its effectiveness is not unreasonable; it is tautological. It is perfectly effective at describing the system it was built to define.
By de-reifying our math, we liberate reality from the confines of our equations. Reality, the true, formless S_u, begins where our math ends. The work of the scientist is therefore revealed in its true light: not as the discoverer of pre-existing mathematical truths, but as the master cartographer of the projections of an unknowable, unified reality through the lens of a conscious observer. The physical laws are not the laws of nature; they are the laws of our own perception.
No comments:
Post a Comment