Core Principle
This axiom establishes three fundamental properties of space, which is one of the three constituents of physical reality identified in Axiom 1:
- Three-dimensional — Space has exactly three dimensions of extension (length, width, height) that correspond to our direct sensory experience
- Infinite in extent — Space extends without limit in all directions; there is no boundary or edge to space
- Infinitely divisible — Space is continuous and can be subdivided without limit; there is no smallest unit of space
These properties arise from basic reasoning about physical extension and our direct experience of reality. Space is the infinite, continuous container within which all matter exists and moves.
What This Axiom Rejects
Axiom 2 stands in direct opposition to three major pillars of modern theoretical physics, each of which contradicts either our direct experience, basic reasoning, or both.
1. Higher Dimensions (String Theory and Beyond)
Conventional View:
Modern theoretical physics, particularly string theory and M-theory, proposes that space has more than three dimensions:
- String theory requires 10 or 11 dimensions
- These "extra dimensions" are supposedly "compactified" or "curled up" at sub-microscopic scales
- Some formulations propose even more dimensions
- These dimensions are purely mathematical constructs with no observational evidence
AAM Position:
The AAM categorically rejects all dimensions beyond the three we directly experience:
- Direct Experience: We observe exactly three dimensions of spatial extension—no more, no fewer
- Mathematical Abstraction vs. Physical Reality: Higher dimensions are mathematical conveniences that arose from manipulating equations, not from observing nature
- Occam's Razor: Adding invisible, unobservable dimensions to make mathematical models work is unnecessary complexity
- Sensory Evidence: All measuring instruments, all observations, all experiments occur in three-dimensional space
Why the Confusion Arose:
- Theoretical physicists found they could make certain equations work by adding mathematical dimensions
- Time is sometimes mistakenly called a "fourth dimension" (it is not - see Axiom 9)
- Success in solving abstract mathematical problems was mistaken for insight into physical reality
- The prestige of quantum mechanics and relativity gave credibility to increasingly abstract formulations
The AAM View:
Space has exactly three dimensions because extension itself is inherently three-dimensional. Any "dimension" beyond these three is either:
- A mathematical parameter in an equation (not a physical dimension)
- A confused conflation of time with spatial dimensions
- An abstract concept that has no correspondence to physical reality
2. Finite Space and Big Bang Cosmology
Conventional View:
Modern cosmology, dominated by Big Bang theory, proposes:
- The universe began from a singularity approximately 13.8 billion years ago
- Space itself is expanding, carrying galaxies apart
- Space may be finite but unbounded (like the surface of a sphere)
- There may be an "edge" to space, or space may curve back on itself
- Before the Big Bang, space did not exist
AAM Position:
The AAM rejects the concept of finite space entirely. Space is infinite in extent—it goes on forever in all directions.
Why Infinite Space Makes Sense:
- No Logical Boundary: What would be at the "edge" of space? More space. Any proposed boundary leads to logical contradiction.
- Conservation Principles: If space were finite and expanding, into what would it expand? The concept of "expanding space" requires pre-existing space to expand into.
- Observational Evidence Misinterpreted:
- Redshift could be explained by light losing energy over vast distances (tired light)
- Cosmic microwave background could be residual radiation from ongoing processes, not a primordial remnant
- Galaxy recession could result from local motions in infinite space, not universal expansion
- Philosophical Problems with Creation:
- The Big Bang requires something from nothing - a logical impossibility
- It violates causality by proposing an uncaused first cause
- It anthropomorphizes the universe by giving it a "birth"
- Infinity is Necessary: As Axiom 5 establishes, there is an infinite amount of matter in the universe. Infinite matter requires infinite space to contain it.
Why the Big Bang Theory Arose:
- Hubble's redshift observations were immediately interpreted as expansion
- Einstein's equations seemed to require either expansion or contraction
- The steady-state alternative fell out of favor
- Theological implications (a "creation event") appealed to many scientists
- Once established as orthodoxy, contradictory evidence was dismissed or reinterpreted
The AAM View:
Space has always existed and will always exist. It is the infinite, unchanging container within which matter moves. The universe has no beginning, no end, no boundary, and no edge. What we observe are local phenomena within an infinite space, not the "expansion of space itself."
3. Discrete/Quantized Space
Conventional View:
Several modern theories propose that space is fundamentally discrete rather than continuous:
Planck Length as Fundamental Limit:
- Max Planck calculated a length (~10⁻³⁵ meters) below which quantum effects supposedly dominate
- Many physicists treat this as a fundamental "pixel" of space
- Below this scale, space is thought to lose meaning or become quantized
Stephen Wolfram's Computational Universe:
- Proposes the universe is a cellular automaton running on discrete computational rules
- Space consists of discrete "nodes" in a computational network
- Physical laws emerge from simple computational rules acting on these nodes
Loop Quantum Gravity:
- Proposes space has a discrete "atomic" structure
- "Space atoms" have a minimum volume (the Planck volume)
- Space is woven from discrete loops forming a network or "spin foam"
AAM Position:
The AAM categorically rejects all forms of discrete space. Space is continuous and infinitely divisible—there is no smallest unit of space.
Why Continuous Space Makes Sense:
- Direct Experience: We observe smooth, continuous motion, not jerky jumps between discrete positions
- Mathematical Convenience Mistaken for Reality: The Planck length is where certain equations break down, not where space itself becomes granular
- Infinite Divisibility: If space had a smallest unit, we could ask: "What's between two adjacent units?" The answer must be: more space. Any "smallest unit" can conceptually be subdivided.
- Motion Requires Continuity: Smooth motion through space requires that space be continuous. Discrete space would require matter to "teleport" between positions, violating basic mechanics.
- Arbitrary Limits: The Planck length is a quantum mechanical calculation, not an observed phenomenon. Using it as a fundamental limit is circular reasoning - it assumes quantum mechanics is fundamental, which the AAM challenges.
- Computational Metaphor: Wolfram's approach treats the universe as a computer program - but this is metaphor, not physics. The universe doesn't "compute" - it simply is. Matter moves through space according to mechanical principles, not computational rules.
Why Discrete Space Theories Arose:
- Quantum mechanics suggested "quantization" of energy levels, leading to speculation about quantized space
- Computer simulations became powerful, leading to computational metaphors
- Mathematical convenience: discrete systems are easier to model computationally
- Difficulty resolving infinities in quantum field theory led to speculation about fundamental length scales
- Wolfram's success with cellular automata led him to generalize this to physics
The AAM View:
Space is infinitely divisible - a continuum that can be subdivided without limit. The Planck length, computational nodes, and space atoms are mathematical constructs that have no physical reality. Just as Axiom 3 will establish that matter is infinitely divisible, space too admits no fundamental limit to its subdivision. This infinite divisibility is essential for the self-similar structure of the universe across all similarity levels.
Supporting Reasoning
Three Dimensions: Why Not More, Why Not Fewer?
Two Dimensions Would Be Insufficient:
- No way to have crossing paths that don't intersect
- No way to have complex mechanical structures (no knots possible)
- Rotation would require leaving the 2D plane
Four or More Dimensions Are Unnecessary:
- Every physical phenomenon can be described and occurs in three dimensions
- No observation has ever required a fourth spatial dimension
- Adding dimensions doesn't solve real physical problems—it only complicates mathematical equations
Three Dimensions Are Natural:
- Three perpendicular directions exhaust the possibilities for independent spatial extension
- All mechanical motion and structure naturally exist in three dimensions
- Our sensory apparatus, measuring instruments, and all observations confirm three dimensions
Infinity: Why Space Cannot Have Limits
The Boundary Problem:
If space had a boundary, what would be on the other side of the boundary? Three possibilities:
- More space—but then the boundary wasn't real
- "Nothing"—but "nothing" is incoherent; it cannot "be" anywhere
- A "wall"—but a wall is something, made of matter, existing in... space
All three lead to contradictions. The only logically consistent answer is that space has no boundary—it is infinite.
The Container Problem:
If space were finite, we could ask: "What contains this finite space?"
- If something contains it, that container exists in... space
- If nothing contains it, then it has no boundary, and we're back to infinite space
Historical Recognition:
Philosophers and scientists have long recognized this:
- Giordano Bruno argued for infinite space in the 16th century
- Newton assumed absolute infinite space
- The logical necessity of spatial infinity was recognized before modern cosmology's finite universe models
Infinite Divisibility: Why Space Must Be Continuous
The Subdivision Thought Experiment:
Suppose space had a smallest unit—call it a "space atom." Consider two adjacent space atoms. We can meaningfully ask:
- "What separates them?"
- "What is between them?"
- "How far apart are they?"
Any answer to these questions requires space. The "gap" between space atoms would itself be space, which we could subdivide. Therefore, there can be no smallest unit.
Motion Requires Continuity:
Consider an object moving from point A to point B:
- In continuous space, an object passes smoothly through all intermediate points
- In discrete space, it would have to "jump" from one space atom to the next
- Such jumps would violate conservation of momentum and energy
- We observe smooth motion, not discontinuous jumps
Mathematical vs. Physical:
- Mathematics can work with discrete systems (like the integers)
- But physical space is not a mathematical construct - it's the actual container of reality
- Just because we can mathematically model something as discrete doesn't make it physically discrete
- We can just as easily model space as the infitite and infinitesimal set of real numbers.
Connection to Matter
Just as Axiom 3 will establish that matter is infinitely divisible (no smallest particle), space too must be infinitely divisible to accommodate matter at all scales. The self-similar structure of the universe (Axiom 10) requires both matter and space to be infinitely divisible.
Implications for the AAM
1. Same Physics at All Scales
Because space is infinitely divisible and uniform at all scales, the same mechanical principles that operate at our experiential level ($SL_{0}$) also operate at the atomic level ($SL_{-1}$) and the galactic level ($SL_{1}$). There are no special quantum rules at small scales or relativistic rules at cosmic scales—just mechanics operating in three-dimensional infinite continuous space.
2. No Spacetime
Time is not a dimension of space. Time and space are separate aspects of reality:
- Space is the container (Axiom 2)
- Time is the occurrence of matter in motion (Axiom 9)
Conflating them into "spacetime" is a mathematical convenience that obscures physical reality.
3. Self-Similarity Across Scales
The infinite divisibility of space allows for the hierarchical structure of the AAM:
- $SL_{-2}$ (sub-atomic) exists in space
- $SL_{-1}$ (atomic) exists in space
- $SL_{0}$ (human scale) exists in space
- $SL_{1}$ (galactic) exists in space
- And so on, infinitely in both directions
Each level exists in the same continuous, infinite, three-dimensional space—just at different scales.
4. No Edge Effects or Boundary Conditions
Physics doesn't change at some extreme distance. There is no "edge of the universe" where physics breaks down or becomes undefined. This simplifies the AAM enormously - we don't need special rules for:
- What happens "at infinity"
- Boundary conditions in our equations
- Edge cases or cosmic horizons
5. Light Propagation Through Space
Light is wave motion through the aether medium (Axiom 1). The infinite extent of space means:
- Aether fills all of space
- Light can propagate indefinitely
- No need for "expanding space" to explain cosmological phenomena
6. No Singularities
The Big Bang requires a singularity—infinite density at zero volume. Such a concept is physically meaningless. In the AAM:
- Space has always existed
- Matter has always existed
- No creation event, no singularity, no beginning
Common Objections and Responses
Objection 1: "Doesn't Einsteinian relativity prove spacetime is real?"
Response: Einstein's mathematical formalism treats time as if it were a dimension for calculational convenience. But mathematical convenience doesn't establish physical reality. Time is fundamentally different from space:
- You can move freely in any direction in space
- You cannot move freely in time - you are carried along by it
- Space is where things are; time is when things happen
- Conflating them obscures rather than clarifies physical understanding
Objection 2: "What about quantum entanglement and non-locality?"
Response: Quantum entanglement will be explained mechanically in the AAM through aether connections. There is no true "non-locality" - all influences propagate through a medium (the aether). What appears as instantaneous correlation is actually mechanical connection through the aether structure.
Objection 3: "How do you explain cosmic microwave background radiation without the Big Bang?"
Response: The CMB can be explained as residual radiation from ongoing processes in an infinite universe rather than primordial radiation from a creation event. An infinite universe in dynamic equilibrium would naturally produce background radiation. The interpretation of CMB as "echo of the Big Bang" is not the only possible interpretation - it's the interpretation that fits the Big Bang model.
Objection 4: "Doesn't the redshift of distant galaxies prove space is expanding?"
Response: Redshift proves that light from distant objects is shifted toward longer wavelengths. The interpretation of this shift as "expanding space" is one hypothesis. Alternative explanations include:
- Tired light (light losing energy over vast distances)
- Gravitational effects over cosmic distances
- Local motions in infinite space
The expansion interpretation was adopted early and became orthodox, but it's not the only mechanical explanation for the observations.
Objection 5: "What about Planck-scale quantum foam and spacetime fluctuations?"
Response: These are mathematical artifacts of quantum field theory attempting to handle infinities in its calculations. They are not observed phenomena - they are theoretical constructs derived from equations that the AAM considers fundamentally flawed. Space is not a quantum field that fluctuates - it is the continuous container within which matter exists and moves.
Objection 6: "If space is infinite, doesn't that lead to paradoxes like Olbers' Paradox?"
Response: Olbers' Paradox (why isn't the night sky uniformly bright if there are infinite stars?) is resolved by:
- Light loses energy over vast distances (tired light effect)
- Matter is not uniformly distributed in infinite space
- Not every line of sight necessarily intersects a star in infinite space with local clustering
The paradox only arises if you assume uniform distribution and no energy loss - both questionable assumptions.
Objection 7: "How can you have infinite space but still have local structure?"
Response: Infinity doesn't mean uniformity. An infinite space can have:
- Local clustering of matter
- Regions of high and low density
- Structure at all scales
- Dynamic processes creating and dissolving structures
Just as there are infinitely many integers but they still have local structure and patterns, infinite space can have rich local structure.
Open Questions for Future Investigation
Theoretical Development
- Tired Light Mechanism:
Since aether is the medium for light propagation, energy transfer to the medium is inevitable. A perfectly rigid or inelastic medium would be non-physical. As light waves propagate through the aether over vast distances, they continuously transfer small amounts of energy to the medium, causing amplitude to decrease. This provides a mechanical explanation for:
- Cosmological redshift (light losing energy over cosmic distances)
- Olbers' Paradox resolution (distant starlight dims significantly)
- Why the night sky isn't uniformly bright despite infinite space and stars
- Matter Distribution and Organization:
This will be addressed comprehensively in Axiom 10. Key insight: As similarity levels increase ($SL_0$ $\rightarrow$ $SL_{1}$ $\rightarrow$ $SL_{2}$...), organization and order decrease because higher levels are still settling into stable configurations. As similarity levels decrease ($SL_0$ $\rightarrow$ $SL_{-1}$ $\rightarrow$ $SL_{-2}$...), organization increases because lower levels have had sufficient time to reach stable equilibrium states. Each level builds upon the stability of the next lower level - robust atomic structure at $SL_{-1}$ depends on stable sub-atomic particles at $SL_{-2}$. Observable example: In galaxies (approaching $SL_{1}$), globular clusters are just beginning to become the "planetrons" of atoms at the next higher level, while most galaxy cores will eventually become the single nucleus/sun for higher-level "solar systems."
- Cosmic Microwave Background:
The mechanical processes maintaining the CMB in an infinite steady-state universe will be addressed in future discussions. These axioms establish the groundwork; detailed mechanisms will follow.
Observational Predictions
- The Fundamental Organization Question
A profound question that requires exploration after completing the axioms: Are all similarity levels simultaneously undergoing constant breakdown and reorganization (supernovae, virtual particle formation/annihilation, etc.)? Or must each lower level reach near-complete stability before the next higher level can organize? If the latter is correct, then perhaps only one level at a time undergoes major reorganization, with lower levels providing stable "building blocks" for higher levels. This raises the intriguing possibility that levels beyond $SL_{+3}$ or $SL_{+4}$ may not yet be sufficiently organized to qualify as distinct "levels" - or may not exist at all if there isn't enough organizational pattern. This question will be explored after the axioms establish the foundational framework.
- Big Bang Theory's Shifting Predictions:
As telescopes improve and observations extend further, Big Bang cosmology continually adjusts its parameters and "rules" to accommodate discoveries that don't fit the original model - such as the discovery of extremely distant, ancient galaxies that "shouldn't exist" according to BBT timelines. An infinite steady-state universe would predict mature structures at all observable distances without requiring constant theoretical adjustments.
- Direct Tired Light Evidence:
Further investigation needed to identify observational signatures that distinguish tired light energy loss from "expanding space" interpretations. This will be developed in future discussions.
Conceptual Clarification
- Rigorous Definition of Similarity Levels:
The concept of similarity levels in continuous infinite space has been introduced through our previous discussions and will be more precisely defined as subsequent axioms are developed. The relationship between levels, their organizational states, and the time-scaling between levels will become clearer as the framework develops.
- Local Density Variations and Cosmic Structure:
The relationship between local density variations and cosmic-scale structure in infinite space will be addressed comprehensively in Axiom 10, which deals with self-similarity and the organization of matter across scales.
- Conservation Laws in Infinite Space:
How conservation laws (energy, momentum, angular momentum) operate in infinite space with infinite matter will be addressed in Axiom 8, which deals with the nature of motion.
Relationship to Other Axioms
Axiom 2 directly builds upon Axiom 1's foundation:
Axiom 1 established:
All phenomena can be reduced to space, matter, and the motion of matter.
Axiom 2 specifies:
The nature of space - one of those three fundamental constituents.
Logical Flow:
- Axiom 1 tells us that space exists as a fundamental constituent
- Axiom 2 tells us what space is: three-dimensional, infinite in extent, infinitely divisible
Supporting Axiom 1:
The properties of space established in Axiom 2 support the mechanical worldview of Axiom 1:
- Three-dimensional space allows for all mechanical interactions
- Infinite space provides room for infinite matter (supporting gravitational mechanisms)
- Continuous space allows for smooth motion of matter (no quantum jumps needed)
- Infinite divisibility allows for aether at all scales (supporting wave propagation and pressure gradients)
Foundation for Future Axioms:
With space now characterized, we can proceed to: