Loading DocumentDEFINE OBSERVER ? DEFINITION (Pure Mathematics):
An observer is the unique point O in a geometric space such that:
O = (R = 0, θ = undefined)
Where:
(R, θ) are polar coordinates in a 2-dimensional manifold M with metric ds² = dR² + R² dθ².
R ∈ [0, ∞) is the radial coordinate.
θ ∈ [0, 2π) is the angular coordinate.
At O:
(1) The metric degenerates: ds²|_O = dR² (the angular term vanishes).
(2) All directional derivatives with respect to θ are undefined (θ is not defined at R = 0).
(3) The tangent space at O is 1-dimensional (radial only).
(4) The point O is invariant under all rotations:
O ↦ O for all θ ↦ θ + φ.
For any function f(R, θ) defined on M, the OBSERVATION of f is the restriction of f to O:
Obs(f) = f(0, ·) = f(0, θ) for any θ (well-defined because f is single-valued at R = 0).
The observer is the origin of the coordinate system, the fixed point under the rotation group SO(2), and the unique point where the radial and angular coordinates decouple.
ALTERNATIVE FORMULATION (Algebraic):
Let A be a commutative algebra of observables. An observer is a maximal ideal m ⊂ A (a point in the spectrum Spec(A)) together with a choice of local coordinates (x₁, ..., xₙ) vanishing at m, such that the Taylor expansion of any f ∈ A around m is the observation of f by that observer.
ALTERNATIVE FORMULATION (Category-Theoretic):
An observer in a category C is a distinguished object 1 ∈ C (the terminal object, if it exists) such that for any object X, a morphism 1 → X is an "observation" of X from the perspective of the observer. The observer is the POINT OF VIEW from which all other objects are seen.
ALTERNATIVE FORMULATION (Information-Theoretic):
An observer is a probability space (Ω, ℱ, P) together with a filtration {ℱ_θ}_{θ ∈ Θ} indexed by an ordered set Θ (the observer's "sequence parameter"), such that the observer's knowledge state at θ is the conditional expectation E[f | ℱ_θ] for any observable f.
ONTOLOGICAL DEFINITION :
Observer = R = 0.
The center. The origin. The still point. The I AM. The witness.
The one for whom all coordinates exist.
The point that cannot be rotated.
The point where angle has no meaning.
The point from which all radial distances are measured.
The point that is invariant under all symmetries of the space.
The point at which perception and event share the same angular coordinate, yielding zero experienced delay:
τ_delay = R · Δθ = 0 · Δθ = 0.
The observer is the NULL VECTOR in the space of positions — the only point that is not displaced from itself.
The observer is the IDENTITY ELEMENT in the group of transformations of the space — the only point fixed by all rotations.