Interactive Visualizations
Explore the Mathematics of Observer-Dependent Reality
ΔxΔp ≥ ℏ/2
↓ uncertainty!
↓ uncertainty!
Same particles,
8 different views →
8 different views →
dS/dt ≥ 0
(entropy always ↑)
(entropy always ↑)
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Framework Benchmark
8 observers • Relativistic causality • Complete physics
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All 8 Observers Grid
Same particles through 8 lenses simultaneously
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Entropy Evolution
2nd Law in real-time • Watch disorder increase
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Black Holes
Where time stops • Gravitational lensing • Event horizons
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Special Relativity
Simultaneity • Twin paradox • Length contraction
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Quantum Measurement
Wave function collapse • Observer effect
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Echo Chamber Formation
Network polarization • Homophily dynamics
1. Shannon Entropy
Shannon entropy measures the uncertainty or information content in a probability distribution.
It's fundamental to information theory and appears across physics, consciousness theories, and social dynamics.
Formula:
$$H(X) = -\sum_{i=1}^{n} p_i \log_2 p_i$$where \(p_i\) is the probability of outcome \(i\)
Interactive Visualization:
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2. Kullback-Leibler Divergence
KL divergence measures how one probability distribution differs from another.
It's crucial for understanding how observers' models differ from reality and appears in the Free Energy Principle.
Formula:
$$D_{KL}(P||Q) = \sum_{i} P(i) \log\frac{P(i)}{Q(i)}$$Measures the divergence of distribution \(Q\) from \(P\)
Interactive Visualization:
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3. Semantic Embeddings & Cosine Similarity
Semantic embeddings map concepts to vectors in high-dimensional space.
Cosine similarity measures the directional alignment between concepts, quantifying conceptual distance.
Cosine Similarity:
$$\cos(\theta) = \frac{A \cdot B}{||A|| \times ||B||} = \frac{\sum A_i B_i}{\sqrt{\sum A_i^2} \times \sqrt{\sum B_i^2}}$$Euclidean Distance:
$$d(A,B) = ||A-B|| = \sqrt{\sum (A_i - B_i)^2}$$Interactive 2D Embedding Space:
4. Echo Chamber Network Dynamics
Network clustering creates information bubbles where similar ideas reinforce each other.
Modularity and clustering coefficients quantify echo chamber formation.
Clustering Coefficient:
$$C_i = \frac{2T_i}{k_i(k_i-1)}$$where \(T_i\) = triangles through node \(i\), \(k_i\) = node degree
Modularity:
$$Q = \frac{1}{2m}\sum_{ij}\left[A_{ij} - \frac{k_i k_j}{2m}\right]\delta(c_i, c_j)$$Interactive Network:
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5. Free Energy Principle
The Free Energy Principle unifies perception, action, and learning as minimization of surprise.
It bridges quantum mechanics, neural dynamics, and social cognition.
Free Energy:
$$F = E_q[\ln q(x) - \ln p(x,o)]$$Decomposition:
$$F = D_{KL}[q(x)||p(x|o)] - \ln p(o)$$ $$F = \text{Complexity} - \text{Accuracy}$$Accuracy-Complexity Trade-off:
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6. Heisenberg Uncertainty Principle
Fundamental limits on observer knowledge encoded in quantum mechanics.
Certain observables cannot be simultaneously known with arbitrary precision.
Position-Momentum Uncertainty:
$$\Delta x \Delta p \geq \frac{\hbar}{2}$$Energy-Time Uncertainty:
$$\Delta E \Delta t \geq \frac{\hbar}{2}$$where \(\hbar = 1.054 \times 10^{-34}\) J·s
Uncertainty Visualization:
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7. Ruliad: Observer-Dependent Computational Paths
Wolfram's ruliad contains all possible computations. Different observers sample different paths through this computational space.
Computational boundedness forces observers to perform "equivalencing" - creating observer-dependent reality slices.
Ruliad Definition:
$$\mathcal{R} = \{U_r \mid r \in \mathcal{R}\}$$where \(\mathcal{R}\) is all possible computational rules and \(U_r\) is the universe generated by rule set \(r\)
Observer Equivalencing:
Computationally bounded observers map many microscale states → fewer macroscale states
Interactive Multiway Graph:
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8. Integrated Information Theory (Φ)
IIT proposes that consciousness is identical to integrated information (Φ). A system is conscious to the degree that it integrates information beyond its parts.
Φ measures how much a system constrains its own past and future states.
Intrinsic Information:
$$ii(m,z) = \pi(z|m) \times \log_2\frac{\pi(z|m)}{\pi(z)}$$Integrated Information:
$$\phi_s(s,\theta) = \min[\phi_c(s,s'_c,\theta), \phi_e(s,s'_e,\theta)]$$where \(\phi_c\) and \(\phi_e\) measure cause and effect integrated information
System Connectivity & Φ:
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9. Quantum Measurement & Observer-Dependent Collapse
Before measurement, quantum systems exist in superposition of states. Measurement "collapses" the wavefunction to a definite state.
This demonstrates radical observer-dependence: the act of observation fundamentally changes reality.
Superposition State:
$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$$where \(|\alpha|^2 + |\beta|^2 = 1\)
Measurement Probability:
$$P(j|\rho) = \text{Tr}(\rho \Pi_j)$$Born rule: probability from trace with projection operator
Animated Wavefunction Collapse:
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10. Worldview Embeddings & Paradigm Distance
Semantic embeddings enable quantitative measurement of conceptual incommensurability. Different worldviews occupy distant regions in embedding space,
making translation between paradigms computationally difficult. This formalizes Kuhn's philosophical insights.
Mahalanobis Distance:
$$d^2(x,y) = (x-y)^T A (x-y)$$where \(A\) captures semantic-change-aware dimensional weighting