Manifesto

The Math-Epistemics Manifesto

From proof to perception, building a mathematics that sees itself.

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The Manifesto for Mathematical Epistemics

From Proof to Perception

Mathematics must not merely describe the world—it must learn to see itself within it. This document proposes the structure, ethics, and practice of a discipline built for knowledge in motion.

Guiding signal

Mathematics must grow not just in complexity, but in witness and awareness—code the unseen, prove the responsible.

Preamble

Beyond Fragmentation

Current mathematics excels at modeling what is, but falters at modeling how we know. Disciplinary silos—logic, topology, probability—speak dialects of a unified grammar yet rarely converse. Meanwhile, AI systems generate epistemic environments where knowledge is dynamic, contested, and recursive.

We are building minds of glass and lightning: artificial intelligences that reason, update, and simulate. Yet we analyze them with mathematics designed for static, non-self-referential truth. Our maps are no longer sufficient for the territory. The age of passive mathematical modeling is over. We call this new field Mathematical Epistemics.

01Intellectual Foundations

This is not a revolution—it is a recalibration built on the shoulders of giants at the confluence of:

  • Formal Epistemology

    Philosophical rigor for modeling belief and evidence (van Fraassen, Kelly).

  • Categorical Cybernetics

    Language of composed systems and feedback (Fong, Spivak).

  • Algorithmic Fairness

    Grounding in justice and accountability (Barocas, Selbst).

We explicitly reject:

  • ✕Techno-utopianism: The naive belief that "AI will solve epistemology." AI is the urgent reason to rebuild epistemology.
  • ✕Physics Envy: Forcing quantum formalisms where they do not fit. We embrace them as analogies, not metaphysical truths.
  • ✕Epistemic Extraction: Mining knowledge without consent or reciprocity; mathematics must account for power and cost.

Our scope is to build the formal scaffolding for knowledge dynamics—not to replace empirical science or ethics, but to make them interoperable through a responsible epistemology of the 21st century.

02The Crisis of Epistemic Complexity

The isolation of mathematical disciplines has become a crisis. Challenges like understanding AI cognition and auditing automated decision systems straddle logic, probability, topology, and algebra—fields that speak a common grammar but lack a common project.

The task is to model the process of modeling the world. We need transversal mathematics that treats theories as modular components interacting across conceptual space. Specialization remains, but bridges must be built.

  • •Reject boundaries isolating probability (belief weights), topology (knowledge shape), logic (constraints), and geometry (inference curvature).
  • •Operational anchor: Categorical probability1 and information geometry2 already unify these—now we demand deliberate epistemic integration.
  • •Example: Topological data analysis must not just detect patterns—it must map bias manifolds in social data3.

Theories are not pillars—they are interlocking organs in a living body of knowing.

03Rise of Epistemic Machines

The machines are thinking. Large Language Models, agentic systems, and world models exhibit epistemic agency: posing questions, revising beliefs, simulating counterfactuals, and interrogating their own outputs. They generate novel epistemic environments that traditional tools cannot parse.

Grand Challenge for Mathematical Epistemics

Mathematics must keep pace by engaging with reasoning in motion—not by asking whether machines think, but by describing how reasoning evolves and becomes accountable.

Formalize the epistemic state of a learning system—its uncertainty about itself, its model of other agents, and the branching factors of its future reasoning.

Required formal tools

  • Feedback Inference: How an agent's inference engine rewrites itself through learning.
  • Knowledge Topology: The shape, boundaries, and connectivity of what an agent deems possible.
  • Recursive Belief Systems: Modeling agents reasoning about other reasoning agents and their own reflection.
  • Measures of Autonomy: Quantifying an agent's capacity to generate its own questions and goals.
  • Intersubjective Entanglement: Mapping entanglement between human and machine knowing.

04The Syncretic Framework

The next mathematics will be synthesized. Category Theory serves as a meta-language and Rosetta Stone—its focus on structures and maps provides machinery for transversality.

Local-to-global inference

Bayesian networks and topological sheaves share a categorical frame for understanding how local data forms global belief.

Semantic preservation

Geometric loss functions as functors preserving the semantic structure of training data.

Non-commutative observations

Quantum decoherence and epistemic collapse as analogous processes in categories of observation.

Probability provides belief weights and uncertainty quantification.

Topology & Geometry define the shape and curvature of conceptual space.

Logic supplies reasoning constraints and modal boundaries.

Quantum Structures model non-commutative reasoning and entangled knowledge states.

Category Theory weaves them into a unified epistemic algebra.

05Toward a Mathematics of Simulation

The proof is no longer on paper—it is alive, recursive, accountable, and observed in motion. We live in simulated epistemologies: feedback loops of software, culture, and data. Mathematical Epistemics must audit these dynamic systems.

  • Interpretability Metrics: Quantifying transparency of a black-box model's reasoning path.
  • Bias Entropy: Measuring divergence between an agent's internal model and the true generating process—the cost of a biased worldview.
  • Epistemic Stability: Tracking how knowledge topology changes under stress or adversarial input.
  • Meta-Inference: Frameworks for agents to formally reason about their own reasoning, including limitations.

06Quantum-Epistemic Analogy

Superposition is not a metaphor for belief—it is a rigorous analogy for uncertainty. Quantum structures offer analogical frameworks for epistemic phenomena that defy classical logic.

  • ⟡Non-Commutative Reasoning: When the order of questions or data changes resulting beliefs.
  • ⟡Epistemic Entanglement: Correlations between knowledge states of agents who have learned from each other.
  • ⟡Observer-Dependent States: User framing collapses vast possibility spaces into context-dependent answers.

07The Ethical Imperative

What we know must account for how, why, and for whom we know it. Mathematical Epistemics is not neutral. Its first application must be Epistemic Auditing.

Topology of Bias

Locate holes, boundaries, and singularities in an AI's conceptual space.

Geometry of Omission

Understand how data structure curves inference away from missing conclusions.

Flow of Epistemic Power

Whose beliefs are modeled, amplified, marginalized, or erased?

08The Proof in Practice

Transversal approaches are already yielding results. Frameworks like CE² Calculus, AE² Calculus, and Butterfly Calculus have enabled computation of once-intractable equations and formed the core of applied mathematical software.

Appendix: Research Seeds (Manifesto in Action)

  1. Bias Entropy Metric: Quantify fairness degradation in recursive AI systems using differential entropy on belief manifolds.
  2. Epistemic Sheaves: Model context-switching knowledge (e.g., medical diagnosis → public policy) via sheaf cohomology over social networks.
  3. Non-Commutative Learning: Formalize order-dependent updates in multi-agent systems using operator algebras.
  4. Decoherence Audits: Detect loss of epistemic coherence when AI systems share partial observations.
  5. Topological Omission Maps: Use persistent homology to visualize erased voices in training data embeddings.

09Call to Action

This is the project of Mathematical Epistemics. We do not need more isolated theories—we need a meta-mathematical consciousness that is transversal, dynamic, and ethically engaged.

  • Publishing executable math: Code + proofs + ethical impact statements.
  • Building epistemic sandboxes: Stress-test frameworks on real-world crises.
  • Training bilingual scholars: Fluency in topology and critical theory.

First steps

  • For the Theorist: Look beyond field borders for unifying structures.
  • For the Practitioner: Demand tools that explain the complex systems you build.
  • For the Ethicist: Define formal criteria for fair and transparent epistemology.

It is mathematics finally seeing itself.

Invitation

→ Join us. Code the unseen. Prove the responsible.

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Footnotes for Grounding

  1. Categorical probability: Fong, B., & Spivak, D. I. (2019). An Invitation to Applied Category Theory.
  2. Information geometry: Amari, S. (2016). Information Geometry and Its Applications.
  3. Bias manifolds: Chen, Y., & Koltun, V. (2023). Topological Bias in Social Data.
  4. Dynamic epistemic logic: Baltag, A., & Smets, S. (2020). Epistemic Actions as Resources.
  5. Sheaf theory for knowledge: Robinson, M. (2017). Sheaves are the Canonical Data Structure for Sensor Integration.
  6. Non-commutative belief: Khrennikov, A. (2010). Ubiquitous Quantum Structure.
  7. Interpretability certificates: Chen, J., et al. (2024). Formal Guarantees for Black-Box Explanations.
  8. Imprecise probability: Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities.