Type-Safe Physics

A comprehensive framework for expressing physics with mathematical rigor using type theory, category theory, and formal verification methods.

Overview

This repository contains the academic paper, Haskell implementations, and educational resources for Type-Safe Physics - a revolutionary paradigm that transforms how we formulate and verify physical theories.

What is Type-Safe Physics?

Type-safe physics leverages advanced mathematical structures from type theory and category theory to construct physical theories where:

• Dimensional errors become impossible - Adding meters to seconds is a compile-time error
• Gauge artifacts are eliminated - Only physically meaningful quantities can be expressed
• Conservation laws are enforced structurally - Violations are type errors
• Compositional reasoning is guaranteed - Complex theories build from verified components

Repository Structure

type-safe-physics/
├── tex/                    # LaTeX source files
│   └── type_safe_physics.tex
├── pdf/                    # Compiled papers
│   └── type_safe_physics.pdf
├── haskell/               # Haskell proof implementations
│   ├── src/               # Source code
│   └── README.md          # Haskell-specific documentation
├── docs/                  # GitHub Pages site
│   ├── index.html
│   ├── assets/            # CSS, JS
│   └── images/            # Social media and diagrams
└── README.md              # This file

The Paper

"The Rise of Type-Safe Physics: A Global Revolution in Theoretical Foundations"

The paper surveys the emerging paradigm of type-safe physics, covering:

1. Mathematical Foundations - Type theory, category theory, Curry-Howard correspondence
2. Historical Development - From dimensional analysis to dependent types
3. Framework - Physical theories as categories, functorial semantics
4. Applications:
   • Quantum Mechanics (categorical formulation)
   • General Relativity (synthetic differential geometry)
   • Quantum Field Theory (TQFTs, factorization algebras)
   • Quantum Gravity (emergent spacetime conjecture)
5. Computational Tools - Proof assistants (Lean, Coq, Agda)
6. Philosophical Implications - Structural realism, constructivism

Key Concepts

Categories in Physics

Physical theories are formalized as structured categories:

• Objects = Physical systems
• Morphisms = Physical processes
• Composition = Sequential processes
• Tensor products = Parallel/composite systems

Type Safety Prevents

Error Type	Traditional	Type-Safe
Dimensional mismatch	Runtime surprise	Compile error
Invalid tensor contraction	Subtle bug	Type error
Gauge-dependent "observable"	Conceptual confusion	Impossible to write
Negative probability	Unphysical result	Structurally prevented

The No-Cloning Theorem

One of the key demonstrations: the no-cloning theorem follows purely from categorical structure (†-compact closure), not from specifics of Hilbert spaces or complex numbers.

Getting Started

Reading the Paper

open pdf/type_safe_physics.pdf

Running the Haskell Code

cd haskell
stack build
stack exec type-safe-physics

Viewing the Documentation Site

Visit the GitHub Pages site or run locally:

cd docs
python -m http.server 8000
# Open http://localhost:8000

Mathematical Prerequisites

To fully appreciate this work, familiarity with:

• Basic Category Theory: Categories, functors, natural transformations
• Type Theory: Dependent types, Curry-Howard correspondence
• Quantum Mechanics: Hilbert spaces, observables, measurement
• Differential Geometry: Manifolds, fiber bundles (for GR sections)

References

Key foundational works:

• Abramsky & Coecke (2004) - Categorical semantics of quantum protocols
• Baez & Dolan (1995) - Higher-dimensional algebra and TQFT
• Coecke & Kissinger (2017) - Picturing Quantum Processes
• HoTT Book (2013) - Homotopy Type Theory: Univalent Foundations

Contributing

Contributions welcome! Areas of interest:

• Additional Haskell formalizations
• Translations to other proof assistants (Lean 4, Agda)
• Educational materials and visualizations
• Corrections and improvements to the paper

License

This work is licensed under CC BY-SA 4.0.

Authors

• Matthew B. Crawford - Independent Researcher, Chicago, IL
• The YonedaAI Collaboration

Acknowledgments

Thanks to the global community working at the intersection of physics, category theory, and type theory.