00. General Mathematics

This volume establishes the meta-layer of mathematics. It covers language, structure, methodology, and cross-cutting tools that apply across all branches. The scope follows MSC 00: classification, exposition, philosophy, and general methods.

Part I. What Mathematics Is

Chapter 1. Nature of Mathematical Objects

1.1 Abstract objects and structures 1.2 Sets, types, and universes 1.3 Equality, identity, and equivalence 1.4 Finite vs infinite objects 1.5 Constructive vs classical viewpoints

Chapter 2. Mathematical Truth

2.1 Truth vs provability

2.2 Formal systems and semantics

2.3 Consistency and completeness 2.4 Independence phenomena 2.5 Examples across fields

Chapter 3. Mathematical Language

3.1 Symbols and notation design 3.2 Formal vs informal language 3.3 Definitions and naming conventions 3.4 Precision vs readability 3.5 Notation as an interface

Part II. Structures and Abstraction

Chapter 4. Structural Thinking

4.1 Structure vs instance 4.2 Morphisms and mappings 4.3 Isomorphism as sameness 4.4 Invariants and classification 4.5 Examples: groups, spaces, graphs

Chapter 5. Levels of Abstraction

5.1 Concrete computation 5.2 Algebraic abstraction 5.3 Categorical abstraction 5.4 Meta-mathematical abstraction 5.5 Trade-offs in abstraction

Chapter 6. Patterns Across Mathematics

6.1 Duality 6.2 Symmetry and invariance 6.3 Local-to-global principles 6.4 Decomposition and composition 6.5 Recursion and induction

Part III. Methods of Reasoning

Chapter 7. Proof Techniques

7.1 Direct proof 7.2 Proof by contradiction 7.3 Induction and recursion 7.4 Constructive proofs 7.5 Probabilistic and combinatorial proofs

Chapter 8. Problem Solving Strategies

8.1 Reduction and transformation 8.2 Generalization and specialization 8.3 Analogy and transfer 8.4 Heuristics and experimentation 8.5 Counterexamples and edge cases

Chapter 9. Computation and Algorithms

9.1 Algorithmic thinking 9.2 Complexity awareness 9.3 Exact vs approximate computation 9.4 Symbolic vs numeric methods 9.5 Reproducibility and verification

Part IV. Mathematical Communication

Chapter 10. Writing Mathematics

10.1 Structure of a paper 10.2 Definitions, theorems, proofs 10.3 Clarity and minimalism 10.4 Common pitfalls 10.5 Style guidelines

Chapter 11. Visualizing Mathematics

11.1 Diagrams and graphs 11.2 Geometric intuition 11.3 Visual proofs 11.4 Limits of visualization 11.5 Tools and software

Chapter 12. Teaching and Learning

12.1 Cognitive models 12.2 Concept vs procedure 12.3 Common misconceptions 12.4 Designing exercises 12.5 Assessment strategies

Part V. Organization of Mathematics

Chapter 13. Classification Systems

13.1 MSC structure 13.2 Subject boundaries 13.3 Interdisciplinary links 13.4 Evolution of fields 13.5 Indexing and retrieval

Chapter 14. Mathematical Libraries and Data

14.1 Theorems as data 14.2 Formal libraries 14.3 Knowledge graphs 14.4 Search and indexing 14.5 Open datasets

Chapter 15. Notation and Standards

15.1 Symbol standardization 15.2 Units and conventions 15.3 File formats (LaTeX, MathML) 15.4 Interoperability 15.5 Versioning of knowledge

Part VI. Foundations and Meta-Mathematics

Chapter 16. Formal Systems

16.1 Syntax and inference rules 16.2 Axiomatic systems 16.3 Models and interpretations 16.4 Gödel-type phenomena 16.5 Limits of formalization

Chapter 17. Constructive and Computational Mathematics

17.1 Constructivism 17.2 Type theory 17.3 Proof assistants 17.4 Verified mathematics 17.5 Programs as proofs

Chapter 18. Philosophy of Mathematics

18.1 Platonism 18.2 Formalism 18.3 Intuitionism 18.4 Structuralism 18.5 Pragmatic perspectives

Part VII. Practice and Workflow

Chapter 19. Research Process

19.1 Problem selection 19.2 Literature review 19.3 Experimentation 19.4 Writing and revision 19.5 Publication process

Chapter 20. Mathematical Software

20.1 Computer algebra systems 20.2 Numerical libraries 20.3 Proof assistants 20.4 Data tools (DuckDB, etc.) 20.5 Reproducible pipelines

Chapter 21. Collaboration

21.1 Co-authorship models 21.2 Version control for math 21.3 Open science practices 21.4 Peer review 21.5 Community norms

Part VIII. Mathematics in Context

Chapter 22. Mathematics and Science

22.1 Modeling physical systems 22.2 Interaction with physics 22.3 Data-driven mathematics 22.4 Limits of models 22.5 Case studies

Chapter 23. Mathematics and Engineering

23.1 Approximation and constraints 23.2 Robustness and error 23.3 Optimization pipelines 23.4 Systems design 23.5 Real-world trade-offs

Chapter 24. Mathematics and Society

24.1 Economics and decision theory 24.2 Cryptography and security 24.3 Ethics in mathematics 24.4 Accessibility 24.5 Future directions

Appendix

A. Common notation reference B. Proof templates C. Problem-solving checklist D. Software and tools index E. MSC quick reference table

This volume acts as the entry point to all other branches. It defines how to think, write, organize, and operationalize mathematics before specializing.