processed #processing #toprocess #important #short #long #casual #focus¶

Tags¶

Metadata: #topic
Part of: Metamathematics Mathematics Philosophy of mathematics Logic Philosophy of mathematics
Related: [[Russel’s paradox]]
Includes: Philosophy of mathematics Set theory [[ZFC]] Category theory [[Classical logic]] [[Constructivism]] [[Computationalism]] Finitism [[Topos theory]]
Additional:

Significance¶

Intuitive summaries¶

Definitions¶

Study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense
Mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics

Technical summaries¶

Main resources¶

Foundations of mathematics - Wikipedia

Landscapes¶

Contents¶

Deep dives¶

Brain storming¶

Additional resources¶

AI¶

The foundations of mathematics is a field of study that focuses on the underlying theoretical frameworks and logical structures of mathematics. It seeks to understand and formalize the rules and principles that form the basis of mathematical reasoning and knowledge. Here's a comprehensive list of various branches and topics within the foundations of mathematics:

1. Mathematical Logic ¶

Propositional Logic
Predicate Logic
Modal Logic
Temporal Logic
Fuzzy Logic
Intuitionistic Logic
Non-classical Logics

2. Set Theory ¶

Naive Set Theory
Axiomatic Set Theory (Zermelo-Fraenkel, Von Neumann-Bernays-Gödel)
Ordinal Theory
Cardinal Theory
Infinitary Combinatorics
Large Cardinals

3. Model theory ¶

First-Order Model Theory
Model-Theoretic Algebra
Model-Theoretic Semantics
Nonstandard Models
Ultrafilters and Ultraproducts

4. Proof Theory¶

Formal Proofs and Deduction
Constructive Mathematics
Cut-Elimination and Normalization
Reverse Mathematics
Consistency Proofs
Computational Proof Theory

5. Category Theory ¶

Abstract Categories
Functors and Natural Transformations
Topos Theory
Monoidal Categories
Categorical Algebra

6. Computability and Complexity Theory¶

Turing Machines
Recursive and Recursively Enumerable Sets
Church-Turing Thesis
Complexity Classes
NP-Completeness
Decidability and Undecidability

7. Formal Semantics¶

Formal Languages
Syntax-Semantics Interface
Semantics of Programming Languages
Formal Specification Languages

8. Philosophy of Mathematics¶

Platonism
Formalism
Intuitionism
Logicism
Constructivism
Nominalism
Structuralism
Mathematical Realism and Anti-Realism

9. History of Foundations of Mathematics¶

Development of Axiomatic Systems
Historical Analysis of Foundational Crises
Contributions of Key Mathematicians (e.g., Cantor, Gödel, Hilbert)

10. Non-Standard Analysis¶

Hyperreal Numbers
Internal Set Theory
Applications to Calculus and Analysis

11. Type Theory¶

Lambda Calculus
Intuitionistic Type Theory
Dependent Type Theory
Homotopy Type Theory

12. Formal Number Theory¶

Peano Arithmetic
Diophantine Equations
Gödel's Incompleteness Theorems

13. Foundations of Geometry¶

Axiomatic Systems for Geometry
Non-Euclidean Geometry
Projective and Affine Geometry

14. Axiomatic Foundations¶

Axiomatization of Various Mathematical Theories
Consistency and Independence Proofs
Inter-Axiomatic Relations

15. Algorithmic Foundations¶

Algorithm Theory
Foundations of Computer Science
Theoretical Aspects of Algorithms

The foundations of mathematics is an intellectually rich and challenging field that not only provides the framework for mathematical thought but also intersects with computer science, philosophy, and logic. It is fundamental for understanding the nature and limitations of mathematical knowledge.

Additional metadata¶