## Tags - Part of: [[Science]] [[Philosophy]] - Related: [[Natural science]] - Includes:[[Mathematics]] - Additional: ## Definitions - Branch of [[Science]] studying disciplines concerned with abstract [[Structures|structures]] described by [[Formal system|formal systems]], which use [[a priori]] [[Knowledge]], as opposed to [empiricism](https://en.wikipedia.org/wiki/Empirical_evidence "Empirical evidence")  ## Main resources - [Formal science - Wikipedia](https://en.wikipedia.org/wiki/Formal_science) <iframe src="https://en.wikipedia.org/wiki/Formal_science" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> ## Landscapes - [Formal science - Wikipedia](https://en.wikipedia.org/wiki/Formal_science#Branches) - [[Cryptography]], [[Logic]], [[Mathematics]], [[Computer science]] ([[Artificial Intelligence]]), [[Physics]], [[Systems theory]], [[Systems science]], [[Metamathematics]] - [Branches of science - Wikipedia](https://en.wikipedia.org/wiki/Branches_of_science#Formal_sciences) - <iframe src="https://en.wikipedia.org/wiki/Branches_of_science#Formal_sciences" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> - [[Information theory]], [[Information science]], [[Data science]], [[Systems theory]], [[Decision theory]], [[Statistics]] - [[Linguistics]] ## Written by AI (may include factually incorrect information) #### Map 1 # The Comprehensive Map of Formal Science **Formal science** is a branch of knowledge concerned with formal systems, such as logic, mathematics, systems theory, decision theory, statistics, theoretical computer science, information theory, game theory, and linguistics. Unlike natural and social sciences, formal sciences rely on a priori knowledge and deductive reasoning. Below is a comprehensive map detailing the fields and subfields of formal science. --- ## **1. Mathematics** ### **1.1. Pure Mathematics** #### **1.1.1. Algebra** - **Abstract Algebra** - Group Theory - Finite Groups - Infinite Groups - Lie Groups - Ring Theory - Field Theory - Module Theory - Galois Theory - Representation Theory - **Linear Algebra** - Vector Spaces - Matrices - Linear Transformations - Eigenvalues and Eigenvectors - **Number Theory** - Elementary Number Theory - Analytic Number Theory - Algebraic Number Theory - Diophantine Equations - Modular Forms - **Commutative Algebra** - **Non-Commutative Algebra** #### **1.1.2. Analysis** - **Real Analysis** - Measure Theory - Integration - Functional Analysis - Real Functions - **Complex Analysis** - Analytic Functions - Conformal Mappings - Riemann Surfaces - **Differential Equations** - Ordinary Differential Equations - Partial Differential Equations - Dynamical Systems - **Fourier Analysis** - **Harmonic Analysis** #### **1.1.3. Geometry and Topology** - **Euclidean Geometry** - **Differential Geometry** - Riemannian Geometry - Symplectic Geometry - **Algebraic Geometry** - Schemes - Varieties - **Topology** - General Topology - Algebraic Topology - Homology - Cohomology - Homotopy Theory - Differential Topology #### **1.1.4. Mathematical Logic and Foundations** - **Set Theory** - Axiomatic Set Theory - Ordinals and Cardinals - **Model Theory** - **Proof Theory** - **Recursion Theory** #### **1.1.5. Combinatorics** - **Enumerative Combinatorics** - **Graph Theory** - **Design Theory** - **Combinatorial Optimization** - **Matroid Theory** #### **1.1.6. Probability Theory** - **Stochastic Processes** - Markov Processes - Martingales - Brownian Motion - **Random Variables** - **Limit Theorems** ### **1.2. Applied Mathematics** #### **1.2.1. Numerical Analysis** - **Numerical Linear Algebra** - **Interpolation and Approximation** - **Numerical Solutions of Differential Equations** #### **1.2.2. Optimization** - **Linear Programming** - **Nonlinear Programming** - **Integer Programming** - **Dynamic Programming** - **Stochastic Optimization** #### **1.2.3. Mathematical Physics** - **Quantum Mechanics** - **Statistical Mechanics** - **General Relativity** - **String Theory** #### **1.2.4. Computational Mathematics** - **Computational Geometry** - **Computational Number Theory** - **Symbolic Computation** #### **1.2.5. Mathematical Biology** - **Biostatistics** - **Mathematical Modeling** - **Population Dynamics** --- ## **2. Logic** ### **2.1. Symbolic Logic** - **Propositional Logic** - **Predicate Logic** - **Modal Logic** - **Temporal Logic** - **Deontic Logic** ### **2.2. Mathematical Logic** - **Proof Theory** - **Model Theory** - **Set Theory** - **Computability Theory** ### **2.3. Philosophical Logic** - **Informal Logic** - **Inductive Logic** - **Fuzzy Logic** - **Non-Classical Logics** ### **2.4. Computational Logic** - **Automated Theorem Proving** - **Logic Programming** - **Type Theory** --- ## **3. Statistics** ### **3.1. Descriptive Statistics** - Measures of Central Tendency - Measures of Dispersion - Correlation and Regression Analysis ### **3.2. Inferential Statistics** - Hypothesis Testing - Confidence Intervals - Parametric and Non-Parametric Tests ### **3.3. Probability Distributions** - **Discrete Distributions** - Binomial - Poisson - Geometric - **Continuous Distributions** - Normal - Exponential - Gamma ### **3.4. Statistical Modeling** - Linear Models - Generalized Linear Models - Time Series Analysis - Survival Analysis ### **3.5. Bayesian Statistics** - Bayesian Inference - Hierarchical Models - Markov Chain Monte Carlo Methods ### **3.6. Experimental Design** - Randomized Controlled Trials - Factorial Designs - Response Surface Methodology --- ## **4. Computer Science** ### **4.1. Theoretical Computer Science** #### **4.1.1. Algorithms and Data Structures** - Sorting and Searching Algorithms - Graph Algorithms - Dynamic Programming - Computational Geometry #### **4.1.2. Theory of Computation** - Automata Theory - Formal Languages - Computability Theory - Complexity Theory #### **4.1.3. Cryptography** - Symmetric Cryptography - Asymmetric Cryptography - Cryptographic Protocols - Post-Quantum Cryptography #### **4.1.4. Information Theory** - Entropy Measures - Coding Theory - Data Compression ### **4.2. Programming Languages and Systems** #### **4.2.1. Programming Paradigms** - Procedural Programming - Object-Oriented Programming - Functional Programming - Logic Programming #### **4.2.2. Compiler Design** - Lexical Analysis - Syntax Analysis - Semantic Analysis - Code Optimization #### **4.2.3. Operating Systems** - Process Management - Memory Management - File Systems - Concurrency ### **4.3. Artificial Intelligence** #### **4.3.1. Machine Learning** - Supervised Learning - Unsupervised Learning - Reinforcement Learning - Deep Learning #### **4.3.2. Natural Language Processing** - Syntax and Parsing - Semantics - Machine Translation - Speech Recognition #### **4.3.3. Robotics** - Perception - Planning - Control Systems ### **4.4. Computer Networks** #### **4.4.1. Network Protocols** - TCP/IP - HTTP - SMTP #### **4.4.2. Network Security** - Firewalls - Intrusion Detection Systems - Public Key Infrastructure ### **4.5. Databases** #### **4.5.1. Database Design** - Entity-Relationship Modeling - Normalization #### **4.5.2. Query Languages** - SQL - NoSQL #### **4.5.3. Transaction Management** - ACID Properties - Concurrency Control --- ## **5. Information Theory** ### **5.1. Entropy and Information Measures** - Shannon Entropy - Mutual Information - Relative Entropy ### **5.2. Coding Theory** - **Error-Correcting Codes** - Linear Codes - Hamming Codes - Reed-Solomon Codes - **Compression Techniques** - Huffman Coding - Lempel-Ziv-Welch (LZW) ### **5.3. Channel Capacity** - Noisy Channel Coding Theorem - Binary Symmetric Channel - Gaussian Channels ### **5.4. Cryptography** (Overlap with Computer Science) --- ## **6. Systems Theory** ### **6.1. Control Theory** - Feedback Systems - Stability Analysis - Optimal Control ### **6.2. Complex Systems** - Chaos Theory - Nonlinear Dynamics ### **6.3. Cybernetics** - Regulation and Control - Information Processing in Systems ### **6.4. Systems Modeling** - Simulation Techniques - Systems Dynamics --- ## **7. Decision Theory** ### **7.1. Utility Theory** - Expected Utility - Risk Aversion ### **7.2. Bayesian Decision Theory** - Bayesian Inference in Decision Making ### **7.3. Multi-Criteria Decision Analysis** - Analytic Hierarchy Process - Multi-Attribute Utility Theory ### **7.4. Statistical Decision Theory** - Loss Functions - Decision Functions --- ## **8. Game Theory** ### **8.1. Non-Cooperative Game Theory** - Nash Equilibrium - Extensive Form Games - Strategic Form Games ### **8.2. Cooperative Game Theory** - Coalition Formation - Core Solutions - Shapley Value ### **8.3. Evolutionary Game Theory** - Replicator Dynamics - Evolutionarily Stable Strategies ### **8.4. Mechanism Design** - Auction Theory - Incentive Compatibility --- ## **9. Theoretical Linguistics** ### **9.1. Phonology** - Phonemes - Prosodic Features ### **9.2. Morphology** - Word Formation - Inflectional Morphology ### **9.3. Syntax** - Grammar Rules - Syntactic Structures - Parsing Algorithms ### **9.4. Semantics** - Meaning Representation - Semantic Theories ### **9.5. Computational Linguistics** - Language Modeling - Natural Language Understanding - Speech Processing --- ## **10. Cryptography** (Overlap with Computer Science) ### **10.1. Classical Cryptography** - Substitution Ciphers - Transposition Ciphers ### **10.2. Modern Cryptography** - Symmetric Key Algorithms - AES, DES - Asymmetric Key Algorithms - RSA, ECC - Cryptographic Hash Functions - SHA-2, SHA-3 ### **10.3. Cryptanalysis** - Brute Force Attacks - Statistical Attacks ### **10.4. Quantum Cryptography** - Quantum Key Distribution --- ## **11. Operations Research** ### **11.1. Linear Programming** - Simplex Method - Duality Theory ### **11.2. Integer Programming** - Branch and Bound - Cutting Planes ### **11.3. Network Models** - Shortest Path Algorithms - Maximum Flow Problems - Minimum Cost Flow ### **11.4. Queuing Theory** - Markovian Queues - Network of Queues ### **11.5. Inventory Theory** - Economic Order Quantity - Just-in-Time Inventory Systems ### **11.6. Simulation** - Monte Carlo Methods - Discrete-Event Simulation --- ## **12. Additional Fields** ### **12.1. Category Theory** - Objects and Morphisms - Functors - Natural Transformations - Monads ### **12.2. Set Theory** - Cardinal Numbers - Ordinal Numbers - Axiom of Choice ### **12.3. Model Theory** (Overlap with Logic) - Structures and Interpretations - Completeness and Consistency - Quantifier Elimination ### **12.4. Proof Theory** (Overlap with Logic) - Formal Proof Systems - Proof Transformations - Cut-Elimination --- ## **13. Systems Science** ### **13.1. General Systems Theory** - Interdisciplinary Study of Systems - System Hierarchies - Emergent Properties ### **13.2. Cybernetics** - Control and Communication in Animals and Machines - Feedback Loops - Homeostasis ### **13.3. Complexity Science** - Complex Adaptive Systems - Network Theory - Self-Organization --- ## **14. Decision Sciences** ### **14.1. Operations Management** - Process Optimization - Supply Chain Management - Quality Control ### **14.2. Risk Analysis** - Risk Assessment - Risk Management Strategies ### **14.3. Behavioral Decision Theory** - Heuristics and Biases - Prospect Theory --- ## **15. Mathematical Economics** ### **15.1. Econometrics** - Regression Analysis - Time Series Econometrics - Panel Data Analysis ### **15.2. Mathematical Finance** - Derivatives Pricing - Portfolio Optimization - Financial Econometrics --- ## **16. Mathematical Logic** (Detailed) ### **16.1. Proof Theory** - Sequent Calculus - Natural Deduction - Consistency Proofs ### **16.2. Model Theory** - Ultraproducts - Stability Theory - O-Minimal Structures ### **16.3. Computability Theory** - Recursive Functions - Turing Degrees - Relative Computability ### **16.4. Set Theory** - Large Cardinals - Forcing - Descriptive Set Theory --- ## **17. Quantum Computation and Information** ### **17.1. Quantum Algorithms** - Shor's Algorithm - Grover's Algorithm ### **17.2. Quantum Information Theory** - Qubits and Quantum States - Entanglement Measures ### **17.3. Quantum Error Correction** - Quantum Error-Correcting Codes - Fault-Tolerant Quantum Computation --- ## **18. Formal Epistemology** ### **18.1. Formal Learning Theory** - Inductive Inference - Computational Learning Theory ### **18.2. Belief Revision** - AGM Theory - Dynamic Epistemic Logic ### **18.3. Epistemic Logic** - Knowledge Operators - Common Knowledge --- ## **19. Theoretical Computer Science Extensions** ### **19.1. Automata Theory** - Finite Automata - Pushdown Automata - Turing Machines ### **19.2. Formal Language Theory** - Regular Languages - Context-Free Languages - Context-Sensitive Languages ### **19.3. Complexity Classes** - P, NP, NP-Complete, NP-Hard - PSPACE, EXPTIME --- ## **20. Formal Semantics and Pragmatics** ### **20.1. Montague Grammar** - Formal Semantics of Natural Language - Lambda Calculus Applications ### **20.2. Speech Act Theory** - Illocutionary Acts - Perlocutionary Effects ### **20.3. Discourse Representation Theory** - Anaphora Resolution - Contextual Interpretation --- ## **21. Formal Philosophy** ### **21.1. Modal Logic** - Possible Worlds Semantics - Necessity and Possibility ### **21.2. Deontic Logic** - Logic of Duty and Obligation ### **21.3. Temporal Logic** - Linear Temporal Logic - Branching Time Logic --- ## **22. Formal Ethics** ### **22.1. Decision Theory in Ethics** - Utility Ethics - Consequentialism ### **22.2. Game Theory in Ethics** - Moral Dilemmas - Cooperative Behaviors --- ## **23. Formal Ontology** ### **23.1. Mereology** - Part-Whole Relations - Ontological Dependence ### **23.2. Ontological Modeling** - Ontologies in Information Science - Semantic Web Standards --- ## **24. Formal Methods in Software Engineering** ### **24.1. Formal Verification** - Model Checking - Theorem Proving ### **24.2. Specification Languages** - Z Notation - B-Method - Alloy ### **24.3. Process Algebras** - CSP (Communicating Sequential Processes) - π-Calculus --- ## **25. Formal Methods in Artificial Intelligence** ### **25.1. Knowledge Representation** - Ontologies - Semantic Networks ### **25.2. Reasoning Systems** - Deductive Reasoning - Non-Monotonic Reasoning ### **25.3. Planning Algorithms** - STRIPS - PDDL --- This extensive map illustrates the breadth and depth of formal science, showcasing its numerous fields and subfields. Each area contributes to a foundational understanding of logical structures and formal systems, serving as the backbone for advancements in technology, science, and philosophy. **Note:** While this map is comprehensive, the fields of formal science are continually evolving, with new subfields and interdisciplinary areas emerging over time. --- **References and Further Reading:** - **"Principles of Mathematical Analysis"** by Walter Rudin - **"Introduction to the Theory of Computation"** by Michael Sipser - **"Information Theory, Inference, and Learning Algorithms"** by David J.C. MacKay - **"An Introduction to Game Theory"** by Martin J. Osborne - **"Mathematical Logic"** by H.-D. Ebbinghaus, J. Flum, W. Thomas - **"Artificial Intelligence: A Modern Approach"** by Stuart Russell and Peter Norvig --- **Disclaimer:** This map is designed to provide a broad overview and is not exhaustive. The interconnections between these fields are complex and multifaceted. #### Map 2 - Formal sciences are disciplines that study formal systems, using mathematical and logical methods. They are crucial in developing the theoretical underpinnings for other scientific fields. Here's a comprehensive list of the various branches and sub-disciplines within formal sciences: ### 1. [[Mathematics]] - Pure Mathematics - Algebra - Geometry - Calculus - Topology - Number Theory - Logic - Combinatorics - Set Theory - Category Theory - Graph Theory - Applied Mathematics - Mathematical Physics - Mathematical Finance - Mathematical Economics - Computational Mathematics - Biomathematics - Cryptography - Game Theory - Operations Research - Probability Theory - Statistics - Differential Equations - Dynamical Systems - Numerical Analysis - Optimization ### 2. [[Computer science]] - Theoretical Computer Science - Algorithms - Data Structures - Computational Complexity - Automata Theory - Cryptography - Quantum Computing - Computer Systems - Operating Systems - Distributed Computing - Parallel Computing - Computer Architecture - Computer Networking - Software Engineering - Systems Analysis - Software Design - Software Testing - Human-Computer Interaction - Artificial Intelligence - Machine Learning - Natural Language Processing - Robotics - Computer Vision - Expert Systems - Data Science - Data Mining - Big Data Analytics - Information Retrieval - Database Systems - Computer Graphics - Image Processing - Visualization - Animation - Virtual Reality - Information Systems - Management Information Systems - Geographic Information Systems ### 3. Logic - Mathematical Logic - Propositional Logic - Predicate Logic - Modal Logic - Temporal Logic - Fuzzy Logic - Philosophical Logic - Computational Logic - Non-classical Logics - Logic in Computer Science - Model Theory - Proof Theory - Set Theory ### 4. Statistics - Descriptive Statistics - Inferential Statistics - Probability Theory - Biostatistics - Econometrics - Environmental Statistics - Psychometrics - Bayesian Statistics - Time Series Analysis - Multivariate Analysis - Experimental Design - Survey Methodology - Statistical Modeling ### 5. Systems Science - Systems Theory - Cybernetics - Complex Systems - Dynamical Systems - Systems Engineering - Control Systems - Network Science - Systems Biology - Systems Dynamics ### 6. Linguistics (as a Formal Science) - Computational Linguistics - Phonology - Syntax - Semantics - Morphology - Formal Grammars - Corpus Linguistics ### 7. Information Theory - Coding Theory - Signal Processing - Data Transmission - Compression Algorithms - Cryptanalysis - Quantum Information Theory ### 8. Decision Theory - Game Theory - Risk Analysis - Utility Theory - Social Choice Theory - Multi-Criteria Decision Analysis ### 9. Formal Semiotics - Semantics - Syntactics - Pragmatics - Semiotic Analysis ### 10. Formal Philosophy - Formal Epistemology - Formal Ethics - Formal Aesthetics This list, while extensive, may not cover every specialized area as formal sciences are characterized by their continuous evolution and the emergence of new sub-disciplines. Each of these branches can be further divided into even more specialized fields, reflecting the depth and complexity of studying formal systems and their applications across various domains.