Formal Mathematical System
(Redirected from Mathematical Theory)
A Formal Mathematical System is a formal system that provides a structured framework of symbols, axioms, and inference rules (designed to enable rigorous mathematical reasoning and proof generation within specific mathematical domains).
- AKA: Mathematical System, Math Theory, Mathematical Formalism, Mathematical Formal System, Mathematical Framework, Mathematical Theory, Math Formalism.
- Context:
- It can typically generate Mathematical Theorems through formal mathematical inference rules.
- It can typically express Mathematical Statements using formal mathematical language symbols.
- It can typically derive Logical Consequences from formal mathematical axioms.
- It can typically establish Mathematical Truths through syntactic manipulations.
- It can typically support Proof Verification through mechanical procedures.
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- It can often enable Consistency Analysis through metamathematical methods.
- It can often facilitate Completeness Investigation through logical analysis techniques.
- It can often permit Decidability Study through computational complexity analysiss.
- It can often allow Independence Proofs through model-theoretic methods.
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- It can range from being a Simple Formal Mathematical System to being a Complex Formal Mathematical System, depending on its formal mathematical logical expressiveness.
- It can range from being a First-Order Formal Mathematical System to being a Higher-Order Formal Mathematical System, depending on its formal mathematical quantification level.
- It can range from being a Complete Formal Mathematical System to being an Incomplete Formal Mathematical System, depending on its formal mathematical theorem-proving capability.
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- It can include Mathematical Operation Sets for valid operation specification.
- It can utilize Mathematical Languages for formal expression construction.
- It can be implemented in Mathematical Software Programs for computational verification.
- It can be analyzed by Theoretical Mathematics Disciplines for foundational study.
- It can be distinguished from Mathematical Structures by syntactic versus semantic focus.
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- Examples:
- Arithmetic Formal Mathematical Systems, such as:
- Peano Arithmetic, with successor function axioms and mathematical induction principle.
- Robinson Arithmetic, lacking mathematical induction schema.
- Presburger Arithmetic, restricted to addition operation.
- Geometric Formal Mathematical Systems, such as:
- Euclidean Geometry, based on Euclid's axioms.
- Hilbert's Geometry, with formalized geometric axioms.
- Tarski's Geometry, using first-order logic framework.
- Set-Theoretic Formal Mathematical Systems, such as:
- Algebraic Formal Mathematical Systems, such as:
- Group Theory, with group axioms.
- Ring Theory, with ring operation axioms.
- Field Theory, with field structure axioms.
- Logic Formal Mathematical Systems, such as:
- Propositional Logic, dealing with sentential connectives.
- First-Order Logic, including quantifier symbols.
- Modal Logic, incorporating necessity operators.
- Rewriting Formal Mathematical Systems, such as:
- String Rewriting System, manipulating formal strings.
- Term Rewriting System, transforming algebraic terms.
- Graph Rewriting System, modifying graph structures.
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- Arithmetic Formal Mathematical Systems, such as:
- Counter-Examples:
- Informal Mathematical Theory, which lacks precise syntactic rules.
- Mathematical Structure, which provides semantic interpretation rather than syntactic framework.
- Mathematical Notation System, which lacks inference rules and formal axioms.
- Physical System, which models physical phenomena rather than abstract mathematical reasoning.
- Computational System, which focuses on algorithmic execution rather than formal proof generation.
- See: Mathematical Structure, Model Theory, Proof Theory, Mathematical Logic, Metamathematics, Axiomatic System, Abstract System, Logic System, Mathematical Language, Mathematical Operation Set, Formal System.