[TOC]
↗ Logic (and Critical Thinking) (logic of natural languages and formal languages) ↗ Analytic Philosophy ↗ Philosophy of Language ↗ Linguistics
↗ Algebraic Structure & Abstract Algebra & Modern Algebra
↗ Formal Methods & Formal Verification (FV) ↗ Computer Languages & Programming Methodology
↗ Programming Language Processing & Program Execution ↗ Natural Language Processing (NLP) & Computational Linguistics
↗ Static Code Analysis Tools (SCAT)
↗ Game Theory & Decision Making in Multi-Agents Environments
https://web.ntnu.edu.tw/~algo/Logic.html 「邏輯學」。釐清前因後果，把整件事情想清楚的學問。
https://ljmw.readthedocs.io 本网站是逻辑思维普及性网站。 主要收录各类常见的逻辑谬误，并且加以简要分析与介绍。 本网站不是任何类型的逻辑学教程、教材，亦不会系统性地介绍现代逻辑学知识。 想要系统性学习逻辑学、逻辑思维的读者应购买逻辑学教材进行仔细研读。
- 违反逻辑学的基本准则
- 命题逻辑的基本错误
- 错误的三段论
- 非形式谬误
wikipedia
- Gödel's completeness and incompleteness theorems
- Tarski's undefinability
- Banach–Tarski paradox
- Cantor's theorem, paradox and diagonal argument
- Compactness
- Halting problem
- Lindström's
- Löwenheim–Skolem
- Russell's paradox
- This blog serves – or at least is intended to serve – as a forum for discussing substructural logics, dependent types, type systems, comonads, and whatever strikes my fancy.
- The name is based on “The Monad.Reader,” which serves as a place for publishing articles of not-quite-journal quality that pertain to Haskell. As my musings are likely to be similarly suspect, incomplete, or unoriginal the comparison seemed appropriate.
- In my case, I am interested in things that haven't caught on in the Haskell mainstream (comonads) or that cannot readily be expressed in the Haskell language (linearity, uniqueness, etc). Hence the slightly esoteric title of this blog, “The Comonad.Reader.”
- At other times I'll likely veer off onto a tangent and start talking about 3d graphics, the Plücker quadric, or compiling ecmascript 4.
[!lnks] ↗ Logic (and Critical Thinking) ↗ Formal System, Formal Logics, and Its Semantics
数理逻辑（英語：Mathematical logic）是数学的一个分支，其研究对象是对证明和计算这两个直观概念进行符号化以后的形式系统。数理逻辑是数学基础的一个不可缺少的组成部分。主要的子研究领域有模型论，证明论，集合论和可计算性理论。
数理逻辑的研究范围是逻辑中可被数学模式化的部分。以前称为符号逻辑（相对于哲学逻辑），又称元数学。数理逻辑一般着重于研究公理系统的推断能力和表达能力。它也包括分析正确的数学推断来构筑数学基础。
Mathematical logic is a branch of metamathematics that studies formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.
↗ Formal System, Formal Logics, and Its Semantics
↗ Set Theory & Axiomatic Set Theory
A Map of Mathematical Structures for AI
Posted on December 30, 2022 (https://mentalmodels4life.net/2022/12/30/a-map-of-mathematical-structures/) by Kee Siong Ng (https://mentalmodels4life.net/author/keesiongng/)
Generally speaking, each arrow involves the addition of some new symbols and the axioms that provide their definitions and / or properties. Some boxes have multiple incoming arrows; these are systems constructed from the union of multiple sets of new symbols and axioms. Note also that the relationships represented by the arrows are, in general, transitive.
[!links] ↗ Math History & Development /Foundational Crisis of Mathematics
Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study of the relation of this framework with reality.
The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's Elements. A mathematical assertion is considered as truth only if it is a theorem that is proved from true premises by means of a sequence of syllogisms (inference rules), the premises being either already proved theorems or self-evident assertions called axioms or postulates.
These foundations were tacitly assumed to be definitive until the introduction of infinitesimal calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. This new area of mathematics involved new methods of reasoning and new basic concepts (continuous functions, derivatives, limits) that were not well founded, but had astonishing consequences, such as the deduction from Newton's law of gravitation that the orbits of the planets are ellipses.
During the 19th century, progress was made towards elaborating precise definitions of the basic concepts of infinitesimal calculus, notably the natural and real numbers. This led to a series of seemingly paradoxical mathematical results near the end of the 19th century that challenged the general confidence in the reliability and truth of mathematical results. This has been called the foundational crisis of mathematics.
The resolution of this crisis involved the rise of a new mathematical discipline called mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently, parts of computer science. Subsequent discoveries in the 20th century then stabilized the foundations of mathematics into a coherent framework valid for all mathematics. This framework is based on a systematic use of axiomatic method and on set theory, specifically Zermelo–Fraenkel set theory with the axiom of choice. Foundations based on type theory have also gained prevalence, being commonly used in computer proof assistants.
It results from this that the basic mathematical concepts, such as numbers, points, lines, and geometrical spaces are not defined as abstractions from reality but from basic properties (axioms). Their adequation with their physical origins does not belong to mathematics anymore, although their relation with reality is still used for guiding mathematical intuition: physical reality is still used by mathematicians to choose axioms, find which theorems are interesting to prove, and obtain indications of possible proofs.
[!links] ↗ Set Theory & Axiomatic Set Theory ↗ Type Theory (类型论) ↗ Category Theory (范畴论)
🤖 GPT-5.0 🔗 https://chatgpt.com/share/69738171-49e4-800f-9dbf-654b7c9ecaa1
Mathematical logic is not itself a single foundation of mathematics; rather, it is the discipline that develops and studies formal systems capable of serving as foundations, such as set theory, type theory, and categorical frameworks.
// Final Mental Model (Keep This)
Mathematics
│
├── Mathematical Logic (field)
│ ├── Logic (languages, proof systems, semantics)
│ ├── Set Theory (ZFC, ZF, etc.)
│ ├── Model Theory
│ ├── Proof Theory
│ ├── Computability Theory
│
├── Foundations (choices)
│ ├── Set-theoretic
│ ├── Type-theoretic
│ ├── Categorical
│
└── Mathematical Practice
When people say: “Mathematical logic is the foundation of mathematics” they usually do not mean:
- There exists a single, ultimate theory
- That theory is provably the foundation
- All mathematics is derived from it in a uniquely correct way Instead, it means something much weaker and more pragmatic:
- Mathematical logic studies the formal languages, proof systems, and semantic frameworks in which mathematics can be precisely formulated, analyzed, and justified. So “foundation” here is methodological, not metaphysical.
Major foundational frameworks
- Set theory (ZFC, ZF + large cardinals)
- Type theory (Martin-Löf, HoTT)
- Category theory (topos theory)
- Higher-order logic Each:
- Can formalize large parts of mathematics
- Makes different philosophical commitments
- Has different strengths No single one is provably “the best”.
| Category | Examples |
|---|---|
| Field | Mathematical logic |
| Subfields | Set theory, model theory, proof theory, computability theory |
| Theories | ZFC, ZF, NBG, MLTT, HOL |
| Frameworks | First-order logic, type theory |
==There is no theorem that proves any system is the foundation of mathematics.== Even more strongly:
- The very notion of “absolute foundation” is not well-defined
- Foundations are chosen, not discovered
- What matters is relative consistency, interpretability, and usefulness Reasons:
- Gödel’s incompleteness theorems
- Independence results (CH, AC)
- Multiple non-equivalent foundations exist
- Foundations depend on goals (constructive vs classical) So “foundation” is:
- Historical
- Pragmatic
- Philosophical Not absolute. It supplies:
- Formal languages
- Proof systems
- Semantic frameworks
- Metatheorems (soundness, completeness, consistency) Set theory is one powerful construction using this toolkit.
[!links] ↗ Logic (and Critical Thinking) / Logical Reasoning ↗ Formal System, Formal Logics, and Its Semantics
↗ Classical Logic (Standard Formal Logic) ↗ Intuitionistic (Constructive) Logic
Among various methods of logical reasoning, deductive reasoning is deemed to be the best. Among various methods of deductive reasoning, classical logic (standard formal logic) lays the foundation of all other methods of deductions.