柯里-霍华德同构（Curry–Howard）为什么把“数学证明”和“计算机程序”看成同一种东西？
Why does the Curry–Howard isomorphism treat 'a mathematical proof' and 'a computer program' as the same kind of thing?
柯里-霍华德同构指出：逻辑命题对应程序类型，证明对应构造该类型的程序，推理规则对应函数组合。于是“证明一个定理”等价于“写出具有某类型的程序”，类型检查器就是定理证明器；这把直觉主义逻辑与 λ 演算深刻统一，奠定了证明辅助工具（如 Coq）的基础。
The Curry–Howard isomorphism says a logic proposition corresponds to a program type, a proof to a program inhabiting that type, and inference rules to function composition. Thus 'proving a theorem' equals 'writing a program of a given type,' and a type checker is a theorem prover—unifying intuitionistic logic with the λ-calculus and underpinning proof assistants like Coq.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Sørensen, M. H. 与 Urzyczyn, P., 《柯里-霍华德同构讲义》, Elsevier
Sørensen, M. H. & Urzyczyn, P., 'Lectures on the Curry–Howard Isomorphism', Elsevier
独立核验依据 · Independent verification: 核验依据 1 CROSSREF 标识 10.1016/s0049-237x(06)x8001-1：Lectures on th
#柯里-霍华德同构 #命题即类型 #证明即程序 #直觉主义
λ 演算是什么？
What is the λ-calculus?
λ 演算是用“匿名函数”和“应用”构成的极简计算模型，仅用变量、抽象与应用三种形式就能表达所有可计算函数。它是函数式编程语言的理论基础。
The λ-calculus is a minimal model of computation built from anonymous functions and application; it can express all computable functions and underlies functional programming.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #λ演算
停机问题是什么？
What is the halting problem?
它问是否存在一个程序能判断任意程序是否会在有限步内停止。图灵证明不存在这样的通用判定程序，说明某些问题本质上不可计算。
The halting problem asks if a program can decide whether any program stops; Turing proved no such universal decider exists—some problems are uncomputable.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #计算理论
P 与 NP 问题是什么？
What is the P vs NP problem?
P 是能在多项式时间内求解的问题类，NP 是能在多项式时间内“验证”解的问题类。P=NP 吗？若相等，许多难题（如调度、密码破解）将迎刃而解，是千禧年难题之一。
P = problems solvable in polynomial time; NP = problems whose solutions are verifiable in polynomial time. Whether P=NP is a Millennium problem with huge consequences.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #P对NP
邱奇-图灵论题是什么？
What is the Church–Turing thesis?
它认为任何“可有效计算”的函数都能被图灵机（或 λ 演算）计算。它并非可证明的定理，而是对“计算”本质被广泛接受的刻画。
The Church–Turing thesis holds that anything effectively computable is computable by a Turing machine (or λ-calculus); it's a widely accepted characterization, not a provable theorem.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #邱奇图灵
什么是哥德尔不完备定理？
What is Gödel's incompleteness theorem?
哥德尔不完备定理表明：任何足够强、包含算术且自洽的形式系统，都存在既不能被证明也不能被否证的命题。它深刻限制了用纯逻辑彻底公理化的梦想。
Gödel's incompleteness theorem shows that any consistent formal system strong enough to include arithmetic contains statements that can neither be proved nor disproved within it. It fundamentally limits the dream of fully axiomatizing mathematics by pure logic.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#哥德尔 #不完备 #逻辑
忙海狸函数是什么？
What is the busy beaver function?
忙海狸函数 Σ(n) 给出 n 状态图灵机在停机前最多能写下的 1 的个数。它增长比任何可计算函数都快，且具体值（如 Σ(5)）极难确定，体现了可计算性的边界。
The busy beaver function Σ(n) is the max ones an n-state Turing machine can write before halting; it grows faster than any computable function and is non-computable.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #忙海狸
直觉主义逻辑是什么？
What is intuitionistic logic?
直觉主义（构造性）逻辑拒绝“排中律”的无限应用：它只承认能“构造出证明”的命题为真。这影响了类型论与计算机科学中的“可证明即程序”。
Intuitionistic logic rejects unrestricted excluded middle; a statement is true only if a proof can be constructed—influencing type theory and 'proofs-as-programs'.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #直觉主义
罗素悖论如何动摇了早期集合论的基础？
How did Russell's paradox shake the foundations of early set theory?
罗素在 1901 年提出：考虑「所有不包含自身的集合」所组成的集合 R——若 R 包含自身则按定义不应包含，若不包含则又应当包含，矛盾。这一悖论击中了朴素集合论中「任意性质定义一个集合」的漏洞，直接推动了公理化集合论（如 ZF 系统）的建立，以避免自指带来的不一致。
Russell (1901) asked about the set R of all sets that do not contain themselves—if R contains itself it shouldn't, and if it doesn't it should, a contradiction. This exposed the flaw in naive set theory's 'any property defines a set' and directly spurred axiomatic set theory (e.g., ZF) to avoid self-referential inconsistency.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#罗素悖论 #集合论 #自指 #公理化
什么是图灵机？
What is a Turing machine?
图灵机是一种抽象计算模型，用一条可无限延伸的纸带和读写头模拟任何可算法计算的过程，是现代计算机的理论基础。
A Turing machine is an abstract model—a tape and head—that captures any algorithmic computation, basis of modern computers.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #计算理论
哥德尔不完备定理是什么？
What are Gödel's incompleteness theorems?
它表明：任何包含算术且自洽的形式系统，都存在既不能被证明也不能被否定的命题。也就是说，数学不可能被完全公理化。
Gödel's theorems show any consistent formal system containing arithmetic has true-but-unprovable statements; math can't be fully axiomatized.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #不完备
什么是可判定性？
What is decidability?
一个问题是可判定的，若存在算法能在有限步内对任一输入给出“是/否”答案。否则不可判定，如停机问题。
A problem is decidable if an algorithm always answers yes/no in finite time; otherwise it's undecidable, like the halting problem.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #计算理论
柯里-霍华德同构是什么？
What is the Curry-Howard correspondence?
它揭示“命题即类型、证明即程序”：逻辑证明与计算机程序在结构上一一对应。这为用编程语言做形式化证明奠定基础。
It identifies propositions with types and proofs with programs, linking logic to computation and enabling formal proof assistants.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #类型论
什么是命题逻辑？
What is propositional logic?
命题逻辑把能判真假的陈述当作不可再拆的单位，用「且、或、非、如果…则、当且仅当」拼出复合命题；复合式的真假完全由各部分的真假决定，所以一张真值表就能穷尽所有情形，n 个变元也不过 2 的 n 次方行。它的限度也出在这一步：命题被当成原子，「所有」「存在」这类量词根本表达不出来，「所有人都会死」在这里只是一个 p。要看穿命题内部的主谓结构，就得上升到谓词逻辑。
Propositional logic takes truth-evaluable statements as indivisible units and combines them with "and", "or", "not", "if-then" and "if and only if". The truth of the compound is fixed entirely by the truths of its parts, so one truth table settles every case: n variables give just 2-to-the-n rows. Its limit comes from that same move, since propositions stay atoms: quantifiers like "all" and "some" cannot be expressed at all, and "every man is mortal" is only a single p here. Seeing the subject-predicate structure inside a proposition means ascending to predicate logic.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #命题
1900 年希尔伯特列出的 23 个问题，为什么被称为“数学家的北极星”？
Why are Hilbert's 23 problems, posed in 1900, called a guiding star for mathematicians?
希尔伯特在 1900 年巴黎国际数学家大会上提出 23 个重大问题，像一份“待办清单”，指明了 20 世纪数学的研究方向。一个世纪里它们被逐个攻克：第 7 题（2^√2 的超越性）1934 年解决，第 10 题被证明不可判定，第 1 题（连续统假设）被证明独立于现有公理体系。
At the 1900 International Congress of Mathematicians in Paris, Hilbert listed 23 deep problems as a to-do list that steered twentieth-century research. Over the century they fell one by one: problem 7 (transcendence of 2^√2) was solved in 1934, problem 10 was shown undecidable, and problem 1 (the continuum hypothesis) was proved independent of the standard axioms.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#希尔伯特问题 #数学史 #开放问题
什么是归纳推理？
What is inductive reasoning?
归纳推理是从许多具体观察中总结一般规律，结论具有或然性，不同于必然成立的演绎推理。
Inductive reasoning generalizes from observations; conclusions are probable, unlike deductive certainty.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #推理
什么是悖论？
What is a paradox?
悖论是从看似合理的前提出发却推出自相矛盾或反直觉结论的情形，例如“这句话是假的”。
A paradox derives a contradiction or counterintuitive result from plausible premises, e.g. “this sentence is false”.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #悖论
模态逻辑（modal logic）是怎样用“必然”和“可能”扩展普通命题逻辑的？
How does modal logic extend ordinary propositional logic with 'necessity' and 'possibility'?
模态逻辑在“真/假”之外加入算子 □（必然）与 ◇（可能），并引入“可能世界”语义：□P 表示“在所有可达的可能世界 P 都真”，◇P 表示“在某一可能世界 P 真”。它用于哲学、程序验证与知识表示（如“知道”“相信”），是命题逻辑的自然推广。
Modal logic adds operators □ (necessarily) and ◇ (possibly) beyond true/false, with 'possible-world' semantics: □P means P holds in all accessible worlds, ◇P in some world. Used in philosophy, program verification, and knowledge representation (knowing, believing), it naturally extends propositional logic.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
Blackburn, P., de Rijke, M., Venema, Y., 《模态逻辑》, Cambridge University Press
Blackburn, P., de Rijke, M., Venema, Y., 'Modal Logic', Cambridge University Press
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL20992032W：Modal Logic（首版 2002）按标
#模态逻辑 #必然 #可能 #可能世界
什么样的函数在原理上是“可计算”的？
What functions are “computable” in principle?
图灵机把“算法”精确化：任何能用有限条规则描述、能得出确定结果的计算，都可以在一台理想化的图灵机上完成。丘奇-图灵论题断言“可计算”就等于“图灵可计算”。它把计算的概念从具体机器中抽象出来，成为理论计算机科学的基石，也揭示了不可计算问题的存在。
The Turing machine formalizes the notion of algorithm: any computation describable by finitely many rules with a definite outcome can be carried out on an idealized Turing machine. The Church–Turing thesis asserts that “computable” means “Turing-computable”. This abstracts computation away from any physical machine, founding theoretical computer science and exposing the existence of non-computable problems.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#可计算性 #图灵机 #丘奇-图灵论题
什么是反证法？
What is proof by contradiction?
反证法先假设结论不成立，再推出矛盾，从而证明原结论必然成立。
Proof by contradiction assumes the negation, derives a contradiction, and thus proves the original claim.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #证明
为什么维克里（次价）拍卖能鼓励买家如实出价？
Why does the Vickrey (second-price) auction encourage bidders to bid truthfully?
在次价拍卖中，出价最高者成交，但只支付第二高的价格。此时如实报价是最优策略：报高不会改变成交价、却可能让你多付；报低则可能输掉本来能赢的交易。出价只影响能否获胜，不影响支付额，所以说真话是占优策略。维克里因此共享 1996 年诺贝尔经济学奖，这一结果是机制设计的基础。
In a second-price auction the highest bidder wins but pays only the second-highest bid. Truthful bidding is then a dominant strategy: overbidding never changes the price yet risks overpaying, and underbidding risks losing a deal you would have won. The bid affects only whether you win, not what you pay. Vickrey shared the 1996 Nobel Prize in Economics, and this result is foundational to mechanism design.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
美国数学会(AMS)《博弈论与经济模型》
American Mathematical Society (AMS), Game Theory and Economic Modeling
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787100047876（豆瓣书目条目: 博弈论与经济模型）
#维克里拍卖 #机制设计 #占优策略
一阶逻辑和命题逻辑有什么区别？
How do first-order and propositional logic differ?
命题逻辑只处理整体命题的真假；一阶逻辑还能使用量词（∀、∃）和个体、关系，能表达“所有”“存在”等更精细的推理。
Propositional logic handles whole statements' truth; first-order logic adds quantifiers (∀,∃) and relations for finer reasoning about 'all' or 'some'.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Enderton, H. B.，《A Mathematical Introduction to Logic》
Enderton, H. B., A Mathematical Introduction to Logic
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780122384523 (2001)
#逻辑 #逻辑系统
什么是说谎者悖论？
What is the liar paradox?
说谎者悖论是指像“这句话是假的”这样自相矛盾的话：如果它为真，则它为假；如果它为假，则它为真。它揭示了语言和逻辑系统中自我指涉可能带来的难题。
The liar paradox is a self-contradictory statement such as 'this sentence is false': if it is true then it is false, and if it is false then it is true. It reveals the difficulties that self-reference can create in language and logical systems.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#悖论 #自我指涉 #逻辑
什么是命题？
What is a proposition?
命题是可以判断真假的陈述句，例如“雪是白的”为真，“2 + 2 = 5”为假。命题逻辑研究命题之间如何通过“与”“或”“非”等连接。
A proposition is a declarative sentence that is either true or false, for example 'snow is white' is true and '2 + 2 = 5' is false. Propositional logic studies how propositions combine through connectives like 'and', 'or', and 'not'.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#命题 #真值 #逻辑
什么是可满足性（SAT）问题？
What is the satisfiability (SAT) problem?
SAT 问是否存在一组真值赋值使给定逻辑公式为真。它是首个被证明为 NP 完全的问题，许多难题都可归约到它。
SAT asks if some truth assignment makes a logical formula true; it was the first proved NP-complete problem, central to complexity.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-04
来源 · Source
S. A. Cook,《The complexity of theorem-proving procedures》,Proceedings of the third annual ACM symposium on Theory of c,1971 (DOI: 10.1145/800157.805047)
S. A. Cook, 'The complexity of theorem-proving procedures', Proceedings of the third annual ACM symposium on Theory of c, 1971, DOI: 10.1145/800157.805047
独立核验依据 · Independent verification: 核验依据 1 Crossref DOI 10.1145/800157.805047
#逻辑 #可满足性
布尔代数和现代计算机有什么关系？
What is the connection between Boolean algebra and modern computers?
布尔代数由乔治·布尔在 1854 年提出，用「与、或、非」等逻辑运算处理真/假命题。克劳德·香农在 1937 年证明电路开关可用布尔代数描述，从而使逻辑代数成为数字电路与计算机硬件设计的数学基础。今天 CPU 里的每一个门电路本质上都是布尔函数的物理实现。
Boolean algebra, introduced by George Boole in 1854, manipulates true/false propositions with AND, OR, NOT operations. Claude Shannon showed in 1937 that switching circuits could be described by Boolean algebra, making it the mathematical basis of digital circuits and computer hardware. Every logic gate in a modern CPU is essentially a physical implementation of a Boolean function.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
大英百科全书（Britannica）；香农 1937 年论文《继电器与开关电路的符号分析》
Encyclopaedia Britannica; Shannon, A Symbolic Analysis of Relay and Switching Circuits (1937)
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL2296211W：A symbolic analysis of
#布尔代数 #逻辑门 #香农 #数字电路
什么是布尔代数？
What is Boolean algebra?
布尔代数是研究真（1）与假（0）以及“与”“或”“非”运算的代数系统。它是数字电路和计算机逻辑门设计的数学基础。
Boolean algebra is the algebraic system dealing with true (1) and false (0) together with the operations 'and', 'or', and 'not'. It is the mathematical foundation of digital circuits and computer logic gates.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#布尔代数 #逻辑门 #计算机
数学为什么能“从几个公理推出整个大厦”？公理化方法是谁开创的？
How can mathematics build a whole edifice from a few axioms, and who pioneered the axiomatic method?
欧几里得在《几何原本》里首创这种方法：先列出少数不加证明的出发点——公设与公理，再只凭逻辑规则一步步推出所有定理。它的威力在于，只要前提被接受，结论就完全确定，不需要依赖实验或直觉。现代数学的每个分支都按这套方式组织，包括实数系统与概率论。
Euclid pioneered it in the Elements: state a few unproved starting points — postulates and axioms — then derive every theorem by logical rules alone. Its power is that once the premises are accepted, the conclusions are fully determined, with no appeal to experiment or intuition. Every modern branch of mathematics, including the real numbers and probability, is organized this way.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-03
来源 · Source
《什么是数学》(柯朗)（数学的哲学与方法）
What Is Mathematics? (Courant), philosophy and method of mathematics
#公理化方法 #欧几里得 #演绎推理
停机问题为什么是不可判定的？
Why is the halting problem undecidable?
停机问题问：能否写一个程序，判断任意程序对给定输入是否会停下来？图灵 1936 年证明不能。若存在这样的判定程序 H，就可构造程序 P，它在“H 判断 P 会停”时反而死循环，反之则立刻停止，形成矛盾。这说明有些数学问题是任何算法都无法解决的，可计算的边界由此被严格刻画。
The halting problem asks whether one program can decide if any other program will halt on a given input. Turing proved in 1936 that it cannot. If such a decider H existed, one could build a program P that loops forever exactly when H predicts P halts, and stops otherwise—a contradiction. So some mathematical problems are beyond any algorithm, sharply delimiting what is computable.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#停机问题 #不可判定 #图灵机
我知道你在想什么这类游戏背后的逻辑推理是怎么工作的?什么是排除法?
How does mind-reading guessing work behind logic games? What is elimination?
这类游戏的秘密不是读心而是排除:每问一个问题就划掉一批不可能的答案。比如猜数字1到100,每次问比50大吗,答案无论真假都能砍掉一半范围——七次之内必中。这就是二分法,计算机查找数据用的正是同一招。好的提问顺序能让效率翻倍再翻倍。
The secret is elimination, not telepathy: every question eliminates impossible answers. Guessing one to one hundred, ask bigger than fifty? Either reply halves the range — seven questions guarantee success. Computers search data with this same binary method. Good question order doubles efficiency again and again.
难度 进阶 · 适合 小学高段(9-12)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
《数学花园漫游记》,马希文,中国少年儿童出版社,2004
Ma Xiwen, Adventures in the Math Garden, China Children Press, 2004
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787500767558（豆瓣书目条目: 数学花园漫游记）
#排除法 #二分查找 #逻辑推理
地湿了，就一定是下雨了吗？
If the ground is wet, must it have rained?
不一定。地湿可能是下雨，也可能是洒水、泼水等原因。不能由结果反推唯一原因（这是“肯定后件”的谬误）。
Not necessarily; wet ground could come from sprinklers or spills, not only rain (affirming the consequent fallacy).
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #谬误
「这句话是假的」这类说谎者悖论为什么让逻辑学家头疼？
Why does the liar paradox ('this sentence is false') trouble logicians?
说谎者悖论是一句自称虚假的命题：若它为真则它假，若它为假则它真，陷入循环矛盾。它揭示了自然语言中「自指」与「真值」概念结合会产生不一致，促使逻辑学家发展分层语言、修正语义等理论来规避。它是自指悖论家族最古老、最典型的代表。
The liar paradox is a statement that asserts its own falsity: if true it is false, if false it is true—a circular contradiction. It shows that self-reference combined with the concept of truth in natural language yields inconsistency, pushing logicians toward stratified languages and revised semantics. It is the oldest and most typical representative of self-referential paradoxes.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#说谎者悖论 #自指 #真值 #语义悖论
什么是逻辑？
What is logic?
逻辑是研究推理是否正确的规律和方法，帮助我们由已知前提得出可靠的结论。它既是数学的基础，也用于日常思考和计算机科学。
Logic is the study of the principles and methods of correct reasoning, helping us draw reliable conclusions from given premises. It is a foundation of mathematics and is also used in everyday thinking and computer science.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#逻辑 #推理 #思维
什么是“肯定前件”和“否定后件”这类有效的演绎推理形式？
What are valid deductive forms like 'modus ponens' (affirming the antecedent) and 'modus tollens' (denying the consequent)?
“肯定前件”：若 P→Q 且 P 成立，则 Q 成立；“否定后件”：若 P→Q 且 Q 不成立，则 P 不成立。二者是命题逻辑里最基本的有效推理规则，保证从真前提必然得到真结论，区别于“肯定后件”这种常见的无效推理。
Modus ponens: from P→Q and P, infer Q. Modus tollens: from P→Q and not-Q, infer not-P. These are the fundamental valid inference rules of propositional logic, guaranteeing true conclusions from true premises—unlike the fallacious 'affirming the consequent.'
难度 入门 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Enderton, H. B., 《逻辑的数学导引》(第2版), Academic Press
Enderton, H. B., 'A Mathematical Introduction to Logic' (2nd ed.), Academic Press
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780122384523 (2001)
#肯定前件 #否定后件 #命题逻辑 #有效推理