William P. Thurston
In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.
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Remarks on mathematical proof and the practice of mathematics.
Teaching Proof to Future Mathematics Teachers in Mathematical Analysis Classes
本研究聚焦师范院校数学分析课程构建，以培养未来数学教师发现和解决证明相关问题的能力为目标，涉及数学教学、教师培训等多个相关领域。
While proof is a central component of postsecondary mathematical study, proof construction has historically posed significant difficulties for students who intend to earn mathematics degrees at the undergraduate level. This work is comprised of detailed literature reviews, an argument for a methodological approach, and a descriptive research plan to study how university students use their mathematical discourse to interpret proof tasks and ultimately engage in constructing proofs in response to these tasks. The theoretical outcomes of using the approach outlined therein are: (a) theorized mode…
The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, as a methodology for the study of mathematica…
Large-Scale Formal Proof for the Working Mathematician -- Lessons learnt from the ALEXANDRIA Project
ALEXANDRIA项目旨在将形式化验证引入数学，过去六年的密集活动表明，当今工具已能形式化借鉴多领域的最先进数学成果。
AI for Mathematics: Progress, Challenges, and Prospects
本文综述了AI4Math领域，探讨其在数学研究中的应用及发展，强调数学作为严格推理的测试场，推动通用推理能力的进步。
The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.
Notes for a study of the didactic transposition of mathematical proof
本文按时间顺序梳理数学证明从过去缺席到成为各年级教学内容的历程，分析其教学转化过程所受的认识论、教学法、逻辑与数学约束，以理解当前研究面临的教学问题。
The unique and beautiful character of certain mathematical results and proofs is often considered one of the most gratifying aspects of engaging with mathematics. We study whether this perception of mathematical arguments having an intrinsic 'character' is subjective or universal -- this was done by having test subjects with varying degrees of mathematical experience match mathematical arguments with paintings and music: 'does this proof feel more like Bach or Schubert?' The results suggest that such a universal connection indeed exists.
We report on a pedagogical experiment to make mathematics easy by changing its philosophy. The Western philosophy of math originated in religious beliefs about mathesis, cursed by the church. Later, mathematics was "reinterpreted", in a theologically-correct way, using the myth of "Euclid" and his deductive proofs. The fact of the empirical proofs in the Elements, was, however, contrary to this myth. The discrepancy was resolved by Hilbert and Russell who rejected empirical proofs as unsound, reducing all mathematics to metaphysics. We explain why that formalist metaphysics is anti-utilitarian…
Reflective Groupwork for Introductory Proof-Writing Courses
本文介绍在本科证明写作入门课中设计的两种反思性小组证明评估活动，分别通过同伴互评和网络资源质量评价，培养学生批判性讨论、形成共识及判断证明质量的专业数学实践能力。
Correctness, Artificial Intelligence, and the Epistemic Value of Mathematical Proof
本文探讨了数学证明的正确性与认知价值之间的关系，提出形式正确性在数学中的作用，并讨论其对人工智能在数学中的应用的影响。
This short "education note" was inspired by Zvi Artstein's masterpiece Mathematics and the Real World, the Remarkable Role of Evolution in the Making of Mathematics (p. 53, and p. 400)
Mathematical Proof Between Generations
本文探讨数学证明的理论定义与实践使用之间的鸿沟，并结合计算机证明助手与形式化验证的发展，提出连接精确性与直觉、不同社群及代际的可能路径。
There is no trivial mathematics, there are only trivial mathematicians! A mathematician is trivial if he or she believes that there exists trivial mathematics. Being a non-trivial mathematician myself, I will describe ten different proofs of the seemingly trivial fact that the number of ways of choosing k people out of n people is less than or equal to the number of ways of choosing k+1 people out of n people, provided that k is less than half of n.
Automated Mathematics and the Reconfiguration of Proof and Labor
本文探讨自动化对数学证明和劳动的影响，分析其对证明标准、自动化形式及劳动力经济的重构作用。
Mathematics and the formal turn
本文围绕20世纪初确立的数学形式化基础，介绍计算证明助手实现数学知识数字化编码的作用，并列举相关技术助力数学研究的多种途径。
This paper contains a discussion of a library of formalized mathematics for the proof assistant Coq which the author worked on in 2011-13.
This survey paper is an expanded version of an invited keynote at the ThEdu'22 workshop, August 2022, in Haifa (Israel). After a short introduction on the developments of CAS, DGS and other useful technologies, we show implications in Mathematics Education, and in the broader frame of STEAM Education. In particular, we discuss the transformation of Mathematics Education into exploration-discovery-conjecture-proof scheme, avoiding usage as a black box . This scheme fits well into the so-called 4 C's of 21st Century Education. Communication and Collaboration are emphasized not only between human…
After a brief flirtation with logicism in 1917-1920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Goettingen between 1917 and 1923, a…
The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how…
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