How to Apply Mathematics

This paper presents a novel account of applied mathematics. It shows how we can distinguish the physical content from the mathematical form of a scientific theory even in cases where the mathematics applied is indispensable and cannot be eliminated by paraphrase.     

How Mathematics Isn't Logic

If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting : nonuniform term substitution in logical sentences. 'Televisions are televisions' and 'TVs are televisions' neither sound alike nor are used interchangeably. Interception synonymy gets assumed because logical sentences and their synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The '=' of mathematics is an objectual relation between numbers; the '=' of logic marks a syntactic relation of coreferring terms.     

How Knowledge Works

The doctrine that knowledge is a species of belief has encouraged philosophers to confuse the question of what knowledge is and the question of how it can be acquired. But we can form a conception of knowledge by asking how knowledge gets expressed in our mental lives and in our conduct, instead of asking where it comes from. Accordingly, knowledge can be defined as the ability to do things, or refrain from doing things, or believe, or want, or doubt things, for reasons that are facts. I examine the nature of reasons, and the relationship between reasons, facts and beliefs; I consider the question of whether animals without language are capable of knowledge; and I briefly criticize Wittgenstein's doctrine that I cannot be said to know that I am in pain.     

学科领域知识与数学学习的知识表征研究

根据领域知识的结构特点对初中数学学科知识作出改编和重组,形成数学学科领域知识单元进行教学实验,以215名普通初中二年级学生为被试,通过自编问卷和深度访谈考察学科领域知识对知识表征的影响,并探讨了知识表征与数学学业成绩之间的关系。研究发现:(1)学科领域知识可以显著提高学生数学陈述性知识表征全面性和总体水平;(2)学科领域知识可以显著提高学生的程序性知识表征全面性、自动化、组织性和总体水平;(3)认知结构三要素、陈述性知识表征及程序性知识表征与数学成绩显著相关;(4)学优生的陈述性知识表征准确性、总体水平和程序性知识表征组织性、总体水平显著高于中等生和学困生。     

A Role for Mathematics in the Physical Sciences

Conflicting accounts of the role of mathematics in our physical theories can be traced to two principles. Mathematics appears to be both (1) theoretically indispensable, as we have no acceptable non‐mathematical versions of our theories, and (2) metaphysically dispensable, as mathematical entities, if they existed, would lack a relevant causal role in the physical world. I offer a new account of a role for mathematics in the physical sciences that emphasizes the epistemic benefits of having mathematics around when we do science. This account successfully reconciles theoretical indispensability and metaphysical dispensability and has important consequences for both advocates and critics of indispensability arguments for platonism about mathematics.     

婴儿关于物体支撑的直觉性物理知识

支撑问题是儿童朴素物理理论知识结构中的重要部分, 是指关于一个物体放在固体平面上是否稳定以及如何才能稳定的问题。支撑问题目前有两种主要实验方法。此领域中的主要实验研究使用这两种方法, 探索了婴儿获得与支撑相关的直觉性物理知识的发展过程及“先建立核心性的初始概念, 再以此为基础增加各类变量”的规律。目前研究中存在“范式单一、缺乏各类研究相互印证”的问题, 有待未来的研究进一步发展。     

Two Examples of How the History of Mathematics Can Inform Theology

In several of his writings, Bernard Lonergan emphasized the study of mathematics and especially its history as being important for philosophy and theology. In this article, I offer two examples of how the history of mathematics might inform theology. My first example explores the disparities between Euclid's Elements and Aquinas' Summa Contra Gentiles. My second example applies the idea of incompleteness to the “good and necessary consequence” clause of the Westminster Confession of Faith.     

中国逻辑如何影响中国传统数学(英文)

中国传统数学在中国古代逻辑特别是墨家逻辑的影响下,形成了自己的推理体系,即:以"以类合类"为基本方法,以"类"和"分类"为核心成分,以"推类"为主导推理形式。正是由于这一推理体系,使得中国传统数学在相当长的时期内保持世界的领先地位。因此,我们可以得出这样的结论:中国逻辑不是阻碍了中国传统数学进一步发展的因素,换句话说,中国传统数学所蕴含的逻辑方法并不是阻碍其进一步发展的原因。     

Computers as a Source of A Posteriori Knowledge in Mathematics

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to be surveyed by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions.     

On the Path to Knowledge Development

I consider the facilitation of knowledge development to be my central focus as editor. The editor makes the final decision on whether a specific research effort significantly advances our understanding of a substantive phenomenon. To that extent, the editor assumes a leadership role and responsibility in deciding the future direction of the field. Such important decisions should reflect the prevailing norms of the field and transcend the editor's individual research agenda or his/her selective worldview. I strive to make decisions that are shared by reviewers and associate editors with the common goal of moving the field forward in productive ways. As a result, articles published in JCP represent a collective view of what is considered significant theoretical contribution and rigorous methodology. In this note, I highlight some central elements of my philosophy related to knowledge development and share a few observations about policies that I have employed in managing the consensual review process.     

中小学生数学知识观的调查研究

本研究采用12道假设性情境题目和包含数学涉及运算、思考、实用性三个维度的数学认识问卷,选取90名六年级小学生和106名初二学生作为被试,探讨中小学生的数学知识观。结果表明:1)中小学生数学知识观形成及发展与学校数学课程内容紧密相关。2)中小学生数学知识观存在差异,主要表现在初二学生对假设性情境的认同程度显著地高于六年级学生;在数学认识问卷的数学实用性维度上,六年级学生的肯定程度显著地高于初二学生。3)从总体上看,学生的学业水平与数学知识观的关系不大。     

分布式知识的研究进展

分布式知识是经典认知逻辑群体知识中的一个重要概念。群体知识主要有普遍知识(general knowledge,universal knowledge)、公共知识(common knowledge)和分布式知识(distrib-uted knowledge,group implicit knowledge)三种知识。群体知识在认知逻辑的研究中占有很重要的地位,因为多主体之间的信息交流和互动都是建立在群体知识的基础之上的。普遍知识因为可以用群体中的个体知识定义,所以对于群体知识的研究主要集中在公共知识和分布式知识上。20世纪90年代,哈尔彭(J.Y.Halpern)、摩西(Y.Moses)、瓦迪(M.Vardi)等人对公共知识进行了深入…     

Students' Coordination of Knowledge When Learning to Model Physical Situations

In this article, I present a study in which 12 pairs of 8th-grade students solved problems about a physical device with algebra. The device, called a winch, instantiates motions that can be modeled by pairs of simultaneous linear functions. The following question motivated the study: How can students generate algebraic models without direct instruction from more experienced others? The first main result of the study is that students have and can use criteria for judging when 1 algebraic expression is better than another. Thus, students can use criteria to regulate their problem-solving activity. The second main result is that constructing knowledge for modeling with algebra can require students to coordinate criteria for algebraic representations with several other types of knowledge that I also identify in the article. These results contribute to research on students' algebraic modeling, cognitive processes and knowledge structures for using mathematical representations, and the development of mathematical knowledge.     

小学数学教师职业知识的结构与内在关系

本研究以小学数学分数起始教学为媒介,通过自编的小学数学教师职业知识测验,考察了162名小学数学教师在学科知识、一般教育学知识和教育实践知识等三类知识上的表现,并据此探讨了这三类职业知识间的相互关系。研究结果表明,自编的小学数学职业知识测验具有良好的信度和结构效度,教师的学科知识和一般教育学知识能显著地预测其教育实践知识。     

How Not To Have an Ontology of Physical OBJECTS

Philosophical Studies -     

The Knowledge of Child Development Inventory

The need for an instrument that measures knowledge of child development and is comprehensible to lay populations and young adolescents has led to the development of the Knowledge of Child Development Inventory (KCDI). The KCDI is a 56-item multiple-choice test of knowledge of child development from birth to age three in the areas of emotional, cognitive, physical, and social development. Normative data were obtained on the KCDI from a sample population of 434 adolescent and adult females from varying racial backgrounds. The instrument has a reading level of 8.0 (Fry Reading Index), a minimum of technical terminology, reliability of .93 (Cronback's alpha coefficient of internal consistency), and criterion validity of .83. The KCDI would be relevant in educational settings where pre- and post-assessment of knowledge of child development is desired, i.e., high school courses in child development, prenatal classes, Head Start parent-training programs, and early childhood/special education parent-training programs. In addition, the KCDI would be relevant in clinical settings where the assessment of a person's concepts of child development are of importance i.e., adoption agencies, abuse centers, and teenage pregnancy centers.     

The Development of Attribute Dominance in the Knowledge Base

Two cuing, free-recall studies were conducted to test Bach and Underwood's (1970) hypothesis that acoustic encoding is dominant among second graders and semantic encoding is dominant among sixth graders. When retrieval cues were presented with to-be-remembered items at both input and output (Experiment 1), and when cues were presented only at output (Experiment 2), semantic cues were more efficient in elevating recall than were acoustic cues for both second and sixth graders. When these and other results generally found using recognition, sorting, incidental learning, and free-recall experimental designs are compared, it seems plausible that item presentation and memory-testing formats interact with age, and that these factors account for the different patterns of attribute dominance found in the literature. The knowledge base cannot be understood by focusing on either subject or task analyses, but only by focusing on interactions between subject and task variables as they change over time. The educational implications for young grade-school children are discussed.