Mathematical Spandrels

The aim of this paper is to open a new front in the debate between platonism and nominalism by arguing that the degree of explanatory entanglement of mathematics in science is much more extensive than has been hitherto acknowledged. Even standard examples, such as the prime life cycles of periodical cicadas, involve a penumbra of mathematical features whose presence can only be explained using relatively sophisticated mathematics. I introduce the term ‘mathematical spandrel’ to describe these penumbral properties, and focus on the property that cicada period lengths are expressible as the sum of two perfect squares. I argue that mathematical spandrels pose a particular problem for nominalism because of the way in which they are entangled with scientific explanations.     

学前儿童抑制控制的发展趋势及其对言语理解和数学认知的预测作用

以长春市某幼儿园的122名3~6岁幼儿为被试,采用2(测试任务:无嵌套规则白天黑夜任务,有嵌套规则的白天黑夜任务)×2(测试时间:间隔9个月的第一次测试和第二次测试)×4(年龄分组:3.00~3.75岁组,3.76~4.50岁组,4.51~5.25岁组,5.26~6.00岁组)的三因素混合设计,来分析不同复杂程度的抑制控制在学前阶段的发展趋势,并在控制幼儿年龄和工作记忆水平后,分析不同复杂程度抑制控制对言语理解和数学认知的独立预测作用。结果发现:(1)对3~6岁儿童来说,有嵌套规则的白天黑夜任务的难度水平要显著高于无嵌套规则的白天黑夜任务;(2)较简单的抑制控制(无嵌套规则的白天黑夜任务)在学前阶段的发展速度先快后慢,但较复杂的抑制控制(有嵌套规则的白天黑夜任务)在3~6岁期间持续快速发展;(3)独立于年龄和工作记忆,较复杂的抑制控制能够显著预测幼儿的数学认知和言语理解,但较简单的抑制控制对言语理解和数学认知的预测作用不显著。根据上述结果,在描述学前儿童抑制控制发展趋势时要注意任务难度的设定,而且难度适当的抑制控制任务成绩更能预测幼儿的言语理解和数学认知。     

认知压力与建构主义数学教学的关系

与通常在消极意义上使用的“精神压力”概念不同,“认知压力”是学习环境促使学习者进行不同类型认知活动的积极力量。根据这种环境导致学习者使用一级还是二级程序性知识达成目标,可将认知压力分成一级认知压力和二级认知压力。本研究以325名小学高年级和初中一二年级学生为被试,采用问卷法考察了数学课堂上认知压力的现状及其与建构主义教学的关系。结果表明：（1）对于数学课堂而言,被试报告的一级和二级认知压力处于中等偏上水平,而且二级认知压力显著高于一级认知压力;（2）建构主义数学教学有助于产生认知压力,特别是二级认知压力;（3）修订后的认知压力问卷有良好的信度和效度。     

Models for the statistics and mechanisms of response speed and accuracy

Van Breukelen offers a promising method for modeling both response speed and response accuracy. However, the underlying conception of both dependent measures is somewhat flawed, leading the author to conclude that the approach possesses limitations that, under revised assumptions, may not hold. The central misconception, and a set of related misconceptions, is addressed, and it is suggested that this approach holds a good deal of promise for application in the perceptual and cognitive sciences.     

Multidimensional adaptive testing with constraints on test content

The case of adaptive testing under a multidimensional response model with large numbers of constraints on the content of the test is addressed. The items in the test are selected using a shadow test approach. The 0–1 linear programming model that assembles the shadow tests maximizes posterior expected Kullback-Leibler information in the test. The procedure is illustrated for five different cases of multidimensionality. These cases differ in (a) the numbers of ability dimensions that are intentional or should be considered as nuisance dimensions and (b) whether the test should or should not display a simple structure with respect to the intentional ability dimensions.     

Philosophical virtues

It has been suggested that philosophers should adopt a methodology largely inspired by mathematics and that the “mathematical” virtues of rigor, clarity, and precision are also fundamental philosophical virtues. In reply, this paper argues that some excellent philosophy lacks these virtues and that too much emphasis on the mathematical virtues excludes potentially valuable forms of philosophical discourse and makes the profession less diverse than it should be. Unduly restrictive conceptions of philosophical argumentation should be avoided. On a contributory conception, philosophy should try to make a positive contribution to human emancipation where possible. The paper argues that it is possible and desirable for epistemology to contribute in this way and that the mathematical virtues are less significant in this context than the emancipatory virtues of what one might call “liberation philosophy.” These include irony, reflectiveness, imagination, contrarianism, and worldliness.     

Conceptual engineering for mathematical concepts

In this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory.