Foundations of Mathematics: Metaphysics, Epistemology, Structure

Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to provide an arena for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics.     

Process Ontology: Conversations and Argumentations,Controversies in Mathematics and Mathematics as Socialisation

To better conceive the socializing and pragmatic aspects of mathematics, it can be useful to use a process ontology, which allows, starting from an analysis of the processes of conversations, to compare their recourse, from degree to degree, to supposedly common “virtualities”, in particular in argumentative conversations, with the construction of more complex mathematical entities that allow new symmetries, but also with controversies between mathematicians on the use of these entities.

     

流畅体验、内/外动机、数学焦虑及数学成绩的路径分析

目的：研究旨在考察数学学习中的流畅体验,探讨流畅体验、内/外动机、数学焦虑及数学成绩的关系。方法：招募296名高中生,在数学月考后完成自编考试流畅体验量表、学习动机量表、数学焦虑量表。结果：路径分析表明,存在四条显著路径,即内部动机→流畅状态→数学成绩;数学焦虑→流畅状态→数学成绩;外部动机→流畅状态→数学成绩;外部动机→数学成绩。结论：只有通过考试过程中的流畅体验,内部动机和数学焦虑才能影响考试成绩。     

Mathematics,science and ontology

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible. The second section examines the problem as it was posed by Benacerraf in ‘Mathematical Truth’ and the next section presents a way of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them. Finally, I argue that all objects are abstract objects. Abstract objects should be viewed as the most general class of objects. The arguments derive from Quine. If all objects are abstract, and if we can have knowledge of any objects, then we can have knowledge of abstract objects and the question of mathematical knowledge is solved. A strict adherence to Quine's philosophy leads to a curious combination of the Platonism of Frege with the empiricism of Mill.     

Women,Education, Mathematics,and Computers: An Instance of Vocational Incongruence

A course in computer literacy was taken by 16 female elementary education majors specializing in mathematics. Results indicated that the students were dissimilar to the typical female college sample and to predicted occupational groups.     

Carnap,Tarski, and Quine at Harvard: Conversations on Logic,Mathematics, and Science

This paper is both a survey and a review of the current state of the debate concerning verisimilitude. As a survey it is intended for the interested outsider who wants both easy access to and some comparison between the respective approaches. As a review it covers the first three books on the topic: those of Oddie. Niiniluoto and Kuipers.     

Mathematics,Morality, and Self‐Effacement

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman‐style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman‐style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden‐shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are vulnerable to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman‐style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman‐style skeptical arguments against logical, mathematical, and normative beliefs are self‐effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality.     

Quine, Analyticity and Philosophy of Mathematics

Quine correctly argues that Carnap's distinction between internal and external questions rests on a distinction between analytic and synthetic, which Quine rejects. I argue that Quine needs something like Carnap's distinction to enable him to explain the obviousness of elementary mathematics, while at the same time continuing to maintain as he does that the ultimate ground for holding mathematics to be a body of truths lies in the contribution that mathematics makes to our overall scientific theory of the world. Quine's arguments against the analytic/synthetic distinction, even if fully accepted, still leave room for a notion of pragmatic analyticity sufficient for the indicated purpose.     

The Mathematics Anxiety Rating Scale,a brief version: psychometric data

As a measure of mathematics anxiety, the Mathematics Anxiety Rating Scale (MARS) has been a major scale used for research and clinical studies since 1972. Despite the usefulness of the original scale, researchers have sought a shorter version of the scale partly to reduce the administration time of the original 98-item scale. This study created a shorter version of the MARS and provides reliability and validity information for the new version. The Cronbach alpha of .96 indicated high internal consistency, while the test-retest reliability for the MARS 30-item was .90 (p<.001). The validity data confirm that the MARS 30-item test is comparable to the original MARS 98-item scale.     

Mathematics Participation and Mathematics Achievement Across Secondary School: The Role of Gender

The present study investigated whether the relationship between mathematics participation and mathematics achievement is reciprocal for boys and girls. In Years 1, 2, 4 and 6 (US grades 7, 8, 10 and 12), we administered mathematics achievement tests to a cohort of 1,495 Flemish students and collected data on the number of classroom hours allocated to mathematics. A cross-lagged panel design was used to analyze the data. Evidence was found for a reciprocal relationship between mathematics participation and mathematics achievement, particularly in Years 4 and 6 (US grades 10 and 12). The results suggest that boys’ better performance in mathematics is related to their higher participation in math, whereas other factors—in addition to gender differences in math achievement—play a role in explaining why boys participate more in mathematics than girls.