The Objectivity of Mathematics

The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.     

The Necessity of Mathematics

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.     

The values of work success

This study attempted to examine the work value correlates on various occupational scale dimensions like service, sales and management. In all 1458 people completed two validated questionnaires: one measuring six occupational scales (HPI: Hogan Personality Inventory) and the other measuring the values and preferences that indicate the type of work that an individual would like to do, and is best suited for (MVPI: Motives and Values Preferences Inventory). There were many sex differences on the values measured. Hierarchical regressions showed some values (Affiliation, Power, Recognition) related to many of the occupational scales but often in the opposite direction. Factor analysis suggested three overall value/vocation factors (Enterprising, Traditional and Social). Implications for vocational guidance and limitations of the study are considered.     

The effects of objective career success on subsequent subjective career success

We use a sample of working adults (N = 638) to explore the effects of past objective career success (mobility, promotions, and salary change) on current subjective success (human capital assessments by one's managers, core self evaluations, satisfaction with one's career) by gender, across an economic cycle (2004–2011), controlling for career stage. Results support a strong influence of past promotions, and less so for salary changes, on subjective career success. These effects were stronger for men and during the economic contraction, with managers being affected in their assessments based on the employees' past promotions. In contrast, past job mobility did not relate to subjective career success for either gender in periods of economic expansion or contraction. Evidence for an interactive perspective of career success whereby past objective success affects current subjective success is presented, as well as potential implications of the findings.     

The effects of objective career success on subsequent subjective career success

We use a sample of working adults (N = 638) to explore the effects of past objective career success (mobility, promotions, and salary change) on current subjective success (human capital assessments by one's managers, core self evaluations, satisfaction with one's career) by gender, across an economic cycle (2004–2011), controlling for career stage. Results support a strong influence of past promotions, and less so for salary changes, on subjective career success. These effects were stronger for men and during the economic contraction, with managers being affected in their assessments based on the employees' past promotions. In contrast, past job mobility did not relate to subjective career success for either gender in periods of economic expansion or contraction. Evidence for an interactive perspective of career success whereby past objective success affects current subjective success is presented, as well as potential implications of the findings.     

The Miracle of Applied Mathematics

Synthese - Mathematics has a great variety ofapplications in the physical sciences.This simple, undeniable fact, however,gives rise to an interestingphilosophical problem:why should physical...     

The Uses of Argument in Mathematics

Stephen Toulmin once observed that ‘it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate’ [Toulmin et al., 1979, An Introduction to Reasoning, Macmillan, London, p. 89]. Might the application of Toulmin’s layout of arguments to mathematics remedy this oversight? Toulmin’s critics fault the layout as requiring so much abstraction as to permit incompatible reconstructions. Mathematical proofs may indeed be represented by fundamentally distinct layouts. However, cases of genuine conflict characteristically reflect an underlying disagreement about the nature of the proof in question.     

The Role of Notations in Mathematics

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways.

     

经典数学的逻辑基础

<正>经典数学理论的逻辑完全是一阶逻辑还是也需要二阶逻辑?逻辑学家对此一直有争议。这种争议大约开始于20世纪20年代,但是似乎直到现在还未尘埃落定。在普遍接受反基础主义的前提下,数学基础的研究任务不再是为数学的各个分支寻找最大程度上免于理性怀疑的基础,而是在重构数学分支的过程中给出各个数学分支间的关系,描绘出数学的大图景。在这样的背景下,数学基础的研究不     

The Role of Axioms in Mathematics

To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support.

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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.     

Alternative Mathematics: The Vague Way

Synthese - Is alternative mathematics possible? More specifically,is it possible to imagine that mathematics could havedeveloped in any other than the actual direction? Theanswer defended in this...     

The success of Caucasian mothers in guiding adolescents

The purpose of this study was to determine how 391 Caucasian American mothers of 10- to 14-year-olds viewed their assets and limitations as parents and to find out how 352 adolescents perceived the parenting performance of their mothers. Both generations completed the Parent Success Indicator. It was found that the amount of time mothers spent talking to, and doing things with, their adolescent children had the greatest impact on how both generations rated mother success.     

The success of Taiwanese fathers in guiding adolescents

The purpose of this study was to examine similarities and differences in Taiwanese fathers' and adolescents' perceptions of paternal competencies. A multi-source (fathers (n = 176) and 10- 14-year-old adolescent children (n = 176), single-method (both generations completed the Parent Success Indicator) investigation was employed. Generational assessments were compared, and effects of independent variables were examined. Impressions from both generations were significantly different by child school grades and the amount of time fathers spent talking to and doing things with their adolescents.     

The Nctm Standards and the Philosophy of Mathematics

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy.