A theory of truth based on a medieval solution to the liar paradox

After the 1930s, the research into the foundations of mathematics changed.None of its main directions (logicism, formalism and intuitionism) had any longer the pretension to be the only true mathematics.Usually, the determining factor in the change is considered to be Gödel’s work, while Heyting’s role is neglected.In contrast, in this paper I first describe how Heyting directly suggested the abandonment of the big foundational questions and the putting forward of a new kind of foundational research consisting in the isolation of formal, intuitive, logical and platonistic elements within classical mathematics.Furthermore, I describe how Heyting indirectly influenced the abandon‐ment of the old directions of foundational research by making out some lists of degrees of evidence that exist within intuitionism     

The influence of Platonism on mathematics research and theological beliefs

An age-old debate in the philosophy of mathematics is whether mathematics is discovered or invented. There are four popular viewpoints in this debate, namely Platonism, formalism, intuitionism, and logicism. A natural question that arises is whether belief in one of these viewpoints affects the mathematician’s research? In particular, does subscribing to a Platonist or a formalist viewpoint influence how a mathematician conducts research? Does the area of research influence a mathematician’s beliefs on the nature of mathematics? How are the beliefs regarding the nature of mathematics connected to theological beliefs? In order to investigate these questions, five professional research mathematicians were interviewed. The mathematicians worked in diverse areas within analysis, algebra, and within applied mathematics, and had a combined 160 years of research experience. Although none of the mathematicians wanted to be pigeonholed into any one category of beliefs, the study revealed that four of the mathematicians leaned towards Platonism, which runs contrary to the popular notion that Platonism is an exception today. This study revealed that beliefs regarding the nature of mathematics influenced how mathematicians’ conducted research and were deeply connected to their theological beliefs. The findings are presented in the form of vignettes that give an insight into the mathematical and theological belief structures of the mathematicians.     

An Elementary System of Axioms for Euclidean Geometry Based on Symmetry Principles

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (1) it supports the thesis that Euclidean geometry is a priori, (2) it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, (3) it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics.     

Cognitive correlates of performance in advanced mathematics

Background. Much research has been devoted to understanding cognitive correlates of elementary mathematics performance, but little such research has been done for advanced mathematics (e.g., modern algebra, statistics, and mathematical logic). Aims. To promote mathematical knowledge among college students, it is necessary to understand what factors (including cognitive factors) are important for acquiring advanced mathematics. Samples. We recruited 80 undergraduates from four universities in Beijing. Methods. The current study investigated the associations between students’ performance on a test of advanced mathematics and a battery of 17 cognitive tasks on basic numerical processing, complex numerical processing, spatial abilities, language abilities, and general cognitive processing. Results. The results showed that spatial abilities were significantly correlated with performance in advanced mathematics after controlling for other factors. In addition, certain language abilities (i.e., comprehension of words and sentences) also made unique contributions. In contrast, basic numerical processing and computation were generally not correlated with performance in advanced mathematics. Conclusions. Results suggest that spatial abilities and language comprehension, but not basic numerical processing, may play an important role in advanced mathematics. These results are discussed in terms of their theoretical significance and practical implications.     

Students’ perception of instrumental support and effort in mathematics: the mediating role of subjective task values

Recent research shows that teacher support is predictive of student outcomes, such as engagement and effort. In this study, we explored the relation between students’ perceptions of teacher instrumental support in mathematics lessons and their effort in mathematics. We also tested whether this relation was mediated through students’ perception of the utility value, cost, and intrinsic value of mathematics. The study was designed as a cross-sectional survey. The participants included 309 9th and 10th grade Norwegian high school students. Three dimensions of mathematics task values were measured: utility value, intrinsic value, and the cost of working with mathematics. The data were analyzed by means of structural equation modeling. Instrumental support was directly and positively related to both the utility value and the intrinsic value of mathematics. However, it was only indirectly related to the perceived cost of working with mathematics, mediated by the utility value and the intrinsic value. Instrumental support was also both directly and indirectly related to effort. The indirect relation was mediated by the students’ perception of mathematics task values.     

Integrating Power to Advance the Study of Connective and Productive Disciplinary Engagement in Mathematics and Science

Abstract

Engle and Conant’s productive disciplinary engagement (PDE) framework has significantly advanced the study of learning in mathematics and science. This artilce revisits PDE through the lens of critical education research. Our analysis synthesizes two themes of power: epistemic diversity, and historicity and identity. We argue that these themes, when integrated into PDE, strengthen it as a tool for design and analysis of disciplinary learning in relation to power and personhood, and describe the broadened framework of connective and productive disciplinary engagement (CPDE). By comparing and contrasting the use of PDE and CPDE in relation to two cases of classroom learning—for science, Warren et al.’s metamorphosis and for mathematics, Godfrey and O’Connor’s measurement—we demonstrate how CPDE surfaces issues of history, power, and culture that may otherwise be overlooked by PDE alone. In particular, we analyze how CPDE makes visible unseen identities and generative resources of disciplinary knowing and doing among minoritized students. We discuss how the revised framework redresses epistemic injustice experienced by minoritized learners held to the narrow rubric of western epistemologies and compels close attention to the diversity of human activity in mathematics and science. Further, we elaborate how it provides a structure for teachers, teacher educators, and researchers to design and analyze learning environments as safeguarding the rightful presence of minoritized learners in STEM classrooms and beyond.     

Some Naturalistic Comments on Frege’s Philosophy of Mathematics

This paper compares Frege’s philosophy of mathematics with a naturalistic and nominalistic philosophy of mathematics developed in Ye (2010a, 2010b, 2010c, 2011), and it defends the latter against the former. The paper focuses on Frege’s account of the applicability of mathematics in the sciences and his conceptual realism. It argues that the naturalistic and nominalistic approach fares better than the Fregean approach in terms of its logical accuracy and clarity in explaining the applicability of mathematics in the sciences, its ability to reveal the real issues in explaining human epistemic and semantic access to objects, its prospect for resolving internal difficulties and developing into a full-fledged theory with rich details, as well its consistency with other areas of our scientific knowledge. Trivial criticisms such as “Frege is against naturalism here and therefore he is wrong” will be avoided as the paper tries to evaluate the two approaches on a neutral ground by focusing on meta-theoretical features such as accuracy, richness of detail, prospects for resolving internal issues, and consistency with other knowledge. The arguments in this paper apply not merely to Frege’s philosophy. They apply as well to all philosophies that accept a Fregean account of the applicability of mathematics or accept conceptual realism. Some of these philosophies profess to endorse naturalism.     

Children’s implicit and explicit gender stereotypes about mathematics and reading ability

Study objectives were to clarify children’s gender-based implicit and explicit mathematics and reading stereotypes, and to determine if implicit and explicit measures were related or represented distinct constructs. One hundred and fifty-six boys and girls (mean age 11.3 years) from six elementary schools completed math or reading stereotype measures. Results for the implicit measures showed that children believed their own gender was superior in mathematics ability, and that girls but not boys believed that girls were better in reading. Explicit measures revealed that girls but not boys believed they were superior at math, and that girls and boys believed girls were better readers than boys. Implicit and explicit measures were not related. Results are discussed in relation to previous studies on children’s mathematics and reading gender stereotypes and large scale tests of mathematics and reading achievement. Educational and research implications are discussed.     

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.     

Putnam, Peano, and the Malin Génie: could we possibly bewrong about elementary number-theory?

This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Gödel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Löwenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following (along with Saul Kripke's ‘scepticalsolution’), Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also adopts a thought-experimental approach – a variant of Descartes' dream scenario – in order to establish the in-principle possibility that we might be deceived by the apparent self-evidence of basic arithmetical truths or that it might be ‘rational’ to doubt them under some conceivable (even if imaginary) set of circumstances. Thus Putnam assumes that mathematical realism involves a self-contradictory ‘Platonist’ idea of our somehow having quasi-perceptual epistemic ‘contact’ with truths that in their very nature transcend the utmost reach of human cognitive grasp. On this account, quite simply, ‘nothing works’ in philosophy of mathematics since wecan either cling to that unworkable notion of objective (recognition-transcendent) truth or abandon mathematical realism in favour of a verificationist approach that restricts the range of admissible statements to those for which we happen to possess some means of proof or ascertainment. My essay puts the case, conversely, that these hyperbolic doubts are not forced upon us but result from a false understanding of mathematical realism – a curious mixture of idealist and empiricist themes – which effectively skews the debate toward a preordained sceptical conclusion. I then go on to mount a defence of mathematical realism with reference to recent work in this field and also to indicate some problems – as I seethem – with Putnam's thought-experimental approach as well ashis use of anti-realist arguments from Dummett, Kripke, Wittgenstein, and others.     

Avicenna on the Primary Propositions

Avicenna introduces the primary propositions (or the primaries, for short) as the most fundamental principles of knowledge. (In this paper, we are not primarily concerned with the primary/first intelligibles as concepts/conceptions.) However, as far as we are aware, Avicenna’s primaries have not yet been independently studied. Nor do Avicenna scholars agree on how to characterize them in the language of contemporary philosophy. It is well-known that the primaries are indemonstrable; nonetheless, it is not clear what the genealogy of the primaries is (§2), how, epistemologically speaking, they can be distinguished from other principles (§3), what their phenomenology is (§4), what the cause of the assent to them is (§5), how to explain the relationship between the ‘innate [nature] of the intellect’ and the primaries (§6) and, finally, back to their indemonstrability, in what sense they are ‘indemonstrable’ (§7). We will try to fill this gap. As a corollary, we will explain why Gutas’s view [Gutas, Dimitri. 2012. ‘The empiricism of Avicenna’, Oriens, 40, 391–436], among others, according to which the primaries are analytic (in the Kantian sense) is not true in general (§8). More particularly, we will argue that some primary propositions can be categorized under Kantian synthetic a priori, consistent with Black’s and Ardeshir’s conjecture [Black, Deborah L. 2013. ‘Certitude, justification, and the principles of knowledge in Avicenna’s epistemology’, in Peter Adamson, Interpreting Avicenna: Critical Essays, New York: Cambridge University Press; Ardeshir, Mohammad. 2008. ‘Ibn Sīnā’s philosophy of mathematics’, in S. Rahman, T. Street, and H. Tahiri, The Unity of Science in the Arabic Tradition, New York: Springer]. We hope that this work opens up some space to study Avicenna’s philosophy of mathematics and logic in connection with his epistemology, philosophy of mind and metaphysics.     

Wittgenstein and the Real Numbers

When it comes to Wittgenstein's philosophy of mathematics, even sympathetic admirers are cowed into submission by the many criticisms of influential authors in that field. They say something to the effect that Wittgenstein does not know enough about or have enough respect for mathematics, to take him as a serious philosopher of mathematics. They claim to catch Wittgenstein pooh-poohing the modern set-theoretic extensional conception of a real number. This article, however, will show that Wittgenstein's criticism is well grounded. A real number, as an ‘extension’, is a homeless fiction; ‘homeless’ in that it neither is supported by anything nor supports anything. The picture of a real number as an ‘extension’ is not supported by actual practice in calculus; calculus has nothing to do with ‘extensions’. The extensional, set-theoretic conception of a real number does not give a foundation for real analysis, either. The so-called complete theory of real numbers, which is essentially an extensional approach, does not define (in any sense of the word) the set of real numbers so as to justify their completeness, despite the common belief to the contrary. The only correct foundation of real analysis consists in its being ‘existential axiomatics’. And in real analysis, as existential axiomatics, a point on the real line need not be an ‘extension’.     

Brief Report: Is Gender a Factor in Mathematics Performance Among Nigerian Preservice Teachers?

The mathematics performance of graduating preservice teachers over a period of 3 years was examined for gender differences. Data were drawn from students’ (170 men and 202 women) final year results from a College of Education in Nigeria. Findings revealed that the gender gap in mathematics achievement among the sample data could be disappearing. This is a source of hope for the country because results such as those reported here are contrary to the general Nigerian stereotypical belief about men’s and women’s performance in the subject. Although it is difficult to generalise to other geographical areas, it is anticipated that the study would be replicated inthe rest of the country for a more meaningful and informative national picture.     

Mathematics and the Mystical in the Thought of Simone Weil

On Simone Weil’s “Pythagorean” view, mathematics has a mystical significance. In this paper, the nature of this significance and the coherence of Weil’s view are explored. To sharpen the discussion, consideration is given to both Rush Rhees’ criticism of Weil and Vance Morgan’s rebuttal of Rhees. It is argued here that while Morgan underestimates the force of Rhees’ criticism, Rhees’ take on Weil is, nevertheless, flawed for two reasons. First, Rhees fails to engage adequately with either the assumptions underlying Weil’s religious conception of philosophy or its dialectical method. Second, Rhees’ reading of Weil reflects an anti-Platonist conception of mathematics his justification of which is unsound and whose influence impedes recognition of the coherence of Weil’s position.     

弗协调的弗雷格(英文)

本文表明,二阶弗协调概括与弗雷格的第五公理是足道的。也表明,如果等数关系是初始符号,那么通过弗协调推理可以从第五公理可以推出休谟原则。最后表明,弗协调的休谟原则不能用作逻辑主义数学的基础。     

Hintikka’s Logical Revolution

Hintikka thinks that second-order logic is not pure logic, and because of Gödel’s incompleteness theorems, he suggests that we should liberate ourselves from the mistaken idea that first-order logic is the foundational logic of mathematics. With this background he introduces his independence friendly logic (IFL). In this paper, I argue that approaches taking Hintikka’s IFL as a foundational logic of mathematics face serious challenges. First, the quantifiers in Hintikka’s IFL are not distinguishable from Linström’s general quantifiers, which means that the quantifiers in IFL involve higher order entities. Second, if we take Wright’s interpretation of quantifiers or if we take Hale’s criterion for the identity of concepts, Quine’s thesis that second-order logic is set theory will be rejected. Third, Hintikka’s definition of truth itself cannot be expressed in the extension of language of IFL. Since second-order logic can do what IFL does, the significance of IFL for the foundations of mathematics is weakened.     

Proof vs Provability: On Brouwer’s Time Problem

Is a mathematical theorem proved because provable, or provable because proved? If Brouwer’s intuitionism is accepted, we’re committed, it seems, to the latter, which is highly problematic. Or so I will argue. This and other consequences of Brouwer’s attempt to found mathematics on the intuition of a move of time have heretofore been insufficiently appreciated. Whereas the mathematical anomalies of intuitionism have received enormous attention, too little time, I’ll try to show, has been devoted to some of the temporal anomalies that Brouwer has invited us to introduce into mathematics.     

Symbolic magnitude processing in elementary school children: A group administered paper-and-pencil measure (SYMP Test)

The ability to compare symbolic numerical magnitudes correlates with children’s concurrent and future mathematics achievement. We developed and evaluated a quick timed paper-and-pencil measure that can easily be used, for example in large-scale research, in which children have to cross out the numerically larger of two Arabic one- and two-digit numbers (SYMP Test). We investigated performance on this test in 1,588 primary school children (Grades 1–6) and examined in each grade its associations with mathematics achievement. The SYMP Test had satisfactory test-retest reliability. The SYMP Test showed significant and stable correlations with mathematics achievement for both one-digit and two-digit comparison, across all grades. This replicates the previously observed association between symbolic numerical magnitude processing and mathematics achievement, but extends it by showing that the association is observed in all grades in primary education and occurs for single- as well as multi-digit processing. Children with mathematical learning difficulties performed significantly lower on one-digit comparison and two-digit comparison in all grades. This all suggests satisfactory construct and criterion-related validity of the SYMP Test, which can be used in research, when performing large-scale (intervention) studies, and by practitioners, as screening measure to identify children at risk for mathematical difficulties or dyscalculia.