The purpose of this study was to determine the form of teacher readiness and difficulty when implementing the mathematics curriculum in high school, measured from four teacher competency assessments. Schools in Indonesia are still 50% learning from home until 2022, this situation has an impact on the achievement of student learning outcomes. In 2020, the ministry conducted a survey of 4000 students and the results of the survey and out of 100% of the participants, 58% were of (...) the opinion that online learning was ineffective, 38% lacked attention from the teacher, 35% had poor communication, 62% had poor internet access, and 80% had difficulty learning mathematics online. The method in this research is descriptive qualitative. The technique of collecting data is by means of field observations, interviews and collecting documentary evidence. Data analysis techniques by presenting data, data reduction and intersecting data are drawn into conclusions. The number of people who were observed was 2 with a total of 58 and 13 people interviewed sources. Findings, difficulties arise from unprepared pedagogic competence and professional competence. Teachers have no self-constructed modules, difficulty writing symbols and mathematical proofs online. Students provide an assessment of teacher competence online, from 100% of respondents 80% disagree with pedagogic and 90% disagree with professional competence. This is due to the lack of readiness of the two competencies. In conclusion, online learning is expected by students and teachers to have their own modules. Students argue that the teacher's difficulties when writing symbols, providing examples of questions and proving mathematical formulas can be written in modules and modules can be given to students to study at home. (shrink)

The study determined the influence of innate mathematical characteristics on the number sense competencies of junior high school students in a Philippine public school. The descriptive-correlational research design was used to accomplish the study involving a nonrandom sample of sixty 7th-grade students attending synchronous math sessions. Data obtained from the math-specific Learning Style and Self-Efficacy questionnaires and the modified Number Sense Test (NST) were analyzed and interpreted using descriptive statistics, Pearson’s Chi-Square, and Simple Linear Regression analysis. The research instruments and (...) statistics were all validated and tested for reliability. The analysis revealed that the students are visual learners, had no or slight self-efficacy, and their number sense competency level is poor. They encountered difficulty in all the components and domains of the NST. Moreover, the students’ mathematical self-efficacy is significantly related and may influence their number sense competency level. Building upon the learners’ self-efficacy to further their understanding and skills in number sense is necessary. (shrink)

This study aimed to investigate the longitudinal trends in mathematical competencies of Grade 8 students in a public high school located in Zamboanga del Sur, Philippines. The study collected data over a period of six academic years, allowing for a comprehensive analysis of students' performance in 16 distinct mathematical competences of basic education curriculum. These topics include, but are not limited to, special products and factors, factoring, and basic concepts of probability. Using a quantitative research design, the study analyzed both (...) numerical and secondary data to identify patterns or trends in students' mathematical competencies. Trends were analyzed through regression analysis using slopes. The significance of these trends was then evaluated using t-tests for the standard error of slopes. The statistical analysis revealed significant upward trends in certain mathematical topics, indicating an improvement in the students' competencies over the six academic years. However, certain topics displayed downward or stable trends, indicating areas for improvement in the teaching methods used for these topics. The study highlights the importance of flexible and adaptable approaches to mathematics education. It emphasizes the need for constant evaluation and improvement of teaching methods to enhance student learning outcomes. The findings of this study provide valuable insights for educators and policymakers to guide evidence-based decision-making and improve the quality of mathematics education in the new normal setup. Implications and future research directions are incorporated in the study. (shrink)

Today’s societies place challenging demands on individuals, who are confronted with complexity in many aspects of their lives. Individuals need to acquire a wide range of competencies in order to overcome the complex challenges of today’s world. Using real-world problems is important not only to hone students’ mathematical thinking and competency but also to prepare them in making well-grounded decisions that involve logical and mathematical reasoning. Thus, this study explored the competence of the students in applying mathematics in (...) real world problems and determined the predictors using the selected profile variables. Regression analysis was used to identify if there are predictors that determine the results of the assessment. The main tool of the study was the Programme for International Student Assessment (PISA) test in mathematics. Findings revealed that the predictors include confidence level, repeated kinder, mother work status and highest level of education to complete. This study could be a guide for determining the strengths and weaknesses of students in applying mathematics in the real world as framework in helping them handle the rigors of connecting math with their lives. (shrink)

Competence structure that pre-service teachers need to develop to become a future teacher has been defined since the 1930s. For pre-service mathematics teachers, their competence has its own characteristics. ICT integration in developing competence for pre-service mathematics teachers has been been proved to be effective in many previous studies. In Vietnam, the Ministry of Education and Training (MOET) has recommended the use of ICT to enhance teaching-learning activities in schools and universities, therefore, there have been many studies on (...) ICT intergration at different educational levels. However, there are only a few studies on the use ICT integration in developing competence for pre-service mathematics teachers. This paper presents results from a research on the feasibility of ICT integration in developing competence of Vietnamese pre-service mathematics teachers. The research was conducted by surveying on a randomly selected of 297 pre-service mathematics teachers and 40 mathematics lecturers in six universities which provide mathematics training programs. Results show that despite of the availability of technologies, ICT integration in training pre-service teachers is still limited. In addition, ICT integration in teaching activities of mathematics teachers is not really effective because the levels of their ICT skills are very different. Pre-service mathematics teachers have the ability to use ICT equipment, and good accessibility to learning systems, but their use of the learning management systems (LMS) is limited. This study can be a reference for future studies to test the effective of ICT integration in training pre-service teachers when designing activities to develop feasibly and effectively their competence. (shrink)

Mathematical modeling offers a promising approach to improving mathematics education. This study aims to determine if the concept of metacognitive awareness in the learning process is associated with mathematical modeling. This study also considers the interaction effect of sex and academic year level on both variables. Focusing the study on preservice elementary teachers might address potential issues and targeted intervention in their preparation program concerning their ability to teach and guide young learners in modeling activities. The research sample includes (...) 140 preservice elementary teachers at Central Luzon State University, Philippines. Data collection used an adapted metacognitive awareness inventory and a validated researcher-made mathematical modeling competency test aligned with the K-12 mathematics curriculum in the Philippines. Results revealed that the preservice elementary teachers had a high metacognitive awareness and mathematical modeling competency, ranging from 22 to 31 out of 36 points. Besides, Factorial ANOVA indicates that academic year level positively affects both variables regardless of sex, and stepwise regression analysis unveiled that information management, declarative knowledge, and planning significantly predict 41.4% of the mathematical modeling competency variance. This suggests that developing metacognitive awareness supports preservice elementary teachers in performing modeling tasks that improve their competency level in mathematics. (shrink)

This study aimed to construct structural equation model of students’ competence in mathematics through selected students profile variables. The structural model revealed interesting influence of the profile variables to the competency in mathematics. It can be conveyed that better mother’s work status, higher educational level expected to complete, more confident and did not repeat kinder, have better competency in mathematics. The four variables that directly influenced the competence variables were also influenced with other profile variables (...) such as family background. The family background and confidence level consistently had the highest total effect and indirect effect to the competence in mathematics. Hence, this model can serve as a guide in making programs in the classroom or curriculum in mathematics. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

To help clarify the concept of «teaching practice», this article analyzes a repository of professional skills of the teacher. It then examines how two groups of teacher educators attempting to use this repository to evaluate two teaching sessions (one secondary and one primary). Analyses revealed the difficulty in assessing a session from elements determined a priori from a list of skills. The evaluation of a teaching session seems to require the inclusion of its organic whole. Afin de contribuer à (...) préciser la notion de « pratique enseignante », le présent article analyse un référentiel de compétences professionnelles de l’enseignant. Il examine ensuite la manière dont deux groupes de formateurs d’enseignants tentent d’utiliser ce référentiel pour évaluer deux séances d’enseignement (une dans le secondaire et une dans le primaire). Les analyses révèlent la difficulté à évaluer une séance à partir d’éléments déterminés a priori dans une liste de compétences. L’évaluation d’une séance d’enseignement semble requérir la prise en compte de sa totalité organique. (shrink)

This paper aims to address the difficulties of high school students in bridging their computational understanding with their visualization skills in understanding the notion of the limits in their calculus class. This research used a pre-experimental one-group pretest-posttest design research on 62 grade 10 students enrolled in the Science, Technology, and Engineering Program (STEP) in one of the public high schools in Zamboanga del Sur, Philippines. A series of remedial sessions were given to help them understand the function values, one-sided (...) limits, limits of a function, and its discontinuities with the aid of the GeoGebra. Results revealed that there is a significant improvement in their mathematics performance before and after the remediation, t(61) = -19.99, p<.05. The real-time feedback provided by the drawing interface of GeoGebra has provided the students the insights into how objects behave in geometrical space, bridging their understanding in locating the values shown by the computational algebraic processes. Additionally, the significant difference in the achievement reveals a large effect size (Cohen’s =2.54) which could be accounted for by the remediation using interactive activities in the GeoGebra. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors (...) tried to naturalize mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

An inquiry on the training needs in Mathematics was conducted to Laura Vicuña Center - Palawan (LVC-P) learners. Specifically, this aimed to determine their level of performance in numbers, measurement, geometry, algebra, and statistics, identify the difficulties they encountered in solving word problems and enumerate topics where they needed coaching. -/- To identify specific training needs, the study employed a descriptive research design where 36 participants were sampled purposively. The data were gathered through a problem set test and focus (...) group discussion. Findings revealed that LVC-P learners had an unsatisfactory performance in numbers, measurement, and statistics while alarmingly poor in geometry and algebra. They also faced difficulties in remembering, understanding, applying, and analyzing mathematical concepts when solving problems. Further, the learners were able to name certain topics subjected to tutorials. -/- The above facts and observations suggest that learners of LVC-P have urgent training needs in Mathematics. It is recommended that the Western Philippines University - College of Education RDE Unit continue its research, development, and extension services to LVC-P. More importantly, the results of this inquiry regarding the training needs of the learners will serve as bases for conducting an extension project and development program for LVC-P. Series of tutorial sessions is deemed necessary to address the needs and difficulties of the learners. (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could (...) mathematics be knowledge about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

The ability to solve problems is a prerequisite in preparing mathematics preservice teachers. This study assessed preservice teachers’ problem-solving difficulties and performance, particularly in worded problems on number sense, measurement, geometry, algebra, and probability. Also, academic profile differences in the preservice teacher’s problem-solving performance and common errors were determined. A descriptive-comparative research design was employed with 158 random respondents. Data were gathered face-to-face during the first semester of the school year 2022-2023, and data were analyzed with the aid of (...) jamovi software, ensuring ethical measures. Overall findings revealed that the preservice teachers experienced average difficulty in solving problems. The low performance of the preservice teachers on the given problems was also demonstrated. Further analysis revealed a significant difference between the preservice teachers’ problem-solving performance based on their subject preference and program. Moreover, the error analysis revealed that the preservice teachers incurred comprehension errors in misrepresentation, misinterpretation, and miscalculation. These results will serve as a measure for policymakers and curriculum developers of the teacher education institution concerned to make relevant enhancements to the math courses offered in the elementary and secondary education programs. (shrink)

The sudden shift in the education system during the pandemic and its subsequent evolution during the post-pandemic era have been pivotal in fostering significant educational development and growth. However, this paradigm shift has not been without challenges. This paper aims to investigate the challenges faced by 68 mathematics teachers in four public elementary schools in the Philippines. The respondents were purposively selected to answer the study’s instrument. Using a descriptive survey research methodology, this study explored the five domains in (...) teaching mathematics, namely planning lessons, executing lessons, assessment, mathematical knowledge, and using technology tools. This paper revealed that novice, experienced, and highly experienced teachers alike had moderate difficulty enriching mathematical knowledge and utilizing technology in teaching the subject matter. Moreover, teachers with more than five years’ experience had moderate difficulty in the five domains of teaching mathematics in the post-pandemic era. This finding emphasizes the widespread impact of the education shift for teachers of all levels of experience. The paper further emphasizes the importance of supporting experienced teachers during this transformative phase of formulating innovative learning activities and adapting to reimagined pedagogical concepts. Consequently, this study underscores the need for additional training interventions to enhance their mathematical knowledge and use of technology as an integral teaching tool. (shrink)

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when (...) one tries to naturalize in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method. (shrink)

Mathematics is considered one of the most challenging courses for many elementary pre-service teachers (EPTs). In hopes for improvement, teacher educators have been incorporating a jigsaw strategy to equip future teachers with the necessary competencies. Hence, the study aimed to determine the effectiveness of the jigsaw strategy with EPTs’ level of mathematics achievement and interest in an online cooperative learning environment. The research utilized a pre-experimental design with 40 participants. The researchers used pretest and posttest questions covering fraction (...) operations and a mathematics interest questionnaire before and after the intervention. With the implementation of the jigsaw strategy, results showed that the EPTs’ mathematics achievement improved while their interest in mathematics improved to some extent. Likewise, a significant difference between the pre-and post-intervention concerning the EPTs’ mathematics achievement and interest in mathematics was found. Furthermore, correlation analysis revealed that study time is negatively related to mathematics achievement, as is the overall level of interest in mathematics. These findings imply that a jigsaw strategy is a useful tool in teaching mathematics online. However, several modifications may be considered to recognize that the EPTs’ interest in mathematics varies from one person to another. Teacher education institutions are encouraged to apply this strategy not only for EPTs to increase their mathematical know-hows but also to serve as an example for them to employ in their future teaching careers. (shrink)

We argue that the mathematization of science should be understood as a normative activity of advocating for a particular methodology with its own criteria for evaluating good research. As a case study, we examine the mathematization of taxonomic classification in systematic biology. We show how mathematization is a normative activity by contrasting its distinctive features in numerical taxonomy in the 1960s with an earlier reform advocated by Ernst Mayr starting in the 1940s. Both Mayr and the numerical taxonomists sought to (...) formalize the work of classification, but Mayr introduced a qualitative formalism based on human judgment for determining the taxonomic rank of populations, while the numerical taxonomists introduced a quantitative formalism based on automated procedures for computing classifications. The key contrast between Mayr and the numerical taxonomists is how they conceptualized the temporal structure of the workflow of classification, specifically where they allowed meta-level discourse about difficulties in producing the classification. (shrink)

As part of the development of an epistemology for mathematics, some Platonists have defended the view that we have (i) intuition that certain mathematical principles hold, and (ii) intuition of the properties of some mathematical objects. In this paper, I discuss some difficulties that this view faces to accommodate some salient features of mathematical practice. I then offer an alternative, agnostic nominalist proposal in which, despite the role played by mathematical intuition, these difficulties do not emerge.

This thesis starts with three challenges to the structuralist accounts of applied mathematics. Structuralism views applied mathematics as a matter of building mapping functions between mathematical and target-ended structures. The first challenge concerns how it is possible for a non-mathematical target to be represented mathematically when the mapping functions per se are mathematical objects. The second challenge arises out of inconsistent early calculus, which suggests that mathematical representation does not require rigorous mathematical structures. The third challenge comes from (...) renormalisation group (RG) explanations of universality. It is argued that the structural mapping between the world and a highly abstract minimal model adds little value to our understanding of how RG obtains its explanatory force. I will address the first and second challenges from the similarity perspective. The similarity account captures representations as similarity relations, providing a more flexible and broader conception of representation than structuralism. It is the specification of the respect and degree of similarity that forges mathematics into a context of representation and directs it to represent a specific system in reality. Structuralism is treatable as a tool for explicating similarity rela-tions set-theoretically. The similarity account, combined with other approaches (e.g., Nguyen and Frigg’s extensional abstraction account and van Fraassen’s pragmatic equivalence), can dissolve the first challenge. Additionally, I will make a structuralist response to the second challenge, and suggestions regarding the role of infinitesimals from the similarity perspective. In light of the similarity account, I will propose the “hotchpotch picture” as a method-ological reflection of our study of representation and explanation. Its central insight is to dissect a representation or an explanation into several aspects and use different theories (that are usually thought of competing) to appropriate each of them. Based on the hotchpotch picture, RG explanations can be dissected to the “indexing” and “inferential” conceptions of explanation, which are captured or characterised by structural mappings. Therefore, structuralism accommodates RG explanations, and the third challenge is resolved. (shrink)

I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological account that combines the theoretical frameworks of enculturation and cumulative cultural evolution. I show that this account can explain why arithmetical knowledge appears to be objective and has scientific applications. Finally, I will argue (...) that, while this account is compatible with platonist metaphysics, it does not require postulating mind-independent mathematical objects. (shrink)

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods from (...) computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way. (shrink)

It has become a fact that fluency and competency in utilizing the advancement of technology, specifically the computer and the internet is one way that could help in facilitating learning in mathematics. This study investigated the effects of Dynamic Geometry Software (DGS) and Computer Algebra Systems (CAS) in teaching Mathematics. This was conducted in Zamboanga del Sur National High School (ZSNHS) during the third grading period of the school year 2015-2016. The study compared the achievement and attitude (...) towards Mathematics between those students taught with DGS and CAS and those students who were taught without DGS and CAS. A quasi-experimental pretest-posttest control group design was used with two groups which was matched according to their mathematical level. An achievement test was developed and administered to assess the students’ achievement in Mathematics. Aiken’s attitude scale was also used to assess the students’ attitude towards Mathematics. The data were treated with One-Way Analysis of Covariance (ANCOVA) and t-test at 0.05 level of significance. The study revealed a significant difference between the mathematics achievement between students taught with DGS and CAS and those who were taught without it. The integration of DGS and CAS in Mathematics lessons leads to better academic achievement result compared with that of the conventional method of teaching. Moreover, there was no significant difference on the attitude level of the students between the two groups. (shrink)

The current study evaluates qualitatively how the mathematical dimension of gambling is reflected in the content of gambling websites. A number of gambling websites have been reviewed for their content in that respect. A statistical analysis recorded the presence of the mathematical dimension of gambling and its forms in the content of the participating websites, and a qualitative research study analyzed and assessed the quality of the content with respect to that dimension. The technical reports associated with this study describe (...) the methods used in the processes of review and content analysis, details of the processes, and the specific challenges and difficulties that they posed. (shrink)

A new hypothesis on the basic features characterising the Foundations of Mathematics is suggested. By means of them the entire historical development of Mathematics before the 20th Century is summarised through a table. Also the several programs, launched around the year 1900, on the Foundations of Mathematics are characterised by a corresponding table. The major difficulty that these programs met was to recognize an alternative to the basic feature of the deductive organization of a theory - (...) more precisely, to Hilbert’s main tenet. Ironically, already half a century before the births of these programs the alternative organization has been substantially represented by Lobachevsky's theory on parallel lines. Moreover, although each program’s founder recognised the basic features in a partial way only, all together these programs represented just the four possible foundational approaches. (shrink)

The objective of this case study concentrated on examining the learning gap, going through some components of the transformation process, and coming up with some ways for aiding students who were experiencing lost learning. A qualitative research design was utilized by the researchers to understand and solve the cases related to Mathematics learning difficulties. Creswell (2008) asserts that qualitative research can be used to discover and comprehend the significance that certain people or groups assign to social or human issues. (...) The researchers used purposive sampling to determine 15 student participants to understand the cases of learning loss and prescribe better interventions to the students. To solve the problems associated with loss of learning experienced by the learners after remote learning, a case study will be utilized as the strategy of inquiry since it fits the situation that is to focus the said key cases, discuss some transforming process, and develop interventions. After analyzing the interviews, researchers identified three common themes across the three cases which serve to clarify the learners' experiences after the post-remote learning: the learners' learning experiences during remote learning, factors that caused learning loss in Mathematics, and learners' coping strategies. In a deeper analysis, the impact of distance learning on students during the pandemic goes beyond just the difficulties in learning and understanding lessons which resulted in difficulty in learning math during in-person classes. The lack of in-person interaction with classmates and teachers also contributes to a sense of isolation and boredom, negatively affecting students' overall well-being and mental health. (shrink)

Descartes' philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this article, I (...) show why there is no numerical infinity in Cartesian mathematics, as such a concept would be inconsistent with the main fundamental attribute of numbers: to be comparable with each other. In the second part, I analyze the indefinite in the context of Descartes' mathematical physics. It is my contention that, even with no trace of infinite in his mathematics, Descartes does refer to an actual indefinite because of its application to the material world within the system of his physics. This fact underlines a discrepancy between his mathematics and physics of the infinite, but does not lead a difficulty in his mathematical physics. Thus, in Descartes' physics, the indefinite refers to an actual dimension of the world rather than to an Aristotelian mathematical potential infinity. In fact, Descartes establishes the reality and limitlessness of the extension of the cosmos and, by extension, the actual nature of his indefinite world. This indefinite has a physical dimension, even if it is not measurable. La filosofía de Descartes contiene una noción intrigante de lo infinito, un concepto nombrado por el filósofo como indefinido. Aunque en varias ocasiones Descartes definió claramente este término en su correspondencia con sus contemporáneos y en sus Principios de filosofía, han surgido muchos problemas acerca de su significado a lo largo de los años. La mayoría de comentaristas rechaza la idea de que indefinido podría significar una cosa real y, en cambio, la identifica con un infinito potencial aristotélico. En la primera parte de este artículo muestro por qué no hay infinito numérico en las matemáticas cartesianas, en la medida en que tal concepto sería inconsistente con el principal atributo fundamental de los números: ser comparables entre sí. En la segunda parte analizo lo indefinido en el contexto de la física matemática de Descartes. Mi argumento es que, aunque no hay rastro de infinito en sus matemáticas, Descartes se refiere a un indefinido real a causa de sus aplicaciones al mundo material dentro del sistema de su física. Este hecho subraya una discrepancia entre sus matemáticas y su física de lo infinito, pero no implica ninguna dificultad en su física matemática. Así pues, en la física de Descartes, lo indefinido se refiere a una dimensión real del mundo más que a una infinitud potencial matemática aristotélica. De hecho, Descartes establece la realidad e infinitud de la extensión del cosmos y, por extensión, la naturaleza real de su mundo indefinido. Esta indefinición tiene una dimensión física aunque no sea medible. (shrink)

In their recent article, Graber and Golemon (_Sophia_ 1–18, 2019 ) argue that any attempted evolutionary debunking of naturalism faces a dilemma. First, in order to be evolutionarily plausible, the skeptical implications must not be too broad. Second, in order to constitute a genuine challenge to scientific realism, the skeptical implications must not be too narrow. Graber and Golemon further develop an evolutionary debunking argument that avoids both horns of this dilemma. De Ray (_Erkenntnis_ 1–21, 2020 ) criticizes Graber and (...) Golemon’s debunking argument then develops his own, competing debunking argument. We first respond to de Ray’s criticisms by showing how Graber and Golemon’s argument can be extended so as to avoid de Ray’s critique. We then show that de Ray’s argument debunking argument is impaled by both horns of Graber and Golemon’s dilemma. The ultimate result will be both a better understanding of the desiderata and the argumentative contours of a successful evolutionary debunking argument. (shrink)

This article examines the possibility of philosophizing about mathematics with children. It aims at outlining the nature of the practice of philosophy of mathematics with children in a mainly theoretical and exploratory way. First, an attempt at a definition is proposed. Second, I suggest some reasons that might motivate such a practice. My thesis is that one can identify an intrinsic as well as two extrinsic goals of philosophizing about mathematics with children. The intrinsic goal is related (...) to a presumed inherent importance of presenting children with some philosophical questions about mathematics. The extrinsic goals consist of first the positive effects such a practice can have on mathematical learning and abilities and second the fostering of children's understanding of philosophical method of inquiry and thinking and therefore of their philosophical thinking competences. Third, some examples found in the literature of previously developed ways of practising philosophy of mathematics with children are presented. This article aims at giving a general outlining picture of the issues surrounding the practice of philosophy of mathematics with children and should therefore be read as an encouragement to further development and studies. (shrink)

Mathematics has a long track record of refining the concepts by which we make sense of the world. For example, mathematics allows one to speak about different senses of "sameness", depending on the larger context. Phenomenology is the name of a philosophical discipline that tries to systematically investigate the first-personal perspective on reality and how it is constituted. Together, mathematics and phenomenology seem to be a good fit to derive statements about our experience that are, at the (...) same time, well-defined, precise, and significant to our inner lives. However, there are difficulties stemming from the fact that phenomenology deals with inherently subjective things. Phenomenological investigations seem to lose their appeal when trying to approach them in the clear-cut way afforded by mathematics. How to overcome this obstacle? We argue that the Arts play a special role in mediating between the precise statements of mathematics and the sometimes fuzzy nature of our experience. Mathematics and art are complementary ways to come to a comprehensive understanding of reality. (shrink)

This research article was championed as a way of providing discourses pertaining to the concept of "Critical Realism (CR)" approach, which is amongst many othe forms of competing postmodern philosophical concepts for the engagement of dialogical discourses in the area of established econonetric methodologies for effective policy prescription in the economic science discipline. On the the whole, there is no doubt surrounding the value of empirical endeavours in econometrics to address real world economic problems, but equally so, the heavy weighted (...) use and reliance on mathematical contents as a way of justifying its scientific base seemed to be loosing traction of the intended focus of economics when it comes to confronting real world problems in the domain of social interaction. In this vein, the construction of mixed methods discourse(s), which favour that of CR philosophy is hereby suggested in this article as a way forward in confronting with issues raised by critics of mainstream economics and other professionals in the postmodern era. (shrink)

Various changes and advances have taken place in the ways of teaching and learning. With the encountered global pandemic crisis, flexible learning has been practiced particularly in state universities and colleges. This study sought to determine the level of procedural fluency in College Algebra as well as the difficulties and interventions applied by tertiary students of Cotabato State University (CSU) in the flexible learning set-up. The period covered by the study was during the first semester of the school year 2021-2022. (...) The respondents were a sample of 30 first-year mathematics major students of CSU. It used mixed methods; the quantitative approach was utilized to know the students’ knowledge of procedures in solving mathematical problems while the qualitative approach was utilized to explore the difficulties of the students in learning mathematics as well as the coping strategies that can serve as basis for mathematics intervention program. The findings revealed that most of the respondents have a low level of procedural fluency in College Algebra, particularly emerging in the understanding the problem, devising a strategy, and solving steps while inept in the problem-solving checking process. On the part of new normal learning, difficulties were experienced such as poor knowledge of the prerequisite skills and unstable internet connection. As part of their interventions, facing the encountered problems, practicing repeatedly, and watching online video tutorials were applied. Based on the findings of the study, it is concluded that the first-year students have low level of procedural fluency in College Algebra problem-solving particularly in the Polya’s principles of understanding the problem, devising a plan, carrying out the plan, and checking the answer. Because of the learning challenges encountered by the students, it is recommended to develop step-by-step video tutorials that can address their problem-solving difficulties and supplement their learning in the flexible mode of delivery. (shrink)

The researcher was prompted to conduct this study to give intervention of the alarming situation which there is a low performance in solving problems related to geometry in Grade IV Mathematics. This study was about on how to enhance the mathematical competencies of the grade IV pupils using a downloaded worksheets as a learning activity. This study focused in giving remediation applying the intervention materials. These resources give several approaches to attain mastery using distinct drill cards. The investigation was (...) carried out utilizing quasi-experimental design. During administering the pre-test and post-test, the researcher used the weighted mean and t-test to determine the mean percentage scores of the respondents. The respondents were 56 fourth grade learners enrolled in Marangog Elementary School and Tagnate Elementary School in Hilongos East District for the school year 2019 – 2020. At the end of the study, it was found out the use of downloaded math worksheets were found effective means of improving the mastery level. In addition, based on the results gathered, the downloaded math worksheets which were used in the drills or practice greatly contributed to the elimination of non-numerates, and give significantly increasing the number of pupils who mastered the competencies related to geometry. As a result, a proposed innovative teaching strategy is recommended. -/- . (shrink)

There have been several recent attempts to think about psychological kinds as homologies. Nevertheless, there are serious epistemic challenges for individuating homologous psychological kinds, or cognitive homologies. Some of these challenges are revealed when we look at competing claims of cognitive homology. This paper considers two competing homology claims that compare human anger with putative aggression systems of nonhuman animals. The competition between these hypotheses has been difficult to resolve in part because of what I call the boundary problem: boundaries (...) between instances of psychological kinds (e.g., anger and fear) cannot be directly observed. Thus, there are distinctive difficulties for individuating psychological kinds across lineages. I draw four conclusions from this case study: First, recent evidence from the neuroscience of fear suggests that one of the proposed homologies involves a straightforward conflation of anger and fear. Second, this conflation arises because of the boundary problem. Third, there is an implicit constraint on the operational criteria that is easy to overlook in the psychological case. In this case, ignoring the constraint is part of the problem. Fourth, this is a clear case in which knowledge of homology cannot be accumulated piecemeal. Identifying homologs of human anger requires identifying homologs of fear. (shrink)

This study aimed to trace the 2015 to 2019 MAME graduates in the College of Teacher Education for Graduate Studies of the University of Northern Philippines in terms of their personal profile, their work-related profile before and after taking their master’s degree, reasons of taking up the program, competency level before and after taking the program, appraisal of the most useful courses offered in the program, evaluation on the contribution of the program to their personal and professional growth and (...) assessment on the features of the program and suggestions of how to improve the program. The researcher utilized mixed method of research. The subjects were the MAME graduates, and a questionnaire and open-ended questions were sent to them thru various online flat forms. Based on the findings of the study, the following conclusions are derived: pursuing graduate studies chooses no sex, civil status or age as long as there is a will to improve one's self; students have math- related educational background and finished the program within the prescribed number of years; having master’s degree could contribute greatly in landing a permanent teaching job in public schools; professional and personal growth and development is the strongest drive to pursue advance education; the program offers subjects that are useful which enhanced the teaching competencies of the graduates and consequently contribute to the holistic development of the graduates; and that there are areas needing improvement in the MAME program. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are (...) devoted. At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

In an attempt to curtail examination malpractice, the Joint Admission and Matriculation Board (JAMB) has been generating different paper types with a different order of test items in the Unified Tertiary Matriculation Examination (UTME). However, the permutation of test items may compromise students’ performance unintentionally because constructive suggestions in theory and practice recommend that test items be sequenced in ascending order of difficulty. This study used data collected from a random sample of 1,226 SSIII students to ascertain whether the (...) permutation of test items has any effect on the performance of students in two different subjects (Use of English and Mathematics). The study adopted the Equivalent Groups Quasi-Experimental Research Design with three independent groups. Findings emerged, amongst others, that there is a significant difference in the performance scores of prospective university students' in use of English and Mathematics examintaions arranged in three different orders (ED, DE, R). There are no significant gender differences in the performance of students in Use of English and Mathematics based on test item permutation. However female students perform better than male students when test items are arranged in ascending order of difficulty while males perform better when test items are arranged in descending order of difficulty. It was concluded that the permutation of test items in UTME examination tends to affect the performance of students in Use of English and Mathematics. This finding has implications for the future conduct of UTME examinations and enrolment into higher education as the randomization of UTME test items changes the difficulty order of different paper types. It was recommended that other measures of curtailing examination malpractices that would not affect students’ academic performance should be adopted. (shrink)

This study explores the needs of Senior High School (SHS) Mathematics Teachers in Cabanatuan City’s Schools Division Office, focusing on professional qualifications, familiarity with essential Mathematics knowledge, and employed pedagogical strategies. Using descriptive research approach, the study examines the correlation between respondent profiles, Mathematics knowledge, pedagogical approaches, and the adequacy of educational infrastructure. Results indicate that most respondents demonstrate excellent familiarity with essential Mathematics knowledge, predominantly employing assessment as a pedagogical strategy. No significant relationship exists between (...) respondent profiles and General Mathematics essential knowledge competence. Respondents highly rate the DepEd-issued Teacher’s Guide, and Mathematics materials’ availability and sufficiency in classrooms are satisfactory. The study recommends providing more usable Mathematics tools and materials, emphasizing targeted Mathematics-related seminars for SHS Mathematics teachers in public secondary schools within Cabanatuan City. (shrink)

The philosopher and builder Ludwig Wittgenstein remarks that architecture is more difficult than philosophy. He suggests an exhaustion criterion for how difficult a discipline is: a field is more difficult the more exhausting it is. I make a case against this claim. There was once a demand to prevent the Greek myths from establishing themselves in the curriculum by means of “our own rival myths.” It is difficult to compete with a renowned Greek myth, but if one does produce a (...) rival, it may well not be exhausting to do so. (shrink)

The purpose of this study was to evaluate the differentiated learning training program at the mathematics subject teachers' meeting (MGMP). A descriptive quantitative approach was used to identify the successes of the program and areas that require improvement. The sample included 21 mathematics teachers in Sub Rayon 2 of Lebak District. The instruments used were questionnaires in which data on participants' responses to resource persons, materials, and suggestions for future activities were collected, and the results of direct observations. (...) Data analysis was carried out using descriptive statistical methods to describe the overall evaluation results. Based on the findings, it is expected to provide an in-depth view of the differentiated learning program, contribute to the development of the program in the future, and become a guideline for improving the quality of mathematics learning in the Lebak district. The results of this training program evaluation show that the differentiated learning training activities have been well implemented in MGMP Mathematics Sub Rayon 2 Lebak District. This is evidenced by the level of teacher satisfaction in participating in the training, the majority of which were at scores 4 and 5, namely agreeing and strongly agreeing. This was also the case with the resource persons, materials, and training facilities, and organization. It is hoped that in the future continue to carry out ongoing training related to differentiated learning by current needs and to improve the competence of mathematics teachers in particular, as well as teachers of other subjects in general. (shrink)

The quantification theory of propositions in Russell’s Principles of Mathematics has been the subject of an intensive study and in reconstruction has been found to be complete with respect to analogs of the truths of modern quantification theory. A difficulty arises in the reconstruction, however, because it presents universally quantified exportations of five of Russell’s axioms. This paper investigates whether a formal system can be found that is more faithful to Russell’s original prose. Russell offers axioms that are (...) universally quantified implications that have antecedent clauses that are conjunctions. The presence of conjunctions as antecedent clauses seems to doom the theory from the onset, it will be found that there is no way to prove conjunctions so that, after universal instantiation, one can detach the needed antecedent clauses. Amalgamating two of Russell’s axioms, this paper overcomes the difficulty. (shrink)

This study focused on the significant relationship between Creative Thinking Skills and Problem-Solving Skills in Mathematics among Grade 6 pupils. The study's respondents were thirty (30) Grade 6 pupils from one of the elementary schools in the District of Sta. Ana, Pampanga. The respondents were selected using simple random sampling techniques. The study used a survey questionnaire comprising the Creative Thinking Skills Test (Pentang et al., 2022) and a researcher-made word problem-solving written test and rubrics. The researchers found a (...) high level of Creative Thinking Skills among the Grade 6 pupils, while the respondents' problem-solving skills were categorized as competent. The study revealed that the Creative Thinking Skills and Problem-Solving Skills in Mathematics of the Grade 6 pupils were very lowly correlated. Still, creative thinking and problem-solving in the applied subject are recommended to help students acquire the 21st-century skills that empower them to develop holistically and career ready. (shrink)

In philosophy of language, a distinction has been proposed by Diego Marconi between two aspects of lexical competence, i.e. referential and inferential competence. The former accounts for the relation-ship of words to the world, the latter for the relationship of words among themselves. The aim of the pa-per is to offer a critical discussion of the kind of formalisms and computational techniques that can be used in Artificial Intelligence to model the two aspects of lexical competence, and of the main (...) difficulties related to the use of these computational techniques. The first conclusion of our discussion is that the dis-tinction between inferential and referential semantics is instantiated in the literature of Artificial Intelli-gence by the distinction between symbolic and connectionist approaches. The second conclusion of our discussion is that the modelling of lexical competence needs the advent of hybrid models integrating symbolic and connectionist frameworks. Our hypothesis is that Conceptual Spaces, a framework devel-oped by Gärdenfors more than fifteen years ago, can offer a lingua franca that allows to unify and gener-alize many aspects of the representational approaches mentioned above and to integrate “inferential” (=symbolic) and “referential” (=connectionist) computational approaches on common ground. (shrink)

A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind of view: its (...) proponents rely on a distinction between "essential" and "nonessential" features of mathematical objects, and there's no good way to articulate this distinction which is compatible with basic structuralist commitments. But all is not lost. For I further argue that the insights motivating structuralism (or at least those worth preserving) can be preserved without formulating the view in ontologically committal terms. (shrink)

Elective mathematics has been an extra mathematics subject for pilot students of Eduardo Barretto Sr. National High School for quite some time now. Through this, many alumni testified how this helped them understand senior high school and college math. However, the teachers have also been struggling with the resources for specific areas of mathematics, such as Business Math, Statistics, Analytic Geometry, Trigonometry, and Calculus. When the pandemic hit the Philippines, contextualized learning material aligned with the Most Essential (...) Learning Competencies (MELC) helped elective mathematics teachers. The contextualized learning material was developed from April to August 2020 and was first validated in February 2021 by the students who participated in the pilot testing. However, following the due process, it was suggested that I revalidate the IDEA-formatted Analytic Geometry and Trigonometry instructional material. It examines Eduardo Barretto Sr. National High School mathematics teachers and SDO Calamba City experts' evaluation of the content's topics, objectives, concepts, directions, and activities and the format's prints, illustrations, design, and layout. Participants' perceptions of learning material were assessed using a 4-point Likert Scale survey. Hence, descriptive quantitative design. Participants were surveyed using a 4-point Likert Scale to assess their learning content impression. The findings reveal that the participants found the format and content of the developed learning material to be very satisfactory. The indicators "localized" and "can be done independently" were viewed as Fair and Satisfactory, respectively. The learning material was enhanced by localizing the activities and giving directions that could facilitate self-learning. (shrink)

The development of symbolic logic is often presented in terms of a cumulative story of consecutive innovations that led to what is known as modern logic. This narrative hides the difficulties that this new logic faced at first, which shaped its history. Indeed, negative reactions to the emergence of the new logic in the second half of the nineteenth century were numerous and we study here one case, namely logic at Oxford, where one finds Lewis Carroll, a mathematical teacher who (...) promoted symbolic logic, and John Cook Wilson, the Wykeham Professor of Logic who notoriously opposed it. An analysis of their disputes on the topic of logical symbolism shows that their opposition was not as sharp as it might look at first, as Cook Wilson was not so much opposed to the « symbolic » character of logic, but the intrusion of mathematics and what he perceived to be the futility of some of its problems, for logicians and philosophers alike. (shrink)

This study aims to show a similarity of Kant’s and Jung’s approaches to an issue of the possibility of scientific psychology, hence to explicate what they thought about the future of psychology. Therefore, the article contains heuristic material, which can contribute in a resolving of such methodological task as searching of promising directions to improve philosophical and scientific psychology. To achieve the aim the author attempts to clarify an entity of Kant’s and Jung’s objections against even the possibility of scientific (...) psychology and to find out ways to overcome those objections in Kant’s and Jung’s works. The main methods were explication, reconstruction and comparative analysis of Kant’s and Jung’s views. As a result it was found, that Kant and Jung allocated one and the same obstacles, which, on their opinion, prevent psychology to become a science in the strict sense. They are: 1) coincidence of subject and object in psychology; 2) impossibility to apply quantitative mathematic methods in psychology; 3) pendency of the issue of psychophysical parallelism. However, Kant and Jung indicated ways to resolve formulated by them fundamental difficulties. All those ways lay through the searching a principle of interaction and connection between the psychic and the physical. (shrink)

We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this short paper I will pose the (...) difficulty, and then propose a solution. (shrink)

These remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre’s theory of pure reflection, the linchpin of the works of Sartre’s early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Gödel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception through (...) the discovery of the diagonal theorems can be recognized as a particularly clear instance of the drama of reflection according to Sartre, especially in the positing and overcoming of its proper value-ideal, viz. the synthesis of consistency and completeness. Conversely, this translation solves a number of systematic problems about pure reflection’s relations to accessory reflection, phenomenological reflection, pre-reflective self-consciousness, conversion, and value. Negative foundations, the metaphysical position emerging from this translation between existential philosophy and metalogic, concurs by different paths with Badiou’s Being and Event in rejecting both ontotheological foundationalisms and constructivist antifoundationalisms. (shrink)

My aim in this chapter is to extend the Realist account of the foundations of linguistics offered by Postal, Katz and others. I first argue against the idea that naive Platonism can capture the necessary requirements on what I call a ‘mixed realist’ view of linguistics, which takes aspects of Platonism, Nominalism and Mentalism into consideration. I then advocate three desiderata for an appropriate ‘mixed realist’ account of linguistic ontology and foundations, namely (1) linguistic creativity and infinity, (2) linguistics as (...) a theory of types (and not tokens) and (3) independence but structural respect between language and the linguistic competence thereof. My own brand of mixed realism, what I call ante rem realism, is defended along the lines of an ante rem or non-eliminative structuralism, the likes of which has been offered for mathematics by Resnik (1997) and Shapiro (1997). In other words, grammars describe a mind-independent (but not necessarily unconnected) linguistic reality in terms of linguistic patterns or structures also known as natural languages. I further amend this picture to allow for the possibility of a naturalistic account of language acquisition and evolution by arguing against a particular view of the type-token distinction. (shrink)