Showing Mathematical Flies the Way Out of Foundational Bottles: The Later Wittgenstein as a Forerunner of Lakatos and the Philosophy of Mathematical Practice

This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as precursors for the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy.

The Relationship Between Proof and Certainty in Mathematical Practice

Many mathematics educators believe a goal of instruction is for students to obtain conviction and certainty in mathematical statements using the same types of evidence that mathematicians do. However, few empirical studies have examined how mathematicians use proofs to obtain conviction and certainty. We report on a study in which 16 advanced mathematics doctoral students were given a task-based interview in which they were presented with various sources of evidence in support of a specific mathematical claim and were asked how convinced they were that the claim was true after reviewing this evidence. In particular, we explore why our participants retained doubts about our claim after reading its proof and how they used empirical evidence to reduce those doubts.

Coarse Geometry of Topological Groups

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

MATHEMATICAL PRACTICE AND MATHEMATICAL LOGIC: TO THE HISTORY OF RELATIONSHIP

The article annuls the role of practice in the development of mathematics in the 19th century in the formation of mathematical logic. It is shown that the revolutionary transformations of mathematics of the 19th century, which led to an increase in the abstractness of mathematical theories and concepts, was accompanied by an increase in uncertainty regarding the standards of proof, which led to the universal spread of anxiety (J. Gray) as an element of mathematical practice. It is argued that this element of practice was one of the sources of the emergence of mathematical logic, which claims to give rigor and accuracy to mathematics. The article argues that the socio- epistemological analysis of the practice of mathematics and the formation of mathematical logic will clarify the specifics of the development of relations between mathematics and mathematical logic.

Three Roles of Empirical Information in Philosophy: Intuitions on Mathematics do Not Come for Free

This work gives a new argument for ‘Empirical Philosophy of Mathematical Practice’. It analyses different modalities on how empirical information can influence philosophical endeavours. We evoke the classical dichotomy between “armchair” philosophy and empirical/experimental philosophy, and claim that the latter should in turn be subdivided in three distinct styles: Apostate speculator, Informed analyst, and Freeway explorer. This is a shift of focus from the source of the information towards its use by philosophers. We present several examples from philosophy of mind/science and ethics on one side and a case study from philosophy of mathematics on the other. We argue that empirically informed philosophy of mathematics is different from the rest in a way that encourages a Freeway explorer approach, because intuitions about mathematical objects are often unavailable for non-mathematicians (since they are sometimes hard to grasp even for mathematicians). This consideration is supported by a case study in set theory.

Differences in Ethnomathematical Characteristics between Buton Traditional Houses

The mathematical practice of the Buton people can be seen from the ethnomathematics of the Buton traditional house. This qualitative study aims to explore the differences in ethnomathematical characteristics between Buton traditional houses. The subjects of this study were 3 traditional leaders and 3 builders of Buton traditional houses. Data were collected through measurements, interviews, and observations of the traditional Buton house in Baubau city, the capital of the Buton kingdom, Indonesia. The credibility of the data was tested by using triangulation of sources. Data were analyzed descriptively qualitatively from Miles, Huberman, and Saldana in the form of data condensation, data presentation, and conclusions. The results of data analysis showed that the differences in the ethnomathematical characteristics of the three Buton traditional houses can be seen from the size and area of the house, the angle of the roof and stairs, the arrangement of the roof, the number of poles, the number of rooms, the model of the house, and the type, model, and the number of windows used. Teachers need to know the differences in ethnomathematical characteristics so that they can be used to increase students’ active participation in learning mathematics in class.

Ethnomathematics values in Temple of Heaven: An Imperial Sacrificial Altar in Beijing, China

Many studies are proving that learning mathematics with an ethnomathematical approach can improve students’ mathematical skills. Developing and using ethnomathematics concepts are important to raise history and cultural awareness of mathematics. This study aims to analyse the ethnomathematics values of the Temple of Heaven. Temple of Heaven is one of the famous heritage sites in Beijing, China, which bears many ethnomathematics concepts. The researchers applied a qualitative method in this study. The subject of this research is the Temple of Heaven building that is located in Beijing, China. Researchers identified the geometrical concept present in the exterior, interior design, and building structure of the Temple of Heaven building. This research shows the existence of mathematical concepts in the architecture of the Temple of Heaven. This research result can help teachers in making mathematical practice questions with ethnomathematics concepts.

Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning

Proving and refuting are fundamental aspects of mathematical practice that are intertwined in mathematical activity in which conjectures and proofs are often produced and improved through the back-and-forth transition between attempts to prove and disprove. One aspect underexplored in the education literature is the connection between this activity and the construction by students of knowledge, such as mathematical concepts and theorems, that is new to them. This issue is significant to seeking a better integration of mathematical practice and content, emphasised in curricula in several countries. In this paper, we address this issue by exploring how students generate mathematical knowledge through discovering and handling refutations. We first explicate a model depicting the generation of mathematical knowledge through heuristic refutation (revising conjectures/proofs through discovering and addressing counterexamples) and draw on a model representing different types of abductive reasoning. We employed both models, together with the literature on the teachers’ role in orchestrating whole-class discussion, to analyse a series of classroom lessons involving secondary school students (aged 14–15 years, Grade 9). Our analysis uncovers the process by which the students discovered a counterexample invalidating their proof and then worked via creative abduction where a certain theorem was produced to cope with the counterexample. The paper highlights the roles played by the teacher in supporting the students’ work and the importance of careful task design. One implication is better insight into the form of activity in which students learn mathematical content while engaging in mathematical practice.

Mathematics in the middle: The relationship between measurement and metamorphic matter

This paper revisits philosophical questions regarding the relationship between mathematics and matter. I briefly present four contrary and contemporary perspectives on the speculative force of mathematics, as a provocation for further discussion on the subject of sciento-metrics. I first consider the ideas of the philosopher Quentin Meillassoux, as a way of setting the stage for various kinds of materialist philosophies of mathematics. I then turn to the ideas of two mathematicians - Fernando Zalamea and Giuseppe Longo - and a computer scientist - Gregory Chaitin - and explore how their discussions of contemporary mathematical practice offer important insight (and twist) regarding the relationship between mathematics and matter.

Comparing the Mathematical Practices Pre-Service Teachers and Mathematics Teacher Educators Identified as Relevant to Problems and Tasks

In the U.S., state adopted or developed college- and career-ready mathematics standards, including the Common Core State Standards for Mathematics, not only impact districts, students, and their teachers, but also university teacher preparation programs. In order to attain and sustain Common Core’s vision of developing mathematically competent citizens, teacher preparation programs must support pre-service teachers’ development of practical conceptions of the Standards for Mathematical Practice. In this article, we examine the mathematical practices middle grades pre-service teachers (grades 4-9 licensure) and mathematics teacher educators identified as playing a role in attempts to make sense of and work toward solutions to mathematics problems. In addition, we compare the mathematical practices indicated both within and across pre-service teachers and mathematics teacher educators. Results identify pre-service teachers’ potential difficulties operationalizing six specific mathematical habits of mind. Finally, we describe how such comparisons can guide the design of future teacher education and professional learning by describing a process for identifying problems and tasks with the greatest potential to support pre-service teachers’ development of practical conceptions of mathematics or other content-specific habits of mind.