#5: Martin Schmidt, Bharath Sriraman
Nyaya Methodology and Western Mathematical Logic: Origins and Implications
Western Nevada College
University of Montana
Abstract: In this chapter we compare the Nyaya school of logic, namely one of the six major schools of Hindu philosophy to Aristotelian logic. In particular we compare the intent, nature and structure of syllogisms in Nyaya to that in Western mathematical logic to highlight differences in premises and conclusions. In doing so, we draw on the foundational writings and commentaries on Nyaya methodology from both Hindu and Buddhist sources, as well as the major sources of Aristotelian logic. We explore the possibility of Nyaya methodology being influenced by Aristotelian logic before arguing that the Nyaya system developed on its own trajectory. Some modern implications of Nyaya methodology are given.
Keywords: Indian philosophy; Hindu philosophy; Mathematical Logic; Nyaya logic; Aristotelian logic; Syllogisms; Milinda; Law of Contrapositive; Buddhist logic
AMS Subject Classification: 03-03; 03F50
#4: Scott Chamberlin, Bharath Sriraman
Affect in Mathematical Modeling
Scott Chamberlin, University of Wyoming
Bharath Sriraman, University of Montana
Abstract: The relationship between affect and mathematical modeling must be investigated because, as educational psychologists have suggested for decades, affect directly influences achievement. Moreover, given the importance of mathematical modeling and the applications to high level mathematics, providing the field of mathematics psychology with insight regarding affect, in relation to mathematical modeling, might help determine the degree to which aspiring mathematicians facilitate understanding of mathematics and understanding affect in mathematical modeling episodes may have a direct effect on cognition. Though it existed for centuries, mathematical modeling may have first been mentioned in literature by Bell (1972). However, more than anyone else, often Lesh (1981) is credited with bringing the process of mathematical modeling to the attention mathematics educators and mathematical psychologists. In the initial version of the National Council of Teachers of Mathematics document, Principles and Standards for School Mathematics (2000), the process of mathematical modeling is embedded as a secondary task in algebra. It is not even mentioned as one of the five process standards. When the Common Core State Standards-Mathematics were released in 2010 (National Governor’s Association & Council of Chief State School Officers, 2010), modeling with mathematics surfaced as the fourth mathematical practice. One may ask, “Why the sudden interest in mathematical modeling in national standards?” In fact, the models and modelling movement had been rapidly growing as a direct result of Lesh and his colleagues at PME-NA (Psychology of Mathematics-North America) and ICTMA (International Conference on the Teaching of Mathematical Modeling and Applications). Many of those colleagues are represented as authors in this book. The authors selected to contribute to this edited book were identified as a result of their work in mathematical modeling, research in affect, or both. This book will help readers develop understanding of the relationship between affect and mathematical modelling from an international perspective. In so doing, a vast array of authors will be utilized to provide insight on affective states while problem solvers create mathematical models.
Keywords: Affect; Mathematical Modeling; Cognition; AMS Subject Classification: 97
#3: Bharath Sriraman
Handbook of the Mathematics of the Arts and Sciences
Bharath Sriraman, University of Montana- Missoula
Abstract: This Handbook aims to become a definitive source with chapters that show the origins, unification, and points of similarity between different disciplines and mathematics. The seven sections in this book explore: Mathematics and Architecture; Mathematics, Biology and Dynamical Systems; Mathematics, Humanities and the Language Arts; Mathematics, Art and Aesthetics; Mathematics, History and Philosophy; Mathematics in Society; and New Directions. Science and Art are used as umbrella terms to encompass the physical, natural and geological sciences, as well as the visual and performing arts.
Keywords: Mathematics and Architecture; Mathematics and Dynamical Systems; Mathematics and Humanities; Mathematics, Art and Aesthetics; Mathematics, History and Philosophy
AMS Subject Classification: 00
#2: Bharath Sriraman
On Measures of Measurement and Mismeasurement: A commentary on Planning and Assessment
Bharath Sriraman, University of Montana - Missoula
Abstract: In this commentary the six Canadian chapters on planning and assessment in mathematics are critiqued. Some common strands are discussed on what entails a measurable item in mathematics assessments at the secondary school level. Implications are made for teachers.
Keywords: Learning; Teaching; Mathematical Memory; Testing; Rubrics; Assessment
AMS Subject Classification: 097D60
#1: Bharath Sriraman, Per Haavold
Reconciling the Realist/Anti Realist Dichotomy in the Philosophy of Mathematics
University of Montana
University of Tromso, Norway
In the philosophy of mathematics, the realist vs. anti-realist debate continues today with differing positions on the status of mathematical objects. For realists, objects sit in “Plato’s heaven”, immovable, objective, eternal, and we contemplate them, whereas anti-realists (or Constructionists) are the opposite, and emphasize epistemology over ontology, saying that we construct mathematical objects. There are numerous results in mathematics which can be arrived at both from a realist and an anti-realist viewpoint. In other words, they can be contemplated (proved) via methods deemed unsuitable by anti-realists- or simply arrived at it through methods (or construction) as the anti-realist would say. In this chapter, we argue that realism and anti-realism can be seen as two sides of the same coin, or different ways of knowing the same thing, and therefore the so called dichotomy between these positions is reconcilable for particular mathematical objects.
Keywords: Intuitionism - Constructionism - Realism - Platonism - Anti-realism - Brouwer - Constructive mathematics - Philosophy of mathematics
AMS Subject Classification: 03A, 03F