Kant's Mathematical World

Kant's Mathematical World aims to transform our understanding of Kant's philosophy of mathematics and his account of the mathematical character of the world. Daniel Sutherland reconstructs Kant's project of explaining both mathematical cognition and our cognition of the world in terms of our most basic cognitive capacities. He situates Kant in a long mathematical tradition with roots in Euclid's Elements, and thereby recovers the very different way of thinking about mathematics which existed prior to its 'arithmetization' in the nineteenth century. He shows that Kant thought of mathematics as a science of magnitudes and their measurement, and all objects of experience as extensive magnitudes whose real properties have intensive magnitudes, thus tying mathematics directly to the world. His book will appeal to anyone interested in Kant's critical philosophy -- either his account of the world of experience, or his philosophy of mathematics, or how the two inform each other.

Design Mathematical Activity in Mathematics Classroom: Decimal Number

The research was aimed to investigate six-lesson study team members in designing mathematical activities to develop students’ mathematization using Open Approach in the second step of the Lesson Study process in teaching decimal numbers. A total of 16 Grade 4 students participated as the target group. Three instruments were used namely lesson plans, student worksheets, and observation field notes. Researchers employed ethnographic research design to study how the mathematical activities could assist students to develop their mathematical ideas from the real world to the mathematical world through a flow of lessons over the four stages of the Open Approach along with the Lesson Study process. The research results revealed that a series of five research lesson plans encompassing various mathematical activities were successfully encouraging students to elaborate their ideas and transmitting their ideas from the real-world to the mathematical world using semi-concrete aids. Moreover, the results of using the Open Approach have been proved to be relevant as students demonstrated their mathematization in fostering their mathematical thinking to transform their ideas smoothly. Therefore, designing mathematical activities is important to cultivate students’ mathematical thinking in problem-solving instantaneously. A limitation of the research was identified when the Lesson Study team members were reflecting on the teaching practice. This is because they found that the unclear illustration in the student worksheets has raised confusion. In conclusion, the overall results of this research have contributed significantly to our recognition of the practicality of Open Approach treatment in the Lesson Study process in developing students’ mathematization through their participation in mathematical activities.

Shifts in the Foundation: The Continual Modification and Generalization of Axioms and the Search for the Mathematical Principles that Underlie our Reality

The study shall seek to explore the deep, underlying correspondence between the mathematical world of pure numbers and our physical reality. The study begins by pointing out that while the familiar, one-dimensional real numbers quantify many aspects of our day-to-day reality, complex numbers provide the mathematical foundations of quantum mechanics and also describe the behavior of more complicated quantum networks and multi-party correlations, and quaternions underlie Einsteinian special theory of relativity, and then poses the question whether the octonions could play a similar role in constructing a grander theory of our universe. The study then points out that by increasing the level of abstraction and generalization of axiomatic assumptions, we could construct a more powerful number system based on octonions, the seditions, or even other hypercomplex numbers so that we may more accurately describe the universe in its totality.

Srinivas Ramanujan: An Indigenous Mathematician

The paper undertaken for the deliberation of International stature on December 22, 2020 rivets attention on the topic of THE CONTRIBUTION OF RAMANUJAN in the arena of MATHEMATICS. He is remembered for India’s greatest mathematical genii. He made significant contribution to the analytical theory of numbers elliptical functions, continued fractions and infinite series. Ramanujan left a slew of unpublished note books enfolding theorems that future generation of mathematical world have been exploring continuously. He is an icon of a self-studied, self learnt, self-taught mathematical genius who is a living legend and ennobling soul for the posterity. He is known as child prodigy. Owing to his ingenuous acumen and surprising accomplishments in the field of Mathematics, Indian govt. decided to celebrate his birthday 22nd December as National Mathematics Day.

From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.

Trouble down memory lane: Misremembrance reveals encoding differences in arithmetic word problems

Is there a fundamental difference between counting years and kilograms? Marbles and centimeters? Floors and euros? Recent evidence suggests that non-mathematical world knowledge irrelated to the mathematical structure of a problem can nevertheless influence its semantic encoding. To tackle this question, we created arithmetic word problems devised to promote contrasting encodings by featuring different quantities, in French and in English. We designed three experiments investigating the representations constructed and memorized by 302 adult participants when solving the problems. After an initial solving task, participants were given an unexpected task: either recall the problems (Experiments 1 and 2) or identify experimenter-induced changes in target problem sentences (Experiment 3). We predicted that the use of specific quantities in the problem statements was enough to lead participants to erroneously recall mathematical information that was not present in the problems, but that could be inferred from one of the two possible encodings of the situations. Results across all three experiments consistently indicate that participants construct and memorize a different problem encoding depending on the quantities involved. They misremembered problems involving durations, heights, or elevators by including new information into their problem representation. The same recall mistakes were not made for problems involving prices, weights or collections. This supports the claim that knowledge related to daily-life quantities substantially influences arithmetic reasoning, despite such knowledge being irrelevant for abstract reasoning.

Conexões entre grafos e matrizes na modelagem de problemas matemáticos

Graphs theory is very important in the mathematical world as an excellent way of connecting with the real world. By using the theory of directed graphs it is possible to transform many of the everyday problems into mathematical problems, so as to make an exact study in each case. In this work we explore the matrices related to the various types of graphs, such as the vertex matrix, which is associated with a directed graph, and the adjacency matrix. Moreover, matrices of multi-step connections are constructed so as to separate the various blades between the vertices of a directed graph. Then, we will construct some applications of those results in the form of examples.