Obtaining Formal Models through Non-Monotonic Refinement

When designing a model for formal verification, we want to\ud be certain that what we proved about the model also holds for the system we modelled. This raises the question of whether our model represents the system, and what makes us confident about this. By performing so called, non-monotonic refinement in the modelling process, we make the steps and decisions explicit. This helps us to (1) increase the confidence that the model represents the system, (2) structure and organize the communication with domain experts and the problem owner, and (3) identify rational steps made while modelling. We focus on embedded control systems

Applying the proto-theory of design to explain and modify the parameter analysis method of conceptual design

This article reports on the outcomes of applying the notions provided by the reconstructed proto-theory of design, based on Aristotle’s remarks, to the parameter analysis (PA) method of conceptual design. Two research questions are addressed: (1) What further clarification and explanation to the approach of PA is provided by the proto-theory? (2) Which conclusions can be drawn from the study of an empirically derived design approach through the proto-theory regarding usefulness, validity and range of that theory? An overview of PA and an application example illustrate its present model and unique characteristics. Then, seven features of the proto-theory are explained and demonstrated through geometrical problem solving and analogies are drawn between these features and the corresponding ideas in modern design thinking. Historical and current uses of the terms analysis and synthesis in design are also outlined and contrasted, showing that caution should be exercised when applying them. Consequences regarding the design moves, process and strategy of PA allow proposing modifications to its model, while demonstrating how the ancient method of analysis can contribute to better understanding of contemporary design-theoretic issues

Reductionism and the Universal Calculus

In the seminal essay, "On the unreasonable effectiveness of mathematics in the physical sciences," physicist Eugene Wigner poses a fundamental philosophical question concerning the relationship between a physical system and our capacity to model its behavior with the symbolic language of mathematics. In this essay, I examine an ambitious 16th and 17th-century intellectual agenda from the perspective of Wigner's question, namely, what historian Paolo Rossi calls "the quest to create a universal language." While many elite thinkers pursued related ideas, the most inspiring and forceful was Gottfried Leibniz's effort to create a "universal calculus," a pictorial language which would transparently represent the entirety of human knowledge, as well as an associated symbolic calculus with which to model the behavior of physical systems and derive new truths. I suggest that a deeper understanding of why the efforts of Leibniz and others failed could shed light on Wigner's original question. I argue that the notion of reductionism is crucial to characterizing the failure of Leibniz's agenda, but that a decisive argument for the why the promises of this effort did not materialize is still lacking.Comment: 11 pages, 1 figur

An Investigation of the Duality Between Art and Math

The following paper is an investigation of the relationship that exists between math and art. It argues the importance of integration between the two disciplines by shedding light on unrecognized characteristics within both. The paper provides examples of mathematical art and beautiful proofs which help unearth the potential of mixing math and art at an academic level

Project-based high school geometry

Project-based learning (PBL) is an instructional strategy that allows students the autonomy to learn, explore and investigate throughout the learning process by means of projects. Many educators have seen the need for such a strategy in the classroom as a remedy for motivating students, showing relevance of student’s education to everyday life, preparing students for college and the work force, and the dire need for students to develop critical thinking skills to encourage future success. In my thesis I will define project-based learning, discuss its characteristics, compare PBL to traditional teaching methods and reflect on my experiences with project-based learning in the classroom. I will also show how a traditional math problem can become more interesting and applicable to students if project-based elements are incorporated

Mathematics

This chapter aims to provide the reader with a brief introduction to the origins of the various branches of mathematics. While tracing back these origins, an insight will be offered into how the mathematics as a discipline developed throughout many thousands of years and the variety of cultures. Key practices of the discipline of mathematics will be highlighted, followed by a discussion which argues in favour of incorporating these practices into the school mathematics