Bookcast

Werner Heisenberg's previously unpublished 1941/42 essay, "Reality and Its Order," which explores his philosophical reflections on the nature of reality and human knowledge, ranging from physics and chemistry to society and religion. The essay, a comprehensive statement on the epistemological implications of modern physics, is published here for the first time as a separate work and includes commentary by Ernst Peter Fischer. Heisenberg's views are analyzed in relation to the ideas of his mentors, Max Planck and Niels Bohr, highlighting the importance of language in shaping our understanding of the world. The essay also addresses the relationship between science and religion, exploring the role of faith, parables, and symbols in comprehending reality. Finally, a commentary provides context by referencing relevant works by other notable thinkers.

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George Pólya's Mathematics and Plausible Reasoning, Volume 2. It explores plausible reasoning, a type of inference used in mathematics and other fields. The book examines various patterns of plausible reasoning, including induction and analogy, and their relationship to probability. It uses numerous examples to illustrate these patterns, discussing their application in mathematical discovery and problem-solving, as well as in areas like legal reasoning and the natural sciences. The text also considers the limitations of plausible reasoning and contrasts it with demonstrative reasoning.

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George Pólya's book, Mathematics and Plausible Reasoning, Volume 1, focusing on induction and analogy in mathematics. Pólya contrasts demonstrative reasoning (proofs) with plausible reasoning (conjectures and inductive evidence), arguing that plausible reasoning plays a crucial role in mathematical discovery. The book uses numerous examples to illustrate how generalization, specialization, and analogy aid in forming and testing mathematical conjectures. These examples range from elementary geometry and number theory to more advanced topics in calculus and physics. Pólya emphasizes the importance of inductive investigation—testing consequences of a conjecture and comparing them to facts—in building confidence in a mathematical result, even before rigorous proof is achieved.

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Geometry Revisited, a mathematics book aimed at high school students and laypeople. It revisits classical Euclidean geometry, introducing modern theorems and techniques like transformations and projective geometry. The book uses various methods, including trigonometry and analytic geometry, to explore properties of triangles and circles, emphasizing concepts such as collinearity, concurrence, and the power of a point. It also covers more advanced topics like inversive geometry and hyperbolic functions, connecting them to classical results. Finally, it includes numerous exercises to enhance the reader's understanding.

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Roger B. Nelsen's Proofs Without Words: Exercises in Visual Thinking (1993) is a collection of mathematical proofs illustrated using diagrams. The book features proofs from various sources and historical periods, showcasing the power of visual reasoning in mathematics. Many proofs concern geometric theorems, but also include algebraic identities, and other mathematical concepts. The book's introduction discusses the history and nature of visual proofs. The author acknowledges numerous contributors and notes alterations made for uniformity.

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Mueller's Philosophy of Mathematics and Deductive Structure in Euclid’s Elements examines Euclid's geometry, analyzing its logical structure and challenging interpretations that incorporate algebraic concepts. Mueller argues that a strictly geometric reading is more historically accurate and philosophically sound, contrasting Euclid's methods with those of later mathematicians like Hilbert. The text explores Euclid's axioms, postulates, common notions, and proofs, paying close attention to their arrangement and purpose within the Elements. Specific propositions are analyzed in detail, comparing Euclid's geometrical approach to algebraic interpretations. Finally, the author discusses the method of exhaustion used by Euclid in proving certain theorems related to areas and volumes.

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Sir Thomas Heath's translation of Euclid's Elements, specifically focusing on Book X. Book X provides a comprehensive classification of irrational magnitudes, particularly concerning the incommensurability of certain line segments, like the diagonal of a square. Heath's commentary explores the historical context of this challenging material, tracing its development from Pythagorean discoveries through contributions from Theaetetus and others, and further examining its influence on later mathematicians. The excerpt also includes portions of Books XI and XII, covering solid geometry, where Euclid expands on the concepts of planes, solids, and their properties, as well as providing methods to calculate the volumes of pyramids and cones. Finally, Book XIII is discussed, showing the construction and relationships of regular solids inscribed within a sphere.

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Sir Thomas Heath's translation of Euclid's Elements, specifically Volumes II (Books III-IX). The excerpts focus on Euclid's propositions and definitions, including discussions of circles, angles, and proportions. Heath's commentary incorporates insights from other mathematicians, such as Heron and De Morgan, offering alternative proofs and analyses of Euclid's work, highlighting ambiguities and controversies surrounding certain definitions and proofs. The selected passages extensively cover Euclid's treatment of geometrical magnitudes and their ratios and proportions, especially as they pertain to similar figures and numbers.

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T.L. Heath's 1908 edition of Euclid's Elements, specifically focusing on the introduction and Book I. Heath analyzes the historical context of Euclid's work, examining commentaries by Proclus and others, exploring the evolution of Euclidean geometry, and investigating textual variations and interpolations across different manuscripts and translations. He also addresses the challenges posed by Euclid's postulates, particularly the parallel postulate, and various attempts to prove or replace it. Furthermore, the text delves into the definitions and propositions of Euclid, comparing different interpretations and offering alternative proofs where applicable. Finally, Heath touches upon the influence of Euclid's Elements on subsequent mathematical developments, including non-Euclidean geometry.

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Wiggins and McTighe's Understanding by Design (2011) is a guide for creating high-quality educational units. The authors present a three-stage backward design process focusing on desired results, assessment evidence, and learning plans. They emphasize the importance of understanding and transfer of knowledge, distinguishing these from simple knowledge acquisition and skill development. The text provides templates, examples, and exercises to help educators design units aligned with these principles, using a framework that considers different learning goals and assessment methods. Included are discussions of essential questions, understandings, and the six facets of understanding to aid in assessment design.

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