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Sixteen original essays exploring recent developments in the philosophy of mathematics, written in a way mathematicians will understand.
How do mathematics, philosophy, and theology intersect? In Ideas at the Intersection of Mathematics, Philosophy, and Theology, Carlos Bovell proposes a wide range of possibilities. In a series of eleven thought-provoking essays, the author explores such topics as the place of mathematics in the work of Husserl and Heidegger, the importance of infinity for the Christian conception of God, and the impact of Godel's Theorem on the Westminster Confession of Faith. This book will appeal to readers with backgrounds in mathematics, philosophy, and theology and can be used in core, interdisciplinary modules that contain a math component.
This book provides a collection of chapters from prominent mathematics educators in which they each discuss vital issues in mathematics education and what they see as viable directions research in mathematics education could take to address these issues. All of these issues are related to learning and teaching mathematics. The book consists of nine chapters, seven from each of seven scholars who participated in an invited lecture series (Scholars in Mathematics Education) at Brigham Young University, and two chapters from two other scholars who are writing reaction papers that look across the first seven chapters. The recommendations take the form of broad, overarching principles and ideas that cut across the field. In this sense, this book differs from classical “research agenda projects,” which seek to outline specific research questions that the field should address around a central topic.
This is an anthology of contemporary studies from various disciplinary perspectives written by some of the world's most renowned experts in each of the areas of mathematics, neuroscience, psychology, linguistics, semiotics, education, and more. Its purpose is not to add merely to the accumulation of studies, but to show that math cognition is best approached from various disciplinary angles, with the goal of broadening the general understanding of mathematical cognition through the different theoretical threads that can be woven into an overall understanding. This volume will be of interest to mathematicians, cognitive scientists, educators of mathematics, philosophers of mathematics, semioticians, psychologists, linguists, anthropologists, and all other kinds of scholars who are interested in the nature, origin, and development of mathematical cognition.
Are there objects that are “thin” in the sense that not very much is required for their existence? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. Øystein Linnebo aims to do so by drawing on some Fregean ideas. First, to be an object is to be a possible referent of a singular term. Second, singular reference can be achieved by providing a criterion of identity for the would-be referent. The second idea enables a form of easy reference and thus, via the first idea, also a form of easy being. Paradox is avoided by imposing a predicativity restriction on the criteria of identity. But the abstraction based on a criterion of identity may result in an expanded domain. By iterating such expansions, a powerful account of dynamic abstraction is developed. The result is a distinctive approach to ontology. Abstract objects such as numbers and sets are demystified and allowed to exist alongside more familiar physical objects. And Linnebo also offers a novel approach to set theory which takes seriously the idea that sets are “formed” successively.
When debating the need for prophets, Muslim theologians frequently cited an objection from a group called the Barāhima – either a prophet conveys what is in accordance with reason, so they would be superfluous, or a prophet conveys what is contrary to reason, so they would be rejected. The Barāhima did not recognise prophecy or revelation, because they claimed that reason alone could guide them on the right path. But who were these Barāhima exactly? Were they Brahmans, as their title would suggest? And how did they become associated with this highly incisive objection to prophecy? This book traces the genealogy of the Barāhima and explores their profound impact on the evolution of Islamic theology. It also charts the pivotal role that the Kitāb al-Zumurrud played in disseminating the Barāhima’s critiques and in facilitating an epistemological turn in the wider discourse on prophecy (nubuwwa). When faced with the Barāhima, theologians were not only pressed to explain why rational agents required the input of revelation, but to also identify an epistemic gap that only a prophet could fill. A debate about whether humans required prophets thus evolved into a debate about what humans could and could not know by their own means.
How do we get new knowledge? Following the maverick tradition in the philosophy of science, Carlo Cellucci gradually came to the conclusion that logic can only fulfill its role in mathematics, science and philosophy if it helps us to answer this question. He argues that mathematical logic is inadequate and that we need a new logic, framed in a naturalistic conception of knowledge and philosophy – the heuristic conception. This path from logic to a naturalistic conception of knowledge and philosophy explains the title, From a Heuristic Point of View, which recalls the celebrated collection of essays, From a Logical Point of View, by Willard Van Orman Quine, the father of modern naturalized epistemology. The word ‘heuristic’ points to Cellucci’s favorite theme and the main difference between him and Quine: the emphasis on discovery and building a ‘logic’ for generating new knowledge. This book is a collection of essays from leading figures in this field who discuss, criticize, or expand on the main topics in Cellucci’s work, dealing with some of the most challenging questions in logic, science and philosophy.
What questions do you have about Your Life, Your Being, Your Soul, and Your Conscious Awareness? GOD--The Dimensional Revelation is a book revealing reality. Broadly stated, reality is defined as "all that exists." The study of reality is called "metaphysics." Metaphysics is the primary field of philosophy. Metaphysics is divided into two major fields of study. These are cosmology and ontology. GOD--The Dimensional Revelation reveals cosmological facts about our big bang. Our research reconciles relativity theory and quantum mechanics. Ontologically, This Teaching Reveals a clear theology about GOD and Your Relationship with GOD. Learn how to use the scientific method to prove that the Reality You and I Experience is a partial Fusion of our physical universe with our separate Spiritual Universe. This Teaching proves this in the context of 10 Dimensions of Reality, some of which are original to this book. These 10 Dimensions are clearly explained and verified. Dimensional Reasoning is offered as a tool anyone can use to answer all the great questions of Existence. The Source of Being is clearly identified, as well as the Destination of Individual Being. Reading and Participating in the Knowledge Shared in this book will increase your Confidence, Personal Power, and help You Grow Spiritually.
In The Logic of Number, Neil Tennant defines and develops his Natural Logicist account of the foundations of the natural, rational, and real numbers. Based on the logical system free Core Logic, the central method is to formulate rules of natural deduction governing variable-binding number-abstraction operators and other logico-mathematical expressions such as zero and successor. These enable 'single-barreled' abstraction, in contrast with the 'double-barreled' abstraction effected by principles such as Frege's Basic Law V, or Hume's Principle. Natural Logicism imposes upon its account of the numbers four conditions of adequacy: First, one must show how it is that the various kinds of number are applicable in our wider thought and talk about the world. This is achieved by deriving all instances of three respective schemas: Schema N for the naturals, Schema Q for the rationals, and Schema R for the reals. These provide truth-conditions for statements deploying terms referring to numbers of the kind in question. Second, one must show how it is that the naturals sit among the rationals as themselves again, and the rationals likewise among the reals. Third, one should reveal enough of the metaphysical nature of the numbers to be able to derive the mathematician's basic laws governing them. Fourth, one should be able to demonstrate that there are uncountably many reals. Natural Logicism is realistic about the limits of logicism when it comes to treating the real numbers, for which, Tennant argues, one needs recourse to geometric intuition for deeper starting-points, beyond which logic alone will then deliver the sought results, with absolute formal rigor. The resulting program enables one to delimit, in a principled way, those parts of number theory that are produced by the Kantian understanding alone, and those parts that depend on recourse to (very simple) a priori geometric intuitions.