Author: Arkady Plotnitsky
Publisher: Springer
ISBN: 3319320688
Category : Science
Languages : en
Pages : 341
Book Description
The book considers foundational thinking in quantum theory, focusing on the role the fundamental principles and principle thinking there, including thinking that leads to the invention of new principles, which is, the book contends, one of the ultimate achievements of theoretical thinking in physics and beyond. The focus on principles, prominent during the rise and in the immediate aftermath of quantum theory, has been uncommon in more recent discussions and debates concerning it. The book argues, however, that exploring the fundamental principles and principle thinking is exceptionally helpful in addressing the key issues at stake in quantum foundations and the seemingly interminable debates concerning them. Principle thinking led to major breakthroughs throughout the history of quantum theory, beginning with the old quantum theory and quantum mechanics, the first definitive quantum theory, which it remains within its proper (nonrelativistic) scope. It has, the book also argues, been equally important in quantum field theory, which has been the frontier of quantum theory for quite a while now, and more recently, in quantum information theory, where principle thinking was given new prominence. The approach allows the book to develop a new understanding of both the history and philosophy of quantum theory, from Planck’s quantum to the Higgs boson, and beyond, and of the thinking the key founding figures, such as Einstein, Bohr, Heisenberg, Schrödinger, and Dirac, as well as some among more recent theorists. The book also extensively considers the nature of quantum probability, and contains a new interpretation of quantum mechanics, “the statistical Copenhagen interpretation.” Overall, the book’s argument is guided by what Heisenberg called “the spirit of Copenhagen,” which is defined by three great divorces from the preceding foundational thinking in physics—reality from realism, probability from causality, and locality from relativity—and defined the fundamental principles of quantum theory accordingly.
The Principles of Quantum Theory, From Planck's Quanta to the Higgs Boson
Author: Arkady Plotnitsky
Publisher: Springer
ISBN: 3319320688
Category : Science
Languages : en
Pages : 341
Book Description
The book considers foundational thinking in quantum theory, focusing on the role the fundamental principles and principle thinking there, including thinking that leads to the invention of new principles, which is, the book contends, one of the ultimate achievements of theoretical thinking in physics and beyond. The focus on principles, prominent during the rise and in the immediate aftermath of quantum theory, has been uncommon in more recent discussions and debates concerning it. The book argues, however, that exploring the fundamental principles and principle thinking is exceptionally helpful in addressing the key issues at stake in quantum foundations and the seemingly interminable debates concerning them. Principle thinking led to major breakthroughs throughout the history of quantum theory, beginning with the old quantum theory and quantum mechanics, the first definitive quantum theory, which it remains within its proper (nonrelativistic) scope. It has, the book also argues, been equally important in quantum field theory, which has been the frontier of quantum theory for quite a while now, and more recently, in quantum information theory, where principle thinking was given new prominence. The approach allows the book to develop a new understanding of both the history and philosophy of quantum theory, from Planck’s quantum to the Higgs boson, and beyond, and of the thinking the key founding figures, such as Einstein, Bohr, Heisenberg, Schrödinger, and Dirac, as well as some among more recent theorists. The book also extensively considers the nature of quantum probability, and contains a new interpretation of quantum mechanics, “the statistical Copenhagen interpretation.” Overall, the book’s argument is guided by what Heisenberg called “the spirit of Copenhagen,” which is defined by three great divorces from the preceding foundational thinking in physics—reality from realism, probability from causality, and locality from relativity—and defined the fundamental principles of quantum theory accordingly.
Publisher: Springer
ISBN: 3319320688
Category : Science
Languages : en
Pages : 341
Book Description
The book considers foundational thinking in quantum theory, focusing on the role the fundamental principles and principle thinking there, including thinking that leads to the invention of new principles, which is, the book contends, one of the ultimate achievements of theoretical thinking in physics and beyond. The focus on principles, prominent during the rise and in the immediate aftermath of quantum theory, has been uncommon in more recent discussions and debates concerning it. The book argues, however, that exploring the fundamental principles and principle thinking is exceptionally helpful in addressing the key issues at stake in quantum foundations and the seemingly interminable debates concerning them. Principle thinking led to major breakthroughs throughout the history of quantum theory, beginning with the old quantum theory and quantum mechanics, the first definitive quantum theory, which it remains within its proper (nonrelativistic) scope. It has, the book also argues, been equally important in quantum field theory, which has been the frontier of quantum theory for quite a while now, and more recently, in quantum information theory, where principle thinking was given new prominence. The approach allows the book to develop a new understanding of both the history and philosophy of quantum theory, from Planck’s quantum to the Higgs boson, and beyond, and of the thinking the key founding figures, such as Einstein, Bohr, Heisenberg, Schrödinger, and Dirac, as well as some among more recent theorists. The book also extensively considers the nature of quantum probability, and contains a new interpretation of quantum mechanics, “the statistical Copenhagen interpretation.” Overall, the book’s argument is guided by what Heisenberg called “the spirit of Copenhagen,” which is defined by three great divorces from the preceding foundational thinking in physics—reality from realism, probability from causality, and locality from relativity—and defined the fundamental principles of quantum theory accordingly.
Quantum Probability and Randomness
Author: Andrei Khrennikov
Publisher: MDPI
ISBN: 3038977144
Category : Science
Languages : en
Pages : 276
Book Description
The last few years have been characterized by a tremendous development of quantum information and probability and their applications, including quantum computing, quantum cryptography, and quantum random generators. In spite of the successful development of quantum technology, its foundational basis is still not concrete and contains a few sandy and shaky slices. Quantum random generators are one of the most promising outputs of the recent quantum information revolution. Therefore, it is very important to reconsider the foundational basis of this project, starting with the notion of irreducible quantum randomness. Quantum probabilities present a powerful tool to model uncertainty. Interpretations of quantum probability and foundational meaning of its basic tools, starting with the Born rule, are among the topics which will be covered by this issue. Recently, quantum probability has started to play an important role in a few areas of research outside quantum physics—in particular, quantum probabilistic treatment of problems of theory of decision making under uncertainty. Such studies are also among the topics of this issue.
Publisher: MDPI
ISBN: 3038977144
Category : Science
Languages : en
Pages : 276
Book Description
The last few years have been characterized by a tremendous development of quantum information and probability and their applications, including quantum computing, quantum cryptography, and quantum random generators. In spite of the successful development of quantum technology, its foundational basis is still not concrete and contains a few sandy and shaky slices. Quantum random generators are one of the most promising outputs of the recent quantum information revolution. Therefore, it is very important to reconsider the foundational basis of this project, starting with the notion of irreducible quantum randomness. Quantum probabilities present a powerful tool to model uncertainty. Interpretations of quantum probability and foundational meaning of its basic tools, starting with the Born rule, are among the topics which will be covered by this issue. Recently, quantum probability has started to play an important role in a few areas of research outside quantum physics—in particular, quantum probabilistic treatment of problems of theory of decision making under uncertainty. Such studies are also among the topics of this issue.
Quanta and Mind
Author: J. Acacio de Barros
Publisher: Springer Nature
ISBN: 3030219089
Category : Philosophy
Languages : en
Pages : 261
Book Description
This edited volume examines aspects of the mind/consciousness that are relevant to the interpretations of quantum mechanics. In it, an international group of contributors focus on the possible connections between quantum mechanics and consciousness. They look at how consciousness can help us with quantum mechanics as well as how quantum mechanics can contribute to our understanding of consciousness. For example, what do different interpretations aimed at solving the measurement problem in quantum mechanics tell us about the nature of consciousness, such as von Neumann's interpretation? Each interpretation has, associated to it, a corresponding metaphysical framework that helps us think about possible “models” of consciousness. Alternatively, what does the nature of consciousness tell us about the role of the observer and time reversibility in the measurement process? The book features 20 papers on contemporary approaches to quanta and mind. It brings together the work of scholars from different disciplines with diverse views on the connections between quanta and mind, ranging from those who are supportive of a link between consciousness and quantum physics to those who are very skeptical of such link. Coverage includes such topics as free will in a quantum world, contextuality and causality, mind and matter interaction, quantum panpsychism, the quantum and quantum-like brain, and the role of time in brain-mind dynamics.
Publisher: Springer Nature
ISBN: 3030219089
Category : Philosophy
Languages : en
Pages : 261
Book Description
This edited volume examines aspects of the mind/consciousness that are relevant to the interpretations of quantum mechanics. In it, an international group of contributors focus on the possible connections between quantum mechanics and consciousness. They look at how consciousness can help us with quantum mechanics as well as how quantum mechanics can contribute to our understanding of consciousness. For example, what do different interpretations aimed at solving the measurement problem in quantum mechanics tell us about the nature of consciousness, such as von Neumann's interpretation? Each interpretation has, associated to it, a corresponding metaphysical framework that helps us think about possible “models” of consciousness. Alternatively, what does the nature of consciousness tell us about the role of the observer and time reversibility in the measurement process? The book features 20 papers on contemporary approaches to quanta and mind. It brings together the work of scholars from different disciplines with diverse views on the connections between quanta and mind, ranging from those who are supportive of a link between consciousness and quantum physics to those who are very skeptical of such link. Coverage includes such topics as free will in a quantum world, contextuality and causality, mind and matter interaction, quantum panpsychism, the quantum and quantum-like brain, and the role of time in brain-mind dynamics.
From Electrons to Elephants and Elections
Author: Shyam Wuppuluri
Publisher: Springer Nature
ISBN: 3030921921
Category : Science
Languages : en
Pages : 886
Book Description
This highly interdisciplinary book, covering more than six fields, from philosophy and sciences all the way up to the humanities and with contributions from eminent authors, addresses the interplay between content and context, reductionism and holism and their meeting point: the notion of emergence. Much of today’s science is reductionist (bottom-up); in other words, behaviour on one level is explained by reducing it to components on a lower level. Chemistry is reduced to atoms, ecosystems are explained in terms of DNA and proteins, etc. This approach fails quickly since we can’t cannot extrapolate to the properties of atoms solely from Schrödinger's equation, nor figure out protein folding from an amino acid sequence or obtain the phenotype of an organism from its genotype. An alternative approach to this is holism (top-down). Consider an ecosystem or an organism as a whole: seek patterns on the same scale. Model a galaxy not as 400 billion-point masses (stars) but as an object in its own right with its own properties (spiral, elliptic). Or a hurricane as a structured form of moist air and water vapour. Reductionism is largely about content, whereas holistic models are more attuned to context. Reductionism (content) and holism (context) are not opposing philosophies — in fact, they work best in tandem. Join us on a journey to understand the multifaceted dialectic concerning this duo and how they shape the foundations of sciences and humanities, our thoughts and, the very nature of reality itself.
Publisher: Springer Nature
ISBN: 3030921921
Category : Science
Languages : en
Pages : 886
Book Description
This highly interdisciplinary book, covering more than six fields, from philosophy and sciences all the way up to the humanities and with contributions from eminent authors, addresses the interplay between content and context, reductionism and holism and their meeting point: the notion of emergence. Much of today’s science is reductionist (bottom-up); in other words, behaviour on one level is explained by reducing it to components on a lower level. Chemistry is reduced to atoms, ecosystems are explained in terms of DNA and proteins, etc. This approach fails quickly since we can’t cannot extrapolate to the properties of atoms solely from Schrödinger's equation, nor figure out protein folding from an amino acid sequence or obtain the phenotype of an organism from its genotype. An alternative approach to this is holism (top-down). Consider an ecosystem or an organism as a whole: seek patterns on the same scale. Model a galaxy not as 400 billion-point masses (stars) but as an object in its own right with its own properties (spiral, elliptic). Or a hurricane as a structured form of moist air and water vapour. Reductionism is largely about content, whereas holistic models are more attuned to context. Reductionism (content) and holism (context) are not opposing philosophies — in fact, they work best in tandem. Join us on a journey to understand the multifaceted dialectic concerning this duo and how they shape the foundations of sciences and humanities, our thoughts and, the very nature of reality itself.
The Quantum-Like Revolution
Author: Arkady Plotnitsky
Publisher: Springer Nature
ISBN: 3031129865
Category : Science
Languages : en
Pages : 387
Book Description
Over the last ten years, elements of the formalism of quantum mechanics have been successfully applied beyond physics in areas such as psychology (especially cognition), economics and finance (especially in the formalization of so-called ‘decision making’), political science, and molecular biology. An important stream of work along these lines, commonly under the heading of quantum-like modeling, has been published in well regarded scientific journals, and major publishers have devoted entire books to the topic. This Festschrift honors a key figure in this field of research: Andrei Khrennikov, who made momentous contributions to it and to quantum foundations themselves. While honoring these contributions, and in order to do so, this Festschrift orients its reader toward the future rather than focusing on the past: it addresses future challenges and establishes the way forward in both domains, quantum-like modeling and quantum foundations. A while ago, in response to the developments of using the quantum formalism outside of quantum mechanics, the eminent quantum physicist Anton Zeilinger said, ‘Why should it be precisely the quantum mechanics formalism? Maybe its generalization would be more adequate...’ This volume responds to this statement by both showing the reasons for the continuing importance of quantum formalism and yet also considering pathways to such generalizations. Khrennikov’s work has been indispensable in establishing the great promise of quantum and quantum-like thinking in shaping the future of scientific research across the disciplines.
Publisher: Springer Nature
ISBN: 3031129865
Category : Science
Languages : en
Pages : 387
Book Description
Over the last ten years, elements of the formalism of quantum mechanics have been successfully applied beyond physics in areas such as psychology (especially cognition), economics and finance (especially in the formalization of so-called ‘decision making’), political science, and molecular biology. An important stream of work along these lines, commonly under the heading of quantum-like modeling, has been published in well regarded scientific journals, and major publishers have devoted entire books to the topic. This Festschrift honors a key figure in this field of research: Andrei Khrennikov, who made momentous contributions to it and to quantum foundations themselves. While honoring these contributions, and in order to do so, this Festschrift orients its reader toward the future rather than focusing on the past: it addresses future challenges and establishes the way forward in both domains, quantum-like modeling and quantum foundations. A while ago, in response to the developments of using the quantum formalism outside of quantum mechanics, the eminent quantum physicist Anton Zeilinger said, ‘Why should it be precisely the quantum mechanics formalism? Maybe its generalization would be more adequate...’ This volume responds to this statement by both showing the reasons for the continuing importance of quantum formalism and yet also considering pathways to such generalizations. Khrennikov’s work has been indispensable in establishing the great promise of quantum and quantum-like thinking in shaping the future of scientific research across the disciplines.
Amazing Grace of Quantum Physics
Author: Dillard W. Faries
Publisher: Wipf and Stock Publishers
ISBN: 1532614225
Category : Science
Languages : en
Pages : 283
Book Description
Science and faith have had a long intertwined history. The relationship has run the gamut from a total disconnect to an adversarial battleground where proponents of each claim total victory. However, if God created the physical world and remains active in the physical world, we cannot ignore the interaction nor can we assume or expect a world of conflict. While nineteenth-century physics brought classical physics--which quite reasonably divorced God and nature--to a culmination, twentieth-century physics, especially quantum physics, has opened a new realm of possible interactions. Even though one can reasonably say that no one understands quantum physics, the fruits of the discipline overflow the cornucopia. People of faith can share the feast; and people of science are welcome at the table of faith.
Publisher: Wipf and Stock Publishers
ISBN: 1532614225
Category : Science
Languages : en
Pages : 283
Book Description
Science and faith have had a long intertwined history. The relationship has run the gamut from a total disconnect to an adversarial battleground where proponents of each claim total victory. However, if God created the physical world and remains active in the physical world, we cannot ignore the interaction nor can we assume or expect a world of conflict. While nineteenth-century physics brought classical physics--which quite reasonably divorced God and nature--to a culmination, twentieth-century physics, especially quantum physics, has opened a new realm of possible interactions. Even though one can reasonably say that no one understands quantum physics, the fruits of the discipline overflow the cornucopia. People of faith can share the feast; and people of science are welcome at the table of faith.
Reality Without Realism
Author: Arkady Plotnitsky
Publisher: Springer Nature
ISBN: 3030845788
Category : Science
Languages : en
Pages : 358
Book Description
This book presents quantum theory as a theory based on new relationships among matter, thought, and experimental technology, as against those previously found in physics, relationships that also redefine those between mathematics and physics in quantum theory. The argument of the book is based on its title concept, reality without realism (RWR), and in the corresponding view, the RWR view, of quantum theory. The book considers, from this perspective, the thinking of Bohr, Heisenberg, Schrödinger, and Dirac, with the aim of bringing together the philosophy and history of quantum theory. With quantum theory, the book argues, the architecture of thought in theoretical physics was radically changed by the irreducible role of experimental technology in the constitution of physical phenomena, accordingly, no longer defined independently by matter alone, as they were in classical physics or relativity. Or so it appeared. For, quantum theory, the book further argues, made us realize that experimental technology, beginning with that of our bodies, irreducibly shapes all physical phenomena, and thus makes us rethink the relationships among matter, thought, and technology in all of physics.
Publisher: Springer Nature
ISBN: 3030845788
Category : Science
Languages : en
Pages : 358
Book Description
This book presents quantum theory as a theory based on new relationships among matter, thought, and experimental technology, as against those previously found in physics, relationships that also redefine those between mathematics and physics in quantum theory. The argument of the book is based on its title concept, reality without realism (RWR), and in the corresponding view, the RWR view, of quantum theory. The book considers, from this perspective, the thinking of Bohr, Heisenberg, Schrödinger, and Dirac, with the aim of bringing together the philosophy and history of quantum theory. With quantum theory, the book argues, the architecture of thought in theoretical physics was radically changed by the irreducible role of experimental technology in the constitution of physical phenomena, accordingly, no longer defined independently by matter alone, as they were in classical physics or relativity. Or so it appeared. For, quantum theory, the book further argues, made us realize that experimental technology, beginning with that of our bodies, irreducibly shapes all physical phenomena, and thus makes us rethink the relationships among matter, thought, and technology in all of physics.
Introduction to Quantum Computing with Q# and QDK
Author: Filip Wojcieszyn
Publisher: Springer Nature
ISBN: 3030993795
Category : Science
Languages : en
Pages : 286
Book Description
This book introduces the fundamentals of the theory of quantum computing, illustrated with code samples written in Q#, a quantum-specific programming language, and its related Quantum Development Kit. Quantum computing (QC) is a multidisciplinary field that sits at the intersection of quantum physics, quantum information theory, computer science and mathematics, and which may revolutionize the world of computing and software engineering. The book begins by covering historical aspects of quantum theory and quantum computing, as well as offers a gentle, algebra-based, introduction to quantum mechanics, specifically focusing on concepts essential for the field of quantum programming. Quantum state description, state evolution, quantum measurement and the Bell’s theorem are among the topics covered. The readers also get a tour of the features of Q# and familiarize themselves with the QDK. Next, the core QC topics are discussed, complete with the necessary mathematical formalism. This includes the notions of qubit, quantum gates and quantum circuits. In addition to that, the book provides a detailed treatment of a series of important concepts from quantum information theory, in particular entanglement and the no-cloning theorem, followed by discussion about quantum key distribution and its various protocols. Finally, the canon of most important QC algorithms and algorithmic techniques is covered in-depth - from the Deutsch-Jozsa algorithm, through Grover’s search, to Quantum Fourier Transform, quantum phase estimation and Shor’s algorithm. The book is an accessible introduction into the vibrant and fascinating field of quantum computing, offering a blend of academic diligence with pragmatism that is so central to software development world. All of the discussed theoretical aspects of QC are accompanied by runnable code examples, providing the reader with two different angles - mathematical and programmatic - of looking at the same problem space.
Publisher: Springer Nature
ISBN: 3030993795
Category : Science
Languages : en
Pages : 286
Book Description
This book introduces the fundamentals of the theory of quantum computing, illustrated with code samples written in Q#, a quantum-specific programming language, and its related Quantum Development Kit. Quantum computing (QC) is a multidisciplinary field that sits at the intersection of quantum physics, quantum information theory, computer science and mathematics, and which may revolutionize the world of computing and software engineering. The book begins by covering historical aspects of quantum theory and quantum computing, as well as offers a gentle, algebra-based, introduction to quantum mechanics, specifically focusing on concepts essential for the field of quantum programming. Quantum state description, state evolution, quantum measurement and the Bell’s theorem are among the topics covered. The readers also get a tour of the features of Q# and familiarize themselves with the QDK. Next, the core QC topics are discussed, complete with the necessary mathematical formalism. This includes the notions of qubit, quantum gates and quantum circuits. In addition to that, the book provides a detailed treatment of a series of important concepts from quantum information theory, in particular entanglement and the no-cloning theorem, followed by discussion about quantum key distribution and its various protocols. Finally, the canon of most important QC algorithms and algorithmic techniques is covered in-depth - from the Deutsch-Jozsa algorithm, through Grover’s search, to Quantum Fourier Transform, quantum phase estimation and Shor’s algorithm. The book is an accessible introduction into the vibrant and fascinating field of quantum computing, offering a blend of academic diligence with pragmatism that is so central to software development world. All of the discussed theoretical aspects of QC are accompanied by runnable code examples, providing the reader with two different angles - mathematical and programmatic - of looking at the same problem space.
The Palgrave Handbook of German Idealism and Poststructuralism
Author: Tilottama Rajan
Publisher: Springer Nature
ISBN: 3031273451
Category : Philosophy
Languages : en
Pages : 541
Book Description
The Palgrave Handbook of German Idealism and Poststructuralism offers a wide-ranging dialogue between theory and German Idealism, joining up the various lines of influence connecting German Idealist and Romantic philosophies in all their variety to post-'68 European philosophies, from Derrida and Deleuze to Žižek and Malabou. Key features: Provides in-depth reflections on the various conversations between German Idealism and theory, including an expanded canon of Idealist philosophers and a wide range of contemporary anti-foundationalist thinkers. Includes marginalized voices and concepts that reflect both contemporary concerns as well as the sheer abundance of readings of German Idealism undertaken by European theorists over the last fifty years. Expands the existing scholarship by focusing on new, future directions emerging out of the idealism-theory relationship. The Palgrave Handbook of German Idealism and Poststructuralism is essential reading for researchers and students of all levels — from senior scholars to advanced undergraduates — working on the legacy of German Idealist philosophers within philosophy departments, as well as all those interested in theory from across the humanities.
Publisher: Springer Nature
ISBN: 3031273451
Category : Philosophy
Languages : en
Pages : 541
Book Description
The Palgrave Handbook of German Idealism and Poststructuralism offers a wide-ranging dialogue between theory and German Idealism, joining up the various lines of influence connecting German Idealist and Romantic philosophies in all their variety to post-'68 European philosophies, from Derrida and Deleuze to Žižek and Malabou. Key features: Provides in-depth reflections on the various conversations between German Idealism and theory, including an expanded canon of Idealist philosophers and a wide range of contemporary anti-foundationalist thinkers. Includes marginalized voices and concepts that reflect both contemporary concerns as well as the sheer abundance of readings of German Idealism undertaken by European theorists over the last fifty years. Expands the existing scholarship by focusing on new, future directions emerging out of the idealism-theory relationship. The Palgrave Handbook of German Idealism and Poststructuralism is essential reading for researchers and students of all levels — from senior scholars to advanced undergraduates — working on the legacy of German Idealist philosophers within philosophy departments, as well as all those interested in theory from across the humanities.
Logos and Alogon
Author: Arkady Plotnitsky
Publisher: Springer Nature
ISBN: 3031136780
Category : Mathematics
Languages : en
Pages : 307
Book Description
This book is a philosophical study of mathematics, pursued by considering and relating two aspects of mathematical thinking and practice, especially in modern mathematics, which, having emerged around 1800, consolidated around 1900 and extends to our own time, while also tracing both aspects to earlier periods, beginning with the ancient Greek mathematics. The first aspect is conceptual, which characterizes mathematics as the invention of and working with concepts, rather than only by its logical nature. The second, Pythagorean, aspect is grounded, first, in the interplay of geometry and algebra in modern mathematics, and secondly, in the epistemologically most radical form of modern mathematics, designated in this study as radical Pythagorean mathematics. This form of mathematics is defined by the role of that which beyond the limits of thought in mathematical thinking, or in ancient Greek terms, used in the book’s title, an alogon in the logos of mathematics. The outcome of this investigation is a new philosophical and historical understanding of the nature of modern mathematics and mathematics in general. The book is addressed to mathematicians, mathematical physicists, and philosophers and historians of mathematics, and graduate students in these fields.
Publisher: Springer Nature
ISBN: 3031136780
Category : Mathematics
Languages : en
Pages : 307
Book Description
This book is a philosophical study of mathematics, pursued by considering and relating two aspects of mathematical thinking and practice, especially in modern mathematics, which, having emerged around 1800, consolidated around 1900 and extends to our own time, while also tracing both aspects to earlier periods, beginning with the ancient Greek mathematics. The first aspect is conceptual, which characterizes mathematics as the invention of and working with concepts, rather than only by its logical nature. The second, Pythagorean, aspect is grounded, first, in the interplay of geometry and algebra in modern mathematics, and secondly, in the epistemologically most radical form of modern mathematics, designated in this study as radical Pythagorean mathematics. This form of mathematics is defined by the role of that which beyond the limits of thought in mathematical thinking, or in ancient Greek terms, used in the book’s title, an alogon in the logos of mathematics. The outcome of this investigation is a new philosophical and historical understanding of the nature of modern mathematics and mathematics in general. The book is addressed to mathematicians, mathematical physicists, and philosophers and historians of mathematics, and graduate students in these fields.