- I am a Distinguished Professor of Literature, Theory, and Cultural Studies Program, Department of English, and in the... moreI am a Distinguished Professor of Literature, Theory, and Cultural Studies Program, Department of English, and in the Philosophy and Literature Program at Purdue University. I earned my M.S. in mathematics from the Leningrad (St. Petersburg) University in Russia, and my Ph.D. in comparative literature and literary theory from the University of Pennsylvania. I am the author of ten books and a coeditor of several volumes of articles and special issues of scholarly and scientific journal, and over one hundred fifty scholarly articles on such subjects as continental philosophy, the philosophy of mathematics and science, Romanticism, modernism, and the relationships among philosophy, literature, and science.edit
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... more
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.
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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... more
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.
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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... more
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.
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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... more
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.
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.
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This book offers a discussion of Niels Bohr’s conception of “complementarity,” arguably his greatest contribution to physics and philosophy. By tracing Bohr’s work from his 1913 atomic theory to the introduction and then refinement of the... more
This book offers a discussion of Niels Bohr’s conception of “complementarity,” arguably his greatest contribution to physics and philosophy. By tracing Bohr’s work from his 1913 atomic theory to the introduction and then refinement of the idea of complementarity, and by explicating different meanings of “complementarity” in Bohr and the relationships between it and Bohr’s other concepts, the book aims to offer a contained and accessible, and yet sufficiently comprehensive account of Bohr’s work on complementarity and its significance.
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Quantum mechanics, discovered by Werner Heisenberg and Erwin Schrödinger in 1925-1926, is famous for its radical implications for our conception of physics and for our view of human knowledge in general. While these implications have been... more
Quantum mechanics, discovered by Werner Heisenberg and Erwin Schrödinger in 1925-1926, is famous for its radical implications for our conception of physics and for our view of human knowledge in general. While these implications have been seen as scientifically productive and intellectually liberating to some, Niels Bohr and Heisenberg, among them, they have been troublesome to many others, including Schrödinger and, most famously, Albert Einstein. The situation led to the intense debate that started in the wake of its discovery and has continued into our own time, with no end appearing to be in sight.
Epistemology and Probability aims to contribute to our understanding of quantum mechanics and of the reasons for its extraordinary impact by reconsidering, under the rubric of "nonclassical epistemology," the nature of epistemology and probability, and their relationships in quantum theory. The book brings together the thought of the three figures most responsible for the rise of quantum mechanics—Heisenberg and Schrödinger, on the physical side, and Bohr, on the philosophical side—in order to develop a deeper sense of the physical, mathematical, and philosophical workings of quantum-theoretical thinking. Reciprocally, giving a special emphasis on probability and specifically to the Bayesian concept of probability allows the book to gain new insights into the thought of these figures. The book reconsiders, from this perspective, the Bohr-Einstein debate on the epistemology of quantum physics and, in particular, offers a new treatment of the famous experiment of Einstein, Podolsky, and Rosen (EPR), and of the Bohr-Einstein exchange concerning the subject. It also addresses the relevant aspects of quantum information theory and considers the implications of its epistemological argument for higher-level quantum theories, such as quantum field theory and string and brane theories. One of the main contributions of the book is its analysis of the role of mathematics in quantum theory and in the thinking of Bohr, Heisenberg, and Schrödinger, in particular an examination of the new (vis-à-vis classical physics and relativity) type of the relationships between mathematics and physics introduced by Heisenberg in the course of his discovery of quantum mechanics.
Although Epistemology and Probability is aimed at physicists, philosophers and historians of science, and graduate and advanced undergraduate students in these fields, it is also written with a broader audience in mind and is accessible to readers unfamiliar with the higher-level mathematics used in quantum theory.
Epistemology and Probability aims to contribute to our understanding of quantum mechanics and of the reasons for its extraordinary impact by reconsidering, under the rubric of "nonclassical epistemology," the nature of epistemology and probability, and their relationships in quantum theory. The book brings together the thought of the three figures most responsible for the rise of quantum mechanics—Heisenberg and Schrödinger, on the physical side, and Bohr, on the philosophical side—in order to develop a deeper sense of the physical, mathematical, and philosophical workings of quantum-theoretical thinking. Reciprocally, giving a special emphasis on probability and specifically to the Bayesian concept of probability allows the book to gain new insights into the thought of these figures. The book reconsiders, from this perspective, the Bohr-Einstein debate on the epistemology of quantum physics and, in particular, offers a new treatment of the famous experiment of Einstein, Podolsky, and Rosen (EPR), and of the Bohr-Einstein exchange concerning the subject. It also addresses the relevant aspects of quantum information theory and considers the implications of its epistemological argument for higher-level quantum theories, such as quantum field theory and string and brane theories. One of the main contributions of the book is its analysis of the role of mathematics in quantum theory and in the thinking of Bohr, Heisenberg, and Schrödinger, in particular an examination of the new (vis-à-vis classical physics and relativity) type of the relationships between mathematics and physics introduced by Heisenberg in the course of his discovery of quantum mechanics.
Although Epistemology and Probability is aimed at physicists, philosophers and historians of science, and graduate and advanced undergraduate students in these fields, it is also written with a broader audience in mind and is accessible to readers unfamiliar with the higher-level mathematics used in quantum theory.
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Series • Studies in Literature and Science This book investigates the relationships between modern mathematics and science (in particular, quantum mechanics) and the mode of theorizing that Arkady Plotnitsky defines as "nonclassical" and... more
Series • Studies in Literature and Science This book investigates the relationships between modern mathematics and science (in particular, quantum mechanics) and the mode of theorizing that Arkady Plotnitsky defines as "nonclassical" and identifies in the work of Bohr, Heisenberg, Lacan, and Derrida. Plotnitsky argues that their scientific and philosophical works radically redefined the nature and scope of our knowledge. Building upon their ideas, the book finds a new, nonclassical character in the "dream of great interconnections" Bohr described, thereby engaging with recent debates about the "two cultures" (the humanities and the sciences). Plotnitsky highlights those points at which the known gives way to the unknown (and unknowable). These points are significant, he argues, because they push the boundaries of thought and challenge the boundaries of disciplinarity. One of the book's most interesting observations is that key figures in science, in order to push toward a framing of the unknown, actually retreated into a conservative disciplinarity. Plotnitsky's informed, interdisciplinary approach is more productive than the disparaging attacks on postmodernism or scientism that have hitherto characterized this discourse.
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Many commentators have remarked in passing on the resonance between deconstructionist theory and certain ideas of quantum physics. In this book, Arkady Plotnitsky rigorously elaborates the similarities and differences between the two by... more
Many commentators have remarked in passing on the resonance between deconstructionist theory and certain ideas of quantum physics. In this book, Arkady Plotnitsky rigorously elaborates the similarities and differences between the two by focusing on the work of Niels Bohr and Jacques Derrida. In detailed considerations of Bohr’s notion of complementarity and his debates with Einstein, and in analysis of Derrida’s work via Georges Bataille’s concept of general economy, Plotnitsky demonstrates the value of exploring these theories in relation to each other.
Bohr’s term complementarity describes a situation, unavoidable in quantum physics, in which two theories thought to be mutually exclusive are required to explain a single phenomenon. Light, for example, can only be explained as both wave and particle, but no synthesis of the two is possible. This theoretical transformation is then examined in relation to the ways that Derrida sets his work against or outside of Hegel, also resisting a similar kind of synthesis and enacting a transformation of its own.
Though concerned primarily with Bohr and Derrida, Plotnitsky also considers a wide range of anti-epistemological endeavors including the work of Nietzsche, Bataille, and the mathematician Kurt Gödel. Under the rubric of complementarity he develops a theoretical framework that raises new possibilities for students and scholars of literary theory, philosophy, and philosophy of science.
Bohr’s term complementarity describes a situation, unavoidable in quantum physics, in which two theories thought to be mutually exclusive are required to explain a single phenomenon. Light, for example, can only be explained as both wave and particle, but no synthesis of the two is possible. This theoretical transformation is then examined in relation to the ways that Derrida sets his work against or outside of Hegel, also resisting a similar kind of synthesis and enacting a transformation of its own.
Though concerned primarily with Bohr and Derrida, Plotnitsky also considers a wide range of anti-epistemological endeavors including the work of Nietzsche, Bataille, and the mathematician Kurt Gödel. Under the rubric of complementarity he develops a theoretical framework that raises new possibilities for students and scholars of literary theory, philosophy, and philosophy of science.
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Reading Bohr: Physics and Philosophy offers a new perspective on Niels Bohr's interpretation of quantum mechanics as complementarity, and on the relationships between physics and philosophy in Bohr's work, which has had momentous... more
Reading Bohr: Physics and Philosophy offers a new perspective on Niels Bohr's interpretation of quantum mechanics as complementarity, and on the relationships between physics and philosophy in Bohr's work, which has had momentous significance for our understanding of quantum theory and of the nature of knowledge in general. Philosophically, the book reassesses Bohr's place in the Western philosophical tradition, from Kant and Hegel on. Physically, it reconsiders the main issues at stake in the Bohr-Einstein confrontation and in the ongoing debates concerning quantum physics. It also devotes greater attention than in most commentaries on Bohr to the key developments and transformations of his thinking concerning complementarity.
Most significant among them were those that occurred, first, under the impact of Bohr's exchanges with Einstein and, second, under the impact of developments in quantum theory itself, both quantum mechanics and quantum field theory. The importance of quantum field theory for Bohr's thinking has not been adequately addressed in the literature on Bohr, to the considerable detriment to our understanding of the history of quantum physics. Filling this lacuna is one of the main contributions of the book, which also enables us to show why quantum field theory compels us to move beyond Bohr without, however, simply leaving him behind.
Most significant among them were those that occurred, first, under the impact of Bohr's exchanges with Einstein and, second, under the impact of developments in quantum theory itself, both quantum mechanics and quantum field theory. The importance of quantum field theory for Bohr's thinking has not been adequately addressed in the literature on Bohr, to the considerable detriment to our understanding of the history of quantum physics. Filling this lacuna is one of the main contributions of the book, which also enables us to show why quantum field theory compels us to move beyond Bohr without, however, simply leaving him behind.
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Mathematics, Science, and Postclassical Theory is a unique collection of essays dealing with the intersections between science and mathematics and the radical reconceptions of knowledge, language, proof, truth, and reality currently... more
Mathematics, Science, and Postclassical Theory is a unique collection of essays dealing with the intersections between science and mathematics and the radical reconceptions of knowledge, language, proof, truth, and reality currently emerging from poststructuralist literary theory, constructivist history and sociology of science, and related work in contemporary philosophy. Featuring a distinguished group of international contributors, this volume engages themes and issues central to current theoretical debates in virtually all disciplines: agency, causality, determinacy, representation, and the social dynamics of knowledge.
In a substantive introductory essay, the editors explain the notion of "postclassical theory" and discuss the significance of ideas such as emergence and undecidability in current work in and on science and mathematics. Other essays include a witty examination of the relations among mathematical thinking, writing, and the technologies of virtual reality; an essay that reconstructs the conceptual practices that led to a crucial mathematical discovery—or construction—in the 19th century; a discussion of the implications of Bohr’s complementarity principle for classical ideas of reality; an examination of scientific laboratories as "hybrid" communities of humans and nonhumans; an analysis of metaphors of control, purpose, and necessity in contemporary biology; an exploration of truth and lies, and the play of words and numbers in Shakespeare, Frege, Wittgenstein, and Beckett; and a final chapter on recent engagements, or nonengagements, between rationalist/realist philosophy of science and contemporary science studies.
Contributors. Malcolm Ashmore, Michel Callon, Owen Flanagan, John Law, Susan Oyama, Andrew Pickering, Arkady Plotnitsky, Brian Rotman, Barbara Herrnstein Smith, John Vignaux Smyth, E. Roy Weintraub
In a substantive introductory essay, the editors explain the notion of "postclassical theory" and discuss the significance of ideas such as emergence and undecidability in current work in and on science and mathematics. Other essays include a witty examination of the relations among mathematical thinking, writing, and the technologies of virtual reality; an essay that reconstructs the conceptual practices that led to a crucial mathematical discovery—or construction—in the 19th century; a discussion of the implications of Bohr’s complementarity principle for classical ideas of reality; an examination of scientific laboratories as "hybrid" communities of humans and nonhumans; an analysis of metaphors of control, purpose, and necessity in contemporary biology; an exploration of truth and lies, and the play of words and numbers in Shakespeare, Frege, Wittgenstein, and Beckett; and a final chapter on recent engagements, or nonengagements, between rationalist/realist philosophy of science and contemporary science studies.
Contributors. Malcolm Ashmore, Michel Callon, Owen Flanagan, John Law, Susan Oyama, Andrew Pickering, Arkady Plotnitsky, Brian Rotman, Barbara Herrnstein Smith, John Vignaux Smyth, E. Roy Weintraub
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This article reconsiders E. Schrödinger’s cat paradox experiment from a new perspective, grounded in the interpretation of quantum mechanics that belongs to the class of interpretations designated as “reality without realism” (RWR)... more
This article reconsiders E. Schrödinger’s cat paradox experiment from a new perspective, grounded in the interpretation of quantum mechanics that belongs to the class of interpretations designated as “reality without realism” (RWR) interpretations. These interpretations assume that the reality ultimately responsible for quantum phenomena is beyond conception, an assumption designated as the Heisenberg postulate. Accordingly, in these interpretations, quantum physics is understood in terms of the relationships between what is thinkable and what is unthinkable, with, physical, classical, and quantum, corresponding to thinkable and unthinkable, respectively. The
role of classical physics becomes unavoidable in quantum physics, the circumstance designated as the Bohr postulate, which restores to classical physics its position as part of fundamental physics, a position commonly reserved for quantum physics and relativity. This view of quantum physics
and relativity is maintained by this article as well but is argued to be sufficient for understanding fundamental physics. Establishing this role of classical physics is a distinctive contribution of the article, which allows it to reconsider Schrödinger’s cat experiment, but has a broader significance
for understanding fundamental physics. RWR interpretations have not been previously applied to the cat experiment, including by N. Bohr, whose interpretation, in its ultimate form (he changed it a few times), was an RWR interpretation. The interpretation adopted in this article follows Bohr’s
interpretation, based on the Heisenberg and Bohr postulates, but it adds the Dirac postulate, stating that the concept of a quantum object only applies at the time of observation and not independently.
role of classical physics becomes unavoidable in quantum physics, the circumstance designated as the Bohr postulate, which restores to classical physics its position as part of fundamental physics, a position commonly reserved for quantum physics and relativity. This view of quantum physics
and relativity is maintained by this article as well but is argued to be sufficient for understanding fundamental physics. Establishing this role of classical physics is a distinctive contribution of the article, which allows it to reconsider Schrödinger’s cat experiment, but has a broader significance
for understanding fundamental physics. RWR interpretations have not been previously applied to the cat experiment, including by N. Bohr, whose interpretation, in its ultimate form (he changed it a few times), was an RWR interpretation. The interpretation adopted in this article follows Bohr’s
interpretation, based on the Heisenberg and Bohr postulates, but it adds the Dirac postulate, stating that the concept of a quantum object only applies at the time of observation and not independently.
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This chapter considers the nature of radical transformations of mathematics, enabled by minority mathematics. It will be particularly concerned with modern mathematics, which emerged roughly around 1800, as abstract mathematics –... more
This chapter considers the nature of radical transformations of mathematics,
enabled by minority mathematics. It will be particularly concerned with modern mathematics, which emerged roughly around 1800, as abstract mathematics – abstracted from mathematics’ relations to the natural world and physics, relations that previously dominated mathematics. As, however, defined here (transferring Gilles Deleuze and Félix Guattari’s concept of a minor(ity) literature, as exemplified by F. Kafka’s work), a minority mathematics is not something that exists entirely outside a major mathematics, to be distinguished here from a majority mathematics. Instead, it is a mathematics that, while still exterior to the major mathematics to which it juxtaposes, constructs itself within and even at the very core of this major mathematics. This may be seen as merely a special form of
revolutionary vis-à-vis normal mathematical practice in T. Kuhn’s sense. I shall argue, however, by using the work of N. H. Abel, É. Galois, E. Noether, and A. Grothendieck, as my main cases, that this “special” type of revolutionary practice is the primary and even the only form of revolutionary practice possible in mathematics. I designate this mathematics Abelian mathematics, the term commonly associated with formal mathematical properties (such as commutative group or abelian categories), because Abel’s work was, arguably, the first manifested case of a minority mathematics in this sense in modern mathematics.
enabled by minority mathematics. It will be particularly concerned with modern mathematics, which emerged roughly around 1800, as abstract mathematics – abstracted from mathematics’ relations to the natural world and physics, relations that previously dominated mathematics. As, however, defined here (transferring Gilles Deleuze and Félix Guattari’s concept of a minor(ity) literature, as exemplified by F. Kafka’s work), a minority mathematics is not something that exists entirely outside a major mathematics, to be distinguished here from a majority mathematics. Instead, it is a mathematics that, while still exterior to the major mathematics to which it juxtaposes, constructs itself within and even at the very core of this major mathematics. This may be seen as merely a special form of
revolutionary vis-à-vis normal mathematical practice in T. Kuhn’s sense. I shall argue, however, by using the work of N. H. Abel, É. Galois, E. Noether, and A. Grothendieck, as my main cases, that this “special” type of revolutionary practice is the primary and even the only form of revolutionary practice possible in mathematics. I designate this mathematics Abelian mathematics, the term commonly associated with formal mathematical properties (such as commutative group or abelian categories), because Abel’s work was, arguably, the first manifested case of a minority mathematics in this sense in modern mathematics.
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The starting point of this article is the uncircumventable interference of observational instruments in our observations of nature in quantum physics and, thus, in the constitution of quantum phenomena vs. classical physics or relativity,... more
The starting point of this article is the uncircumventable interference of observational instruments in our observations of nature in quantum physics and, thus, in the constitution of quantum phenomena vs. classical physics or relativity, where this interference can be disregarded or controlled, enabling one to represent the independent behavior of the objects considered. This difference, seen by N. Bohr as the principal difference between quantum and classical physics, grounded his interpretation of quantum phenomena and quantum mechanics, developed by him through his concept of complementarity and, in the ultimate version of his interpretation, introduced in the late 1930s, his concept of (quantum) phenomenon. Bohr's ultimate interpretation belongs to the class of interpretations defined in this article as realitywithout-realism (RWR) interpretations. The interpretation offered in this article follows Bohr's ultimate interpretation but adds several new concepts. The article reconsiders, from the standpoint of this interpretation, the concepts of event, temporality, and causality in quantum physics, by introducing the concepts of quantum causality, juxtaposed to classical causality grounding classical physics and relativity, and the arrow of events, which replaces the concept of the arrow of time, commonly used in this context.
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The argument of this article is grounded in the irreducible interference of observational instruments in our interactions with nature in quantum physics and, thus, in the constitution of quantum phenomena versus classical physics, where... more
The argument of this article is grounded in the irreducible interference of observational instruments in our interactions with nature in quantum physics
and, thus, in the constitution of quantum phenomena versus classical physics, where this interference can, in principle, be disregarded. The irreducible character of this interference was seen by N. Bohr as the principal distinction between classical and quantum physics and grounded his interpretation of quantum phenomena and quantum theory. Bohr saw
complementarity as a generalization of the classical ideal of causality, which defined classical physics and relativity. While intimated by Bohr, the relationships among observational technology, complementarity, causality and the arrow of events (a new concept that replaces the arrow of time commonly used in this context) were not addressed by him either. The article introduces another new concept, that of quantum causality, as a form of probabilistic causality. The argument of the article is based on a particular
interpretation of quantum phenomena and quantum theory, defined by the concept of ‘reality without realism (RWR)’. This interpretation follows Bohr’s
interpretation but contains certain additional features, in particular the Dirac postulate. The article also considers quantum-like (Q-L) theories (based
in the mathematics of QM) from the perspective it develops.
and, thus, in the constitution of quantum phenomena versus classical physics, where this interference can, in principle, be disregarded. The irreducible character of this interference was seen by N. Bohr as the principal distinction between classical and quantum physics and grounded his interpretation of quantum phenomena and quantum theory. Bohr saw
complementarity as a generalization of the classical ideal of causality, which defined classical physics and relativity. While intimated by Bohr, the relationships among observational technology, complementarity, causality and the arrow of events (a new concept that replaces the arrow of time commonly used in this context) were not addressed by him either. The article introduces another new concept, that of quantum causality, as a form of probabilistic causality. The argument of the article is based on a particular
interpretation of quantum phenomena and quantum theory, defined by the concept of ‘reality without realism (RWR)’. This interpretation follows Bohr’s
interpretation but contains certain additional features, in particular the Dirac postulate. The article also considers quantum-like (Q-L) theories (based
in the mathematics of QM) from the perspective it develops.
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The argument of this chapter is governed by the double meaning of its title: (a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and... more
The argument of this chapter is governed by the double meaning of its title:
(a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and discontinuous, between mathematics and physics, as, from Galileo on, a mathematical-experimental science, with mathematics coming first in this conjunction. The project of modern physics was, thus, defined by an essential continuity between mathematics and physics, as a mathematical representation of nature, principally by means of continuous functions and then calculus, until the rise of quantum theory, specifically as quantum
mechanics (QM) and quantum field theory (QFT). With QM, at least in certain interpretations, such as the one adopted in this chapter, quantum theory disconnected the ultimate constitution of nature from any mathematical representation, thus, making physics discontinuous with mathematics. As this chapter will argue, however, this new epistemological situation did not
disconnect quantum physics from mathematics, but, on the contrary, led to relating abstract mathematics, such as that of Hilbert spaces over C and operator algebra there, to discrete physical phenomena in terms of probabilities, thus reestablishing the connection, which was no longer representational, between mathematics and physics. This chapter will also argue that the development of the concepts of continuity and discontinuity,
and the relationships between them, has acquired a great richness and complexity through the nineteenth and twentieth centuries in mathematics itself, richness and complexity that found their way into physics, in particular relativity and quantum theory. One of the intriguing aspects of this development is the idea, advanced by, among others, A. Grothendieck,
following B. Riemann’s comment on the subject, that the continuous may serve as an approximation of the discontinuous, rather than seeing, as is more common, the discontinuous as a mode of technical approach to the continuous. This chapter will discuss this idea, and the relationships
between continuity and discontinuity in, and between, mathematics and physics in terms of two new concepts: reality without realism (RWR), applicable in both mathematics and physics, and the relationships between them, and ideality without idealism (IWI), the version of the concept of RWR, applicable specifically to mathematics.
(a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and discontinuous, between mathematics and physics, as, from Galileo on, a mathematical-experimental science, with mathematics coming first in this conjunction. The project of modern physics was, thus, defined by an essential continuity between mathematics and physics, as a mathematical representation of nature, principally by means of continuous functions and then calculus, until the rise of quantum theory, specifically as quantum
mechanics (QM) and quantum field theory (QFT). With QM, at least in certain interpretations, such as the one adopted in this chapter, quantum theory disconnected the ultimate constitution of nature from any mathematical representation, thus, making physics discontinuous with mathematics. As this chapter will argue, however, this new epistemological situation did not
disconnect quantum physics from mathematics, but, on the contrary, led to relating abstract mathematics, such as that of Hilbert spaces over C and operator algebra there, to discrete physical phenomena in terms of probabilities, thus reestablishing the connection, which was no longer representational, between mathematics and physics. This chapter will also argue that the development of the concepts of continuity and discontinuity,
and the relationships between them, has acquired a great richness and complexity through the nineteenth and twentieth centuries in mathematics itself, richness and complexity that found their way into physics, in particular relativity and quantum theory. One of the intriguing aspects of this development is the idea, advanced by, among others, A. Grothendieck,
following B. Riemann’s comment on the subject, that the continuous may serve as an approximation of the discontinuous, rather than seeing, as is more common, the discontinuous as a mode of technical approach to the continuous. This chapter will discuss this idea, and the relationships
between continuity and discontinuity in, and between, mathematics and physics in terms of two new concepts: reality without realism (RWR), applicable in both mathematics and physics, and the relationships between them, and ideality without idealism (IWI), the version of the concept of RWR, applicable specifically to mathematics.
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The primary aim of this chapter is to consider mathematicians’ working philosophy of mathematics as emerging in and defining actual mathematical thinking and practice, as exemplified in three cases stated in my subtitle: Niels Henrik Abel... more
The primary aim of this chapter is to consider mathematicians’ working philosophy of mathematics as emerging in and defining actual mathematical thinking and practice, as exemplified in three cases stated in my subtitle: Niels Henrik Abel and Évariste Galois, Nikolai Lobachevsky and Bernhard Riemann, and André Weil and Alexander Grothendieck. I speak of “and” and hence “conjunction,” rather than the disjunctive “or,” in all three paired cases considered in this chapter because, while my primary concern is on the (historically) second figure – Galois, Riemann, and Grothendieck – in each case, my aim is not merely to juxtapose these figures, especially given the significance of their thinking for transforming mathematics. The work of Abel, Lobachevsky, and Weil were revolutionary events as well. Instead, while granting the differences between their thinking, my aim is to explore the shared grounding that gives rise to these differences, defining, and defined by, their mathematical practice as philosophy. The chapter’s approach to their mathematical practice and to creative mathematical practice in
general is parallel to that of Deleuze and Guattari in creative philosophical
practice, which or even true philosophy itself is defined by them as the practice of the invention of new concepts, with philosophical concepts given a particular definition, in part in juxtaposition to mathematical and scientific concepts. The working philosophy of mathematics this chapter considers, under the heading of “mathematical practice as philosophy,” is, analogously, defined as the practice of the invention of new mathematical concepts, defined, against the grain of Deleuze and Guattari’s argument, in affinity with (although not identically to) philosophical concepts in their definition.
general is parallel to that of Deleuze and Guattari in creative philosophical
practice, which or even true philosophy itself is defined by them as the practice of the invention of new concepts, with philosophical concepts given a particular definition, in part in juxtaposition to mathematical and scientific concepts. The working philosophy of mathematics this chapter considers, under the heading of “mathematical practice as philosophy,” is, analogously, defined as the practice of the invention of new mathematical concepts, defined, against the grain of Deleuze and Guattari’s argument, in affinity with (although not identically to) philosophical concepts in their definition.
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This article considers a rarely discussed aspect, the no-cloning principle or postulate, recast as the uniqueness postulate, of the mathematical modeling known as quantum-like, Q-L, modeling (vs. classical-like, C-L, modeling, based in... more
This article considers a rarely discussed aspect, the no-cloning principle or postulate, recast as the uniqueness postulate, of the mathematical modeling known as quantum-like, Q-L, modeling (vs. classical-like, C-L, modeling, based in the mathematics adopted from classical physics) and the
corresponding Q-L theories beyond physics. The principle is a transfer of the no-cloning principle (arising from the no-cloning theorem) in quantum mechanics (QM) to Q-L theories. My interest in this principle, to be related to several other key features of QM and Q-L theories, such as the irreducible
role of observation, complementarity, and probabilistic causality, is connected to a more general question: What are the ontological and epistemological reasons for using Q-L models vs. C-L ones? I shall argue that adopting the uniqueness postulate is justified in Q-L theories and adds an important new motivation for doing so and a new venue for considering this question. In order to properly ground this argument, the article also offers a discussion along similar lines of QM, providing a new angle on Bohr’s concept of complementarity via the uniqueness postulate.
corresponding Q-L theories beyond physics. The principle is a transfer of the no-cloning principle (arising from the no-cloning theorem) in quantum mechanics (QM) to Q-L theories. My interest in this principle, to be related to several other key features of QM and Q-L theories, such as the irreducible
role of observation, complementarity, and probabilistic causality, is connected to a more general question: What are the ontological and epistemological reasons for using Q-L models vs. C-L ones? I shall argue that adopting the uniqueness postulate is justified in Q-L theories and adds an important new motivation for doing so and a new venue for considering this question. In order to properly ground this argument, the article also offers a discussion along similar lines of QM, providing a new angle on Bohr’s concept of complementarity via the uniqueness postulate.
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This chapter considers the question of difference and its implications in the context of the relationships between German Idealism and contemporary philosophy-a context important and even indispensable for our understanding of this... more
This chapter considers the question of difference and its implications in the context of the relationships between German Idealism and contemporary philosophy-a context important and even indispensable for our understanding of this question and its implications. Indeed, one might argue that what is at stake in this conjunction and what still defines our own thinking concerning difference and our philosophical thinking in general is enormous, as Jacques Derrida said referring specifically to G. W. F. Hegel and Derrida's own rethinking of the question of difference: "What is at stake here is enormous [L'enjeu est ici énorme]." 1 The signifier jeu [play] in French is worth registering: the concept of play is important for Derrida and is, like it is in Hegel ("the play [Spiel] of forces"), a concept of difference. 2 Derrida here expressly juxtaposes his most famous concept or, as he saw it, neither a term nor a concept, that of différance, to Hegel's concept of Aufhebung (an operation combining a negation, conservation, and sublation of a given concept in a new concept). Specifically, Derrida sees différance as "the limit, the interruption, the destruction of the Hegelian [Aufhebung] wherever it operates." 3 Derrida qualifies: "I emphasize the Hegelian Aufhebung, such as it is interpreted by a certain Hegelian discourse, for it goes without saying that the double [technically, triple] meaning of Aufhebung could be written otherwise." 4 It follows that at stake in the question of différance is also a reading of Hegel, and hence Immanuel Kant and German Idealism, where, at least in its dominant reception-history, the confrontation between Kant and Hegel occupies center stage.
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This article reconsiders the double-slit experiment from the nonrealist or, in terms of this article, "reality-without-realism" (RWR) perspective, grounded in the combination of three forms of quantum discontinuity: (1) "Heisenberg... more
This article reconsiders the double-slit experiment from the nonrealist or, in terms of this article, "reality-without-realism" (RWR) perspective, grounded in the combination of three forms of quantum discontinuity: (1) "Heisenberg discontinuity", defined by the impossibility of a representation or even conception of how quantum phenomena come about, even though quantum theory (such as quantum mechanics or quantum field theory) predicts the data in question strictly in accord with what is observed in quantum experiments); (2) "Bohr discontinuity", defined, under the assumption of Heisenberg discontinuity, by the view that quantum phenomena and the data observed therein are described by classical and not quantum theory, even though classical physics cannot predict them; and (3) "Dirac discontinuity" (not considered by Dirac himself, but suggested by his equation), according to which the concept of a quantum object, such as a photon or electron, is an idealization only applicable at the time of observation and not to something that exists independently in nature. Dirac discontinuity is of particular importance for the article's foundational argument and its analysis of the double-slit experiment.
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This article aims to reconsider E. Schrödinger's famous thought experiment, the cat-paradox experiment, and its place in quantum foundations from a new perspective, grounded in the type of interpretation of quantum phenomena and quantum... more
This article aims to reconsider E. Schrödinger's famous thought experiment, the cat-paradox experiment, and its place in quantum foundations from a new perspective, grounded in the type of interpretation of quantum phenomena and quantum mechanics, which belongs to the class of interpretations designated here as "reality without realism" (RWR) interpretations. Such interpretations have not been previously brought to bear on the cat experiment, including by N. Bohr, whose interpretation in its ultimate form (as he changed his interpretation a few times) is an RWR interpretation, but who does not appear to have commented on the cat experiment. The interpretation adopted in this article follows Bohr's interpretation, as based on two assumptions or postulates, the Heisenberg and Bohr postulates, but it adds a third postulate, the Dirac postulate. The article also introduces, in conjunction with the concept of reality without realism, the concepts of visible and invisible to thought and considers their role in the cat-paradox experiment.
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This article addresses the effectiveness of the predictive modeling of cognition and behavior based on quantum principles and some of the reasons for this effectiveness. It also aims, however, to explore the limitations of mathematical... more
This article addresses the effectiveness of the predictive modeling of cognition and behavior based on quantum principles and some of the reasons for this effectiveness. It also aims, however, to explore the limitations of mathematical modeling so based, quantum-like (Q-L) modeling, and all mathematical modeling, including classical-like (C-L), in considering human cognition and behavior. It will discuss certain alternative approaches to both, essentially philosophical in nature, although sometimes found in literary works, approaches that, while not quantitative, may help compensate for limitations of mathematical modeling there. Most Q-L and C-L approaches beyond physics are realist, insofar as they offer representations of human thinking by the formalism of quantum or classical physical theories. The position adopted in this article is based on the non-realist assumption that such a representation may not be possible, which is not the same as that it is impossible. I designate interpretations that do not make this assumption reality-without-realism, RWR, interpretations, and in considering mental processes as ideality-without-idealism, IWI, interpretations.
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Although Niels Bohr’s most famous conceptual contribution to physics and philosophy is his concept of complementarity, introduced in 1927, two of his later concepts, correlative to each other, phenomenon and atomicity, introduced in the... more
Although Niels Bohr’s most famous conceptual contribution to physics and philosophy is his concept of complementarity, introduced in 1927, two of his later concepts, correlative to each other, phenomenon and atomicity, introduced in the late 1930s, are as innovative and important. According to Bohr (writing in 1949): “Planck’s discovery of the quantum of action [h] disclosed a novel feature of atomicity in the laws of nature supplementing in such unsuspected manner the old doctrine of the limited divisibility of matter” (Bohr 1987, v. 2, p. 34). The realization of this disclosure, however, took three decades after Max Planck’s discovery of quantum theory in 1900. It was shaped by several subsequent developments, in particular by Bohr (1913). Philosophical Magazine 26, 1–25 atomic theory, the discovery of quantum mechanics (QM) in 1925, and Bohr’s interpretation of QM, developed in its ultimate form, grounded in his concepts of phenomena and atomicity, in1930s.
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This article reconsiders the concept of physical reality in quantum theory and the concept of quantum measurement, following Bohr, whose analysis of quantum measurement led him to his concept of a (quantum) “phenomenon,” referring to “the... more
This article reconsiders the concept of physical reality in quantum theory and the concept of quantum measurement, following Bohr, whose analysis of quantum measurement led him to his concept of a (quantum) “phenomenon,” referring to “the observations obtained under the specified circumstances,” in the interaction between quantum objects and measuring instruments. This situation makes the terms “observation” and “measurement,” as conventionally understood, inapplicable. These terms are remnants of classical physics or still earlier history, from which classical physics inherited it. As defined here, a quantum measurement does not measure any preexisting property of the ultimate constitution of the reality responsible for quantum phenomena. An act of measurement establishes a quantum phenomenon by an interaction between the instrument and the
quantum object or in the present view the ultimate constitution of the reality responsible for quantum phenomena and, at the time of measurement, also quantum objects. In the view advanced in this
article, in contrast to that of Bohr, quantum objects, such as electrons or photons, are assumed to exist only at the time of measurement and not independently, a view that redefines the concept of quantum object as well. This redefinition becomes especially important in high-energy quantum regimes and quantum field theory and allows this article to define a new concept of quantum field. The article also considers, now following Bohr, the quantum measurement as the entanglement
between quantum objects and measurement instruments. The argument of the article is grounded in the concept “reality without realism” (RWR), as underlying quantum measurement thus understood, and the view, the RWR view, of quantum theory defined by this concept. The RWR view places a stratum of physical reality thus designated, here the reality ultimately responsible for quantum phenomena, beyond representation or knowledge, or even conception, and defines the corresponding set of interpretations quantum mechanics or quantum field theory, such as the one assumed in this article, in which, again, not only quantum phenomena but also quantum objects are (idealizations) defined by measurement. As such, the article also offers a broadly conceived response to J. Bell’s argument “against ‘measurement’”.
quantum object or in the present view the ultimate constitution of the reality responsible for quantum phenomena and, at the time of measurement, also quantum objects. In the view advanced in this
article, in contrast to that of Bohr, quantum objects, such as electrons or photons, are assumed to exist only at the time of measurement and not independently, a view that redefines the concept of quantum object as well. This redefinition becomes especially important in high-energy quantum regimes and quantum field theory and allows this article to define a new concept of quantum field. The article also considers, now following Bohr, the quantum measurement as the entanglement
between quantum objects and measurement instruments. The argument of the article is grounded in the concept “reality without realism” (RWR), as underlying quantum measurement thus understood, and the view, the RWR view, of quantum theory defined by this concept. The RWR view places a stratum of physical reality thus designated, here the reality ultimately responsible for quantum phenomena, beyond representation or knowledge, or even conception, and defines the corresponding set of interpretations quantum mechanics or quantum field theory, such as the one assumed in this article, in which, again, not only quantum phenomena but also quantum objects are (idealizations) defined by measurement. As such, the article also offers a broadly conceived response to J. Bell’s argument “against ‘measurement’”.
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This article considers a partly philosophical question: What are the ontological and epistemological reasons for using quantum-like models or theories (models and theories based on the mathematical formalism of quantum theory) vs.... more
This article considers a partly philosophical question: What are the ontological and epistemological reasons for using quantum-like models or theories (models and theories based on the mathematical formalism of quantum theory) vs. classical-like ones (based on the mathematics of classical physics), in considering human thinking and decision making? This question is only partly philosophical because it also concerns the scientific understanding of the phenomena considered by the theories that use mathematical models of either type, just as in physics itself, where this question also arises as a physical question. This is because this question is in effect: What are the physical reasons for using, even if not requiring, these types of theories in considering quantum phenomena, which these theories predict fully in accord with the experiment? This is clearly also a physical, rather than only philosophical, question and so is, accordingly, the question of whether one needs classical-like or quantum-like theories or both (just as in physics we use both classical and
quantum theories) in considering human thinking in psychology and related fields, such as decision science. It comes as no surprise that many of these reasons are parallel to those that are responsible for the use of QM and QFT in the case of quantum phenomena. Still, the corresponding situations should be understood and justified in terms of the phenomena considered, phenomena defined by human thinking, because there are important differences between these phenomena and quantum
phenomena, which this article aims to address. In order to do so, this article will first consider quantum phenomena and quantum theory, before turning to human thinking and decision making,
in addressing which it will also discuss two recent quantum-like approaches to human thinking, that by M. G. D’Ariano and F. Faggin and that by A. Khrennikov. Both approaches are ontological in the sense of offering representations, different in character in each approach, of human thinking by the formalism of quantum theory. Whether such a representation, as opposed to only predicting the outcomes of relevant experiments, is possible either in quantum theory or in quantum-like theories of human thinking is one of the questions addressed in this article. The philosophical position adopted in it is that it may not be possible to make this assumption, which, however, is not the same as
saying that it is impossible. I designate this view as the reality-without-realism, RWR, view and in considering strictly mental processes as the ideality-without-idealism, IWI, view, in the second case in part following, but also moving beyond, I. Kant’s philosophy.
quantum theories) in considering human thinking in psychology and related fields, such as decision science. It comes as no surprise that many of these reasons are parallel to those that are responsible for the use of QM and QFT in the case of quantum phenomena. Still, the corresponding situations should be understood and justified in terms of the phenomena considered, phenomena defined by human thinking, because there are important differences between these phenomena and quantum
phenomena, which this article aims to address. In order to do so, this article will first consider quantum phenomena and quantum theory, before turning to human thinking and decision making,
in addressing which it will also discuss two recent quantum-like approaches to human thinking, that by M. G. D’Ariano and F. Faggin and that by A. Khrennikov. Both approaches are ontological in the sense of offering representations, different in character in each approach, of human thinking by the formalism of quantum theory. Whether such a representation, as opposed to only predicting the outcomes of relevant experiments, is possible either in quantum theory or in quantum-like theories of human thinking is one of the questions addressed in this article. The philosophical position adopted in it is that it may not be possible to make this assumption, which, however, is not the same as
saying that it is impossible. I designate this view as the reality-without-realism, RWR, view and in considering strictly mental processes as the ideality-without-idealism, IWI, view, in the second case in part following, but also moving beyond, I. Kant’s philosophy.
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Throughout the history of quantum theory, "an elementary particle" has been a problem to which only fragments of a possible solution could be offered. At a certain point of this history, "a virtual particle" has become part of this... more
Throughout the history of quantum theory, "an elementary particle" has been a problem to which only fragments of a possible solution could be offered. At a certain point of this history, "a virtual particle" has become part of this problem. While still considering this problem as a problem, this article, in contrast to most approaches to it, which are realist in nature, offers a nonrealist one. This approach is grounded in the concept of reality without realism, RWR, introduced by this author previously, and an interpretation of high-energy quantum phenomena and quantum field theory (QFT), grounded in this concept, which allows for a range of (RWR-type) interpretations. The status of real and virtual particles is different in the present interpretation. While, in this interpretation, both concepts are idealizations, that of real particles is required by it. By contrast, the concept of virtual particles is not required, although, because of certain observable effects in high-energy quantum regimes, this interpretation assumes that there exists something in nature, especially manifested in high-energy quantum regimes, that may be handled by means of the concept of virtual particles. RWR-type interpretations do, however, permit an idealization defined by this concept and may adopt it, because of its practical effectiveness. This article will do so, while still maintaining the essential difference in the nature of the idealizations defined by the concepts of real and virtual particles.
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This article introduces the concept of radical Pythagorean mathematics, as part of mathematical modernism, which emerged around 1900 and extends to our own time. Arguably, the greatest discovery of Pythagorean mathematics of (sixth-fifth... more
This article introduces the concept of radical Pythagorean mathematics, as part of mathematical modernism, which emerged around 1900 and extends to our own time. Arguably, the greatest discovery of Pythagorean mathematics of (sixth-fifth century BC) was that of the incommensurability of the side and the diagonal of the square. This discovery established the complex relationships between geometry and arithmetic and, correlatively, the possibility of a type of logos, a logos containing the alogon within it – a logical thought containing the unthinkable within it. I shall call the mathematics defined by this combination “Pythagorean mathematics.” I shall argue that, as part of mathematical modernism, there emerged a new attitude toward the unthinkable in thought, an attitude that accepts
this situation as a positive, enabling condition of thinking and knowledge. I shall call the mathematical thinking defined by this attitude, while keeping the irreducible relationship between geometry and algebra (which subsumes arithmetic within it), radical Pythagorean mathematics. The original Pythagorean attitude, at least in one reconstitution of Pythagorean thought, was, as against the Platonist attitude, tolerant of this condition, but still sought to overcome it by finding a logos without the alogon within it. This attitude, often sliding into a Platonist one, has remained dominant throughout the history of mathematics, physics, and philosophy, including during the modernist period, notable as much for the presence of the radical Pythagorean thinking as for a resistance to it.
this situation as a positive, enabling condition of thinking and knowledge. I shall call the mathematical thinking defined by this attitude, while keeping the irreducible relationship between geometry and algebra (which subsumes arithmetic within it), radical Pythagorean mathematics. The original Pythagorean attitude, at least in one reconstitution of Pythagorean thought, was, as against the Platonist attitude, tolerant of this condition, but still sought to overcome it by finding a logos without the alogon within it. This attitude, often sliding into a Platonist one, has remained dominant throughout the history of mathematics, physics, and philosophy, including during the modernist period, notable as much for the presence of the radical Pythagorean thinking as for a resistance to it.
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This article aims to contribute to the ongoing task of clarifying the relationships between reality, probability, and nonlocality in quantum physics. It is in part stimulated by A. Khrennikov’s argument, in several communications, for... more
This article aims to contribute to the ongoing task of clarifying the relationships between reality, probability, and nonlocality in quantum physics. It is in part stimulated by A. Khrennikov’s argument, in several communications, for “eliminating the issue of quantum nonlocality” from the analysis of quantum entanglement. I argue, however, that the question may not be that of eliminating but instead that of further illuminating this issue, a task that can be pursued by relating quantum nonlocality to other key features of quantum phenomena. I suggest that the following features of quantum phenomena and quantum mechanics, distinguishing them from classical phenomena and classical physics—(1) the irreducible role of measuring instruments in defining quantum phenomena, (2) discreteness, (3) complementarity, (4) entanglement, (5) quantum nonlocality, and (6) the irreducibly probabilistic nature of quantum predictions—are all interconnected, so that it is difficult to give an unconditional priority to any one of them. To argue this case, I shall consider quantum phenomena and quantum mechanics from a nonrealist or, in terms adopted here, “reality-without-realism” (RWR) perspective. This perspective extends Bohr’s view,
grounded in his analysis of the irreducible role of measuring instruments in the constitution of quantum phenomena.
grounded in his analysis of the irreducible role of measuring instruments in the constitution of quantum phenomena.
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Following the view of several leading quantum-information theorists, this paper argues that quantum phenomena, including those exhibiting quantum correlations (one of their most enigmatic features), and quantum mechanics may be best... more
Following the view of several leading quantum-information theorists, this paper argues that quantum phenomena, including those exhibiting quantum correlations (one of their most enigmatic features), and quantum mechanics may be best understood in quantum-informational terms. It also argues that this understanding is implicit already in the work of some among the founding figures of quantum mechanics, in particular W. Heisenberg and N. Bohr, half a century before quantum information theory emerged and confirmed, and gave a deeper meaning to, to their insights. These insights, I further argue, still help this understanding, which is the main reason for considering them here. My argument is grounded in a particular interpretation of quantum phenomena and quantum mechanics, in part arising from these insights as well. This interpretation is based on the concept of reality without realism, RWR (which places the reality considered beyond representation or even conception) , introduced by this author previously, in turn, following Heisenberg and Bohr, and in response to quantum information theory.
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This chapter addresses the relationships between modernist literature, a long-established denomination, and modernist mathematics, a recent and infrequent denomination. Historically, both phenomena are commonly understood as belonging to... more
This chapter addresses the relationships between modernist literature, a long-established denomination, and modernist mathematics, a recent and infrequent denomination. Historically, both phenomena are commonly understood as belonging to the same period, roughly, from 1900 to the 1940s, but as extending to or continuing to impact literature and mathematics of our own time. Conceptually, the thinking concerning each (or modernism elsewhere) is more diverse and, in each case, only partially reflects the nature of each and the relationships between them. 1 This is an unavoidable limitation, and the conception of modernism to be offered here cannot circumvent it either. This conception is grounded in the double movement, found, I argue, in modernist literature and modernist mathematics alike. The first is the movement from ontology to technology, specifically the technology of composition, which can place the ultimate nature of reality (referring, roughly, to what exits) beyond a representation or even conception, and thus beyond ontology (referring to such a representation or at least conception). While, generally, technology is a means of doing something, of getting "from here to there," as it were, this article extends the concept of "experimental technology" in modern, post-Galilean physics, defined by its jointly experimental and mathematical character, to mathematics and literature, and to theoretical thinking elsewhere. Thus, this extension, when it comes to abandoning ontological thinking, also defines (modernist) theoretical physics, beginning with quantum mechanics, as a fundamentally mathematical project. The second movement is toward independence and self-determination of either field, which does not preclude the relationships between them.
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This chapter continues the examination of the relationships between modernist literature and modernist mathematics begun in chapter "Modernist Literature and Modernist Mathematics I: Mathematics and Composition, with Mallarmé, Heisenberg,... more
This chapter continues the examination of the relationships between modernist literature and modernist mathematics begun in chapter "Modernist Literature and Modernist Mathematics I: Mathematics and Composition, with Mallarmé, Heisenberg, and Derrida." The previous chapter discussed a movement in literature and mathematics toward independence and self-determination and proposed that the technology of composition can place the ultimate nature of reality beyond a representation or even conception. This chapter continues to address the question of ontology and mathematics in modernism, with a focus on Stéphane Mallarmé's (1842-1898) theoretical ideas and Alain Badiou's (1937-) philosophical work. Mallarmé and Badiou, for whom Mallarmé is a major literary inspiration, share the mathematical groundings of their thinking concerning, respectively, literature, and philosophy. There are, however, significant differences between them in this regard, in the present reading (Badiou reads Mallarmé differently), because Mallarmé's appeal to and use of mathematics are primarily technological while for Badiou, they are ontological. Badiou contends that any rigorous philosophical ontology can only be mathematical, at least as "a thesis … about discourse": "mathematics … pronounces what is expressible about being qua being" (Badiou 2007, p. 8). On the other hand, as discussed in "Modernist Literature and Modernist Mathematics I," Badiou's interest in Mallarmé is equally shaped by Mallarmé's thinking concerning "the power of chance,” and also concerns Badiou’s concept of “event,” always an event of trans-Being placed beyond ontology (in his sense) and thus mathematics.
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This article discusses the relationships between Thomas Pynchon’s novels and the philosophy of chance and probability, especially in connection with quantum theory, which radically redefined our thinking concerning both concepts, and to... more
This article discusses the relationships between Thomas Pynchon’s novels and the philosophy of chance and probability, especially in connection with quantum theory, which radically redefined our thinking concerning both concepts, and to begin with, the nature of physical reality. The article considers how different scientific theories dealing with chance and probability figure in Pynchon’s major novels, which, the article argues, helps us to think more deeply about Pynchon’s use of these theories or other mathematical and scientific theories, and about the relationships among literature, philosophy, and mathematics and science in general.
Cet essai s’interroge sur les relations que tissent les romans de Thomas Pynchon avec la philosophie du hasard et de la probabilité, notamment dans son rapport avec la théorie quantique, qui a radicalement redéfini la façon dont nous pensons ces deux concepts, ainsi que, pour commencer, la nature même de la réalité physique. Cet article interroge la présence de diverses théories scientifiques traitant du hasard et de la probabilité dans les principaux romans de Pynchon, et cherche à approfondir la réflexion autour de l’utilisation que fait le romancier de ces théories et d’autres, qu’elles soient mathématiques ou scientifiques. Ce sont donc les articulations entre littérature, philosophie, mathématiques et, plus largement, la science que cet article vise à interroger.
Cet essai s’interroge sur les relations que tissent les romans de Thomas Pynchon avec la philosophie du hasard et de la probabilité, notamment dans son rapport avec la théorie quantique, qui a radicalement redéfini la façon dont nous pensons ces deux concepts, ainsi que, pour commencer, la nature même de la réalité physique. Cet article interroge la présence de diverses théories scientifiques traitant du hasard et de la probabilité dans les principaux romans de Pynchon, et cherche à approfondir la réflexion autour de l’utilisation que fait le romancier de ces théories et d’autres, qu’elles soient mathématiques ou scientifiques. Ce sont donc les articulations entre littérature, philosophie, mathématiques et, plus largement, la science que cet article vise à interroger.
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One contribution of 16 to a theme issue 'Contextuality and probability in quantum mechanics and beyond'. Subject Areas: quantum physics This article brings together reality, complementarity and contextuality in quantum theory. It... more
One contribution of 16 to a theme issue 'Contextuality and probability in quantum mechanics and beyond'. Subject Areas: quantum physics This article brings together reality, complementarity and contextuality in quantum theory. It clarifies Bohr's concept of complementarity by considering the non-realist epistemology and the corresponding interpretations of quantum mechanics, based in the concept of 'reality without realism'. Finally, as its main novel contribution, it establishes the connections between complementarity and contextuality.
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We consider the concept of curve in the context of the transition from mathematical "modernity" to mathematical "modernism," the transition defined, the article argues, by the movement from the primacy of geometrical to the primacy of... more
We consider the concept of curve in the context of the transition from mathematical "modernity" to mathematical "modernism," the transition defined, the article argues, by the movement from the primacy of geometrical to the primacy of algebraic thinking. The article also explores the ontological and epistemological aspects of this transition and the connections between modernist mathematics and modernist physics, especially quantum theory, in this set of contexts.
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This chapter is an essay on the conceptual nature of Riemann's thinking and its impact, as conceptual thinking, on mathematics, physics, and philosophy. In order to fully appreciate the revolutionary nature of this thinking and of... more
This chapter is an essay on the conceptual nature of Riemann's thinking and its impact, as conceptual thinking, on mathematics, physics, and philosophy. In order to fully appreciate the revolutionary nature of this thinking and of Riemann's practice of mathematics, one must, this chapter argues, rethink the nature of mathematical or scientific concepts in Riemann and beyond. The chapter will attempt to do so with the help of Deleuze and Guattari's concept of philosophical concept. The chapter will argue that a fundamentally analogous concept of concept is also applicable in mathematics and science, specifically and most pertinently to Riemann, in physics, and that this concept is exceptionally helpful and even necessary for understanding Riemann's thinking and practice, and creative mathematical and scientific thinking and practice in general.
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In his recent paper, L. Freidel noted that instead of representing the motion of electrons in terms of oscillators and predicting their future states on the basis on this representation, as in the previous, classical, electron theory of... more
In his recent paper, L. Freidel noted that instead of representing the motion of electrons in terms of oscillators and predicting their future states on the basis on this representation, as in the previous, classical, electron theory of H. Lorentz, quantum theory was, beginning nearly with its inception, concerned with the probabilities of transitions between states of electrons, without necessarily representing how these transitions come about. Taking N. Bohr's and then W. Heisenberg's thinking along these lines in, respectively, Bohr's 1913 atomic theory and Heisenberg's quantum mechanics of 1925 as a point of departure, this article reconsiders, from a nonrealist perspective (which suspends or even precludes this representation of the mechanism behind these transitions), the concept of quantum state, as a physical concept, in contradistinction to the mathematical concept of quantum state, a vector in the Hilbert-space formalism of quantum mechanics. Transitions between quantum states appear, from this perspective, as "transitions without connections," because, while one can register the change from one quantum phenomena to another, observed in measuring instruments , we have no means of representing or possibly even conceiving of how this change comes about. The article will also discuss quantum field theory and, in closing, briefly quantum information theory as confirming, and giving additional dimensions to, these concepts of quantum state and transitions between them.
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The article reconsiders quantum theory in terms of the following principle, which can be symbolically represented as QUANTUMNESS → PROBABILITY → ALGEBRA and will be referred to as the QPA principle. The principle states that the... more
The article reconsiders quantum theory in terms of the following principle, which can be symbolically represented as QUANTUMNESS → PROBABILITY → ALGEBRA and will be referred to as the QPA principle. The principle states that the quantumness of physical phenomena, that is, the specific character of physical phenomena known as quantum, implies that our predictions concerning them are irreducibly probabilistic, even in dealing with quantum phenomena resulting from the elementary individual quantum behavior (such as that of elementary particles), which in turn implies that our theories concerning these phenomena are fundamentally algebraic, in contrast to more geometrical classical or relativistic theories, although these theories, too, have an algebraic component to them. It follows that one needs to find an algebraic scheme able make these predictions in a given quantum regime. Heisenberg was first to accomplish this in the case of quantum mechanics, as matrix mechanics, whose matrix character testified to his algebraic method, as Einstein characterized it. The article explores the implications of the Heisenberg method and of the QPA principle for quantum theory, and for the relationships between mathematics and physics there, from a nonrealist or, in terms of this article, "reality-without-realism" or RWR perspective, defining the RWR principle, thus joined to the QPA principle.
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This article reconsiders the three concepts stated in its title—reality, phenomena, and atomicity—in terms of the concept of “reality without realism,” which follows and in part radicalizes Bohr’s view of quantum reality. It also... more
This article reconsiders the three concepts stated in its title—reality, phenomena, and atomicity—in terms of the concept of
“reality without realism,” which follows and in part radicalizes Bohr’s view of quantum reality. It also addresses the work ofA. N.
Whitehead and H. Stapp from this perspective.
“reality without realism,” which follows and in part radicalizes Bohr’s view of quantum reality. It also addresses the work ofA. N.
Whitehead and H. Stapp from this perspective.
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The article introduces a new concept of structure, defined, echoing Wheeler's concept of "law without law," as a "structure without law", and a new philosophical viewpoint, that of structural nonrealism, both of which, the article argues,... more
The article introduces a new concept of structure, defined, echoing Wheeler's concept of "law without law," as a "structure without law", and a new philosophical viewpoint, that of structural nonrealism, both of which, the article argues, emerged with Heisenberg's discovery of quantum mechanics and Bohr's interpretation of it in terms of complementarity. The article takes advantage of the circumstance that any instance of quantum data or, in presentday terms, quantum information is a "structure"-an organization of elements, ultimately bits, of classical information, manifested in measuring instruments. While, however, this organization can, along with the observed behavior of measuring instruments, be described by means of classical physics, it cannot be predicted by means of classical physics, but only probabilistically or statistically by means of quantum mechanics or quantum field theory (or possibly some alternative theories within their scope). By contrast, the emergence of this information and of this structure cannot, in the present view, be described by either classical or quantum theory, or possibly by any other means, which leads to the concept of "structure without law" and the viewpoint of structural nonrealism.
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The article introduces a new concept of structure, defined, echoing J. A. Wheeler's concept of "law without law", as a "structure without law"; and a new philosophical viewpoint, that of structural nonrealism, and considers their... more
The article introduces a new concept of structure, defined, echoing J. A. Wheeler's concept of "law without law", as a "structure without law"; and a new philosophical viewpoint, that of structural nonrealism, and considers their applications to quantum theory in general and quantum information theory in particular. It takes as its historical point of departure W. Heisenberg's discovery of quantum mechanics, which, the article argues, could, in retrospect, be considered in quantum-informational terms. The article takes advantage of the circumstance that any quantum information is a structure, an organization of elements, ultimately bits, of classical information, manifested in measuring instruments. While, however, this organization can, along with the observed behavior of measuring instruments, be described by means of classical physics, it cannot be predicted by means of classical physics, but only, probabilistically or statistically, by means of quantum physics.
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This article considers the relationships between the character of physical law in quantum theory and Bohr’s concept of complementarity, under the assumption of the unrepresentable and possibly inconceivable nature of quantum objects and... more
This article considers the relationships between the character of physical law in quantum theory and Bohr’s concept of complementarity, under the assumption of the unrepresentable and possibly inconceivable nature of quantum objects and
processes, an assumption that may be seen as the most radical departure from realism currently available. Complementarity, the article argues, is a reflection of the fact that, as against classical physics or relativity, the behavior of quantum objects of the same type, say, all electrons, is not governed by the same physical law in all contexts, specifically in complementary contexts. On the other hand, the mathematical formalism of quantum mechanics offers correct probabilistic or statistical predictions (no other predictions are possible on experimental grounds) in all contexts, here, again, under the assumption that quantum objects themselves and their behavior are beyond representation or even conception. Bohr, in this connection, spoke of “an entirely new situation as regards the description of physical phenomena that, the notion of complementarity aims at characterizing.” The article also considers the relationships among complementarity, entanglement, and quantum information, by basing these
relationships on this understanding of complementarity.
processes, an assumption that may be seen as the most radical departure from realism currently available. Complementarity, the article argues, is a reflection of the fact that, as against classical physics or relativity, the behavior of quantum objects of the same type, say, all electrons, is not governed by the same physical law in all contexts, specifically in complementary contexts. On the other hand, the mathematical formalism of quantum mechanics offers correct probabilistic or statistical predictions (no other predictions are possible on experimental grounds) in all contexts, here, again, under the assumption that quantum objects themselves and their behavior are beyond representation or even conception. Bohr, in this connection, spoke of “an entirely new situation as regards the description of physical phenomena that, the notion of complementarity aims at characterizing.” The article also considers the relationships among complementarity, entanglement, and quantum information, by basing these
relationships on this understanding of complementarity.
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First, this article considers the nature of quantum reality (the reality responsible for quantum phenomena) and the concept of realism (our ability to represent this reality) in quantum theory, in conjunction with the roles of locality,... more
First, this article considers the nature of quantum reality (the reality responsible for quantum phenomena) and the concept of realism (our ability to represent this reality) in quantum theory, in conjunction with the roles of locality, causality, and probability and statistics there. Second, it offers two interpretations of quantum mechanics, developed by the authors of this article, the second of which is also a different (from quantum mechanics) theory of quantum phenomena. Both of these interpretations are statistical. The first interpretation, by A. Plotnitsky, "the statistical Copenhagen interpretation ," is nonrealist, insofar as the description or even conception of the nature of quantum objects and processes is precluded. The second, by A. Khrennikov, is ultimately realist, because it assumes that the quantum-mechanical level of reality is underlain by a deeper level of reality, described, in a realist fashion, by a model, based in the pre-quantum classical statistical field theory, the predictions of which reproduce those of quantum mechanics. Moreover, because the continuous fields considered in this model are transformed into discrete clicks of detectors, experimental outcomes in this model depend on the context of measurement in accordance with N. Bohr's interpretation and the statistical Copenhagen interpretation, which coincides with N. Bohr's interpretation in this regard.
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This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be addressed as well, in view of their role in quantum electrodynamics and... more
This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be addressed as well, in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac's work, which, in particular Dirac's derivation of his relativistic equation of the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall also consider Heisenberg's earlier work leading him to the discovery of quantum mechanics, which inspired Dirac's work. I argue that Heisenberg's and Dirac's work was guided by their adherence to and their confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by D'Ariano and coworkers on the principles of quantum information theory, which extend quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac's equations from these principles alone, without using the principles of relativity.
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The project of this article is twofold. First, it aims to offer a new perspective on, and a new argument concerning, realist and non-realist mathematical models, and differences and affinities between them, using physics as a paradigmatic... more
The project of this article is twofold. First, it aims to offer a new perspective on, and a new argument concerning, realist and non-realist mathematical models, and differences and affinities between them, using physics as a paradigmatic field of mathematical modeling in science. Most of the article is devoted
to this topic. Second, the article aims to explore the implications of this argument for mathematical modeling in other fields, in particular in cognitive psychology and economics.
to this topic. Second, the article aims to explore the implications of this argument for mathematical modeling in other fields, in particular in cognitive psychology and economics.
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Beginning with its introduction by W. Heisenberg, quantum mechanics was often seen as an overly abstract theory, mathematically and physically, vis-à-vis classical physics or relativity. This perception was amplified by the fact that,... more
Beginning with its introduction by W. Heisenberg, quantum mechanics was often seen as an overly abstract theory, mathematically and physically, vis-à-vis classical physics or relativity. This perception was amplified by the fact that, while the quantum-mechanical formalism provided effective predictive algorithms for the probabilistic predictions concerning quantum experiments, it appeared unable to describe, even by way idealization, quantum processes themselves in space and time, in the way classical mechanics or relativity did. The aim of the present paper is to reconsider the nature of mathematical and physical abstraction in modern physics by offering an analysis of the concept of ``physical fact'' and of the concept of ``physical concept,'' in part by following G. W. F. Hegel's and G. Deleuze's arguments concerning the nature of conceptual thinking. In classical physics, relativity, and quantum physics alike, I argue, physical concepts are defined...
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The point of departure for this article is Werner Heisenberg’s remark, made in 1929: “It is not surprising that our language [or conceptuality] should be incapable of describing processes occurring within atoms, for . . . it was invented... more
The point of departure for this article is Werner Heisenberg’s remark, made in 1929: “It is not surprising that our language [or conceptuality] should be incapable of describing processes occurring within atoms, for . . . it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. . . . Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme—the quantum theory [quantum mechanics]—which seems entirely adequate for the treatment of atomic processes.” The cost of this discovery, at least in Heisenberg’s and related interpretations of quantum mechanics (such as that of Niels Bohr), is that, in contrast to classical mechanics, the mathematical scheme in question no longer offers a description, even an idealized one, of quantum objects and processes. This scheme only enables predictions, in general, probabilistic in character, of the outcomes of quantum experiments. As a result, a new type of the relationships between mathematics and physics is established, which, in the language of Eugene Wigner adopted in my title, indeed makes the effectiveness of mathematics unreasonable in quantum but, as I shall explain, not in classical physics. The article discusses these new relationships between mathematics and physics in quantum theory and their implications for theoretical physics—past, present, and future.
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Taking as its point of departure the discovery of the Higgs boson, this article considers quantum theory, including quantum field theory, which predicted the Higgs boson, through the combined perspective of quantum information theory and... more
Taking as its point of departure the discovery of the Higgs boson, this article considers quantum theory, including quantum field theory, which predicted the Higgs boson, through the combined perspective of quantum information theory and the idea of technology, while also adopting a non-realist interpretation, in ‘the spirit of Copenhagen’, of quantum theory and quantum phenomena themselves. The article argues that the ‘events’ in question in fundamental physics, such as the discovery of the Higgs boson (a particularly complex and dramatic, but not essentially different, case), are made possible by the joint workings of three technologies: experimental technology, mathematical technology and, more recently, digital computer technology. The article will consider the role of and the relationships among these technologies, focusing on experimental and mathematical technologies, in quantum mechanics (QM), quantum field theory (QFT) and finite-dimensional quantum theory, with which quant...
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The history of mathematical modeling outside physics has been dominated by the use of classical mathematical models, C-models, primarily those of a probabilistic or statistical nature. More recently, however, quantum mathematical models,... more
The history of mathematical modeling outside physics has been dominated by the use of classical mathematical models, C-models, primarily those of a probabilistic or statistical nature. More recently, however, quantum mathematical models, Q-models, based in the mathematical formalism of quantum theory have become more prominent in psychology, economics, and decision science. The use of Q-models in these fields remains controversial, in part because it is not entirely clear whether Q-models are necessary for dealing with the phenomena in question or whether C-models would still suffice. My aim, however, is not to assess the necessity of Q-models in these fields, but instead to reflect on what the possible applicability of Q-models may tell us about the corresponding phenomena there, vis-à-vis quantum phenomena in physics. In order to do so, I shall first discuss the key reasons for the use of Q-models in physics. In particular, I shall examine the fundamental principles that led to the development of quantum mechanics. Then I shall consider a possible role of similar principles in using Q-models outside physics. Psychology, economics, and decision science borrow already available Q-models from quantum theory, rather than derive them from their own internal principles, while quantum mechanics was derived from such principles, because there was no readily available mathematical model to handle quantum phenomena, although the mathematics ultimately used in quantum did in fact exist then. I shall argue, however, that the principle perspective on mathematical modeling outside physics might help us to understand better the role of Q-models in these fields and possibly to envision new models, conceptually analogous to but mathematically different from those of quantum theory, that may be helpful or even necessary there or in physics itself. I shall, in closing, suggest one possible type of such models, singularized probabilistic models, SP-models, some of which are time-dependent, TDSP-models. The necessity of using such models may change the nature of mathematical modeling in science and, thus, the nature of science, as it happened in the case of Q-models, which not only led to a revolutionary transformation of physics but also opened new possibilities for scientific thinking and mathematical modeling beyond physics.
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This paper discusses a particular type of quantum-like literary models, which are conceptual, rather than mathematical, in character. These models share with quantum mechanics the difficulties of applying the concepts of reality and... more
This paper discusses a particular type of quantum-like literary models, which are conceptual, rather than mathematical, in character. These models share with quantum mechanics the difficulties of applying the concepts of reality and causality at the ultimately ontological levels they consider, analogous to the level of quantum objects and processes in quantum mechanics. They respond to this difficulty by suspending and even precluding the application of both concepts, as do certain interpretations of quantum mechanics. I call such models and such interpretations "nonclassical," in juxtaposition to "classical" models, which retain realism and causality at the ultimate level of description, even when considering random events. While I offer a sketch of Western thinking concerning the subject, I focus on certain philosophical and literary quantum-like thinking of the late-eighteenth and early-nineteenth centuries, associated with Romantic literature, which shows particular affinities with quantum-theoretical thinking later on. I also consider, in closing, the literary model, found in Beckett's plays, that was developed after quantum mechanics and that shares with it features that earlier literary quantum-like models do not possess.
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The paper explores the ontology and logic of the irreducibly multiple in set theory and in topos theory by considering the differences between Badiou's logical and Grothendieck's ontological approach to topos theory. It argues that... more
The paper explores the ontology and logic of the irreducibly multiple in set theory and in topos theory by considering the differences between Badiou's logical and Grothendieck's ontological approach to topos theory. It argues that Grothendieck's ontological program for topos theory leads to a more radical concept of the multiple than does the set-theoretical ontology, which defines Badiou's view of ontology even in his later, more topos theoretically oriented work. Extending Grothendieck's way of thinking to other fields enables one to give ontological multiplicities--no longer bound by the set-theoretical ontology or ultimately by any mathematical ontology, even in mathematics--a great diversity and richness. It follows that the set-theoretical ontology is not sufficiently rich to accomplish what Badiou thinks it could accomplish even in mathematics itself, let alone elsewhere; and Badiou wants it to work elsewhere--indeed, wherever it is possible to speak of ontology. I shall also consider, in closing, some implications of the arguments for the workings of the multiple in ethics, politics, and culture.
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This article approaches the problem of organizing Romanticism and, with it, the nineteenth century by way of exploring a new type of ontological thinking, introduced by the Romantics, specifically Hölderlin, and designated here as... more
This article approaches the problem of organizing Romanticism and, with it, the nineteenth century by way of exploring a new type of ontological thinking, introduced by the Romantics, specifically Hölderlin, and designated here as “Romantic ontology.” This ontology is defined by the impossibility of capturing the ultimate workings of the world by any given concept, which only allows us
an access to certain effects of these ultimate workings. The radical limitations thus established do not inhibit but instead help to advance thought and knowledge and make them succeed where they failed previously, including in developing a better understanding of Romantic and nineteenth-century literature
and culture. The nature of thought and knowledge changes, however: the unthinkable and the unknowable are now irreducible at any stage of our thinking and knowledge. As a result, we are irreducibly deprived of certainty, and the recourse to probability becomes an unavoidable aspect of our thinking and knowledge. Accordingly, we can only estimate and argue for the probability of
our historical and theoretical claims, for example, those concerning how Romanticism and the nineteenth century were organized by a given figure or community at the time, or how they could be organized by us. This, however, is thought and knowledge, too, and they may be the best we can have.
an access to certain effects of these ultimate workings. The radical limitations thus established do not inhibit but instead help to advance thought and knowledge and make them succeed where they failed previously, including in developing a better understanding of Romantic and nineteenth-century literature
and culture. The nature of thought and knowledge changes, however: the unthinkable and the unknowable are now irreducible at any stage of our thinking and knowledge. As a result, we are irreducibly deprived of certainty, and the recourse to probability becomes an unavoidable aspect of our thinking and knowledge. Accordingly, we can only estimate and argue for the probability of
our historical and theoretical claims, for example, those concerning how Romanticism and the nineteenth century were organized by a given figure or community at the time, or how they could be organized by us. This, however, is thought and knowledge, too, and they may be the best we can have.
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The unthinkable, in its customary (essentially, metaphorical) sense of something terrible or horrible, is the bread and bu er of, and even the ultimate raison d’être for, psychoanalysis, beginning with Sigmund Freud, or indeed Sophocles,... more
The unthinkable, in its customary (essentially, metaphorical)
sense of something terrible or horrible, is the bread and bu er
of, and even the ultimate raison d’être for, psychoanalysis, beginning with Sigmund Freud, or indeed Sophocles, who put this unthinkable in play (in either sense) in his tragedies, Oedipus Tyrannus most psychoanalytically famous among them. This, however, is not the “unthinkable” with which I will be concerned is this essay, although the psychoanalytic connections between these two senses of the unthinkable are intriguing, and I will indicate some of them as I proceed. My concern is the unthinkable in the literal sense of the term, the unthinkable, as that which is beyond the reach of thought altogether, closer to the ancient Greek sense of chaos as areton or alogon, which is at stake in the ancient Greek tragedy as well. This conception of the unthinkable implies that it cannot have a direct or literal sense either, any more than any other sense. Ultimately, it is unthinkable even as unthinkable.
sense of something terrible or horrible, is the bread and bu er
of, and even the ultimate raison d’être for, psychoanalysis, beginning with Sigmund Freud, or indeed Sophocles, who put this unthinkable in play (in either sense) in his tragedies, Oedipus Tyrannus most psychoanalytically famous among them. This, however, is not the “unthinkable” with which I will be concerned is this essay, although the psychoanalytic connections between these two senses of the unthinkable are intriguing, and I will indicate some of them as I proceed. My concern is the unthinkable in the literal sense of the term, the unthinkable, as that which is beyond the reach of thought altogether, closer to the ancient Greek sense of chaos as areton or alogon, which is at stake in the ancient Greek tragedy as well. This conception of the unthinkable implies that it cannot have a direct or literal sense either, any more than any other sense. Ultimately, it is unthinkable even as unthinkable.
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This statement by Jacques Derrida has been endlessly circulated in recent discussions around the so-called "Science Wars," in the wake of Paul R. Gross and Norman Levitt's Higher Superstition , and then Alan Sokal's... more
This statement by Jacques Derrida has been endlessly circulated in recent discussions around the so-called "Science Wars," in the wake of Paul R. Gross and Norman Levitt's Higher Superstition , and then Alan Sokal's "hoax ...
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Building on Martin Heidegger’s insight that, beginning with René Descartes and Galileo Galilei, “modern [physics] is experimental because of its mathematical project” [Heidegger 1967, p. 93], I argue that the advancement of modern physics... more
Building on Martin Heidegger’s insight that, beginning with René Descartes and Galileo Galilei, “modern [physics] is experimental because of its mathematical project” [Heidegger 1967, p. 93], I argue that the advancement of modern physics has been defined by the invention of new mathematical schemes (possibly borrowing them from mathematics itself). Among the greatest such inventions, all using differential equations, are:
*Classical physics, based on calculus;
*Maxwell’s electromagnetic theory, based on the ideal of (classical) field and its mathematization, as represented by Maxwell’s equations;
*Relativity, SR and especially GR, based on Riemannian geometry;
*Quantum mechanics (QM) and quantum field theory (QFT), based on the mathematics of Hilbert spaces over C, and the operator algebras.
I further argue that QM and QFT (to either of which the term quantum theory will refer hereafter) gave this thesis a radically new meaning:
Quantum phenomena are defined physically, as essentially different from all previous physics, as is manifested in paradigmatic experiments such as the double-slit experiment or those dealing with quantum correlations.
On the other hand, quantum theory, at least QM or QFT, is defined as different from classical physics on the basis of purely mathematical postulates, which connect it to quantum phenomena strictly in terms of probabilities, without, at least in the interpretation adopted here, representing or otherwise relating to how these phenomena come about.
*Classical physics, based on calculus;
*Maxwell’s electromagnetic theory, based on the ideal of (classical) field and its mathematization, as represented by Maxwell’s equations;
*Relativity, SR and especially GR, based on Riemannian geometry;
*Quantum mechanics (QM) and quantum field theory (QFT), based on the mathematics of Hilbert spaces over C, and the operator algebras.
I further argue that QM and QFT (to either of which the term quantum theory will refer hereafter) gave this thesis a radically new meaning:
Quantum phenomena are defined physically, as essentially different from all previous physics, as is manifested in paradigmatic experiments such as the double-slit experiment or those dealing with quantum correlations.
On the other hand, quantum theory, at least QM or QFT, is defined as different from classical physics on the basis of purely mathematical postulates, which connect it to quantum phenomena strictly in terms of probabilities, without, at least in the interpretation adopted here, representing or otherwise relating to how these phenomena come about.