Books by Arkady Plotnitsky
Springer/Nature, Heidelberg, 2022
This book is a philosophical study of mathematics, pursued by considering and relating two aspect... 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.
Springer/Nature: Heidelberg; ISBN: 978-3-030-84577-3, 2021
This book presents quantum theory as a theory based on new relationships among matter, thought, a... 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.
Springer/Nature, 2023
Over the last ten years, elements of the formalism of quantum mechanics have been successfully ap... 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.
Springer/Nature, 2016
The book considers foundational thinking in quantum theory, focusing on the role the fundamental ... 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.
New York/Berlin: Spinger/ Nature, 2012
This book offers a discussion of Niels Bohr’s conception of “complementarity,” arguably his great... 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.
Berlin& New York: Springer/Nature; Fundamental Theories in Physics Series, 2009
Quantum mechanics, discovered by Werner Heisenberg and Erwin Schrödinger in 1925-1926, is famous ... 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.
University of MIchigan Press, Ann Arbor,, MI M, 2002
Series • Studies in Literature and Science This book investigates the relationships between moder... 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.
Duke University Press, 1994
Many commentators have remarked in passing on the resonance between deconstructionist theory and ... 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.
Springer/Nature; Fundamental Theories in Physics, 2006
Reading Bohr: Physics and Philosophy offers a new perspective on Niels Bohr's interpretation of q... 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.
Duke University Press, 1997
Mathematics, Science, and Postclassical Theory is a unique collection of essays dealing with the ... 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
Papers by Arkady Plotnitsky
Entropy 2024, 26, 418, 2024
This article reconsiders E. Schrödinger’s cat paradox experiment from a new perspective, grounded... 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.
B. Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice (Springer Nature, Cham, Switzerland, AG. 2023), 2023
This chapter considers the nature of radical transformations of mathematics,
enabled by minority ... 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.
The European Physical Journal , 2023
The starting point of this article is the uncircumventable interference of observational instrume... 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.
Phil. Trans. R. Soc. A 381: 20220295., 2023
The argument of this article is grounded in the irreducible interference of observational instrum... 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.
in A. Papadopoulos, ed. Essays in Geometry Dedicated to Norbert A’Campo IRMA, Strasbourg, France , 2023
The argument of this chapter is governed by the double meaning of its title:
(a) the relationship... 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.
B. Sriraman (ed.) Handbook of the History and Philosophy of Mathematical Practice, 2023
The primary aim of this chapter is to consider mathematicians’ working philosophy of mathematics ... 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.
Entropy 2023, 25, 706, 2023
This article considers a rarely discussed aspect, the no-cloning principle or postulate, recast a... 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.
Rajan, T. Whistler, D. (eds.) The Palgrave Handbook of German Idealism and Poststructuralism, Palgrave Handbooks in German Idealism,, 2023
This chapter considers the question of difference and its implications in the context of the rela... 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.
Plotnitsky, A. “Yet Once More”: The Double‐Slit Experiment and Quantum Discontinuity. Entropy 2022, 24, 1455., 2022
This article reconsiders the double-slit experiment from the nonrealist or, in terms of this arti... 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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Books by Arkady Plotnitsky
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.
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.
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.
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.
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
Papers by Arkady Plotnitsky
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.
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.
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.
(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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
(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.
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.
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.
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 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.
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.
grounded in his analysis of the irreducible role of measuring instruments in the constitution of quantum phenomena.
*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.