This Element is a concise, high-level introduction to the philosophy of physical symmetry. It begins with the notion of `physical representation' (the kind of empirical representation of nature that we effect in doing physics), and then lays out the historically and conceptually central case of physical symmetry that frequently falls under the rubric of 'the Relativity Principle', or 'Galileo's Ship. This material is then used as a point of departure to explore the key hermeneutic challenge concerning physical symmetry in the (...) last century, namely understanding the physical significance of the notion of 'local' gauge symmetry. The approach taken stresses both the continuity with historically important themes such as the Relativity Principle, as well as novel insights earned by working with contemporary media such as the covariant phase space and derived critical locus formalisms. (shrink)

In this paper, I discuss one form of the idea that spacetime and gravity might ‘emerge’ from quantum theory, i.e. via a holographic duality, and in particular via AdS/CFT duality. I begin by giving a survey of the general notion of duality, as well as its connection to emergence. I then review the AdS/CFT duality and proceed to discuss emergence in this context. We will see that it is difficult to find compelling arguments for the emergence of full quantum gravity (...) from gauge theory via AdS/CFT, i.e. for the boundary theory's being metaphysically more fundamental than the bulk theory. (shrink)

This article investigates and resolves the question whether gauge symmetry can display analogs of the famous Galileo’s ship scenario. In doing so, it builds on and clarifies the work of Greaves and Wallace on this subject.

As a prolegomenon to understanding the sense in which dualities are theoretical equivalences, we investigate the intuitive `equivalence' of hyper-regular Lagrangian and Hamiltonian classical mechanics. We show that the symplectification of these theories provides a sense in which they are isomorphic, and mutually and canonically definable through an analog of `common definitional extension'.

This article uncovers a foundational relationship between the ‘gauge symmetry’ of a Newton-Cartan theory and the celebrated Trautman Recovery Theorem and explores its implications for recent philosophical work on Newton-Cartan gravitation.

Philosophers of physics and physicists have long been intrigued by the analogies and disanalogies between gravitational theories and gauge theories. Indeed, repeated attempts to collapse these disanalogies have made us acutely aware that there are fairly general obstacles to doing so. Nonetheless, there is a special case space-time dimensions) in which gravity is often claimed to be identical to a gauge theory. I subject this claim to philosophical scrutiny in this article. In particular, I analyse how the standard disanalogies can (...) be overcome in dimensions, and consider whether really licenses the interpretation of gravity as a gauge theory. Our conceptual analysis reveals more subtle disanalogies between gravity and gauge, and connects these to interpretive issues in classical and quantum gravity. 1 Introduction1.1 Motivation1.2 Prospectus2 Disanalogies3 Three-dimensional gravity and gauge3.1 gravity3.2 Chern–Simons3.2.1 Cartan geometry3.2.2 Overcoming obst-gauge via Cartan connections3.3 Disanalogies collapsed4 Two More Disanalogies4.1 What about the symmetries?4.2 The phase spaces of the two theories5 Summary and Conclusion. (shrink)

Rovelli’s “Why Gauge?” offers a parable to show that gauge-dependent quantities have a modal and relational physical significance. We subject the morals of this parable to philosophical scrutiny and argue that, while Rovelli’s main point stands, there are important disanalogies between his parable and Yang-Mills type gauge theory.

The idea that gauge theory has `surplus' structure poses a puzzle: in one much discussed sense, this structure is redundant; but on the other hand, it is also widely held to play an essential role in the theory. In this paper, we employ category-theoretic tools to illuminate an aspect of this puzzle. We precisify what is meant by `surplus' structure by means of functorial comparisons with equivalence classes of gauge fields, and then show that such structure is essential for any (...) theory that represents a rich collection of physically relevant fields which are `local' in nature. (shrink)

We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the ‘flavour’ that two dual theories are ‘closer in content’ than you might think. For both points, we adopt a simple conception of a duality as an ‘isomorphism’ between theories: more precisely, as appropriate bijections between the two theories’ sets of states (...) and sets of quantities. The first point is that this conception of duality meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be ‘gauge’. The second point is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary theory. (shrink)

Rovelli's "Why Gauge?" offers a parable to show that gauge-dependent quantities have a modal and relational physical significance. We subject the morals of this parable to philosophical scrutiny and argue that, while his overarching point stands, there are subtle disanalogies between his parable and our best gauge theories, e.g. Yang-Mills theory and General Relativity.

It is part of information theory folklore that, while quantum theory prohibits the generic cloning of states, such cloning is allowed by classical information theory. Indeed, many take the phenomenon of no-cloning to be one of the features that distinguishes quantum mechanics from classical mechanics. In this paper, we use symplectic geometry to argue that pace conventional wisdom, in the case where one does not include a machine system, there is an analog of the no-cloning theorem for classical systems. However, (...) upon adjoining a non-trivial machine system one finds that, pace the quantum case, the obstruction to cloning disappears for pure states. We then discuss the difference between this result and the quantum case, and show that it can be explained in terms of the rigidity of the theories' respective geometries. Finally, we discuss the relationship between this result and classical no-cloning arguments in the context of symmetric monoidal categories and statistical classical mechanics. (shrink)

Foundations of Relational Realism: A Topological Approach to Quantum Mechanics and the Philosophy of Nature by Michael Epperson and Elias Zafiris sets out to achieve three goals: to develop a version of Whiteheadian metaphysics that the authors call “relational realism”; to formalize relational realism in terms of category theory, in particular sheaf theory; and to use relational realism to solve the interpretative problems of quantum mechanics. These goals are ambitious, to say the least, and all this is leaving aside those (...) sections of FRR which argue that relational realism yields the key to understanding quantum gravity!The text is 388 pages long and comprises two parts. Part I, by Epperson, introduces relational realism and its application to quantum mechanics. Part II, by Zafiris, develops the sheaf theory formalism for relational realism. As the authors say, their exposition is “nonlinear”, and this feature of FRR, alon .. (shrink)

Category theory has become central to certain aspects of theoretical physics. Bain has recently argued that this has significance for ontic structural realism. We argue against this claim. In so doing, we uncover two pervasive forms of category-theoretic generalization. We call these ‘generalization by duality’ and ‘generalization by categorifying physical processes’. We describe in detail how these arise, and explain their significance using detailed examples. We show that their significance is two-fold: the articulation of high-level physical concepts, and the generation (...) of new models. 1 Introduction2 Categories and Structuralism 2.1 Categories: abstract and concrete 2.2 Structuralism: simple and ontic3 Bain’s Two Strategies 3.1 A first strategy for defending Objectless 3.2 A second strategy for defending Objectless4 Two Forms of Categorical Generalization 4.1 Generalization by duality 4.2 Generalization by categorification 4.3 The role of category theory in physics5 Conclusion. (shrink)

We make two points about dualities in string theory. The first point is that the conception of duality, which we will discuss, meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. The second point is a result about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than one might expect: each of a certain class of gauge symmetries in the gravity theory, that is, diffeomorphisms, is related to a (...) position-dependent symmetry of the gauge theory. (shrink)

We construct a notion of teleparallelization for Newton-Cartan theory, and show that the teleparallel equivalent of this theory is Newtonian gravity; furthermore, we show that this result is consistent with teleparallelization in general relativity, and can be obtained by null-reducing the teleparallel equivalent of a five-dimensional gravitational wave solution. This work thus strengthens substantially the connections between four theories: Newton-Cartan theory, Newtonian gravitation theory, general relativity, and teleparallel gravity.

The recent debate about whether gauge symmetries can be empirically significant has focused on the possibility of 'Galileo's ship' types of scenarios, where the symmetries effect relational differences between a subsystem and the environment. However, it has gone largely unremarked that apart from such Galileo's ship scenarios, Greaves and Wallace (2014) proposed that gauge transformations can also be empirically significant in a 'non-relational' manner that is analogous to a Faraday-cage scenario, where the subsystem symmetry is related to a change in (...) a charged boundary state. In this paper, we investigate the question of whether such non-relational scenarios are possible for gauge theories. Remarkably, the answer to this question turns out to be closely related to a foundational puzzle that has driven a host of recent developments at the frontiers of theoretical physics. By drawing on these recent developments, we show that a very natural way of elaborating on Greaves and Wallace's claim of non-relational empirical significance for gauge symmetry is incoherent. However, we also argue that much of what they suggest is correct in spirit: one can indeed construct non-relational models of the kind they sketch, albeit ones where the empirical significance is not witnessed by a gauge symmetry but instead by a superficially similar boundary symmetry. Furthermore, the latter casts doubt on whether one really abandons Galileo's ship in such scenarios. (shrink)

The equivalence principle has constituted one of the cornerstones of discussions in the foundations of spacetime theories over the past century. However, up to this point the principle has been considered overwhelmingly only within the context of relativistic physics. In this article, we demonstrate that the principle has much broader, super-theoretic significance: to do so, we present a unified framework for understanding the principle in its various guises, applicable to both relativistic and Newtonian contexts. We thereby deepen significantly our understanding (...) of the role played by the equivalence principle in a broad class of spacetime theories. (shrink)

DBS Think Tank IX was held on August 25–27, 2021 in Orlando FL with US based participants largely in person and overseas participants joining by video conferencing technology. The DBS Think Tank was founded in 2012 and provides an open platform where clinicians, engineers and researchers can freely discuss current and emerging deep brain stimulation technologies as well as the logistical and ethical issues facing the field. The consensus among the DBS Think Tank IX speakers was that DBS expanded in (...) its scope and has been applied to multiple brain disorders in an effort to modulate neural circuitry. After collectively sharing our experiences, it was estimated that globally more than 230,000 DBS devices have been implanted for neurological and neuropsychiatric disorders. As such, this year’s meeting was focused on advances in the following areas: neuromodulation in Europe, Asia and Australia; cutting-edge technologies, neuroethics, interventional psychiatry, adaptive DBS, neuromodulation for pain, network neuromodulation for epilepsy and neuromodulation for traumatic brain injury. (shrink)

Recently, there has been some discussion of how Dutch Book arguments might be used to demonstrate the rational incoherence of certain hidden variable models of quantum theory. In this paper, we argue that the 'form of inconsistency' underlying this alleged irrationality is deeply and comprehensively related to the more familiar 'inconsistency' phenomenon of contextuality. Our main result is that the hierarchy of contextuality due to Abramsky and Brandenburger corresponds to a hierarchy of additivity/convexity-violations which yields formal Dutch Books of different (...) strengths. We then use this result to provide a partial assessment of whether these formal Dutch Books can be interpreted normatively. (shrink)

The last two decades have seen two significant trends emerging within the philosophy of science: the rapid development and focus on the philosophy of the specialised sciences, and a resurgence of Aristotelian metaphysics, much of which is concerned with the possibility of emergence, as well as the ontological status and indispensability of dispositions and powers in science. Despite these recent trends, few Aristotelian metaphysicians have engaged directly with the philosophy of the specialised sciences. Additionally, the relationship between fundamental Aristotelian concepts—such (...) as "hylomorphism", "substance", and "faculties"—and contemporary science has yet to receive a critical and systematic treatment. _Neo-Aristotelian Perspectives on Contemporary Science _aims to fill this gap in the literature by bringing together essays on the relationship between Aristotelianism and science that cut across interdisciplinary boundaries. The chapters in this volume are divided into two main sections covering the philosophy of physics and the philosophy of the life sciences. Featuring original contributions from distinguished and early-career scholars, this book will be of interest to specialists in analytical metaphysics and the philosophy of science. (shrink)

It is now well-known that Newton–Cartan theory is the correct geometrical setting for modelling the quantum Hall effect. In addition, in recent years edge modes for the Newton–Cartan quantum Hall effect have been derived. However, the existence of these edge modes has, as of yet, been derived using only orthodox methodologies involving the breaking of gauge-invariance; it would be preferable to derive the existence of such edge modes in a gauge-invariant manner. In this article, we employ recent work by Donnelly (...) and Freidel in order to accomplish exactly this task. Our results agree with known physics, but afford greater conceptual insight into the existence of these edge modes: in particular, they connect them to subtle aspects of Newton–Cartan geometry and pave the way for further applications of Newton–Cartan theory in condensed matter physics. (shrink)

Although there are various studies comparing Greek and Indian philosophy and religion, and Chinese and Western philosophy and religion, Brahman and Dao: Comparatives Studies in Indian and Chinese Philosophy and Religion is a first of its kind that brings together Indian and Chinese philosophies and religions. Brahman and Dao helps close the gap on a much needed examination on the rich history of Buddhist transmission to China, and the many generations of Indian Buddhist missionaries to China and Chinese Buddhist pilgrims (...) to India, including the legendary Bodhidharma, and Faxian and Xuanzang. (shrink)

This volume brings together eminent theologians, philosophers and political theorists to discuss such questions as how religious understandings have shaped the moral landscape of contemporary culture; the possible contributions of theology and theologically informed moral argument to contemporary public life; the problem of religious and moral discourse in a pluralistic society; and the proper relationship between religion and culture.

In his writing, Gilles Deleuze drew on a vast array of source material, from philosophy and psychoanalysis to science and art. Yet scholars have largely neglected one of the intellectual currents underlying his work: Western esotericism, specifically the lineage of hermetic thought that extends from Late Antiquity into the Renaissance through the work of figures such as Iamblichus, Nicholas of Cusa, Pico della Mirandola, and Giordano Bruno. In this book, Joshua Ramey examines the extent to which Deleuze's ethics, (...) metaphysics, and politics were informed by, and can only be fully understood through, this hermetic tradition. Identifying key hermetic moments in Deleuze's thought, including his theories of art, subjectivity, and immanence, Ramey argues that the philosopher's work represents a kind of contemporary hermeticism, a consistent experiment in unifying thought and affect, percept and concept, and mind and nature in order to engender new relations between knowledge, power, and desire. By uncovering and clarifying the hermetic strand in Deleuze's work, Ramey offers both a new interpretation of Deleuze, particularly his insistence that the development of thought demands a spiritual ordeal, and a framework for retrieving the pre-Kantian paradigm of philosophy as spiritual practice. (shrink)

This essay explores the historiographic and ethnographic valence of video in Indonesia since 1998, against the backdrop of transition from an authoritarian to a neoliberal regime, and the concurrent renewal of the country’s public sphere. The first section takes Joshua Oppenheimer’s controversial film The Act of Killing (2012) as exemplary of the moving image’s purchase on national trauma, emphasizing its role in the production (and perversion) of official history. The second section concerns the state of video discourse in Indonesia (...) as reflected in a conference held in Yogyakarta in 2011, addressing the diversity of contemporary video practices and situating them in local art and media histories. In a third and final section, we return to the epistemological quandaries raised by Oppenheimer’s film, mooting video’s capacity as a means of reconciliation with the past – a reconciliation that would seem to require both the documentary and a ‘revelatory’ function of the image. (shrink)

In the last decade there arose a debate between William P. Alston and Nicholas Wolterstorff on the subject of Thomas Aquinas’s doctrine of analogia—that is, the position that perfection terms, when properly predicated of God and of creatures, are distinct, yet related in meaning. Whereas Alston interprets Aquinas to hold this well-known position before criticizing it, Wolterstorff argues that Aquinas actually did not hold the position as it is usually presented. In this paper, I show why Alston’s “orthodox” interpretation (...) is more faithful to the letter of Aquinas’s text than is Wolterstorff’s “heterodoxy” and attempt to defuse Alston’s criticisms. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 42.2 (2004) 219-220 [Access article in PDF] Jack Zupko. John Buridan: Portrait of a Fourteenth-Century Arts Master. Notre Dame: University of Notre Dame Press, 2003. Pp. xix + 446. Cloth, $70.00. Paper, $40.00. What does the name "John Buridan" call to mind? For many, including medievalists, not much at all—at best, perhaps, a set of apparently unrelated ideas: nominalism; an impetus theory of (...) projectile motion; and a peculiar account of the will (although "Buridan's ass," the indecisive donkey caught between two equally appetizing bales of hay, is not found in Buridan's writing). In correcting this ignorance—an injustice to arguably the most famous and influential philosopher of the early-to-mid-fourteenth century—the temptation might have been to articulate Buridan's "positions" on "the big questions," allowing readers to fit Buridan into received categories. Thankfully, the present work, the first book-length study of Buridan in any language, resists this temptation. The John Buridan that emerges from Jack Zupko's skillful "exercise in philosophical portraiture" doesn't serve to simply fill gaps in our awareness of the historical record, but to enlighten and challenge our understanding of the practice of philosophy and the study of its history.Zupko emphasizes throughout that Buridan was a teacher, a master of arts in Paris. Buridan's pedagogical concerns help make sense of his central work in logic. As the basis for his arts lectures, Buridan chose Peter of Spain's Summulae Logicales, but in commenting on Peter, Buridan moves beyond creative exegisis to comprehensive renovation. Buridan (c. 1300-c. 1361) naturally wanted to accommodate terminist or via moderna supposition theory; but more importantly, he wanted a logic text that would set out a general plan of philosophical inquiry, "a compendium of methods, a 'how-to' book for the philosopher" (135). Buridan thus designed his own Summulae de Dialectica as a new authoritative logic text, and Zupko persuasively argues that its "nominalism" should be understood as less a set of doctrines in theoretical logic, than a program of logical practice, an education in logic as the ars artium, the universal intellectual tool.Zupko's portrait of "Buridan's vision of the philosophical enterprise" is thus appropriately divided into two parts. The first summarizes Buridan's method as described in the individual treatises of his Summulae. The second displays the method in practice, rehearsing and evaluating some of Buridan's arguments on various subjects. Many scholars will gravitate first to the latter part, where Zupko offers important interpretations and clarifications of Buridan's views. In these chapters, Zupko helps us to appreciate, among other things: how the problem of universals served to help Buridan define the scope of philosophy (ch. 10); the philosophical modesty of Buridan's account of the intellectual soul's presence in the body (ch. 11); the relationship between Buridan's epistemology and the skepticism of his Parisian contemporary Nicholas of Autrecourt (ch. 12); the sense in which Buridan is an "empiricist," viz an epistemic naturalist and reliabilist (ch. 13); how Buridan regarded moral psychology as applied physics, treating virtue as a kind of impetus (ch. 14); how Buridan's account of freedom can be regarded as a "perfection" of Thomistic intellectualism (ch. 15).Throughout these discussions, Zupko always keeps in sight his larger purpose of illustrating Buridan's conception of philosophical inquiry. Zupko makes much of the fact that [End Page 219] Buridan never wrote as a theologian, remaining his entire career in the arts faculty, where he wanted logic to organize secular inquiry. As Zupko notes, in this way Buridan's scholastic project seems to prefigure the "modern." But Buridan seems modern in another way as well. As Zupko points out in his "Introduction," though a cleric, Buridan never joined a religious order, so that his ideas developed independently of the Franciscan and Dominican traditions. He was, then, a teacher without a tradition—at least without deliberate allegiance to tradition, although he was in many ways the product of traditional scholastic intellectual discipline.Perhaps this... (shrink)

Thirteen dwarves and a wizard invade the quiet abode of Bilbo Baggins in an effort to recruit him for an expedition, the purported purpose of which is to recover stolen treasure and exact vengeance on Smaug the dragon, the robber who had cruelly killed a large portion of Thorin's family and friends. Although most readers and critics approach J. R. R. Tolkien's The Hobbit as a children's story, an unserious dress-rehearsal-sketch of The Lord of the Rings at best, and in (...) spite of the fact that, as Nicholas Boyle notes, "there is something embarrassing" about discussing Tolkien's work in an academic context, for "it is so obviously not real literature," The Hobbit, with its "Secondary World" of fantastical creatures... (shrink)

Neuroscience and psychology have recently turned their attention to the study of the subpersonal underpinnings of moral judgment. In this article we critically examine an influential strand of research originating in Greene's neuroimaging studies of ‘utilitarian’ and ‘non-utilitarian’ moral judgement. We argue that given that the explananda of this research are specific personal-level states—moral judgments with certain propositional contents—its methodology has to be sensitive to criteria for ascribing states with such contents to subjects. We argue that current research has often (...) failed to meet this constraint by failing to correctly ‘fix’ key aspects of moral judgment, criticism we support by detailed examples from the scientific literature. (shrink)

Good book. See this review's final paragraph for my conspiracy theory defending reflectance physicalism.

The authors focus on four major thematic areas – the reform of church, the reform of theology, the reform of perspective, and the reform of method – which together encompasses the breadth and depth of Cusanus’ own reform initiatives.

Mystical Theology and Platonism in the Time of Cusanus engages with the history of mystical theology and Neoplatonic philosophy through the lens of the 15th century philosopher and theologian, Nicholas of Cusa. The volume comprises nineteen essays that break down the barriers between medieval and Renaissance studies, reinterpreting Cusanus’ place in the history of thought by exploring the archive that informed his thinking, while also interrogating his works by exploring them from the standpoint of their later reception by modern (...) philosophers and theologians. The volume also offers tribute to the career of Donald F. Duclow, a leading scholar in the field of Cusanus studies in particular and of the history of mystical theology and Neoplatonic philosophy more generally. (shrink)

The Heythrop Journal, Volume 63, Issue 1, Page 142-143, January 2022.

Ethical naturalism holds that ethical facts about such matters as good and bad, right and wrong, are part of a purely natural world — the world studied by the sciences. It is supported by the apparent reasonableness of many moral explanations. It has been thought to face an epistemological challenge because of the existence of an “is-ought gap”; it also faces metaphysical objections from philosophers who hold that ethical facts would have to be supernatural or “nonnatural,” sometimes on the grounds (...) that ethical thought has a practical role that no thought about purely natural facts could have. Its defenders have argued resourcefully against these challenges. (shrink)

The correspondence theory of truth is a precise and innovative account of how the truth of a proposition depends upon that proposition's connection to a piece of reality. Joshua Rasmussen refines and defends the correspondence theory of truth, proposing new accounts of facts, propositions, and the correspondence between them. With these theories in hand, he then offers original solutions to the toughest objections facing correspondence theorists. Addressing the Problem of Funny Facts, Liar Paradoxes, and traditional epistemological questions concerning how (...) our minds can access reality, he challenges recent objections, and defends what has traditionally been the most popular theory of truth. Written with clarity, precision, and sensitivity to a range of philosophical backgrounds, his book will appeal to advanced students and scholars seeking a deeper understanding of the relationship between truth and reality. (shrink)

In mainstream analytic epistemology, Reformed theology has made its presence prominently felt in Reformed epistemology, the view of religious belief according to which religious beliefs can be properly basic and warranted when formed by the proper functioning of the sensus divinitatis, an inborn capacity or faculty for belief in God that can be prompted to generate certain religious beliefs when presented with things (e.g., certain majestic aspects of creation). A major competitor to Reformed epistemology is Wittgensteinian quasi-fideism, a position drawn (...) from the epistemology of Wittgenstein’s On Certainty, according to which some foundational religious commitments are not properly basic beliefs but some kind of apodictic, unshakeable certainties that form part of the structure of one’s rationality and worldview. I argue that the Reformed tradition has contained elements of quasi-fideism at least as far back as Calvin’s account of faith from the Institutes of the Christian Religion, book 3, chapter 2. (shrink)

The increasingly lively controversy over scientific realism has become one of the principal themes of recent philosophy. 1 In watching this controversy unfold in the rather technical way currently in vogue, it has seemed to me that it would be useful to view these contemporary disputes against the background of such older epistemological issues as fallibilism, scepticism, relativism, and the traditional realism/idealism debate. This, then, is the object of the present book, which will recon sider the newer concerns about scientific (...) realism in the context of these older philosophical themes. Historically, realism concerns itself with the real existence of things that do not "meet the eye" - with suprasensible entities that lie beyond the reach of human perception. In medieval times, discussions about realism focused upon universals. Recognizing that there are physical objects such as cats and triangular objects and red tomatoes, the medievels debated whether such "abstract objects" as cathood and triangularity and redness also exist by way of having a reality indepen dent of the concretely real things that exhibit them. Three fundamen tally different positions were defended: (1) Nominalism. Abstracta have no independent existence as such: they only "exist" in and through the objects that exhibit them. Only particulars (individual substances) exist. Abstract "objects" are existents in name only, mere thought fictions by whose means we address concrete particular things. (2) Realism. Abstracta have an independent existence as such. (shrink)