Abstractionism is a philosophical account of the ontology of mathematics according to which abstract objects are grounded in a process of abstraction. Abstraction involves arranging a domain of underlying objects into classes and then identifying … Continue reading Abstractionism →.

Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, the volume is split into 4 sections. The contributions within these sections explore the semantics and meta-ontology of Abstractionism, abstractionist epistemology, the mathematics of Abstractionis, and finally, Frege's application constraint within (...) an abstractionist setting. (shrink)

I discuss certain representative accounts of metaphysical emergence falling into three broad categories, assessing their prospects for satisfying certain criteria; the ensuing dialectic has a bit of the Goldilocks fable about it. At one end of the spectrum are what I call ‘scientistic’ accounts, which characterize metaphysical emergence by appeal to one or another specific feature commonly registered in scientific descriptions of seeming cases of emergence; such accounts, I argue, typically fail to provide a clear basis for ensuring incompatibility with (...) ontological reduction, and thus fail to guarantee satisfaction of the criterion of appropriate contrast (§2). At the other end of the spectrum are what I call ‘abstractionist’ accounts, which characterize emergence in terms floating free of scientific notions; such accounts, I’ll argue, typically fail to guarantee satisfaction of the criterion of illuminating contrast (§3). I’ll then look at several accounts of metaphysical emergence that are what I call ‘substantive’, in appealing to familiar relations or posits which are properly metaphysical, while being intelligibly connected to scientific relations and posits. As we’ll see, the resources of such accounts are up to the tasks of blocking ontological reduction and of enabling investigations into metaphysical emergence to proceed in an illuminating way (§4). I close with some methodological morals (§5). (shrink)

Mainstream psychologists have not only ignored the unique and radical character of practice; they have generally misunderstood it. A major reason for this ignorance and misunderstanding is mainstream psychology's assumption of a particular ontology--abstractionism. With abstractionism, psychologists have generally assumed that abstractions, such as theories, techniques, and principles, capture and embody the fundamentally real. Most pertinently, abstractions are believed to precede and lay the foundation for good and thoughtful practice. Indeed, practices do not exist, in an important ontological sense, (...) except in relation to the concrete and particular situations and cultures that give rise to them, implying what we might call a relational ontology. The purpose of this article is to explore the possible meanings and implications of a relational ontology for psychology. I first attempt to clarify the important features of relational conceptions in general, distinguishing most importantly between their weak and strong forms. Then, I situate strong or ontological relationality historically by describing its initial embrace and eventual rejection by early scientists. To bring these historical lessons into the present, ontological relationality is next compared to the abstractionist understandings of the real and fundamental in mainstream psychology. These two forms of ontology are then illustrated by exploring their contrasting implications for individual and community identity. 2012 APA, all rights reserved). (shrink)

The aim of this article is to offer a method for determining the ontological commitments of a formalized theory. The second section shows that determining the consequence relation of a language model-theoretically entails that the ontology of a theory is tied very closely to the variables that feature in that theory. The third section develops an alternative way of determining the ontological commitments of a theory given a proof-theoretic account of the consequence relation for the language that theory is (...) in. It is shown that the proof-theoretic account of ontological commitment does not entail that the ontological commitments of a theory depend on the variables of that theory. The last section of the article discusses how this account of ontological commitment can be used in other philosophical projects such as Wright’s (Frege’s conception of numbers as objects, Aberdeen University Press, Aberdeen, 1983) abstractionism. The article concludes with a discussion of the upshots of adopting the proof-theoretic account of ontological commitment for ontology generally. (shrink)

In this paper I argue that Aristotle's understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. On my reading, Aristotle can only be a mathematical abstractionist of a certain sort. To show this, I first present an analysis of Aristotle's notion of continuity by bringing together texts from his Metaphysica and Physica, to show that continuity is, for Aristotle, a certain kind of per se unity, and that upon this rests his distinction between continuity and (...) contiguity. Next I argue briefly that Aristotle intends for his discussion of continuity to apply to pure mathematical objects such as lines and figures, as well as to extended bodies. I show that this leads him to a difficulty, for it does not at first appear that the distinction between continuity and contiguity can be preserved for abstract mathematicals. Finally, I present a solution according to which Aristotle's understanding of continuity can only be saved if he holds a certain kind of mathematical ontology. (shrink)

"The main aim of the paper is to compare two types of abstractionistic accounts of fictional objects, and to analyze their consequences for interpretation of existential quantification. According to a proponent of general abstractionistic theory, fictional objects have abstract nature in a way similar to contracts, marriages, and the likes. This view is an alternative to strongly realistic accounts of fictional objects, defended by Terence Parsons or David Lewis. Within abstractionistic theories, as in all philosophical areas, one can find divergences (...) of opinions. The main differences between two of them – Peter van Inwagen's and Edward Zalta's – are connected with the interpretation of existential quantification. According to van Inwagen, “being” is the same as “existence” and its sense is captured by the existential quantifier. Edward Zalta's theory is much closer to the Meinongian Theory of Objects. He argues for the need of distinguishing between “being” and “existence” and for invoking nonexistent objects. Because of that he suggests an alternative interpretation of quantification. Admittedly, there is one abstractionistic source for both theories, but their ontological consequences are different - van Inwagen is a staunch opponent of nonexistent objects, and Zalta describes his own theory as "Meinongian".". (shrink)

The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one:. The principal aim of this article is to use the notion of grounding to develop (...) this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of metaphysical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege's Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction. (shrink)

Abstract objects are widely held to pose a formidable epistemological challenge. It has seemed mysterious to many how we can have access to such strange and intangible entities. The article considers five influential ways to meet the challenge: Transcendental arguments, the indispensability argument, insisting that we just are able to grasp abstract objects and that no further explanation is needed, abstractionist accounts, and ontological reduction. None of these approaches is by itself sufficient or completely convincing, but together they make (...) out a strong cumulative case for the accessibility of abstract objects. (shrink)

The prevailing approach to the problem of the ontological status of mathematical entities such as numbers and sets is to ask in what sense it is legitimate to ascribe a reference to abstract singular terms; those expressions of our language which, taken at face value, denote abstract objects. On the basis of this approach, neo‐Fregean Abstractionists such as Hale and Wright have argued that abstract singular terms may be taken to effect genuine reference towards objects, whereas nominalists such as Field (...) have asserted that these apparent ontological commitments should not be taken at face value. In this article I argue for an intermediate position which upholds the legitimacy of ascribing a reference to abstract singular terms in an attenuated sense relative to the more robust ascription of reference applicable to names denoting concrete entities. In so doing I seek to clear up some confusions regarding the ramifications of such a thin notion of reference for ontological claims about mathematical objects. (shrink)

Oystein Linnebo's book Thin Objects is partly devoted to defending the view that some objects are “thin” in that their existence does not impose any substantive demands on the world. In this paper, I discuss the concern that the defense relies on there being entities that serve as the referents of predicates. Linnebo thus seems to assume the thinness of those entities. In the course of my discussion, I also discuss what Linnebo says about the role of criteria of identity (...) in his discussion of reference and existence. (shrink)

This text, touching on the problem of the ontology of the image from a political-theological perspective, focuses on Reiner Schürmann’s philosophical reading of the pictorial art of Louis Comtois. While placing it in the context of post-World War II modernism, he nevertheless underlines its special spiritual quality that, far from any theorization of messianic abstractionism or any affirmation of artistic sovereignty, shows all the anarchic simplicity of painting. It is a form of modernity still to be imagined but one (...) that is already in some way active in the unpredictable meeting of matter and light, which Comtois’s pictorial act and Schürmann’s thought contemplate in the inseparable connection between history and immortality. (shrink)

I examine one aspect of the central role which mathematics plays in Proclus's ontology and epistemology, with particular reference to his Elements of Theology. I focus on his peculiar views about the ontological status of mathematical objects and the special faculties of the soul that are involved in understanding them. If they are merely abstract objects that are "stripped away" from sensible things, then they are unlikely to reorient the mind towards the intelligible realm, as envisioned by Plato in (...) the Republic. Thus, in order to defend the function of mathematics as a prodaideutic to dialectic, Proclus rejects Aristotelian abstractionism in favor of an elaborate account in terms of Nous projecting images of its Forms through the medium of the imagination. In metaphorical terms, he replaces the Aristotelian image of the soul as a blank tablet with that of a tablet that has always been inscribed and is always writing itself, while also being written on by Nous. The mediating function of mathematics for understanding the higher realities is grounded in the fact that its central principles of Limit and Unlimited have a universal provenance in Proclus's whole system of reality. (shrink)

This is a technical supplement to "Abstraction and Grounding", forthcoming in /Philosophy and Public Affairs/.

In the present paper, the author looks at the political dimension of some trends in the visual arts within twentieth-century avant-garde groups (cubism, expressionism, fauvism, Dada, abstractionism, surrealism) through George Santayana’s idea of vital liberty. Santayana accused the avant-gardists of social and political escapism, and of becoming unintentionally involved in secondary issues. In his view, the emphasis they placed on the medium (or diverse media) and on treating it as an aim in itself, not, as it should be, as a (...) transmitter through which a stimulating relationship with the environment can be had, was accompanied by a focus on fragments of life and on parts of existence, and, on the other hand, by a de facto rejection of ontology and cosmology as being crucial to understanding life and the place of human beings in the universe. The avant-gardists became involved in political life by responding excessively to the events of the time, instead of to the everlasting problems that are the human lot. (shrink)

In his development of an ontology for mathematics in line with Fregean abstractionism, Øystein Linnebo proposes criteria of identity as a mechanism for reference. This proposed mechanism appears to conflict with Saul Kripke's historical theory of names. However, Linnebo, distinguishing between semantic and metasemantic domains, asserts that no real conflict exists; while Kripke's theory is semantic, his is metasemantic. This paper posits a counter-argument, contending that Linnebo's account, contrary to his claim, indeed conflicts with Kripke's historical theory of names. (...) The paper underscores that, in Kripke's framework, the reference relation transcends the individual, inhabiting a social domain. (shrink)

According to Daniel Flage, Berkeley thinks that all necessary truths are founded on acts of will that assign meanings to words. After briefly commenting on the air of paradox contained in the title of Flage’s paper, and on the historical accuracy of Berkeley’s understanding of the abstractionist tradition, I make some remarks on two points made by Flage. Firstly, I discuss Flage’s distinction between the ontological ground of a necessary truth and our knowledge of a necessary truth. Secondly, I (...) discuss Flage’s attempt to show that, according to Berkeley, the resemblance relation does not constitute a necessary connection. (shrink)

A considerable number of philosophers are attracted to what might be called ‘causal type-abstractionism’ – the view that photographs, symphonies, models of cars, novels, flags, and other multiply i...

Reification is to abstraction as disease is to health. Whereas abstraction is singling out, symbolizing, and systematizing, reification is neglecting abstractive context, especially functional, historical, and analytical-level context. William James and John Dewey provide similar and nuanced arguments regarding the perils and promises of abstraction. They share an abstraction-reification account. The stages of abstraction and the concepts of “vicious abstractionism,” “/the/ psychologist’s fallacy,” and “the philosophic fallacy” in the works of these pragmatists are here analyzed in detail. For instance, in (...) 1896 Dewey exposes various fallacies associated with reifying dualistic reflex arc theory. The conclusion prescribes treatments (pluralism and assumption archaeology) for de-reifying ill models (i.e., universalized, narrowed, and ontologized models) in contemporary scientific fields such as cognitive science and biology. (shrink)

A philosophical inquiry into the nature of information and communication media raises conceptual and ontological questions. This analysis provides conceptual mappings and also raises the question of what is involved in the emergence of media out of some prior state from which they were absent, and again in subsequent phases of higher-level emergent phenomena. “Emergence” can be understood in a number of ways: epistemically, causally, or metaphysically, and there is a danger of equivocating between these different senses in general or (...) in considering particular levels of media emergence alone, and drawing the wrong conclusions. In describing and attempting to understand recent developments in “emerging media” it may be useful to consider an analogy with developments in the visual arts involving the transition from figurative naturalism to abstractionism, then to minimalism, and from these to performance art and conceptualism. (shrink)

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of (...) contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

The ideas of fixed points (Kripke in Recent essays on truth and the liar paradox. Clarendon Press, London, pp 53–81, 1975; Martin and Woodruff in Recent essays on truth and the liar paradox. Clarendon Press, London, pp 47–51, 1984) and revision sequences (Gupta and Belnap in The revision theory of truth. MIT, London, 1993; Gupta in The Blackwell guide to philosophical logic. Blackwell, London, pp 90–114, 2001) have been exploited to provide solutions to the semantic paradox and have achieved admirable (...) success. This happy situation naturally encourages one to look for other philosophical areas of their further applications where paradoxical results seem to follow from intuitively acceptable principles. In this paper, I propose to extend the use of these ideas to give two new treatments of abstract objects. Sections 1 and 2 below check several abstractionist theories and their main defects. Section 3 shows how the two ideas can be applied to generate consistent theories of abstract objects without any ad hoc restriction on any principle. (shrink)

This paper argues in the first place that nominalists are right in insisting against ontological realists that semantic universality does not require commitment to universal entities. However, Ockham, in his zeal to get rid of Scotus’s universal entities, swept under the carpet the issue of universal representational content of genuinely universal symbols, conflating it with the mere indifference of the information content of non-distinctive singular representations. Buridan did come up with an abstractionist theory of the formation of genuinely universal (...) representational content to solve the resulting issues, however, the paper argues further, his solution is committed to attributing the sort of “aspectuality” to universal absolute concepts that his Ockhamist semantics denies to them. The conclusion of the paper suggests how Aquinas’s “moderate realism” can provide a consistent solution without the ontological extravaganza of ontological realists, without conflating the mere indifference of singular representation with genuine universality, and without having to deny aspectuality to our quidditative universal concepts formed by abstraction. (shrink)

Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more clearly the primary meta-ontological (...) and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics. (shrink)

According to neo-Fregeans, an expression that is syntactically singular and figures in a true sentence is guaranteed to have some existing thing in the world to pick out. But this approach is confronted by a dilemma. If reality is crystalline, has a structure fixed independently of language, then the view that reality is guaranteed to contain a sufficient plenitude of objects to supply referents for the relevant expressions is left hostage to cosmological fortune. Whereas if reality is plastic then it (...) becomes dubiously coherent to conceive of our ordinary, scientific, and mathematical claims about a diversity of objects as being genuinely true or false of an independent reality. To avoid this dilemma, the neo-Fregeans must abandon the dichotomy that says either our true sentences must ‘mirror’ the structure of a crystalline reality or ‘impose’ structure upon a plastic one. If we can legitimately deny that there is an intelligible question to be raised of how language hooks onto reality, then prima facie we can grant that a sentence S is true without thereby becoming embroiled in the uncomfortable consequences of having to say what makes S true. So ontological quietism holds out the last, best hope for neo-Fregeanism. (shrink)

Existence questions have been topics for heated debates in metaphysics, but this book argues that they can often be answered easily, by trivial inferences from uncontroversial premises. This 'easy' approach to ontology leads to realism about disputed entities, and to the view that metaphysical disputes about existence questions are misguided.

A systematic treatment of Hume's conception of imagination in all the main topics of his philosophy.

Key elements of John Searle’s articulation of the Standard Model of Social Ontology can be found within Max Weber’s ideal type of legal-rational authority. However, the fact that, for Weber, legal-rational authority is just one of three types of legitimate authority, along with traditional and charismatic authority, suggests limitations to the Standard Model’s scope of applicability. Where Searle takes himself to have provided an account of “the structure of human civilization,” Weber’s taxonomy suggests that Searle has only given us (...) an account of a way of being a civilization. This understanding of traditional authority also reveals why the Standard Model misconstrues the structure of ordinary, informal statuses, such as friendship. (shrink)

This chapter presents a displacement of the organism as a privileged level of analysis in evolutionary biology. It is concerned with the ontology of biology systems, with particular reference to hierarchical organization. It argues that the concept of a rank-free hierarchy can be transposed to the major transitions hierarchy, with interesting consequences. This chapter shows that the idea of rank freedom makes good sense of a number of facets of the recent discussion of evolutionary transitions and multilevel selection. It (...) suggests that the idea of rank freedom is already at work, implicitly, in much theorizing about evolutionary transitions and/or multilevel selection. (shrink)

Though science and philosophy take different approaches to ontology, metaphysical inferences are relevant to interpreting scientific work, and empirical investigations are relevant to philosophy. This book argues that there is no uniquely rational way to determine which domains of ontology are appropriate for belief, making room for choice in a transformative account of scientific ontology.

The French philosopher Renaud Barbaras remarked that late in Maurice Merleau-Ponty’s career, “The phenomenology of perception fulfills itself as a philosophy of expression.” In _Tracing Expression in Merleau-Ponty: Aesthetics, Philosophy of Biology, and Ontology, _Véronique M. Fóti_ _addresses the guiding yet neglected theme of expression in Merleau-Ponty’s thought. She traces Merleau-Ponty’s ideas about how individuals express creative or artistic impulses through his three essays on aesthetics, his engagement with animality and the “new biology” in the second of his lecture (...) courses on nature of 1957–58, and in his late ontology, articulated in 1964 in the fragmentary text of _Le visible et l’invisible _. With the exception of a discussion of Merleau-Ponty’s 1945 essay “Cezanne’s Doubt,” Fóti engages with Merleau-Ponty’s late and final thought, with close attention to both his scientific and philosophical interlocutors, especially the continental rationalists. Expression shows itself, in Merleau-Ponty’s thought, to be primordial, and this innate and fundamental nature of expression has implications for his understanding of artistic creation, science, and philosophy. (shrink)

Helen Steward puts forward a radical critique of the foundations of contemporary philosophy of mind, arguing that it relies too heavily on insecure assumptions about the sorts of things there are in the mind--events, processes, and states. She offers a fresh investigation of these three categories, clarifying the distinctions between them, and argues that the category of state has been very widely and seriously misunderstood.

"... an attempt to revive ontology —indeed philosophy itself—by means of a two-sided conception of being.... This is a remarkable idea which has produced a powerful book." —Leonard Lawlor "... a major philosophical study: rich, brilliant... a tour de force, a seminal study that will be a starting-point for future research in this area." —Robert Bernasconi In Truth and Genesis, Miguel de Beistegui considers the role and meaning of philosophy today. Calling for a new departure for philosophy, one that (...) brings together philosophy’s scattered identities, de Beistegui proposes a robust and unified philosophy that would find itself equally at home in artistic and scientific disciplines. To build this renewed philosophy, de Beistegui turns to Aristotle and the earliest foundations of thought. He traces philosophy’s development through the medieval and modern periods before comparing and investigating the work of two of the 20th century’s most influential thinkers, Martin Heidegger and Gilles Deleuze. In particular, de Beistegui focuses on Deleuze’s Difference and Repetition and Heidegger’s Contributions to Philosophy for their handling of the concept of difference. De Beistegui concludes that Deleuze and Heidegger are irreconcilable, but it is in their disagreements that he sees a way to liberate philosophy from its current crisis. (shrink)

D. M. Armstrong is an eminent Australian philosopher whose work over many years has dealt with such subjects as: the nature of possibility, concepts of the particular and the general, causes and laws of nature, and the nature of human consciousness. This collection of essays explores the many facets of Armstrong's work, concentrating on his more recent interests. There are four sections to the book: possibility and identity, universals, laws and causality, and philosophy of mind. The contributors comprise an international (...) group of philosophers from the United States, England and Australia. An interesting feature of the volume is that Armstrong himself has written responses to each of the essays. There is also a complete bibliography of Armstrong's writings. (shrink)

First Published in 2005. Routledge is an imprint of Taylor & Francis, an informa company.

This paper argues that the conditions of the Anthropocene render the four ontologies described by Philippe Descola obsolete, and begins the search for a fifth ontology that speaks to the meaning of...

An overview of current issues in musical ontology, including debates about "fundamental" vs. "higher-order" musical ontology and skepticism about both kinds.

The aim of this paper is to summarize a particular approach of doing metaphysics through physics - the primitive ontology approach. The idea is that any fundamental physical theory has a well-defined architecture, to the foundation of which there is the primitive ontology, which represents matter. According to the framework provided by this approach when applied to quantum mechanics, the wave function is not suitable to represent matter. Rather, the wave function has a nomological character, given that its (...) role in the theory is to implement the law of evolution for the primitive ontology. (shrink)