In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal (...) axiom that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

This book pursues the question of how and whether natural language allows for reference to abstract objects in a fully systematic way. By making full use of contemporary linguistic semantics, it presents a much greater range of linguistic generalizations than has previously been taken into consideration in philosophical discussions, and it argues for an ontological picture is very different from that generally taken for granted by philosophers and semanticists alike. Reference to abstract objects such as properties, (...) numbers, propositions, and degrees is considerably more marginal than generally held. (shrink)

Part of Frege's concern about whether number words are properties of objects was that if they could be construed as such it would lend support to the view that truths of arithmetic were empirical truths. Such concern is ill-founded. Even if number words do apply to objects as predicates, this does not entail that numerical truths would be empirical, any more than the fact that ‘bachelor’ and ‘unmarried’ are predicates of objects entails that their relationship is an (...) empirical one. The account of number words as predicates given here, consequently, should not be taken as supportive of any one particular theory concerning arithmetic truth rather than another.The view of number words that emerges above from the criticism of Frege's arguments is that sentences containing number words can and do say something about objects in a collective and syncategorematic way. That is, they say something about classes of objects, when it is clear what is to count as an object. Whether this view commits us to accepting certain ontological entities, classes of individuals, above and beyond individuals themselves, I am not prepared to say, and at any rate this question falls beyond the intended scope of this essay. To those who would eschew abstract entities, however, it may be noted that the ontological problems that arise are not peculiar to this view; they seem no more or less severe than those that arise in connection with other collective predicate statements, as, for example, “Blue whales are becoming extinct.”The view presented here also does not pretend to be a fatal criticism of the logicist program. Taking our cue from the above noted fact that syncategorematic predicates share in a common ‘partial’ abstract sense, it seems reasonable to say that it is just the set of such partial senses of number words, and the relationships that hold between such senses, that the logicist and the pure number theorist investigate. (shrink)

Philosophers often debate the existence of such things as numbers and propositions, and say that if these objects exist, they are abstract. But what does it mean to call something 'abstract'? And do we have good reason to believe in the existence of abstract objects? This Element addresses those questions, putting newcomers to these debates in a position to understand what they concern and what are the most influential considerations at work in this area (...) of metaphysics. It also provides advice on which lines of discussion promise to be the most fruitful. (shrink)

Philosophers frequently draw on natural language to motivate properties, numbers, and propositions as objects, and it is generally taken for granted that abstract objects of this sort are well-reflected in natural language and in fact that reference to them in natural language is pervasive In this paper, I will review and modify in a certain way the view I had advanced in Abstract Objects and the Semantics of Natural Language (Moltmann 2013a). This is the (...) view that natural language permits reference to abstract objects only in its periphery, not in its core. Even though reference to abstract objects of the sort of properties, propositions, and numbers is available only in the periphery of language, natural language permits reference to other sorts of abstract objects in its core, in particular abstract artifacts of various sorts. (shrink)

Art and Abstract Objects presents a lively philosophical exchange between the philosophy of art and the core areas of philosophy. The standard way of thinking about non-repeatable (single-instance) artworks such as paintings, drawings, and non-cast sculpture is that they are concrete (i.e., material, causally efficacious, located in space and time). Da Vinci's Mona Lisa is currently located in Paris. Richard Serra's Tilted Arc is 73 tonnes of solid steel. Johannes Vermeer's The Concert was stolen in 1990 and remains (...) missing. Michaelangelo's David was attacked with a hammer in 1991. By contrast, the standard way of thinking about repeatable (multiple-instance) artworks such as novels, poems, plays, operas, films, symphonies is that they must be abstract (i.e., immaterial, causally inert, outside space-time): consider the current location of Melville's Moby Dick, the weight of Yeats' "Sailing to Byzantium", or how one might go about stealing Puccini's La Boheme or vandalizing Mozart's Piano Concerto No. 9. Although novels, poems, and symphonies may appear radically unlike stock abstract objects such as numbers, sets, and propositions, most philosophers of art think that for the basic intuitions, practices, and conventions surrounding such works to be preserved, repeatable artworks must be abstracta. -/- This volume examines how philosophical enquiry into art might itself productively inform or be productively informed by enquiry into abstracta taking place within not just metaphysics but also the philosophy of mathematics, epistemology, philosophy of science, and philosophy of mind and language. While the contributors chiefly focus on the relationship between philosophy of art and contemporary metaphysics with respect to the overlap issue of abstracta, they provide a methodological blueprint from which scholars working both within and beyond philosophy of art can begin building responsible, mutually informative, and productive relationships between their respective fields. (shrink)

ABSTRACT Michael Rescorla has argued that it makes sense to compute directly with numbers, and he faulted Turing for not giving an analysis of number-theoretic computability. However, in line with a later paper of his, it only makes sense to compute directly with syntactic entities, such as strings on a given alphabet. Computing with numbers goes via notation. This raises broader issues involving de re propositional attitudes towards numbers and other non-syntactic abstract entities.

This is a dialogue in which five characters are involved. Various issues in the philosophy of mathematics are discussed. Among those issues are these: numbers as abstract objects, our knowledge of numbers as abstract objects, a proof as showing a mathematical statement to be true as opposed to the statement being true in virtue of having a proof.

Abstract objects are standardly taken to be causally inert, however principled arguments for this claim are rarely given. As a result, a number of recent authors have claimed that abstract objects are causally efficacious. These authors take abstracta to be temporally located in order to enter into causal relations but lack a spatial location. In this paper, I argue that such a position is untenable by showing first that causation requires its relata to have a temporal (...) location, but second, that if an entity is temporally located then it is spatiotemporally located since this follows from the theory of Relativity. Since abstract objects lack a spatiotemporal location, then if something is causally efficacious, it is not abstract. (shrink)

What sorts of things are there in the world? Clearly enough, there are concrete, material things; but are there other things too, perhaps nonconcrete or non-material things? Some people believe that there are such things, which are often called abstract ; purported examples of such objects include numbers, properties, possible but non-actual states of affairs, propositions, and sets. Following a long-standing tradition, I shall describe persons who believe that there are abstract objects as ‘platonists’. In (...) this paper, I shall not directly address the plausibility of platonism, as compared with its rivals; instead, I shall confine my attention to one way in which some people have tried to combine platonism and theism. More specifically, I shall concentrate upon the claim that abstract objects depend upon God ontologically ; I shall argue that platonistic theists should reject DEP in favour of the claim that abstract objects exist independently of God. In order to evaluate the relative merits of DEP versus IND, it will be helpful to examine in some detail a particular articulation of DEP. When it comes to recent work on DEP, we can do no better in this regard than to examine the recent work of Thomas V. Morris and Christopher H. Menzel. According Morris and Menzel, there is a sense in which God literally creates such abstracta through engaging in intellective activities. (shrink)

This books offers a novel account of the nature of numbers firmly grounded in results from numerical cognition and the philosophy of mathematics. Drawing on empirical data on the human experience of what we call “numbers,” the author shows that numbers do not exist as abstract objects, but that the idea that they do is a useful cognitive tool. Contrary to the platonist view, according to which arithmetic is true of a realm of abstract (...) entities, the nominalistic account presented in this book shows arithmetic to be true of descriptions of structural properties of techniques such as counting and calculating procedures. This book is of interest to both philosophers and cognitive scientists who want to have a deeper understanding of what numbers are. (shrink)

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, (...) class='Hi'>numbers are not primarily treated abstract objects, but rather 'aspects' of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. (shrink)

According to platonists, entities such as numbers, sets, propositions and properties are abstract objects. But abstract objects lack causal powers and a location in space and time, so how could we ever come to know of the existence of such impotent and remote objects? In Knowledge, Cause, and Abstract Objects, Colin Cheyne presents the first systematic and detailed account of this epistemological objection to the platonist doctrine that abstract objects exist (...) and can be known. Since mathematics has such a central role in the acquisition of scientific knowledge, he concentrates on mathematical platonism. He also concentrates on our knowledge of what exists, and argues for a causal constraint on such existential knowledge. Finally, he exposes the weaknesses of recent attempts by platonists to account for our supposed platonic knowledge. This book will be of particular interest to researchers and advanced students of epistemology and of the philosophy of mathematics and science. It will also be of interest to all philosophers with a general interest in metaphysics and ontology. (shrink)

In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in (...) a system treating apparent terms for numbers using these definitions. (shrink)

Linnebo in 2018 argues that abstract objects like numbers are “thin” because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress (...) is avoided in the account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations. (shrink)

The question whether numbers are objects is a central question in the philosophy of mathematics. Frege made use of a syntactic criterion for objethood: numbers are objects because there are singular terms that stand for them, and not just singular terms in some formal language, but in natural language in particular. In particular, Frege (1884) thought that both noun phrases like the number of planets and simple numerals like eight as in (1) are singular terms referring (...) to numbers as abstract objects. (shrink)

I present a deflationary account of abstract objects, focusing on numbers and properties. This account can be seen as a natural extension of the deflationary account of truth. Very roughly, the key idea is that (1) our talk about numbers and properties serves a quasi-logical function analogous to our talk about truth, and (2) key questions about the metaphysics and epistemology of numbers and properties can be answered with reference to that function. In contrast to (...) many other deflationary approaches in metaphysics, the present account does not assume that the principles governing our talk about numbers and properties is analytically true. (shrink)

Numbers have many puzzling features. Their properties are mostly essential to them, but they exist in all possible worlds. Number theory seems a priori, yet it makes existence claims and existence (setting aside the Cogito) is not supposed to be a priori knowable. If-thenism can perhaps explain the felt a priority, but it makes numerical truth relative where it seems absolute. A figuralist solution is proposed: ‘2 + 3 = 5’ seems necessary, a priori, and absolute because it has (...) a logical truth as its assertive content. A rule that associates logical truths with each arithmetical truth is given, and also a rule that associates a logical truth (modulo concrete combinatorics) with each truth about hereditarily finite impure sets. The view that emerges takes something from Frege and something from Kant; one might call it Kantian logicism. The view is Kantian because it sees mathematics as arising out of our representations. Numbers and sets are ‘there’ because they are inscribed on the spectacles through which we see other things. It is logicist because the facts seen through our numerical spectacles are facts of first-order logic. (shrink)

A realist view of numbers often rests on the following thesis: statements like ‘The number of moons of Jupiter is four’ are identity statements in which the copula is flanked by singular terms whose semantic function consists in referring to a number (henceforth: Identity). On the basis of Identity the realists argue that the assertive use of such statements commits us to numbers. Recently, some anti-realists have disputed this argument. According to them, Identity is false, and, thus, we (...) may deny that the relevant statements commit us to numbers. The present paper argues that the correct linguistic analysis of the relevant number statements supports the anti-realist view that Identity is false. However, as will further be shown, pace the anti-realist, this analysis does not establish that such statements do not commit us to numbers after all. (shrink)

Are there objects that are “thin” in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. This book (...) attempts to develop the needed explanations by drawing on some Fregean ideas. First, to be an object is to be a possible referent of a singular term. Second, singular reference can be achieved by providing a criterion of identity for the would-be referent. The second idea enables a form of easy reference and thus, via the first idea, also a form of easy being. Paradox is avoided by imposing a predicativity restriction on the criteria of identity. But the abstraction based on a criterion of identity may result in an expanded domain. By iterating such expansions, a powerful account of dynamic abstraction is developed. (shrink)

Defining the real numbers by abstraction as ratios of quantities gives prominence to then- applications in just the way that Frege thought we should. But if all the reals are to be obtained in this way, it is necessary to presuppose a rich domain of quantities of a land we cannot reasonably assume to be exemplified by any physical or other empirically measurable quantities. In consequence, an explanation of the applications of the reals, defined in this way, must proceed (...) indirectly. This paper explains the main complications involved and answers the main objections advanced in Batitsky's paper in this issue. (shrink)

According to Augustine, abstract objects are ideas in the mind of God. Because numbers are a type of abstract object, it would follow that numbers are ideas in the mind of God. Call such a view the “Augustinian View of Numbers” (AVN). In this paper, I present a formal theory for AVN. The theory stems from the symmetry conception of God as it appears in Studtmann (2021). I show that the theory in Studtmann’s paper (...) can interpret the axioms of Peano Arithmetic minus the induction schema. This fact allows for the development of arithmetic in a natural way. The development eventuates in a theory that can interpret second-order arithmetic. The conception of God that emerges by the end of the discussion is a conception of an infinite, ineffable, self-cause that contains objects that not only serve as numbers but also encode information about each other. (shrink)

The formal ontological concept of an _object_ is explicated and contrasted with that of a _property_. F. P. Ramsey’s objections to this distinction are challenged. The sense in which objects possess an _individuality_ not exhibited by entities of certain other types is discussed. The object/property distinction is distinguished from that between universals and particulars. The ontological status of events and processes, and that of abstract entities such as numbers, are examined. Gottlob Frege’s treatment of number and his (...) object/concept distinction are criticized, and an alternative account of the ontological status of concepts is advanced. (shrink)

In this article, we evaluate various responses to a noteworthy objection, namely, the infinite God objection to the kalām cosmological argument. As regards this objection, the proponents of the kalām argument face a dilemma—either an actual infinite cannot exist or God cannot be infinite. More precisely, this objection claims that God’s omniscience entails the existence of an actual infinite with God knowing an actually infinite number of future events or abstract objects, such as mathematical truths. We argue, however, (...) that the infinite God objection is based on two questionable assumptions, namely, that it is possible for an omniscient being to know an actually infinite number of things and that there exist an actually infinite number of abstract objects for God to know. (shrink)

In his latest book, Realistic Rationalism (Cambridge, MA: MIT Press, 1998), Jerrold J. Katz proposes an ontology designed to handle putative counterexamples to the traditional abstract/concrete distinction. Objects like the equator and impure sets, which appear to have both abstract and concrete components, are problematic for classical Platonism, whose exclusive categories of objects with spatiotemporal location and objects lacking spatial or temporal location leave no room for them. Katz proposes to add a “composite” category to (...) Plato’s dualistic ontology, which is supposed to include all those objects with both abstract and concrete components.But every concrete object stands in an indefinite number of relations to abstract ones. Thus, Katz must offer principled criteria describing just those relations that produce a composite object, lest all concrete objects turn out to be composite. The trouble that he has in specifying such a “creative” relationship results from his clinging to the traditional definitions of “abstract” and “concrete.” The substance dualism that results renders the articulation of any relations between abstract and concrete difficult, and a category such as Katz’s “composite objects” impossible. (shrink)

In this paper, I defend traditional Platonic mathematical realism from its contemporary detractors, arguing that numbers, understood as abstract, non-physical objects of rational intuition, are indispensable for the act of counting.

Chapter 7 provides a noneist account of mathematical and other abstract objects, and of worlds. It then discusses a number of objections, such as that this is just a form of platonism in disguise.

This book proposes a different perspective on actual queries within the field of ontology. Focusing on non-standard objects, it offers original answers to classic problems in metaphysics, such as individuation, reference, existence and non-existence. The chosen ontological fields are, for this purpose, ontologies that essentially involve human social practices such as intentional objects, fictions, mental illnesses or social entities. Even though the papers can be read independently, readers will discover a number of original intersections that provide fresh points (...) of entry to contemporary metaphysical issues. By dealing with entities that are not so frequently addressed in related works, the book also contributes to broaden their perspective on ontology. Both scholars and students will find it particularly useful to have a survey about non-standard abstract entities that can provide new topics for their research. (shrink)

This is a book about numbers– what they are as concepts and how and why they originate–as viewed through the material devices used to represent and manipulate them. Fingers, tallies, tokens, and written notations, invented in both ancestral and contemporary societies, explain what numbers are, why they are the way they are, and how we get them. Cognitive archaeologist Karenleigh A. Overmann is the first to explore how material devices contribute to numerical thinking, initially by helping us to (...) visualize and manipulate the perceptual experience of quantity that we share with other species. She explores how and why numbers are conceptualized and then elaborated, as well as the central role that material objects play in both processes. Overmann’s volume thus offers a view of numerical cognition that is based on an alternative set of assumptions about numbers, their material component, and the nature of the human mind and thinking. Karim Zahidi (“Radicalizing numerical cognition,” Synthese, 2021, p. 530) called this reconstruction, as presented in journal articles, a “naturalistically plausible account of the emergence of the modern natural number concept," while César Dos Santos ("Review of: 'The Materiality of Numbers'," Qeios, 2023, p. 1) called it "a 'Copernican Revolution' in the way we understand the relationship between numbers and the material devices we use to record and manipulate them." . (shrink)

Platonism is the view according to which entities such as numbers and properties, characterized as abstract objects, exist. Agustín Rayo argues that the truth conditions for statements about mathematical objects and properties make no demands on the world and are, in this way, trivial. The result is trivialist platonism. In this paper, we question the stability of the two central features of the view: it cannot be both trivialist and platonist.

This chapter investigates the second developmental stage in number concept acquisition–calculation–and its role in the ontogenetic and historical reification of numbers. While early numerical competence is grounded in the mastery of number words and counting, calculation introduces new cognitive and linguistic structures that promote treating numerals as referring to abstract objects. The chapter argues that the widespread tendency to speak of numbers as objects arises not from their ontological status but from the projection of object-based (...) linguistic frameworks onto numerical discourse. This reification is traced through developmental, historical, and philosophical perspectives, offering a cognitive account of how numbers come to be treated as independent entities within mathematics and everyday language. (shrink)

In recent discussions concerning the definition of argument, it has been maintained that the word ‘argument’ exhibits the process-product ambiguity, or an act/object ambigu-ity. Drawing on literature on lexical ambiguity we argue that ‘argument’ is not ambiguous. The term ‘argu-ment’ refers to an object, not to a speech act. We also examine some of the important implications of our argument by considering the question: what sort of abstract objects are arguments?

Abstract -/- Libertarians typically believe that we are morally responsible for the choices (or decisions) we make only if those choices are free, and our choices are free only if they are neither caused nor nomically necessitated by antecedent events. Recently, there have been a number of attempts by philosophers to refute libertarianism by arguing that because a libertarianly free decision (choice) is both causally and nomically undetermined, which decision an agent makes in a deliberative situation is a matter (...) of luck, which implies (due to the way these philosophers use 'luck') that the agent does not have control over which decision he makes. This argument has been dubbed "the Argument from Luck" or "the Luck Objection" against libertarianism – henceforth 'LO'. In this paper, we examine some versions of LO as reflected in the works of Alfred Mele (2006), Neil Levy (2011), and Peter van Inwagen (2000, 2011). We argue that libertarians have nothing to fear from LO. Deep down the objection reflects a failure, on the part of its proponents, really to come to grips with the libertarian position. (shrink)

A central thesis of neo-Fregean abstractionism is that numerical expressions of the form ‘the number of Fs’, introduced by Hume’s Principle, should be read as genuine singular terms whose semantic function is to refer to particular objects. This paper explores the prospects of a variant of abstractionism in which such expressions have existential assertoric content, as in Russell’s analysis of definite descriptions. The neo-Russellian abstractionist faces three initial challenges: (i) the Russellian rendering of Hume’s Principle does not retain the (...) ontological modesty that any admissible abstraction principle must respect. (ii) It defuses the objectual character of natural numbers, and thereby fails to ground their infinity. And (iii) it does not involve the means for engaging in identifying reference to numbers as particular objects, thereby constituting a derogation of arithmetical platonism. I shall investigate these challenges and propose solutions to address them. (shrink)

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question of the (...) independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

In recent discussions concerning the definition of argument, it has been maintained that the word ‘argument’ exhibits the process-product ambiguity, or an act/object ambi-guity. Drawing on literature on lexical ambiguity we argue that ‘argument’ is not ambiguous. The term ‘argument’ refers to an object, not to a speech act. We also examine some of the important implications of our argument by considering the question: what sort of abstract objects are arguments?

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 (...) class='Hi'>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)

Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s (...) fundamental razor. (shrink)

Abstract The debate about the ethics of uninhabited aerial vehicles (UAVs) is failing to keep pace with the rise of the technology. Therefore, all the key players, including ethicists, lawyers, and roboticists, are keen to offer their views on the use of these drone aircraft. Some are opposed to their use, citing a range of ethical, legal and operational issues, while others argue for their ethically mandated use. B.J. Strawser fits into this latter category. He develops a principle of (...) ?unnecessary risk?, from which he argues that we have an ethical obligation to employ UAVs if we can do so without incurring a loss of capability. Strawser defends his argument against a number of potential objections, most questionably, the argument that the use of distance weaponry such as drones, against another state without distance weaponry, crosses some moral threshold that makes the combat immoral. Utilising Jeff McMahan's work on the inequality of combatants, Strawser essentially argues that there are no grounds for a ?fair fight?. However, this paper will argue that it is not so easy to overturn the doctrine of the moral equality of combatants, nor dismiss the problem with asymmetry. It will demonstrate that if the asymmetry reaches a certain level, the justification for resorting to war may be removed and some sort of policing action may remain the only option. (shrink)

This chapter advances a nominalistic and empirically grounded account of the semantics and epistemology of arithmetic, challenging the traditional view that the truth and objectivity of arithmetical statements require the existence of numbers as abstract objects. Building on the discussion from previous chapters, it argues that numerical reference, truth, and epistemic features such as objectivity, necessity, and apriority can be explained by the cognitive and social practices of counting and calculation. Drawing on a Peircean framework of reference, (...) the chapter proposes that numerals can function referentially even in the absence of ontologically robust numerical entities. The core thesis is that the epistemic roles ascribed to numbers are instead fulfilled by procedural structures inherent in arithmetical practices. This approach preserves the standard interpretation of arithmetic while avoiding ontological commitment to abstract mathematical objects. (shrink)

Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not to assess Williamson’s (...) arguments either for necessitism (although discussion of these arguments will play a central role in the dialectic) or for necessitism’s two famous corollaries. Instead, the focus shall be a general principle governing abstract objects—the abstract of principle (or AOP) —instances of which seems to be at work in some of Williamson’s central arguments for necessitism. The AOP can be straightforwardly formulated and applied within the neo-logicist framework—in fact, arguably the principle is most naturally formulated in neo-logicist terms. After closely examining, and carefully formalizing, the AOP, the remainder of the paper focuses on arguments for necessitism-like claims (the exact meaning of “necessitism-like” will become clearer as the essay progresses) based on the AOP. In particular, we shall focus on the instance of the AOP that applies to the abstract objects governed by the most well-known and most fully studied abstraction principle: Hume’s Principle (HP). It turns out that, although we cannot reconstruct a valid argument for necessitism based on this numerical instance of the AOP, we can obtain valid arguments for weaker, but equally interesting conclusions. In particular, we shall show that, although HP combined with the AOP (and some additional, related assumptions) allows the contents of the domains of possible worlds to vary, the size of those domains must remain constant. The paper concludes by developing and critiquing some related arguments for necessitism based on applying relevant instances of the AOP to abstraction principles governing sets (or extensions), and to a simple objectual abstraction principle. (shrink)

(1) Abstract objects. The nominalist (as the label is used today) denies that there exist abstract objects. The platonist holds that there are abstract objects. One example is numbers. The nominalist denies that there are numbers; the platonist typically affirms it.

Ordinary language and scientific language enable us to speak about, in a singular way (using demonstratives and names), what we recognize not to exist: fictions, the contents of our hallucinations, abstract objects, and various idealized but nonexistent objects that our scientific theories are often couched in terms of. Indeed, references to such nonexistent items-especially in the case of the application of mathematics to the sciences-are indispensable. We cannot avoid talking about such things. Scientific and ordinary languages thus (...) enable us to say things about Pegasus or about hallucinated objects that are true (or false), such as "Pegasus was believed by the ancient Greeks to be a flying horse," or "That elf I'm now hallucinating over there is wearing blue shoes." Standard contemporary metaphysical views and semantic analyses of singular idioms on offer in contemporary philosophy of language have not successfully accommodated these routine practices of saying true and false things about the nonexistent while simultaneously honoring the insight that such things do not exist in any way at all (and have no properties). That is, philosophers often feel driven to claim that such objects do exist, or they claim that all our talk isn't genuine truth-apt talk, but only pretence. This book reconfigures metaphysics (and the role of metaphysics in semantics) in radical ways that allow the accommodation of our ordinary ways of speaking of what does not exist while retaining the absolutely crucial presupposition that such objects exist in no way at all, have no properties, and so are not the truth-makers for the truths and falsities that are about them. (shrink)

Contemporary philosophers of mathematics are deadlocked between two alternative ontologies for numbers: Platonism and nominalism. According to contemporary mathematical Platonism, numbers are real abstract objects, i.e. particulars which are nonetheless “wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal.” While this view does justice to intuitions about numbers and mathematical semantics, it leaves unclear how we could ever learn anything by mathematical inquiry. Mathematical nominalism, by contrast, holds that numbers do not exist extra-mentally, which raises difficulties (...) about how mathematical statements could be true or false. Both theories, moreover, leave inexplicable how mathematics could have such a close relationship with natural science, since neither abstract nor mental objects can influence concrete physical objects. The Thomist understanding of quantity as an accident inhering in concrete substances breaks this deadlock by granting numbers a foundation in extra-mental reality which explains why numeric expressions are relevant in natural science and how we come to know the truth or falsity of mathematical statements. Further, this kind of moderate realism captures the semantic advantages of the Platonists and the ontological parsimony of the nominalists (since for Thomists quantity is not a separate, free-standing ontological addition). Despite these advantages, the Thomistic understanding of number has been out of favor since the work of Cantor, who argued that actual infinities (regarded by Aristotelians as impossible) are required for modern developments in mathematics. This paper frees philosophers of mathematics to embrace the advantages of Thomistic realism by showing that Cantor’s understanding of actual and potential infinity is not the same as Aquinas’s, and that potential infinities (in Aquinas and Aristotle’s sense) can do all the work Cantor claims for actual infinities in modern mathematics. (shrink)

A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on (...) the size of the continuum hold. As a result, whether or not these higher-order versions of Hume’s Principle are acceptable seems to be independent of standard (ZFC) set theory. This places the abstractionist in an uncomfortable dilemma: Either there is some inherent difference between counting objects and counting concepts, or new criteria for acceptability will need to be found. It is argued that neither horn looks promising. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:THE PROBLEM OF THING AND OBJECT IN MARITAIN JOHN c. CAHALAN Methuen, Massachusetts I N THE essay, "Critical Realism," Jacques Maritain said, "The problem of thing and object is the crux of the problem of realism." 1 Since then, the distinction between thing and object has received little attention, except for some helpful discussions by Yves Simon. Either Maritain and Simon were very mistaken, or we have been (...) missing something very important. This study will attempt to explain the importance of the thing/object distinction by showing how it applies to a wide range of questions about human knowledge, from the nature of metaphysics to the problem of conceptual relativity in the philosophy of science and hermeneutics. Skeptics and idealists grant that our awareness has objects. Skeptics question whether we can make extramentally existing things objects of awareness as they are. Idealists deny that things have an existence other than being objects of awareness.2 Maritain considered the presuppositions of calling something an " object " of awareness: what are we doing when we call something an object of awareness and what conditions are necessary for knowing that something is such an object. He found that the conditions necessary for knowing that something is an object of awareness provide the basis for refutations of skepticism and 1 Jacques Maritain, The Degrees of Knowledge, trans. Gerald B. Phelan (New York: Scribners, 1959) 107 (translation corrected). Hereafter, I refer to this work as DK, followed by a page number. 2 Maritain uses " idealism " for both ways of denying that we directly attain extramental existents, that is, for any position holding that we must begin from awareness of our mental states. 21 22 JOHN C. CAHALAN idealism. But his analysis also gives us a tool, the distinction between things as things and things as objects, capable of illuminating many other epistemological problems. I will begin by explaining the thing/object distinction and how it enables us to refute idealism. Then, I will apply the distinction to problems concerning the knowledge of being in metaphysics. Next, I will show how the distinction reveals, for the first time, the nature of that " conformity" between the mind and things which we call truth. Finally, I will apply the distinction to the problems of skepticism and conceptual relativity. I How do the conditions required for knowing that something is an object enable us to defend realism? We can ask epistemological questions because we are reflexively aware of our own awareness. The reality we call "awareness" is a way of relating to terms that we call its "objects." "Object" is a relational term here; to call something an "object" of awareness is to describe it as a term of a knowledge relation (where " knowledge " means cognition in general, not just knowledge of truth). " We must distinguish," Maritain says, "between the thing as thingas existing or able to exist for itself-and the thing as objectwhen it is set before the faculty of knowing and made present to it" (DK, 91). "The object is the correlative of a knowing subject... which precisely takes the name 'object' from the fact that it is presented to the mind" (DK, 93; my emphasis). When we describe something as known, conceived, imagined, heard, referred to, expressed, meant, thought about, remembered, etc., we are describing it as an object, in Maritain's sense. The subjectobject polarity is a fundamental fact that reflexive self-awareness reveals to us: A state of awareness has an object that is distinct from itself at least to the extent that the term of any relation is distinct from the relation, and just as we can describe awareness as a relation to something, we can describe that something as a term of a relation of awareness, an object. But we cannot describe objects solely as objects; for whatever we are first aware of, we must be aware of other than as an ob- THING AND OBJECT IN MARITAIN 23 ject. To be an object is to be a term of a knowledge relation; consequently, to be aware that X is an object is to have a knowledge relation whose term is the existence of a knowledge relation to X. But what are... (shrink)

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

Abstract:A number of philosophers of language have proposed that people do not have conceptual access to‘bare particulars’, or attribute‐free individuals (e.g. Wiggins, 1980). Individuals can only be picked out under some sortal, a concept which provides principles of individuation and identity. Many advocates of this view have argued thatobjectis not a genuine sortal concept. I will argue in this paper that a narrow sense of‘object’, namely the concept of any bounded, coherent, three‐dimensional physical object that moves as a whole (...) (Spelke, 1990) is a sortal for both infants and adults. Furthermore,objectmay be the infant's first sortal and more specific sortals such ascupanddogmay be acquired later in the first year of life. I will discuss the implications for infant categorization studies, trying to draw a conceptual distinction between a perceptual category and a sortal, and I will speculate on how a child may construct sortal concepts such ascupanddog. (shrink)

Bob Hale championed the view that some objects exist of necessity, most prominently, mathematical objects like numbers. In contrast, this chapter upholds radical contingentism, the view that no object exists necessarily, and seeks to undermine the idea that the best possible candidates for necessary existence, the natural numbers, exist necessarily, despite there being in fact many contingent objects. Even the best neo-Fregean arguments for the existence of natural numbers depend on assumptions a nominalist may (...) reject. A positive account of cardinalities as belonging to multitudes shows that every finite cardinality is exemplified only if there are two or more individuals, but that there are at least two individuals is not a necessary truth. Hence, even if numbers were admitted to abstract existence contingently upon their being exemplified—which the chapter denies—they would not exist necessarily. (shrink)

In this paper, we propose a conceptual-spaces model of numerical cognition, and more precisely, of representations generated by Approximate Number System. The model is an extended and improved version of our earlier result (Gemel A, Quinon P: The approximate numbers system and the treatment of vagueness in conceptual spaces. In: Lukowski L, Gemel A, Zukowski B (eds) Cognition, meaning and action. Jagiellonian-Lodz University Press, Kraków, pp 87–108, 2015), where only purely quantitative information was accounted for. We focused on the (...) idea that ANS evolved to detect numerosity in the input, and to abstract this information from all possible magnitude-related cues, such as size of compared objects, aggregate area of those objects, or density of their location. The idea is that when one sees a pile of apples, ANS acts as a “number sense”, informing one of the approximate quantity of apples. Consequently, the original model was very simplified, accounting only for one, uniform perceptual discrete visual input. With inspiration from computational models (ex. Dehaene S, Changeux JP: J Cogn Neurosci 5:390–407, 1993;, Lourenco SF, Longo MR: Psychol Sci 21(6):873–881, 2010) that can process more complex stimuli, we propose in this paper a conceptual-spaces model for non-uniform input. The improved version of the model accounts additionally for magnitude-related cues related to both “number sense” and “magnitude sense”. (shrink)