This essay explores the use of platonist and nominalist concepts, derived from the philosophy of mathematics and metaphysics, as a means of elucidating the debate on spacetime ontology and the spatial structures endorsed by scientific realists. Although the disputes associated with platonism and nominalism often mirror the complexities involved with substantivalism and relationism, it will be argued that a more refined three-part distinction among platonist/nominalist categories can nonetheless provide unique insights into the core assumptions that underlie spatial (...) ontologies, but it also assists in critiquing alternative uses of nominalism, platonism, and both ontic and epistemic structural realism. (shrink)

I introduce an argument for Platonism based on intra-mathematical explanation: the explanation of one mathematical fact by another. The argument is important for two reasons. First, if the argument succeeds then it provides a basis for Platonism that does not proceed via standard indispensability considerations. Second, if the argument fails it can only do so for one of three reasons: either because there are no intra-mathematical explanations, or because not all explanations are backed by dependence relations, or because (...) some form of noneism-the view according to which non-existent entities possess properties and stand in relations-is true. The argument thus forces a choice between nominalism without noneism, intra-mathematical explanation and a backing conception of explanation. You can have any two, but not all three. (shrink)

As part of the development of an epistemology for mathematics, some Platonists have defended the view that we have (i) intuition that certain mathematical principles hold, and (ii) intuition of the properties of some mathematical objects. In this paper, I discuss some difficulties that this view faces to accommodate some salient features of mathematical practice. I then offer an alternative, agnostic nominalist proposal in which, despite the role played by mathematical intuition, these difficulties do not emerge.

I here investigate whether there is any version of the principle of charity both strong enough to conflict with an error-theoretic version of nominalism (EN) about abstract objects, and supported by the considerations adduced in favour of interpretive charity in the literature. I argue that in order to be strong enough, the principle, which I call (Charity), would have to read, “For all expressions e, an acceptable interpretation must make true a sufficiently high ratio of accepted sentences containing e”. I (...) next consider arguments based on Davidson's intuitive cases for interpretive charity, the reliability of perceptual beliefs, and the reliability of “non-abstractive inference modes”, and conclude that none support (Charity). I then propose a diagnosis of the view that there must be some universal principle of charity ruling out (EN). Finally, I present a reason to think (Charity) is false, namely, that it seems to exclude the possibility of such disagreements as that between nominalists and realists. (shrink)

Nominalists are confronted with a grave difficulty: if abstract objects do not exist, what explains the success of theories that invoke them? In this paper, I make headway on this problem. I develop a formal language in which certain platonistic claims about properties and certain nominalistic claims can be expressed, develop a formal language in which only certain nominalistic claims can be expressed, describe a function mapping sentences of the first language to sentences of the second language, and prove some (...) facts about that function and facts about some sound logics for those languages. In doing so, I prove that, given some plausible metaphysical assumptions, a large class of sentences about properties of concrete objects are “safe” on nominalistic grounds. Whenever some true sentences about concrete objects and some sentences belonging to this class that are true according to platonists collectively entail a conclusion about concrete objects, some nominalistically acceptable sentences are true and entail the same conclusion. Because the proof can itself be formulated without abstract objects, it provides a nominalistic explanation of the success of theories that invoke properties of concrete objects. (shrink)

This rich book differs from much contemporary philosophy of mathematics in the author’s witty, down to earth style, and his extensive experience as a working mathematician. It accords with the field in focusing on whether mathematical entities are real. Franklin holds that recent discussion of this has oscillated between various forms of Platonism, and various forms of nominalism. He denies nominalism by holding that universals exist and denies Platonism by holding that they are concrete, not abstract - looking (...) to Aristotle for inspiration. (shrink)

A nominalist account of statements of number (e.g., ‘There are exactly two moons of Mars’) is rebutted.

A problem for Aristotelian realist accounts of universals (neither Platonist nor nominalist) is the status of those universals that happen not to be realised in the physical (or any other) world. They perhaps include uninstantiated shades of blue and huge infinite cardinals. Should they be altogether excluded (as in D.M. Armstrong's theory of universals) or accorded some sort of reality? Surely truths about ratios are true even of ratios that are too big to be instantiated - what is the (...) truthmaker of such truths? It is argued that Aristotelianism can answer the question, but only a semi-Platonist form of it. (shrink)

In this paper, I introduce an intrinsic account of the quantum state. This account contains three desirable features that the standard platonistic account lacks: (1) it does not refer to any abstract mathematical objects such as complex numbers, (2) it is independent of the usual arbitrary conventions in the wave function representation, and (3) it explains why the quantum state has its amplitude and phase degrees of freedom. -/- Consequently, this account extends Hartry Field’s program outlined in Science Without Numbers (...) (1980), responds to David Malament’s long-standing impossibility conjecture (1982), and establishes an important first step towards a genuinely intrinsic and nominalistic account of quantum mechanics. I will also compare the present account to Mark Balaguer’s (1996) nominalization of quantum mechanics and discuss how it might bear on the debate about “wave function realism.” In closing, I will suggest some possible ways to extend this account to accommodate spinorial degrees of freedom and a variable number of particles (e.g. for particle creation and annihilation). -/- Along the way, I axiomatize the quantum phase structure as what I shall call a “periodic difference structure” and prove a representation theorem as well as a uniqueness theorem. These formal results could prove fruitful for further investigation into the metaphysics of phase and theoretical structure. -/- (For a more recent version of this paper, please see "The Intrinsic Structure of Quantum Mechanics" available on PhilPapers.). (shrink)

Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti- platonist argument proposed by Hartry Field avoids both horns (...) of their dilemma. (shrink)

In this paper, I argue that all expressions for abstract objects are meaningless. My argument closely follows David Lewis’ argument against the intelligibility of certain theories of possible worlds, but modifies it in order to yield a general conclusion about language pertaining to abstract objects. If my Lewisian argument is sound, not only can we not know that abstract objects exist, we cannot even refer to or think about them. However, while the Lewisian argument strongly motivates nominalism, it also undermines (...) certain nominalist theories. (shrink)

Should philosophers prefer simpler theories? Huemer (Philos Q 59:216–236, 2009) argues that the reasons to prefer simpler theories in science do not apply in philosophy. I will argue that Huemer is mistaken—the arguments he marshals for preferring simpler theories in science can also be applied in philosophy. Like Huemer, I will focus on the philosophy of mind and the nominalism/Platonism debate. But I want to engage with the broader issue of whether simplicity is relevant to philosophy.

Recent debates about mathematical ontology are guided by the view that Platonism's prospects depend on mathematics' explanatory role in science. If mathematics plays an explanatory role, and in the right kind of way, this carries ontological commitment to mathematical objects. Conversely, the assumption goes, if mathematics merely plays a representational role then our world-oriented uses of mathematics fail to commit us to mathematical objects. I argue that it is a mistake to think that mathematical representation is necessarily ontologically innocent (...) and that there is an argument from mathematics' representational capacity to Platonism. Given that it is common ground between the Platonist and nominalist that mathematics plays a representational role in science, this representationalist argument is to be preferred over the explanatory, or enhanced, indispensability argument. (shrink)

In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers (...) to exist and the teleological why-question asks for what purpose numbers exist. I argue that in a psychologist constructivist account, in which numbers are understood to exist as referents of a particular type of culturally shared concepts, both why-questions can get plausible answers. (shrink)

Platonists affirm the existence of abstract mathematical objects, and Nominalists deny the existence of abstract mathematical objects. While there are standard arguments in favor of Nominalism, these arguments fail to account for the necessity of Nominalism. Furthermore, these arguments do nothing to explain why Nominalism is true. They only point to certain theoretical vices that might befall the Platonist. The goal of this paper is to formulate and defend a simple, valid argument for the necessity of Nominalism that seeks to (...) precisify the widespread intuition that mathematical objects are somehow ‘spooky’ or ‘mysterious’. (shrink)

Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for (...) example, is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to (...) Baker :223–238, 2005; Br J Philos Sci 60:611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position :779–793, 2017a). We pick up the circularity problem brought up by Leng Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu :13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic. (shrink)

Nicholas of Cusa (1401-1464) was active during the Renaissance, developing adventurous ideas even while serving as a churchman. The religious issues with which he engaged – spiritual, apocalyptic and institutional – were to play out in the Reformation. These essays reflect the interests of Cusanus but also those of Gerald Christianson, who has studied church history, the Renaissance and the Reformation. The book places Nicholas into his times but also looks at his later reception. The first part addresses institutional issues, (...) including Schism, conciliarism, indulgences and the possibility of dialogue with Muslims. The second treats theological and philosophical themes, including nominalism, time, faith, religious metaphor, and prediction of the end times. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world (...) and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

Realism about musical works is often tied to some type of Platonism. Nominalism, which posits that musical works exist and that they are concrete objects, goes with ontological realism much less often than Platonism: there is a long tradition which holds human-created objects (artifacts) to be mind-dependent. Musical Platonism leads to the well-known paradox of the impossibility of creating abstract objects, and so it has been suggested that only some form of nominalism becoming dominant in the ontology (...) of art could cause a great change in the field and open up new possibilities. This paper aims to develop a new metaontological view starting from the widely accepted claim that musical works are created. It contends that musical works must be concrete and created objects of some sort, but, nevertheless, they are mind-independent, and we should take the revisionary methodological stance. Although musical works are artifacts, what people think about them does not determine what musical works are. Musical works are similar to natural objects in the following sense: semantic externalism applies to the term ‘musical work’ because, firstly, they possess a shared nature, and, secondly, we can be mistaken about what they are. (shrink)

In the ontology of music the Aristotelian theory of musical works is the view that musical works are immanent universals. The Aristotelian theory (hereafter Musical Aristotelianism) is an attractive and serviceable hypothesis. However, it is overlooked as a genuine competitor to the more well-known theories of Musical Platonism and nominalism. Worse still, there is no detailed account in the literature of the nature of the universals that the Aristotelian identifies musical works with. In this paper, I argue that the (...) best version of Musical Aristotelianism identifies musical works with structural universals. I first motivate the view by outlining its explanatory benefits. I then argue that Musical Aristotelianism is preferable to Musical Platonism and present a novel account of musical works as structural universals by developing D. M. Armstrong’s theory of structural universals. I discuss the consequences of Musical Aristotelianism with respect to on-going issues in debates about musical works and defend the view against an influential objection, concluding that Musical Aristotelianism is a genuine competitor in debates about the nature of musical works. (shrink)

In a lot of domains in metaphysics the tacit assumption has been that whichever metaphysical principles turn out to be true, these will be necessarily true. Let us call necessitarianism about some domain the thesis that the right metaphysics of that domain is necessary. Necessitarianism has flourished. In the philosophy of maths we find it held that if mathematical objects exist, then they do of necessity. Mathematical Platonists affirm the necessary existence of mathematical objects (see for instance Hale and Wright (...) 1992 and 1994; Wright 1983 and 1988; Schiffer 1996; Resnik 1997; Shapiro 1997 and Zalta 1988) while mathematical nominalists, usually in the form of fictionalists, hold that necessarily such objects fail to exist (see for instance Balaguer 1996 and 1998; Rosen 2001 and Yablo 2005). In metaphysics more generally, until recently it was more or less assumed that whatever the right account of composition—the account of under what conditions some xs compose a y—that account will be necessarily true (for a discussion of theories of composition see Simons 1987 and van Inwagen 1987 and 1990; the modal status of the composition relation is explicitly addressed in Schaffer 2007; Parsons 2006 and Cameron 2007). Similarly, it has generally been assumed that whatever the right account is of the nature of properties, whether they be universals, tropes, or whether nominalism is true, that account will be necessarily true (though see Rosen 2006 for a recent suggestion to the contrary). In considering theories of persistence it has been widely held that whether objects endure or perdure through time is a matter of necessity (Sider 2001; though see Lewis 1999 p227 who defends contingent perdurantism). And with respect to theories of time it is frequently held that whichever of the A- or B-theory is true is necessarily true. A-theorists often argue that there is time in a world only if the A-theory is true at that world (see for instance McTaggart 1903; Markosian 2004; Bigelow 1996; Craig 2001) while B-theorists often argue that the A-theory is internally inconsistent (Smart 1987; Mellor 1998; Savitt 2000 and Le Poidevin 1991). Once again, we find a few recent contingentist dissenters. Bourne (2006) suggests that it is a contingent feature of time that it is tensed, and thus that the A-theory is contingently true. Worlds in which there exist only B-theoretic properties are worlds with time, it is just that time in those worlds time is radically different to the way it is actually. Other defenders of the B-theory, though not expressly contingentists, do offer arguments against versions of the A-theory that try to show that such A-theories theories are inconsistent with the actual laws of nature (see for instance Saunders 2002 and Callender 2000); these arguments, at least, leave room for the possibility that the A-theories in question are contingently false (at least on the assumption that the laws of nature are themselves contingent, an assumption that not everyone accepts). Despite some notable exceptions, necessitarianism has flourished in many, if not most, domains in metaphysics. One such exception is Lewis’ famous defence of Humean supervenience as a contingent claim about our world. Lewis does not argue that necessarily, the supervenience base for all matters of fact in a world is nothing but a vast mosaic of local matters of particular fact. Rather, he thinks that we have reason to think that our world is one in which Humean supervenience holds (see Lewis 1986 p9-10 and 1994). Another exception to the necessitarian orthodoxy is to be found in the lively debate about the modal status of the laws of nature. Here, if anything, contingentism has been the dominating force, with it generally being held that there are possible worlds in which different laws of nature hold (this view is defended by, among others, Lewis 1986 and 2010; Schaffer 2005 and Sidelle 2002). Necessitarian dissenters hold that the laws of nature are necessary, frequently because they think it is necessary that fundamental properties have the causal or nomic profiles they do (see for instance Shoemaker 1980 and 1988; Swoyer 1982; Bird 1995; Ellis and Lierse 1994). Nevertheless, when it comes to thinking about the nature of the laws themselves, the necessitarian presumption is back on firm footing. Though there is disagreement about whether the laws are generalisations that feature in the most virtuous true axiomatisation of all the particular matters of fact (often known as the Humean view of laws and defended by Ramsey 1978; Lewis 1986 and Beebee 2000) or whether laws are relations of necessity that hold between universals (a view defended by Armstrong 1983; Dretske 1977; Tooley 1977 and Carroll 1990) no one has seriously suggested that it might be a contingent matter which of these is the right account of laws. The necessitarian orthodoxy is not surprising since metaphysics is largely an a priori process. While a priori reasoning may be used to determine whether a proposition is necessary or contingent, it is not well placed (in the absence of a posteriori evidence) to determine whether a contingent proposition is actually true or false. Since metaphysicians aim to tell us which principles are true in which worlds, on the face of it the discovery that metaphysical principles are contingent seems to make part of the task of metaphysics epistemically intractable. In what follows I consider two reasons one might end up embracing contingentism and whether this would lead one into epistemic difficulty. The following section considers a route to contingent metaphysical truths that proceeds via a combination of conceptual necessities and empirical discoveries. Section 3 considers whether there might be synthetic contingent metaphysical truths, and the final section raises the question of whether if there were such truths we would be well placed to come to know them. (shrink)

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...) this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. (shrink)

My aim in this chapter is to extend the Realist account of the foundations of linguistics offered by Postal, Katz and others. I first argue against the idea that naive Platonism can capture the necessary requirements on what I call a ‘mixed realist’ view of linguistics, which takes aspects of Platonism, Nominalism and Mentalism into consideration. I then advocate three desiderata for an appropriate ‘mixed realist’ account of linguistic ontology and foundations, namely (1) linguistic creativity and infinity, (2) (...) linguistics as a theory of types (and not tokens) and (3) independence but structural respect between language and the linguistic competence thereof. My own brand of mixed realism, what I call ante rem realism, is defended along the lines of an ante rem or non-eliminative structuralism, the likes of which has been offered for mathematics by Resnik (1997) and Shapiro (1997). In other words, grammars describe a mind-independent (but not necessarily unconnected) linguistic reality in terms of linguistic patterns or structures also known as natural languages. I further amend this picture to allow for the possibility of a naturalistic account of language acquisition and evolution by arguing against a particular view of the type-token distinction. (shrink)

The answer I shall sketch is not mine. Nor, as far as I can tell, is it an answer to be found in the voluminous literature inspired by Kripke’s work. Many of the elements of the answer are to be found in the writings of Wittgenstein and his Austro-German predecessors, Martinak, Husserl, Marty, Landgrebe and Bühler. Within this Austro-German tradition we may distinguish between a strand which is Platonist and anti-naturalist and a strand which is nominalist and naturalist. Thus (...) Husserl’s account of what he calls “directly referring” uses of singular terms invokes senses or individual concepts, albeit simple, not descriptive senses. But the account of reference fixing and reference given by Landgrebe, Bühler and Wittgenstein rejects senses.1 I confine further reference to these writers to footnotes since my aim here is to develop and unify some of their suggestions, in particular by comparing them with more recent work (cf. Mulligan 1997). (shrink)

Mathematics appears to play an explanatory role in science. This, in turn, is thought to pave a way toward mathematical Platonism. A central challenge for mathematical Platonists, however, is to provide an account of how mathematical explanations work. I propose a property-based account: physical systems possess mathematical properties, which either guarantee the presence of other mathematical properties and, by extension, the physical states that possess them; or rule out other mathematical properties, and their associated physical states. I explain why (...) Platonists should accept that physical systems have mathematical properties, and why a property based account is better than existing accounts of mathematical explanation. I close by considering whether nominalists can accept the view I propose here. I argue that they cannot. (shrink)

This essay presents an alternative to contemporary substantivalist and relationist interpretations of quantum gravity hypotheses by means of an historical comparison with the ontology of space in the seventeenth century. Utilizing differences in the spatial geometry between the foundational theory and the theory derived from the foundational, in conjunction with nominalism and platonism, it will be argued that there are crucial similarities between seventeenth century and contemporary theories of space, and that these similarities reveal a host of underlying conceptual (...) issues that the substantival/relational dichotomy fails to distinguish. (shrink)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key (...) intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

Noam Chomsky’s well-known claim that linguistics is a “branch of cognitive psychology” has generated a great deal of dissent—not from linguists or psychologists, but from philosophers. Jerrold Katz, Scott Soames, Michael Devitt, and Kim Sterelny have presented a number of arguments, intended to show that this Chomskian hypothesis is incorrect. On both sides of this debate, two distinct issues are often conflated: (1) the ontological status of language and (2) the relation between psychology and linguistics. The ontological issue is, I (...) will argue, not the relevant issue in the debate. Even if this Chomskian position on the ontology of language is false, linguistics may still be a subfield of psychology if the relevant methods in linguistic theory construction are psychological. Two options are open to the philosopher who denies Chomskian conceptualism: linguistic nominalism or linguistic platonism. The former position holds that syntactic, semantic, and phonological properties are primarily properties, not of mental representations, but rather of public languagesentence tokens; The latter position holds that the linguistic properties are properties of public language sentence types. I will argue that both of these positions are compatible with Chomsky’s claim that linguistics is a branch of psychology, and the arguments that have been given for nominalism and platonism do not establish that linguistics and psychology are distinct disciplines. (shrink)

Plato's philosophy is in line with the pre-Socratics, sophists and artistic traditions that underlie Greek education, in a new framework, defined by dialectics and the theory of Ideas. For Plato, knowledge is an activity of the soul, affected by sensible objects, and by internal processes. Platonism has its origins in Plato's philosophy, although it is not to be confused with it. According to Platonism, there are abstract objects (a notion different from that of modern philosophy that exists in (...) another realm distinct from both the external sensible world and the internal world of consciousness, and is the opposite of nominalism). An essential distinction for Plato in his philosophy is the theory of Forms, the distinction between perceptible but unintelligible reality (science) and imperceptible but intelligible reality (mathematics) Geometry was Plato's main motivation, and this shows the influence of Pythagoras. Forms are perfect archetypes whose real objects are imperfect copies. -/- DOI: 10.13140/RG.2.2.21536.05121. (shrink)

For Rudolf Carnap the question ‘Do numbers exist?’ does not have just one sense. Asked from within mathematics, it has a trivial answer that could not possibly divide philosophers of mathematics. Asked from outside of mathematics, it lacks meaning. This paper discusses Carnap ’s distinction and defends much of what he has to say.

What is the role of mathematics in scientific explanations? Does it/can it play an explanatory part? This question is at the core of the recent ontological debate in the philosophy of mathematics. My aim in this paper is to argue that the two main approaches to this problem found in recent literature (i.e. the top-down and the bottom-up approaches) are both deeply problematic. This has an important implication for the dispute over the existence of mathematical entities: to make progress possible (...) in this debate, we either have to find a different approach to the problem of the role of mathematics in scientific explanations (one that is not affected by the problems that, as I argue in this paper, plague the existing approaches), or we need to recast it in different terms. (shrink)

Gardeners, poets, lovers, and philosophers are all interested in the redness of roses; but only philosophers wonder how it is that two different roses can share the same property. Are red things red because they resemble each other? Or do they resemble each other because they are red? Since the 1970s philosophers have tended to favour the latter view, and held that a satisfactory account of properties must involve the postulation of either universals or tropes. But Gonzalo Rodriguez-Pereyra revives the (...) dormant alternative theory of resemblance nominalism, showing first that it can withstand the attacks of such eminent opponents as Goodman and Armstrong, and then that there are reasons to prefer it to its rival theories. The clarity and rigour of his arguments will challenge metaphysicians to rethink their views on properties. (shrink)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

Dispositional Essentialism, as commonly conceived, consists in the claims that at least some of the fundamental properties essentially confer certain causal-nomological roles on their bearers, and that these properties give rise to the natural modalities. As such, the view is generally taken to be committed to a realist conception of properties as either universals or tropes, and to be thus incompatible with nominalism as understood in the strict sense. Pace this common assumption of the ontological import of Dispositional Essentialism, the (...) aim of this paper is to explore a nominalist version of the view, Austere Nominalist Dispositional Essentialism. The core features of the proposed account are that it eschews all kinds of properties, takes certain predicative truths as fundamental, and employs the so-called generic notion of essence. As I will argue, the account is significantly closer to the core idea behind Dispositional Essentialism than the only nominalist account in the vicinity of Dispositional Essentialism that has been offered so far—Ann Whittle’s Causal Nominalism—and is immune to crucial problems that affect this view. (shrink)

‘Nominalism’ refers to a family of views about what there is. The objects we are familiar with (e.g. hands, laptops, cookies, and trees) can be characterized as concrete and particular. Nominalists agree that there are such things. But one group of nominalists denies that anything is non-particular and another group denies that anything is non-concrete. These two sorts of nominalism, referred to as ‘nominalism about universals’ and ‘nominalism about abstract objects’, have common motivations in contemporary philosophy.

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about (...) natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism. (shrink)

The paper focuses on Nominalism in history, its application, and its historiographical implications. By engaging with recent scholarship as well as classic works, a survey of Nominalism’s role in the discipline of history is made; such examination is timely, since it has been done but scantily in a purely historical context. In the light of recent theoretical works, which often display aporias over the nature and method of historical enquiry, the paper offers new considerations on historical theory, which in the (...) author’s view may solve some of the contradictions that have surfaced in recent times. The Nominalistic stance is argued against by disputing theorists such as Paul Veyne, who has made a case for Nominalism in history. A brief philosophical section introduces Nominalism in its metaphysical dimension and the discussion is speedily brought to its significance for history. The paper also proposes a solution to the misconstrued yet too often vague application of scientism in history, and offers theoretical grounds that might solve some of the ‘stormy grounds’ historiography finds itself today. Articles by Marcel Gauchet and History and Theory’s Anton Froeyman and Bert Leuridan are engaged with, as well as Murray Murphy’s books on the philosophy of history. Works by Georg Gadamer, Marc Bloch, Benedetto Croce, Hyppolite Taine, and Anthony Grafton crucially inform the discussion and brace the consequential conclusion. (shrink)

Many philosophers of mathematics are attracted by nominalism – the doctrine that there are no sets, numbers, functions, or other mathematical objects. John Burgess and Gideon Rosen have put forward an intriguing argument against nominalism, based on the thought that philosophy cannot overrule internal mathematical and scientific standards of acceptability. I argue that Burgess and Rosen’s argument fails because it relies on a mistaken view of what the standards of mathematics require.

Bertrand Russell argued that any attempt to get rid of universals in favor of resemblances fails. He argued that no resemblance theory could avoid postulating a universal of resemblance without falling prey to a vicious infinite regress. He added that admitting such a universal of resemblance made it pointless to avoid other universals. In this paper I defend resemblance nominalism from both of Russell's points by arguing that (a) resemblance nominalism can avoid the postulation of a universal of resemblance without (...) falling into a vicious infinite regress, and (b) even if resemblance nominalism had to admit a universal of resemblance, this would not make it pointless to avoid postulating other universals. (shrink)

In the opening chapter of the Monologion, Anselm offers an intriguing proof for the existence of a Platonic form of goodness. This proof is extremely interesting, both in itself and for its place in the broader argument for God’s existence that Anselm develops in the Monologion as a whole. Even so, it has yet to receive the scholarly attention that it deserves. My aim in this article is to begin correcting this state of affairs by examining Anslem’s proof in some (...) detail. In particular, I aim to clarify the proof’s structure, motivate and explain its central premises, and begin the larger project of evaluating its overall success as an argument for Platonism about goodness. (shrink)

In this dissertation I consider the merits of certain nominalist accounts of phenomena related to the character of ordinary objects. What these accounts have in common is the fact that none of them is an error theory about standard cases of predication and none of them deploys God or uniquely theistic resources in its explanatory framework. -/- The aim of the dissertation is to answer the following questions: -/- • What is the best nominalist account on offer? • (...) How might it be improved? • Does it ultimately succeed? -/- I will argue that while so-called trope theory is the best account on offer, it can be significantly improved—or replaced—by a novel version of nominalism that is modeled after trope theory. Ultimately, however, I will argue that even the novel version fails. -/- The dissertation unfolds as follows. In Chapter 1, I introduce Austere Nominalism (AN), which is perhaps the most extreme version of nominalism that falls within the scope of the dissertation. AN is often described as the view that there exist only concrete particulars. According to AN, it is unnecessary to posit any entities other than ordinary objects—turkeys, tables, and the like—in order to account for explananda related to the character of those objects. (Such explananda include the phenomenon of attribute agreement, of attribute possession, of true subject-predicate sentences, etc.) In this chapter I argue that AN fails to provide an adequate account of these explananda. In addition, introducing and criticizing AN serves an important heuristic role for the rest of the dissertation. To understand this role, we must distinguish between the basic explanatory strategy deployed by the austere nominalist and the type of explananda for which she deploys that strategy. The austere nominalist deploys the strategy to account for the character of ordinary objects. As I argue in Chapter 1, this deployment is a failure. As I go on to show in Chapter 2, the widespread rejection AN has led to a variety of rival accounts of the character of ordinary objects. In rejecting AN, however, these accounts also tacitly reject its basic explanatory strategy. Thus goes the baby with the bathwater, since, arguably, there are some attractive features of AN’s basic explanatory strategy. Indeed, those who defend the most prominent version of nominalism—trope theory—seem to overlook the advantages of AN’s basic strategy, and by so doing, make an unnecessary concession to the realist. Or so I argue in Chapter 3. And, as I will argue in Chapter 4, the strongest version of nominalism is a novel account, modeled after trope theory, that deploys AN’s basic strategy at a more fundamental level than that of ordinary objects. This novel account—troper theory—is closer in spirit to AN than is traditional trope theory. (Thus, AN serves as a foil for the discussion of other nominalist views.) Finally, in the Afterword I indicate how troper theory is equally vulnerable to some of the traditional objections that plague trope theory. Thus, if you are not convinced that traditional objections to trope theory are conclusive and you want to be a nominalist, then you should abandon trope theory and adopt troper theory. If you take traditional objections against trope theory to have significant force, then you should reject both theories. (shrink)

In my book *Resemblance Nominalism* I argued that the truthmakers of ´a and b resemble each other´ are just a and b. In his "Resemblance Nominalism and counterparts" Alexander Bird objects to my claim that the truthmakers of ´a and b resemble each other´ are just a and b. In this paper I respond to Bird´s objections.

Platonism and Christian Thought in Late Antiquity examines the various ways in which Christian intellectuals engaged with Platonism both as a pagan competitor and as a source of philosophical material useful to the Christian faith. The chapters are united in their goal to explore transformations that took place in the reception and interaction process between Platonism and Christianity in this period. -/- The contributions in this volume explore the reception of Platonic material in Christian thought, showing that (...) the transmission of cultural content is always mediated, and ought to be studied as a transformative process by way of selection and interpretation. Some chapters also deal with various aspects of the wider discussion on how Platonic, and Hellenic, philosophy and early Christian thought related to each other, examining the differences and common ground between these traditions. -/- Platonism and Christian Thought in Late Antiquity offers an insightful and broad ranging study on the subject, which will be of interest to students of both philosophy and theology in the Late Antique period, as well as anyone working on the reception and history of Platonic thought, and the development of Christian thought. (shrink)

The idea of “material plenitude” has been gaining traction in recent discussions of the metaphysics of material objects. My main goal here is to show that this idea may have important dialectical implications for the metaphysics of properties – more specifically, that it provides nominalists with new resources in their attempt to reject an ontology of universals. I will recapitulate one of the main arguments against nominalism – due to David Armstrong – and show how plenitude helps the nominalist (...) overcome the argument. (shrink)

ABSTRACT: Sellars formulated his thesis of 'psychological nominalism' in two very different ways: (1) most famously as the thesis that 'all awareness of sorts…is a linguistic affair', but also (2) as a certain thesis about the 'psychology of the higher processes'. The latter thesis denies the standard view that relations to abstract entities are required in order to explain human thought and intentionality, and asserts to the contrary that all such mental phenomena can in principle ‘be accounted for causally' without (...) any use of normative terms in the explanation. Recent 'Hegelian Sellarsians' such as Rorty, McDowell, and Brandom have argued that the holistic, normative themes in (1) support various non-realist or rather (German) 'idealist but common-sense realist' outlooks. By contrast, Sellars' own defenses of (2) reveal psychological nominalism itself to be a naturalistic empiricism intended to sustain the normative-holistic themes in (1) within an exhaustively scientific naturalist conception of reality. (shrink)

Being able to apply grue-like predicates would allow one to instantly solve an infinite number of mysteries (historical, scientific, etc.). In this paper I’ll give a couple of examples. It is still a mystery whether George Mallory and Andrew Irvine managed to reach the summit of Mount Everest in 1924. The predicate “greverest” applies to an object if either the object is green and Mount Everest was scaled in 1924, or the object is not green and Mount Everest was not (...) scaled in 1924. The predicate “greverest” is interdefinable with the predicate “green”: the predicate "green" applies to an object if either the object is greverest and Mount Everest was scaled in 1924, or the object is not greverest and Mount Everest was not scaled in 1924. The predicate "non-greverest" applies to an object if either the object is non-green and Mount Everest was scaled in 1924, or the object is green and Mount Everest was not scaled in 1924. The predicate "non-green" applies to an object if either the object is non-greverest and Mount Everest was scaled in 1924, or the object is greverest and Mount Everest was not scaled in 1924. We know that the famous diamond called Golden Jubilee is not green. If someone (a greverest-speaker) informed us that the Golden Jubilee diamond is greverest, we would automatically come to know Mount Everest was not scaled in 1924. According to the definitions, a greverest object can only be either 1) green (in case Mount Everest was scaled in 1924), or 2) non-green (in case Mount Everest was not scaled in 1924). Since we know that the Golden Jubilee diamond is not green, if we come to know that the Golden Jubilee diamond is also greverest, we would automatically know from the definition of “greverest” that we are faced with case 2, the case in which the object is both non-green and greverest, and Mount Everest was not scaled in 1924. Vice versa, if someone informed us that the Golden Jubilee diamond is not greverest, we would automatically come to know that Mount Everest was scaled in 1924. According to the definitions, a non-greverest object can only be either 1) non-green (in case Mount Everest was scaled in 1924), or 2) green (in the case Mount Everest was not scaled in 1924). Since we know that the Golden Jubilee diamond is not green, if we come to know that the Golden Jubilee diamond is also non-greverest, we would automatically know from the definition of “non-greverest” that we are faced with case 1, the case in which the object is both non-green and non-greverest, and Mount Everest was scaled in 1924. At the moment no one has yet been able to convince the scientific community that he can correctly determine whether the Golden Jubilee diamond is greverest. I am skeptical that anyone will demonstrate to the scientific community that he can correctly determine whether a specific object is greverest; but I would be happy if someone could demonstrate to the scientific community that he can correctly determine whether an object is greverest: that would be a good news for historical studies. The paper continues with a second example. (shrink)

The object of this paper is to provide a solution to Nelson Goodman’s Imperfect Community difficulty as it arises for Resemblance Nominalism, the view that properties are classes of resembling particulars. The Imperfect Community difficulty consists in that every two members of a class resembling each other is not sufficient for it to be a class such that there is some property common to all their members, even if ‘x resembles y’ is understood as ‘x and y share some property’. (...) In the paper I explain and criticise several solutions to the difficulty. Then I develop my own solution, which is not subject to the objections I make to the other solutions, and which accords completely with the basic tenets of Resemblance Nominalism. (shrink)