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)

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

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

ontology metaontology wright platonism fregean existence epistemology.

Fictionalism has long presented an attractive alternative to both heavy-duty realist and simple eliminativist views about entities such as properties, propositions, numbers, and possible worlds. More recently, a different alternative to these traditional views has been gaining popularity: a form of deflationism that holds that trivial arguments may lead us from uncontroversial premisses to conclude that the relevant entities exist — but where commitment to the entities is a trivial consequence of other claims we accept, not a posit to explain (...) what makes the relevant claims true. The deflationist’s trivial arguments, however, have been attacked by fictionalists, who suggest that the ontological conclusions we get from these arguments should not be taken as serious ontological assertions at all, but rather as implicitly in the context of a fiction or simulation. This paper examines the fictionalist’s criticisms of ‘easy’ arguments for numbers, properties, and other entities, and concludes that they beg the question against the deflationist and so do not undermine the deflationist’s position. Close attention to the argument also reveals a crucial disanalogy between overtly fictional discourse and discourse about numbers, properties, and so on, which undermines the case for fictionalism. Finally, I argue that the motivations for fictionalism (particularly those based in its ability to offer a good account of the discourse) are served as well or better by deflationism. Overall, this gives us reason to think that deflationism may provide a preferable approach for those looking for an alternative to both traditional realism and traditional eliminativism. (shrink)

In order to perfectly describe the world, it is not enough to speak truly. One must also use the right concepts - including the right logical concepts. One must use concepts that "carve at the joints", that give the world's "structure". There is an objectively correct way to "write the book of the world". Much of metaphysics, as traditionally conceived, is about the fundamental nature of reality; in the present terms, this is about the world's structure. Metametaphysics - inquiry into (...) the status of metaphysical questions - turns on structure. The question of whether ontological, causal, or modal questions are "substantive" is in large part a question of whether the world has ontological, causal, and modal structure - whether quantifiers, causal relations, and modal operators carve at the joints. (shrink)

In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...

Arguments that ordinary inanimate objects such as tables and chairs, sticks and stones, simply do not exist have become increasingly common and increasingly prominent. Some are based on demands for parsimony or for a non-arbitrary answer to the special composition question; others arise from prohibitions against causal redundancy, ontological vagueness, or co-location; and others still come from worries that a common sense ontology would be a rival to a scientific one. Until now, little has been done to address these arguments (...) in a unified and systematic way. Ordinary Objects is designed to fill this gap, demonstrating that the mistakes behind all of these superficially diverse eliminativist arguments may be traced to a common source. It aims to develop an ontology of ordinary objects subject to no such problems, providing perhaps the first sustained defense of a common sense ontology in two generations. The work done along the way addresses a number of major issues in philosophy of language and metaphysics, contributing to debates about analyticity, identity conditions, co-location and the grounding problem, vagueness, overdetermination, parsimony, and ontological commitment. In the end, the most important result of addressing these eliminativist arguments is not merely avoiding their conclusions; examining their failings also gives us reason to suspect that many apparent disputes in ontology are pseudo-debates. For it brings into question widely-held assumptions about which uses of metaphysical principles are appropriate, which metaphysical demands are answerable, and how we should go about addressing such fundamental questions as "What exists?". As a result, the work of Ordinary Objects promises to provide not only the route to a reflective understanding of our unreflective common-sense view, but also a better understanding of the proper methods and limits of metaphysics. (shrink)

This paper investigates the meta-ontological problem, what is the Julius Caesar objection? I distinguish epistemic, metaphysical and semantic versions. I argue that neo-Fregean and supervaluationist solutions to the Caesar objection fails because, amongst other flaws, they fail to determine which version of the problem is in play.

Stephen Schiffer presents a groundbreaking account of meaning and belief, and shows how it can illuminate a range of crucial problems regarding language, mind, knowledge, and ontology. He introduces the new doctrine of 'pleonastic propositions' to explain what the things we mean and believe are. He discusses the relation between semantic and psychological facts, on the one hand, and physical facts, on the other; vagueness and indeterminacy; moral truth; conditionals; and the role of propositional content in information acquisition and explanation. (...) This radical new treatment of meaning will command the attention of everyone who works on fundamental questions about language, and will attract much interest from other areas of philosophy. (shrink)

Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...) most powerful presentation yet of a neo-Fregean program. (shrink)

Recasting important questions about truth and objectivity in new and helpful terms, his book will become a focus in the contemporary debates over realism, and ...

In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ...

On the now dominant Quinean view, metaphysics is about what there is. Metaphysics so conceived is concerned with such questions as whether properties exist, whether meanings exist, and whether numbers exist. I will argue for the revival of a more traditional Aristotelian view, on which metaphysics is about what grounds what. Metaphysics so revived does not bother asking whether properties, meanings, and numbers exist (of course they do!) The question is whether or not they are fundamental.

It is my aim in this paper to show that the contemporary assimilation of essence to modality is fundamentally misguided and that, as a consequence, the corresponding conception of metaphysics should be given up. It is not my view that the modal account fails to capture anything which might reasonably be called a concept of essence. My point, rather, is that the notion of essence which is of central importance to the metaphysics of identity is not to be understood in (...) modal terms or even to be regarded as extensionally equivalent to a modal notion. The one notion is, if I am right, a highly refined version of the other; it is like a sieve which performs a similar function but with a much finer mesh. (shrink)

In recent analytic metaphysics, the view that ‘ordinary inanimate objects such as sticks and stones, tables and chairs, simply do not exist’ has been defended by some noteworthy writers. Thomasson opposes such revisionary ontology in favour of an ontology that is conservative with respect to common sense. The book is written in a straightforward, methodical and down-to-earth style. It is also relatively non-specialized, enabling the author and her readers to approach problems that are often dealt with in isolation in a (...) more unified way.Thomasson's arguments are mainly counter-attacks on six ‘eliminativist’ arguments against ordinary objects. A causal redundancy argument espoused by Trenton Merricks holds that to suppose that there are ordinary objects is to suppose that these objects have distinctive causal powers. However, the casual efficacy of a baseball, for example, is exhausted by that of some suitably arranged …. (shrink)

According to Gottlob Frege, his logicism died when it was discovered that the underlying theory of extensions is inconsistent. The neo-logicist attempts to found mathematics on other abstraction principles, such as the so-called Hume’s principle that two concepts have the same number if and only if they are equinumerous. This chapter discusses the state of neo-logicism, responding to various objections that have been raised against it.

First published in 1903, _Principles of Mathematics_ was Bertrand Russell’s first major work in print. It was this title which saw him begin his ascent towards eminence. In this groundbreaking and important work, Bertrand Russell argues that mathematics and logic are, in fact, identical and what is commonly called mathematics is simply later deductions from logical premises. Highly influential and engaging, this important work led to Russell’s dominance of analytical logic on western philosophy in the twentieth century.

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...) was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many. (shrink)

Early analytic philosophers like Carnap, Wittgenstein and Ryle regarded ontology as a branch of metaphysics that is either trivial or meaningless. But at present it is generally assumed that philosophy can make substantial discoveries about what kinds of things exist and about the essence of these kinds. My paper challenges this ontological turn. The currently predominant conceptions of the subject, at any rate, do not license the idea that ontology can provide distinctively philosophical insights into the constituents of reality. I (...) distinguish four main sources of analytic ontology—Strawson's descriptive metaphysics, Kripke's realist semantics, the Austro-Australian truth-maker principle, Quine's naturalistic conception of ontology—and indicate briefly why the first three do not rehabilitate ontology. In the remainder, I concentrate on the most influential and promising position. Quinean ontology seeks to bring out and reduce the ontological commitments of our best scientific theories through logical paraphrase. Against this programme, I argue that Quine's conception of ontological commitment is inadequate, and that his logical paraphrase cannot contribute to the exploration of reality, but at most to the clarification of our conceptual framework. (shrink)

Fictionalism and deflationism are two moderate meta-ontological positions that try to occupy a middle ground between the extremes of heavy-duty realism and hard-line eliminativism. Deflationists believe that the existence of certain entities (e.g.: numbers) can be established by means of ‘easy’ arguments—arguments that, supposedly, rely solely on uncontroversial premises and trivial inferences. Fictionalists, however, find easy arguments unconvincing. Amie Thomasson has recently argued that, in their criticism of easy arguments, fictionalists beg the question against deflationism and that the fictionalist alternative (...) interpretation of easy arguments is untenable. In this paper, I argue that both charges are unsubstantiated. Properly understood, the fictionalist’s objection to ‘easy’ arguments takes the form of a dilemma—either the premises of ‘easy’ arguments are not truly uncontroversial or the inferences on which they rely are not truly trivial. Moreover, I argue not only that, contrary to what Thomasson claims, the fictionalist’s interpretation of easy argument is tenable but that the fictionalist might, in fact, have a better explanation of the seemingly trivial nature of the inferences involved in easy arguments than the deflationist’s. (shrink)

How are philosophical questions about what kinds of things there are to be understood and how are they to be answered? This paper defends broadly Fregean answers to these questions. Ontological categories—such as object , property , and relation —are explained in terms of a prior logical categorization of expressions, as singular terms, predicates of varying degree and level, etc. Questions about what kinds of object, property, etc., there are are, on this approach, reduce to questions about truth and logical (...) form: for example, the question whether there are numbers is the question whether there are true atomic statements in which expressions function as singular terms which, if they have reference at all, stand for numbers, and the question whether there are properties of a given type is a question about whether there are meaningful predicates of an appropriate degree and level. This approach is defended against the objection that it must be wrong because makes what there depend on us or our language. Some problems confronting the Fregean approach—including Frege’s notorious paradox of the concept horse—are addressed. It is argued that the approach results in a modest and sober deflationary understanding of ontological commitments. (shrink)

This paper aims to determine what kind of problem Frege's famous “Julius Caesar problem” is. whether it is to be understood as the metaphysical problem of determining what kind of things abstract objects like numbers or value‐courses are, or as the epistemological problem of providing a means of recognizing these objects as the same again, or as the logical problem of providing abstract sortal concepts with a sharp delimitation in order to fulfill the law of excluded middle, or as the (...) semantic problem of fixing the referents of the corresponding abstract singular terms. It is argued that, for Frege, the Caesar problem is a bundle of related problems of which the semantic problem is the most basic one. (shrink)

Neo-Fregeanism in the philosophy of mathematics consists of two main parts: the logicist thesis, that mathematics (or at least branches thereof, like arithmetic) all but reduce to logic, and the platonist thesis, that there are abstract, mathematical objects. I will here focus on the ontological thesis, platonism. Neo-Fregeanism has been widely discussed in recent years. Mostly the discussion has focused on issues specific to mathematics. I will here single out for special attention the view on ontology which underlies the neo-Fregeans’ (...) claims about mathematical objects, and discuss this view in a broader setting. (shrink)

In an earlier paper in this journal I argued that deflationism is preferable to fictionalism as an alternative to both traditional realism and eliminativism. Gabriele Contessa questions this conclusion, denying that fictionalist arguments beg the question against easy ontological arguments, presenting a new argument against easy ontology, and suggesting a response to the challenge I raise for fictionalists. Below I respond to these points in turn. In so doing, I hope to clarify the broader theoretic orientation of easy ontology—in particular, (...) its rejection of a Quinean criterion of ontological commitment and its commitment to a form of functional pluralism about language. (shrink)

pt. 1. The proposition and its sense.--pt. 2. On logic and mathematics.

According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second-order logic is set theory in disguise. (shrink)

Ontology and the Ambitions of Metaphysics By HofweberThomasOxford University Press, 2016. xvi + 366 pp. £50.00.

The neo-Fregeans have argued that definition by abstraction allows us to introduce abstract concepts such as direction and number in terms of equivalence relations such as parallelism between lines and one-one correspondence between concepts. This paper argues that definition by abstraction suffers from the fact that an equivalence relation may not be sufficient to determine a unique concept. Frege’s original verdict against definition by abstraction is thus reinstated.

INTRODUCTION MATHEMATICAL logic differs from the traditional formal logic so markedly in method, and so far surpasses it in power and subtlety, ...