Online research in philosophy

Can a musical work be created? Some say ‘no’. But, we argue, there is no handbook of universally accepted metaphysical truths that they can use to justify their answer. Others say ‘yes’. They have to find abstract objects that can plausibly be identified with musical works, show that abstract objects of this sort can be created, and show that such abstract objects can persist. But, we argue, none of the standard views about what a musical work is allows musical works both to be created and to persist.

In this paper, the author shows how one can independently prove, within the theory of abstract objects, some of the most significant claims, hypotheses, and background assumptions found in Kripke's logical and philosophical work. Moreover, many of the semantic features of theory of abstract objects are consistent with Kripke's views — the successful representation, in the system, of the truth conditions and entailments of philosophically puzzling sentences of natural language validates certain Kripkean semantic claims about natural language.

In this paper, the authors discuss Frege''s theory of logical objects (extensions, numbers, truth-values) and the recent attempts to rehabilitate it. We show that the eta relation George Boolos deployed on Frege''s behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the eta relation to assert the existence of logical objects under certain highly restricted conditions, the theory of abstract objects uses unrestricted Comprehension for Logical Objects and banishes encoding (eta) formulas from Comprehension for Properties. The relative mathematical and philosophical strengths of the two theories are discussed. Along the way, new results in the theory of abstract objects are described, involving: (a) the theory of extensions, (b) the theory of directions and shapes, and (c) the theory of truth values.

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.

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible.The second section examines the problem as it was posed by Benacerraf in Mathematical Truth and the next section presents a way of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them.

Whether or not there are non-existent objects seems to be one of the more mysterious and speculative issues in ontology.1 To affirm that there are non-existent objects is to affirm that reality consists of two kinds of things, the existing and the non-existing. The existing contains all of what is in our space-time world, plus all abstract objects, if there are any. Most people, it seems fair to say, would think that this is all there is. For them the only real question in ontology can be what kinds of existing things there are. However, followers of Meinong maintain that this isn’t all there is. There is also another kind of things, those that do not exist. And to say this, the Meinongians continue, is to accept that reality is divided into two basic kinds of things, the existing and the non-existing. Whether or not reality contains two basic categories of things, existing and non-existing, or only one, existing, is what the debate about non-existent objects is all about. And as such it seems to be the most speculative of the debates in ontology. How could we human beings possibly decide it? One might think that to find out whether or not there are abstract objects is hard to decide, since they are not in space and time, causally inaccessible, unobservable, etc.. But whatever difficulty there might be to answer the question whether or not there are abstract objects, it has to be even harder to decide whether or not there are non-existent objects. Abstract objects, if there are any, at least..

Some recently-proposed counterexamples to the traditional definition of essential property do not require a separate logic of essence. Instead, the examples can be analysed in terms of the logic and theory of abstract objects. This theory distinguishes between abstract and ordinary objects, and provides a general analysis of the essential properties of both kinds of object. The claim ‘x has F necessarily’ becomes ambiguous in the case of abstract objects, and in the case of ordinary objects there are various ways to make the definition of ‘F is essential to x’ more fine-grained. Consequently, the traditional definition of essential property for abstract objects in terms of modal notions is not correct, and for ordinary objects the relationship between essential properties and modality, once properly understood, addresses the counterexample.

(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.

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.

Awareness is a two-place determinable relation some determinates of which are seeing, hearing, etc. Abstract objects are items such as universals and functions, which contrast with concrete objects such as solids and liquids. It is uncontroversial that we are sometimes aware of concrete objects. In this paper I explore the more controversial topic of awareness of abstract objects. I distinguish two questions. First, the Existence Question: are there any experiences that make their subjects aware of abstract objects? Second, the Grounding Question: if an experience makes its subject aware of an abstract object, in virtue of what does it do so? I defend the view that intuitions, specifically mathematical intuitions, sometimes make their subjects aware of abstract objects. In defending this view, I develop an account of the ground of intuitive awareness.