Abstract Objects

  It is widely supposed that every entity falls into one of two categories: Some are concrete; the rest abstract. The distinction is supposed to be of fundamental significance for metaphysics and epistemology.

 

 The abstract/concrete distinction has a curious status in contemporary philosophy. It is widely agreed that the distinction is of fundamental importance. And yet there is no standard account of how it should be drawn. There is a great deal of agreement about how to classify certain paradigm cases. Thus it is universally acknowledged that numbers and the other objects of pure mathematics are abstract (if they exist), whereas rocks and trees and human beings are concrete. Some clear cases of abstracta are classes, propositions, concepts, the letter ‘A’, and Dante’s Inferno. Some clear cases of concreta are stars, protons, electromagnetic fields, the chalk tokens of the letter ‘A’ written on a certain blackboard, and James Joyce’s copy of Dante’s Inferno.

 

The challenge is to say what underlies this dichotomy, either by defining the terms explicitly, or by embedding them in a theory that makes their connections to other important categories more explicit. In the absence of such an account, the philosophical significance of the contrast remains uncertain. We may know how to classify things as abstract or concrete by appeal to intuition. But in the absence of theoretical articulation, it will be hard to know what (if anything) hangs on the classification.

 

It should be stressed that there need not be one single “correct” way of explaining the abstract/concrete distinction. Any plausible account will classify the paradigm cases in the standard way, and any interesting account will draw a clear and philosophically significant line in the domain of objects. Yet there may be many equally interesting ways of accomplishing these two goals, and if we find ourselves with two or more accounts that do the job rather well, there will be no point in asking which corresponds to the real abstract/concrete distinction. This illustrates a general point: when technical terminology is introduced in philosophy by means of examples but without explicit definition or theoretical elaboration, the resulting vocabulary is often vague or indeterminate in reference. In such cases, it is normally pointless to seek a single correct account. A philosopher may find himself asking questions like, ‘What is idealism?’ or ‘What is a substance?’ and treating these questions as difficult questions about the underlying nature of a certain determinate philosophical category. A better approach is to recognize that in many cases of this sort, we simply have not made up our minds about how the term is to be understood, and that what we seek is not a precise account of what this term already means, but rather a proposal for how it might fruitfully be used in the future. Anyone who believes that something in the vicinity of the abstract/concrete distinction matters for philosophy would be well advised to approach the project of explaining the distinction with this in mind.

 

2. Historical Remarks

The contemporary distinction between abstract and concrete is not an ancient one. Indeed, there is a strong case for the view that despite occasional anticipations, it played no significant role in philosophy before the 20th century. The modern distinction bears some resemblance to Plato’s distinction between Forms and Sensibles. But Plato’s Forms were supposed to be causes par excellence, whereas abstract objects are generally supposed to be causally inert in every sense. The original ‘abstract’/‘concrete’ distinction was a distinction among words or terms. Traditional grammar distinguishes the abstract noun ‘whiteness’ from the concrete noun ‘white’ without implying that this linguistic contrast corresponds to a metaphysical distinction in what these words stand for. In the 17th century this grammatical distinction was transposed to the domain of ideas. Locke speaks of the general idea of a triangle which is “neither Oblique nor Rectangle, neither Equilateral, Equicrural nor Scalenon [Scalene]; but all and none of these at once,” remarking that even this idea is not among the most “abstract, comprehensive and difficult” (Essay IV.vii.9). Locke’s conception of an abstract idea as one that is formed from concrete ideas by the omission of distinguishing detail was immediately rejected by Berkeley and then by Hume. But even for Locke there was no suggestion that the distinction between abstract ideas and concrete or particular ideas corresponds to a distinction among objects. “It is plain, …” Locke writes, “that General and Universal, belong not to the real existence of things; but are Inventions and Creatures of the Understanding, made by it for its own use, and concern only signs, whether Words or Ideas”

 

The abstract/concrete distinction in its modern form is meant to mark a line in the domain of objects or entities. So conceived, the distinction becomes a central focus for philosophical discussion only in the 20th century. The origins of this development are obscure, but one crucial factor appears to have been the breakdown of the allegedly exhaustive distinction between the mental and the material that had formed the main division for ontologically minded philosophers since Descartes. One signal event in this development is Frege’s insistence that the objectivity and aprioricity of the truths of mathematics entail that numbers are neither material beings nor ideas in the mind. If numbers were material things (or properties of material things), the laws of arithmetic would have the status of empirical generalizations. If numbers were ideas in the mind, then the same difficulty would arise, as would countless others. (Whose mind contains the number 17? Is there one 17 in your mind and another in mine? In that case, the appearance of a common mathematical subject matter is an illusion.) In The Foundations of Arithmetic (1884), Frege concludes that numbers are neither external ‘concrete’ things nor mental entities of any sort. Later, in his essay “The Thought” (Frege 1918), he claims the same status for the items he calls thoughts—the senses of declarative sentences—and also, by implication, for their constituents, the senses of subsentential expressions. Frege does not say that senses are ‘abstract’. He says that they belong to a ‘third realm’ distinct both from the sensible external world and from the internal world of consciousness. Similar claims had been made by Bolzano (1837), and later by Brentano (1874) and his pupils, including Meinong and Husserl. The common theme in these developments is the felt need in semantics and psychology as well as in mathematics for a class of objective (i.e., non-mental) supersensible entities. As this new ‘realism’ was absorbed into English speaking philosophy, the traditional term ‘abstract’ was enlisted to apply to the denizens of this ‘third realm’.

 

Philosophers who affirm the existence of abstract objects are sometimes called platonists; those who deny their existence are sometimes called nominalists. This terminology is lamentable, since these words have established senses in the history of philosophy, where they denote positions that have little to do with the modern notion of an abstract object. However, the contemporary senses of these terms are now established, and so the reader should be aware of them. (In Anglophone philosophy, the most important source for this terminological innovation is Quine. See especially Goodman and Quine 1947.) In this connection, it is essential to bear in mind that modern platonists (with a small ‘p’) need not accept any of the distinctive metaphysical and epistemological doctrines of Plato, just as modern nominalists need not accept the distinctive doctrines of the medieval nominalists. Insofar as these terms are useful in a contemporary setting, they stand for thin doctrines: platonism is the thesis that there is at least one abstract object; nominalism is the thesis that the number of abstract objects is exactly zero (Field 1980). The details of this dispute are discussed in the article on nominalism in metaphysics. (See also the entry on platonism in metaphysics.) The aim of the present article is not to describe the case for or against the existence of abstract objects, but rather to say what an abstract object would be if such things existed.