Abstract Entities

Entities Chris Swoyer Penultimate draft of a paper for Contemporary Debates in Metaphysics, eds., Theodore Sider, John Hawthorne, and Dean W. Zimmerman, Blackwell, 2008; pp 1-31. One of the most puzzling topics for newcomers to metaphysics is the debate over the existence and nature of abstract entities, things like numbers (seven), sets (the set of even numbers), properties (triangularity), and so on. The major questions about abstract entities are whether there are any, if so which ones there are, and if any do exist, what they are like. My aim here is to provide an accessible overview of the debates about abstract objects. I will try to explain what abstract entities are and to say why they are important, not only in contemporary metaphysics but also in other areas of philosophy. Like many significant philosophical debates, those involving the nature and existence abstract objects are especially interesting, and difficult, because there are strong motivations for the views on each side. In §1 I discuss what abstract entities are and how they differ from concrete entities and in §2 consider the most compelling kinds of arguments for believing that abstract entities exist. In §3 I consider two examples, focusing on numbers (which will be more familiar to newcomers than other types of abstract objects) and properties (to illustrate a less familiar sort of abstract entity). In §4 I turn to criticisms of arguments for the existence of abstract entities and consider questions about what phenomena need, or are even amenable to, philosophical explanation. In §5 I examine the costs and benefits of philosophical accounts that employ abstract entities and in §6 draw some conclusions. 1 What are Abstract Entities? Prominent examples of abstract entities (also known as abstract objects) include numbers, sets, properties and relations, propositions, facts and states-of-affairs, possible worlds, and merely-possible individuals (we’ll see what some of these are in a bit). Such entities are typically contrasted with concrete entities, things like trees, dogs, tables, the Earth, our Solar System, and Hoboken. I won’t discuss all of these examples, but will consider a few of the more accessible ones as case studies to help orient the reader. Numbers and Sets Thought and talk about numbers are familiar; we spend hundreds of hours learning to do it right. We learn about the natural numbers (like three, four and four billion), about fractions (rational numbers, like 2 3 and 5 9 ), and about irrational numbers (like the square root of 2 and π). And we learned a bit about sets in school, for example the empty set, the set containing just 3 and 4, and the set of even numbers; we even learned to write names of sets using notation like ‘{3,4}’. But what are numbers and sets? We cannot see them or point to them; they do not seem to have any location, nor do they interact with us or any of our instruments for detection or measurement in any discernible way. This may lead us to wonder whether there really are any such things as numbers, and whether, when we say things like ‘there is exactly one prime number between four and six’ we are literally and truly asserting that such a number exists (after all, what could it be?). But as we will see in §3.1, there are also strong philosophical arguments that numbers do exist. Hence a philosophical problem: do they or don’t they? Properties and Relations The world is full of resemblances, recurrences, repetitions, similarities. Tom and Ann are the same height. Tom is the same height now as John was a year ago. All electrons have a charge of 1.6022× 10−19 coulomb. The examples are endless. There are also recurrences in relations and patterns and structures. Bob and Carrol are married, and so are Ted and Alice; the identity relation is symmetrical and so is that of similarity. Resemblance and similarity are also central features of our experience and thought; indeed not just classifications, but all the higher cognitive processes involve general concepts. Philosophers call these attributes of qualities or features of things (like their color and shape and electrical charge) properties. Properties as the ways things can be; similarly, relations are the ways things can be related. Assuming for the moment that there are properties and relations, it appears that many things have them. Physical objects: The table weighs six pounds, is brown, is a poor conductor of electricity, and is heavier than the chair. Events: World War I was bloody and was fought mainly in Europe. People: Wilbur is six feet tall, an accountant, irascible, and married to Jane. Numbers: three is odd, prime, and greater than two. All of these ways things can be and ways they can be related are repeatable; two tables can have the same weight, two wars can both be bloody. The two adjacent diamonds in Figure 1 are the same size, orientation, and uniform shade of gray. Champions of properties hold that things like grayness (or being a b Figure 1: Resemblance and Ways Things Can Be gray) and triangularity (or being triangular) are properties and that things like being adjacent and being a quarter of an inch apart are relations. Since the goal here is just to give on prominent example of a (putative) sort of abstract object, I will think of properties as universals (as many, but not all philosophers do). On this construal, there is a single, universal entity, the property of being gray that is possessed or exemplified by the two diamonds in our figure. It is wholly present both a and b, and will be as long as each remains gray. Philosophers who concur that properties exist may disagree about which properties there are and what they are like, but at least many properties (according to numerous philosophers, all) are abstract entities. Perhaps a property like redness is located in those things that are red, but where is justice, or the property of being a prime number, or the relation of live a century before? Such properties and relations exist outside space and time and the causal order, so they are rather mysterious. But, as we will see, there are also good reasons for thinking that properties and relations can do serious philosophical work, helping explain otherwise puzzling philosophical phenomena. This is a reason to think that they do exist. Another problem. Propositions Two people can use different words to say the same thing; indeed, they can even use different languages. When Tom says ‘Snow is white’ and his friend Hans says ‘Schnee ist weiss’ there is an obvious sense in which they say the same thing. So whatever this thing is, it seems to be independent of any particular language. Philosophers call these entities propositions. They are abstract objects that exist independently of language and even thought, though many of them are expressed in language. Propositions have been said to be the basic things that are true or false, the basic truth-bearers, with the sentences or statements that express them being derivatively true or false. Tom also believes that snow is white and Hans, who speaks no English, believes that Schnee ist weiss. Again there is an obvious sense in which they believe the same thing (though of course Hans couldn’t express his belief by talking about snow). Some philosophers urge that the best way to explain this is to conclude that there is some thing that Tom and Hans both believe. On this view propositions are said to be the contents or meanings of beliefs, desires, hopes and the like. They as also said to be the objects of beliefs. Thus the object of Tom’s belief that red is a bright color is the proposition, that red is a bright color. On this view propositions are abstract objects that express the meanings of sentences, serve as the bearers of truth values (truth and falsehood), and are the objects of belief. But like numbers, propositions are somewhat mysterious things. We can’t see them, hear them, point to them. They don’t seem to do anything at all. This gives us reason to doubt their existence, but, there are also reasons to think that they exist. Problems, problems, problems. 1.1 What Abstract Entities Are (Nearly Enough) Debates about abstract objects play a central role in contemporary metaphysics. There is wide agreement about the paradigm examples of abstract entities, though there is also disagreement about the exact way to characterize what counts as abstractness. Perhaps this shouldn’t come as a surprise; if any two things are so dissimilar that their difference is brute and primitive and hard to pin down, abstract entities and concrete entities (abstracta and concreta) are certainly plausible candidates. Even so, the philosophically important features of the paradigm examples of abstracta (like those listed above) are pretty clear. They are atemporal, non-spatial and acausal, i.e., they do not exist in time or space (or space-time), they cannot make anything happen, nothing can affect them, and they are incapable of change. They are causally inert, impotent, inefficacious; neither they, their properties or aspects, nor events involving them can make anything happen here in the natural world. We don’t see them, feel them, taste them, or see their traces in the world around us. Still, according to a familiar metaphor of some philosophers, they exist “out there,” independent of human language and thought. Taken alone, being atemporal, non-spatial, and acausal are probably not necessary for being abstract in the sense many philosophers have in mind, and others aren’t sufficient. Not necessary because many things that seem to be abstract also seem to have a beginning (and ending) in time, among them natural languages like Urdu, forms of government like democracy, melodies like “La Marseillaise,” fictional characters like Sherlock Holmes, and dance styles like the Charleston. It may seem tempting to say that such things exist in time but not in space, but where exactly? Moreover, this claim can’t be literally true in a relativistic world (like ours certainly seems to be), where space and time are (framework-dependent) aspects of a single, more basic thing, namely space-time. And not sufficient because an elementary particle (e.g., an electron) that is not in an eigenstate for a definite spatial location is typically thought to lack any definite position in space. Indeed, if it is in an eigenstate for a specific momentum, it has a zero probability of turning up in any given region of space when measured for position—though it will turn up somewhere. The technicalities don’t matter here; the point is just that although such particles may seem odd, they do have causal powers, and so virtually no one would classify them as abstract. Again, according to many religious traditions God exists outside of space and time, but he brought everything else into existence, He is the “first cause,” and so many would be reluctant to classify Him as an abstract object. All this suggests that the division into concrete and abstract may be too restrictive, or that abstractness may come in degrees. I won’t consider such possibilities here, however, because the puzzles about abstract entities that most worry philosophers concern those entities that are, if they exist, atemporal, nonspatial, and acausal (how could such things ever affect us?). And we don’t need a sharp bright line between abstracta and concreta to examine these. A philosopher who believes in the existence of a given sort of abstract entity is called a realist about that sort of entity, and a philosopher who disbelieves is called an anti-realist about it. Abstract entities are not a package deal; it is quite consistent, and not uncommon, for a philosopher to be a realist about some kinds of abstract entities (e.g., numbers) and an anti-realist about others (e.g., properties). Not-Quite Existence Finally, some champions of abstract entities claim that there are such things, but grant them a lower grade of being than the normal, straightforward sort of existence enjoyed by Al Gore, Hoboken, and the Eiffel Tower. They often devise rather esoteric labels for this state; for example, numbers, properties, and the like have been said to have being, to subsist, to exist but not be actual, or partake of one or another of the bewildering varieties of not-quite-full existence contrived by philosophers. Such claims are rarely very clear, but frequently they at least mean that a given sort of entity is real in some sense, but doesn’t exist in the spatio-temporal-causal order. Which is pretty much just to say it is abstract. Appealing to notions like subsistence can look like trying to have one’s cake and eat it too, thus allowing a philosopher to invoke properties or propositions without having to grant them genuine existence. We will not pursue such matters here, however, since many of the same problems arise whether the issue about the status of abstracta is framed in terms of the existence or merely the subsistence or being of such things. Whatever mode of being the number two or the property of having a mass of 2kg possesses, we still cannot perceive it, pick it out in any way, and it seems to make no difference to anything here in the natural world. Because many of the most debated issues arise for all the proposed modes of being of abstract objects, I will focus on existence. Why Questions about Abstracta Matter Explicit discussion of abstract entities is a relatively recent philosophical phenomenon. Plato’s Forms (his version of universal properties) have many of the features of abstract objects. They exist outside of space and time, but they seem to have some causal efficacy. We can learn about them, perhaps even do something like perceive them, though perhaps only in an earlier life (this is Plato’s doctrine of recollection). Soon after Plato, properties and other candidate abstracta, e.g., merely possible individuals (individual things, e.g., persons, that could have existed but don’t), were reconstrued as ideas in the mind of God. This occurred through the influence of Augustine and others, partly under the influence of Plotinus and partly under that of Christianity. Human beings were thought to have access to these ideas because of divine illumination, wherein God somehow transfered his ideas into our minds. In later accounts like Descartes’ we had access to such ideas because God placed them in our minds at birth (they are innate). Such views persisted though Medieval philosophy and well into the modern period. In this period philosophers like Locke began to view what we thought of above as properties (e.g., redness, justice) as ideas or concepts in individual human minds. It was really only in the nineteenth century with work on logic and linguistic meaning by figures like Bernard Bolzano and Gottlob Frege that abstract entities began to come into their own. They emerged with a vengeance around the turn of the twentieth century, with work in logic, the theory of meaning, and the philosophy of mathematics, and, more generally because of a strongly realist reorientation of much of philosophy at this time in the English and German speaking worlds. After a few decades interest in abstract entities subsided, but by the end of the twentieth century, there was perhaps more discussion of a wider array of abstract objects than ever before. Although explicit discussion of abstract entities has a fairly recent history, they are central to debates over venerable philosophical issues, including the nature of mathematical truth, the meanings of words and sentences, the features of causation, and the nature of cognitive states like belief and desire. These debates also lie at the center of many perennial disputes over realism and anti-realism, particularly standard flavors of nominalism. Discussions about the existence of abstract objects may also illuminate the nature of human beings and our place in the world. If there are no abstract objects, nothing that transcends the spatio-temporal-causal order, then there may well be no transcendent values or standards (e.g., no eternal moral properties) to ground our practices and evaluations. And if there is also no God, it looks like truth and value must instead be somehow rooted here in the natural order; we are more on our own. 2 Why Believe there are Abstract Objects? We now come to the central questions about abstract objects. Are there any? How can we decide such issues? If at least some kinds of abstract objects exist, can we discover what they are like (this is a problem because it seems to be difficult to make contact with them in order to learn about their nature). In this section I will offer an answer to the first question that also suggests an answer to the second. A good way to get a handle on the issues involving abstract entities is to begin by focusing on the point of introducing them in the first place. The precise point is somewhat different in the case of different sorts of abstracta, but the general motivations are similar. Philosophers who champion one or another type of abstract object almost always do so because they think those objects are needed to solve certain philosophical problems, and their views about the nature of these abstracta are strongly influenced by the problems they think they are needed to solve and the ways in which they solve them. Hence, our discussion here will be organized around the tasks abstracta have been introduced to perform. These tasks are typically explanatory, to explain various features of philosophically interesting phenomena, so to understand such accounts we need to ask about the legitimacy, role, and nature of explanation in metaphysics. 2.1 Philosophical Explanations and Existence Ontology is the branch of metaphysics that deals with the most general issues about existence. Of course we know a great deal about what sorts of things exist just from daily life: things like trees, cats, cars, other people, the moon. And science tells us more about what sorts of things there are: electrons, molecules of table salt, genes. But ontology attempts to get at the most general categories or sorts of things there are, e.g., physical objects, persons, numbers, properties, and the like. Some philosophers doubt that the very enterprise of ontology makes sense, but we will begin by assuming that it does. For many centuries ontology aspired to be a demonstrative enterprise, one based on valid arguments. A valid argument is one which must have a true conclusion if all of its premises are true. Such arguments are sometimes called demonstrative. A valid argument with all true premises will have a true conclusion (by the definition of validity); such arguments are sometimes called deductively sound. On this traditional conception of ontology, it employs valid arguments to establish conclusions about what the most general and fundamental things in the universe are. It proceeds from obviously secure premises, step by deductively valid step, to obviously secure conclusions. The traditional standards for security were very high, requiring unassailable, necessary, self-evident “first principles.” These were supposed to be claims that couldn’t possibly be false and that no reasonable person could doubt. The chief problem with this picture is that when we judge classical arguments in ontology by such standards most not only fail—many fail miserably. There is, among other things, no consensus about which candidates for first principles are even true, much less necessarily so, and and in many cases demanding valid arguments seems to be asking for too much. By these standards even the best the greatest philosophers could manage comes up far short. Nowadays many philosophers would gladly settle for premises that are uncontroversially true—or even just fairly plausible. But they still devote a good deal of time distilling arguments for (or against) the existence of one or another sort of abstract object down to a few numbered premises and a conclusion to write on the board, check for validity, then (most often) dismiss then. This approach is often invaluable, but it has limitations. For one thing, few philosophical arguments survive long when judged by the passfail standards of deductive validity (how likely is it, after all these centuries of inconclusive results, that Jones has just devised a twelve line demonstration that properties exist?). Indeed, it is quite possible that there are no deductively sound arguments beginning from true premises which do not mention abstracta and end with conclusions that abstracta exist (“no abstracta in, no abstracta out”). However that may be, we often miss things of value if we write arguments off simply because they are not deductively valid. But if traditional and contemporary versions of the demonstrative ideal set the bar too high, how should we think about arguments and disagreements in ontology? When we turn to the ways philosophers actually evaluate views about abstract objects, we typically find things turning on the pluses and minuses of one view compared to those of its competitors. And a very common feature of the (putative) pluses is that they often mention explanation. For example, we are told that the existence of numbers would explain mathematical truth or that the existence or properties (like triangularity) would explain why it is that various objects are triangular and that it would also help explain how we recognize newly encountered triangles as triangles. Moreover, even when the word ‘explain’ is absent we frequently hear that some phenomenon holds in virtue of, or because of, this or that property, that a property is the ground or foundation or most enlightening account of some phenomenon, or that a property is (in part) the truth maker, the fundamentum in re (as the Medievals would have said) for the phenomenon. For example, it has been urged that the exemplification of a single, common property grounds the fact that our two items in Figure 1 (p. 3) are triangular; it makes it true that each is a triangle. The same property also helps to explain how we recognize that they are triangular and why the world ‘triangle’ applies to them. Similar claims have been made on behalf of other abstracta. The role of expressions like ‘explain’ is to give reasons, to answer why-questions, which is a central point of explanation. My suggestion is that we should (re)construe arguments for the existence of abstract entities as inferences to the best over-all available ontological explanation (we’ll return to this in §§3-4; see also Swoyer 1982, 1983, 1999a). I will develop this idea in the course of examining several examples, but first let’s see what morals we can draw from the view that arguments for the existence of abstract objects are ampliative (i.e., deductively invalid but capable of offering good, though not conclusive, support for their conclusions). First, we should acknowledge at the outset that there will rarely (probably never) be knock-down arguments for (or against) the existence of any type of abstract entity. On this approach, metaphysics (including ontology) is a fallibilistic enterprise, one in which we may need to revise even our currently best-supported views as we acquire new knowledge. By way of example, twentieth-century physics presents us with a very surprising picture of physical reality, and it may well call for innovations in ontology. To note just one case, quantum field theory, that branch of physics that deals with things at a very small scale (quarks, electrons, etc.), strongly suggests that there are (at the fundamental level) no individual, particular things; there may be no fact about how many “particles” of a given kind there are in a particular region of space-time. If so, the traditional view that individuals or substances are a fundamental category of reality may be overthrown. Or, to take a rather different example, philosophers may discover new, entirely-unanticipated arguments for, or against, the existence of various philosophical entities, including abstracta. Metaphysics was once thought of as “first philosophy,” coming first in the sense that it not only described the most fundamental features of reality but provided the basis for the rest of our knowledge about that reality. On the current view there is no first philosophy; nor is there any point in calling metaphysics secondor thirdphilosophy, because nothing, neither philosophy nor anything else, comes first. Almost any aspect of our knowledge can be affected by any other part, and far from being a mere chapter out of history, metaphysics is a living, ongoing enterprise and should be responsive to intellectual change and novel information. Second, although each specific argument for the existence of a certain kind of abstract entity may not be fully compelling, if there are a number of independent arguments that a given sort of entity exists, the claim that they do could receive cumulative confirmation by helping to explain a variety of phenomena. Third, if some type of abstract entity is postulated to play particular explanatory roles, this affords a principled way to learn about its nature. We ask what such an entity would have to be like in order to play the roles it is postulated to fill. What, to take a question considered below, would the existence or identity conditions of properties have to be for them to serve as the meanings of predicates like ‘round’ or ‘red’? Explanatory Targets and Target Ranges An explanation requires at least two components. First, something to be explained, an explanation target. Second, something to explain it. In ontology, it is a philosophical theory (though ‘theory’ is often a bit grandiose) like Plato’s theory of forms that does the explaining. We will be concerned with those theories that employ abstract objects in their explanation. Explanation targets for ontology can come from anywhere. From the everyday world around us (e.g., different objects can be the same color, and a single object can change color over time). From mathematics (e.g., it is necessarily the case that the area of a Euclidean triangle is the length of the base times one-half its height’; there is simply no way that something could be a Euclidean triangle without having its area determined in this way). From natural languages (e.g., the word ‘triangle’ is true of many different individual figures). From science (e.g., objects attract one another because of their gravitational mass but may repel one another if they are different charges). And many from almost any area of philosophy (e.g., many moral values seem to be objective, but it’s a bit mysterious how this can be so). I will call a more-or-less unified collection of explanation targets a target domain. In the next section I briefly discuss several target domains that have led some philosophers to postulate abstract entities. Although I believe that arguments in ontology are usually best construed as ampliative, much of what follows can be adapted fairly straightforwardly to the view that philosophical arguments should aim to be deductively sound. 3 Examples of Work Abstracta Might Do When we turn to actual debates about abstract objects we find few (arguably no) knock-down, iron-clad, settled-once-and-for-all arguments for or against the existence of most of the abstract objects that interest philosophers. Instead, the evaluation of the arguments involves the art of making trade offs, the weighing of philosophical costs and philosophical benefits. I will urge that although there are widely shared, quite sensible criteria for this, but they fall short of providing rules or a recipe that forces a uniquely correct answer to the question of which, if any, abstract entities exist. The chief philosophical benefit claimed for most sorts of abstract entities is that they do philosophical work, perform some “ontic labor.” And the chief work they do is to help explain otherwise puzzling phenomena, phenomena that would be illuminated if those entities really existed, and so that accounts employing them provide philosophical enlightenment. Benefits, however, rarely come without costs, and we will examine some of the costs of abstracta in §4. In this section we will consider some of their benefits. There are many candidate abstracta and there is space to discuss only a few. I will begin with one that will be familiar to readers with little background in philosophy, the natural numbers. The natural numbers (0, 1, 2, and so on up forever) are just the objects of the arithmetic whose rudiments we have been familiar with since grade school.