“Fictions, Universals, and Abstract Entities”

One could, of course, assume that fictions, universals, and so-called “abstract entities” are part of the furniture of the world, or of “what there is.” One could compare and contrast fictions with “realities,” universals with particulars, and “abstract entities” with concrete entities. One could also ostracize fictions, universals, and “abstract entities” either drastically, by refusing even to name them, or benevolently, by suggesting that although we may name them, we are not really talking about them at all.

I will follow neither course. I will assume that what there is can be categorized exhaustively into “physical entities,” “persons,” and “objectifications,” so that fictions, universals, and “abstract entities” will have to find a niche in that categorical framework, or else relinquish whatever rights they may claim. Thus, I am protecting their rights provided that they claim few, or none.

The notions, “physical entity,” “person,” and “objectification,” are not easy to elucidate, and in any case I am afraid that no amount of elucidation would free them entirely from the suspicion that they are sweeping, sketchy, inadequate, or plainly wrong. Exhibiting the wider philosophical framework in which they function would not help much unless it were done under assumptions which would require justification until some ultimate ontological commitment were adopted. I will thus confine myself to pointing out what kinds of beings, entities, or processes each of these notions aims to endorse.

Physical realities include not only spatiotemporal objects (or processes), both inanimate and animate, but also “mental acts,” conceived as “real intentions.” All of these entities are deemed to be “natural,” in a sense not too remote from the omnivorous Aristotelian physiké. Persons are performers of real intentional acts, such as thinking, willing, or feeling, as well as agents, or sources of actions, in so far as “actions” can be distinguished from “events,” even if actions are also events. There is no definitive sufficient reason to restrict the range of persons to humans, and there are a number of plausible reasons to enlarge this range to include some nonhumans. For the time being, however, I will equate 'person' with 'human person.' “Objectifications” are results of intentions, acts, and actions of persons. Thus, both tables and theories are objectifications. We can equate “objectification” with “cultural products” provided that they include activities as well as results. Uttering a sentence is an objectification, but the language in which it is uttered, as well as the syntactic roles of this language are also objectifications.

The distinction between “physical entities,” “persons,” and “objectifications” is not one among three different classes of objects, such as diamonds, daisies, and dolls, or horses, Pennsylvanians, and Roman numerals. “Physical entities,” “persons,” and “objectifications” are neither subjects nor predicates. They are concepts, and specifically second-order concepts. The statements 'x is a physical entity' and 'P is a person' have the same grammatical form as the statement 'x is hard' and 'P is doing his best,' but they belong to different conceptual orders. To say 'x is a physical entity' and 'P is a person' is to categorize x and P within conceptual frameworks; this ontological operation has above all a semantic thrust. To be sure, since P is a person, he is also a physical entity, and since objectifications are the result of intentions, acts, and actions of persons, which have become, as it were, “embodied,” and thus also physical entities, one might wonder whether I can remain as ontologically aseptic as I proclaim; in particular one might wonder whether I am not surreptitiously favoring physical entities as the ultimate bearers. An honest answer to this innuendo would be “Yes”; dishonest answers, of course, would be far more elaborate.

Fictions

Since the constants or, as the case may be, the values of variables which are used to discourse on fictions do not designate anything in the spatiotemporal world, it has often been concluded that fictions are “purely mental,” or else that they belong to a special realm variously named: “the realm of ideal objects,” “the realm of unreal (or a-real) objects,” “the realm of nonexistent, but subsistent objects,” and so on.

I find all of these conclusions unacceptable. The acts by means of which fictions are produced and grasped are mental, and hence in a very liberal meaning of this word, “physical.” Fictions themselves, however, are no more physical than, for example, a theory, which is devised by someone and is expressed in signs and symbols, but which is not describable in terms of the mental-physical acts that the theorist performs in order to grasp it. On the other hand, since there is no special category corresponding to an assumedly subsistent world of ideal objects, it may be concluded that fictions are objectifications. They are produced, reproduced, grasped, and discussed by persons, and at the same time, as with all cultural products, they constitute a network within which persons behave, produce further objectifications, dissolve some of the ones already produced, etc. I say 'persons' in plural because objectifications in general, and fictions in particular, are normally part of the environment of human communities. They are interpersonal, and social, as much as, if not more than, personal stricto sensu.

It has often been asked what kind of “reality” such “things” as mermaids have, and each of the following two answers has received overwhelming support: (1) Fictions are “thoughts,” or “imaginations,” and as such real, but only in so far as they are resolved into mental, or perhaps neurophysiological processes; (2) Fictions are some kind of “possibilities,” “essences,” “essential possibilities,” “ideal objects,” etc. which, although not physically real, can become the intelligible ground for real existence.

Neither answer is satisfactory. Mermaids are not, for the time being, natural beings. They are, however, cultural products, playing a role in a cultural context within which they are “re-presented,” namely, made “present” by means of words, drawings, and all sorts of musings. They have no being apart from their being “represented” or “expressed,” but their representation and expression are not governed solely by mental and neurophysiological rules. Mermaids in literature are not mermaids in the mind even if minds are primarily responsible for producing literature in which mermaids appear.

There are, to be sure, many kinds of fictions. A particularly interesting variety is the fictitious characters described in literary works normally grouped under the term 'fiction.' Such characters are not physical but neither are they exclusively mental or purely subjective. It may happen—and it often does happen—that literary fictitious characters are contrived on the basis of persons who really exist, or have really existed. Thus, the Baron de Charlus was contrived by Proust on the basis of two real persons: the Count Robert de Montesquiou and the Baron Doazan. Nevertheless, Proust's Baron de Charlus is not a sum of a real Count and a real Baron, nor is it a sum of features extracted from both. It has its own “mode of being,” which occasionally becomes “independent” from the author—in the sense, of course, that once he has contrived the character, he cannot help following the “norms” which contributed to its production. This explains why some peculiarly “strong” fictitious literary characters finally “dominate” the author rather than the other way around. Furthermore, a fictitious character, especially when it becomes a “universal character” or “archetype,” such as don Quixote, Hamlet, Don Juan or Faust, may even exert “real influences,” namely, he may influence the behavior of real persons. But the fact that the character is obviously in the mind of the author and of the readers does not allow us to conclude that it can be reduced to a series of mental acts. It remains “out there,” as “represented,” and hence it is objectified, no matter how “spiritual” this objectification may be, to use a Hegelian expression, if we want to keep a trace of the Hegelian Geist in an otherwise rather un-Hegelian approach.

Universals

Although often interesting, fictions have not always been philosophically respectable. Universals, on the other hand, are part and parcel of the philosophical Establishment, including that section of the Establishment intent on exercising or, at least, exorcising fictions.

Universals have frequently been contrasted with particulars. The Official Doctrine has it that whereas the former are abstract the latter are concrete. One version of the Official Doctrine neatly separates universals from particulars, but within this version two subversions occur: according to one, only particulars exist; according to the other, particulars are only instances of truly existing universals. Another version of the Official Doctrine maintains that universals can somehow mingle with particulars; among various subversions of this version one has been particularly successful: according to it, universals are in particulars. There are, however, many ways in which 'in' can be said; Aristotle, who seemed to favor the “in-theory,” acknowledged that there are many ways, eight to be precise, to understand this syncategoreme. Unofficial, or less official doctrines, have cast aside the “universal-abstract” and “particular-concrete” pairings in favor of the idea of concrete universals, but there are also several versions of this Doctrine, as Hegel and Husserl testify, and furthermore there are as many subversions of this theory as there are ways of interpreting what Hegel and Husserl said. To make things more complicated, an enormous list of the so-called “universals” can be extracted from philosophical literature: qualities, properties, relations, functions, numbers, classes, among others, have been candidates for the title of 'universals.' The clause 'among others' is not merely rhetorical; Nicholas Wolterstorff, for example, includes among universals actions, symphonies, and poems—and on that basis I assume that a great many “things” which, to Aristotle for example, would have looked like very good examples of particulars, become universals. One may even wonder whether there are “things” such as particulars, for if a symphony is a universal there is no reason to consider any of its notes differently. If the range of universals becomes so universal, then it may be difficult or at least cumbersome, to establish conditions which identify an ever expanding universe of universals, and utterly impossible to establish conditions which identify universals at large. We could perhaps appeal to some non-theological version of the via remotionis, and call “universal” anything which is not a particular. Unfortunately, the identification of particulars may not be as easy an undertaking as is generally assumed since some rather peculiar particulars, like the summer of '42, are sometimes classified as universals.

I will consider any property or feature of something, like being pleasant (perhaps only retrospectively) for the summer of '42, or being red for one of the shoes used in the filming of “The Red Shoes,” as a perfectly good example of a universal. Such properties or features may include relations, but it is wiser to reserve the latter for more formal company. Whether properties should be divorced from properties, as some maintain, is an interesting point, but one which I will brush aside here. We cannot always afford the price of subtlety.

Despite appearances to the contrary, I will not assume that only particulars exist while universals do not, or that only universals exist as opposed to particulars which do not. Those who support the former claim may be led to believe that only particulars as such, sometimes called “bare particulars,” have a right to existence, but then they have to condone ethereal entities such as uncategorized first substances. Although the existential particular quantifier ranges over one or more x's, it is of one or more of these x's that some F is asserted—or, if the universal quantifier is preferred, it is not the case that for all x's, no F is asserted. Those who support the latter claim may be led to think that what is said of x, constitutes x, thus “constructing” an x out of F's (or G's, or H's). To avoid both of these consequences it is necessary to qualify the ontology of particulars and universals in various ways. When this is done, however, there are no longer only bare particulars or bare universals; their shadowy being has vanished in favor of a sturdy entity for which Aristotle found the right, still untranslatable, name, a synolon. Thus, there is never a thing which is only a thing and there is never a property which is only a property. A black horse is not just an x that happens to be black; it is a black horse. From this point of view the battle between the abstract and the concrete soon abates, and we do not have to postulate any “concrete universals.” We do not have to say that if a feather is lighter than air, being lighter than air is an abstraction, or that temperature is an abstraction. Indeed, we will be wise to dispense with all talk of abstract and concrete unless, as I have shown elsewhere, they are understood as limiting notions.

The universals I am talking about are whatever is said to be the case by means of common terms used predicatively. This restricts the area of our inquiry in two ways: because of the narrow scope given to universals, and also because there are many other ways of saying what the case is besides saying it predicatively. Some restrictions, however, are necessary to avoid being lost in an ontological labyrinth. Now, questions have been asked again and again about whether or nor these universals have some kind of “reality,” whether they are “separable&##8221; from their “subjects”; and whether, under the guise of abstract singular names, they can function as subjects and, indeed, as the ontological sources of realities.

Answers to these questions have been grouped under various classical labels. The most illustrious of these is “realism,” succinctly expressed by the medieval formula universalia ante rem. This formula is vague because it does not give us any inkling of how ante is used. It is commonly held that this formula asserts the ontological priority of universals, but there are many ways of understanding priority—a fact that explains why there are uncountable kinds of realistic doctrines, including some which, by dint of moderation and/or qualification, can scarcely be called “realist.” In its most extreme form, realism claims, grosso modo, that particulars are instances—generally imperfect ones—of universals, which are often expressed by means of abstract singular names. Universals fare better than particulars when separated; the marriage of universals and particulars is always a story of long quarrels and short provisional reconciliations. Against realists it has been contended that if we have a red shoe, its redness cannot be separated from the red shoe, as if there was the shoe plus the red color; furthermore, if x is red, and y is red, then there are two red things and not two things plus a red color common to both. But realists are somewhat impatient with such quibbles, for being a shoe is no less a universal than being red. What else can a red shoe be if it is not a shoe, and if it is not red? Unfortunately, being a shoe is what this particular shoe is, and being red is what this particular red shoe is, and this is not a universal, even if it appears wrapped, as it were, in universal predicates. Well and good, the realist counters, but, in the last resort, this is nothing if it is neither a shoe nor is red nor has any other property or belongs to a class which is intensionally expressable as a property. The discussion could go on sempiternally, and this seems the way it actually does; but the objections of antirealists, especially those whom we may call “particularists,” often make realists think twice before giving a straightforward literal interpretation of the ante rem formula. It has been said, for example, that although universals are not ontological forms, either under the guise of formal causes or of exemplary models, they may be units of meaning. No metaphysical sin such as “hypostasizing concepts” is then committed. A price, however, must be paid for this virtuous accomplishment: the reintroduction of a “universe of meanings,” which, furthermore, must be declared ontologically autonomous. Now, universals do not become more palatable when they are saddled with meanings, since most of the problems concerning the ontological status of the former reappear when inquiring about the ontological status of the latter.

At this point a nominalistic interpretation of universals can be proposed which may extricate us from some of these troubles. Since under this interpretation universals are equated with names or terms whose function is to eliminate any ontological or, for that matter, epistemological intermediaries, universals cease to be sui generis entities, distinct from particulars. Universals are then not universals at all, but particulars of some kind.

Unfortunately, a number of new difficulties, emerge. How one can entirely dispense with universals in any reasonably tractable language, whether informal or formal, is only one of the many difficulties. A more annoying difficulty concerns the problem of relating various occurrences of signs to the same type of sign. A nonphysical criterion of relation is incompatible with a strictly nominalistic view. A physical criterion, on the other hand, must be spelled out in a non-physical manner. A behavioristic interpretation can circumvent some of these difficulties, but at a price: it must be assumed that structural connections between stimuli and linguistic reactions never depart from minimal deviations.

Conceptualist interpretations look rather plausible in so far as they maintain that universals are general (not necessarily abstract) ideas signified by certain terms. Unfortunately, it is not clear what these “general ideas” are. If it is claimed that they are “in” realities, the “in problem” arises again. It is agreed that general ideas are not in particulars in the same way in which a thing may be in, or inside of, another. It is also agreed that they are not distributed among particulars. They seem to be “in” in so far as they are mentally conceived. Now, as terms of a postulated “mental language”—which is itself controversial enough—it must be assumed that universals as “general ideas” serve to talk about particulars. And they may, indeed, serve this purpose, but how they manage to do so is not altogether clear. A certain amount of confusion can be removed by means of an analysis of 'about,' but puzzlement soon replaces confusion, for the term “about” seems to dissolve into a myriad of functions, about which we can, paradoxically, talk a lot even if we do not know what 'aboutness' is about.

All of the classical interpretations of universals suffer from a common disease: they try to ground universals on a level of reality, either previously accepted or especially postulated. Thus, universals become, as the case may be, sets of intelligible models, series of mental acts, meanings, or collections of physical signs. To be sure, all these interpretations strongly emphasize the function that universals play: they function as models in the realist interpretation; as meanings or as items of a mental language in most of the conceptualist views; as “suppositions” (or relations of the type supponere pro) in the vast majority of nominalistic theories. However, all these interpretations seem to claim that the functions which universals perform are a result of the “nature” of universals and not the other way around. The various behaviorist interpretations stand or fall on psychological grounds, and it would seem that, unless qualified to death, they fall too many times not to deserve the dubious honor of being indicted. It should be acknowledged, however, that they rightly insist upon the function of words in general, and universals in particular.

The function of universals—or of the terms which are called “universals”—is primarily to order and classify what there is. Thus, 'red' classifies things in so far as they are red; 'nimble' classifies things (or rather acts) in so far as they are nimble; 'just' classifies beings or actions in so far as they are just, etc. It has been argued that once beings or actions are classified by a universal, such a universal denotes the class of things of which it is true. Accordingly, 'red' denotes the class of red things, whereas redness is connoted by 'red.' This argument is not to be dismissed lightly; after all, there is a problem concerning abstract entities, notably classes. For the moment, however, there seems to be no need to give universals the supplementary task of naming.

To say that universals are terms used predicatively is a categorical statement which does not necessarily turn universals into any kind of beings. Realism and conceptualism have transformed universals into intelligible models or into meanings. This may seem to be quite different from, and even opposite to, saying that they are some kind of entities; neither an intelligible model nor a meaning is a thing, and, therefore, neither is entity. Yet, both intelligible models and meanings are conceived as modes of being, and even, in the case of realism at least, perhaps basic modes. As a consequence, both have become “reified.” The possibility of existentially quantifying predicates tends to reinforce some of these reifying tendencies. If it can be affirmed that there is a quality Q, such that Q belongs to x, then it seems that the quality is, in fact, “something.” Paradoxically, nominalism is not less, but more “reifying” than realism and conceptualism, although, to be sure here the reification is completely candid; especially in its terministic and inscriptionistic versions, nominalism has no qualms about admitting universals as concrete singular entitles.

Should we go on asking what universals “are”? Of course, we may go on asking this question, but then it is most probable that we will, in the end, answer it by adhering to one or other of the positions outlined, with all the qualifications which more and more refined techniques allow. I prefer to take a different course; rather than what universals are or are not, or what universal terms may or may not designate, I want to argue what function, or functions, universal terms perform in a language in so far as this language is used to talk about the world.

It may then not be so consequential to pair universals with terms, with meanings, or with “ideas,” for any of these pairings will become a particular, and not necessarily indispensable, interpretation of the ways in which universals, regardless of what they are supposed to be, function. The so-called “universals” serve to ascertain varying similarities among realities. To be sure, they could not fulfil this purpose adequately unless realities themselves could be classified, distributed, and graded. In this sense one could again bring to life the old conceptualist dictum that universals have a fundamentum in re. This is, however, a far cry from saying that universals are “in” things. They are neither in things nor in minds—either in the form of mental processes or “ideas.” Universals are produced by rule-guided linguistic behavior when this behavior is primarily “descriptive,”—although it can be secondarily nondescriptive, and in particular ascriptive. If universals are “in” anything at all, they are “in” language.

I have tried to clear the ground for the claim that universals are objectifications in two senses of this word: as objectifying activities performed by persons, and as a result of such activities. The two senses are related, at least in the way in which “language as an activity” and language as a structure'' are related. This may explain why there is no “realm of universals” established once and for all and necessarily acknowledged by every rational person. In so far as they are “cultural constructions,” universals undergo semantic changes. There may be a greater or lesser tendency towards the use of universals in general, or specific universals in particulars, as well as a greater or lesser tendency towards abstractly expressing their “universality,” according to peoples and times. In principle, any name serving to predicate something of something can engender an abstract singular name with universal scope. Within a given language, or a given period in the history of a language, however, certain adjectives lend themselves more easily than others to such linguistic formations. Common examples in English are 'humanity,' from 'human,' 'whiteness,' from 'white,' 'justice,' from 'just,' 'roundness,' from 'round.' The more often an adjective is used as a universal, the more “natural” it seems to derive an abstract superior name from it. Thus, in English there is talk of poverty and richness, of generosity and stinginess, but does anyone talk much about pluckiness? There is talk about mankind and womankind, but what about wooliness or cottonness? People talk about nervousness and impotence, but how often do they talk about frutescency or imprecatoriness? Yet, such linguistic devices are always at hand, and in some languages, as in German and, I suspect, Russian, they proliferate; some time ago few people spoke of credibility; suddenly there were credibility gaps everywhere, and one may wonder whether there is some gapness of credibility. 'Doghood' looks somewhat awkward, but only because it is sparsely used, if used at all; on the other hand, 'caninity' seems to be academically acceptable. Now, in no case does the use of a singular abstract name prove that there is a greater need for classificatory abbreviations of the objects, processes, or acts of which the corresponding adjective is said to be true. There is wisdom in saying that there is a lot of madness even if, strictly speaking, there is none of both.

I am emphasizing here the (relative) freedom in the linguistic formation of universals. The latter include the various guises of adjectives taken universally, as common nouns, and as abstract singular names. This freedom also explains why some terms which have the form of universals from a grammatical point of view may not be used as such. 'This white' is an expression where 'white' may not be used universally when, for example, some disappointed owner of a house just painted white muses, 'I don't like this white.' Now, this does not mean that universal names are always completely “arbitrary.” In so far as they are accepted by a majority of those who speak a language, they constitute a more or less “fixed” vocabulary, which is ready at hand. The number, scope, and use of universal terms is also determined to a great extent by the function which they play within a civilization and a language. There are civilizations and languages where universals are singled out as terms, or concepts, by means of which operations such as classification, definition, division, and comparison, are effected. Such operations can serve, at the same time, as explanations, or at least as bases for explanations. For example, classification of biological species provided a conceptual ground for an explanation in terms of organ evolution. However, classifications are not always sufficient, not even as starting points of explanations. This may be one of the reasons why universal terms play a relatively minor role in scientific explanation, and in the formulation of scientific laws, as compared to the role they play in ordinary linguistic usage.

Just as all objectifications, universals can exhibit varying degrees of recurrence. They also exhibit varying degrees of influence on human behavior. The universal term 'snub-nosed' serves to classify shapes of human noses; there is scarcely any need here to denote the class of snub-nosed things, although if we want to, we could appeal to the singular abstract name 'snub-nosedness'—a name which I may be the first person to use. The universal term 'just' serves to classify certain human acts, but even if we deny that there is any such thing as 'justice,' or at any rate that justice is a thing, there is little doubt that the term 'justice' can be used to express an ideal of human behavior. In this sense it may exert a considerable influence on what humans do, or abstain from doing. It may well happen that 'justice' is not a universal in the plain sense of this term and that it is more than a universal, as Plato seems to have surmised.

Abstract entities

The distinction between universals and so-called “abstract entities” is not always sharp, particularly when abstract singular names are used in lieu of, or even to talk about, “universals.” Expressions such as '68,' 'mankind,' and 'velocity' may have fundamentally the same semantic status, for all seem to name, aim to name, or fail to name, some kind of “being.”

Nevertheless, I think that the distinction is useful because, despite obvious similarities, universals and abstract entities play different functions.

Let me consider as examples of abstract entities relations, functions, numbers, and classes. The problem raised in ontology—as opposed to what the case is, or is supposed to be, in “straight logic”—is whether there are any reasons to introduce them as “entities,” in our discourse.

We could, of course, introduce them as “nonentities,” but this would modify the problem only slightly. We could also distinguish between 'intensionally abstract' and 'extensionally abstract' entities, but whether we ground the former in the latter, or deny the latter only to admit the former, we still have to cope with the problem of how to categorize any of these so-called, for good or for ill, “entities.” Let us again sacrifice subtlety, and assume that although we may not know the status, semantical or ontological, or both, of these “entities,” we do know what we are saying when some parts of our discourse contain one or more of their names.

Whereas the objectifications called “universals” tend to proliferate or, at any rate, multiply as much as needed to fulfil the task which speakers assign to them, it is characteristic of “abstract entities,” whatever they are, to offer possibilities of “mutual reduction.” Consider numbers of various “types”: integers, fractions, real numbers, irrational numbers, transcendental numbers, complex numbers, algebraic numbers, quaternions, matrices, possibly also transfinite numbers. Two operations can be performed. One consists in starting with positive integers, and in subsequently introducing other types of numbers, such as negative integers and zero. On the basis of the notion of “ordered pairs,” further types of numbers are eventually admitted. The other operation consists in beginning with all types of numbers, and in trying to see whether or not they “reduce” to integers.

This does not mean that all abstract entities can be reduced to just one type. Several cases are possible. First, numbers themselves can be treated as ordered pairs, and ordered pairs, in turn, can be viewed as classes of ordered pairs. If relations are also treated as classes of ordered pairs, then we are on the way to “reducing,” and thus “explaining away,” a number of types of abstract entities in terms of classes. The problem raised will then be the problem of the ontological status of classes. Second, it may be claimed that to define, or explain, numbers in terms of classes is only one among many possible definitions (or explanations). Appeal can be made to set theory—particularly if the distinction between class and set is introduced—; or to some notion of “order;” so that an integer is assigned to a “slot” within a progression; to “cuts” in a continuum, etc. Third, it may be held that numbers are not to be defined, or explained, in terms of classes (and classes of classes), since numbers and classes do not have the same properties. Finally, it may be said that numbers are, indeed, “irreducible,” in so far as they are the result of some “intuitive” operation called “counting,” which is previous to any linguistic frame. Indeed, it may be claimed that no type of abstract entities should be defined, or explained, in terms of any other type, because there are as many types of such entities as kinds of abstract structures.

Since abstract entities are not physical, and since they are not persons, we may wonder whether they belong to some “third world.” In a way, they do, but this “third world” is not necessarily a Platonic one—which, if correctly, or respectfully, interpreted, would be the first and not the second or third—; it is, again, the “world” of objectifications, constantly intertwined with that of persons and, of course, with that of physical entities. To be sure, universals—expressed in common terms, and even in abstract singular names—are also considered as objectifications. On the other hand, a poem and an economic system are also objectifications, so the question arises concerning the specific ways in which abstract entities as objectifications differ from economic systems, poems, or universals.

Two points should be considered: one is the type of relation between abstract entities and their “producers,” namely, the persons performing objectifying activities; the other is the type of relations among abstract entities themselves. To avoid any possible confusion, I will henceforth give abstract entities the special name of “objectivities.”

All objectifications, except the special case called “objectivities,” are historical products, not only because they emerge in the course of the history of a human community—a fate that also befalls objectivities—but also, and above all, because their historical origin is an integral part of their “reality.” This must not be construed to mean that objectifications are always relative, subjective, or “arbitrary.” Although produced by people, at one time or another, they tend to become “independent” from their producers, principally in so far as they fit into, or start to produce, a context—a network of beliefs, concepts, purposes, and what not—which conditions and shapes further objectifications. Objectifications do not exist or occur by themselves, as natural processes do, nor are they produced by agents. Yet, at the same time, they are mutually conditioned. Scientific theory, which is an example of an objectification, does not “exist” until it has been produced; it is not a property of the objects and processes which it purports to describe or explain, and thus it is not “discovered,” but invented—as well as, of course, tested, verified, confirmed, falsified, etc. Since these operations would become futile if theories were contrasted only with other theories, as some radical “internalists” have surmised, it must be admitted that some kind of “external” conditions are not alien to the formulation of a theory. In this manner we end up with a considerable number of factors which seem to estrange the theory from its personal, as well as its interpersonal, and social origins; however, only through these origins can the theory come into existence, no matter how independent it may later become from the process of its discovery.

In so far as they are objectifications, abstract entities do not, as it were, “preexist” their discovery and formulation; numbers or classes, for example are constructions—possibly along lines suggested by mathematical intuitionists. In contrast with other objectifications, however, abstract entities and, in general, what I have called “objecivities” (among which we may include values) are not, except from the genetic point of view, “historical.” I will call them “recurrent constructions,” in the sense that they recur in the same way every time that they are produced. Abstract entities are neither eternal nor contingent. The number 4, for example, does not exist in an eternal intelligible world from which it is, as it were, “extracted,” although whenever the number 4 is constructed, or used, it always has the same stucture—or abides by the abstract structure of which the number 4 is an element. Time makes no difference here. Objectivities are constructed in the only way in which they can be constructed. This does not prevent the possibility of alternative systems of objectivities, but it does mean that each one of these systems exhibits a recurrent structure of its own.

It would seem that abstract entities are, indeed, abstract, and that they have little, if anything, to do with anything which we call “concrete.” On the other hand, I am partial to the idea that both “abstract” and “concrete” are limiting notions—Grenzbegriffe—so that to say that x is abstract only means that it tends to be so, not that it is plainly and simply abstract. These two contentions appear to be in conflict, but only when we realize that abstract structures are not necessarily alien to concrete structures—which, of course, are never plainly and simply concrete. In any case, the former can be correlated with the latter. This correlation is similar to the one which we find between the unit of measurement called “yard,” and a given distance which is measured by yards. It is also similar to the one which we find between the season called “winter,” and the set of things and processes which we normally associate with the winter.

I must acknowledge that abstract entities are somewhat resilient, and that classes are particularly resilient, namely, they seem to resist being pigeonholed within a categorial structure which admits only physical entities, persons, and objectifications. In other words, abstract entities, and in particular, classes, are hard to swallow unless we make special room for them. I have tried to deny them a special niche, but I am ready to give it back to them if “swallowing” them is going to produce a severe case of philosophic indigestion.