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Abstract objects

Many consider that any essence falls under one of two categories: one of them are specific, others — are abstract. It is supposed that this distinction has fundamental value for metaphysics and an epistemologiya. In this article the review of a number of recent attempts to show how this distinction has to be carried out is given.

1. Introduction
2. Historical remarks
3. Way of denial
   3.1 Criterion not - spatiality
  3.2 Criterion of a causal inefficiency
4. Way of an example
5. Way of association
6. Way of abstraction
7. For further reading
Bibliography

 

1. Introduction

In modern philosophy distinction of the abstract and concrete has quite curious status. The majority agrees that this distinction is fundamentally important. And at the same time there is no standard representation as it has to be carried out. Many agree in that, properly to classify certain paradigmatic examples. So, it is conventional that numbers and other objects of an abstract mathematics are abstract (if they exist) whereas stones, trees and people — are specific. As obvious examples of abstractions classes, pro-positions, concepts, a letter "A" and "Hell" of Dante act. Obvious examples of concrete objects are the stars, protons, electromagnetic fields written by chalk on a certain board symbols of a letter "A" and the copy of "Hell" of Dante belonging to James Joyce.

The task consists in showing that is the cornerstone of this dichotomy — whether by means of obvious definition of these terms, or by means of their inclusion in the theory which will allow to reveal better their communication with other important categories. For lack of such clearing, the philosophical importance of this distinction remains doubtful. We can be able to classify things as abstract or as concrete, relying on intuition. But for lack of theoretical expression it will be difficult to understand that (if at all something) depends on this classification.

It is worth emphasizing that not necessarily there has to be the only "correct" way of an explanation of distinction abstract and concrete. Any plausible explanation will classify paradigmatic examples standardly, and any interesting explanation will carry out clear and philosophically significant division of area of objects. However there can be a set equally of interesting ways of achievement of both these is more whole and if we find two or more explanations which cope with this task rather well, will senselessly ask what of them corresponds to "real" distinction abstract and concrete. This case serves as an illustration of a general comment: when technical terminology is entered into philosophy by means of examples for lack of obvious definition or theoretical development, the dictionary which is turning out as a result often is indistinct or uncertain in respect of a reference. In such cases it is usually senseless to look for only a right explanation. The philosopher can ask look questions "That there is an idealism?" or "That there is a substance?" and to treat these questions as to difficult questions of the nature of that lies behind strictly certain philosophical category. But it is more preferable to recognize that in many cases of this sort we simply have no ready decision on that, properly to understand this term, and we look for not an exact explanation of that is the term already means, but it is rather an offer of how it could be fruitfully used in the future. To everyone who believes that the subjects connected with distinction abstract and concrete, are significant for philosophy, it is necessary to remember it, addressing to the project of an explanation of this distinction.

 

2. Historical remarks

Modern distinction of the abstract and concrete arose rather recently. Really, there are weighty arguments of that, except for rare anticipations, it didn't play a significant role philosophies before the XX century. Modern distinction has some similarity to platonovsky distinction of idea and sensually perceived. However platonovsky ideas were considered as the reason par excellence whereas abstract objects are usually considered as in every respect causally the inert. Originally distinction of "abstract" and "concrete" belonged to words or terms. The traditional grammar distinguishes abstract "whiteness" and concrete "white color" (the abstract noun ‘whiteness’ from the concrete noun ‘white’) that doesn't assume as if this linguistic opposition corresponds to some metaphysical distinction which would be expressed by the specified words. In the XVII century this grammatical distinction was postponed for area of ideas. Locke speaks about the general idea of a triangle which "shouldn't be idea neither oblique-angled, nor rectangular, neither equilateral, nor isosceles, unequalateral triangles; she has to be all of them and any of them at the same time" (Essay IV.vii.9). Lokkovsky understanding of abstract idea as idea which is formed of concrete ideas by means of derivation from the details distinguishing them, it was immediately rejected by Berkeley, and then and Yum. But even Locke had no hint that distinction between abstract and concrete or civil (particular) ideas corresponds to some distinction among objects. "Clearly — Locke writes — that the general and general don't treat the valid existence of things, and are invented and created by reason for his own use and concern only signs — words or ideas" (III.iii.11).

Distinction abstract and concrete in his modern look is urged to designate the border lying in the field of objects or sushchnost. In such understanding this distinction becomes central in philosophical discussions only in the XX century. Sources of such strategy of consideration aren't clear, however overcoming, appear, of exhaustive division of objects on mental and material, having fundamental value for ontologically focused philosophers since the time of Descartes could become one of key factors. Frege's statement that from objectivity and apriority of mathematical truth follows that numbers aren't neither material sushchnost, nor ideas of mind became one of sign events in this direction. If numbers were material things (or properties of material things), laws of arithmetics would have the status of empirical generalizations. If numbers were ideas of mind, there would be the same problem, as well as an incalculable great number of others. (Whose mind contains number 17? Whether there is one 17 in your mind and another — in mine? In that case the community of mathematical subjects would appear illusion.) In "Osnovopolozheniyakh of arithmetics" [Frege 1884] of Frege comes to a conclusion that numbers aren't neither external "concrete" things, nor mental sushchnost of any sort. Subsequently, in work "Thought" [Frege 1918], he claims the same status by that he calls thoughts — sense of declarative offers — and also, as a result, behind their constituents, sense of subpropositional (subsentential) expressions. Frege doesn't say that meanings are "abstractions". He says that they belong to "the third world", excellent both from the sensual outside world, and from an inner world of consciousness. Similar statements expressed Bolzano [Bolzano 1837], and later — Brentano [Brentano 1874] and its pupils, including Meynonga and Gusserl. The general motive of such reasonings — felt in semantics and psychology, it is equal as in mathematics, need for a class objective (i.e. not - mental) supersensual sushchnost. And when such new "realism" was apprehended by English-speaking philosophy, the traditional term "abstract" began to be used for designation of inhabitants of it of "the third world".

Philosophers who recognize existence of abstract objects, sometimes call platonist; those who denies their existence, sometimes call nominalists. Such word usage is extremely unsuccessful as these terms already have settled values in the history of philosophy where they designate the positions far from modern ideas of abstract objects. Nevertheless, new values were already assigned to these terms and therefore the reader has to be acquainted with them. (In English-speaking philosophy the most influential source of such terminological innovation is Quine — see in particular [Goodman and Quine 1947].) In this regard it must be kept in mind that modern platonist (from a small "p") aren't obliged to accept any metaphysical or epistemological doctrines, characteristic for Platon, as well as modern nominalists aren't obliged to accept the doctrine of actually medieval Nominalists. In that measure in what these terms are useful in a modern context, they are used for designation of narrow theories: the platonizm — is a position according to which there is at least one abstract object; the nominalism — is a position according to which the amount of abstract objects is equal in accuracy to zero [Field 1980]. Details of the discussions which are conducted between their supporters are discussed in article about nominalism in metaphysics (en). The purpose of this article isn't consideration of arguments in advantage or against existence of abstract objects, but it is rather a demonstration of that, than there would be abstract objects if they existed.

 

3. Way of denial

Fregeansky strategy of carrying out distinction abstract and concrete — an example of that Lewis [Lewis 1986a] called the Way of denial. According to him, abstract objects are defined as such which have no certain properties inherent in paradigmatic concrete objects. Practically all obvious definitions abstract, meeting in literature, follow this way. We will consider some their variations.

According to the position assumed in Frege's works

The object is abstract in only case when when it at the same time is not - mental and not - sensual.

Here the priority consists in explaining that means to a thing to be "not - mental" or as speak usually, "independent of consciousness". The simplest answer — to tell that the thing depends on consciousness in that case when it wouldn't exist (or couldn't exist) if there was no consciousness. But from this it follows that tables and chairs are dependent on consciousness, and it by no means not that philosophers mean, using this concept. To call object "dependent on consciousness" in a metaphysical context — means to assume that he is somehow obliged by the existence of mental activity, however not in that banal "causal" sense in what usual artifacts are obliged by the existence to consciousness. But what it can mean? It agrees to one of perspective approaches, the object has to be considered dependent on consciousness when by the most nature he exists in a certain timepoint if only if it is object or the maintenance of some mental condition or process in this timepoint. It allows to consider tables and chairs independent of consciousness, so far as they can remain at destruction of the conceiving things. However paradigmatically mental objects — as, for example, a purple residual image which I now realize — will be considered dependent on consciousness as they as it is supposed, are objects which by the most nature are realized always when exist. Not clearly, however, such treatment grasps the specified concept fully. We will consider, for example, the mereologichesky sum of my residual image and your headache. It, undoubtedly, mental essence if those in general exist. But she it won't be obligatory object of a mental condition. (This mereologichesky sum can exist even if anybody doesn't think of it.) Wider concept probably would consider such objects depending on consciousness which exist at some moment only thanks to the mental activity which is carrying out at this moment even if this object isn't object of any single mental condition or the act. The sum of my residual image and your headache depends on consciousness in this second sense, but not in the first. Also it is an occasion to prefer the second treatment of dependence on consciousness.

If we understand concept of dependence on consciousness thus, by a mistake will claim that abstract objects are dependent on consciousness. Concerning a question to which we will come back still: many believe that sets and classes are abstract sushchnost — even dirty sets, whose praelement are concrete objects. Any treatment of distinction abstract and concrete which will place set-theoretic designs — such as { Alfred, { Bettie, { Wendy, George } } } — on the party concrete, will disperse from the accepted use. Remembering it, we will consider a set which only elements are my residual image and your headache, or some more difficult set-theoretic object constructed on the basis of these elements. If to assume (that is quite plausible) that the dirty set exists at some moment, only when his elements exist at this moment, it will be dependent — in a broad sense — from consciousness essence. But it as it is represented, also is also abstract essence.

The similar problem arises with so-called abstract artifacts, such as Jane Austen's novels and the characters occupying them. Some philosophers consider them as eternally existing abstract essence which in our frail world "are only "described" or "coded" by authors, but aren't created by them. Though, of course, the common sense prompts what exactly Austen created "Pride and prejudice" and Elisabeth Bennet and to deny it are no bases ([Thomasson 1999]; cf. with [Sainsbury 2009]; see also article about fiction (en)). If we accept this position of common sense, we will find that their existence will depend on mental activity of Austen and, probably, on mental activity of the subsequent readers in quite certain sense. They can be considered independent of consciousness in that sense in what this dependence was discussed above, so far as "Pride and prejudice" can as it is represented to exist at some moment even if at this moment will be led up nobody to be conceiving. (If the mankind is captured suddenly by a short collective sleepiness, "Pride and prejudice" won't drop out of area of the existing.) But they are obviously dependent on consciousness in some not - simply - causal sense. And nevertheless they remain, most likely, abstract objects. Thereof it is, seemingly, wrong to claim that abstract objects aren't dependent on consciousness (in more detail about dependence on consciousness see Rosen 1994.)

Frege's position in his initial formulation is unsatisfactory as well on other bases. Quarks and electrons aren't neither sensual, nor dependent on consciousness. And nevertheless they aren't abstract objects. In more successful formulation Frege's position would say that

The object is abstract if only if it at the same time is not - physical and not - mental.

Such approach can successfully carry out significant distinction; however he inherits a familiar problem of an explanation that means to a thing to be physical object [Crane and Mellor 1990]. Discussion of this question see in article about a fizikalizm.

3.1 Criterion not - spatiality

Modern followers of the Way of denial usually improve Frege's criterion, introducing for abstract objects the requirement or not - spatiality, or a causal inefficiency, or both at the same time. Really, if some definition abstract can be also considered as standard — the following:

The object is abstract if only if it is a не-пространственен and is causally inefficient.

Such standard approach, nevertheless, faces a number of difficulties.

We will consider first of all the requirement according to which abstract objects have to be not - spatial (or not - existential). Some paradigmatic examples of abstractness of a не-пространственны literally. It is senseless to ask where there was a function of a cosine last Tuesday. Or, if this question makes sense, the only reasonable answer will be that she was anywhere. Similarly, there is no special sense to ask when Pythagorean theorem comes. Or, if in it also there is a sense, the only reasonable answer will be that she existed always or that, probably, she doesn't exist "in time" at all. These paradigmatic "pure abstractions" don't possess uncommon spatial or temporary properties. They have no spatial localization, and they don't exist in any separate timepoint.

However some abstract objects, apparently, stand in more interesting relation to space. We will consider, for example, game in chess. Some philosophers will tell that chess, like mathematical objects, exists not "where" and not "when" — or is eternal, or entirely out of time. But it is not the most natural position. The position claiming is natural that chess was invented in certain time in a certain place (though to establish precisely when and where exactly they were invented, can be difficult); that before they were invented, they didn't exist at all; that they were delivered from India to Persia in the VII century; that with a current of years they underwent changes and so on. The only basis to oppose to this natural approach is the thought that, so far as chess definitely is abstract object — not physical it is object, eventually! — and so far as abstract objects don't exist in space and time — by definition! — chess in the relation to space and time has to resemble function of a cosine. However with the same right the example with chess and other abstract artifacts can be considered as a counterexample to a rash statement that abstract objects possess only trivial spatial and temporary properties.

Do we have to refuse criterion of not-space-temporariness (non-spatiotemporality) now? It isn't obligatory. Even if some abstract essence in some sense possess uncommon existential properties, it is possible to tell, nevertheless, that concrete essence exist in space time in a special way. If we could explain in what this special way of existential existence, characteristic for concrete objects consists, we could tell: the object is abstract (if) only if it can't exist in space time thus.

Using this approach, it is possible to note that paradigmatic concrete objects tend to occupy rather certain volume of space at every moment of the existence, or a certain volume of space time during the existence. It makes sense to ask about such object where he now is and how many places it takes even if the answer sometimes will be quite uncertain. On the contrary, even if game in chess "is somehow involved" in space and time, it is senseless to ask, how many places it takes. (In that measure, in what this question it can make sense, the only reasonable answer will be that it doesn't take a place at all that doesn't mean that it occupies a point in space.) Thus, it is possible to tell:

The object is abstract (if) only if it can't borrow as - or a certain area of space (or spaces times).

This perspective idea faces a number of questions. First, quite representable, that some of those objects which can be considered as abstract, nevertheless occupied a certain volume of space and time. We will consider, for example, various sets which elements are Pyotr and Pavel: { Pyotr, Pavel }, { Pyotr, { Pyotr, {{Pavel}} } }, etc. Usually we don't ask a question where there are similar things or how many spaces they occupy. And it is valid, many philosophers will tell that this question is senseless, or that it is worthy only the scornful answer "anywhere, at all". But nothing compels us to give such answer neither in the theory of sets, nor in metaphysics. Even if we will assume that pure sets are only in the most trivial relations to space, we still will have an opportunity to claim, as well as some philosophers did that dirty sets exist there and then, where and when there are their elements [Lewis 1986a]. Won't be unnatural to tell that the set of books is on a certain shelf in library — and is valid, there are some theoretical bases to wish to tell so [Maddy 1990]. Such approach presents us with a choice: we can tell that as dirty sets exist in space, all of them aren't abstract objects; or we can tell that as dirty sets are abstract, to believe that abstract objects can't take a place in space, was a mistake.

One of ways of round of this difficulty — to note that even if dirty sets and take a place in space, they occupy him with a derivative image. The set { Pyotr, Pavel } takes some place of that his concrete elements, Pyotr and Pavel, in common take this place. A set as that doesn't take this place. In this plan it is possible to tell that

The object is abstract (if) only if it or can't take a place in space at all, or occupies it only that some other objects — in this case, his praelement — occupy this area of space.

Of course, Pyotr himself occupies a certain area of space of that his parts — his head, hands, etc. — in common occupy this area. So more successful option of the offered definition would say:

The object is abstract (if) only if it or can't take a place in space at all, or occupies it only that some other objects which aren't his parts, occupy this area of space.

Such approach, apparently, quite well allows to classify the mentioned cases, and still it is in many respects artificial. Moreover, he causes a number of questions. What to tell us about a statue which occupies some area of space not because her parts are located in space, but it is rather because material of which she is made, occupies this area? And how to be with not observed electron which, according to some interpretations of quantum mechanics, actually doesn't occupy a certain area of space at all, but costs in some more exotic relation to space time in which it lives rather? It is enough to tell that philosophers who consider "not - spatiality" as criterion abstract, but thus assume that some abstract objects can have uncommon spatial properties, have to provide still the concept of a certain relation to space and space time which would allow to distinguish the paradigmatically concrete.

Probably, the most important question of criterion "not - spatiality" concerns classification of parts of the space. Let's say the space or space time exist not only as objects of an abstract mathematics, but as the arena on which physical objects and events in one way or another settle down. Physical objects are "in" this or that area of space and therefore, according to criterion not - spatiality, is considered as the concrete. But how to be with points and areas of the space? Discussions about, whether acceptance of an existential substantsionalizm will be coordinated with nominalistsky refusal of abstract sushchnost, [Field 1980, 1989 already took place; Malament 1982]. If we define abstract as "not - spatial", these discussions are reduced to a question of whether to consider space "spatial". But it, of course, only question of word usage. We can expand the existing use so that to allow points and areas of space to be "in" to ourselves, or not to do it is it already a matter of taste. To the philosopher who believes that with a question of, whether it is possible to consider parts of space as something concrete, serious difficulties are connected, it is worth addressing to different ways of the description of distinction abstract and concrete.

3.2 Criterion of a causal inefficiency

According to the most widespread version of the Way of denial,

The object is abstract (if) only if it is causally inefficient.

Concrete objects, whether it be mental or physical, possess causal abilities; numbers, functions, etc. generate nothing. Game in chess as that can't be the participant of the causal relations (unlike her concrete realization). And even if dirty sets somewhat really exist in space, it is easy to be convinced that they don't make any own causal contribution that occurs. Pyotr and Pavel can make impacts separately. Together they can make even such impacts which any of them couldn't render independently. However such in common made impacts naturally speak rather as the actions caused by two concrete objects operating in common, or, probably as the impacts made by their mereologichesky association (which itself is paradigmatically concrete), than as influences of a certain set-theoretic design. We will assume, Pyotr and Pavel together moved some freight on scales. If we assume opportunity that this event was prichinno generated by a set, we will have to ask, what set was his cause: the set including only Pyotr and Pavel? or some more intricate design constructed on their basis? or, perhaps, the set including molecules of which Pyotr and Pavel consist? Such proliferation of possible answers first of all suggests an idea of an attributing inaccuracy to sets of causal abilities. It is good news to those who would like to recognize all sets abstract.

(We will note, however, that some authors identify ordinary physical events — being causally effective par excellence — with sets. According to David Lewis, for example, such event as falling of Rome — is the ordered couple which first member is the space time area, and the second — a set of such areas [Lewis 1986b]. From this point of view it will be absolutely incorrect to tell that dirty sets are abstract objects, as well as that abstract objects can't be the reasons.)

The idea according to which the causal inefficiency makes a sufficient condition of abstractness, disperses from the accepted use a little. Some philosophers are convinced of existence "the epifenomenalnykh of a kvali" — objects of conscious perception (sensual data) or the kvalitativnykh of conscious states which can be generated by physical processes in a brain but which in itself have no descending (downstream) of causal consequences [Jackson 1982; Chalmers 1996]. Similar to essence if they exist, are causally inefficient, however usually they aren't considered as the abstract. The supporter of criterion of a causal inefficiency can answer with the statement that abstract objects differ in that aren't neither the reasons, nor actions. But it risky. The abstract artifacts similar to Jane Austen's novels, (as we usually understand them) arise as result of human activity. The same belongs and to dirty sets which arise when their concrete praelement are created. From a position of common sense, they obviously are some generations; nevertheless, they remain abstract if at all exist. Not clearly, as supporters of the tough version of criterion of a causal inefficiency (who consider a causal inefficiency at the same time as a necessary and sufficient condition of abstractness) can well answer this difficulty.

Besides these fears, evident counterexamples which would have crucial importance, in relation to this approach of distinction abstract and concrete it isn't found. The main difficulty — and it hardly solving — rather conceptual. Usually it is considered that the causal relation, strictly speaking, is the relation between events or situations. When we say that the stone (object) is the reason of the broken window, we mean that some event or a state (either the fact, or situation) with participation of a stone, became the reason of that the window broke. If the stone itself is the reason, it is the reason in some derivative sense. But this derivative sense is imperceptible. Blow of a stone to a window — this event in which the stone definitely "participates" and as the stone participates in an event thus so far as we also attribute to the stone causal efficiency. But what means to object to participate in an event? We will present that John reflects on Pythagorean theorem, and you ask him of what he thinks. His answer — this event: pronouncing phrase; and one of his reasons — an event of reflection of John about the theorem. Whether Pythagorean theorem "participates" in this event? Undoubtedly, somewhat — participates. The event is that John becomes in a certain relation to the theorem — in the same way as dissecting by a stone of a window is that the stone becomes in a certain relation to glass. But we don't attribute to Pythagorean theorem causal efficiency only because in this sense she participates in an event which is the reason. The task, thus, consists in defining a specific way of "participation in causal sequence", characteristic for concrete objects. A little attention was paid to this problem. There are no bases to believe that it can't be solved. However in the absence of her any decision, this standard version of the Way of denial has to be submitted incomplete.

 

4. Way of an example

Besides the Way of denial, Lewis allocates three main strategy of an explanation of distinction abstract and concrete. Following the Way of an example, it is enough to make the list of paradigmatic examples of abstract and concrete sushchnost in hope that the sense of their distinction somehow will be investigated. If this distinction was initial and not analyzed, it, perhaps, there would be an only way to explain it. However, as we already noted, such approach calls into question the importance of this distinction. Distinction of the abstract and concrete is significant as abstract objects in general, apparently, present a certain common problem to epistemologiya and philosophies of language. It is represented not clear as we learn abstract objects in that sense in which it is clear as we learn concrete objects [Benacerraf 1973]. It is represented not clear as at us it turns out to refer precisely (refer) on abstract essence in that sense in which it is clear as at us it turns out to refer to other things [Benacerraf 1973, Hodes 1984] precisely. But if it is original problems, there has to be an explanation why in this regard abstract objects as those cause special difficulties. It is difficult to believe that everything put — in their only one elementary abstractness. It is much simpler to believe that business in them not - spatiality, a causal inefficiency or something similar. It isn't excluded that distinction of the abstract and concrete is fundamental and that the Way of an example — the best of possible ways to clear him. But if it so, becomes absolutely not clear why this distinction has to be significant.

 

5. Way of association

According to the Way of association, distinction of the abstract and concrete is identified with this or that metaphysical distinction already known under other name: for example, with distinction between sets and individuals or with distinction between universaliya and partikulyariya. Certainly, some authors used these terms thus. (So, Quine [Quine 1953] uses "abstract essence" and "universaliya" as interchangeable.) Such association, however, seldom meet in modern philosophy.

 

6. Way of abstraction

As the most important alternative of the Way of denial that Lewis called the Way of abstraction acts. It agrees one of long traditions of philosophical psychology, the abstraction is understood as a certain mental process, during which by means of consideration of some objects or ideas and derivation from those properties by which they differ, formed new ideas or concepts. Let a number of white things of various forms and the sizes be given; we will distract or "abstract" from with what they differ and by that we will come to abstract idea of a whiteness. Doesn't contain in the considered tradition requirements that the ideas created thus represented special type of objects or corresponded to them. But it is possible to claim that distinction between abstract and concrete objects has to speak the link to psychological process of abstraction or something similar. The simplest version of this strategy will say that the object is abstract if it is (or can be) the reviewer of abstract idea, i.e. the idea created by means of abstraction.

Presented in such form, the Way of abstraction is directly connected with the consciousness philosophy which got out of fashion. However in recent years the wide circulation was got by approach close to him. Crispin Wright [Wright 1983] and Bob Hale [Hale 1987] developed the theory of abstract objects which makes a start from a number of the hints which are found at Frege [Frege 1884]. Frege notes (in fact) that many of singular terms which, apparently, send to abstract sushchnost, are created by means of functional expressions. We speak about a building form, about the direction of a straight line, about number of books on the shelf. Of course, many singular terms created by means of functional expressions designate usual concrete objects: "Platon's father", "capital of France". However functional terms which grasp abstract essence, differ in the following relation. If "f (a)" is such expression, equality of a look usually takes place

f (a) = f (b), if only if Rab,

where R is the equivalence relation. (The relation of equivalence is the relation reflexive, symmetric and transitive). For example,

The direction a = the direction b, if only if a parallel to b.

F-ov number = G-ov number if only if is available in the accuracy of so many F-ov, how many and G-ov.

Moreover, these equalities (or the principles of abstraction) as it is represented, have the special semantic status. Though they, strictly speaking, aren't definitions of the functional expression standing in the left part, they are apparently fair owing to value of this expression. To understand the term "direction" means (in particular) the nobility that "direction a" and "direction b" send to the same essence if only if direct an and b are parallel. Moreover, the equivalence relation standing in the right member of equation, apparently, is semantic and, probably, epistemological primary in relation to functional expression in the left part [Noonan 1978]. Possession of concept of the direction assumes possession of concept of parallelism, but not on the contrary.

Existence of the principles of abstraction meeting these conditions can be used for a distinction explanation between abstract and concrete objects. If "f" — the functional expression submitting to the principle of abstraction, is such sort (kind) of Kf corresponding to him that

x Kf if only if for some y it is faithful that x = to f (y) is.

For example, x is a cardinal number if only if for some concept F it is right that x = F-ov number. The simplest version of such approach on the Way of abstraction will say then that

x is abstract object if (and only if) x is a case (instance) of some sort Kf — such that the related functional expression of "f" submits to the corresponding principle of abstraction.

The strong version of such approach — applying for establishing a necessary condition of abstractness — significantly disperses from the accepted use. As we already noted, pure sets are paradigmatic abstract objects. Not clearly, however, they satisfy to the offered criterion. According to the naive theory of sets, functional expression "the set (something)" is really described by the estimated principle of abstraction.

The set of F-ov = a set of G-ov if only if for all x it is right that x is F in only case when when x G is.

However this principle is insolvent and therefore it isn't capable to describe interesting concepts. In modern mathematics the concept of a set isn't entered through abstraction. Open is a question, whether something can be described like mathematical concept of a set by certain narrower principle of abstraction. (See the review of the recent attempts made in this direction, in [Burgess 2005].) But even if such principle will be found, the condition of epistemological superiority will hardly manage to be satisfied. (That is possession of concept of a set will hardly assume possession of the equivalence relation presented in the right part.) Therefore it isn't clear, whether it is possible to classify on the Way of abstraction understood thus objects of the theory of pure sets as abstract essence (that as it is supposed, it is required).

Similarly, as it was noted by Dammit [Dummett 1973], in many cases usual names of paradigmatically abstract objects have no functional expression about which it is told in definition. Chess — it is abstract essence. But we don't understand a synonym of expression of the f (x) form where "f" submits to the principle of abstraction as the word "chess". Similar remarks as it is represented, are applicable to such things as English, social justice, architecture and Charlie Parker's style. But if it is right, abstraktsionistsky approach doesn't give a necessary condition of abstractness in that sense in what this concept can be understood.

Even more important that there are bases to believe that he doesn't give also a sufficient condition. The Mereologichesky sum of concrete objects itself is concrete object. However the concept of the mereologichesky sum, most likely, submits to the principle of abstraction:

F-ov sum = G-ov sum if only if F-y and G-y block each other (cover),

where F-y block G-y if only if each part of each G has part, the general with F. Respectively, let us assume, that the train — is the maximum chain of trains, each of which is linked to another. We can define functional expression "train x" by means of the principle of "abstraction": The train x = the train at, if only if x and y — the linked structures. In that case we can tell that x — the train if only if for some structure at truly that x — is the train at. Simple consideration, thus, leads to a conclusion that trains have to be considered as abstract sushchnost.

Not it is clear, whether these objections are applicable to Wright and Hale's thinner abstraktsionistsky provisions, however one of aspects of the simple consideration outlined above definitely is applicable to these provisions and can form the basis of criticism of such variation of the Way of abstraction. Neo-fregeansky approach tries to explain distinction abstract and concrete in semantic terms. We said that abstract object is such object which falls under the functional expression submitting to the principle of abstraction where "f" submits to the principle of abstraction when this principle is fair owing to f value. The concept of justice of the statement owing to a word meaning is notorious for the problematical character (see article about distinction analytical and synthetic). But even if such concept is justified, it is all the same possible to declare that distinction of the abstract and concrete has to be metaphysical distinction; that abstract objects have to differ from other objects in some important ontologic relation. So, has to be possible to carry out this distinction directly in metaphysical terms: to explain that is in objects such that does one things abstract, and others — concrete. As Lewis in response to the similar project of Dammit wrote:

"Even if on it … the way can succeed in carrying out distinction (and as far as I know, it is possible), we learn nothing about with what essence, divorced on the opposite sides, differ by the nature. It all the same what to tell that snakes are animals of whom we instinctively are afraid most of all — probably, it and so, but it tells nothing us about the nature of snakes" [Lewis 1986a: 82].

The task consists in developing not - the semantic version of abstraktsionistsky criterion which in metaphysical terms strictly and directly would define that objects have the general, whose initial names submit to the principle of abstraction.

One of possible solutions of this problem consists in transposing abstraktsionistsky situation in more metaphysical tonality. We will begin with idea according to which any fregeansky number by the most nature is number of some fregeansky concept, in the same way as any fregeansky direction by the most nature is (at least, potentially) the direction of some concrete straight line. In each case the abstract object appreciably (essentially) is value (value) of function of abstraction for a certain class of arguments. The speech doesn't go about values (meanings) of linguistic expressions. It is about essence or the nature of objects. (About the corresponding concept of essence see [Fine 1994].) So, for example, the fregeansky number two (if such thing exists) appreciably, by the most nature is the number corresponding to concept F if only if there are in accuracy two F-and. More generally: for each fregeansky abstract object x there is such function of abstraction of f that x appreciably is value f for each argument of a certain sort.

Function of abstraction has two key properties. First, for each function of abstraction of f there is such relation of equivalence of R that in the nature of f is put that f (x) = f (y), if only if Rxy. Intuitively we are inclined to believe that R metaphysically initially in relation to f and that function of abstraction of f is defined (entirely or partly) by this biconditional (biconditional). Secondly, each function of abstraction is the making function: its values appreciably are values of this function. Many functions aren't the making functions. Paris is the capital of France, but it isn't the capital appreciably. On the contrary, the number of planets of Solar system appreciably is number. The concept of function of abstraction can be defined through two of these properties:

f is function of abstraction, if only if

a. for some relation of equivalence of R it is right that in the nature of f it is put that f (x) = f (y), if only if Rxy; and

b. for everyone x it is right that if x is value f, in the nature x it is put that there is (or can exist) some object of y — such that x = f (y).

We, thus, can tell that

x is abstraction if only if for some function of abstraction of f there exists or can be such object of y that x = f (y);

and

x is abstract object if (and only if) x is abstraction.

Such treatment tells a lot of things to us about the distinctive nature of these the fregeanskikh (in a broad sense) of abstract objects. She says to us that each of them by the most nature is special type of value of function which nature quite simply is defined by the equivalence relation inherent in her. It is worth emphasizing, however, that such treatment doesn't give about them rich metaphysical information. She doesn't speak about, whether there are they in space, whether they can consist in the causal relations and so on. The question of, whether will be coordinated so unusual way of carrying out distinction abstract and concrete with any of more traditional ways outlined above remains open.

 

7. For further reading

Patnem [Putnam 1975] gives scientific arguments in favor of abstract objects. Fild [Field 1980, Field 1989] gives arguments against abstract objects. Biler [Bealer 1993] and Tennant [Tennant 1997] represent aprioristic arguments in favor of necessary existence of abstract sushchnost. Balager [Balaguer 1998] claims that any argument for or against existence of abstract objects isn't convincing and that a question of, whether there are abstract things, in general isn't necessary. The review of discussion about existence of the abstract is submitted at Byordzhess and Rosen [Burgess and Rosen 1997]. Fayn [Fine 2002] offers systematic research of the principle of abstraction in the mathematics bases. The general theory of abstract objects is developed Zalta in [by Zalta 1983; Zalta 1999 (in "Other Internet resources")].

 

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A. V. Mertsalov Lane

 

How to quote this article

Rosen, Gideon. Abstract objects//Stenfordsky encyclopedia of philosophy (version of fall of 2014) / Edition Edward N. Zalta. The lane with English A.V. Mertsalova. URL =

Original: Rosen, Gideon, "Abstract Objects", The Stanford Encyclopedia of Philosophy (Fall 2014 Edition), Edward N. Zalta (ed.), URL =

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