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Judah Abrabanel

4. Philosophy of Beauty

4.2 Cosmology and Metaphysics

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

According to the first model, Judah argues—citing Ghazali, Avicenna and Maimonides—that God in His complete simplicity and “by the love of His infinite beauty produces out of Himself alone the first intelligence and mover of the first heaven” (con l’amore de la sua immense bellezza immediate da sé sola la prima intelligenzia movtrice del primo cielo produce).[25] The first intelligence, in turn, contemplates (1) the beauty of its cause to produce the second intellect, and (2) its own beauty to produce the first heavenly sphere. This theory of emanation, based on the love of beauty,[26] pervades the entire universe (both supra- and sub-lunar). The Active Intellect, the lowest of the ten heavenly intellects and associated with the sphere of the moon, becomes the intellect of the corporeal world. By contemplating its own beauty it produces the forms found in this world, and in contemplating the beauty of its cause, it produces human intellects. Following this, Judah offers a Plotinian account based on a celestial triad.[27] He now distinguishes between three types of beauty that pervade the cosmos. The first is God qua the Source of beauty (l’attore di bellezza), the second is beauty itself (bellezza; i.e., intelligible beauty), and the third is the physical universe produced by this idea in the intellect of God (il participante di bellezza).[28]

Judah Abrabanel

4. Philosophy of Beauty

4.2 Cosmology and Metaphysics

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Judah subsequently uses the latter model, combined with kabbalistic embellishment, to interpret the first creation account in Genesis, where the physical world is now described as the offspring between God, the male principle, and intelligible beauty, now personified as a female.[29] Corporeal beauty, according to this model, becomes the primogeniture of God’s love for His female consort, wisdom. Since this physical world is intimately connected to God, it cannot be negated. Rather, this world becomes necessary, and this is a leitmotiv that runs throughout the work, for humans to make physical beauty “spiritual in our intellect.”[30] One must, in other words, orientate oneself towards sensible beauty in such a manner that one reverses the ontological chain.

Judah Abrabanel

4. Philosophy of Beauty

4.3 Psychology and Prophecy

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

The above discussion is directly related to the way in which Judah’s envisages both psychology and prophecy. According to him, the five external senses divide into (1) those that are primarily material (materiali): touch (tatto), taste (gusto), and smell (odorato); and (2) those that are increasingly spiritual (spirituali): hearing (per l’audito) and sight (per l’occhi).[31] It seems that only the latter are able to penetrate behind the purely physical so as to abstract the spiritual from the corporeal. Hearing, intimately connected to the Renaissance ideal of rhetoric, consists of the ability to discern “fine speeches, excellent reasoning, beautiful verses, sweet music, and beautiful and harmonious melodies.”[32] Sight, ranked just above the faculty of hearing, owing to the primacy that Judah puts on vision, deals with “beautiful colors, regular patterns, and light in all its varied splendor.”[33] The senses, thus, function hierarchically as a prolegomenon to any form of higher knowledge, with the imagination forming the threshold (mezzo) between the senses and the intellect.

Judah Abrabanel

4. Philosophy of Beauty

4.3 Psychology and Prophecy

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Central to the unfolding argument in the Dialoghi is the concept of ocular power (forza oculare). In the first dialogue, Judah describes two modes of apprehending spiritual matters. The first is through the faculty of sight and the second through the intellect.[34] For the eye, like the intellect, is illumined by means of light, thereby establishing a relationship between the eye, the object seen, and the space that separates them.[35] Just as the sun supplies light to the eye, the divine intellect illumines the human intellect during the act of intellection. It is light, then, that enables us to “comprehend all the beautiful shining objects of the corporeal world.”[36] Sight becomes the model by which we engage the universe: it is what makes knowledge possible since it is only through vision of tangible particulars that we acquire knowledge of intelligibles.

Judah Abrabanel

4. Philosophy of Beauty

4.3 Psychology and Prophecy

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Judah subsequently distinguishes between three types of vision. The highest type is that of God’s visual apprehension of himself; following this is that associated with the angelic world, which sees God directly though not on equal terms; finally, there is human vision, which is the weakest of the three types and can only visualize the divine indirectly.[37] The best that most humans can do in this world is to obtain knowledge of incorporeal essences through corporeal particulars. Judah does admit, though, that some special individuals are able to unite with the angelic world, which he describes as the Agent Intellect (intelleto agente). When such unification (copulare) occurs, the individual “sees and desires divine beauty as in a crystal or a clear mirror, but not directly” (vede e desia la bellezza divine come in uno mezzo cristiallino, o sia chiaro specchio, ma non in se stessa immediate).[38] Judah refers to this act as prophecy. Like Maimonides, Judah claims that Moses did not prophesy through the imagination, but only through the intellect, which he nevertheless describes in highly visual terms as beholding “the most beautiful figure of God” (la bellissima figura di Dio).[39]

Judah Abrabanel

4. Philosophy of Beauty

4.3 Psychology and Prophecy

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

In the third dialogue, Abrabanel further divides the human into a tripartite structure consisting of the body (il corpo), the soul (l’anima), and the intellect (l’intelletto).[40] The soul, which I have interpreted as the imaginative faculty because of the properties assigned to it,[41] is once again the intermediary (mezzo) between the body (and the senses) and the intellect.[42] Although he does not come right out and define the functioning of the soul in any detail, he does claim that it is indispensable to the proper working of the body and the intellect. Moreover, it is this faculty that is in constant danger of being corrupted by unhealthy corporeal desire and, most importantly for the present, it is ultimately responsible for translating the corporeal into the incorporeal and vice versa. This soul, in turn,

Judah Abrabanel

4. Philosophy of Beauty

4.3 Psychology and Prophecy

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

has two faces (due faccie) , like those of the moon turned towards the sun and the earth respectively, the one being turned towards the intellect above it, and the other toward the body below. The first face looking towards the intellect is the understanding with which the soul reasons of universals and spiritual knowledge, extracting the forms and intellectual essences from particular and sensible bodies (estraendo le forme ed essenzie intellettuali da li particulari e sensibili corpi)…the second face turned towards the body is sense, which is particular knowledge of corporeal things known … These two faces have contrary or opposed notions; and as our soul with its upper face or understanding makes the corporeal incorporeal (l’incorporeo al corporeo), so the lower face, or sensible cognition, approaching the objects of sense and mingling with them, draws the incorporeal to the corporeal.[43]

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

The traditional philosophical notion of love, going back at least to the time of Plato, is that love results from the imperfection and privation of that which loves. One loves, in other words, what one does not possess (see, e.g., Plato, Symposium 200a-201e).[44] Accordingly, that which is imperfect loves that which is perfect, and, that which is perfect (i.e., God) neither loves nor desires. Aristotle likewise claimed that that which is less perfect (e.g., slaves, children, wives and ruled) should have more love for that which is more perfect (e.g., freeborn, parents, husbands, and rulers) than vice versa. The First Cause, then, is loved but does not love. This discussion would predominate from Late Antiquity to the Middle Ages.

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Judah’s theory of love, by contrast, was intimately connected to the literary interests of humanism and the aesthetic sensibilities of Renaissance artists.[45] As a consequence, Judah faults previous thinkers for (1) not ascribing love to God and (2) confining their discussion of love primarily to that between humans, thereby ignoring the dynamic role of God in relationships based on this principle. Using the name of Plato as a metonym for other philosophers, he is critical of this approach:

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Plato in his Symposium discusses only the kind of love that is found in men, which has its final cause in the lover but not in the beloved (terminato ne l’amante ma non ne l’amato), for this kind mainly is called love, since that which ends in the loved one is called friendship and benevolence (ché quel che si termina ne l’amato si chiama amicizia e benivolenzia). He rightly defines this love as a desire of beauty (desiderio di bellezza). He says that such love is not found in God, because that which desires beauty and doesn’t have it is not beautiful, and God, who is the highest beauty, does not lack beauty nor can he desire it, whence he cannot have love, that is, of such a kind (Tale amore dice che non si truova in Dio, però che quel che desia bellezza non l’ha né è bello, e a Dio, che è sommo bello, non gli manca bellezza né la può desiare, onde non può avere amore, cioè di tal sorte).[46]

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Judah seeks to provide a corrective to this and, in the process, offer what he considers to be a more comprehensive theory of love. In particular, he intertwines love and beauty such that the lover of beauty seeks to unite with the source of beauty, something that the lover subsequently seeks to reproduce himself (the lover is always male, according to Judah because it is responsible for impregnating the passive and receptive female principle). This can take the form of God’s creation of the universe, the artist’s creation of a work of art, and the philosopher’s composition of a pleasing work of philosophy. In his discussion of love, Judah also departs from other Renaissance thinkers. Whereas Ficino had equated human love with sensual love between humans, Judah, drawing upon Maimonidean precedents, resignifies human love as, on one level, that which the intellect has for God.[47]

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Perhaps, Judah’s biggest departure from earlier thinkers is his ascription of love to God. Here, however, his discussion is not entirely new; rather, he picks up on a number of issues discussed in the work of Hasdai Crescas (ca. 1340–ca. 1410), the important critic of Aristotelianism and Maimonideanism. For Crescas, breaking from both the Platonic and Aristotelian positions, love need not be associated with privation or imperfection:

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Inasmuch as it is known that God, may He be blessed, is the sources and fountain of all perfection, and in virtue of his perfection, which is His essence, He loves the good, as may be seen from His actions in bringing into existence the entire universe, sustaining it perpetually, and continuously creating it anew, and all be means of His simple will, it must necessarily be that the love of the good is an essential property of perfection. It follows from this that the greater perfection [of the lover] the greater will be the love and the pleasure in the desire.[48]

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Love, for Crescas, is tantamount to God’s creative activity. In the third book of the Dialoghi Judah picks up this theme:

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Divine Love (L’amor divino) is the inclination (tendenzia) of God’s most beautiful wisdom toward the likeness of His own beauty, to wit, the universe created by Him, together with its return to union with His supreme wisdom; and His pleasure is the perfect union of His image with Himself (la perfetta unione di sua immagine in se stesso), and of His created universe with Himself as Creator.[49]

Judah Abrabanel

5. Philosophy of Love

5.1 Judah’s Recasting of Love as a Philosophical Principle

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Judah’s subsequent discussion of love, however, departs significantly from either Crescas or other Renaissance thinkers. This is especially the case when it comes to Judah’s refusal to abnegate sensual love. Rather, he celebrates such love as the gateway to cosmic or spiritual love. Sensual love, for him, becomes that which orientates the individual towards the Divine. Unlike Crescas, however, Judah does not reject the Maimonidean concept of God as divine intellect. Furthermore, Judah’s discussion finds no homolog in the thought of Crescas or even Maimonides when it comes to the concept of God’s beauty.

Judah Abrabanel

5. Philosophy of Love

5.2 “The Circle of Love” (il circulo degli amari)

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Near the end of the third dialogue, Judah introduces the “circle of love,” il circulo degli amari, as follows:

Judah Abrabanel

5. Philosophy of Love

5.2 “The Circle of Love” (il circulo degli amari)

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

The circle of all things (il circulo di tutte) begins from their first origin, and passing successively through each thing in turn, returns to its first origin as to its ultimate end (ultimo fine), thus containing every degree of being in its circular form (comprendendo tutti li gradi de le cose a modo circulare), so that the point which is the beginning also comes to be the end. This circle has two halves (due mezzi), the first from the beginning to the point most distant from it, the mid-point, and the second from this mid-point to the beginning again.[50]

Judah Abrabanel

5. Philosophy of Love

5.2 “The Circle of Love” (il circulo degli amari)

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

This circle, in other words, begins with the divine, whose love creates and sustains the universe: “each degree of being with paternal love procreates its immediate inferior, imparting its being or paternal beauty to it, although in a lesser degree as is only fitting.”[51] This emanative framework, the love of that which is more beautiful for that which is less beautiful, comprises the first half of the circle. Every thing in the universe exists on a hierarchical chain of being, from the pure actuality of the divine to the pure potentiality of prime matter. Just as the superior desires the perfection of the inferior, the inferior desires to unite with the superior. The first half of the circle spans from God to utter chaos, whereas, the second half of this circle works in reverse. It is the love of the inferior for the superior, predicated on the former’s privation and subsequent desire to unite with the superior.

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

The telos of Judah’s system occurs when the intellect is “absorbed in the science of the Divine and of things abstracted from matter, rejoicing in and becoming enamored of the highest grace and beauty which is in the creator and artificer of all things, so that it attains its ultimate happiness.”[52] Within this context, Judah employs a well-known and well-used metaphor in his discussion of the imaginative faculty, that of the mirror (specchio).[53] In the medieval Islamicate philosophical tradition, following Plotinus, the mirror (Ar. mir’â), is frequently used as an image for the perfected soul, at whose vanguard resides the intellect.[54] Just as the mirror reflects what is placed in front of it, the soul, in a state of perfection, reflects the higher principles of the universe. Early on in the Dialoghi, Philo asks Sophia, “Do you not see how the form of man is impressed on and received by a mirror, not as a complete human being, but within the limits of the mirror’s powers and capabilities, which reflect the figure only and not the essence.”[55] Later, in dialogue three, he argues that it is through our own “intellectual mirror” that we apprehend the divine:

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

It is enough for our intellectual mirror (specchio intellettuale) to receive and image (figurare) the infinite divine essence (l’immensa essenzia divina) according to the capacity of its intellectual nature; though there is a measureless gulf fixed between them, so far does its nature fall short of that of the object of its understanding.[56]

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

By corporealizing the spiritual and spiritualizing the corporeal, the intellect, in tandem with the imagination, enables individuals to gain knowledge of the divine world. Unlike Maimonides and other medieval Aristotelians who equated the natural world with impermanence and evil, Judah argues that this world is the natural receptacle of heavenly powers: “Hence earth is the proper and regular consort of heaven, whereof the other elements are but paramours. For it is upon the earth that heaven begets all on the greater part of its progeny.”[57] In an interesting passage, Judah, like Maimonides before him, compares matter to a harlot.[58] Yet, unlike Maimonides, he reaches a radically different conclusion. For Judah, “it is this adulterous love that beautifies the lower world with such wondrous variety of the fair-formed things.”[59] In keeping with Judah’s claim that “the lower can be found in the higher,”[60] this world becomes one gigantic mirror that reflects spiritual beauty, and in which one can grasp divine intelligibles.[61] Just as Philo is enthralled by Sophia’s beauty, the individual, upon contemplating physical objects that are beautiful, apprehends the divine:

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

God has implanted His image and likeness in His creatures through the finite beauty imparted to them from His surpassing beauty. And the image of the infinite must be finite, otherwise it would not be a copy, but that of which it is the image. The infinite beauty of the Creator is depicted and reflected in finite created beauty like a beautiful face in a mirror and although the image is not commensurate with its divine pattern, nonetheless it will be its copy, portrait, and true likeness (Si depinge e immagina la bellezza infinita del creatore ne la bellezza finita creata come una bella figura in uno specchio: non però commisura l’immagine il divino immaginato, ma bene gli sará simulacro similitudine e immagine).[62]

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

It is ultimately the love of beauty in the soul of the individual, combined with the cognizance that one lacks it in its entirety, that moves not only the soul of the individual, but also the entire cosmos. Virtuous love, which Philo answers in response to Sophia’s fourth question concerning the parents of love, is the highest form of love and, significantly, one can have this for either corporeal or spiritual things. Indeed, Philo intimates that such virtuous love can only emerge from sensible phenomena:[63]

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

When [the soul] perceives a beautiful person whose beauty is in harmony with itself, it recognizes in and through this beauty, divine beauty, in the image of which this person is also made.[64]

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

The goal of Judah’s system is to ascend through this hierarchy, that gateway to which is the sensual enjoyment that one derives from physical objects. Only after this can one appreciate spiritual beauty, an appreciation of which culminates in basking in the divine presence. Judah discusses this process in the following manner:

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

We ought principally to love the higher forms of beauty separated from formless matter and gross corporeality (amiamo le grandi bellezze separate da la deforme material e brutto corpe) , such as the virtues and the sciences, which are ever beautiful and devoid of all ugliness and defect. Here again we may ascend through a hierarchy of beauty from the lesser to the greater (ascendiamo per le minori a le maggiori bellezze) and from the pure to the purest leading to the knowledge and love, not only of the most beautiful intelligences, souls and motors of the heavenly bodies, but also of the highest beauty and of the supremely beautiful, the giver of all beauty, life, intelligence and being. We may scale this ladder only when we put away earthly garments and material affection (potremo fare quando noi abbandonaremo le vesti corporee e le passioni materiali)…[65]

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Even though the corporeal was, at the outset of this journey, indispensable; the higher one moves up the hierarchy the less important the material becomes. As far as the individual is concerned, the highest felicity resides in the union with God, which the Italian describes erotically as felice coppulativa:

Judah Abrabanel

6. The End of Human Life

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Because the love of the human soul is twofold, it is directed not only towards the beauty of the intellect, but also towards the image of that beauty in the body. It happens that at times the love of intellectual beauty is so strong that it draws the soul to cast off all affection for the body; thus the body and soul in man fall apart, and there follows the joyful death in union with the divine (la morte felice coppulativa).[66]

Judah Abrabanel

7. Reception History and Influence

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

One of the most surprising features concerning the reception history of the Dialoghi is that a work of Jewish philosophy would subsequently become a European bestseller among non-Jews. In the years immediately following its Italian publication, the Dialoghi was translated into virtually every European vernacular. This popularity might be the result of the prominent role that grace (grazia) plays in the Dialoghi or the fact that Judah frequently stresses the interlocking relationship between the corporeal and the spiritual, something that seems to have resonated with contemporaneous Christian treatments of the incarnation in literary fiction.[67]

Judah Abrabanel

7. Reception History and Influence

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

This popularity of the work has led some to posit that Judah Abrabanel’s thought is epigonic, responsible for disseminating the thought of “great thinkers” such as Marsilio Ficino (1433–1499) and Giovanni Pico della Mirandola (1463–1494) to a wider audience. This essay’s point of departure, however, has been that the thought of Judah Abrabanel, while dependent upon certain features of these earlier thinkers, nevertheless makes significant departures from them in terms of his construction of an overarching system that revolves around the twin principles of love and beauty

Judah Abrabanel

7. Reception History and Influence

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

Paradoxically, the initial response of Jews to the Dialoghi was for the most part negative. Some of the earliest criticisms, especially those of Saul ha-Kohen Ashkenazi, fault Judah with rationalizing kabbalistic principles. Ashkenazi, in a letter to Don Isaac Abrabanel, criticizes Judah for his lack of philosophical esotericism, and for spending too much time on linguistic matters, such as riddles and eloquence. Such antagonism reflects the broader context of the Maimonidean controversies, in which philosophers sought to make philosophy known to a broader Jewish public often by means of dramatic dialogues or philosophical novels. Those opposed to the Aristotelian-Maimonidean paradigm of philosophy often blamed such treatises for weakening the faith of Jews by diminishing their commitment to the halakhah (law) and, thus, making them more susceptible to conversion. Despite such initial criticisms, however, subsequent generations stressed the interrelationship between Platonism and kabbalah on the one hand, and philosophy and aesthetics on the other. Notable individuals include Azariah de Rossi (d. ca. 1578) and Judah Moscato (d. ca. 1594).

Judah Abrabanel

7. Reception History and Influence

abrabanel

First published Fri Dec 2, 2005; substantive revision Wed Jun 8, 2022

The actual influence that the Dialoghi would have on subsequent thinkers is more difficult to judge. Essentially, Judah adopted certain trajectories of medieval cosmology and psychology, combined them with Renaissance notions of beauty, and thereby created a full-blown aesthetics of Judaism. This led him to conceive of the universe as a living, dynamic structure, in which all levels share in a symbiotic and organic relationship. Unlike many of his medieval predecessors, he did not define this world, the world of form and matter, in terms of privation or its distance from the divine. On the contrary, he envisages this world as the arena wherein individuals encounter, through sensual particulars, the beauty and love of the divine. Abrabanel’s emphasis, like that of many of his Renaissance contemporaries, on aesthetics and the phenomenal world would eventually become an important dimension of 16th- and 17th -century natural philosophy. We do know, for example, that Baruch Spinoza had a copy of the Dialoghi in his library.

Abstract Objects

1. Introduction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

The abstract/concrete distinction has a curious status in contemporary philosophy. It is widely agreed that the ontological distinction is of fundamental importance, but as yet, there is no standard account of how it should be drawn. There is a consensus about how to classify certain paradigm cases. For example, it is usually acknowledged that numbers and the other objects of pure mathematics, like pure sets, are abstract (if they exist), whereas rocks, trees, and human beings are concrete. In everyday language, it is common to use expressions that refer to concrete entities as well as those that apparently refer to abstractions such as democracy, happiness, motherhood, etc. Moreover, formulations of mathematical theories seem to appeal directly to abstract entities, and the use of mathematical expressions in the empirical sciences seems indispensable to the formulation of our best empirical theories (see Quine 1948; Putnam 1971; and the entry on indispensability arguments in the philosophy of mathematics). Finally, apparent reference to abstract entities such as sets, properties, concepts, propositions, types, and possible worlds, among others, is ubiquitous in different areas of philosophy.

Abstract Objects

1. Introduction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Though there is a pervasive appeal to abstract objects, philosophers have nevertheless wondered whether they exist. The alternatives are: platonism, which endorses their existence, and nominalism, which denies the existence of abstract objects across the board. (See the entries on nominalism in metaphysics and platonism in metaphysics.) But the question of how to draw the distinction between abstract and concrete objects is an open one: it is not clear how one should characterize these two categories nor is there a definite list of items that fall under one or the other category (assuming neither is empty).

Abstract Objects

1. Introduction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

The first challenge, then, is to articulate the distinction, either by defining the terms explicitly or by embedding them in a theory that makes their connections to other important categories more explicit. In the absence of such an account, the philosophical significance of the contrast remains uncertain, for the attempt to classify things as abstract or concrete by appeal to intuition is often problematic. Is it clear that scientific theories (e.g., the general theory of relativity), works of fiction (e.g., Dante’s Inferno), fictional characters (e.g., Bilbo Baggins) or conventional entities (e.g., the International Monetary Fund or the Spanish Constitution of 1978) are abstract?

Abstract Objects

1. Introduction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

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

Abstract Objects

1. Introduction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

So before we turn to the discussion of abstract objects in earnest, it will help if we clarify how some of the key terms will be used in what follows.

Abstract Objects

1. Introduction

1.1 About the Expression ‘Object’

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Frege famously distinguished two mutually exclusive ontological domains, functions and objects. According to his view, a function is an ‘incomplete’ entity that maps arguments to values, and is denoted by an incomplete expression, whereas an object is a ‘complete’ entity and can be denoted by a singular term. Frege reduced properties and relations to functions and so these entities are not included among the objects. Some authors make use of Frege’s notion of ‘object’ when discussing abstract objects (e.g., Hale 1987). But though Frege’s sense of ‘object’ is important, it is not the only way to use the term. Other philosophers include properties and relations among the abstract objects. And when the background context for discussing objects is type theory, properties and relations of higher type (e.g., properties of properties, and properties of relations) may be all be considered ‘objects’. This latter use of ‘object’ is interchangeable with ‘entity.’[1] Throughout this entry, we will follow this last usage and treat the expressions ‘object’ and ‘entity’ as having the same meaning. (For further discussion, see the entry on objects.)

Abstract Objects

1. Introduction

1.2 About the Abstract/Concrete Distinction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Though we’ve spoken as if the abstract/concrete distinction must be an exhaustive dichotomy, we should be open to the possibility that the best sharpening of it will entail that some objects are neither abstract nor concrete. Holes and shadows, if they exist, do not clearly belong in either category; nor do ghosts, Cartesian minds, fictional characters,[2] immanent universals, or tropes. The main constraint on an account of the distinction is that it draws a philosophically significant line that classifies at least many of the standard examples in the standard ways. It is not a constraint that everything be shoehorned into one category or the other.

Abstract Objects

1. Introduction

1.2 About the Abstract/Concrete Distinction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Finally, some philosophers see the main distinction not as between abstract and concrete objects but as between abstract objects and ordinary objects, where the distinction is a modal one – ordinary objects are possibly concrete while abstract objects (like the number 1) couldn’t be concrete (Zalta 1983, 1988). In any case, in the following discussion, we shall assume that the abstract/concrete distinction is a division among existing objects, and that any plausible explanation of the distinction should aim to characterize a distinction among such objects.

Abstract Objects

2. Historical Remarks

2.1 The Provenance of the Distinction

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First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

The contemporary distinction between abstract and concrete is not an ancient one. Indeed, there is a strong case for the view that, despite occasional exceptions, it played no significant role in philosophy before the 20th century. The modern distinction bears some resemblance to Plato’s distinction between Forms and Sensibles. But Plato’s Forms were supposed to be causes par excellence, whereas abstract objects are generally supposed to be causally inert. The original ‘abstract’/‘concrete’ distinction was a distinction among words or terms. Traditional grammar distinguishes the abstract noun ‘whiteness’ from the concrete noun ‘white’ without implying that this linguistic contrast corresponds to a metaphysical distinction in what these words stand for. In the 17th century, this grammatical distinction was transposed to the domain of ideas. Locke speaks of the general idea of a triangle which is “neither Oblique nor Rectangle, neither Equilateral, Equicrural nor Scalenon [Scalene]; but all and none of these at once,” remarking that even this idea is not among the most “abstract, comprehensive and difficult” (Essay, IV.vii.9). Locke’s conception of an abstract idea as one that is formed from concrete ideas by the omission of distinguishing detail was immediately rejected by Berkeley and then by Hume. But even for Locke there was no suggestion that the distinction between abstract ideas and concrete or particular ideas corresponds to a distinction among objects. “It is plain, …” Locke writes, “that General and Universal, belong not to the real existence of things; but are Inventions and Creatures of the Understanding, made by it for its own use, and concern only signs, whether Words or Ideas” (III.iii.11).

Abstract Objects

2. Historical Remarks

2.1 The Provenance of the Distinction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

The abstract/concrete distinction in its modern form is meant to mark a line in the domain of objects or entities. So conceived, the distinction becomes a central focus for philosophical discussion primarily in the 20th century. The origins of this development are obscure, but one crucial factor appears to have been the breakdown of the allegedly exhaustive distinction between mental and material objects, which had formed the main division for ontologically-minded philosophers since Descartes. One signal event in this development is Frege’s insistence that the objectivity and aprioricity of the truths of mathematics entail that numbers are neither material beings nor ideas in the mind. If numbers were material things (or properties of material things), the laws of arithmetic would have the status of empirical generalizations. If numbers were ideas in the mind, then the same difficulty would arise, as would countless others. (Whose mind contains the number 17? Is there one 17 in your mind and another in mine? In that case, the appearance of a common mathematical subject matter would be an illusion.) In The Foundations of Arithmetic (1884), Frege concludes that numbers are neither external concrete things nor mental entities of any sort.

Abstract Objects

2. Historical Remarks

2.1 The Provenance of the Distinction

abstract-objects

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Later, in his essay “The Thought” (1918), Frege claims the same status for the items he calls thoughts—the senses of declarative sentences—and also, by implication, for their constituents, the senses of subsentential expressions. Frege does not say that senses are abstract. He says that they belong to a third realm distinct both from the sensible external world and from the internal world of consciousness. Similar claims had been made by Bolzano (1837), and later by Brentano (1874) and his pupils, including Meinong and Husserl. The common theme in these developments is the felt need in semantics and psychology, as well as in mathematics, for a class of objective (i.e., non-mental) non-physical entities. As this new realism was absorbed into English-speaking philosophy, the traditional term ‘abstract’ was enlisted to apply to the denizens of this third realm. In this vein, Popper (1968) spoke of the ‘third world’ of abstract, objective entities, in the broader sense that includes cultural products such as arguments, theories, and works of art.

Abstract Objects

2. Historical Remarks

2.1 The Provenance of the Distinction

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

As we turn to an overview of the current debate, it is therefore important to remember that the use of the terms platonist (for those who affirm the existence of abstract objects) and nominalist (for those who deny existence) is somewhat lamentable, since these words have established senses in the history of philosophy. These terms stood for positions that have little to do with the modern notion of an abstract object. Modern platonists (with a small ‘p’) need not accept any of the distinctive metaphysical and epistemological doctrines of Plato, just as modern nominalists need not accept the distinctive doctrines of the medieval nominalists. Moreover, the literature also contains mention of anti-platonists, many of whom see themselves as fictionalists about abstracta, though this doesn’t help if it turns out that the best analysis of fictions is to regard them as abstract objects. So the reader should therefore be aware that terminology is not always well-chosen and that the terms so used sometimes stand for doctrines that are more restricted than the traditional doctrines that go by the same name. Henceforth, we simply use platonism for the thesis that there exists at least one abstract object, and nominalism for the thesis that the number of abstract objects is exactly zero (Field 1980).

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Before we survey the various proposals for drawing the abstract/concrete distinction, we should briefly say why the distinction has been thought to matter. Among philosophers who take the distinction seriously, it is generally supposed that while concrete objects clearly exist, abstract entities are problematic in distinctive ways and deny the existence of abstract entities altogether. In this section we briefly survey the arguments for nominalism and the responses that platonists have offered. If the abstract objects are unified as a class, it is because they have some feature that generates what seems to be a distinctive problem—a problem that nominalists deem unsolvable and which platonists aim to solve. Before we ask what the unifying feature might be, it may therefore help to characterize the various problems it has been thought to generate.

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

The contemporary debate about platonism developed in earnest when Quine argued (1948) that mathematical objects exist, having changed his mind about the nominalist approach he had defended earlier (Goodman & Quine 1947). Quine’s 1948 argument involves three key premises, all of which exerted significant influence on the subsequent debate: (i) mathematics is indispensable for empirical science; (ii) we should be ontologically committed to the entities required for the truth of our best empirical theories (all of which should be expressible in a first-order language); and (iii) the entities required for the truth on an empirical theory are those in the range of the variables bounded by its first-order quantifiers (i.e., the entities in the domain of the existential quantifier ‘\(\exists x\)’ and the universal quantifier ‘\(\forall x\)’). He concluded that in addition to the concrete entities contemplated by our best empirical science, we must accept the existence of mathematical entities, even if they are abstract (see also Quine 1960, 1969, 1976).

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Quine’s argument initiated a debate that is still alive. Various nominalist responses questioned one or another of the premises in his argument. For instance, Field (1980) challenged the idea that mathematics is indispensable for our best scientific theories—i.e., rejecting (i) above—and thus faced the task of rewriting classical and modern physics in nominalistic terms in order to sustain the challenge. Others have taken on the somewhat less daunting task of accepting (i) but rejecting (ii) and (iii); they’ve argued that even if our best scientific theories, in regimented form, quantify over mathematical entities, this doesn’t entail a commitment to mathematical entities (see Azzouni 1997a, 1997b, 2004; Balaguer 1996, 1998; Maddy 1995, 1997; Melia 2000, 2002; Yablo 1998, 2002, 2005, 2009; Leng 2010.) Colyvan (2010) coined the expression ‘easy-roaders’ for this second group, since they avoided the ‘hard road’ of paraphrasing our best scientific theories in non-mathematical terms.

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

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By contrast, some mathematical platonists (Colyvan 2001; Baker 2005, 2009) have refined Quine’s view by advancing the so-called ‘Enhanced Indispensability Argument’ (though see Saatsi 2011 for a response). Some participants describe the debate in terms of a stalemate they hope to resolve (see Baker 2017, Baron 2016, 2020, Knowles & Saatsi 2019, and Martínez-Vidal & Rivas-de-Castro 2020, for discussion).[3]

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Aside from the debate over Quine’s argument, both platonism and nominalism give rise to hard questions. Platonists not only need to provide a theory of what abstract objects exist, but also an account of how we cognitively access and come to know non-causal, abstract entities. This latter question has been the subject of a debate that began in earnest in Benacerraf (1973), which posed just such a dilemma for mathematical objects. Benacerraf noted that the causal theory of reference doesn’t seem to make it possible to know the truth conditions of mathematical statements, and his argument applies to abstract entities more generally. On the other hand, nominalists need to explain the linguistic uses in which we seem to appeal to such entities, especially those uses in what appear to be good explanations, such as those in scientific, mathematical, linguistic, and philosophical pursuits (see Wetzel 2009, 1–22, for a discussion of the many places where abstract types are used in scientific explanations). Even though nominalists argue that there are no abstracta, the very fact that there is disagreement about their existence suggests that both platonists and nominalists acknowledge the distinction between the abstract and concrete to be a meaningful one.

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

On the platonist side, various proposals have been raised to address the challenge of explaining epistemic access to abstract entities, mostly in connection with mathematical objects. Some, including Gödel (1964), allege that we access abstract objects in virtue of a unique kind of perception (intuition). Maddy (1990, 1997) developed two rather different ways of understanding our knowledge of mathematics in naturalistic ways. Other platonists have argued that abstract objects are connected to empirical entities, either via abstraction (Steiner 1975; Resnik 1982; Shapiro 1997) or via abstraction principles (Wright 1983; Hale 1987); we’ll discuss some of these views below. There are also those who speak of existent and intersubjective abstract entities as a kind of mental representation (Katz 1980).

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

A rather different line of approach to the epistemological problem was proposed in Linsky & Zalta 1995, where it is suggested that one shouldn’t attempt to explain knowledge of abstracta on the same model that is used to explain knowledge of concrete objects. They argue that not only a certain plenitude principle for abstract objects (namely, the comprehension principle for abstract objects put forward in Zalta 1983, 1988—see below) yields unproblematic ‘acquaintance by description’ to unique abstract objects but also that their approach actually comports with naturalist beliefs. Balaguer (1995, 1998) also suggests that a plenitude principle is the best way forward for the platonist, and that our knowledge of the consistency of mathematical theories suffices for knowledge of mathematical objects. And there are views that conceive of abstract objects as constituted by human—or, in general, intelligent—subjects, or as abstract artifacts (see Popper 1968; Thomasson 1999).

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

A number of nominalists have been persuaded by Benacerraf’s (1973) epistemological challenge about reference to abstract objects and concluded that sentences with terms making apparent reference to them—such as mathematical statements—are either false or lack a truth value. They argue that those sentences must be paraphrasable without vocabulary that commits one to any sort of abstract entity. These proposals sometimes suggest that statements about abstract objects are merely instrumental; they serve only to help us establish conclusions about concrete objects. Field’s fictionalism (1980, 1989) has been influential in this regard. Field reconstructed Newtonian physics using second-order logic and quantification over (concrete) regions of space-time. A completely different tactic for avoiding the commitment to abstract, mathematical objects is put forward in Putnam (1967) and Hellman (1989), who separately reconstructed various mathematical theories in second-order modal logic. On their view, abstract objects aren’t in the range of the existential quantifier at the actual world (hence, we can’t say that they exist), but they do occur in the range of the quantifier at other possible worlds, where the axioms of the mathematical theory in question are true.

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

These nominalistic approaches must contend with various issues, of course. At the very least, they have to successfully argue that the tools they use to avoid commitments to abstract objects don’t themselves involve such commitment. For example, Field must argue that space-time regions are concrete entities, while Putnam and Hellman must argue that by relying on logical possibility and modal logic, there is no commitment to possible worlds considered as abstract objects. In general, any nominalist account that makes essential use of set theory or model-theoretic structures must convincingly argue that the very use of such analytic tools doesn’t commit them to abstract objects. (See Burgess & Rosen 1997 for a systematic survey of different proposals about the existence of abstract objects.)

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

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Another nominalistic thread in the literature concerns the suggestion that sentences about (posited) abstract objects are quasi-assertions, i.e., not evaluable as to truth or falsehood (see Yablo 2001 and Kalderon 2005). Still others argue that we should not believe sentences about abstracta since their function, much like the instrumentalism discussed earlier, is to ensure empirical adequacy for observational sentences (Yablo 1998). This may involve differentiating between apparent content, which involves posited abstract objects, and real content, which only concerns concrete objects (Yablo 2001, 2002, 2010, 2014). (For more on these fictionalist accounts, see Kalderon 2005, Ch. 3, and the entry on fictionalism.)

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

A final group of views in the literature represents a kind of agnosticism about what exists or about what it is to be an object, be it abstract or concrete. These views don’t reject an external material world, but rather begin with some question as to whether we can have experience, observation, and knowledge of objects directly, i.e., independent of our theoretical frameworks. Carnap (1950 [1956]), for example, started with the idea that our scientific knowledge has to be expressed with respect to a linguistic framework and that when we wish to put forward a theory about a new kind of entity, we must have a linguistic framework for talking about those entities. He then distinguished two kinds of existence questions: internal questions within the framework about the existence of the new entities and external questions about the reality of the framework itself. If the framework deals with abstract entities such as numbers, sets, propositions, etc., then the internal question can be answered by logical analysis of the rules of the language, such as whether it includes terms for, or implies claims that quantify over, abstract objects. But, for Carnap, the external question, about whether the abstract entities really exist, is a pseudo-question and should be regarded as nothing more than the pragmatic question of whether the framework is a useful one to adopt, for scientific or other forms of enquiry. We’ll discuss Carnap’s view in more detail in subsection 3.7.1.

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Some have thought that Carnap’s view offers a deflationist view of objects, since it appears that the existence of objects is not language independent. After Carnap’s seminal article, several other deflationist approaches were put forward (Putnam 1987, 1990; Hirsch 2002, 2011; Sider 2007, 2009; Thomasson 2015), many of them claiming to be a vindication of Carnap’s view. However, there are deflationist proposals that run counter to Carnap’s approach, among them, deflationary nominalism (Azzouni 2010) or agnosticism about abstract objects (Bueno 2008a, 2008b, 2020). Additionally, philosophers inspired by Frege’s work have argued for a minimal notion of an object (Rayo 2013, Rayo 2020 [Other Internet Resources]; and Linnebo 2018). We’ll discuss some of these in greater detail below, in subsection 3.7.2. A final agnostic position that has emerged is one that rejects a strict version of platonism, but suggests that neither a careful analysis of mathematical practice (Maddy 2011), nor the enhanced version of the indispensability argument (Leng 2020) suffice to decide between nominalism and moderate versions of platonism. Along these lines, Balaguer (1998) concluded that the question doesn’t have an answer, since the arguments for ‘full-blooded’ platonism can be matched one-for-one by equally good arguments by the anti-platonist.

Abstract Objects

2. Historical Remarks

2.2 An Initial Overview of the Contemporary Debate

abstract-objects

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For additional discussion about the basic positions in the debate about abstract and concrete objects, see Szabó 2003 and the entries on nominalism in metaphysics and platonism in metaphysics, nominalism in the philosophy of mathematics and platonism in the philosophy of mathematics.

Abstract Objects

3. What is an Abstract Object?

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As part of his attempt to understand the nature of possible worlds, Lewis (1986a, 81–86) categorizes different ways by which one can draw the abstract/concrete distinction.[4] These include: the way of example (which is simply to list the paradigm cases of abstract and concrete objects in the hope that the sense of the distinction will somehow emerge); the way of conflation (i.e., identifying abstract and concrete objects with some already-known distinction); the way of negation (i.e., saying what abstract objects are by saying what they are not, e.g., non-spatiotemporal, non-causal, etc.); and the way of abstraction (i.e., saying that abstract objects are conceptualized by a process of considering some known objects and omitting certain distinguishing features). He gives a detailed examination of the different proposals that typify these ways and then attempts to show that none of them quite succeeds in classifying the paradigms in accord with prevailing usage. Given the problems he encountered when analyzing the various ways, Lewis became pessimistic about our ability to draw the distinction cleanly.

Abstract Objects

3. What is an Abstract Object?

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Despite Lewis’s pessimism about clarifying the abstract/concrete distinction, his approach for categorizing the various proposals, when extended, is a useful one. Indeed, in what follows, we’ll see that there are a number of additional ways that categorize attempts to characterize the abstract/concrete distinction and theorize about abstract objects. Even if there is no single, acceptable account, these various ways of drawing the distinction and theorizing about abstract objects do often cast light on the questions we’ve been discussing, especially when the specific proposals are integrated into a supplementary (meta-)ontological project. For each method of drawing the distinction and specific proposal adopting that method acquires a certain amount of explanatory power, and this will help us to compare and contrast the various ideas that are now found in the literature.

Abstract Objects

3. What is an Abstract Object?

3.1 The Way of Example and the Way of Primitivism

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According to the way of example, it suffices to list paradigm cases of abstract and concrete entities in the hope that the sense of the distinction will somehow emerge. Clearly, a list of examples for each category would be a heuristically promising start in the search for some criterion (or list of criteria) that would be fruitful for drawing the distinction. However, a simple list would be of limited significance since there are too many ways of extrapolating from the paradigm cases to a distinction that would cover the unclear cases, with the result that no clear notion has been explained.

Abstract Objects

3. What is an Abstract Object?

3.1 The Way of Example and the Way of Primitivism

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For example, pure sets are paradigm examples of abstract entities. But the case of impure sets is far from clear. Consider the unit set whose sole member is Joe Biden (i.e., {Joe Biden}), the Undergraduate Class of 2020 or The Ethics Committee, etc. They are sets, but it is not clear that they are abstract given that Joe Biden, the members of the class and committee are concrete. Similarly, if one offers the characters of Sherlock Holmes stories as examples to help motivate the primitive concept abstract object, then one has to wonder about the object London that appears in the novels.

Abstract Objects

3. What is an Abstract Object?

3.1 The Way of Example and the Way of Primitivism

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The refusal to characterize the abstract/concrete distinction while maintaining that both categories have instances might be called the way of primitivism, whenever the following condition obtains: a few predicates are distinguished as primitive and unanalyzable, and the explanatory power rests on the fact that other interesting predicates can be defined in terms of the primitives and that interesting claims can be judged true on the basis of our intuitive understanding of the primitive and defined notions. Thus, one might take abstract and concrete as primitive notions. It wouldn’t be an insignificant result if one could use this strategy to explain why abstract objects are necessarily existent, causally ineffiacious, non-spatiotemporal, intersubjective, etc. (see Cowling 2017: 92–97).

Abstract Objects

3. What is an Abstract Object?

3.1 The Way of Example and the Way of Primitivism

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But closer inspection of this method reveals some significant concerns. To start with, when a distinction is taken as basic and unanalyzable, one typically has to offer some intuitive instances of the primitive predicates. But it is not always so easy to do this. For example, when mathematicians take set and membership as primitives and then assert some principles of set theory, they often illustrate their primitives by offering some examples of sets, such as The Undergraduate Class of 2020 or The Ethics Committee, etc. But these, of course, aren’t quite right, since the members of the class and committee may change while the class and committee remain the same, whereas if the members of a set change, one has a different set. A similar concern affects the present proposal. If one offers sets or the characters of the Sherlock Holmes novels as examples to help motivate the primitive concept abstract object, then one has to wonder about impure sets such as the unit set whose sole member is Aristotle (i.e., \(\{\textrm{Aristotle}\}\)) and the object London that appears in the novels.

Abstract Objects

3. What is an Abstract Object?

3.2 The Way of Conflation

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According to the way of conflation, the abstract/concrete distinction is to be identified with one or another metaphysical distinction already familiar under another name: as it might be, the distinction between sets and individuals, or the distinction between universals and particulars. There is no doubt that some authors have used the terms in this way. (Thus Quine 1948 uses ‘abstract entity’ and ‘universal’ interchangeably.) This sort of conflation is however rare in recent philosophy.

Abstract Objects

3. What is an Abstract Object?

3.3 The Way of Abstraction

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Another methodology is what Lewis calls the way of abstraction. According to a longstanding tradition in philosophical psychology, abstraction is a distinctive mental process in which new ideas or conceptions are formed by considering the common features of several objects or ideas and ignoring the irrelevant features that distinguish those objects. For example, if one is given a range of white things of varying shapes and sizes; one ignores or abstracts from the respects in which they differ, and thereby attains the abstract idea of whiteness. Nothing in this tradition requires that ideas formed in this way represent or correspond to a distinctive kind of object. But it might be maintained that the distinction between abstract and concrete objects should be explained by reference to the psychological process of abstraction or something like it. The simplest version of this strategy would be to say that an object is abstract if it is (or might be) the referent of an abstract idea; i.e., an idea formed by abstraction. So conceived, the way of abstraction is wedded to an outmoded philosophy of mind.

Abstract Objects

3. What is an Abstract Object?

3.3 The Way of Abstraction

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It should be mentioned, though, that the key idea behind the way of abstraction has resurfaced (though transformed) in the structuralist views about mathematics that trace back to Dedekind. Dedekind thought of numbers by the way of abstraction. Dedekind suggested that when defining a number-theoretic structure, “we entirely neglect the special character of the elements, merely retaining their distinguishability and taking into account only the relations to one another” (Dedekind 1888 [1963, 68]). This view has led some structuralists to deny that numbers are abstract objects. For example, Benacerraf concluded that “numbers are not objects at all, because in giving the properties (that is, necessary and sufficient) of numbers you merely characterize an abstract structure—and the distinction lies in the fact that the ‘elements’ of the structure have no properties other than those relating them to other ‘elements’ of the same structure” (1965, 70). We shall therefore turn our attention to a variant of the way of abstraction, one that has led a number of philosophers to conclude that numbers are indeed abstract objects.

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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In the contemporary philosophical literature, a number of books and papers have investigated a form of abstraction that doesn’t depend on mental processes. We may call this the way of abstraction principles. Wright (1983) and Hale (1987) have developed an account of abstract objects on the basis of an idea they trace back to certain suggestive remarks in Frege (1884). Frege notes (in effect) that many of the singular terms that appear to refer to abstract entities are formed by means of functional expressions. We speak of the shape of a building, the direction of a line, the number of books on the shelf. Of course, many singular terms formed by means of functional expressions denote ordinary concrete objects: ‘the father of Plato’, ‘the capital of France’. But the functional terms that pick out abstract entities are distinctive in the following respect: where \(f(a)\) is such an expression, there is typically an equation of the form:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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where \(R\) is an equivalence relation, i.e., a relation that is reflexive, symmetric and transitive, relative to some domain. For example:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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The direction of \(a\) = the direction of \(b\) if and only if \(a\) is parallel to \(b\)

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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The number of \(F\text{s}\) = the number of \(G\text{s}\) if and only if there are just as many \(F\text{s}\) as \(G\text{s}\)

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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These biconditionals (or abstraction principles) appear to have a special semantic status. While they are not strictly speaking definitions of the functional expression that occurs on the left hand side, they would appear to hold in virtue of the meaning of that expression. To understand the term ‘direction’ is (in part) to know that the direction of \(a\) and the direction of \(b\) are the same entity if and only if the lines \(a\) and \(b\) are parallel. Moreover, the equivalence relation that appears on the right hand side of the biconditional would appear to be semantically and perhaps epistemologically prior to the functional expressions on the left (Noonan 1978). Mastery of the concept of a direction presupposes mastery of the concept of parallelism, but not vice versa.

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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The availability of abstraction principles meeting these conditions may be exploited to yield an account of the distinction between abstract and concrete objects. When ‘\(f\)’ is a functional expression governed by an abstraction principle, there will be a corresponding kind \(K_{f}\) such that:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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\(x\) is a \(K_{f}\) if and only if, for some \(y\), \(x\! =\! f(y)\).

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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For example,

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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\(x\) is a cardinal number if and only if for some concept \(F\), \(x\) = the number of \(F\text{s}\).

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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The simplest version of the way of abstraction principles is then to say that:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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\(x\) is an abstract object if (and only if) \(x\) is an instance of some kind \(K_{f}\) whose associated functional expression ‘\(f\)’ is governed by a suitable abstraction principle.

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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The strong version of this account—which purports to identify a necessary condition for abstractness—is seriously at odds with standard usage. Pure sets are usually considered paradigmatic abstract objects. But it is not clear that they satisfy the proposed criterion. According to a version of naïve set theory, the functional expression ‘set of’ is indeed characterized by a putative abstraction principle.

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

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The set of \(F\text{s}\) = the set of \(G\text{s}\) if and only if, for all \(x\), \(x\) is \(F\) if and only if \(x\) is \(G\).

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

But this principle, which is a version of Frege’s Basic Law V, is inconsistent and so fails to characterize an interesting concept. In contemporary mathematics, the concept of a set is not introduced by an abstraction principle, but rather axiomatically. Though attempts have been made to investigate abstraction principles for sets (Cook 2003), it remains an open question whether something like the mathematical concept of a set can be characterized by a suitably restricted abstraction principle. (See Burgess 2005 for a survey of recent efforts in this direction.) Even if such a principle is available, however, it is unlikely that the epistemological priority condition will be satisfied. That is, it is unlikely that mastery of the concept of set will presuppose mastery of the equivalence relation that figures on the right hand side. It is therefore uncertain whether the way of abstraction principles will classify the objects of pure set theory as abstract entities (as it presumably must).

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

On the other hand, as Dummett (1973) has noted, in many cases the standard names for paradigmatically abstract objects do not assume the functional form to which the definition adverts. Chess is an abstract entity, but we do not understand the word ‘chess’ as synonymous with an expression of the form ‘\(f(x)\)’, where ‘\(f\)’ is governed by an abstraction principle. Similar remarks would seem to apply to such things as the English language, social justice, architecture, and Charlie Parker’s musical style. If so, the abstractionist approach does not provide a necessary condition for abstractness as that notion is standardly understood.

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

More importantly, there is some reason to believe that it fails to supply a sufficient condition. A mereological fusion of concrete objects is itself a concrete object. But the concept of a mereological fusion is governed by what appears to be an abstraction principle:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

The fusion of the \(F\text{s}\) = the fusion of the \(G\text{s}\) if and only if the \(F\text{s}\) and \(G\text{s}\) cover one another,

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

where the \(F\text{s}\) cover the \(G\text{s}\) if and only if every part of every \(G\) has a part in common with an \(F\). Similarly, suppose a train is a maximal string of railroad carriages, all of which are connected to one another. We may define a functional expression, ‘the train of \(x\)’, by means of an ‘abstraction’ principle: The train of \(x\) = the train of \(y\) if and only if \(x\) and \(y\) are connected carriages. We may then say that \(x\) is a train if and only if for some carriage \(y\), \(x\) is the train of \(y\). The simple account thus yields the consequence that trains are to be reckoned abstract entities.

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

It is unclear whether these objections apply to the more sophisticated abstractionist proposals of Wright and Hale, but one feature of the simple account sketched above clearly does apply to these proposals and may serve as the basis for an objection to this version of the way of abstraction principles. The neo-Fregean approach seeks to explain the abstract/concrete distinction in semantic terms: We said that an abstract object is an object that falls in the range of a functional expression governed by an abstraction principle, where ‘\(f\)’ is governed by an abstraction principle when that principle holds in virtue of the meaning of ‘\(f\)’. This notion of a statement’s holding in virtue of the meaning of a word is notoriously problematic (see the entry the analytic/synthetic distinction). But even if this notion makes sense, one may still complain: The abstract/concrete distinction is supposed to be a metaphysical distinction; abstract objects are supposed to differ from concrete objects in some important ontological respect. It should be possible, then, to draw the distinction directly in metaphysical terms: to say what it is in the objects themselves that makes some things abstract and others concrete. As Lewis writes, in response to a related proposal by Dummett:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Even if this … way succeeds in drawing a border, as for all I know it may, it tells us nothing about how the entities on opposite sides of that border differ in their nature. It is like saying that snakes are the animals that we instinctively most fear—maybe so, but it tells us nothing about the nature of snakes. (Lewis 1986a: 82)

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

The challenge is to produce a non-semantic version of the abstractionist criterion that specifies directly, in metaphysical terms, what the objects whose canonical names are governed by abstraction principles all have in common.

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

One response to this difficulty is to transpose the abstractionist proposal into a more metaphysical key (see Rosen & Yablo 2020). The idea is that each Fregean number is, by its very nature, the number of some Fregean concept, just as each Fregean direction is, by its very nature, at least potentially the direction of some concrete line. In each case, the abstract object is essentially the value of an abstraction function for a certain class of arguments. This is not a claim about the meanings of linguistic expressions. It is a claim about the essences or natures of the objects themselves. (For the relevant notion of essence, see Fine 1994.) So for example, the Fregean number two (if there is such a thing) is, essentially, by its very nature, the number that belongs to a concept \(F\) if and only if there are exactly two \(F\text{s}\). More generally, for each Fregean abstract object \(x\), there is an abstraction function \(f\), such that \(x\) is essentially the value of \(f\) for every argument of a certain kind.

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Abstraction functions have two key features. First, for each abstraction function \(f\) there is an equivalence relation \(R\) such that it lies in the nature of \(f\) that \(f(x)\! =\! f(y)\) iff \(Rxy\). Intuitively, we are to think that \(R\) is metaphysically prior to \(f\), and that the abstraction function \(f\) is defined (in whole or in part) by this biconditional. Second, each abstraction function is a generating function: its values are essentially values of that function. Many functions are not generating functions. Paris is the capital of France, but it is not essentially a capital. The number of solar planets, by contrast, is essentially a number. The notion of an abstraction function may be defined in terms of these two features:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

We may then say that:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

\(x\) is an abstraction if and only if, for some abstraction function \(f\), there is or could be an object \(y\) such that \(x\! =\! f(y)\),

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

and that:

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

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

Abstract Objects

3. What is an Abstract Object?

3.4 The Way of Abstraction Principles

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

This account tells us a great deal about the distinctive natures of these broadly Fregean abstract objects. It tells us that each is, by its very nature, the value of a special sort of function, one whose nature is specified in a simple way in terms of an associated equivalence relation. It is worth stressing, however, that it does not supply much metaphysical information about these items. It doesn’t tell us whether they are located in space, whether they can stand in causal relations, and so on. It is an open question whether this somewhat unfamiliar version of the abstract/concrete distinction lines up with any of the more conventional ways of drawing the distinction outlined above. An account along these lines would be at odds with standard usage, but may be philosophically interesting all the same. In any case, the problem remains that this metaphysical version of the way of abstraction principles leaves out paradigmatic cases of abstract objects such as the aforementioned game of chess.

Abstract Objects

3. What is an Abstract Object?

3.5 The Ways of Negation

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

According to the way of negation, abstract objects are defined as those which lack certain features possessed by paradigmatic concrete objects. Many explicit characterizations in the literature follow this model. Let us review some of the options.

Abstract Objects

3. What is an Abstract Object?

3.5 The Ways of Negation

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

According to the account implicit in Frege’s writings:

Abstract Objects

3. What is an Abstract Object?

3.5 The Ways of Negation

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

An object is abstract if and only if it is both non-mental and non-sensible.

Abstract Objects

3. What is an Abstract Object?

3.5 The Ways of Negation

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

Here the first challenge is to say what it means for a thing to be ‘non-mental’, or as we more commonly say, ‘mind-independent’. The simplest approach is to say that a thing depends on the mind when it would not (or could not) have existed if minds had not existed. But this entails that tables and chairs are mind-dependent, and that is not what philosophers who employ this notion have in mind. To call an object ‘mind-dependent’ in a metaphysical context is to suggest that it somehow owes its existence to mental activity, but not in the boring ‘causal’ sense in which ordinary artifacts owe their existence to the mind. What can this mean? One promising approach is to say that an object should be reckoned mind-dependent when, by its very nature, it exists at a time if and only if it is the object or content of some mental state or process at that time. This counts tables and chairs as mind-independent, since they might survive the annihilation of thinking things. But it counts paradigmatically mental items, like a purple afterimage of which a person \(X\) may become aware, as mind-dependent, since it presumably lies in the nature of such items to be objects of conscious awareness whenever they exist. However, it is not clear that this account captures the full force of the intended notion. Consider, for example, the mereological fusion of \(X\)’s afterimage and \(Y\)’s headache. This is surely a mental entity if anything is. But it is not necessarily the object of a mental state. (The fusion can exist even if no one is thinking about it.) A more generous conception would allow for mind-dependent objects that exist at a time in virtue of mental activity at that time, even if the object is not the object of any single mental state or act. The fusion of \(X\)’s afterimage and \(Y\)’s headache is mind-dependent in the second sense but not the first. That is a reason to prefer the second account of mind-dependence.

Abstract Objects

3. What is an Abstract Object?

3.5 The Ways of Negation

abstract-objects

First published Thu Jul 19, 2001; substantive revision Mon Aug 9, 2021

If we understand the notion of mind-dependence in this way, it is a mistake to insist that abstract objects be mind-independent. To strike a theme that will recur, it is widely supposed that sets and classes are abstract entities—even the impure sets whose urelements are concrete objects. Any account of the abstract/concrete distinction that places set-theoretic constructions like \(\{\textrm{Alfred}, \{\textrm{Betty}, \{\textrm{Charlie}, \textrm{Deborah}\}\}\}\) on the concrete side of the line will be seriously at odds with standard usage. With this in mind, consider the set whose sole members are X’s afterimage and Y’s headache, or some more complex set-theoretic object based on these items. If we suppose, as is plausible, that an impure set exists at a time only when its members exist at that time, this will be a mind-dependent entity in the generous sense. But it is also presumably an abstract entity.