CHAPTER 1

A

ABSTRACTION, ABSTRACTA,ABSTRACT ENTITIES

FRENCH abstraction, abstrait

GERMAN Abstraktion, Entbildung

GREEK aphairesis [TEXT NOT REPRODUCIBLE IN ASCII]

LATIN abstractio, ablatio, absolutio, abnegatio; separata,abstracta

CATEGORY, EPOCHÊ, ESSENCE, FICTION, IMAGINATION, INTELLECTUS,INTENTION, NEGATION, NOTHING, REALITY, RES, SEIN, SUBJECT, UNIVERSALS

While the meaning of the term "abstraction" is not a problem in formallogic, where it refers to the operation that makes it possible toconstruct, using an "abstractor," a so-called "abstract" expression onthe basis of another expression containing one or more free variables,the term's semantic field in philosophy and the theory of knowledgeis more difficult to organize. When Condillac (L'Art de penser I.viii) denounces"the abuse of constructed abstract notions," and "in order toavoid this problem" asks that we look back to "the generation of all ourabstract notions, ... a method that has been unknown to philosophers, ...who have sought to make up for it by means of definitions," his aim is differentfrom that of Aristotle when the latter mentions, under therubric "abstract entities" or "things that exist in the abstract [TEXT NOT REPRODUCIBLE IN ASCII],"the forms that mathematical science deals with "byabstracting from their inherent matter" (Aristotle, De anima, 431b.13–17),and from that of Dionysius the Areopagite when he asks to be raised bythought to the superessential "through the aphairesis [TEXT NOT REPRODUCIBLE IN ASCII] of allbeings." Thus when speaking of "abstraction" we must distinguish theproblem of the generation of abstract ideas insofar as it involves that ofuniversals, that of the existence or nonexistence of general objects, andthat of the practice of abstractive negation in the diverse fields—loglcal,epistemological, theological—where it occurs. The broad range of theterm "abstraction" is well illustrated by the modern English usage of theterms "abstracta" and "abstract entities," which are more or less synonymouswith "universals," and whose extension includes mathematicalobjects (numbers, classes, sets), geometrical figures, propositions, properties,and relations. Although English-language historiography has atendency to regard Plato's Ideas or Forms as the first occurrence of real,non-spatio-temporal "abstract" entities, instantiated or participatedin by spatio-temporal objects, it seems more precise to reserve thisterm for "Aristotelian" ontology by distinguishing, as was done duringthe Middle Ages, separate entities (separata) from abstract entities(abstracta).

I. Epagôgê and Aphairesis, Two Models ofAbstraction according to Aristotle

There are two models of abstraction in Aristotelianism. Thefirst is that of "abstractive induction" (epagôgê [TEXT NOT REPRODUCIBLE IN ASCII]),which Aristotle describes this way:

So out of sense-perception comes to be what we callmemory, and out of frequently repeated memories ofthe same thing develops experience; for a number ofmemories constitute a single experience. From experienceagain—i.e. from the universal now stabilized in itsentirety within the soul, the one beside the many whichis a single identity within them all—originate the skill ofthe craftsman and the knowledge of the man of science,skill in the sphere of coming to be and science in thesphere of being.

(trans. G.R.G. Mure, Posterior Analytics, II.§19)

The second model is that of mathematical (chieflygeometrical) abstraction, which consists not in "bringingtogether" (epagein [TEXT NOT REPRODUCIBLE IN ASCII]) similar elements andgrouping them under a single concept, but in "stripping"(aphaireisthai [TEXT NOT REPRODUCIBLE IN ASCII]) the image or representationof a thing of its individualizing characteristics (essentiallymaterial).

The conflict between these two models is a structuralgiven, a major tendency in Aristotelianism, whose effectsmade themselves felt throughout the Middle Ages. Philosophershave never ceased to vacillate between the registrationof resemblances (the basis of "resemblance nominalism")and the neutralization of individualizing characteristics thatare not pertinent for the type, though some have sought tofind unlikely compromises between these poles.

See Box 1.

II. The Peripatetic Theory of Aphairesis and ItsMedieval Extensions: "Abstractionism"

A. The classification of the sciences

In his treatise De caelo (III.§1.299a 15–17), Aristotle uses theterm "abstraction" to distinguish between "mathematicalobjects" (ta ex aphaireseôs [TEXT NOT REPRODUCIBLE IN ASCII], lit. "proceedingfrom a subtraction") and "physical objects" (ta ek prostheseôs[TEXT NOT REPRODUCIBLE IN ASCII], lit. "proceeding from an addition"). Nonetheless,it is only in De anima (III.§7.431b.12–16) that Aristotleexplains how the intellect conceives abstractions:

As for so-called "abstractions" (ta en aphairesei legomena[TEXT NOT REPRODUCIBLE IN ASCII]), the intellect thinksof them as one would think of the snub-nosed (simon[TEXT NOT REPRODUCIBLE IN ASCII]): qua snub-nosed, one would think of it not asseparate (ou kechôrismenôs [TEXT NOT REPRODUCIBLE IN ASCII]) but asconcave (koilon [TEXT NOT REPRODUCIBLE IN ASCII]), if one thought of it in action(energeiai [TEXT NOT REPRODUCIBLE IN ASCII]), one would think of it withoutthe flesh in which the concavity is realized (aneu têssarkos an enoei en hêi to koilon [TEXT NOT REPRODUCIBLE IN ASCII]):so is it when the intellect thinks ofabstract terms, it thinks of mathematical things as ifthey were separate, even though they are not separate(ou kechôrismena hôs kechôrismena [TEXT NOT REPRODUCIBLE IN ASCII]).

In Michael Scot's Latin translation of Averroës's longcommentary on De anima, the expressions used in De animaIII.§4.429b.18–22 and III.§7.431b.12–16 are rendered respectivelyby "things that exist in mathesis" and "things thatare said negatively." Averroës notes that by "things that aresaid negatively"...