Abstraction
Abstraction
(Lat. abs, from trahere, to draw).
Abstraction is a process (or a faculty) by which the mind selects for consideration some one of the attributes of a thing to the exclusion of the rest. With some writers, including the Scholastics, the attributes selected for attention are said to be abstracted; with others, as Kant and Hamilton, the term is applied to the exclusion of the attributes which are ignored; the process, however, is the same in both cases. The simplest-seeming things are complex, i.e. they have various attributes; and the process of abstraction begins with sensation, as sight perceives certain qualities; taste, others; etc. From the dawn of intelligence the activity progresses rapidly, as all of our generalizations depend upon the abstraction from different objects of some phase, or phases, which they have in common. A further and most important step is taken when the mind reaches the stage where it can handle its abstractions such as extension, motion, species, being, cause, as a basis for science and philosophy, in which, to a certain extent at least, the abstracted concepts are manipulated like the symbols in algebra, without immediate reference to the concrete. This process is not without its dangers of fallacy, but human knowledge would not progress far without it. It is, therefore, evident that methods of leading the mind from the concrete to the abstract, as well as the development of a power of handling abstract ideas, are matters of great importance in the science of education.
With this account of the place of abstraction in the process of knowledge, most philosophers — and all who base knowledge on experience — are in substantial agreement. But they differ widely concerning the nature and validity of abstract concepts themselves. A widely prevalent view, best represented by the Associationist school, is that general ideas are formed by the blending or fusing of individual impressions. The most eminent Scholastics, however, following Aristotle, ascribe to the mind in its higher aspect a power (called the Active Intellect) which abstracts from the representations of concrete things or qualities the typical, ideal, essential elements, leaving behind those that are material and particular. The concepts thus formed may be very limited in content, and they vary in number and definiteness with the knowledge of particulars; but the activity of the faculty is always spontaneous and immediate; it is never a process of blending the particular representations into a composite idea, much less a mere grouping of similar things or attributes under a common name. The concept thus obtained represents an element that is universally realized in all members of the class, but it is recognized formally as a universal only by means of further observation and comparison. The arguments for the existence of such a faculty are not drawn from a study of its actual operation, which eludes our powers of introspection, but from an analysis of its results. Its defenders rely mainly on the fact that we possess definite universal concepts, as of a triangle, which transcend the vague floating images that represent the fusion of our individual representations; and also on the element of universality and necessity in our judgments. It is in connection with this latter point that the question is of most importance, as systems of philosophy which reject this power of direct abstraction of the universal idea are naturally more or less sceptical about the objective validity of our universal judgments.
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Porter, The Human Intellect (New York, 1869), 377-430; Maher, Psychology (London and New York, 1900), 294, 307, 310; Spencer, Psychology (New York, 1898), I, viii; Mill, Logic (London and New York, 1898), I, ii; IV, ii; Mivart, The Origin of Human Reason (London, 1889), ii; Van Becelaere, The Philos. Rev., Nov., 1903; Newman, Grammar of Assent (London 1898), viii; Bowne, Theory of Thought and Knowledge (New York, 1897), xi; Bain, Education as a Science (New York, 1879), vii; Sully, Teacher’s Psychology (New York, 1887), xii, xiii.
F.P. DUFFY Transcribed by Michael Mueller
The Catholic Encyclopedia, Volume ICopyright © 1907 by Robert Appleton CompanyOnline Edition Copyright © 2003 by K. KnightNihil Obstat, March 1, 1907. Remy Lafort, S.T.D., CensorImprimatur. +John Cardinal Farley, Archbishop of New York
Fuente: Catholic Encyclopedia
Abstraction
(Lat. ab, from + trahere, to draw) The process of ideally separating a partial aspect or quality from a total object. Also the result or product of mental abstraction. Abstraction, which concentrates its attention on a single aspect, differs from analysis which considers all aspects on a par. — L.W.
In logicGiven a relation R which is transitive, symmetric, and reflexive, we may introduce or postulate “new elements corresponding to the members of the field of R, in such a way that the same new element corresponds to two members x and y of the field of R if and only if xRy (see the article relation). These new elements are then said to be obtained by abstraction with respect to R. Peano calls this a method or kind of definition, and speaks, e.g., of cardinal numbers (q.v.) as obtained from classes by abstraction with respect to the relation of equivalence — two classes having the same cardinal number if and only if they are equivalent.
Given a formula A containing a free variable, say x, the process of forming a corresponding monadic function (q.v.) — defined by the rule that the value of the function for an argument b is that which A denotes if the variable x is taken as denoting b — is also called abstraction, or functional abstraction. In this sense, abstraction is an operation upon a formula A yielding a function, and is relative to a particular system of interpretation for the notations appearing in the formula, and to a particular variable, as x. The requirement that A shall contain x as a free variable is not essentialwhen A does not contain x as a free variable, the function obtained by abstraction relative to x may be taken to be the function whose value, the same for all arguments, is denoted by A.
In articles herein by the present writer, the notation ?x[A] will be employed for the function obtained from A by abstraction relative to (or, as we may also say, with respect to) x. Russell, and Whitehead and Russell in Principia Mathematica, employ for this purpose the formula A with a circumflex placed over each (free) occurrence of x — but only for propositional functions. Frege (1893) uses a Greek vowel, say e, as the variable relative to which abstraction is made, and employs the notation e(A) to denote what is essentially the function in extension (the “Werthverlauf” in his terminology) obtained from A by abstraction relative to e.
There is also an analogous process of functional abstraction relative to two or more variables (taken in a given order), which yields a polyadic function when applied to a formula A.
Closely related to the process of functional abstraction is the process of forming a class by abstraction from a suitable formula A relative to a particular variable, say x. The formula A must be such that (under the given system of interpretation for the notations appearing in A) ?x[A] denotes a propositional function. Then x?(A) (Peano), or x (A) (Russell), denotes the class, determined by this propositional function. Frege’s e (A) also belongs here, when the function corresponding to A (relatively to the variable e) is a propositional function.
Similarly, a relation in extension may be formed by abstraction from a suitable formula A relative to two particular variables taken in a given order. — A.C.
Scholz and Schweitzer,
Die sogenannten Definitionen durch Abstraktion, Leipzig, 1935.
W. V. Quine,
A System of Logistic, Cambridge, Mass., 1934.
A. Church,
review of the preceding, Bulletin of the American Mathematical Society, vol. 4l (1935), pp. 498-603.
W. V. Quine,
Mathematical Logic, New York, 1940.
In psychologythe mental operation by which we proceed from individuals to concepts of classes, from individual dogs to the notion of “the dog.” We abstract features common to several individuals, grouping them thus together under one name.
In Scholasticismthe operation by which the mind becomes cognizant of the universal (q.v.) as represented by the individuals. Aristotle and Thomas ascribe this operation to the active intellect (q.v.) which “illuminates” the image (phantasm) and disengages from it the universal nature to be received and made intelligible by the possible intellect. — R.A.