Abstraction: precisive and nonprecisive

On the distinction between

precisive and nonprecisive abstraction

3 excerpts from Roderick T. Long *

From “Review: The Benefits and Hazards of Dialectical Libertarianism.” (Reviewing Total Freedom: Toward a Dialectical Libertarianism by Chris Matthew Sciabarra) The Journal of Ayn Rand Studies Vol. 2, No. 2, The Aesthetics Symposium (Spring 2001): p.408-418 [JSTOR]

Aristotle's theory of abstraction was formed in response to that of Plato, who, like the Hegelians, had thought that abstractions do not strictly apply to concrete reality. Unlike the Hegelians, however, Plato had concluded, not that abstraction falsifies, but rather that the reality to which abstractions apply is itself an abstract reality, a realm of Platonic Forms. Aristotle rejects Platonic Forms; only concrete particulars are real. But for Aristotle, abstraction applies perfectly adequately to concrete particulars. Plato had maintained that mathematics deals with immaterial entities apart from the physical world; Aristotle, by contrast, maintains that mathematics deals with the same objects that physics does-it simply deals with them qua mathematical rather than qua physical (Physics 409 193b22-36; Metaphysics 1076a33-1078b6).

The medieval Aristotelians drew a helpful distinction between precisive and non-precisive abstraction:

" Precision is a mode of abstraction by which we cut off or exclude something from a notion. Abstraction is the consideration of something without either including or excluding from its notion characteristics joined to it in reality. Abstraction without precision does not exclude anything from what it abstracts, but includes the whole thing, though implicitly and indeterminately. "
(Armand Maurer note to Aquinas 1968, 39n)

On the medieval view, the concept soul and the concept angel are both formed by focusing on the psychological characteristics of human beings in abstraction from their physical characteristics. But angel is a precisive abstraction from which physical characteristics are expressly excluded, while soul is a non-precisive abstraction in which physical characteristics are simply not specified one way or the other. In forming the concept soul we do not thereby commit ourselves to the soul's separability from the body. The scholastics of course believed in the separability of the soul, but they thought this separability has to be argued for; it is not inherent in the very concept of soul. By contrast, separability from matter is inherent in the concept of angel An angel is conceived of as not physical; a soul is conceived of not as physical. In Objectivist terms, the distinction can be seen as one between measurement-omission and measurement-exclusion. Aquinas writes:

Abstraction may occur in two ways. First . . . we may under-stand that one thing does not exist in some other, or that it is separate from it. Secondly . . . we understand one thing without considering another. Thus, for the intellect to abstract one from another things which are not really abstract from one another, does, in the first mode of abstraction, imply falsehood. But, in the second mode of abstraction, for the intellect to abstract things which are not really abstract from one another, does not involve falsehood. . . . If, therefore, the intellect is said to be false when it understands a thing otherwise than as it is, that is so, if the word otherwise refers to the thing understood.... Hence, the intellect would be false if it abstracted the species of a stone its matter in such a way as to think that the species did not exist in matter, as Plato held. But it is not so, if otherwise be taken as referring to the one who understands. (Aquinas 1999, 157; Summa Theologia I. 85.1 ad 1)

When we form the concepts angel and soul, we are considering psychological characteristics otherwise than as we find them in reality, but in two different ways. In the case of angel we are considering psychological characteristics as though they themselves existed differently from the way we find them to exist in our experience, namely, in matter, this is what Aquinas means by "otherwise" referring to the thing understood. But in the case of soul we are not considering the psychological characteristics to exist without matter: rather, we are considering them without considering matter; this is what Aquinas means by "otherwise" referring to the one who understands. The process by which the concept soul is formed guarantees that the concept applies to reality, for there is no conflict between the concept and the concretes from which it was formed; for the concept angel this is not so, and hence the applicability of this concept to reality requires further proof.

One can now see that the Hegelians, in treating all abstraction as falsification, were failing to grasp the distinction between precisive and non-precisive abstraction. There is nothing "provisional" or "approximate" about non-precisive abstraction; it is perfectly accurate. It may not express the entire truth, but what it does express is entirely true. In fact, the Aristotelian approach can be seen as- what else? —a dialectical transcendence of the false alternatives of Platonism and Hegelianism:

     Hegelianism:

1. Precisive abstraction always falsifies.

2. But all abstraction is precisive abstraction.

3. Therefore, abstraction always falsifies.

     Platonism:

1. Abstraction does not always falsify.

2. But all abstraction is precisive abstraction.

3. Therefore, precisive abstraction does not always falsify.

     Aristotelianism:

1. Abstraction does not always falsify.

2. But precisive abstraction always falsifies.

3. Therefore, not all abstraction is precisive abstraction.

The crucial premise that unites Platonism and Hegelianism is the premise that all abstraction is precisive. That is not to say that either side formulated that premise explicitly; to do that, they would have had to grasp the difference between precisive and non-precisive abstraction, and grasping that difference is the solution to the problem. Aristotelianism grasps it, and so is in a position to reject the mistaken assumption that lies at the root of the false dichotomy Another way of making the distinction clear is to recall the criticisms leveled by economists of the Austrian School against the idealized models of neoclassical economics, such as "perfect competition" and homo economicus. The criticism is not simply that these models abstract from what human beings are like in their full concreteness; after all, Austrian theory does that too. (Austrian praxeology, for example, considers the logical features of human action in abstraction from their psychological causes.) The criticism is rather that the neoclassical models distort and falsify reality by prescinding (precisively abstracting) from, and thus treating human beings as lacking, certain features (ignorance, non-monetary goals) that they actually possess. Of course, idealizations —when they are not too inaccurate— can sometimes be useful, but only if we keep in mind that they are merely provisional approximations. By contrast, praxeological principles —e.g, that voluntary action always involves an exchange of what the agent wants less for what the agent wants more — are not "provisional" or "approximate," for although they abstract from concrete human reality, they do so non-precisively, and so do not "idealize" or "falsify" the concretes from which they were formed. Hegelian internalism treats all abstractions as though they were idealizations.

On this issue, Hayek and Rand are both squarely in the Aristotelian rather than the Hegelian camp. Both believe that, metaphysically, only concretes exist; thus they reject Platonism. But both are also concerned to defend the validity of abstraction. . . . Hayek sees the internalist cult of concreteness as implicitly totalitarian, and explicitly identifies Hegel as his antagonist on this issue:

[A]bstract concepts are a means to cope with the complexity of the concrete which our mind is not capable of fully mastering. . . . Abstractness [is] the basis of man's capacity to move successfully in a world very imperfectly known to him — an adaption to his ignorance of most of the particular facts of his surroundings. . . . We never act, and never could act, in full consideration of all the facts of a particular situation, but always by singling out as relevant only some aspects of it. . . . The rationalist revolt against reason, if we may so call it, is usually directed against the abstractness of thought. . . . Although the use of abstraction extends the scope of phenomena which we can master intellectually, it does so by limiting the degree to which we can foresee the effects of our actions, and therefore also by limiting to certain general features the degree to which we can shape the world to our liking. . . . Perhaps nobody has seen this connection between [classical] liberalism and the insight into the limited powers of abstract thinking more clearly than that ultra-rationalist who has become the fountain head of most modern irrationalism and totalitarianism, G. W. F. Hegel. . . . It is the over-estimation of the powers of reason to the revolt against the submission to abstract rules. Constructivist rationalism . . . deceives itself that reason can directly master all the particulars; and it is thereby led to a preference for the concrete over the abstract, the particular over the general. . . . [T]he very over-estimation of those powers of reason of which man is conscious has led him to hold in contempt what has made reason as powerful as it is: its abstract character. It was the failure to recognize that abstractions help our reason go further than it could if it tried to master all the particulars which produced a host of schools of philosophy inimical to abstract reason —philosophies of the concrete, of life' and of 'existence' which extol emotion, the particular and the instinctive, and which are only too ready to support such emotions as those of race, nation, and class. (Hayek 1973, 29-34; cf. Hayek 1976, ch 7)

Discussing Comte, whom he regards as Hegel's spiritual sibling, Hayek writes:

Comte and many others regard social phenomena as given wholes... contending that concrete social phenomena can be understood only by considering the totality of everything that can be found within certain spatio-temporal boundaries, that any attempt to select parts or aspects as systematically connected is bound to fail.... Having been led [to] the view that the individual is "a pure abstraction" and society as a whole a single collective being, [Comte] is of necessity led to... a totalitarian view of society. (Hayek 1979a, 103, 354)

In short, Hayek recognizes the connection between totalitarian concreteness-worship, metaphysical internalism, and the tendency to treat all abstraction as precisive and therefore falsifying.

Rand, like Hayek, sees abstraction as a means of reducing cognitive complexity: "[T]he range of what man can hold in the focus of his conscious awareness at any given moment, is limited. The essence, therefore, of man's incomparable cognitive power is the ability to reduce a vast amount of information to a minimal number of units —which is the task performed by his conceptual faculty" (Rand 1990, 63).

Of course, the Hegelians too can claim to endorse abstract aids to cognitive economy —but only as precisive idealizations and provisional approximations. For Rand, however, abstraction is non- precisive and therefore does not falsify: "If a child considers a match, a pencil and a stick, he observes that length is the attribute they have in common, but their specific lengths differ. . . . In order to form the concept 'length,' the child's mind retains the attribute and omits its particular measurements. . . . Bear firmly in mind that the term ‘measurements omitted’ does not mean, in this context, that measurements are regarded as non-existent; it means that measurements exist, but are not specified. . . . The principle is: the relevant measurements must exist in some quantity, but may exist in any quantity" (11-12).

Rand regards abstraction as contextual, but by this she does not mean that it is provisional, to be overturned as one's context of knowledge broadens. On the contrary:

[A]ll conceptualization is a contextual process; the context is the entire field of a mind's awareness or knowledge at any level of its cognitive development. . . . If [a person's] grasp is non-contradictory, then even if the scope of his knowledge is modest and the content of his concepts is primitive, it will not contradict the content of the same concepts in the mind of the most advanced scientists.

The same is true of definitions. All definitions are contextual, and a more primitive definition does not contradict a more advanced one; the latter merely expands the former. (43)

Rand was not always on the opposite side of the fence from Hegel on this issue. In keeping, perhaps, with her dialectically oriented education, Rand had an early suspicion of abstraction and a preference for concreteness; in a journal entry for 15 May 1934, she writes:

There have been too many philosophical abstractions, too much intellectual "algebra". . . . What we need is an "arithmetic" of the spirit. . . . [It] is only the individual and the particular, concrete problem that counts. Algebraic constructions are only a convenience. In practice, they have no use, unless the proper arithmetical content is inserted into the formula. . . . This... has to be the cornerstone of my philosophy-proving the supremacy of actual living over all other considerations. . . . [M]y "arithmetic" of philosophy has to be philosophy brought up to the realm of actual living. (I say intentionally brought up to it, not down.) . . . That philosophical "algebra" is, to my mind, the greatest crime of metaphysics. . . . It is the result of that underlying error of human thinking—which forgets the distinction between abstraction and reality, thus denying reality. For abstractions are only a convenience, not a fact. (Rand 1997, 71-72)

This attitude toward abstraction is almost diametrically opposed to Rand's mature views. Two decades later, in a journal entry for 6 January 1952, she would write that "the root of all philosophical errors" is "to substitute for an abstraction one of the concrete applications of that abstraction, and at the same time make that concrete contradict and invalidate the abstraction" (640). Whereas before, the great mistake was to substitute the abstract for the concrete, now the great mistake is to substitute the concrete for the abstract. This is because the mature Rand no longer regards abstractions as a mere “convenience," but rather as indispensable to all knowledge: "[W]ithout abstract ideas you would not be able to deal with concrete, particular, real-life problems. You would be in the position of a newborn infant, to whom every object is a unique, unprecedented phenomenon" (Rand 1984, 5). In 1934, Rand was saying that the validity of abstractions depends on which concretes you plug into them, which "arithmetical content is inserted" into the algebraic formula. But her mature view is that an abstraction, if it is valid at all, is valid regardless of which concretes one plugs into it. Employing the algebraic metaphor once again but to an opposite purpose, she explains: "The relationship of concepts to their constituent particulars is the same as the relation of algebraic symbols In the equation  2a  =  a + a, any number may be substituted for the symbol 'a' without affecting the truth of the equation. Let those who try to invalidate concepts by declaring that they cannot find 'manness' in men, try to invalidate algebra by declaring that they cannot find 'a-ness' in 5 or in 5,000,000” (Rand 1990, 18).

This is not to say that Rand gave up her earlier view that one must always be able to relate one's abstractions to the concrete; by no means. But her mature view is that if you can't relate your abstraction to the concrete, you haven't successfully formed the abstraction in the first place. In a journal entry for 4 May 1946, she writes:

In order to think at all, man must be able to perform this cycle: he must know how to see an abstraction in the concrete and the concrete in an abstraction, and always relate one to the other. He must be able to derive an abstraction from the concrete [and] then be able to apply the abstraction. . . . Example: a man who has understood and accepted the abstract principle of unalienable individual rights cannot then go about advocating compulsory labor conscription. . . . Those who do have not performed either part of the cycle: neither the abstraction nor the translating of the abstraction into the concrete. The cycle is unbreakable; no part of it can be of any use, until and unless the cycle is completed. . . . A broken electric circuit does not function in the separate parts; it must be unbroken or there is no current. (Rand 1997, 481)

We don't have the abstraction and then see if we can apply it to the concrete; rather, the ability to apply it to the concrete is part of having the abstraction. It follows as a corollary that “floating abstractions” are not really abstractions, just as counterfeit money is not a kind of money. Hence we never have to worry about having an abstraction we don't know how to apply (though we may have to worry about thinking we have an abstraction when we don't).

For Hegel, Comte, and Marx, the two good things —concreteness and the collective— go together, as do the two bad things —abstraction and the individual. Those individualist thinkers who were influenced by Hegel —e.g., Kierkegaard, Stirner, Nietzsche— accepted his preference for the concrete over the abstract, while reversing his preference for the collective over the individual; and so their views about what goes with what altered accordingly. Whereas the Hegelians had dismissed the individual as abstract and extolled the collective as concrete, these proto-existentialists dismissed the collective as abstract and extolled the individual as concrete. It is perhaps a testimony to Rand's early Nietzschean phase that she initially found herself in this camp. In 1934, Rand associated abstraction with collectivism and concreteness with individualism: "Algebra —spiritually— is too much of the mob, of the masses, the collective, being too general. The individual is the arithmetical quantity of the spirit" (71). But in the 1970s, she was making precisely the opposite association: to the "concrete-bound, anti-conceptual mentality," the chief imperative is "loyalty to the group," and its consequent manifestations are tribalism, racism, and xenophobia (Rand 1982, 39-45). Rand ended up reversing both of Hegel's evaluations, whereas the proto-existentialists had reversed only one, and so now concreteness and collectivism went together again, but as partners in sin rather than partners in virtue. In liberating herself from both the Hegelian and the proto-existentialist versions of the cult of concreteness, Rand had migrated firmly from the proto-existentialist to the Aristotelian camp.

We've seen that Hegel and the dialectical tradition he inspired regard all abstraction as precisive and therefore as falsification, and recognize its utility only as a provisional idealization. Hence they believe that nothing is real or comprehensible apart from the whole and so are committed to a metaphysics of internal relations. Aristotle, Hayek, and Rand, by contrast, all recognize the possibility of non-precisive abstraction, and so have no inherent bias toward internalism.

From “Rejoinder to Bissell, Register, and Sciabarra: Keeping Context in Context: The Limits of Dialectics.” The Journal of Ayn Rand Studies Vol. 3, No. 2 (Spring 2002): p.413-415 [JSTOR]

Register challenges my critique of precisive abstraction by offering the example of a historian constructing an abstract model of the Battle of Gettysburg. But is this historian engaged in precisive or in non-precisive abstraction? Register assumes it must be the former, but this is not so clear. The fact that the model "ignores the overwhelming majority of the facts" does not render it precisive; it need not stipulate the presence of such facts, so long as it does not stipulate their absence. What about the fact that the model treats the average starting time of Pickett's Charge as the starting time? Well, it depends to how many significant figures the starting time is specified. Suppose the earliest soldier to start did so at 2:00:08, and the last soldier to start did so at 2:00:15. In that case, it would not be precisive to say that the charge started at 2:00, but it would be precisive to say that the charge started at 2:00:00. The historian may find a precisive model useful; they often are, so long as they are used with care. But the alternative to a precisive model is a non-precisive one, not "an array of unintegrable facts" (Register 2002, 361).

Register (2002, 364) writes: "In order to understand society and history, it's necessary to abstract parts of a society or historical events from the social or historical whole of which they are parts and then study them in relation to one another. But this is precisive abstraction." Well, it can be; but it needn't be. There is nothing inherently precisive about abstracting parts from wholes; it depends whether the part's connection to the whole is specified as absent, or just not specified as present.

Register also states that precisively abstracting X from Y will falsify only if X is internally related to Y. I disagree. So long as X is related (albeit externally) to Y, if a precisive abstraction treats X as not (rather than merely not as) related to Y, then falsification has occurred.

Since non-precisive abstraction (in Objectivist terms, measurement-omission) is the process by which we form concepts, Register concludes that the product of a non-precisive abstraction is always a concept, but I don't see why. An abstract model of the Battle of Gettysburg is not a concept; but if it leaves features out by failing to specify their presence, rather than by specifying their absence, then it clearly is a product of non-precisive abstraction. Likewise, (not just the concepts but) the assertions of Austrian praxeology are the products of non-precisive abstraction. I'm puzzled at Register's insistence that the products of non-precisive abstraction can never have truth-values, when in the previous paragraph he seems to have granted the status of my "sample praxeological claim" as a "conceptual truth" reached by non-precisive abstraction. I likewise cannot accept Register's assumption that precisive abstraction abstracts parts from wholes, while non-precisive abstraction abstracts features from their bearers. Precisive and non-precisive abstraction are distinguished by how, rather than what, they abstract.

Here is how I understand the distinction. Consider any two relata X and Y. (Their relation might be that of part to whole, that of feature to bearer, or something else.) Consider, further, some way of viewing X that does not include X's relation to Y. (This "way of viewing" might be a concept, a proposition, a model, a theory, or anything else.) If this way of viewing X specifies the absence of X's relation to Y, then it is precisive; if it merely fails to specify the presence of X's relation to Y, then it is non-precisive. Hence, I strongly disagree with Register's claim that whenever we consider parts in abstraction from their social whole we are engaged in precisive abstraction.

From "Realism and Abstraction in Economics: Aristotle and Mises versus Friedman." The Quarterly Journal of Austrian Economics Vol.9, No. 3 (Fall 2006): p.5-9 [here]

Aristotle’s theory of abstraction may be seen as a response to the following worry. It can easily seem that abstract concepts do not strictly apply to reality. The concept horse, for example, is supposed to apply to all horses, of whatever color. But obviously it could not do so if it had as its content a horse of any one definite color; if it were the concept of a brown horse, for example, it could not apply to a black one. In order to apply to all horses, then, the concept horse must have as its content a horse of no determinate color. But in that case the concept still does not apply strictly to any actual horse; for every actual horse has some determinate color. Either the concept horse somehow falsifies reality, then, or else—as Aristotle’s teacher Plato had argued—its actual referent is not any physical horse but the transcendent, immaterial Form of Horse, which indeed has no determinate color, and of which our familiar physical horses are merely an inadequate reflection. Hence abstractions have either mysterious otherworldly referents or no referents at all; in either case, they cannot refer to the familiar objects of ordinary experience.

Aristotle’s solution to this puzzle is to reconceive abstraction as a matter of attending to some aspects of a thing and ignoring others. To think the concept horse, for example, we focus on an ordinary horse—whether a real horse before us or an imagined horse before our mind’s eye—and then attend to the features it shares with other horses while ignoring its distinguishing features, such as its particular color.

In making [geometrical] diagrams . . . although we make no use of the fact that the triangle is determinate in quantity, we nonetheless draw it as determinate in quantity. Likewise also one who thinks, even if what he thinks is not quantitative, sets up before his eyes something quantitative but thinks of it not as quantitative; and if what he thinks is of a quantitative nature but indeterminate, he sets up something determinately quantitative but thinks of it merely as quantitative. (On Memory450a1–7)

Accordingly, Aristotle disagrees with Plato’s view that physics and geometry study different sorts of objects, physical and nonphysical respectively. For Aristotle, geometry studies physical objects just as much as physics does, but it studies them in a nonphysical way; the two sciences deal with the same familiar spatially extended objects, but geometry attends to their shape and position while abstracting from their physical embodiment:

We must consider how the mathematician differs from the physicist; for physical bodies have surfaces and volumes, lengths and points, all of which fall within the mathematician’s purview. . . . Now the mathematician too is concerned with such things, but not qua boundaries of physical bodies. . . . For they are separable in thought from motion, though from this separation no distinction or falsity arises. (Physics193b22–36)

Just as there are many statements characterizing things qua movable only, apart from what each of them is and apart from their accidents, and it does not necessarily follow from this that there is something movable apart from perceptible things, nor yet that there is some distinct nature within them, so too there will be statements and sciences that apply to movable things not qua movable but qua corporeal only, and again qua planes only, and qua lines only, and qua divisible, and qua indivisible but having position, and qua indivisible only. . . . For a human being is indivisible, qua human being. Now the arithmetician treats him as one indivisible thing, and considers what belongs to him qua human, while the geometer considers him neither qua human nor qua indivisible, but rather qua solid; for it’s clear that whatever would hold true of him even if he were somehow not indivisible can hold true of him irrespective of these characteristics. Accordingly, geometers are right in saying that the objects they discuss are real existents. (Metaphysics1077b23–1078a29)

This Aristotelian conception of abstraction was revived by the medieval Scholastics. Pierre Abelard (1079–1142), for example, undertook to “explain why thoughts gained through abstraction are not erroneous . . . even though they conceive things other than they are.” John Marenbon summarizes Abelard’s solution:

When I regard a man only as substance or only as a body, he explains, I am not conceiving anything in his nature which is not there, but I am not attending to all which he has. My thought would be erroneous if I regarded his nature as being only substance or only body. There is nothing erroneous, however, in regarding him only as substance or body; the “only” must apply to the regarding, not to the way in which the man exists. (Marenbon 1997, pp. 166–7)

Essentially the same position was held a century later by Thomas Aquinas (1224/5–1274), who wrote:

Abstraction may occur in two ways. First . . . we may understand that one thing does not exist in some other, or that it is separate from it. Secondly . . . we understand one thing without considering another. Thus, for the intellect to abstract one from another things which are not really abstract from one another, does, in the first mode of abstraction, imply falsehood. But, in the second mode of abstraction, for the intellect to abstract things which are not really abstract from one another, does not involve falsehood. . . . If, therefore, the intellect is said to be false when it understands a thing otherwise than as it is, that is so, if the word otherwise refers to the thing understood. . . . Hence, the intellect would be false if it abstracted the species of a stone from its matter in such a way as to think that the species did not exist in matter, as Plato held. But it is not so, if otherwise be taken as referring to the one who understands. (Summa Theologia I. 85. 1 ad 1; Aquinas 1999, p. 157)

Aquinas is here distinguishing between two different ways in which we might consider, say, a horse in abstraction from its color. We may consider the horse as not having a determinate color, or else we may consider the horse not as having a determinate color. To consider the horse as not having a determinate color is to hold, or attempt to hold, as the object of our thought a horse that simply has no determinate color—a creature never encountered in physical reality, and having its home either in Platonic heaven or nowhere. This sort of abstraction falsifies and contradicts the concretes on which it is based. But to consider the horse not as having a determinate color is simply to consider the horse as a horse without considering its color one way or the other; and here no falsification is involved.

These two types of abstraction are often referred to as precisive and nonprecisive. As Armand Maurer explains:

Precision is a mode of abstraction by which we cut off or exclude something from a notion. Abstraction is the consideration of something without either including or excluding from its notion characteristics joined to it in reality. Abstraction without precision does not exclude anything from what it abstracts, but includes the whole thing, though implicitly and indeterminately. (Note to Aquinas 1968, p. 39n)

In short, a precisive abstraction is one in which certain actual characteristics are specified as absent, while a nonprecisive abstraction is one in which certain actual characteristics are absent from specification. Plato failed to see how abstract concepts could apply strictly to physical reality because he failed to see that abstraction could be nonprecisive; one might say that he mistook an indeterminate way of thinking about something for a way of thinking about something indeterminate.

This is very much how the Austrian Aristotelian Franz Brentano (18381917) describes the contrast between Plato and Aristotle:

Plato thought that we recognize flesh and the being of flesh by apprehending two different things. . . . Aristotle teaches the exact opposite of this. . . . For it would obviously be a ridiculous assertion that someone who wanted to know something and instead apprehended something else with his intellect thereby reached the knowledge he desired. For example, a scientist wants to come to know the crystals and the plants and the other bodies that he finds here on earth; hence if he apprehended the concepts of tetrahedrons and octahedrons, and of trees and grasses belonging to another world, he would not reach his aim in any way. (Brentano 1977, pp. 86–88)

Brentano thus endorses the Aristotelian solution:

Whatever is is fully determinate. . . . But a thing that is completely determinate may yet be thought of without its complete determination. . . . It is an error, then, to affirm that there are universals in the strict sense. But it is also an error to deny that anything real can correspond to a universal idea . . . because a multiplicity of things can correspond to them. . . . When we think of the object as stone and when we think of it as this particular stone, we have the same object of thought in each case; but what we are thinking of it as differs in the two cases. (Brentano 1981, pp. 25–26, 39)

In recent years, this Aristotelian approach to abstraction has been revived by Ayn Rand. On the issue of universals Abelard was a nominalist and Aquinas a realist, while Rand attempted to transcend the nominalist/realist dichotomy altogether; all three thinkers, however, stand in the Aristotelian tradition, and all three appealed to nonprecisive abstraction to explain how concepts apply to reality. Rand does not employ the Scholastic terminology, but her approach follows that of her Aristotelian predecessors. (It’s not clear how far Rand was drawing specifically on the Aristotelian tradition, rather than being led by her generally Aristotelian approach to develop the same solution independently; the same question, for that matter, applies as well to Abelard, who had access to only a fraction of the Aristotelian corpus.) In Introduction to Objectivist Epistemology, Rand writes:

If a child considers a match, a pencil and a stick, he observes that length is the attribute they have in common, but their specific lengths differ. . . . In order to form the concept “length,” the child’s mind retains the attribute and omits its particular measurements. Or, more precisely, if the process were identified in words, it would consist of the following: “Length must exist in some quantity, but may exist in any quantity. I shall identify as ‘length’ that attribute of any existent possessing it which can be quantitatively related to a unit of length, without specifying the quantity. . . . Bear firmly in mind that the term “measurements omitted” does not mean, in this context, that measurements are regarded as non-existent; it means that measurements exist, but are not specified. (Rand 1990, pp. 11–12)

To regard the measurements as nonexistent would be to abstract precisively; to regard the measurements as existent without specifying them is, by contrast, to abstract nonprecisively. If all abstraction were precisive, then “every advance of knowledge” would be “a setback, a demonstration of man’s ignorance.” Since “the savages knew that man possesses a head, a torso, two legs and two arms,” it follows that if absence of specification meant specification of absence, then “when the scientists of the Renaissance began to dissect corpses and discovered the nature of man’s internal organs,” we would have to say that their discoveries “invalidated the savages’ concept ‘man’,” and likewise that “when modern scientists discovered that man possesses internal glands, they invalidated the Renaissance concept ‘man’” (pp. 67–8). On a proper understanding of abstraction, however, so long as whatever one fails to include in one’s concepts is merely unspecified, rather than specified as absent, then “even if the scope of [one’s] knowledge is modest and the content of his concepts is primitive, it will not contradict the content of the same concepts in the mind of the most advanced scientists” (p. 43). Like Abelard, Aquinas, and Brentano before her, Rand thus employs the concept of nonprecisive abstraction to reply to the charge that abstraction falsifies reality.