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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas
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ABSTRACTION IN THE FORMATION OF CONCEPTS
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ABSTRACTION IN THE
FORMATION OF CONCEPTS

The term “abstraction” is the usual expression in medi-
eval philosophical terminology for several processes
distinguished in Aristotle's writings by different terms,
viz., aphairesis (ἄφαιρεσις) and korismos (χωρισμός)
described in different ways. In all probability, it
was Boethius who introduced the Latin abstractio and
abstrahere to translate these Greek nouns and the re-
lated verbs.

The main theories of concept formation in Greek
antiquity were those of Democritus, Plato, and Aris-
totle (Beare, 1906). According to all these theories,
sense perception and intellectual cognition have to be
distinguished both by their objects and by their nature.
For Democritus and the Atomists, knowledge as well
as sense perception arises from effluvia of atoms which
are continually thrown off from the surfaces of physical
objects, and eventually enter the percipient through
the various sense organs. Intellectual cognition depends
on finer and subtler effluvia. This theory was further
developed by the Epicureans.

The difference between sense objects and the objects
of intellectual cognition were also recognized by Plato
but accounted for in a very different way. It is gener-
ally assumed that Plato adopted the Heraclitean view
that the physical world is in continuous flux so that
it never exhibits stable objects for sensory cognition.
Because we know, for example, the objects of moral
ideals and of mathematics, it was necessary to assume
a nonsensory origin of this knowledge. Objects of
knowledge really are; objects of sense are perpetually
becoming. The objects of intellectual cognition, ac-
cordingly, must have been stored up in us from a
previous existence. Knowledge, properly so-called, is
reminiscence.

As the Platonic Forms are separate from the physical
world of flux, the knowledge of Forms can only be
suggested by the approximations to them that the
physical world is able temporarily to manifest. The
theory, which Plato expressly defended in the Meno
(81C), Phaedo (73A), and Phaedrus (247C) and nowhere
expressly abandoned, is that we possess knowledge of
the Forms from a previous existence, and that so-called
learning is really reminiscence. Accordingly, we should
not expect to find anything like a doctrine of abstrac-
tion in Plato's writings. The need for such a doctrine
as we find in Aristotle is occasioned by Aristotle's
insistence that the Forms of material things are not
separate realities, yet we seem to be able to consider
them without considering the matter or without con-
sidering other concrete features of material things.
Separate Forms provide us with difficulties but not
with this particular one. Plato's insistence that we are
acquainted with objects that are nowhere completely
realized in the physical world requires a different ac-
count of our knowledge of such objects, and Plato
found the theory of reminiscence the only suitable
explanation.

But if there is no doctrine of abstraction in Plato's
works, there are passages which might have suggested
the doctrine to his successor, Aristotle. It is sufficient
to mention here only the passage in the Phaedrus
(249B-C) where it is written that “man must needs
understand the language of Forms, passing from a
plurality of perceptions to a unity gathered together
by reasoning” (Hackforth, 1952). Since, in the very next
sentence, we are informed that “this understanding is
the recollection of those things which our souls beheld
aforetime...,” the intention of the passage is clear
enough. But the notion that this unity (ἕν) is somehow
connected with a multitude of perceptions might have
been one of the suggestions which led Aristotle to his
doctrine of abstraction.

It was Aristotle's view that form and matter are
joined in physical objects that made a theory of abstrac-
tion both possible and necessary: possible because
forms otherwise could not be known by way of per-
ception and necessary because now perception is the
only immediate source of cognition. Aristotle uses the
term “abstraction” (ἀφαίρεσις) in connection with the
objects of mathematics, which “Platonists” had held
were separate from the material world (Ross, p. 566).
Aristotle maintained that these mathematical features
were, in fact, inseparable from material things but
could be thought of separately. In the Metaphysics
(1060a 28-1061b 31) the process is described as fol-
lows: in the mathematician's investigations, he takes
away everything that is sensible, e.g., weight and light-
ness, hardness and softness, heat and cold, and all other
sensible contrarieties, and leaves only quantity and
continuity in one, two, or three dimensions, as well
as the affections (πάθη) of these quantities. Elsewhere
(Post. Anal. 81b 3; De anima 431b 12ff.; Nic. Eth.
1142a; De caelo III, 1, 299a 15) we are repeatedly
informed that the objects of mathematics are treated
as separate but cannot exist separately. It is this for-
mulation which is repeated throughout the subsequent
history of abstraction both by those who follow Aris-
totle and by those who reject his views.

A point here is worthy of remark. The authors from
Boethius to modern times speak of abstracting forms
(both accidental and mathematical) from matter,
whereas Aristotle (as Owens has pointed out) in de-
scribing mathematical abstraction speaks of taking
away the sensible qualities, and leaving only the quan-
titative features of physical objects.


002

Although the process of coming to know the univer-
sal from repeated perceptions of particulars is not
called abstraction—Aristotle here uses “separation”
(χω̂ρίζειν)—there is at least one passage which, indi-
rectly, connects these two activities (Post. Anal. 81b 3).
In both these cases, induction (ἐπαγῶγή) is associated
with the process of coming to know the universal,
whether mathematical or physical. In the case of
mathematical abstraction, it is sometimes indicated
that the observed object suggests something which is
not actually presented, but the prevailing impression
is that the mathematical features are literally in the
object, and are discovered by removing from consid-
eration all other sensible qualities.

The description of cognition given in De anima and
in the Parva naturalia is important for the later devel-
opment of the doctrine of abstraction as we find it in
medieval writers. The forms of sensible objects without
their matter enter the soul, so that we know objects
by the presence of their forms in consciousness. The
form as it exists in the soul is, presumably, numerically
different though specifically the same as the form in
the object of perception (De anima III; VIII, 431b
26ff.). The forms of objects existing in the soul are,
Aristotle assumes, the fundamental elements of thought
which are the referents of the verbal symbols of spoken
discourse (De interpretatione, I, 16a 3ff; cf. De anima
III, 6, 430a 26-430b 33). So, in many cases at any
rate, the general terms of discourse stand for isolable
objects of intellectual consideration. There are, how-
ever, exceptions to this, the most important of which
are the analogical or systematically ambiguous terms
of metaphysics. Still, the assumption that verbal terms
usually stand for affections of the soul is one of the
important ingredients of the doctrine of abstraction
which was later developed by the medieval philoso-
phers.

There are two doctrines of Aristotle which throw
some light on his views about abstraction. One is the
contention that human cognition first comprehends the
generic features of physical things and only later comes
to the specific differentiae (Physica I, 1). The other is
the view that the essence of an organism is discovered
regressively by first knowing the activities, then the
powers, and by subsequently discovering the essence
on which such powers depend (De anima II). The
former doctrine indicates that there are generic con-
cepts. The latter suggests that the concepts of essences,
in the case of those of organisms, are really no more
than conjunctions of powers. But the view that an
essence is an essential unity obviously conflicts with
this, because the coexistence of powers expressed by
a conjunction of formulae could not constitute the sort
of unity of essence that Aristotle seemed to have had
in mind. It is, therefore, difficult to understand exactly
how the form of anything comes to exist in the soul
as an essential unity.

Two main features, then, characterize Aristotle's
view of abstraction: formal aspects of physical reality
exist in the soul as separate from matter even though
such a separation is impossible in the physical world
itself. This is true of generic concepts, of mathematical
aspects of things, and, of course, the specific concepts
of things. Cognition occurs when a form exists in the
soul.

That abstraction need not involve any falsification
is insisted upon by the medievals, and the first state-
ment of this is to be found in Aristotle. The mathe-
matician is concerned with the shape and size of objects
such as the sun or the moon, for example; but he does
not consider them as limits of natural bodies, or with
any properties of shape or size insofar as they are
aspects of physical objects. On the other hand, he
separates shape, etc., though without any falsity result-
ing from such conceptual separation (Physica II, 2,
193b 33ff.).

The accounts which have come down concerning
the theories of concept-formation of Stoics and Epi-
cureans contain nothing that can properly be described
as a theory of abstraction. Neither of these schools
accepted the form-matter distinction; they both main-
tained a materialistic view of nature, and the Stoics,
at least, were nominalists in some sense.

For the Stoics, the main function of reason was the
grasp of the conclusion of demonstrations such as the
existence of gods and their providential activity. Gen-
eral notions (νουμένα), they maintained, are gained by
contact or by resemblance; some come from analogy,
still others by composition or contrariety. In another
testimony, general notions are said to arise by way of
enlargement or diminution of what is perceived, or by
privation (Diog. Laërt. VII, 52-53).

Epicurus and his school, in addition to their atomistic
materialism, held that we see, for example, shapes, and
think of shapes by virtue of the entrance into the body
of something coming from external objects. The efflu-
ence of atoms coming from the surfaces of physical
objects enters the sense organs and produces images.
Universal ideas are stored in the mind so that when,
for instance, the word “man” is heard, it calls up the
shape stored in the mind. As all this must reduce to
a physical pattern, it is clear that all notions are ulti-
mately derived from perception by contact, analogy,
resemblance, or conjunction. None of this can be called
abstraction.

Since Plotinus rejected the Aristotelian theory of
sensory cognition, there is no place for a doctrine of
abstraction in his account of our conceptual knowledge


003

(Enneads IV, 6, 1-3). The same remark holds for Au-
gustine. In his account of sensory cognition, the soul
suffers no changes from the sense organs, but is essen-
tially active, taking note of changes in the body by
a kind of vital attention. Hence there can be no taking
of a form into the soul from physical nature. An ab-
straction, therefore, is out of the question in his view
of perception. The doctrine that the laws of numbers
and of wisdom are somehow given to human conscious-
ness by interior illumination from a divine source
takes the place of abstraction. As Augustine's views
on these and other questions were derived from Plo-
tinus and, indirectly, from Plato, this is to be expected.

The commentator, Alexander of Aphrodisias, uses the
phrase ἐξ ἀφαιρέσεως in describing the process of ob-
taining any form in consciousness as separated from
the material which it determines in the external world,
and it is from this source (Alexander, De anima, pp. 107,
34) that Boethius derived his account of abstraction
(In Isagogen Porphyrii commenta I, 11).

According to Boethius, there are many things which
cannot actually be separated but which are separated
by the soul and by thought—e.g., no one can actually
separate a triangle from its material substratum, but
a person can mentally separate the triangle and its
properties from matter, and contemplate it. This sepa-
ration does not involve any falsification because falsifi-
cation can only occur when something is asserted to
exist separately which does not or cannot exist sepa-
rately. Thus the separation achieved by abstraction is
not only not false, but is indispensable to the discovery
of truth. This means, we propose, that abstraction
provides the concepts which are to be united in affirm-
ative propositions which truly state what charac-
teristics things possess.

This account of abstraction follows along lines al-
ready laid down by Aristotle and is repeated, with
elaborations, by the logicians of the twelfth century.
Thus Abelard tells us that, although matter and form
are always together, the mind can consider each sepa-
rately. Thus abstraction does not falsify because there
is no assertion that anything has just the abstracted
property and no others. The mind considers only one
feature but does not assert its separation in fact from
other features. “For the thing does not have only it,
but the thing is considered only as having it” (Logica
ingredientibus
). John of Salisbury provides a similar
account. In abstracting a line or surface, the abstracting
intellect does not conceive it as existing apart from
matter. Abstraction is simply a contemplation of form
without considering its matter even though the form
cannot exist without the matter (Metalogicon II, Ch.
20). Again, some things resemble others and the mind
abstracts from these particular individuals and con
siders only the resemblance. In this way, the concept
of “man” is abstracted from the perceptions of individ-
uals, and the concept of “animal” from man, horse,
etc. (ibid.).

Similar views about abstraction are developed by
the anonymous author of De intellectibus (cf. V. Cousin,
1859) and it is clear that this general agreement can
be accounted for by the fact that all the schoolmen
of this period read Boethius, and, perhaps, also by the
influence of Abelard.

The Arabic translations of Aristotle and some of his
earlier Greek commentators made the doctrine of ab-
straction available to the Islamic and Jewish philoso-
phers. But there were also translations or epitomes of
the writings of Plotinus and Proclus and, even when
there was no confusion between Neo-Platonic and
Aristotelian views, attempts were made to harmonize
Aristotelian and Neo-Platonic doctrine. In particular,
the Neo-Platonic system of emanations was grafted
onto that doctrine of Aristotle which concerned the
connection of the Agent Intellect to individual human
cognitive activities. The Active Intellect in Aristotle's
psychology was identified with the last Intelligence.
In some of these systems, the illuminative activity of
the active intellect consists in the radiation of forms
into the material world and into the human mind.
Attempts to combine this doctrine with the doctrine
of abstraction produced strange consequences. In
Avicenna's (Ibn Sina, 980-1037) treatises on psychol-
ogy, for example, there are various degrees of abstrac-
tion of forms which correspond to the ascending se-
quence of cognitive powers, the sensitive, the
imaginative, the estimative, and finally the intellective.
His account of abstraction of sensible forms seems to
conform to the Aristotelian psychology of taking the
form of a material object apart from the matter
(Avicenna, Psychology, p. 40). But forms which have
no embodiment or which are embodied accidentally
must be received from the Agent Intellect when the
individual human souls have been prepared by the
appropriate sense experience to receive these emana-
tions (Avicenna, De anima 5; cf. Al-Ghazali, Meta-
physics,
pp. 174ff.). This explanation of how we know
the nature of qualities and of things thus combines a
theory of abstraction properly so-called with a doctrine
which accounts for conceptual knowledge by emana-
tions of forms from a suprahuman source. This made
it congenial to many of the earlier scholastics of the
thirteenth century.

Another feature of Avicenna's views must be men-
tioned: the doctrine of distinctions. This became im-
portant for the scholastics of the thirteenth and four-
teenth centuries, and figures in the discussions of the
seventeenth century. One of the important sources is


004

Aristotle's statement in the Topics that “if one thing
is capable of existing without the other, the former
will not be the same as the latter” (Topics Book VII,
Ch. 2; Becker, p. 152 b34). This was taken to be the
test of a real distinction of two things. According to
Avicenna, that which is asserted is other than that
which is not asserted, and what is conceded is different
from what is not conceded (De anima I, 1). So, if
someone can assert or concede that he exists even
though he does not assert or admit that his body exists,
this is sufficient ground for holding a real distinction
between the mind and the body. A similar idea under-
lies Descartes' mind-body distinction as a consequence
of cogito, ergo sum.

Yet another aspect of Avicenna's thought, important
to the history of abstraction, is his doctrine of the
common nature. Although Avicenna vehemently denies
that universals have any extra-mental existence and
although he asseverates that individuals alone exist, he
maintains that a nature can be contemplated which,
in itself, is neither one nor many (numerically) but is
simply the nature that it intrinsically is: horseness is
simply horseness. This theory of natures was to be used
by the thirteenth-century scholastics in diverse ways.
Aquinas draws upon it to avoid the Platonic paradox
about the one and the many in his De ente et essentia,
and it is essential to the views of Duns Scotus. Accord-
ing to the latter, the common nature has a unity less
than numerical unity so that the paradox of one nature
or form in many individuals is again avoided. And it
continues to receive support in the fourteenth century
in the critique of Ockham by Richard of Campsall:
Illa natura... non est pleures nec una (Logica, Ch. 15).

This theory that a nature as such is neither one nor
many is essential to Scotus' doctrine of abstraction. For
although such a nature cannot be separated, even by
divine power, from the individual differences by which
each thing is individuated, it can nonetheless be con-
sidered apart from such individuating features by ab-
straction.

Al-Ghazali (1058-1111) criticized Avicenna's view
of abstraction along lines which immediately call to
mind similar criticisms made later by some fourteenth-
century nominalists (especially Ockham) and by some
of the nominalists of the seventeenth and eighteenth
centuries (especially Hobbes and Berkeley). Against the
view that the intelligible universal in the intellect is
divested of all specifying or individuating determi-
nations, Al-Ghazali urges that everything in the intel-
lect is derived from the senses and retains all the con-
crete determinateness of sense experience. True, the
intellect can separate parts of a composite, but each
part thus separated is just as individual as was the
aggregate from which it was separated. Each wholly
determinate part of an aggregate thus separated func-
tions as a universal insofar as it is conceived as standing
in a relation to all similar individuals, and serves as
an image for all other things similar to it (Tahafut...,
1958; cf. Averroës, Tahafut, 1954). Al-Ghazali may,
therefore, be regarded as a precursor of the sort of
criticism of abstraction which later nominalists in
Christendom were to exploit. There is, of course, no
likelihood of any literary influence because this part
of Al-Ghazali was not accessible in Latin until the
sixteenth century (Zedler, 1961). Moreover, Averroës
opposed Al-Ghazali on this point and continued to
uphold the Aristotelian doctrine. We find that Mai-
monides (1135-1204) also adheres to a doctrine of
abstraction derived mostly from Avicenna's Guide of
the Perplexed.

In the philosophical writings of the early thirteenth
century in Christendom attempts were made to ac-
commodate the views of Aristotle to those of Saint
Augustine. Avicenna's writings on psychology made
this accommodation feasible especially to the Fran-
ciscans. But we should glance at one of the first at-
tempts in this vein by Robert Grosseteste.

In his commentary on the Posterior Analytics, Gros-
seteste taught that the mind is capable of knowledge
without the aid of the senses. Due to its incarceration
in the body, however, the mind is darkened and re-
quires the aid of sensation. Accordingly, abstraction
of forms from the data of sensation is normally re-
quired. So the intellect separates out for special con-
sideration the features of things which are confused
in sensation. Abstraction of forms usually is derived
from many individual objects presented to the senses.
But the knowledge thus attained is not of the highest
grade.

A representative view of the Franciscans can be
found in Matthew of Aquasparta. Because the human
soul is a sort of mean between God and creatures, it
has two aspects, one of which, the superior part, is
turned to God; the other, the inferior part, is turned
toward creatures. According to the doctrine of “the
two faces of the soul,” the correct explanation of
human knowledge is a medium between the position
of Augustine and Aristotle.

Knowledge of the world is generated in man by
sensation, memory, and experience from which the
universal concepts of art and science are derived. But
in order fully to understand the natures of things thus
abstracted from sensation we require an illumination
from the Divine Light. Although we do not see Divine
Light in our earthly existence, we see the natures of
things by its means. The existence of this illumination
is explained as follows: we know eternal truths with
certainty. These truths are immutable yet everything


005

in the world about us and our very minds are mutable.
So, the immutable and necessary features of our
knowledge require the illumination of the Divine
Light.

Matthew adopts the Augustinian theory that the
corporeal world cannot produce changes in the soul
(the inferior cannot affect the superior). Rather the soul
is actively aware of changes occurring anywhere in
the body. The data which the soul makes from its
notice of corporeal changes are rendered intelligible
by the Agent Intellect which, Matthew says, is what
Aristotle calls abstractions. But these abstractions are
understood in the light of the immutable rules provided
by divine illumination. This combination of “abstrac-
tion” and illumination is to be found in a number of
Franciscan thinkers of the thirteenth century.

Saint Thomas Aquinas. Aquinas expounds a theory
of abstraction according to which things (in the sense
of objects of apprehension) can be considered, one
aspect apart from another, in cases in which the two
things cannot exist separated from one another. In cases
in which one thing can exist apart from another we
should speak of “separation” rather than abstraction
(Commentary on Boethius' De Trinitate, Q5, a.3). Since
substance, which is the intelligible matter of quantity,
can exist without quantity, it is possible to consider
substance without quantity. Again, to consider “ani-
mal” without considering “stone” is not to abstract
animal from stone. Thus it is only in cases where things
cannot exist separately but can be considered separately
that we can properly speak of abstraction.

Abstraction is of two kinds: the one, mathematical
abstraction, involves a consideration of form from sen-
sible matter. The other is the abstraction of the univer-
sal from the particular. The possibility of abstraction
depends on the fact that things (features of things) exist
in one way in the realm of matter, but in another in
an intellect which apprehends them. Thus, because the
mind is immaterial, the natures of material things exist
in the mind in a way suitable to the mind, i.e., they
have an immaterial existence in the mind. But the
simple apprehension of the mind does not involve any
assertion that the features of things exist thus in reality,
because simple apprehension is not an act which asserts
or denies anything at all. The mathematical abstraction
which considers only the quantitative features of phys-
ical things does not assert that lines, planes, etc., exist
independently of such objects. It merely considers these
features without attending to other aspects of physical
objects, although the mathematical or quantitative
features cannot exist isolated from physical objects.

In the case of the abstraction of the universal from
particulars, the mind considers the specific nature of,
say, man or dog, apart from the individuating aspects
of individual men or dogs. Again, abstrahentium non
est mendacium
(“abstraction is not falsification”) be-
cause the mind does not assert that the specific nature
of man can exist apart from particular men.

The generic nature common to several species can
be abstracted so that the mind thinks only of the
generic aspect of these several species and ignores the
specific differences. “What is joined in reality, the
intellect can at times receive separately, when one of
the elements is not included in the notion of the other”
(Summa contra gentiles I, c. 54, para. 3). So, because
the concept of the genus “animal” does not explicitly
contain the concept of, say, “rational,” the mind can
consider “animal” without considering any particular
kind of animal. But this “animal” is not something
existing apart from particular kinds of animal any more
than these particular kinds can exist apart from indi-
vidual animals. Only in the mind that apprehends the
form of animal stripped of its individuating and speci-
fying characteristics does animal as such exist (ibid.,
I, c. 26, para. 5).

Nothing exists in a genus which does not exist in
some species of that genus (ibid., I, c. 25, para. 2).
Animal cannot exist in re without the differentia “ra-
tional” or the differentia “irrational.” Still animal can
be considered without these differentiae (ibid., I, c. 26,
para. 11). There is, however, no purely generic ex-
emplar in the divine mind (Summa theol. I, 15a. 3 ad 4).

Duns Scotus adopted from Avicenna the doctrine
of a common nature which is in itself neither one nor
many but simply what is indicated in the definition
or description of such a nature. This nature can be
individuated in the individuals of a species by the
further determination of an individual difference or
“haecceity” (i.e., “thisness” in contrast to “quiddity”
or “whatness”), or it can be rendered a universal con-
cept by the action of the active intellect; but in itself
it is neither one nor many.

The process of abstracting a universal concept from
the common nature so conceived is not a “real action”
because the common nature is already present in the
individuals and formally distinguished from the indi-
vidual differences prior to and independent of any
action of the intellect (Duns Scotus, Quaestiones in
metaphyisicorum libros,
VII, q. 18; Opus oxoniense II
d. 1, q. 5 [q. 6], n. 5). This formal distinction of the
specific nature from the individual differences which
contract it to a numerical unity in the various individ-
uals of the species applies just as well to the distinction
between the specific and the generic features of a
common nature, for these also are formally distinct in
such a way that the mind can think of the generic
nature as such. There is, therefore, no distortion or
falsification in the result of abstraction, because abstrac-


006

tion amounts to considering one aspect of a nature
without considering the others (Opus oxon. III, d. 14,
q. 2, n. 12).

Thus, the distinctive feature of Scotus' contribution
to the doctrine of abstraction depends upon his doc-
trine of the formal distinction between the individu-
ating and the common nature which exists prior to any
action of mind on the data of observation.

William of Ockham. Ockham uses the term “abstrac-
tion” and provides a number of meanings for it, but
he departs from his predecessors on one very important
point: he denies that we can think as separate what
is incapable of existing separately in reality. However,
he allows that we can understand one thing without
understanding another at the same time even though
the two things do, in fact, coexist. Thus he states “To
abstract is to understand one thing without under-
standing another at the same time even though in
reality the one is not separated from the other, e.g.,
sometimes the intellect understands the whiteness
which is in milk and does not understand the sweetness
of milk. Abstraction in this sense can belong even to
a sense, for a sense can apprehend one sensible without
apprehending another” (Expositio physicorum, fol.
111c).

In his commentary on the Sentences (II, qq. 14, 15
xx) Ockham tells us that the abstraction of the agent
intellect is twofold. On the one hand, it produces a
thought (an intellection) which is either intuitive or
abstractive, is wholly abstracted from matter because
it is immaterial in itself, and has its existence in some-
thing immaterial (i.e., in the soul). On the other hand,
the abstraction produces a universal, i.e., a universal
concept of a thing in representative existence.

In still another sense, abstraction occurs when one
predicable is predicated of a subject and another pre-
dicable is not predicable of that subject even though
the latter predicable applies to the subject. This takes
place in mathematics. For the mathematician considers
only such statements as “Every body is divisible, is so
long and so deep,” and ignores statements about bodies
which pertain to motion, to the composition of matter
and form in physical things, etc.

Accordingly, Ockham allows that many things are
really distinct which constitute a unity, as in the case
of matter and form, or substance and accident. Now
it is true that, in such cases, the mind can separate
or divide these from one another so as to understand
one and not understand the other. But if a and b are
one thing and a may not be really distinguished from
b, it is impossible that the mind may divide a from
b so as to understand either without understanding the
other (Sent. I, d. 2, q. 3, H).

Hence, Ockham rejects any abstraction of a common
nature or form from its instances in such a way that
the mind can contemplate the common nature as such.
The only distinction Ockham will allow is the real
distinction of one thing from another thing. A distinc-
tion between the common nature and an individual
difference which Scotus had defended is, for Ockham,
entirely out of the question (Sent. I d, qq. 1-4).

The reason why Ockham can allow the abstraction
of matter and form in an individual physical object
is because, for him, this matter and this form could
exist apart from one another, at least by divine power.
The same is true of accident and substance. An accident
can be thought without its substratum because an
accident and its substratum are two really distinct
things, and one can exist without the other (Sent. II,
q. 5, M; cf. I, d. 30, q. 1, P).

Thus Ockham, as Vignaux observed, adopted the
principle, much later exploited by Hume, that what-
ever is distinguishable is separable. And like Hume,
he practically rejected the distinction of reason. The
result was a rejection of the central tenet of the classi-
cal doctrine of abstraction, set forth by Aristotle and
defended, in one form or another, by many of the
scholastics of the twelfth and later centuries.

Descartes. There were many elaborations of the
Thomistic doctrine among the later scholastics of the
fifteenth and sixteenth centuries by Cajetan, Suarez,
John of St. Thomas and others. Suarez in particular,
was responsible for sharpening the differences between
abstraction and distinction (or “separation” as Saint
Thomas had called it). And this, in turn, was almost
certainly the immediate source of Descartes' views.

While Descartes allows that abstraction takes place
in the mind, he is always at pains to notice that abstrac-
tion renders our concepts inadequate in such a way
that we cannot discover the important distinction of
things. Thus, the distinction of reason by which a
substance is distinguished from its principal attribute
(of thought or extension as the case may be) is effected
by abstracting one from the other. This is accomplished
only with some difficulty and the result does not corre-
spond with anything in the way of a real separation
of a substance from its nature or attribute (Principles
of Philosophy,
I, 63). Thus the valuable operation of
the mind is that which provides us with a real distinc-
tion. This Descartes sometimes calls “exclusion.” The
principal difference which Descartes makes between
abstraction and exclusion is that, in the case of abstrac-
tion we consider one thing without considering that
from which abstraction has been made and so may not
be aware that abstraction has rendered a concept in-
adequate, whereas in distinguishing one thing from
another, we must keep both clearly before us. Consid-
ering an abstraction by itself prevents us from knowing


007

well what it has been abstracted from (Letter to
Clerselier, 12 Jan. 1646).

The influence of Descartes on the so-called Port-
Royal Logic of Antoine Arnauld (1612-94) is obvious.
But this famous treatise presents an account of abstrac-
tion which agrees in essential features with the stand-
ard medieval view. Arnauld had argued, in his critique
of Descartes, that the genus can be conceived without
conceiving its species so that, for example, one can
conceive figure without conceiving any of the charac-
teristics proper to such a particular figure as a circle
(“Fourth Objections”). Again, length can be conceived
without breadth or depth. But such abstraction, prop-
erly so-called, is only between aspects of things which
are only distinct by a distinction of reason. Where
things really distinct are distinguished, abstraction does
not occur (La Logique ou l'art de penser [1662], Part
I, Ch. 5).

John Locke. The discussion of abstraction which is
perhaps most familiar to modern readers is to be found
in Locke's Essay Concerning Human Understanding.

Words become general by being made the signs of general
ideas; and ideas become general by separating from them
the circumstances of time and place, and any other ideas
that may determine them to this or that particular existence.
By this way of abstraction they are made capable of repre-
senting more individuals than one; each of which having
in it a conformity to that abstract idea, is (as we call it)
of that sort

(Book III, Ch. 3, para. 6).

He goes on to suggest immediately that nothing new
is introduced in this process but that it is rather a
process of omitting all individuating features, and re-
taining only what is common to all of a set of resem-
bling particulars. This omission, he explains elsewhere
(Book II, Ch. 13, para. 13), is a kind of partial consid-
eration which does not imply a separation. But Locke
applies the notion of abstraction to cases which go
beyond the mere omission of particular spatiotemporal
determinations. In the famous example of forming the
general idea of a triangle, Locke says that this idea
of triangle in general is “something imperfect that
cannot exist, an idea wherein some parts of several
different and inconsistent ideas are put together” (Book
IV, Ch. 7, para. 9). Whatever Locke may have thought
this “putting together” amounted to, it is certainly not
achieved simply by omitting particularizing features
of several particular triangles. The fact is that no single
doctrine of abstraction can be found in Locke, as
I. A. Aaron has shown (Aaron, 1937).

Berkeley and Hume. Berkeley's critique of abstrac-
tion proceeds along lines which were relatively new
to his readers but which had already been worked out
by Al-Ghazali in the eleventh century and even by
Ockham in the fourteenth century. If two things (in
Berkeley's philosophy, of course, two ideas) can exist
separately, the mind can abstract one from the other.
But if it is granted that two things cannot exist one
apart from the other, i.e., that there would be a con-
tradiction if a were supposed to exist without b (or
conversely), the mind cannot think of a without b or
of b without a. To argue otherwise would be to attrib-
ute to the human mind a power which not even God
can be supposed to have or exercise.

Hume adopted Berkeley's critique and elaborated
a positive theory of the function of general terms which
goes beyond Berkeley. Although every idea is particu-
lar, some ideas can function as general ones by being
associated with a name of a number of particulars
which resemble one another exactly or only approxi-
mately. In the latter case, the name is associated with
a number of qualitatively different but resembling
images. One of these associated images will be domi-
nant, the others relatively recessive but, as Hume puts
it, “present in power to be recalled by design or neces-
sity.” Thus, although a red image may be recalled when
the word “color” is pronounced, heard, read, or re-
called, other color-images less strongly associated with
the word “color” tend to appear in consciousness, are
“present in power,” and will be recalled if there is
danger of a mistakenly narrow use of “color” present-
ing itself. This then, is Hume's alternative to the doc-
trine that there are either genuine images or abstract
general ideas. The traditional explanation of the ori-
gin of abstract concepts persisted, with some modi-
fications, among the philosophers of the eighteenth
century.

A considerable advance in the understanding of the
nature and function of concepts seems to have been
made by Immanuel Kant. The verb, adjective, and noun
frequently occur in Kant's Critique of Pure Reason
(Werke, A54, A70 [B95], A76, A96) without any special
explanation. But Kant's doctrine of pure as well as a
posteriori
concepts leaves no doubt that abstraction
alone cannot account for the existence or employment
of concepts (Werke, VII, 400-01). “The form of a
concept, as a discursive representation is always con-
structed.” As Kant puts it in the Prolegomena to any
Future Metaphysics
(para. 20), empirical concepts
would not be possible unless a pure concept were
added to the particular concept which has been ab-
stracted from intuition. And, finally, in the Critique of
Pure Reason,
the concept is presented as a rule by
means of which the imagination can outline, for exam-
ple, the figure of a certain quadruped (say, a dog)
without limiting it to such a determinate figure as one's
experience or concrete images might present. Kant
calls this a schema. Without such a schema (which is
an application of the pure concepts of the under-


008

standing) neither images nor a conceptualization of
images would be possible.

Kant's doctrine that pure concepts, i.e., the cate-
gories of the understanding, be at the basis of all con-
ceptual thinking thus makes the process of abstraction
subsidiary to and dependent upon faculties which are
logically prior to any process of abstraction from em-
pirical data. As more than one writer has recently
pointed out, empirical concepts are more like disposi-
tions than like static constituents of consciousness.
There is, however, no suggestion in Kant that abstrac-
tion does not occur. That this new view of the activities
of the mind would require an entirely different account
of abstraction is not made very plain in Kant's writings.

In the development of metaphysical Idealism in the
post-Kantian philosophers, the notion of abstraction
becomes very general, so general in fact that the origi-
nal meanings of the term seem almost lost. What makes
the matter even more difficult to discuss is the fact
that, among these Idealists, any separation or isolation
of one content or feature of experience or thought from
another is condemned as falsification, so that “to ab-
stract,” “abstract,” “abstraction,” all acquire a pejora-
tive sense. To separate the cognizing subject from its
object, to attend to one discriminable element apart
from its surrounding, and the like, are all condemned
as falsifications of reality. This condemnation rests on
the Hegelian doctrine that “the Truth is the Whole,”
i.e., that all aspects of thought and reality are dialecti-
cally interconnected.

Other more significant attacks on the doctrine that
general concepts result from abstraction come from
Husserl's thorough critique of Locke and his eight-
eenth-century critics. While insisting on the absurdity
of Locke's doctrine, Husserl attacked with equal vehe-
mence the theories of Berkeley, Hume, and Mill. He
insisted that the general attributes are given to con-
sciousness initially, and thus repudiated the traditional
doctrine of abstraction. There are similar views to be
found in some of the writings of Whitehead and San-
tayana. The “eternal objects” of Whitehead and the
“essences” of Santayana are supposed to be discoveries
rather than constructions; they are not the results of
creations of mental activities, and thus are not the
result of abstraction as it was traditionally expounded,
although the accounts of abstraction in terms of atten-
tion and comparisons would be consistent with such
views.

One of the most significant critiques of abstraction
comes from Gottlob Frege, in his Grundlagen der
Arithmetik
(1884). While Frege appears to allow that
“color, weight, and hardness” are abstracted from ob-
jects, he holds that number is not so abstracted. His
theory of the concept of cardinal number makes it
impossible to obtain the number concept by simply
omitting features of empirically accessible objects.
Because a number is a “property” of properties, it is
not available from empirical inspection of individuals.
And if we examine Frege's definitions of particular
finite cardinals we see at once why the notion of cardi-
nal numbers can hardly be extracted as traditional
abstraction doctrines suggest. The number one, for
example, is a characteristic of any “property F” which
satisfies the following condition: there is that which
is F and which is the same as anything which is F,
more exactly: (∃x)(Fx(y)(Fy. ⊃. y = x). It is readily
seen, if we remove the expression “F” from the above
formula thus obtaining (∃x). (–x. (y) –y. ∃. y = x,
that the property of F is expressible solely in terms
of logical constants. Now because these constants
function in discourse in a manner that is not compara-
ble with the way indicative or descriptive expressions
function, it is hardly surprising that there is nothing
available empirically from which they can be ab-
stracted or upon which attention may be concentrated.
A psychological account of the origin of the notion
of number will doubtless be a very complicated affair
but it will necessarily be radically different from ab-
straction.

The technique employed by Frege, Georg Cantor,
and some others to elucidate the mathematical notions
of cardinal number was recognized by Bertrand Russell
as an application of a general principle which Russell
called “the principle of abstraction.” But he added that
it would have been better called the principle “for the
avoidance of abstraction.” The principle is this: for any
relation S which is transitive and symmetrical there
is a relation R which is a many-one relation such that
whenever xSy, there is a unique term z such that xRz
and yRz; conversely, if there is a many-one relation
R such that there is a unique term z so that xRz and
yRz, there is a relation S which is transitive and sym-
metrical (Principia Mathematica, Vol. 1, °72).

The essential principle to notice here is that, instead
of attempting to account for the concept by a psycho-
logical theory by which the concept is derived some-
how from the data of the senses or from some innate
or at least internal feature of human consciousness, the
concept is constructed by logical means from fairly
simple relational concepts. Thus, a cardinal number
is defined as either a class of those classes whose num-
bers can be bi-uniquely correlated (in a one to one
correspondence) with one another, or, as a property
P of those properties q1, q2,..., qr such that those
things having any one of these properties can be corre-
lated bi-uniquely with the things having any other of
these properties.

The formal definition of cardinal number brings into


009

prominence the fact that it is constructed by means
of variables ranging over individuals and properties,
and by logical connectives and quantifiers. There is
nothing about such a construction which even suggests
that it could have been “abstracted” (in the traditional
sense) from sense given materials or that there is some
“inner” source of the notion. It can be objected to all
this that this logical construction of concepts of cardi-
nal and ordinal numbers does not explain their psycho-
logical origin. Doubtless this is correct. Frege and
Russell probably both supposed that they were eluci-
dating the nature of mathematical objects which are
somehow given (in some very different way from ab-
straction), whereas they were actually recommending
the replacement of obscure notions by clear ones. But
whatever the psychological origin of mathematical
concepts may be, the Frege-Russell construction shows
that it must be far more complex than anything pro-
posed by the traditional abstraction theories. So, while
the psychological question remains a highly interesting
one, the focus of interest has shifted to the logical
content of formal concepts.

BIBLIOGRAPHY

For main developments in Greek thought, see J. I. Beare,
Greek Theories of Elementary Cognition (Oxford, 1906). See
also Diogenes Laertius, Lives of the Philosophers, Loeb
Classical Library (London and New York, 1925), esp. VII,
52-53; R. Hackforth, Plato's Phaedrus (Cambridge, 1952),
p. 86; Joseph Owens, The Doctrine of Being in the Aristote-
lian Metaphysics,
2nd ed. (Toronto, 1963); W. D. Ross,
Aristotle's Prior and Posterior Analytics (Oxford, 1949), p.
566.

For Alexander of Aphrodisias, see his De anima (Berlin,
1887), pp. 107, 34. Boethius is found in In Isagogen Por-
phyrii Commenta, Corpus Scriptorem Ecclesiasticorum Lati-
norum,
Vol. XLVIII (Vienna, 1906), 135-69; also Quomodo
substantiae in eo quod sint bonae sint cum non sint substan-
tialia bona
(London, 1928), pp. 44-45; and in De Trinitate
(London, 1928), Q5, a. 3.

The sources for medieval figures include: Abelard, Logica
ingredientibus,
ed. B. Geyer, Beiträge zur Geschichte der
Philosophie des Mittelalters,
Band XXI (Münster in W.,
1921), Heft L, 25; and P. Abelardi opera hactenus inedita,
ed. V. Cousin (Paris, 1849; Vol. II, 1859), II, 733-45; Al-
Ghazali, Algazal's Metaphysics, ed. J. T. Muckle (Toronto,
1933), Part II (IV, 5), pp. 174ff.; and Tahafut Al-Falasifah
(Destruction of the Philosophers), trans. S. A. Kameli
(Lahore, 1958), pp. 218-20; Averroës, Tahafut, trans. S. van
der Bergh (London, 1954), pp. 345-55; Avicenna, Psychol-
ogy,
trans. F. Rahman (Oxford, 1952), p. 40, also De anima
(Venice, 1508), I, 1 and V, 5; Duns Scotus, Quaestiones in
metaphysicorum libros
(Lyon, 1639), VII, q. 18, also Opus
oxoniense
(Lyon, 1639), and Sentences (Lyon, 1639); John
of Salisbury, Metalogicon (Berkeley, 1955), II, Ch. 20; Mai-
monides, The Guide of the Perplexed, trans. Shlomo Pines
(Chicago, 1963), Part I, Ch. 68, pp. 163-64; J. R. O'Donnell,
ed., Nine Medieval Thinkers (Toronto, 1955), p. 191, is the
source for Richard of Campsall. For Saint Thomas Aquinas,
Summa contra gentiles, the standard Latin text is edited
by Leonina Manvalis (Rome, 1946), and an English transla-
tion is that by Anton Pegis et al. (Garden City, N.Y.,
1955-56); for Summa theologica, the standard Latin text is
edited by M. E. Marietti (Turin, 1952), and an English
version is Basic Writings of Saint Thomas Aquinas, ed. Anton
Pegis (New York, 1945). See also Beatrice H. Zedler, ed.,
Averroës Destructio destructionum (Milwaukee, 1961), pp.
18-31.

Since the Renaissance, principal sources include I. A.
Aaron, John Locke (Oxford, 1937), pp. 194-200; A. Arnauld,
La Logique, ou l'art de penser, 5th ed. (Paris, 1683), Part
I, Ch. 5; René Descartes, Letter to P. Mesland, 2 May 1644,
Principles of Philosophy, ed. Charles Adam and Paul Tan-
nery, 12 vols. (Paris, 1897-1913; 1964), I, 63 and VIII, 31.
See also Replies to First Objections, ed. Charles Adam and
Paul Tannery (Paris, 1964), VII, 120, and Quartae objec-
tiones,
in the Haldane and Ross translation (Cambridge,
1912), II, 82; John Locke, Essay Concerning Human Under-
standing
(London, 1690), Book III, Ch. 3, para. 6. I. A;
Immanuel Kant, Werke, ed. E. Cassirer, 11 vols. (Berlin,
1912-22), VII, 400-01; idem, Kritik der reinen Vernunft
(Leipzig, 1924), A, 1781, B, 1787; H. Scholz and H.
Schweitzer, Die sogenannte Definitionen durch Abstraktion
(Leipzig, 1935).

JULIUS WEINBERG

[See also Analogy; Axiomatization; Experimental Science;
Islamic Conception; Number; Optics and Vision; Organi-
cism; Platonism; Rationality.]