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Dictionary of the History of Ideas | ||
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.
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
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
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
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-
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
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-
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
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.]
Dictionary of the History of Ideas | ||