Biblioteca digitale Progetto Agorà
From the beginning of his long philosophical career to the end of his
life,
Leibniz seems to have never dismissed the intent - which, after various
pauses,
he renewed with the regularity of a recurring dream - to compile
long lists of
definitions. These lists had, in principle, two different aims: to
establish a
firm ground for the building of a new encyclopedia and to re-
form the
traditional doctrine of categories. Often in the same “table of defi-
nitions”
these two aims intertwine. Sometimes the lists are quite short and
heterogeneous
in content; sometimes they are long and homogeneous, but
with many repetitions;
and, in general, they all exhibit an unaccomplished,
tentative quality which,
however, do not diminish their philosophical inter-
est. These lists have been
published in the Vorausedition or elsewhere; only
a
handful of them is as yet unpublished.
In this paper I intend to draw attention to an unpublished list of defini-
tions
which Leibniz wrote after 1700 and which almost exclusively contains
logical
items and reflections on logical ontology. I will first give the original
Latin
text (without the critical apparatus); then I will propose an English
translation
of the most relevant passages; and finally I will present a short
commentary to
the translated passages.
The Latin text (LH 7 C Bl. 76 v. A translation will be given
of the
passages comprised between brackets “<” and “>”).
Alia est rerum alia terminorum divisio.
Terminus est cogitabile quod potest esse subiectum propositionis verae
estque
vel implicans contradictionem, nempe chimaera, vel possibile seu
non implicans,
nempe reale.
Reale est terminus possibilis seu distincte cogitabilis, ut homo, doctus,
causa,
actio.
<Rursus terminus sumitur vel materialiter, et ita est ens, cuius
diversi
termini sunt eadem res ut rex et propheta in Davide, nempe idem est
rex
qui est propheta. Vel terminus sumitur formaliter, ut rex qua rex, nempe
etsi non
sit alius rex quam propheta, tamen aliud est regem esse aliud
prophetam esse.
David qua rex differt a Davide qua propheta. Diversi ergo
sunt termini sed ens
idem. Interim in terminis quoque formaliter spectatis
distinctio est. Interdum
enim in idem resolvuntur veluti figura triangula
et figura trilatera, item
rectangulum aequilaterum et quadrilaterum ae-
quiforme, ubi nodo est eadem;
interdum vero diversa notio est, ut aureum
et rotundum, etsi monetae conveniant
quam ducatum vocamus. Termini
ergo differunt re, cum diversa entia spectant,
forma cum ad diversas no-
tiones spectant, consideratione, cum diversa dicunt seu
cogitanda exhibent.
Idem est quidam rex et quidam propheta, omne triangulum et
omne tri-
laterum. Stemus ergo in rebus ipsasque notiones secundas ut res
considere-
mus>.
Terminus est vel Nihil ut Blitiri, vel aliquid.
Aliquid est subiectum propositionis verae, estque vel impossibile, quod
dicitur
Chimaera, vel ens.
Ens est aliquid possibile, ut Deus, homo, scientia, scriptio, motus.
Ens vel est subsistens quod tantum subiectum esse potest, vel
attribu-tum, quod est constituens praedicati alterius entis ut scientia constituit
ut
aliquis sit sciens. Actio ut agens. Sed nonne datur tertium? <Sic tempus,
lo-
cus, neque est subsistens nec attributum. Idem est de numero, ordine.
Sic
decem non est attributum ullius rei. Neque enim de aggregato neque
de
singulis dici potest numerus denarius. Idem est de relatione quae commu-
nis,
v. g. similitudo duorum. Datur itaque attributum quod est simul in
pluribus
subiectis. Talia ergo sunt ordo, adeoque tempus et locus>.
A possible translation of the first passage between brackets:
Again a term is taken either materially, and thus it is a being the differ-
ent
terms of which are the same thing, as king and prophet in David - i. e.
the same
person who is a king is a prophet as well; or a term is taken for-
mally as a
king insofar as he is a king - i. e. even though the king is not a
different
person from the prophet, yet to be a king is different from being a
prophet.
David, insofar as he is a king, differs from David insofar as he is a
prophet.
Therefore the terms are different, but the being is the same. In
some
circumstances, however, even the terms considered formally admit of
a
distinction. In fact, by means of analysis, they give rise, sometimes, to
the
same - as in the case of a triangular and a trilateral shape or in the case
of
an equilateral rectangle and an equiform quadrilateral, where the notion
is
the same; sometimes the notion is different, as in the case of “golden”
and
“round”, even if both apply to the coin which we call “ducat”.
Therefore
terms differ in the following way: about the thing - insofar as they
concern
different beings; about the form - insofar as they concern different
notions;
about the way of considering - insofar as they mention or show
different
thoughts. A certain king and a certain prophet are the same, and so are
ev-
ery triangle and every trilateral. Thus, we have to stay with things and
to
consider as things all second notions as well.
Commentary
As is well known, Leibniz’s use of “term” is quite puzzling. In our
text (in the
part I have not translated), he defines a term as
«something
which can be thought and which can be the subject in a true
proposition».
This seems to be quite in agreement with the definition of a term
as the
meaning of a word, which we find in a letter to Des Bosses of 1712.
the text
under consideration, Leibniz plainly writes that Blitiri,
which was
traditionally mentioned as a typical linguistic expression with no meaning,
is a term. Without excusing Leibniz’s inaccuracies in the use
of the term
“term”, I think that we can easily explain these inaccuracies if we
take into
account his peculiar attitude towards nominalism. In the period that
dates
from the composition of the Dialogue
New Essays
(1704-5), Leibniz firmly claims that men cannot think without signs
-
whether they be written or spoken words, letters or numbers, or
simply
mental images or perceptions of some sort. At the same time, Leibniz
rec-
ognizes that words and other signs are meaningful only if they express
some
idea or ideas. The point is that men cannot have - with the exception of
a
handful of cases - direct access to ideas (in particular if the ideas are
suffi-
ciently complex): the psychological act of grasping an idea - an act
which
gives rise to what Leibniz calls a “concept” - is always associated in
our
mind with the use of words or other signs.Ibid.
which
interposes itself between the realm of ideas and us; and our thoughts
are, for
the most part, “blind thoughts” (cogitationes caecae) made
of words
(or of other signs).are concepts or,
better said, substi-
tutes for concepts. It is not inappropriate to think that
this point of view in-
duced Leibniz to use the expression “term” in a quite
inaccurate way,
which blurs the differences between conceptual content and its
linguistic
expression. At any rate it seems not too far from the spirit - if not
from the
literal meaning - of Leibniz’s writings to consider a term as a
meaningful
linguistic expression. Therefore we can re-define a term in a
Leibnizian
sense as a linguistic expression associated with a meaning, which can
be the
subject (or predicate) in a sentence. That an expression is meaningful does
not
necessarily imply, however, that it is associated with a concept or an
idea. If I
read or hear the compound name “round-square”, I am certainly
able to understand
the meaning of the two composing words “round” and
“square” and I am also able to
understand that the compound refers to
something which is, simultaneously, round
and square. But in this case nei-
ther a thing nor an idea corresponds to the
compound word, because the
expression “round-square” gives rise to a
contradiction.
Terms - as Leibniz writes in our text - may be considered in three
different
ways: 1) from the extensional point of view, simply taking into ac-
count the
object (or objects) which fall under them; 2) in a formal way,
looking at their
intension or conceptual content; 3) according to the
pecu-
liar way in which they «mention or show different thoughts». Leibniz
char-
acterizes the first way of considering terms by means of the
expression
“materially” (materialiter), which is strongly
reminiscent of the “material
supposition” and of the analogous “taking a term
materially” of the
scholastic logicians.Introductionis in Logicam, sive Totius
Logicae Compendii Absolutis-simi, Libr. VI, Venetiis, apud Io.
Baptistam et Io. Bernardum Sessam, 1597, p. 35: «Est ergo sup-
positio
materialis illa qua terminus significat se ipsum, seu vocem; ut “homo est vox”,
homo dici-
tur supponere materialiter, quia seipsum, seu vocem ipsam
significat: quae suppositio reperitur in
vocibus etiam non significativis».
tic structure or
grammatical form of the term itself - according to its pre-
vailing meaning in
medieval times - but to the subject matter or “thing” to
which the term properly
refers. Even though terms - with the exception of
proper names - name properties
or peculiar aspects of things - as, for in-
stance, “yellow” or “round” or “man”
- they refer, insofar as we consider
them materially, to the things themselves to
which the properties are at-
tributed. As clearly emerges from General Inquiries, for Leibniz there is no
difference between adjectives
and substantives: both have the same logical
structure, the same “predicative
character”, and hence both presuppose a
thing (or several things) of which they
are the predicates.Allgemeine Untersuchungen über die Analyse der Begriffe und
Wahrhei-ten, Lateinisch-Deutsch, Herausgegeben von F. Schupp, Felix
Meiner, Hamburg, 1982, p. 2 (C, p.
356).
structure of
“yellow” or of “man”, for example, is interpreted by Leibniz as
implying
something - a concrete thing - which is respectively yellow or a
man. Thus,
different terms may refer to the same individual. Leibniz’s ex-
ample is that of
“king” and “prophet”, which, in the case of the Biblical
David, have identical
denotation.
Terms, however, even though referring to the same object, may differ
formally - i. e. regarding form, insofar as they mention
different properties
or different aspects of the given object. Even though David,
the individual,
is both a king and a prophet, being a king is a different
property from that
of being a prophet. Here Leibniz uses “formally” (formaliter) in sharp con-
trast to “materially” (materialiter) with the aim of distinguishing the form -
or
intension or conceptual content - of a term, from its matter or extension.
In a
short commentary to a book by the Jesuit Aloys Kümmet (edited in
1706), Leibniz
sketches an analogous distinction between things and
concepts:
Thus things which are really distinct usually are distinguished by
means of the
senses, whereas things which differ in concept - i. e. those
which differ in
their formalities, even though they are not really different -
are distinguished
by the mind. On a plane, for example, the triangle and
the trilateral are not two
different things, but differ in concept only; there-
fore they are the same thing
really, not formally.
test, et
denominatio».
Thus, terms are said to differ formally, because they differ
in their formali-ties. The word “formality” (formalitas) is typical of the Scotist tradition, but
it
cannot be assumed as evidence of any scotistic influence on Leibniz’s
phi-
losophy. As is well known, in Leibniz’s time the current scholastic
doctrines
were characterized by an eclectic attitude which blurred the borders
be-
tween the different philosophical positions. Therefore, even if an
expression
is typical of a certain philosophical milieu, from its use one cannot
automat-
ically infer that the user adhered to the theses commonly held by people
be-
longing to that milieu.
In a revealing passage of the New Essays, which is also very
important for
setting the terminological distinction between extension and intension of a
term, “formality” is
explicitly held by Leibniz as a synonym of the word “idea”:
... when I say Every man is an animal I mean that all men are included
amongst
all the animals; but at the same time I mean that the idea of animal
is included
in the idea of man. “Animal” comprises more individual than
“man” does, but “man”
comprises more ideas or more formalities: one has
more instances, the other more
degrees of reality: one has the greater exten-
sion the other the greater
intension.
University Press,
1981).
Hence we may paraphrase what Leibniz writes on formalities saying that
terms
which differ formally are different in virtue of their conceptual con-
tent and
that not all conceptual distinctions give rise to real distinctions (a
conclusion
which maintains a Scotist flavour as well).
Till now I have attempted to illustrate what Leibniz properly means
by the first
two ways of considering a term, out of the three mentioned
above. The task has
been not particularly difficult, because there are many
other texts in which
Leibniz proposes or discusses the distinction between
these two different ways.
Problems arise, however, with the “third way”,
which seems to have been
explicitly recognized only in our text. As Leibniz
observes, terms considered
formally may express the same conceptual con-
tent in a different guise. And what
the “third way” of considering a term
takes into account is precisely the guise
or manner in which a term “ar-
ranges” or “presents” a given conceptual content.
Clearly, this “arrange-
ment” or “presentation” must depend partly on language - i. e. on the dif-
ferent possibilities we
have of expressing the different aspects of an idea. At
the same time it seems to
depend also on the fact that a term expresses a
complex idea or formality. If two
terms express the same simple idea - i. e.
an idea which cannot be further
analyzable - then they name the same
thought in different
ways and do not «mention or show different thoughts»
(as Leibniz writes).
Leibniz tries to clarify this point with his standard example of the
terms
“triangle” and “trilateral”. “Triangle” and “trilateral” differ from
each
other because they represent different ways of considering the same idea
(i.
e. that of a geometrical figure with three sides and three angles).
“Triangle”
and “trilateral” are not different formalities,
but simply different ways of
thinking of the same formality. That the formality
is the same - Leibniz
claims - can easily be shown by means of conceptual
analysis: if we proceed
to analyse the concept corresponding to “triangle” we
individuate a set of
basic notions or ideas which characterize the concept
corresponding to “tri-
lateral” as well.
In our text the formalities are objective, independent of the
different
ways in which we can consider them, whereas the “way of
considering”
gives rise to different expressions of the
same formality. Thus, the third way
of considering a term seems to be determined
by the way (or ways) in
which the formalities are
considered by the mind. As I observed above, this
is apparently the only text in
which Leibniz makes this claim, and what is
puzzling about it is that it seems to
be in neat contrast with Leibniz’s main
doctrine on formalities. If we look again
at the passage previously quoted
from Leibniz’s commentary on Kümmet’s book, we
see that the two terms
“triangle” and “trilateral” are there assumed plainly to express two differ-ent formalities, not two different ways of considering the
same formality. In
Leibniz’s words, “triangle” and “trilateral” «are the same
thing really, not
formally».
New Essays, in which the “triangle-trilateral” issue is
explicitly considered:
... someone who said The triangle and the trilateral are not the same
would be
wrong, since if we consider it carefully we find that three sides
and three
angles always go together... However, one can still say in the ab-
stract that
triangularity is not trilaterality, or that the formai causes of the
triangle and
of the trilateral are not the same, as the philosophers put it.
They are
different aspects of one and the same thing.
“Triangularity” and “trilaterality” are said to be “the formal causes” (but
a
more litteral translation would be: “the formal reasons”) respectively, of
the
triangle and of the trilateral; and they are “different aspects of one and
the
same thing”. In this case too, Leibniz seems to agree that “triangle”
and
“trilateral” correspond to two different formalities.
What is the same is not
the idea, but the “thing” - presumably all the concrete
instances of trian-
gles, which are, simultaneously, triangular and
trilateral.
As Benson Mates observes, the “triangle-trilateral” issue is also of
some
importance for understanding Leibniz’s attitude towards the
substitutivity
principle salva veritate (and, hence,
Leibniz’s attitude towards identity).The Philosophy of Leibniz. Metaphysics and Language,
Oxford-New York,
Oxford University Press, 1986, pp. 123 ff.
The expressions “triangle” and “trilateral” are Leibniz’s favourite
examples
for showing that, in some peculiar contexts, the substitutivity
principle fails.
And, if we look at the way in which Leibniz characterizes these
contexts,
we can gain precious hints concerning formalities. Consider, for example,
the following passage from a text
written about 1686:
A = B means that A and B are the same - i. e. that one can be substi-
tuted for
the other anywhere. (Unless it is precluded, as happens in those
contexts in
which one states that a given term is considered in a peculiar re-
spect. For
instance: even though the triangle and the trilateral are identical,
if one
states The triangle insofar as it is a triangle has 180 degrees, one can-
not
substitute trilateral for triangle. Here there is something material
which
precludes the substitution).
What is interesting here is, first of all, Leibniz’s use of the expression
“ma-
terial”, which has, in this circumstance, a meaning very near to that of
the
medieval tradition, as opposed to the meaning considered above.
Second,
we may observe that preventing the substitutivity in the given context is
the
fact that a term is considered in a peculiar respect.
Once more it is the wayof considering which puts in
evidence the formal difference between two
terms, which
are (the formai difference notwithstanding) identical. In Gen-eral Inquiries, propositions which do not allow the substitutivity
of equiva-
lents are called “formal” and “reflexive”: they «assume one of the
coinci-
dentials in such a way that it is distinguished from the others» and they
«do
not so much speak about a thing, as about our way of
conceiving it»
Allgemeine
Untersuchungen..., p. 34 (C, pp. 366-367).
phasis mine).
Similarly, in a text on geometry written about the same time
as General Inquiries, Leibniz writes a short remark with the aim of
empha-
sizing that the substitutivity principle for terms holds in all the
propositions
which are “direct” (directae) - i. e. in all
those propositions «which do not
take into account the way of considering (nec in ipsum considerandi modumreflectuntur)».Leibniz’s
Theory of Relations, Stuttgart, F. Steiner Verlag, 1992, p. 147.
sponding to “triangle”
is different from that corresponding to “trilateral”
(their formalities are different); and, consequently, he seems to assume:
1)
that it is the way of considering the denotation of a
term that gives rise to a
formality; 2) that his salva veritate
principle provides a criterion for the
identity of things
not of concepts.
Therefore, the problem arises of how to evaluate the position as-
sumed by
Leibniz in our text. If the “third way” of considering a term is
explicitly
mentioned only here, then one may argue that it is something
which does not
correspond to Leibniz’s prevailing view. And hence it can
be assumed to be a
position held on this occasion and suddenly dismissed.
But it seems to me that
this is not so. I have the impression that what we
see here is Leibniz’s
ambiguous attitude towards nominalism. The ambigu-
ity consists mainly of
Leibniz’s acceptance of the existence of ideas and
essences of things, on the one
hand, and of his simultaneous adhesion, on
the other, to a kind of cautious
nominalism which undermines the impor-
tance of abstract terms and ideas, on the
other.Leibniz’s Theory..., pp. 18 ff.
At the beginning of General Inquiries, for example, Leibniz
claims that
the fact that “triangle” and “trilateral” coincide can easily be
shown by
means of analysis, and states in a odd way the following general principle
about
logical coincidence: if terms «are analysed until it appears a
priori
that they are possible, and if the same terms appear formally, then
different
terms are the same».Allgemeine Untersuchungen..., p. 20 (C, p. 362);
english transl. in: Leibniz,
Logicai Papers, ed. by G. H. R. Parkinson, Oxford,
Clarendon Press, 1966, p. 53. A and
B, they coincide «if the one
can be substituted in
place of the other without loss of truth, or if, on
analysing each of the two by
substitution of their values (i. e. of their defini-
tions) in place of the
terms, the same terms appear on both sides».Ibid.
Leibniz specifies, “the same terms” means “the same formally” - i. e. at the
end of the analysis it
is necessary that on both sides of the copula the same
formalities or ideas appear as component parts of A and B: «the same I
mean, formally
- for example, if L, M and N appear on both sides».Ibid.
Leibniz seems to hold a point of view quite similar to that of our
list of
definitions: terms like “triangle” and “trilateral” have the same ideal
or
conceptual content. Presumably, then, they are not different formalities, but
simply different ways of expressing the same formalities or the same finite
set of ideas.
That “triangle” and “trilateral” are different espressions of the same
idea can
be inferred as well from the following short passage from NewEssays:
... I have remarked earlier that there are redundant ways of expressing ideas,
which add nothing to things. It is as though someone were to say “By
Triquetrum
I mean a trilateral triangle” and to infer from that that some tri-
laterals are
not triangular.
To Des Bosses (about 1712), however, Leibniz clearly writes that «Triangle
and
Trilateral are the same being, but different terms: they differ formally,
not
materially».
Thus, in some texts Leibniz writes that terms like “triangle” and “trilat-
eral”
differ formally, whereas in other texts, written at about the same pe-
riod, he
plainly states that “triangle” and “trilateral” are only different ex-
pressions
of the same formality or of an identical set of ideas. Analogously,
whereas in
most of his writings Leibniz holds that the salva veritate
princi-
ple provides a criterion for the identity of things, there are texts (General
Inquiries, for instance, as we have seen) in which he claims
that terms like
“triangle” and “trilateral” are interchangeable salva veritate, because they
express the same idea (or ideas). All this
can be easily explained if we take
into account that for Leibniz the existing
world splits up, as it were, into
two different realms: that of (existing) things
and that of the ideas, or
essences, corresponding to things. Both realms are
“real” and each existing
thing has an ideal counterpart - an essence which, like
the thing itself, is
one in number. The same essence, however, may be expressed
by several
definitions:
To reinforce the distinction between essence and definition, bear in
mind that
although a thing has only one essence, this can be expressed by
several
definitions, just as the same town can be represented by different
drawings in
perspective depending on the direction from which it is
viewed.
In the same vein, as we have seen, Leibniz observes that «there are redun-
dant
ways of expressing ideas, which add nothing to things».
the one hand, different definitions of the same essence are not
different
essences, but simply different ways of presenting one and the same
essence,
on the other, a conceptual distinction does not correspond to any
distinc-
tion among things.
In Leibniz’s logical ontology, terms considered as linguistic items ex-
press
ideas or concepts, and denote things. Concepts and ideas are objec-
tive,
non-linguistic in nature and independent of their expressions: they do
not exist
in space and time, but they have a kind of existence in what Leib-
niz names “the
realm of ideas” (regio idearum).regio
idearum cf. VE 7, p. 1461: «Ego soleo dicere esse
aliquid in regione
idearum itaque spatium numericum videtur esse tantum consideratione
quadam
existentiam in regione idearum, quasi in spatio aut tempore»).
tries to be coherent with his “provisional
nominalism”,per provisionem cf. Grua, p. 547 and M. Mugnai, Astrazione erealtà. Saggio su Leibniz, Milano,
Feltrinelli, 1976, pp. 133 ff.; B. Mates, The Philosophy of
Leibniz,
p. 170 ff.
that he should look at terms as
directly denoting things; whereas if he em-
phasizes the “realistic” side of his
ontology, he considers terms as repre-
senting first concepts or ideas. That the
nominalistic mood is prevalent,
however, is easily shown by our text, where
Leibniz, after having distin-
guished the formalities from
the ways of considering them, states that, in
the last analysis, what really
matters is “staying with things”.
Another element of interest in our text can be found in the second
passage
between brackets, of which I offer a translation:
Thus rime and place are neither subjects nor attributes. And the same
holds for
numbers and orders. Thus “ten” is not an attribute of any thing.
In fact the
number ten cannot be said either of the aggregate or of the sin-
gle numbered
things. The same holds for relations which are common to
the related things - for
instance, the similarity of two things. Hence, there
are attributes which are at
the same time in several subjects: of this kind
are, therefore, orders and even
time and place.
The most relevant points of the entire passage are two: 1) Leibniz, after
having
asked himself if there is a third kind of “things” besides subjects
and
attributes, answers the question positively; 2) numbers and relations are
said to
share some common feature, which determines their belonging to
this third kind of
entities.
That numbers and relations have something in common is a claim
widely held by
scholastic authors. Consider, for example, what Ockham
says in the following
passage: «Similarity stays for (supponit) two
similar
things considered simultaneously, as in the case of the name of a
number
[...] Thus Socrates is not two, even though Socrates and Plato are
two».Quodlibeta septem, ed. by J. C. Wey in Opera
Philosophica et Theologica, New
York, The Franciscan Institute, St.
Bonaventure, vol. IX, 1980, p. 616.
Relying on this scholastic tradition, Leibniz thinks that numbers are
neither
predicates of single things nor predicates of the aggregate (or set) of
num-
bered things: they are attributes of a special kind, which, like relations
«are
simultaneously in several subjects». What is striking here is that, at
first
glance, Leibniz seems to recognize that relations (and numbers) belong
to
an autonomous “third kind” of beings besides subjects (substances) and
at-
tributes, contrary to his “official” doctrine that relations are not beings (en-tia).Leibniz’s
Theory of Relations..., pp. 18 ff.
some coherence on Leibniz’s part on this
point. In fact, Leibniz does not
explicitly state that numbers and relations are
beings (entia) - a
conclusion
which plainly contrasts with his main ontological views. He writes
that all
beings are substances or attributes; he asks himself if there is
something
“third” besides substances and attributes and answers that numbers and
re-
lations are this third kind of “thing”. Clearly, if the realm of all beings
di-
vides itself into substances and attributes and if relations and numbers
are
something “third”, in addition to substances and attributes, then
relations
and numbers cannot be beings (entia) in the
same sense in which sub-
stances and attributes are. And if they are attributes, they must be
attributes
of a special kind.
In this case we are faced with a typical Leibnizian attitude about rela-
tions:
on the one hand, Leibniz emphasizes that relations inhere
simultane-ously in several subjects; on the other, he claims that because of that very
fact, they cannot be real attributes
of things. As is well known, Leibniz
agrees with the scholastic thesis that the
same accident cannot inhere in
things numerically distinct. Therefore, relations
are not real accidents or
real properties, but “mere ideal things”, as we read in
the fifth letter to
Clarke.
kind of inherence is merely mental, a product of our thinking-together
sev-
eral things at once. Thus, if Leibniz plainly recognizes the true logical
na-
ture of relations - insofar as he recognizes their “multiple inherence” -
this
happens at the expense, as it were, of their ontological reality.
Relations
have a “diminished being” and are not real, as substances and
accidents
are.Leibniz’s Theory of Relations..., pp. 18 ff.
A major element of interest in our text lies in the fact that here
Leibniz
mentions, as a characteristic property of relations, “multiple inherence”
and
states that numbers share with relations this same
property. Clearly, insofar
as numbers inhere simultaneously in several subjects,
they have a mere
mental nature as well. And, properly speaking, their “inherence”
is not real,
but is due to our capacity to think-together a plurality of things
in the same
act of thought. Chauvin’s Lexicon
philosophicum helps us better understand
of what Leibniz says about
numbers.Lexicon Philosophicum, Leeuwarden, 1713 (second edition) -
Re-
printed: Duesseldorf, Stern Verlag, Janssen & Co, 1967, p. 444.
Numerus, Chauvin
writes that a
number «says a plurality of unities» and that it implies a «si-
multaneity of the
same unities» - i. e. that a number expresses a relation
be-
tween our minds and a plurality of things. Therefore a number is made
up
of matter and form: matter is constituted of «things which can be
num-
bered, as, for instance, coins»; form is «the idea by means of which a
plural-
ity of things is reduced to unity» by means of an act of the
understanding.
Hence, whereas form is dependent on mind, matter is not. In this
case too,
the analogy with relations is very strong: whereas for a (real)
relation it is
necessary that some individuate exist and that the mind grasp them
in an
act of thought, for the existence of numbers it is necessary that a
plurality
of things (objectively) exist and that the mind have the possibility of
grasp-
ing them in a single act of thought. As Chauvin observes - and surely
Leib-
niz would agree - the form of a number is not something which
adds itself
really to the numbered things. The idea which constitutes the form of a
given number is a modification of the mind of the
person who is thinking -
it is not a modification of the numbered things. This
form «may be named
an “extrinsic denomination”».Ibid.
This shows once more how deep the influence of scholastic (and late-
scholastic)
doctrines is on Leibniz’s ontology and philosophy of logic.