Biblioteca digitale Progetto Agora
Abbreviations employed only in this paper:
AT Descartes, R. Œuvres. Edited by Ch. Adam and P. Tannery, nouvelle présentation, Paris
1964-1976.
EF G. W. Leibniz. Escritos filosóficos, edited by E. de Olaso, Buenos Aires 1982.
M Sextus Empiricus. Adversus Mathematicos, translation by R. G. Bury, London/Cambridge
(Mass.) 1967.
PH Sextus Empiricus. Outlines of Pyrrhonism, translation by R. G. Bury, London/Cambridge
(Mass.) 1967.
The study of modern scepticism – fathered in the last twenty-five years
by Richard
H. Popkin – constitutes today so far-reaching and so fertile a
field of
investigation, as to be considered one of the most important
branches of the history
of modern philosophy. The lines of research laid
down by Popkin and Charles B.
Schmitt, favoured the Pyrrhonian and the
Ciceronian traditions, respectively; but
for my part I feel that there are
meaningful aspects of modern philosophical
speculation not to be explained
otherwise than in the light of the impact thereon of
the tradition be-
queathed us by Zeno of Elea, particularly as regards certain
derivations of
his continuum-paradoxes R. H. Popkin, The History of
Scepticism from Erasmus to Spinoza, Berkeley/Los
Angeles/London
1979. Ch. B. Schmitt, Cicero Scepticus: A Study of the Influence of the
“Academica”in the Renaissance, The Hague 1972. I have published reviews on both books in “Noûs”,
XVIII
(1984), pp. 135-144, and “International Studies in Philosophy” VII (1975),
pp. 57-68, respec-
tively. A general remark: although Leibniz sometimes
dissociates the problem of the continuum
from that of the infinite, such a difference has not been here taken into
account.
the place occupied
by Leibniz’ philosophy within the history of modern
scepticism. Although it seems to
me that I was the first writer to point
out the importance of Pyrrhonism in the
understanding of certain signifi-
cant features of Leibniz’ work, even so I am
disposed to maintain that it
was Zeno’s paradoxes, and the various shapes they have
assumed in modern
young Leibniz, the most perilous challenge to reason
Leibniz and Greek Scepticism (Bryn Mawr College,
1969,
unpublished).
I began by distinguishing three sceptic traditions – the Pyrrhonian, the
Academic
and the Zenonian. Leibniz identified Zeno’s paradoxes with Pyr-
rhonism, but this
point must be cleared up. Leibniz was acquainted with Sex-
tus’ version of these
paradoxes Accessio ad arithmeticaminfinitorum
(end of 1672) A III, I, 16 and 20 (this is the only edition
that contains references to
the Pyrrhonians). Cf. also De religione magnorum virorum, Grua
42 and the letter to Foucher A II,
I, 238. Such an
assimilitation was not uncommon before Leibniz’ time; cf.
Montaigne,
Apolo-gie de Raimond Sebond, Essais II, 12, edited by M. Rat,
Paris 1948, pp. 277-278; and Foucher’s
letter to Leibniz, GP
I, 400 and 411-412.
flicting
opinions and gives us to understand that it is impossible to determine
where lies
the truth, with the emphatic suggestion that it is not possible to
decide between
them, and seeks to provoke in the reader a state of suspended
judgement. Zeno’s
arguments appear in Sextus as being one of these conflict-
ing arguments (for
example, that of those who maintain that motion does not
exist); and in this, restricted sense, Sextus does not subscribe to them,
inas-
much as to do so would imply the formulation of a dogmatic opinion.
Such
arguments run contrary to those which support another conflicting
opinion
(for example that of those who claim to prove the existence of motion by
the
fact that they are themselves walking). Sextus allows both sides merit
suffi-
cient to save them from being disqualified; but at the same time he
suggests
that it is impossible to attribute the truth to either. Consequently, even
if
Sextus passes on Zeno’s arguments, he does not himself subscribe to them
-
rather does he propound them as the opposite poles of force in a
struggle
whose issue cannot be decided
called
a “dialectic” sense does Pyrrhonism partially coincide with Zenonism
but not making
common cause with it.
However Sextus does not consistently follow that rule. One of the cases
in which he
does not is, precisely, that of the infinite. Sometimes he speaks
of the infinite
(apeiron) as if it were a property, sometimes as if it were
an
extant whole. If he were to apply his rule consistently to the infinite,
he
would have to propound arguments of the following nature:
Some claim that the infinite is (for example) knowable.
Others claim that the infinite is not knowable. The conflict between the two
claims is
not (does not appear to be) determinable.
Hence, it is advisable to suspend judgement upon the question of knowability
of the
infinite.
I am not now going to consider what I believe to be the specific connex-
ion
established by the sceptic, between premises and conclusion in this type
of
reasoning. Sextus’ texts furnish us with no exposition as to the
determinabili-
ty of problems involving the infinite. When he speaks of the
infinite, he
assumes sole responsibility for what he asserts, inasmuch as he
maintains his
own opinions without confronting them with those of others. For
example,
he affirms that we have no experience
is not possible, says Sextus, to examine the
infinite, because if it were so possi-
ble, the infinite would be thereby limited;
in fact, it is science (episteme) that
circumscribes the
undefinedapeiron).
cannot
be grasped
infinite, because if it were infinite, it would not be in any given place;
indeed,
if it were in any given place, such place would not partake of its infinite
char-
acter, and hence it would not itself be infinite
analyse in his greatest
work the Sceptics’ attitude. Cf. R. Mondolfo,
El Infinito en el pensamientode la Antigüedad Clásica, Buenos Aires 1971. It is worthwhile mentioning that the infinite
char-
acter of any process of justification of knowledge, is sufficient for the
sceptics to consider such
process unacceptable; cf. Aenesidemus (PH I, 122) and
Agrippa (PH I, 164 ss.). These argu-
ments are endorsed by Sextus. (In the second
section of this study I refer to Leibniz’ reaction
to this problem). One could
say that the sceptic has a view of, or that he slightly assents to, the
infinite,
but that he neither takes standing nor does he makes any claim about the infinite,
i.e.
he does not gives any strong assent to the infinite. Cf. M.
Frede,
The Sceptic’s Two Kinds of Assentand the Question of the
Possibility of Knowledge, in the collective volume Philosophy in
History, edited by
R. Rorty, J. B. Schneewind and Q. Skinner,
“Ideas in Context” Series, Cambridge 1984,
pp. 255 ss. In another study I intend
to go into this matter in some detail, since it is crucial to
the understanding
of Scepticism.
tions are negative does not
make them any the less assertive, and the sceptic is
commiting himself to them. It
is not, I think, by chance, that Sextus refrains
from advising us to suspend
judgement when faced with the infinite, nor that
his opinions seem to be moves in a
dialectic game. Without going at all deep-
ly into the question for the moment, it
seems to me enough to show that the
notion of the infinite, is one case at least in
which the Pyrrhonian sceptic does
express a definite opinion. The relatively
exceptional (but not unique) char-
acter of this notion, seems to me to explain
something observable since Ren-
tion
of Pyrrhonism to such philosophical conceptions as demonstrate the
impotence of
reason to solve problems involving the infinite – the most noto-
rious example being
that of Zeno.
In the course of his life from youth unto old age Leibniz dealt with the
subject of
scepticism in a number of writings. Nevertheless, this aspect of his
work has
received extremely little attention from the scholars, which is to a
certain extent
understandable inasmuch as if only such of his works as were
solely and explicitly
devoted to the rebuttal of scepticism be taken into
account, the following list will
exhaust all possibilities of study:
The list may be lengthened by the inclusion of various letters written
with the
primary purpose of rebutting scepticism, amongst which I would
mention:
In this first stage of our research, any comparison of the foregoing brief
list with
the massively impressive total of Leibniz’ works will be disappoint-
ing; and if we
go on to consider that none of the writings mentioned was
published in Leibniz’
lifetime, we must draw the conclusion that the historical
importance of Leibniz’
examination of scepticism was almost nil; and finally,
if I am to point out that in
none of those writings does Leibniz systematically
deal with the relationship
between scepticism and the problems arising from
the infinite, it may well be
doubted that the present contribution be pertinent
to this symposium.
But this is a first stage, to which it is not necessary or desirable to limit Cf. my study
out
research. The full scope of the sceptical problem in Leibniz’ thought can
only be
accurately gauged by taking the investigation further, by the light of
two
complementary criteria. One of these is very simple, and amounts to the
examination
of the dozens of passages in which Leibniz briefly or even impli-
citly refers to
the subject. In this way the list is considerably expanded Leibniz and Scepticism in the collective
volume Scepticism from the Renaissance tothe
Enlightenment, edited by R. H. Popkin and Ch. B. Schmitt, Wolfenbütteler Forschungen, Band 33,
Wolfenbüttel 1987, where
several of Leibniz’ writings are commented. An up to now unpub-
lished
Leibniz’ text, relevant to my research, especially in the domain of Leibniz’
mathematical
manuscripts, could be discovered at any time. About certain
problems affecting the Berlin
Academy’s edition, cf. E.
Knobloch, L’édition critique des manuscrits
mathématiques leibniziens, “Edi-
zioni critiche e storia della
matematica”, Atti del Convegno CIRM, Trento 1985, pp. 85-108.
The other is not so simple, nor so easy to express in a few words, but it
is
decisive. Please do accept the following statement which I here offer
without
sufficient evidential support. Leibniz’ conception of the history of
philoso-
phy, and especially of scepticism, was systematic,
which explains how it was that
he held an opinion to be sceptic, or susceptible of
sceptic conclusions, without
troubling about the intentions of those who maintained
such opinion. Such is
the case, in Leibniz’ belief, with some of Galileo’s
opinions
considers sceptical Luis de
Molina’s opinion that the will is not subordinated to the rule that
“nothing is
without reason”, A VI, II, 480.
many of the opinions of
professed sceptics do not, as he sees it, contain any-
thing new, and are sometimes
indiscernible from the ideas of the dogmatists
and Leibniz here included some of the
opinions of no lesser an authority than
Sextus Empiricus
novum aliquid dixisse videantur...”) says Leibniz, at
the end of his Specimen quoted in note 14.
In the same
work, when he is looking at the distinction between “thing” and “phaenomenon”,
he
comments, “and this is to be found in many authors, since the dogmatists also, who
under-
stand about things, hold that many of those that we perceive are not
substances, or fixed quali-
ties of substances, but phaenomena” (“... atque hoc
quidem in multis non male. Nam dogma-
tici quoque rerum intelligentes pleraque
quae percipimus non pro substantiis aut fixis substan-
tiarum qualitatibus, sed
pro phaenomenis habent”). Cf. the letter to Remond (1714) GP III,
606. And with
regard to the impossibility of ataraxia Leibniz emphasises the
lack of originality
of the sceptics and even of Sextus himself: “Besides, the
anxiety of hope or fear is becoming no
more to the dogmatist than to the sceptic”
(“... Caeterum ob spes et metus anxium esse non
dogmatico magis philosopho quam
sceptico convenit”).
such of Leibniz’ writings as refer explicitly or implicitly to scepticism.
Our problem is then that of determining the criterion by which Leibniz
held an
opinion to be conducive to scepticism or plainly sceptical by itself.
He himself has
left us no definition or characterization of such a criterion; but
it would not be
unsafe to say, that he must have had in mind such opinions as
in the short or long
run question the validity of principles. Here we are up
against another difficulty,
inasmuch as principles in Leibniz’ writings are
ranked in several different
waysJ. Ortega y Gassetin his book La idea de principio en Leibniz y los orígenes de la teoría
deductiva, Buenos Aires 1958,
pp.
13-16. Cf. L. Couturat, La logique de
Leibniz, Paris 1901, pp. 216 ss.; and R. C. Sleigh Jr.,
“Leibniz on the Two Great Principles of All
Our Reasonings”, in the collective volume Contem-porary
Perspectives on the History of Philosophy, Midwest Studies in Philosophy,
volume VIII, Minnea-
polis 1983, pp. 193
ss.
classification which may
serve as a first guide-line in this matter.
Without departing from generalities, I would observe that in A (a) he
believes that
the principle of contradiction is indirectly at stake
ties, means, according to Leibniz, saying that “to be and not to be are
the same”, De synthesi etanalysi universali, GP VII,
295.
remaining cases, that of sufficient reason. As our
understanding of the subject
by opinions
or theories which questioned principles. Let us see now only two
concrete examples.
The unique philosophical book which he published in his
life-time,
Essais de Théodicée, consists of a long discussion with Pierre Bayle
about B
(b) with interesting digressions into A (b) and B (a). In his most
important
projected work, Science Générale, one of his principal objectives
was,
visibly, to discuss A (a) with the sceptics
p. 699.
I began by suggesting that the study of scepticism was something to
which Leibniz
devoted little time, and that somewhat secretively; now, how-
ever, we shall see
that it would not be wrong to say that such study was a
leit-motiv of all his
workM. Dascal, Sobre
Leibniz y el escepticismo, “Revista Latinoamericana de Filosofía”
XII
(1986) pp. 55-56 has suggested that I have shown that every phase in Leibniz’
philosophical
development reflects new aspects of his battle with Scepticism.
That is an exaggeration; but
the germ of truth in his observation is that a
sophisticated Zenonism is at work in the conceiv-
ing of his philosophy.
In the first part of my contribution I shall go into some aspects of that
subject,
and draw attention to certain historical peculiarities. In the second
part I shall
be referring to A (b) that is to say Leibniz’ attitude to the problem
of the
infinite, involved in the justification of contingent truths and emphasis-
ing its
systematic character.
I. I do not know of any sceptic writings from the ancient world, in
which what
Leibniz calls the principles of necessary truths (identity, non-con-
tradiction, tertium non datur) are explicitly examined and questioned. We
have
only a few philosophical replies to such questioning, the most famous being
no
doubt those of Aristotle in Book Gamma of his Metaphysics.
How such ques-
tioning had been expressed, and the importance its anonymous authors
had
given it, are matters of conjecture. The well-known answers counsel us,
in
general terms, to refrain from arguing about the principles, or alternatively,
to
proceed ad hominem by showing the sceptic that he too
respects these principles
at a linguistic and a practical level. Leibniz in his
early writings (say, from
1666 to 1672) thought that the sceptics held everything to
be “negotiable”
(that is, subject to proof), except the principle of
non-contradiction. Later on,
in short passages in different parts of his writings,
he does in fact offer justifi-
cation of various kinds of this principle. I have
elsewhere examined some La lógica leibniciana de las controversias,
proceedings of the symposium “Controvérsias
Científicas e Filosóficas”, Evora,
Portugal, 1985.
In any case, we can be
fairly sure of following the lines of reasoning which he
believed indirectly to be
questioning rational principles. Leibniz was particu-
larly sensitive to scientific
propositions susceptible of philosophically sceptical
conclusions. Difficulties
related to the continuum made it clear, in his opin-
ion, that the principles of
pure reason are defenceless, if Euclid’s ninth axiom
(that the whole is greater than
its part) be called in question.
Leibniz studied the problems of the infinite and the continuum in con-
nexion with
scepticism, in various writings at the beginning of the 1670s Theoria motus
abstracti (1670) A VI, II, 258 ss.;
Demonstratiopropositionum primarum (1671-1672) A VI, VI, 479 ss.; and Accessio ad arithmeticam infinitorum (end of
1672), quoted
in note 3. With regard to the sense in which Leibniz uses “infinite”: “I
general-
ly say that there are three grades of the infinite. The infimum, as for example the asymptotics of
hyperbole, which alone is
what I call infinite, greater than it is held to be, which might be said
of other
accepted meanings. The second grade is the maximum of its
kind, as the maximum of
all extensions is total space, and the maximum of all
time is eternity. The third grade of the
infinite, the highest grade of all, is
the infinite itself (everything), which exists in God,
because
that unity which is God is all things, since in Him lies all that is
needful to the existence of all
other things”, A VI, III, 385. Cf. A VI, III,
281-282.
and in these he maintained that although the subject of the
infinite was cer-
tainly not new in mathematical tradition, it had in the
immediately past years
given rise to certain fundamental problems, which
philosophers had not
solved, and had sometimes even recommended should not be
tackled. Leibniz
feared that such situation could be exploited by sceptics Regulae... VIII; AT X, 392; Principia
philosophiae, I,
26 and II, 34-35. On Leibniz’ opinion about Descartes’
substitution of the notion of the inde-
finite for the notion of the infinite
cf. Theoria motus abstracti, A VI, II, 264; LH 56 and GP
IV,
228. On the Sceptics’ attitude about wholes and parts, see note 35.
those writings would seem to have been twofold, his view being, on
the one
hand, that problems stemming from a consideration of the infinite in the
con-
text of such especially conflictive scientific matters, must be taken
seriously,
and on the other, that they are essentially soluble. We have to tread
carefully
in this matter. Leibniz does not share the dogmatism of philosophers
about
the axioms. But the tolerance he thinks should be extended to those who
ask
to be given reasons for the axioms, has its root in his immovable conviction
of
the fertile nature of the principle of contradiction. Leibniz is willing
to
accept the sceptic objections, and insists that even axioms should be proved,
quite sufficient
riences, all
truths that are certain can be indisputably deduced” (“... et ex hoc uno
accedentibus
notionibus experimentisque omnes veritates certae irrefragabiliter
deduci possunt”), Specimen,
quoted in note 14. The interpretation of this passage is difficult. Although
Leibniz says that
only the principle of contradiction allows us to deduce all the
truths that are certain, however the
clause “adding notions and experiences”
makes the uniqueness of that principle somewhat rela-
tive. Furthermore Leibniz
maintains that deductions can only be made from the principle of
contradiction.
On a similar case in Monadology, cf. R. C. Sleight’s study
quoted in note 20.
These preoccupations of Leibniz were first made public in 1670 in hisTheoria motus abstracti. In dedicating the work to Louis XIV,
Leibniz under-
lines the importance of unravelling the labyrinthine threads of the
continuum
and the composition of motion “confundendos Scepticorum triumphos”
After propounding his
theory, Leibniz emphatically states that he has arrived
at the solution of problems
“which are the principal successful weapons in the
hands of the Sceptics”
tric wheels turning on a plane surface, that of
incommensurables, and that of
the angle of contact. Since I have elsewhere referred
to the question of con-
centric wheels
tingency). All what I am going to say is for the sake of readers
who are not
familiar with elementary geometry.
Let us try to form a clear and simple idea of some of the aspects of the
problem,
beginning with the more informal version of Leibniz himself. I am
going to follow,
in part, the exposé which Leibniz drafted for Duchess
Sophia
There are two proofs: firstly, that the common angle ABE is greater than
the angle
of contact ABNCDF; secondly, that that common angle is infinitely
greater than the
angle of contact. The more interesting philosophical prob-
lems arise from the
second proof.
Relying upon Euclid, some mathematicians Euclid’s Elements, III,> 16; cf. Th. L. Heath, The Thirteen Books of Euclid’s Elements, New York
1956 and I. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements, Cambridge,
Mass./London 1981,
pp. 177 ss. Leibniz refers to Proclus among the ancients and,
more recently, to
Christoph Schlüssel or Klau, Clavius (1537-1612), the “modern Euclid”, professor in the Archigim-
nasio della
Sapienza, the College of the Jesuits in Rome. His commentary to Euclid was
published
in Rome in 1574. Possibly Leibniz is also referring
to Cardano and to Clavius’ disciple Gregory
of Saint-Vincent (1584-1667) whose Opus geometricum was published in
Amsterdam in 1647.
er that
angle ABE has two branches or lines, AB and BE, which are straight
lines opening
from the vertex B, which opening we know as the magnitude of
the angle. And in the
same way angle ABNCDF has two branches, that is to
say the straight line AB and the
circular line BNCDF, which lines also open
from the vertex B. Inasmuch as the
opening of the angle, or of the vertex,
does not depend on the length of the
branch-lines, we may take these as being
as short (i.e. as near to the vertex B) as
we please: for example, angle ABE is
equal to angle LBM, since it has the same
opening in the vertex, and also (for
the same reason) angle ABNCDF is equal to angle
LBNC.
Well, inasmuch as the circular line BNC falls between the straight lines
LB and BM,
it may be said that the opening of angle LBM or angle ABE is
greater than that of
angle LBNC or angle ABNCDF. And although all the
circular line BNCDF does not fall
between the straight line AB and BE, if we
take small parts of the three lines near
the vertex B, that is to say, LB, BNC
and BD, we find that BNC does fall between the
other two; and this is suffi-
cient to say that angle ABNCDF or angle LBNC is lesser
than angle ABE or
angle LBM.
Now we have to prove that the ordinary angle LBM (contained between
straight lines
or branches) is infinitely greater than the angle of contact LBNC,
so-called because
it is contained between the circular line BNC and the
straight line LB, which
touches the circle without cutting it. But the straight
line AB or LB continued to
G, does not enter the circle nor cut it, whilst the
within
it and partly without.
To prove that the common angle is infinitely greater than the angle of Professor A. G. Ranea has pointed out to me that points L and M are presupposed
in
contact, it
is enough to prove that however small an extension of the former
be considered – for
example, the thousandth part, or the hundred thousandth,
and so on
ad infinitum it will always be found to be greater than the corre-
sponding
extension of the angle of contact ABNCDF; and hence the ordinary
angle ABE is not only a
thousand times, or a hundred thousand times, or a
million times greater than the angle
of contact ABNCDF, but is infinitely
greater. Let us then place one point of a
compass
the first demonstration without needing to be constructed with compasses.
Their construction
in the second demonstration cannot be justified. Indeed,
either it is superfluous or it should
had been in the first demonstration.
on point C, and taking B as our centre, draw an arc LCM
to measure the
angles of the straight lines; and it will be clear that whether arc
LC be the
hundred thousandth or the millionth part of arc LCM (or however minute
we
may suppose it to be, inasmuch as truly minute quantities cannot be
graphical-
ly represented), the circular line BN will always fall between the
straight lines
LB and BC, since BC is totally contained within the circle. Hence the
angle
of contact LBNC (or LBNCD or LBNCDF) is less than the angle ABC
con-
tained within the straight lines, which is the millionth part (or less) of
angle
LBM; it is clear that the angle of contact LBNCDF is less than the
millionth
part, etc. of the angle LBM or ABE, which is to say that the angle of
contact
is infinitely less than the angle formed by two straight lines. Q.E.D.
The philosophical significance of this demonstration can be expressed in
various
ways. Let us begin with Cardano, who here follows Euclid. Cardano
affirmed that the
quantum of the angle of contact can be continuously and
limitlessly diminished, and
yet that the first quantum, however greatly in-
creased, can still be lesser than
the second quantum, however greatly dimin-
ished. Gregory of Saint-Vincent also
shares Euclid’s view that the angle of
contact is lesser than any finite angle, but
maintains that, although the quan-
tum of the angle of contact be unequal to that of
the straight angle in the
finite domain, it would not necessarily be so in an
infinitesimal domain.
However, the straight angle is held to be the whole, of which
the angle of
contact is the part; and hence, in infinitesimal terms, the whole is
not neces-
sarily greater than its part – which is contrary to Euclid’s ninth
axiom J. E. Hofman, Das Opus Geometricum des Gregorius a S. Vincentio und seine Einwirkung aufLeibniz, Abhandlungen der Preussischen Akademie der Wissenschaften, Berlin 1941, pp. 9, 22-
23. Cf. also A III, I, 12. Leibniz had mentioned him before in his writing on Nizolius, A VI,
II, 432 and in Demonstratio propositionum primarum, A VI, II, 480 in connexion with the problem of
the angle of contact as a counterexample of Euclid’s axiom. Hofman does not examine this
text which was published twenty five years after his monograph. On Cardan cf. Heath’s com-
mentary, quoted in note 32, II, 41. Cardan’s case in connexion with these matters is particular-
ly interesting and I think it has not been studied yet. Leibniz read it passionately since his early
youth (cf. Wilhelm Pacidius, A VI, II, 511) and he thinks that Cardan was a sceptic due to his
ideas on individuality, cf. Specimen quaestionum philosophicarum ex jure collectarum (1664) A VI, I, 87.
The denial that the whole is greater than its part, is something that Leib-
niz
attributes to Scepticism, both in his Demonstration and in his
letter to Gal-
lois; and other authors also have made this historical mistake.
Sextus dialecti-
cally leant upon the validity of that axiom in his dispute with the
Dogmatists,
and never denied it directly
PH III, 45 ss., 88 ss. and 98 ss.; M VII, 276 ss.; M IX, 259 ss., etc. From
these texts it has been
inferred that Sextus denies the axiom that the whole is
greater than its part. For example,
Spinoza says: “Sextus Empiricus and other
sceptics whom you cite say that it is not true that the
whole is greater than its
part, and they have the same view of the other axioms”, letter 56 to
Boxel, Spinoza, Opera, edited by Gebhardt,
Heidelberg 1925, IV, 260 (translation by A. Wolf).
However it is here relevant to
read Sextus’ texts as dialectical moves that do not pretend to end
up in a
negation but rather to suggest that the problem is undecidable; cf. M IX, 262
and
309 ss. The problem of the relationship between whole and part is important
in Sextus,
because the relation “greater than”, as all other relations, has a
special standing in Pyrrhonism.
Spinoza reads Sextus apparently conditioned by
the controversies of his time.
ly makes Scepticism responsible for the consequences flowing therefrom
as
regards the possibility of human knowledge. This is why the violation of
the
axiom which states that the whole is greater than its part allows Leibniz
to
relate the mild version of the problem of the angle of contact with
scepticism.
Now, in his Theoria motus abstracti he sustained
certain theoretical positions
which he was shortly afterwards to modify.
I propose to refer to one of these, which is pertinent to our subject. In
his
“Fundamenta praedemonstrabilia” (§ 13), he maintains that the ratio of
the angle of
contact to rectilinear angle is that of the point to the line, a thesis
which he
later was expressly to criticise
propounds various enigmas, which
I shall limit myself to pointing out without
speculating upon his motives. When he
wrote this letter, his doctrinal posi-
tion was already fully developed, which makes
it all the stranger that he
should have made use of a version of the angle of
contact which he had ques-
tioned from his youth on. Let us well understand that in
his letter Leibniz is
using a version of the case of the angle of contact to
illustrate a metaphysical
thesis which postulates the existence of a substance
infinitely more perfect
than all other, finite, substances, upon which it has
supernatural effects. The
of
contact will be spun of postulates such as these: among all accidents there is
one
which is infinitely greater (more perfect) than others; there is an angle
formed by
two straight lines, which is infinitely greater (more perfect) than
another angle
formed by two other lines; just as there are relations between
accidents which
entail the existence of infinite accidents, so there may be a
substance infinitely
greater (more perfect) than all other substances
This seems to me all the more remarkable, if one takes into account that
we are
dealing with two series of facts in symmetrical opposition. On the one
hand, the
mild conception of the angle of contact, from which Leibniz
believes that sceptic
conclusions may be drawn, and which he makes use of in
his letter, in support of a
metaphysical analogy which, in its turn, is somewhat
similar to Anselm’s and
Descartes’ demonstration of the infinitely perfect
being. On the other hand, the
fact that he not only held to be erroneous such
geometrical conception of the angle
of contact, but also derived from his
reflexions thereon his favourite objections to
such proofs of God’s existence as
involved an informal consideration of the
infinite.
The universality of the axiom that the whole is greater than its part has
been
adversely affected by the counter-example of the angle of contact; and
faced with
this problem Leibniz had recourse to two different solutions, the
one purely
logical, and the other based upon geometrical considerations. We
will consider them
in that order.
The Demonstration of Primary Propositions is the first
philosophical text in
which Leibniz clearly propounds the problem of the angle of
contact as a
counter-example to the axiom. As might have been foreseen, he draws
the
conclusion that if the absolute and rigorous universality of these
propositions
be eliminated, the certainty of all propositions discovered by the
human mind
will be called in question. His strategy is to prove the axiom. The data
of
the problem are as follows:
As we can see, (1) and (4) are mutually contradictory; and Leibniz holds
not only
that this conclusion is inadmissible, but also that an effective solution
must be
found, since he believes that all knowledge rests upon these primary
this
subject.
In his Demonstration, Leibniz apparently seeks to solve the
problem in the
following way:
Proposition: The whole cde is greater than
the part de.
Definition: “Greater” is that of which the part is equal to another whole.
Scholium: On the basis of this definition, he proceeds to a
general consideration
of “greater” and “lesser”. Concretely, he propounds two given
lines,
congruent or at least parallel, for example, ab and cde.
a b
c d e
whence it emerges that cde is greater, inasmuch as a part of it,
namely cd,is equal to ab, and with
its other part, de, it stretches beyond the latter.
Demonstration: The whole whose part is equal to another whole, is
greater than
that other whole, by definition of “greater”. A part of the whole cde
(namely de) is equal to the
whole de (i.e., is equal to itself). Therefore, cde
is greater than de, the whole is
greater than its part. Q.E.D.
Although this demonstration is clearly unsatisfactory (not defined notions
are
presupposed in the definition) this is the type of axiom about which Leib-
niz says
that “a Sceptic must necessarily admit it, however radical he may
be”
conclusion from his demonstration:
I believe that Leibniz thought that he had in this way solved the problem.
The
procedure he follows in his Demonstratio, does not take into
account the
counter-example of the angle of contact, but intends to solve the
dilemma (or
axiom or counter-example) inasmuch as it has the purpose of supplying
absolute
proof of the axiom (that is, of restoring it to its place among the
theorems, asIn Euclidis
Prota, GM V, 191-192. Specific expositions on the angle of contact from
the
amended version of the Elements, and also on the so
called by Leibniz “angle of the kiss” inMeditatio nova de
natura anguli contactus et osculi, horumque usu in practica mathesi ad figuras
faciliores succeda-neas difficilioribus substituendas, GM VII, 326 ss., and
331-337; cf. GM VII, 331-337 and Specimengeometriae
luciferae, GM VII, 287.
absurdity. Now proofs per absurdum are highly effective in the defence of
the
truth, but they do not help us to discover where we have been mistaken;
and in this
case it was necessary to explore the problem of the angle of con-
tact, in order to
determine exactly where the error lay, and such was Leibniz’
other approach to the
question.
He argued, inter alia, that Euclid and Clavius had been informal or lax in
the way
they handled the subject of the quantum of an angle. In his technical
writings, he
maintained that an angle of contact has no quantum susceptible of
calculation in
terms of a rectilinear one, which is to say that these two are not
homogeneous
angles, and thence he necessarily infers that an angle of contact
is not
intermediate, in terms of quantity, between a flat angle and a rectilinear
angle.
Let us take a look at Leibniz’ critique of the Euclidian version of the
problem:
When Euclid held that an angle of contact is less than any rectilinear angle,
he
spoke very carelessly, giving it to be understood that “less” refers to the
quantity
contained within the said angle. Hence we may not take it that he
was attributing
any perfect quantity to the angle of contact in relation to the
rectilinear... It
is, therefore, most important to note this distinction between
quantity in a
perfect, or geometrical, evaluation, and quantity in an imperfect,
or popular,
evaluation, which latter is that of which Euclid was thinking when
he held the angle
of contact to be less than any rectilinear one Th. L. Heath,
Mathematics in Aristotle, Oxford 1949, pp.
239-240, and his commentary
to the Elements,
quoted in note 32, cf. II, 39-43.
The deductive procedure followed by Leibniz in his Demonstratio
takes no
account of the counter-example; and the considerations arising from his
exam-
ination of the angle of contact take no account of the axiom. Perhaps
Leibniz
thought that it is the confluence of both series of arguments, which
provides a
fuller solution of the problem and allows of the rebuttal of the Sceptic
chal-
lenge.
I now proceed to a number of historical references to the question of the
angle of
contact. I do not lay any claim to provide a history of the problem
in modern times,
but simple desire to take advantage of the presence of so
many distinguished
historians of science and philosophy, to stress the impor-
tance of the subject.
This was a very ancient problem, the earliest version of which is to be
found in the
Aristotelian text known as Mechanica (851 b 36-40); and
Thomas
Heath observes that it has been disputed since the Middle Ages. In the
six-
teenth century, Clavius and Jacques Pelletier du Mans had an argument
about
the angle of contact, which is here of interest. The former had published
his
commentaries on Euclid, and Pelletier criticised his treatement of the
subject
of the angle of contact, being of the opinion that there is no such thing as
an
angle of contact (which Euclid had shown to be less that any acute,
rectilinear
angle), and maintaining that since the straight line touches the
circumsference
of the circle, there is no angle formed Clavius, Euclidis elementorum libri XV,
Liber III, Theor. 5, Propos. 16, Roma 1607,
pp. 351 ss. Cf. note 33.
many subsequent philosophical
discussions about the scientific status of mathe-
matics and the limits of reason. I
propose to mention a little-known fact:
Leibniz wrote at least twice about a letter
of the Sceptic Francisco Sánchez to
Clavius who perhaps was his teacher between 1571
and 1573 in Rome
my study
Francisco Sanches e Leibniz, “Análise” IV (Lisbon 1986) pp.
37-74.
Sánchez in his letter promises, among other things, to refer to the
problem of
the angle of contact and says that he is going to consult Clavius on this
sub-
ject
tions, which I have here
omitted for brevity’s sake, are obscure and subject to dispute. On the
other
hand, it seems that some affirm the indivisibility of a certain continuous quantity –
for
example, the contingent angle – against what Aristotle believed and, as it is
thought, demon-
strated. If you are willing, we could go into all this some time.
But you know Aristotle was an
excellent mathematician, and knew about the
contingent angle” [5]. Here is another passage,
in which Sánchez speaks his mind
about methodological questions arising from the different
ways in which the
contingent angle-problem has been tackled: “... there are many questions
about
which you are, quite rightly, in doubt; and unless you bring all your intelligence to
bear
upon them, you will be side-tracked even when taking your stand upon
whatever proofs there
may be, as you yourself have so well shown the learned
Pelletier, in another passage, about the
contingent angle” [6]. Finally, in the
penultimate paragraph of his letter, Sánchez returns to
the disputed question of
the contingent angle. Let us look at the text: “Nevertheless, I do not
here wish
to prove anything against Euclid, but rather against Proclus, whom Euclid, if
he
lived, would not defend, and did not defend in his works, in which there is
nothing to be found
similar to that demonstration, or paralogism, of Proclus. And
if one had to prove something
against Euclid, perhaps one might be able to do so,
not in respect of one passage alone, but in
respect of two propositions in the
third book, and of the contingent angle, which, it seems,
cannot be less than any
rectilinear angle, unless it be of minimum magnitude, against Aristotle
and
Euclid himself in his tenth book. This, however, has never led me into approving
Pelle-
tier’s opinion, that there is no such angle and no such magnitude,
although Pelletier is an
immensely learned man. You yourself have replied to
these paralogisms with great erudition,
and given them their right name, showing
us that paralogisms are committed in mathematics
also... [17]. The letter was
found by Joaquin Iriarte in the archiv of the Pontifical Gregorian
University of
Rome and published in the original Latin in “Gregorianum” XXI (1940) pp.
413-
451. I use for my convenience the division into paragraphs made by Iriarte’s
edition, until now
the only existing one. Sánchez examines in this letter another
question that surely had inter-
ested Leibniz: the construction of a triangle
from a straight line. Sánchez is aware of difficulties
in the traditional
approach. Surely Leibniz would have agreed in this point with Sánchez,
although
they propose solutions of different kind. Sánchez assumes a radical empiricism
while
Leibniz offers a demonstration which fills in the gaps of the standard
treatment. An examina-
tion of Leibniz’ contribution to this subject can be read
in Professor Giusti’s essay published in
this volume.
plan of a
book which Leibniz called Ad praefationem elementorum veritatis
aeternae,
which was to be the first part of the Scientia
generalis, the most ambitious of all
Leibniz’ projected works. The Elements dealt with what was generally calledars iudicandi, that is to say, it was devoted to the demonstration
of truths
already discovered, and to the verification of doubtful or disputable
proposi-
tions. This is the part of Leibniz’ work devoted to the elimination of
the
procedure of Cartesian doubt Animadversiones in partem generalem principiorum cartesianorum, ad artic. 1, GP IV, 354-355.
quoted, and indirectly refuted, by
Sánchez.
defects, the Elements
are, for Leibniz, a model of correct thought. It was not,
perhaps, by chance, that
the planned work was to be called, like Euclid’s, Ele-ments.
And Sánchez’ objections to some doubtful propositions, among others
the ninth axiom
of the Euclidian Elements, must be heeded
the ninth axiom of Euclid is not valid – although this
consequence was not intended by Pro-
clus.
In examining Euclid’s Elements Leibniz takes sides with Pelletier
and
remembers that Clavius’ informality concerning the angle of contact
caused
Hobbes’ invective against geometry
Now – as Professor Garin pointed out when this symposium began – we
have to go
deeply into Leibniz’ relation to Galileo; and we have here a matter
which lends
itself to the putting of this advice into practice. It was Galileo,
certainly, who
re-propounded the great themes which Leibniz picks up in hisTheoria motus abstracti: the incommensurables in the “Giornata prima” of
hisDialogo sopra i Due Massimi Sistemi; the concentric
wheels in his Discorsi e Dimo-strazioni Matematiche; and the
angle of contact in the Dialogo
Galileo and Leibniz is particularly regrettable in the second volume
of Saggi su Galileo Galilei, a
cura di C. Maccagni,
Firenze 1972.
following working
hypothesis: Galileo was, for Leibniz, a source of admiration
but also of fear. It is
he who rescued philosophical thought, he is the “restau-
rator philosophiae”
open to Scepticism, by stirring-up, but not solving,
methodological and gno-
seological polemics which endanger truths held to be
immovable.
Descartes for his part, as we have seen, takes his stand upon a refusal to
discuss
matters involving the infinite; but he holds the truth of axioms, clearly
and
distinctly understood, to be self-evident
the possibility of a deceitful God,
he considers “totum maius sua parte” to be
a very evident common idea
of the modern age, book four of Arnauld and Nicole’s Logique ou l’art de penser,which brings the problem of the
axiom of the whole and its part to the centre
of the stage of debate. Arnauld and
Nicole agree with Descartes about the
necessity of discarding all discussion of the
infinite La logique ou
l’art de penser, edited by P. Clair and F. Girbal, Paris 1965, p. 295 where
are
mentioned several Galileo’s questions.
the
whole and its part, but they accord it a status as fundamental as that of thecogitoIbid., p. 318. Arnauld and Nicole criticize the opinion that one can arrive
inductively
to that axiom. Perhaps they think in Gassendi; cf. note 56.
lem.
Nevertheless, when referring to the great Clavius/Pelletier debate, a
gnoseological
principle predominates: “Tout ce qui est contenu dans l’idée
claire et distincte
d’une chose, se peut affirmer avec vérité de cette chose”Ibid., p. 317.
If we have no clear and distinct idea of the problem (since
the infinite is
involved therein), we must not give the angle of contact problem any
higher
status than that of a purely nominal discussionIbid., p. 313.
criterion of clarity and
distinction and at the same time to discard Descartes’
order of reasons, leads us to
elevate the axiom of the whole and its part to the
status of a principle.
Within having gone into any minute detail, I have come across valuable
references to
this subject in the works of men of great authority in the realm
of modern
philosophy: apparently the matter was the subject of constant
quoting,
that the whole is greater than its parts”Syntagma I, 116 B, 457 B-458 A and 543 A. Quoted by T. Gregory, Scetticismo ed empi-rismo. Studio
su Gassendi, Bari 1961, pp. 150-151; cf. R. Walker,
“Gassendi and Skepticism”, inThe Skeptical Tradition,
Berkeley/Los Angeles/London 1983, p. 331.
takes
the proposition “the whole is not greater than the part” as an example of
something
which, if a man were to believe it, he would have to renounce the
faculty of
judgmentTractatus
politicus III, 8; cf. Spinoza, Opera, edited by C. Gebhardt, Heidelberg 1925, III,
287.
The only modern philosopher, who holds the problems of the infinite
involved in the
question of the angle of contact to be an important challenge
of scepticism, is
Hume, who goes further and considers them insuperable; and
he affirms it as such in
the second part of Section XII of his Enquiry Concerningthe Human
Understanding. Hume’s attitude to scepticism is conditioned by a cri-
terion
based upon the clear and distinct idea, so that, fundamentally, he reiter-
ates
Descartes’ reflections upon “the absolutely incomprehensible” nature of a
clear and
distinct idea “which contains circumstances contradictory to itself or
to any other
clear, distinct idea”. The method recommended by Hume for
the solution of the
problem, consists in taking mathematical points to be phy-
sical points, “that is,
parts of extension, which cannot be divided or lessened,
either by the eye
or imagination” Enquiry
Concerning the Human Understanding, Sect. XII, Part II, edited by
Selby-Bigge,
p. 124; cf.
Treatise, I, IV, edited by Selby-Bigge, p.
53; cf. R. H. Popkin, The History of
Scepti-cism... quoted in note 1, p.
98, who quotes L. Marandé, Jugement des
actions humaines, 1624,
p.
71.
For Leibniz, this would have amounted to return to an intuitivist theory
of
knowledge united to an empiricist philosophy of mathematics, a level of
philosophy
which he always thought should be outgrown.
In sum: the subject of Euclid’s axiom of the whole being greater than its
part, and
the counter-examples in which the infinite is of a decisive impor-
tance, has
dominated the methodological thought of modern philosophers to
an extent as yet
unmeasured.
II. I must refer now to another aspect of the problem of the infinite,
which is
closely related to the sceptic objections and with which Leibniz dealt
in an
original way. I am speaking of the regression or progression (I shall use
both
expressions indistinctly) to the infinite in the justification of knowledge.
The
Pyrrhonians are well-known to have alleged that progression to the infi-
answer to this charge was that there are first truths which are the basis
of all
others, both in the intellectual and in the sensory order, and that such
first
truths have two properties: (1) They form the basis of all others of their
kind,
and are not based upon any other; and also, which is especially important,
(2)
without them there can be no knowledge. This second characteristic united
to
the first, constitutes the strong “foundationalist” thesis which Leibniz
defended on
various occasions De principiis where Leibniz
says
on principles “that all other propositions depend on them, i.e., that if
this two principles [of
reason and experience] are not true, then there would be
absolutely no truth nor knowledge”.
That is to say, one should admit to be a
sceptic.
nite” must be avoided
if one is not to renounce all knowledge, inasmuch as
Leibniz
has defined “knowledge” as that which is conceived of itself. Is so
exacting a
definition necessary? All scientists and many philosophers feel cer-
tain of knowing
some things, and at the same time admit to not having such
an understanding of the
absolute principles of human knowledge as will per-
mit them to grasp fundamental
cognitions like the non plus ultra of all justifica-
tion.
Leibniz many times maintained that real knowledge is knowledge flow-
ing from
principles, and went so to far as argue that an infinite regression in
the
justification of any knowledge does not permit us licitly to affirm that we
have
obtained any such knowledge. In accordance with his doctrine of neces-
sary and
contingent truths, all truths are analytical, but in the case of necessary
truths
the process of analysis allows of a reduction to identities in a finite
number of
stages, that is to say the regression is finite. On the other hand, as
far as
contingent truths are concerned, Leibniz points out that if they be also
analytical
(i.e. if their predicate be included in its subject) then they are neces-
sary. He
is unwilling to abandon his thesis that truth lies where predicate is
inherent in
subject, and at the same time refuses to admit any consequences
thereof which run
counter to human freedom. He has told us that it was his
reflexions upon geometry
and infinitesimal analysis, that allowed him to
understand how notions also are
susceptible of infinite analysis. But previous-
ly to evolving this solution, he had
gone into the problem of infinite regres-
sion from another angle, and sought to
prove that there are thoughts which
are conceived per se.
Such are the irresolvable notions, the indefinables, “ex-
istence”, “I”,
“perception”, etc., and also sensible qualities such as “heat”, De libertate,
in Foucher de Careil,Nouvelles Lettres
et Opuscules Inédits, Paris 1857, pp.
179-180. On notions that are conceived per se,
cf. A VI, III, 275. I am examining De organo sive arte magna
cogitandi, C 429-430.
or bears within it the concept of some other thing;
and hence that either there
is an infinite regression, or all concepts can finally
be reduced to the self-
conceived. He desires to prove that there are thoughts
conceived per se and
his reasoning is as follows (I have
placed in square brackets the implicit steps
in the reasoning):
The petitio seems clear to me, inasmuch as in the implicit conclusion (e),
Leibniz
assumes that there are thoughts conceived per se, whilst this is
precisely
what he has set out to prove. It should be noted that (d) is a definition
of
“conceiving something per se” in terms of “conceiving
something in the very
act of conception”, but it must be clear that such equivalence
is not of itself
sufficient to justify his conclusion.
Maybe Leibniz guessed that his reasoning was not conclusive. The
phrase “at the
moment of conception” has been added by him. This intro-
duces an ambiguity: we do
not know whether Leibniz is still referring to thecontent of
the thought, or whether he has gone on to speak of the act
of
thought. Leibniz sometimes tried to overcome the problem of infinite
regres-
sion by means of a process of reasoning of Cartesian origin: whatever be
the
relation between my thought and its object, it is “at least certain” that the
act
of thought is being performed
it is true, at least, that we feel or perceive” (“... in sensibilibus
sunt ipsae perceptiones, saltern
enim nos sentire, aut percipere verum
est...”), Specimen quoted in note 14. Cf. also GP
I,
372-373 and Dialogue quoted in
note 12, fs. 8r-8v. Cf. Descartes, Méditations Métaphysiques, II;
AT IX, 23.
had in mente in stage (d) of his argument. Nay
rather he reasons as follows: we
conceive something in the very act of conceiving it
because we conceive it perse. But there can be no better way
of demonstrating that we cannot pass
directly from the act of conceiving to the
end-product of a concept. And if to
explain the act I have to fall back upon a
conception per se, then I am just
notion is,
then, superfluous, or beggs the questionCouturat (C 429 note) correctly
suggested that this was an incorrect reasoning.H. Heimsoeth
presents it as a valid one, Die Methode der Erkenntnis bei
Descartes und Leibniz, Gies-
sen 1912-1914, p. 276. J. Ortega y Gasset, who sometimes follows Heimsoeth, was not
aware
of this logical mishap; however he thinks there is a gnoseological
incoherence: there is no proof
of concepts perceived per
se; to be rational is to be proved, therefore, according to
Leibniz,
concepts perceived per se are irrational, cf. Ni vitalismo ni racionalismo, in Obras
Completas, Madrid
1947, III, 275.
Encouraged by his own calculus, Leibniz propounded another philosophic
strategy to
resist regression in the field of contingent truths. In the corre-
sponding texts,
he does not take regression to be an abstract and erratic pro-
cess, but a concrete
mathematical reasoning subject to precise rules. The phi-
losophic text which
enshrines Leibniz’ new attitude to infinite regression in
existential propositions,
is his General Investigations into the Analysis of Notions
andTruths. Here Leibniz admits that regression can be infinite in the
resolution of
the notions of subject and predicate, always provided that it be
“possible to
observe a progression in the resolution if it can be reduced to a
rule”, which
he calls the “rule of progress”. There is guarantee enough, although
the coin-
cidence of subject and predicate remain for ever unproven, when according
to
the rule there can never be any contradiction between subject and
predicate.
Moreover, if the difference be lesser than any given, the proposition
will have
been proved to be true (§§ 63-66): this is the proper truth of existential
propo-
sitions (§ 74), as when the infinite series, the asymptotics, the
incommensura-
bles, are reasoned out (§§ 134-136). In these cases the analysis is
not perfect,
but the residual difference is “less than any given”Generales inquisitiones de
analysi notionum et veritatum C 373-374, 376-377 and 388-389. In a
margin
of this text Leibniz wrote: “Hic egregie progressus sum”.
Leibniz understands that which the investigator, or anyone discussing
the
problem with him, may have assigned. This second infinite, then, is
the
noble one, the one which is subject to rules, the infinite in which there
are
continuous approximations, or convergent or infinite series
is
true, regression is not arrested, that is, the concepts become identical in a
finite
number of steps (this is only possible in the case of necessary truths),
but
convergence puts us on terms with a “virtuous” regression.
Thus took definite shape in the 80s, this second strategy of the Leibniz’
gnoseology
for dealing with the question of infinite regression. This idea of
indefinite
approximation is carried over into the field of philosophic analysis
from the
calculus; and when the latter was beginning to be discussed, Leibniz
already
resorted to the notion of “unassignable error” on the part of the think-
er or any
opponent. In mathematical discussions about the calculus, the
opponent appears to
take the place of the absent notion of the infinitesimalE. M. Barth, “Finite Debates about ‘the
Infinite’” in Argumentation. Approaches to The-ory
Formation, edited by E. M. Barth and J. L. Martens, Amsterdam 1982, pp.
260-261.
Leibniz considered that in this way strict rigour had lost nothing;
and this, in
a text addressed chiefly to Bayle, is what he says:
Mathematicians are rigorous enough in their demonstrations, when instead of
taking
infinitely small magnitudes into account, they take such as are small
enough to
prove that the error is lesser than any that can be assigned by an
opponent, that is
to say that there is no assignable errorRéponse aux reflexions contenues dans la seconde Edition du
Dictionnaire Critique de M. Bayle, art.Rorarius, sur le système de l’Harmonie
préétablie (1702) GP IV, 569. Cf. Essais de Théodicée
§ 70; GP
VI, 90.
And in so far as effective discussion of his infinitesimal calculus is con-
cerned,
Leibniz points out that the latter refutes of itself its opponents:
If any opponent wished to contradict our statement, it would follow in accor-
dance
with our calculus that the error would be lesser than any he himself
could
assignJustification du Calcul des infinitésimales par celuyde
l’Algebre ordinaire (1702) GM IV, 105 and Mémoire de Mr. G.
G. Leibniz touchant son sentiment sur lecalcul différentiel, (1701) GM
V, 350.
Leibniz presents his opponent with a concrete case and invites him to
make a
specific claim. As Leibniz plays the dialogue, the burden of proof
passes to his
opponent; but the latter has no infinity of possibilities; and Leib-
niz’ method
permits the conclusion that the objections raised are weaker than
the method they
would impugn – which method is self-justified by its ability
to solve problems and
the demonstration it carries with it.
Leibniz adopted this strategy of argument, replacing infinite concepts
with
incomparable ones and thus dissociating the problem of the infinite in
metaphysical
discussions from the infinite in mathematics
problem
of how to determine the relationship of the ideal and the real, is not
a matter for
mathematicians but for metaphysiciansDe arcanis sublimium vel de summa rerum, A VI, III, 475.
no solution other than a very
general one – that of the hypothesis of pre-
established harmony. Nevertheless, his
proposed solution, if restricted to the
purely mathematical field, undoubtedly
represents an original response to the
to this
proposition of Leibniz? It would seem that if there is to be no discus-
sion in the
court of sovereign reason, if mathematics are to be dissociated from
physical
reality, if it be admitted that infinitesimal quantities are fictional and
no claim
be made to understand by their means the nature of things
Varignon on Fontenelle’s project of writing some metaphysical
elements derived from Leibniz’
calculus. Leibniz clearly disapproves of such a
project for he considers that his infinites and
infinitesimals are rather ideal
things or well-grounded fictions, GM IV, 110.
the sceptic objections
may well be minimised. Leibniz clearly told the sceptic
Foucher, that one must make
use of the artificial infinite to reach the truth,
and that “certain falsehoods help
us to discover the truth”
have such affirmations, if “truth” no
longer means the adequacy of under-
standing with extra-mental reality? However,
this gnoseological alternative
tempted no one. Pierre Bayle and Hume, for their
part, found the subject
beyond themR.
J. Fogelin’s didactical exposition in Hume’s
Skepticism in the Treatise of HumanNature, London 1985, pp. 25-37; cf.
also A. Flew, Infinite Divisibility in Hume’s
Treatise, “Rivista
Critica di Storia della Filosofia” XXII, fasc. IV
(1967) pp. 457-471.
the pretension of such a calculus to be in any way “scientific”
about infinitesimal calculus, letter to
Varignon, GM IV, 94. Cf. note 42.
precisely in the field of
science that philosophy entrenched itself against the
sceptic pretensions, until
Kant came to lean upon the factum of science and
thus wrought
profound changes in the nature of the weapons of argument
employed against
scepticism.
To sum up: to take into account the infinite is, as has been remarked
several times
in the course of these meetings, one of the characteristics of
modern philosophy. At
the same time, the supposition that the infinite is
beyond the reach of human
understanding has been forcefully sustained either
by philosophers and scientists
prone to scepticism or by sceptics themselves.
The defence of the rights of reason,
but within the realm of the infinite, was
one of Leibniz’ predominant
preoccupations, and is one of his best claims
upon our gratitude.