SCEPTICISM AND THE INFINITE
Ezequiel de Olaso
SCEPTICISM AND THE INFINITE*
95

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-paradoxes1. This becomes very clear, if we take a look at

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

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times, especially in the works of Galileo, that constituted, in the eyes of the

young Leibniz, the most perilous challenge to reason2.

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 paradoxes3. Sextus has the habit of propounding con-

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 decided4. That is to say, that only in what might be

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.

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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 experience5 or knowledge6 of infinite things. It

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 undefined7. Sextus also maintains that an infinite series

cannot be grasped8, and even goes so far as to affirm that nothing existent is

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 infinite9. The fact that such proposi-

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-

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aissance times and prominent in Leibniz, that is to say, the frequent assimila-

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:

  • Dialogus inter Theologum et Misosophum10.
  • Conversation du Marquis de Pianese, Ministre d’Etat de Savoye, et du Pere Emery,

    Eremite, qui a esté suivie d’un grand changement dans la vie de ce ministre, ou Dialogue de

    l’application qu’on doit avoir à son Salut
    11.
  • Dialogue entre un habile Politique et un Ecclésiastique d’une piété reconnue.12
  • De principiis13.
  • Specimen animadversionum in Sextum Empiricum percurso libro Pyrrhoniarum Hypothe-

    sium sic primo datum
    14.
  • The list may be lengthened by the inclusion of various letters written

    with the primary purpose of rebutting scepticism, amongst which I would

    mention:

  • Coniectura cur Anaxagoras nivem nigram dicere potuisse videatur, petenti lac. Thomasio

    in scheda missa, d. 16 Febr. 1666
    15.
  • Letters to Foucher16.
  • 99

    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

    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 expanded17.

    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 opinions18. Conversely,

    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 Empiricus19. Hence it would be a mistaken method to examine only

    such of Leibniz’ writings as refer explicitly or implicitly to scepticism.

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    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 ways20. Nevertheless, I am proposing a tentative

    classification which may serve as a first guide-line in this matter.

  • Theoretical Scepticism.
  • The belief that man cannot justify some axioms (e.g. the whole is greater

    than its part).
  • The belief that man has no acceptable justification for the principles of

    contingent knowledge (for example, infinite regress in the analysis of

    truths).
  • Practical Scepticism.
  • The belief that human decisions are fundamentally arbitrary, because our

    norms lack objectivity.
  • The belief – opposite to the latter — that human actions are not free.
  • Without departing from generalities, I would observe that in A (a) he

    believes that the principle of contradiction is indirectly at stake21, and in the

    remaining cases, that of sufficient reason. As our understanding of the subject

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    deepens, we become aware of how tremendously Leibniz’ work was affected

    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 sceptics22.

    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 work23.

    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

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    kinds of Leibniz’ strategy in such passages24 and I will not now refer to them.

    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 1670s25;

    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 sceptics26. His strategy in

    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,

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    because he holds that the principle of contradiction, suitably supplemented, is

    quite sufficient27.

    These preoccupations of Leibniz were first made public in 1670 in his

    Theoria 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”28.

    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”29. And he mentions three problems: that of concen-

    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 wheels30, let us take a look of that of the angle of contact (or of con-

    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

    Sophia31. Let us take a look at the following figure:

    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 mathematicians32 have shown that the ordi-

    104

    nary angle ABE is greater than the angle of contact ABNCDF. Let us consid-

    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

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    straight lines BDE and BCH do cut it at C and D respectively, and are partly

    within it and partly without.

    To prove that the common angle is infinitely greater than the angle of

    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 compass33 on point B and the other

    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 axiom34.

    106

    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 directly35. Leibniz indeed, as we have seen, systematical-

    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 criticise36. Hence, his letter to the Princess

    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

    107

    whole web of suggestion, or demonstration, of God’s existence from the angle

    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 substances37.

    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:

  • The axiom “the whole is greater than its part” is true.
  • The problem presented by the angle of contact is a counter-example to that

    axiom.
  • If an axiom lacks universality it is false.
  • Then, the axiom “the whole is greater than its part” is false.
  • 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

    108

    propositions. At the beginning of the next section I shall deal more fully with

    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”38. It is not inadmissible to imagine that Leibniz draws the following

    conclusion from his demonstration:

  • To deny the truth of the axiom “the whole is greater than its part”, is of

    the essence of scepticism.
  • But scepticism must accept absolute demonstrations.
  • The axiom “the whole is greater than its part” is susceptible of absolute

    demonstration.
  • Scepticism must accept the axiom “the whole is greater than its part”.
  • 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, as

    109

    Hobbes has shown)39. Hence the counter-example is to be excluded as an

    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 one40.

    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.

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    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 formed41. This polemic has been a “must” for

    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 Rome42.

    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-

    ject43. The first occasion in which Leibniz refers to Sánchez is in the form of

    111

    an unfinished note penned between 1677 and 1690. This formed part of the

    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 called

    ars 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 doubt44. Euclid is not infallible45. But with all their

    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 heeded46.

    In examining Euclid’s Elements Leibniz takes sides with Pelletier and

    remembers that Clavius’ informality concerning the angle of contact caused

    Hobbes’ invective against geometry47.

    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 his

    Theoria motus abstracti: the incommensurables in the “Giornata prima” of his

    Dialogo sopra i Due Massimi Sistemi; the concentric wheels in his Discorsi e Dimo-

    112

    strazioni Matematiche
    ; and the angle of contact in the Dialogo48. I suggest the

    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”49; but he is also one of the writers who have left the door

    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-evident50; and once we have eliminated

    the possibility of a deceitful God, he considers “totum maius sua parte” to be

    a very evident common idea51. But it is the great methodological text-book

    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 infinite52, and exalt the axiom of

    the whole and its part, but they accord it a status as fundamental as that of the

    cogito53. This could not be done without examining the angle of contact prob-

    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”54.

    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 discussion55. But to abide strictly by the

    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

    113

    debate. Gassendi speaks of “the proposition that everyone is continually

    quoting, that the whole is greater than its parts”56. Spinoza, for his part,

    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 judgment57.

    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 Concerning

    the 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”58.

    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-



    114

    nite could not be avoided without falling into other fallacies59. Leibniz’

    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 occasions60. Consequently, the “regression to the infi-

    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”,

    115

    “light”, and others61. Leibniz points out that a thing is either self-conceived

    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):

  • If nothing is perceived per se, nothing is perceived at all.
  • [But in fact we do conceive thoughts].
  • If we conceive thoughts through other thoughts, we conceive them in so

    far as we conceive such other thoughts.
  • It may definitely be said that we conceive something in the very act of

    conception, when we conceive it per se.
  • [There are thoughts which are conceived per se].
  • 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 the

    content 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 performed62. But this does not appear to be what Leibniz

    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 per

    se
    . 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

    116

    where I was before introducing the notion of the “act of thought”. Such

    notion is, then, superfluous, or beggs the question63.

    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 and

    Truths
    . 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”64. By “given” difference

    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 series65. In this way, it

    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

    117

    proposed to eliminate the difficult notion of the infinitesimal. Leibniz had

    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 infinitesimal66.

    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 error67.

    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 assign68.

    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 mathematics69. The inevitable

    problem of how to determine the relationship of the ideal and the real, is not

    a matter for mathematicians but for metaphysicians70. But Leibniz proffered

    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

    118

    sceptic challenge of infinite regression. How would scepticism have reacted

    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 things71, then

    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”72. What value

    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 them73, and no follower of the classical sceptics would have allowed

    the pretension of such a calculus to be in any way “scientific”74. And it was

    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.

    *.

    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.

    1.
    Cf. 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.
    2.
    Cf. my doctoral dissertation Leibniz and Greek Scepticism (Bryn Mawr College, 1969,

    unpublished).
    3.
    Cf. the assimilation of both traditions in the letter to Gallois, Accessio ad arithmeticam

    infinitorum
    (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.
    4.
    PH III, 65 ss.; cf. M X, 45 ss.
    5.
    M I, 66.
    6.
    M I, 86 and 224.
    7.
    M I, 81. Sextus uses “aoriston” but he is discussing the infinite (apeiron).
    8.
    M VIII, 16. Cf. PH II, 78, 85 and 89; PH III, 24.
    9.
    M VII, 69-70. One of the great specialists in this subject in Greek philosophy did not

    analyse in his greatest work the Sceptics’ attitude. Cf. R. Mondolfo, El Infinito en el pensamiento

    de 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 Assent

    and 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.
    10.
    Dialogus inter theologum et misosophum. LH I, VI, 6; Grua 18. A substantially better edi-

    tion is been prepared, cf. VE, Faszikel 1, Münster 1982, pp. 1-6.
    11.
    Conversation du Marquis de Pianese, Ministre d’Etat de Savoye, et du Pere Emery, Eremite, qui a esté

    suivie d’un grand changement dans la vie de ce ministre, ou Dialogue de l’application qu’on doit avoir à son salut
    ,

    LH I, VI, 5. Partially edited by J. Baruzi, Trois dialogues mystiques inédits de Leibniz, “Revue de

    Métaphysique et de Morale”, XIII (1905) pp. 1-38.
    12.
    Dialogue entre un habile Politique et un Ecclésiastique d’une piété reconnue, LH I, VI, 4. Subopti-

    mal edition by Foucher de Careil, Œuvres de Leibniz, Paris 1859-1875, II, 520 ss. Spanish

    translation from the original manuscript, with commentary, in EF, pp.> 218-251.
    13.
    De principiis, LH IV, VI, 12. Bl. 19. C 183-184. A better edition is been prepared, cf.

    VE, Faszikel 5, Münster 1986, pp. 908-909.
    14.
    Specimen animadversionum in Sextum Empiricum percurso libro Pyrrhoniarum Hypothesium sic primo

    datum
    , LH IV, VIII, 26. I am now preparing an edition of this manuscript with a running

    commentary.
    15.
    Coniectura cur Anaxagoras nivem nigram dicere potuisse videatur, petenti lac. Thomasio in scheda

    missa, d. 16 Febr. 1666
    , A II, I, 4-5.
    16.
    A II, I, 245 ss.
    17.

    Cf. my study Leibniz and Scepticism in the collective volume Scepticism from the Renaissance to

    the 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.

    18.
    About Galileo see further down the first section of this study. Unexpectedly Leibniz

    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.
    19.
    “... Although the sceptics may seem to have said something new...” (“... ut sceptici

    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”).
    20.
    This aspect of Leibniz’ philosophy has been strongly emphasized by J. Ortega y Gasset

    in 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.
    21.
    To admit that the ordinary axioms cannot be proven, i.e., cannot be reduced to identi-

    ties, means, according to Leibniz, saying that “to be and not to be are the same”, De synthesi et

    analysi universali
    , GP VII, 295.
    22.
    C. 191. A definitive edition is being prepared, cf. VE, Faszikel 4, Münster 1985,

    p. 699.
    23.
    M. 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.
    24.
    Cf. La lógica leibniciana de las controversias, proceedings of the symposium “Controvérsias

    Científicas e Filosóficas”, Evora, Portugal, 1985.
    25.
    I take especially into account Theoria motus abstracti (1670) A VI, II, 258 ss.; Demonstratio

    propositionum 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.
    26.
    Notoriously, such is Descartes’ case 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.
    27.
    “... and only from this i.e. from the principle of contradiction adding notions and expe-

    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.
    28.
    A VI, II, 262.
    29.
    A VI, II, 267.
    30.
    Cf. my study quoted in note 17.
    31.
    Letter to Duchess Sophia (October 1691) A I, VII, 48-49, note.
    32.
    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.
    33.

    Professor A. G. Ranea has pointed out to me that points L and M are presupposed in

    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.

    34.
    Cf. J. E. Hofman, Das Opus Geometricum des Gregorius a S. Vincentio und seine Einwirkung auf

    Leibniz
    , 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.
    35.
    In Sextus’ works there are plenty discussions about wholes and parts, cf. PH II, 215 ss.,

    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.
    36.
    A I, VII, 47-50.
    37.
    A VI, II, 480 and 482-483.
    38.
    A III, I, 13.
    39.
    In 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” in

    Meditatio 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 Specimen

    geometriae luciferae
    , GM VII, 287.
    40.
    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.
    41.

    Clavius, Euclidis elementorum libri XV, Liber III, Theor. 5, Propos. 16, Roma 1607,

    pp. 351 ss. Cf. note 33.

    42.
    Leibniz’ passages in C 191 and GM IV, 92-93. A general presentation of this theme in

    my study Francisco Sanches e Leibniz, “Análise” IV (Lisbon 1986) pp. 37-74.
    43.
    Here are some relevant passages of this scarcely known text: “... many other ques-

    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.
    44.
    Animadversiones in partem generalem principiorum cartesianorum, ad artic. 1, GP IV, 354-355.
    45.
    “Nunquam contra Euclidem quidquam demonstran posse” Clavius said in a passage

    quoted, and indirectly refuted, by Sánchez.
    46.
    Proclus is criticized by Sánchez because it follows from one of his demonstrations that

    the ninth axiom of Euclid is not valid – although this consequence was not intended by Pro-

    clus.
    47.
    GM V, 191; cf. VE, Faszikel 5, Münster 1986, p. 1051.
    48.
    On Leibniz reader of Galileo, cf. A VI, III, 163-168. The lack of a parallel between

    Galileo and Leibniz is particularly regrettable in the second volume of Saggi su Galileo Galilei, a

    cura di C. Maccagni, Firenze 1972.
    49.
    A III, I, 12.
    50.
    AT III, 64.
    51.
    AT IV, 111.
    52.
    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.
    53.
    Ibid., p. 318. Arnauld and Nicole criticize the opinion that one can arrive inductively

    to that axiom. Perhaps they think in Gassendi; cf. note 56.
    54.
    Ibid., p. 317.
    55.
    Ibid., p. 313.
    56.
    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”, in

    The Skeptical Tradition, Berkeley/Los Angeles/London 1983, p. 331.
    57.
    Tractatus politicus III, 8; cf. Spinoza, Opera, edited by C. Gebhardt, Heidelberg 1925, III,

    287.
    58.
    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.
    59.
    PH I, 166, 171-172, 176 and 179; M VIII, 347.
    60.
    Cf. the text quoted in note 37. See also, among others, 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.
    61.
    Leibniz tells us about his discovery in C 18 and in 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.
    62.
    “[The first truths] in the domain of sensible things are perceptions themselves, because

    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.
    63.
    Couturat (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.
    64.
    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”.
    65.
    Letter to Wallis (1697) GM IV, 24.
    66.
    E. 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.
    67.
    Réponse 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.
    68.
    Letter to Varignon (1702) GM IV, 92. Cf. Justification du Calcul des infinitésimales par celuy

    de l’Algebre ordinaire
    (1702) GM IV, 105 and Mémoire de Mr. G. G. Leibniz touchant son sentiment sur le

    calcul différentiel
    , (1701) GM V, 350.
    69.
    Letter to Varignon, GM IV, 91 and 94; GP IV, 569.
    70.
    De arcanis sublimium vel de summa rerum, A VI, III, 475.
    71.
    GP VI, 629; letter to Tolomei (1705) GP VII, 468. Remember his confidence to

    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.
    72.
    Letter to Foucher (1692) GP I, 404-406.
    73.
    Cf. R. J. Fogelin’s didactical exposition in Hume’s Skepticism in the Treatise of Human

    Nature
    , 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.
    74.
    Sextus Empiricus and Sánchez are the sceptics evoked by Leibniz while discussing

    about infinitesimal calculus, letter to Varignon, GM IV, 94. Cf. note 42.


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