The first chapter begins with the pronoucement that our reasonings are based on two great principles: the principle of contradiction and the principle of sufficient reason. Du Châtelet also speaks about first principles, d.i. “certain principles whose truth is known without even reflecting on it, because they are self-evident.”
The principle of contradiction states that one cannot both affirm and deny something in the same sense at the same time (InstPhy, § 4):
Something is referred to as a contradiction, when it affirms and denies the same thing simultaneously. This principle is the first axiom, upon which all truths are founded. (Copyright © 2009 BZ)
In line with Aristotle, Du Châtelet defends the principle of contradiction as a first principle, or axiom. One cannot argue against this principle without using it, and this, in turn, is self-defeating. Thus, the principle of contradiction is the “foundation of all certainty in human knowledge” (“fondement de toute certitude”). Even the Phyrrhonists, well known for their radical skepticism, never denied this principle. Descartes even used it in his philosophy to prove that we exist.
The violation of the principle of contradiction would mean that anything can be proven, e.g., the sum of 2 and 2 would be 4 as well as 6. However, we all know that it is impossible for two and two to make six. Thus, some philosophers have argued that the principle of contradiction defines what is possible and what is impossible (InstPhy, § 5):
It follows from this that the impossible is that which implies contradiction; and the possible does not imply it at all. (Copyright © 2009 BZ)
According to this definition, possibility means the freedom from contradiction. However, there is an alternative definition of the possible and the impossible: Impossible is that which does not give a clear and distinct idea; possible is that which one can conceive, and which corresponds to a clear idea. Du Châtelet criticizes this definition. Clear notions might be in fact erroneous. Why? Du Châtelet refers to the same example which Leibniz gave in order to argue against Decartes’ idea entis perfectissimi. In “Mediationes de cognitione, veritate et ideis”, which appeared in the November issue of the journal Acta eruditorum in 1684, Leibniz argued: For let us suppose some wheel turning with the fastest motion. Everyone can see that any spoke of the wheel extended beyond the edge would move faster than a nail on the rim of the wheel. Therefore the nail’s motion is not the fastest, contrary to the hypothesis. Wolff discusses this example in Philosophia prima sive ontologia (1730, § 51), too. It illustrates that we can think about a most perfect being; we might have an idea of it. But this is not a sufficient reason to argue for the existence of God. One must examine whether the notion of God, as the most perfect being, contains any contradiction.
Whenever it is claimed that something is impossible, we must be able to prove it. The crux is: Our inability to prove a contradiction in a finite number of steps does not yet provide us with absolute certainty. Hidden contradictions remain possible. The principle of contradiction suffices for necessary truths («vérités nécessaire»), i.e., truths which can be only determined by a single way («ne sont déterminables que d’une seule manière«). Concerning contingent truths («vérites contingents») one needs the principle of sufficient reason. Contingent truths mean that a thing can exist in various ways («différentes manieres») and none of its determination is more necessary than another. This definition might borrowed from Christian Wolff’s German Metaphysics: Das Nothwendige lässet sich nur auf einerley Art determiniren und kann daher nicht anders seyn. Das Zufällige lässet sich auf vielerley Art determiniren (Wolff 1720, § 175).
To give an example: A triangle can exist in only one way, i.e., it is a figure whose three angles, when added together, are equal to the sum of two right angles. We cannot conceive it in any other way. Conversely, it is a contingent truth that I can stand, sit, lie down etc. All these determinations are equally possible. But, at the moment when I stand up, there must be a sufficient reason for my standing instead of sitting or lying down.
Like Christian Wolff, Du Châtelet uses this definition in order to explain the different functions of the principle of contradiction and the principle of sufficient reason. The principle of contradiction governs the consecutive trains of all necessary truth. Contingent truths, by contrast, are defined in terms of the principle of sufficient reason. The latter paves the way to our freedom of choice. Because, unlike God, who knows the cause and reason for everything, our knowledge is limited.
According to Du Châtelet the principle of sufficient reason is as fundamental and universal as the principle of contradiction. The negation of this principle would result in great absurdities. Firstly, things could exist only for an instant and one could no longer be sure that a thing is the same as it was a moment before. Thus, truths could only exist for an instant. Secondly, if one left a room, to be sure that no one had entered since one had left, everything could be disordered in the room. Thirdly, without the principle of sufficient reason, it would be impossible to account for a basic notion like identity unless there was a sufficient reason why the weight on a scale, for example, with particular properties at a given time, would be identical with the weight immediately before weighing. In a similar sense, Archimedes used this principle in mechanics. Archimedes postulated that equal weights at equal distances are in equilibrium and that equal weights at unequal distances are not in equilibrium, which provides the sufficient condition for equilibrium. Having explained the meaning and role of the principle of sufficient reason, Du Châtelet distinguishes between “possibility” and its fulfillment, or “actuality” (InstPhy, § 9):
So in order that a thing might be, it is not sufficient for it to be possible; this possibility must also be actualized [more literal: ” this possibility must have its fulfillment”], and this is called existence. (Copyright © 2009 BZ)
For Du Châtelet, actuality is a fulfillment, i.e. the realization of possibility, and it happens due to action. In his German Metaphysics, Wolff discussed existence (understood as a complement or “fulfilling” of possibility) in terms of actuality (“Würklichkeit”). This use of the term is in fact merely a translation of the Latin actualitas (in German: “Erfüllung”).
Two further principles follow from the principle of sufficient reason: the principle of the identity of indiscernibles and of continuity. The identity of indiscernibles was a central principle in Leibniz’s philosophy. In the fith letter to Samuel Clarke (see Des Maizeaux 1720), Leibniz used the principle of sufficient reason in order to derive the identity of indiscernibles. Leibniz’s argumentation hinges on the question of the extent to which God is capable of arbitrarily choosing between qualitatively identical alternatives. The indifference of space to the placement of matter would imply the existence of distinct but qualitatively identical particles. Du Châtelet summarizes Leibniz’s argumentation and refers to Leibniz’s often presented story, which took place in the garden of Princess Sophie at Herrenhausen around the year 1695. Conversing with Leibniz and another “ingenious gentleman,” the princess observed that she did not believe there were two identical leaves. Yet the gentleman thought he could find some; after an extensive search, he failed because he was convinced by his eyes that he could always note the difference. Leibniz told the story of the leaves as if it exemplified the futility of opposing his principle not only through argument but also through experience.
For Leibniz, all nature is of one piece, a smooth, seamless whole of individual’s uniqueness in which there are no breaks or discontinuities. Exactly this is what the principle of continuity says. In slogan form: “nature never makes leaps.” According to Du Châtelet, the law of continuity states (InstPhy, § 13):
That nothing happens at one jump in nature, and a being does not pass from one state to another without passing through all the different states that one can conceive between them. (Copyright © 2009 BZ)
c’est lui qui nous enseigne que rien ne se fait par saut dans la Nature, & qu’un Etre ne passe point d’un état à un autre, sans passer par tous les différens états que’on peut concevoir entre eux. (Amsterdam 1742)
Du Châtelet presents several examples from geometry in order to show the problem-solving effectiveness of the principle of continuity. For example, a concave line could not become convex without passing through a series of points in between. And she rightly describes the inflection point as the point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. To give another example: By the principle of continuity, the properties that are preserved in the continuous transformation from ellipse to parabola will also be continuous.
For Du Châtelet, the principle of continuity is an ontological and methodological principle. One can find continuity not only in geometry, but elsewhere in nature. This once inspired Plato to famously claim, as Du Châtelet remarks, that God was an eternal geometrist who employed mathematical principles to construct the world in an orderly way.
If the principle of continuity is true of the natural order, the range of plausible scientific theories is reduced, since the theories that violate continuity can be easily rejected. In the same manner Leibniz used this principle against the Cartesian physics of his day. Assume a case in which B and C are of equal mass and are moving in opposite directions, B slower than C. Then imagine the series of cases in which the velocity of C is continuously diminished until it is equal to B. There will be a point at which an infinitesimal difference creates a substantial result. The disproportion of the difference in cause and effect violates the principle of continuity, and the two laws cannot be simplified. This is enough to show that Descartes’ principles are wrong, and the result can be confirmed by experiment. Leibniz had used this argumentation in Specimen Dynamicum (Acta Eruditorum 1695, 145-157) to which Du Châtelet refers here.
See the entries Continuité, (loi de), Contradiction and Suffisante raison in d’Alembert’s and Diderot’s Encyclopédie. The articles “Continuité, (loi de)” and “Contradiction” are signed by d’Alembert, including the additional remark “Nous devons cet article à M. Formey.” The entry “Continuité, (loi de)” includes the reference “Lisez le chap. j. des instit. de Physiq. de Mad. Duchatelet, depuis le § 13 jusqu’à la fin.” See also “Contradiction”; “Loi de continuité” in: Johann Heinrich Samuel Formey (ed.): Dictionnaire instructif, où l’on trouve les principaux termes des sciences et des arts dont l’explication peut être utile ou agréable aux personnes qui n’ont pas fait des études approfondies. Halle: Gebauer 1767.