Preface
That space is actually constituted by points, though
here abstractly meant as terms of situational relations, is perhaps the highest
result of Leibniz’s geometrical investigation and, at the same time, it
also marks the core of Leibniz’s theory of phenomenal expression. It shows
in fact that a set of non-spatial relations (such as those occurring between
monads) can be isomorphic to (“expressed by”) a set of situational relations
that per se suffice to produce phenomenal extension and thus, ultimately,
faithfully represent the supersensible through the sensible. (p. xii)
Chapter I. Historical Survey
"Having thoroughly inquired, I have found that two things are perfectly similar when they cannot be discerned other than by compresence, for example, two unequal circles of the same matter could not be discerned other than by seeing them together, for in this way we can well see that the one is bigger than the other. You may object: I shall measure the one today, the other tomorrow, and thus I will discern them even without seeing the two of them together. But I say that this still is a way of discerning them not by memory, but by compresence: because the measure of the first one is not stored in your memory, for magnitudes cannot be retained by memory, but in a material measure marked off on a ruler or some other thing. In fact, if all of the things in the world affecting us were diminished by one and the same proportion, it is evident that nobody could make out the change." (Letter to Jean Gallois, September 1677; GM I.180, A III.ii.227-28, A II.i.380) (pp. 58-59)
Chapter II. Geometry
He [Leibniz] says that two figures are congruent not only provided they are only distinguishable through the simultaneous perception of them (which would be the similarity definition), but also provided such a perception requires the presence of a third object (which was not a condition for similarity). For example, two spheres are similar, for you can distinguish them only by seeing them both simultaneously (otherwise, seen one at a time, they prove totally indiscernible). Furthermore, if they are also congruent, you can distinguish them only through their reciprocal situation (one being on the right, the other on the left), which calls in the use of a reference element external to the two spheres such as, for instance, a third object that fixes right and left, or proximity and distance, or an ideal object such as given set of coordinates (Specimen Geometriae luciferae, GM VII.275) (p. 143)
On the notion of homogony and in general on the importance of motion in the foundation of the continuum, Cassirer has forcefully insisted in his Leibniz' System: «Man begreift jedoch, dass es für
Leibniz schwierig sein musste, die reine Auffassung der Kontinuität, die er zunächst am Problem der Veränderung gewonnen
hatte, auch dann festzuhalten, wenn er für die Definition vom
Sein der Ausdehnung ausging. Denn gerade dies erwies sich
als das Originale von Leibniz' Gedanken, dass er die Unmöglichkeit zeigt, die Stetigkeit als Eigenschaft an einem gegebenen
Sein erschöpfend zu bestimmen. Erst aus dem Gesetz des Werdens verstehen wir das Kontinuum. Die Unbestimmtheit, die wir
auch in Kants Definition noch fanden, lässt sich gleichfalls darauf
zurückführen, dass Kant die Kontinuität hier noch als Eigenschaft der Grösse, nicht als Prinzip ihrer Entstehung sucht.» [It is understood, however, that it would be difficult for Leibniz to maintain the pure notion of continuity, which he had first gained from the problem of change, even if he had assumed the definition of being as extension. For precisely this proved to be the originality of Leibniz's thinking: that he showed the impossibility of exhaustively determining continuity as a property of a given being. Only from the law of Becoming can we understand the continuum. The indeterminacy which we have also found in Kant's definition can also be attributed to the fact that Kant seeks continuity here as a property of magnitude rather than a principle of its origin.] (Werke, vol. 1, p. 167). Here we need however to remark that, no matter how much you look into kinematics, never will you extract from it nor from its metaphysical foundation even the least proof that it should be continuous. Nor let us be deceived by the fact that Leibniz has assumed it to be true since his young years for definition purposes («motus est mutatio situs continua»), as he should then demonstrate (and he does not) that a continuous change of situation is also possible, which is untrue if one assumes space to be discontinuous. The answer Cassirer seems to suggest to such a puzzle is that continuity of motion is however dynamically founded, i.e. founded on the continuity of force, which in turn may be founded on the continuity of monadic activity. Yet, it is precisely in this way that one goes out of kinematics and thus geometry, so that all objections about using motion in geometry present themselves again—in fact motion cannot be used in geometry because the continuity of changes of a monad is (perhaps) only a contingent one. Hence, it is not motion in itself, let alone the motion generated by the system of forces, that can guarantee continuity of space, but rather, as already suggested, some transcendental operation of production of space. (pp. 184-85n)
Chapter III. Phenomenology
"That is said to express a thing in which there are relations [habitudines] which correspond to the relations of the thing expressed. But there are various kinds of expression; for example, the model of a machine expresses the machine itself, the projective delineation on a plane expresses a solid, speech expresses thoughts and truths, characters express numbers, and an algebraic equation expresses a circle or some other figure. What is common to all these expressions is that we can pass from a consideration of the relations in the expression to a knowledge of the corresponding properties of the thing expressed. Hence it is clearly not necessary for that which expresses to be similar to the thing expressed, if only a certain analogy is maintained between the relations." (G VII: 263–64) (p. 298)
What most matters here about the relational character of the distinctive properties of monads is that an isomorphism which fully or partly expresses intermonadic relations as relations occurring between representational elements whatsoever is clearly a faithful isomorphism, in the sense that beyond such relations there is nothing else that may further characterize monads. In other words, saying that a perceptual isomorphism expresses the relations between monads is equivalent to saying that it expresses monads themselves, because monads consist in those relations or, at least, are perfectly individuated by them. Leibniz's substance, as it were, ends up as a function. (p. 323)
As however supersensible and phenomenal elements, at least in their distinctive features, are reducible to two sets of relations, an isomorphism for which all relations between monads are preserved would only express the logical identity between the noumenal world and the phenomenal one—hence, it would radically deny any phenomenalism. As we can see, some relations must exist that are not preserved by the situational isomorphism or at least (distributive) by the totality of homomorphisms. (p. 326)
[M]ost divergences between Critical Idealism and Leibnizean Phenomenalism can be reduced to the following. In Leibniz, the relation between phenomena and noumena is regulated by a structural morphism, whereas in Kant by a simply functional relation (not a determination, but only the ground of it, the Bestimmungsgrund). Thus, for example, no real intentionality can be recognized in Berkeley's idealism, because here, owing to the lack of any partial isomorphism, the noumenal world is simply overruled (i.e., identified with the phenomenal one): «But, say you, though the ideas themselves do not exist without
the mind, yet there may be things like them, whereof they are copies
or resemblances, which things exist without the mind in an
unthinking substance. I answer, an idea can be like nothing but an
idea; a colour or figure can be like nothing but another colour or
figure.» (Treatise, §8). It is evident that here Berkeley deals with similarity in a most ingenious way and without having the notion of a structural identity that can be preserved beyond material distinctions of genus between phenomena and noumena. What he misses is the very concept of expression. Relevantly, even though no morphism is involved in Kantian philosophy, some functional relationship however subsists there linking a phenomenon to the thing-in-itself. (p. 327)
[T]he limited expressivity of the perceptual isomorphism—which is required in order to establish a distance between phenomena and noumena—finds its further specifications in certain limits of aesthetic comprehension. Furthermore, as it is the only representational content distinguishing monads from one another, only the limitedness of aesthetic comprehension guarantees the possibility for a plurality of substances to be determined, which substances at this point can be distinguished through the different limits of their aesthetic comprehension. In effect, since monads also express such a difference phenomenally through each one's different situs, the most accurate specification of the limits of their finite aesthetic comprehension is to be found in the situation of the phenomenon that expresses the representing monad. Clearly, a monad superordinated to another one will have a broader aesthetic comprehension. We have arrived thus at Leibniz's well-known simile of a town multiplied perspectively. ... [E]ach sight sees the whole town (however variously deformed according to different points of view)—which means that it is not different isomorphisms we are dealing with here, but just one isomorphism variously specified into different homomorphisms. (p. 334)
[I]n order for the various (real) homomorphisms to be determined as genuinely expressive, i.e., referred to a unique intentional object (the noumenal world as a given set of relations), or also, say, as harmoniously perceptual homomorphisms, the ideal isomorphism needs to be determined as a regulative limit of the set of such homomorphisms. It plays actually no other role, in Leibniz's system, than that of an indispensable ideal unity that makes universal harmony possible. (pp. 339-40)
"In fact, even though a circle of a foot [pedalis], one of half a foot [semipedalis], etc., all are different between them, no definition of a foot can nevertheless be given, but there must be some fixed and permanent sample; this is why usually measures are made of a long-lasting material, and it has even been suggested that the Egypt Pyramids, which have lasted for many centuries and presumably will still last for a very long time, be employed for this purpose. Thus, as long as we assume that neither the globe of the Earth nor the motions of stars are significantly to change, our descendants will be able to recognize the same quantity of the Earth's degrees as we have. And if any forms keep their sizes throughout the world and the centuries, as is the case, according to many, with the cells of a honeycomb, these may also provide a constant measure. And as long as we assume that nothing is going significantly to change in the causes of gravity and the motions of the stars, our descendants will be able to learn about our measures through a pendulum. If on the contrary, as I have already said elsewhere, God changed everything and yet preserved the proportions, every measure would fail us, nor could we ever know exactly how much things have changed, because measure cannot by any means be defined, nor can it be kept by memory: and its real preservation is necessary. From this argument, I think, the difference between size and form, or between quantity and quality, appears most clearly." (Specimen Geometriae luciferae, GM VII.276) (pp. 356-57)
"If we conceive of two points existing together, and wonder why we say that they exist together, we will think that this is so either because they are perceived together or because it is just possible to perceive them together. When we perceive an object as existent, for this reason we perceive it in space, that is, we perceive that an indefinite number of other objects absolutely indiscernible from it can exist. Or, still in other words: we perceive that such an object can move, and thus come to be either in one place or in another one; but, since it cannot exist in a plurality of places at the same time, nor can it move in just one instant, this is why we perceive this place as continuous." (Characteristica geomterica §108) (p. 412)
→ もし運動がなければ空間もない（？）
"There is, moreover, a definite order in the transition of our perceptions when we
pass from one to the other through intervening ones. This order, too, we can call a
path. But since it can vary in infinite ways, we must necessarily conceive of one that is
most simple, in which the order of proceeding through determinate intermediate states
follows from the nature of the thing itself, that is, the intermediate stages are related in
the simplest way to both extremes. If this were not the case, there would be no order
and no reason for distinguishing among coexisting things, since one could pass from
one given thing to another by any path whatever. It is this minimal path from one
thing to another whose magnitude is called distance."(Initia rerum mathematicarum metaphysica, GM VII.25) (p. 423)
→ 二点間の移行のしやすさの度合いを「距離」と呼ぶ。
Chapter IV. Metaphysics
In many places we read that the possible must possess an internal tendency to existence, without which nothing would ever exist (De rerum Originatione radicali G VII.303) (p.440)
