Journal of the History of Philosophy

Leibnizian Expression CHRIS SWOYER "I HAVESAID NOTHING," Leibniz wrote de Voider in 1703, "that does not follow from my doctrine that every body expresses all other things, that every soul expresses its own body, and that through its body each soul also expresses all other things" (G.ii, 253 = PPL, 531)? Perhaps this passage overstates the importance of Leibniz's concept of expression in his philosophy, but it is a very special sort of representation that figures prominently in his accounts of a number of phenomena, including sensory ideas, linguistic representation, and the pre-established harmony of monads. Yet, despite the importance of the notion of expression in Leibniz's work, it has received surprisingly little attention from commentators. My aim in this paper is to provide a precise characterization of Leibniz's concept of expression and to show how this illuminates several aspects of his philosophical work. In w1 I set out the relevant data for this project, including Leibniz's terse characterizations of expression, his examples of it, and, what is often neglected by those commentators who do explore the notion, his views about the role of expression in human reasoning. In w I present an interpretation of expression that explains how it works in Leibniz's most frequent illustration of the notion, the perspectival projection of a geometrical figure onto a plane, and in w I show how a simple and natural generalization of this account accommodates Leibniz's nongeometrical examples as well. In w 1compare my interpretation with those of several recent commentators, and in the final section I show how it explains a number of Leibniz's examples of expression. 1. LEIBNIZ ON EXPRESSION I.z. Leibniz's Characterizations of Expression. Leihniz often talks about expression , but he rarely pauses to explain it in any detail. However, in "What Is an Idea?," a paper of t678 that probably contains his most sustained discussion of the notion, he does say: ' I will use the abbreviations in the left margin of the bibliography in citing Leibniz's works. [651 66 JOURNAL OF THE HISTORY OF PHILOSOPHY 33:1 jANUARY 1995 (l) That is said to express a thing [exprimere aliquam rein dicitur illud] in which there are relations that correspond to the relations of the thing expressed. (G. vii, 263 = PPL, 907) Nine years later, in response to Arnauld's puzzlement about the notion, he offers this: (2) One thing expresses another [une chose exprime une autre] (in my language ), when there is a constant and ordered relation between what can be asserted of the one and what can be asserted of the other. (G.ii, 1 12 = PW, 71) And in the "Metaphysical Consequences of the Principle of Sufficient Reason ," probably written around 1712, he tells us that: (3) it is sufficient for the expression of one thing in another [sufficit enim ad expressionem unius in alio] that there should be a certain constant relational law, by which particulars in the one can be referred to corresponding particulars in the other. (C, 15 = PW, 176-77) These passages, which exhibit at least a superficial similarity, span a period of nearly thirty-five years. Leibniz's examples and use of the concept of expression over this period remain strikingly similar as well, so it appears that his views about the notion remained relatively constant throughout his mature career. 1.2. Leibniz's Examples of Expression. Passages (1)-(3), which contain the most detailed characterizations of expression the Leibniz provides, are too brief to tell us much about the notion, but he does supplement them with a number of examples that provide additional information about it. His favorite illustration is that of a perspectival projection of a figure onto a plane; for example, when a circle is projected onto an ellipse, the latter is an expression of the former (e.g., T, w NE, ~31 ; C, 15 = PW, 177; G.ii, 1~2 = PPL, 339; G.i, 383 = R, 259; G.vii, 263 = PPL, 2o7; GM.v, 14a). Many of his other examples are also drawn from mathematics. In analytic geometry, numbers (GM.vii, 141 ) and equations (G.vii, 263 = PPL, 2o7) express geometrical figures , while in algebra...