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Handbook of Mereology Total Simons, P., 1987. Part I is a standard The notion of structure is of central im- reference for extensional mereology. portance to mereology (for the following Parts II and III treat further topics and see also Koslicki 2008, ch. IX). Histori- question some assumptions of extension- cal contributions had this clearly in view; ality. Has an extensive bibliography. for example, as is brought out in Harte 2002, Plato in numerous dialogues grap- ples with the question of how a whole References and further readings which has many parts can nevertheless be a single unified object and ultimately Cartwright, R., 1975, “Scattered Ob- endorses a structure-based response to jects”, as reprinted in his Philosophical this question (see Plato). Wholes, ac- Essays, 1987, Cambridge, MIT Press: cording to Plato’s mature views (as de- 171-86. veloped primarily in the Sophist, Par- Lewis, D., 1991, Parts of Classes, Ox- menides, Philebus and Timaeus), have a ford: Blackwell. dichotomous nature, consisting of both material as well as structural compo- Sanford, D. H., 2003, “Fusion Confu- nents; it is the job of structure to unify sion”, Analysis 63: 1-4. and organize the plurality of material Sanford, D. H., 2011, “Can a sum change parts that are present in a unified whole. its parts?”, Analysis 71: 235-9. A similar conception is taken up and worked out further by Aristotle who Simons, P., 1987, Parts: A Study in On- famously believed that ordinary material tology, Oxford: Clarendon Press. objects, such as houses, are compounds Tarski, A., 1929, “Foundations of the of matter (viz., the bricks, wood, etc.) Geometry of Solids” as reprinted in his and form (viz., the arrangement exhibited Logic, Semantics, Metamathematics, by the material components for the pur- Oxford: Oxford University Press, 1959: pose of providing shelter). 24-29. In contrast, due to the development in the Van Cleve, J., 2008, “The Moon and early 20th century of a theory often re- Sixpence: A Defence of Mereological ferred to as ‘standard mereology’, based Universalism”, in Contemporary De- on the work of Stanislaw Leśniewski and bates in Metaphysics, edited by Sider, T. Alfred North Whitehead (see also Tarski et. al., Oxford: Blackwell: 321-40. 1937, 1956; Leonard and Goodman 1940), the notion of structure has been van Inwagen, P., 1990, Material Beings, largely absent from more recent mereo- Ithaca: Cornell U. P. logical frameworks. (A notable excep- van Inwagen, P., 2006, “Can Mereologi- tion, however, is the Third Logical In- cal Sums Change their Parts?”, The vestigation of Husserl 1900-1.) Because Journal of Philosophy 103: 614-30. the founders of standard mereology were primarily interested in providing a nomi- Wiggins, D., 1980, Sameness and Sub- nalistically acceptable alternative to set stance, Cambridge, Harvard U. P. theory, according to standard mereology David H. Sanford wholes (also known as ‘mereological sums’, ‘fusions’ or ‘aggregates’) are conceived of as completely unstructured entities. On analogy with the axiom of extensionality in set theory, the existence Structure and identity of a mereological sum is determined exclusively on the basis of Koslicki 2008; Simons 1987). To these the existence and identity of its parts; the theorists, it seems quite clear that the arrangement or configuration of these material objects we encounter in ordinary parts is immaterial to the existence and life and scientific practice cannot have identity of the sum they compose. In fact, the conditions of identity and individua- because standard mereology does not tion that are attributed to mereological recognize a distinction analogous to that sums by standard mereology: for, unlike between subset and membership, mereo- mereological sums, not only are these logical sums are, if anything, even more objects quite obviously capable of sur- unstructured than sets, since all the enti- viving changes with respect to their ties recognized by standard mereology parts, while mereological sums (like sets) are of the same ontological type, viz., so- have their parts essentially; but, in con- called ‘individuals’. Finally, as a result trast to the completely unstructured na- of its endorsement of the now controver- ture of mereological sums, the existence sial principle of Unrestricted Composi- and identity of these objects is also evi- tion (according to which any plurality of dently tied to the arrangement or config- objects itself composes a further object, uration of their parts. For example, as is viz., their mereological sum), standard pointed out in Fine 1999, a ham sand- mereology is committed to a plenitude of wich does not in fact come into existence potentially gerry-mandered objects, such until a slice of ham is placed between as David Lewis’ notorious ‘trout-turkey’, two slices of bread; and the ham sand- an object composed of, say, the (still wich does not remain in the existence attached) upper half of a trout and the unless the parts in question continue to (still attached) lower half of a turkey (see exhibit this arrangement. Given the ap- Lewis 1986). parent clash between the conditions of identity and individuation of material Because standard mereology has been objects, as we ordinarily conceive of and perhaps still is the most well-worked them, and those of mereological sums in out and widely accepted conception of the standard sense, there seems to be parthood and composition in recent his- plenty of room, then, for the develop- tory, it was thought that, insofar as ordi- ment of alternative structure-based mer- nary material objects are wholes (i.e., eologies. composite objects made up of parts), they must therefore be conceptualized as One obstacle that has stood in the way of mereological sums in the standard sense. the pursuit of such alternative systems is This seemingly universal consensus that the notion of structure, given its among contemporary metaphysicians, traditional affiliation with Platonic forms however, is now beginning to be called or Aristotelian essences, in the minds of into question by the arrival of some dis- many contemporary metaphysicians and senting voices, who have turned their mereologists inherits much of the philo- attention to the development of alterna- sophical baggage that is associated with tive non-standard mereological frame- its historical precursors. Aristotle already works, and in particular to the re- intro- criticized Platonic forms for being so far duction of the notion of structure into the removed from the sensible particulars analysis of parthood and composition, whose characteristics they were sup- especially as it aims to capture the mere- posed to explain that they became, in his ological characteristics of ordinary mate- view, causally inert and explanatorily rial objects (see for example Fine 1982, useless. Plato’s invocation of the partici- 1994, 1999; Harte 2002; Johnston 2002; pation relation, which was meant to con- nect Platonic forms to sensible particu- or nodes for other objects to occupy; in lars, did not improve the situation, in order to be admissible occupants of these Aristotle’s mind since he found this positions, the objects in question must relation to be utterly unexplained and satisfy two different sorts of constraints: mysterious. In reaction to the Platonic (i) constraints concerning the type of model, Aristotle made an effort to con- object which may occupy the position in nect his own explanatory and causal question; and (ii) constraints concerning principles much more intimately to the the configuration or arrangement which matter/form compounds whose behavior must be exhibited by the occupants of the and characteristics they were supposed to positions made available by the structure. make comprehensible. However, in the Secondly, a particularly noteworthy course of doing so, Aristotle’s own con- characteristic of structures or structural ception of form or essence became asso- features across different domains is that ciated with philosophically loaded no- the numerical identity of the particular tions such as his actuality/potentiality objects occupying the positions made distinction and the accompanying Ho- available within a structure inevitably monymy Principle (according to which tends to be immaterial to the question of an ‘axe’ that cannot cut, for example, is whether the structure or structural feature an ‘axe’ in name alone), which in turn in question is implemented; as long as made Aristotelian forms or essences the occupants in question satisfy the two acceptable only to philosophers who constraints just mentioned, they are con- share his general teleological outlook. sidered indistinguishable and hence in- When we look more closely at the vari- terchangeable from the point of view of ous disciplines in which the notion of the structure. Thus, the notion of a struc- structure obviously plays a central and ture or structural feature should be significant role, however, we realize that thought of as going along with a distinc- Aristotle’s notion of structure as form tion between what is considered to be need not be conceived of as the causally variable and what is considered to be and explanatorily inert metaphysical invariable within a given domain or invention ridiculed by Descartes and context; variability, in this connection, others. Rather, in such disciplines as amounts to the interchangeability of mathematics, logic, chemistry, linguistics objects in the domain relative to certain and music, for example, we find that the admissible transformations which leave notion of structure is alive and well, the structural features at issue un- whatever exactly its metaphysical status changed. turns out to be. Although the notion of Finally, in each case, the discipline in structure, as it is applied in each case, is question is interested in particular in tailored to the particular concerns of each capturing, usually by means of a system such discipline, we can nevertheless of laws, axioms, and the like, the charac- recognize certain general characteristics teristics and behavior of those features that go along with any such domain- that are taken as invariable, i.e., the specific conception of structure.
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