Mereology

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MEREOLOGY

"Mereology" (from Greek meros, "part") is the theory (often formalized) of part, whole, and cognate concepts. The notion of part is almost ubiquitous in domain of application, and for this reason Edmund Husserl assigned its investigation to formal ontology. Aristotle observed that the term part was used in various ways, as for a subquantity, a physical part (leg of an animal), a part in definition (animal is part of man), a part in extension (man is part of animal). Part concepts had obvious applications in geometry and were among Euclid's undefined terms. Several senses of "part" are expressible using the preposition "in," but not all uses of "in" express parthood.

Until the twentieth century it was generally assumed that the concept of part was sufficiently clear not to require elucidation, but gradually the need for a formal treatment became apparent. Euclid's maxim that the whole is greater than the part appeared to be contradicted by infinite classes, for example. In 1901 Husserl proposed a general theory of part and whole and distinguished several kinds of parts, notably dependent and independent parts. Explicit formal theories of part and whole were developed around 1914 to 1916 by Alfred North Whitehead and Stanisław Leśniewski, who worked independently of each other. They had different motivations: Whitehead wanted an empirical basis for geometry, whereas Leśniewski wished to offer a paradox-free class theory. Mereology was later formulated within first-order predicate logic by H. S. Leonard and Nelson Goodman, who called it "the calculus of individuals." Mereology has often been employed by nominalists as a partial substitute for set theory, but it is not intrinsically a nominalistic theory: Part relations are definable via endomorphisms in many mathematical domains.

The most natural basic concept of mereology is that of a (proper) part to its (larger) whole. A coincident of an object is the object itself or something that shares all parts with it. An ingredient of an object is a part or coincident of it. Two objects overlap if and only if they share an ingredient, and they are disjoint if and only if they do not. The relation of part to whole has some minimal formal properties: It is (1) existence entailing; (2) asymmetrical; (3) transitive; and (4) supplementative. That means (1) that if one thing is part of another, if either the part or the whole exists, so does the other; (2) that if one thing is part of another, the second is not part of the first; (3) that a part of a part of a whole is itself a part of the whole; and (4) that if an object has a part, it has another part disjoint from the first. Principles (3) and (4) have occasionally been doubted, (4) unconvincingly. Some meanings of "part" are not transitive; for example, a hand is said to be part of the body, but an arbitrary chunk of flesh is not, and for such concepts counterexamples to (3) may sound plausible, but only because they restrict the general (and transitive) concept, to mean, for example, organ, functional part, immediate part, assembly component.

Beyond such minimal properties mereologists often make further assumptions. Very often it is assumed that objects with the same ingredients are identical: Such a mereology is extensional. Extensionality makes good sense for homogeneous domains such as regions of space or masses of matter, but some objects of distinct sorts seem to be able to coincide, at least temporarily, without identity. Another assumption often made is that any two objects make up a third, indeed that any nonempty collection of objects constitutes a single object, their mereological sum. The minimal properties together with extensionality and this general-sum principle constitute the classical mereology of Leśniewski and Leonard/Goodman: It is as rich in parts as an extensional theory can be, differing algebraically from Boolean algebra only in lacking a null element. It does, however, have an ontologically maximal object or universe, the sum of all there is, which by extensionality is unique. Whitehead denied that there was a universe: For him every object is part of something greater, so he rejected the sum principle. Whitehead also denied there are atoms, that is, objects without parts: For him, every object has a part. This antiatomism, together with supplementarity, ensures that every object has nondenumerably many parts. Whitehead thus denies geometrical points, and his method of extensive abstraction is directed to logically constructing substitutes for points out of classes of extended objects, an idea also carried through by Alfred Tarski. As the examples indicate, the issue whether atomism or antiatomism holds is independent of general mereology. Formally, the best worked-out forms of mereology are those of Leśniewski and his followers; they have shown that any of a wide range of mereological concepts may be taken as sole primitive of the classical theory.

Beyond extensional mereology attention has focused on the combination of mereological notions with those of space, time, and modality. Thus, Whitehead and a number of more recent authors combine mereological with topological concepts to define such notions as two regions' being connected, or their abutting (externally or internally), using mereology as its modern authors intended, as an alternative framework to set theory. When time is considered, matters become more complex. Some objects have temporal parts, including phases, and perhaps momentary temporal sections. States, processes, and events (occurrents) are uncontroversial cases of objects that are temporally extended, but many modern metaphysicians apply the same analysis to ordinary things such as bodies and organisms, giving them a fourth, temporal dimension, though this view is not uncontested. Whether or not continuants (spatially extended objects with a history but not themselves temporally extended) are thus reduced to occurrents, a number of chronomereological concepts may be defined and applied, such as temporary part, initial part, final part, permanent part, temporary overlapping, growth, diminution, and others, though their formulation will vary as applying to occurrents or continuants.

Embedding mereological notions within a modal framework likewise opens up a wider range of concepts such as essential part, accidental part, dependent part, accidental overlapping. Combining these in their turn with temporal notions allows the definition of concepts such as accidental permanent part, essential initial part, and so on. In general, where mereological notions are enriched with others, their interactions become multifarious and lose the algebraic elegance of the classical theory while gaining in applicability and usefulness.

In modal mereology much attention has been paid to R. M. Chisholm's thesis of mereological essentialism, which states that every part of a continuant is both essential and permanent to that continuant (though, conversely, a part may outlast the whole and need not have it as whole). Chisholm's position is presaged in Gottfried Wilhelm Leibniz and Franz Brentano. Since it appears to be contradicted by everyday experience of such things as rivers, mountains, organisms, and artifacts, it is natural for Chisholm to regard such mereologically fluctuating things as not "real" continuants but as entia successiva, supervenient upon successions of continuants for which mereological essentialism holds.

The ubiquity and importance of mereological concepts ensure them a growing place within cognitive science and formal representations of commonsense knowledge, and there is no doubt that mereology is firmly established as a part of formal ontology.

Bibliography

Chisholm, R. M. Person and Object: A Metaphysical Study. London: Allen and Unwin, 1976.

Husserl, E. "On the Theory of Wholes and Parts." In Logical Investigations, 2 vols. London: Routledge, 1970.

Leonard, H. S., and N. Goodman. "The Calculus of Individuals and Its Uses." Journal of Symbolic Logic 5 (1940): 45–55.

Leśniewski, S. "On the Foundations of Mathematics." In Collected Works. Dordrecht: Kluwer Academic, 1992.

Simons, P. M. Parts: A Study in Ontology. Oxford: Clarendon Press, 1987.

Whitehead, A. N. "Principles of the Method of Extensive Abstraction." In An Enquiry concerning the Principles of Natural Knowledge. Cambridge, U.K.: Cambridge University Press, 1919.

Peter Simons (1996)

Encyclopedia of Philosophy