Part-whole theory

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Part-whole theory is the name of a loose collection of historical theories, all informal and nearly all unwitting, relating wholes to their parts via inclusion. Part-whole theory has been overtaken by mereology.

Metaphysics, especially ontology, has invoked part-whole concepts ever since Aristotle founded the subject.

Husserl (1970) (German original first published in 1901) was the first to consciously elaborate a part-whole theory (on which see Tieszen 1995). However he employed no symbolism or logic, even though his doctorate was in mathematics and Cantor was his friend and colleague; Husserl wrote only for his fellow philosophers.

19th century mathematicians became dimly aware that they were invoking a part-whole theory of sorts only after Cantor and Peano first articulated set theory. Before then, mathematicians often confused inclusion and membership. Grattan-Guinness (2000) appears to have been the first to draw attention to this unwitting part-whole theory.

Peano was among the first to articulate clearly the distinction between membership in a given set, and being a subset of that set. A subset of a set is usually not also a member of that set. However, the members of a subset are all members of the set. In set theory, a singleton cannot be identified with its member. In part-whole theory and mereology, this identification necessarily holds.

The Cantor-Peano concept of set did not become canonical until about 1910, when the first volume of Principia Mathematica appeared, and right after Ernst Zermelo proposed the first axiomatization of set theory in 1908.

Starting in 1916, and culminating in his 1929 Process and Reality, A. N. Whitehead published several books invoking part-whole concepts of varying degrees of formality; see Whitehead's point-free geometry.

Part-whole theory has been superseded by a collection of fully formal theories called mereology, Stanislaw Lesniewski's term for a formal part-whole theory he began expositing in 1912. Over the course of the 20th century, a number of Polish logicians and mathematicians contributed to this "Polish mereology." Even though Polish mereology is now only of historical interest, the word "mereology" endures as the name of a collection of first order theories relating parts to their respective wholes. These theories, unlike set theory, can be proved sound and complete.

Nearly all work that has appeared since 1970 under the heading of mereology descends from the 1940 calculus of individuals of Henry Leonard and Nelson Goodman. Simons (1987) is a survey of mereology aimed at philosophers.

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[edit] References

Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots. Princeton Univ. Press.
Edmund Husserl, 1970. Logical Investigations, Vol. 2, John Findlay, trans. Yale Univ. Press.
Simons, Peter, 1987. Parts: A Study in Ontology. Oxford University Press.
Tieszen, Richard, 1995. "Mathematics" in David W. Smith & Barry Smith, eds., The Cambridge Companion to Husserl. Cambridge University Press.