HYPOSTATIC ABSTRACTION IN EMPIRICAL SCIENCE

T.L. SHORT Kenyon College

"The logics of today mostly confound abstraction with generalization. " (Peirce c. 1901)

Peirce formulated the concept of hypostatic abstraction with reference particularly to mathematics, but with hints about how it might be applied to the mind's understanding of itself and to the analysis of empirical science. I will adapt the concept specifically to this last subject. I believe its careful application has several important implications for the analysis of empirical science, but I will concen• trate on just one: concepts of natural kinds cannot be vague in the philosopher's standard sense of that term, because they are vague in a quite different and incompatible sense, hitherto insufficiently recog• nized. I

I. Hypostatic abstraction in mathematics

Peirce distinguishes hypostatic from precissive abstraction. The latter consists in selecting one feature from a complex of features or from a concrete object, as when we remark of a building that it is large. The former makes this feature into an object of reference, as when we speak of the largeness of the building (4.332, 2.364).2

1. This paper is a revised version of one read at a conference on "The Background of Contemporary Philosophical Logic," at the University of Miami, Coral Gables, Florida in March 1987. I would like to thank the other participants for their helpful suggestions, particularly Jay Zeman, Arthur Burks, and Risto Hilpinen. Subsequently, my colleague, Beth Cohen, saved me from a serious error in Section IV. 2. Numerical references of the form n.m are to the mth paragraph of the nth volume of the Collected Papers of Charles Sanders Peirce, Hartshorne, 52 In this sense hypostatic abstraction is identical with abstraction as Quine and some others use that term: it is the transition from first to second order predicate logic. By the introduction of appropriate operators binding the variables of a general description so as to form a singular term, we can designate classes, attributes, and relations. From uses of 'is the father or we can abstract the class, fathers, the attribute, fatherhood, and so on. As we shall use the term, hypostatic abstraction is an act. Sometimes, as in the case of classes and attributes, it is natural to describe the entities abstracted as abstractions. But, as we will see in Section II, not everything introduced into discourse by abstraction need be abstract. As far as mathematics is concerned, however, all the entities introduced by abstraction are abstract. Peirce remarked that hypostatic abstraction " ... may be called the hprinciple engine of mathematical thought" (2.364, see also 4.234 and 5.534). "It is by abstraction that a matematician conceives the particle as occuppying a point. The mere place is now made a subject of thought" (NEM 4.11). "When the mathematician regards an opera• tion as itself the subject of operations, he is using abstraction ... " (NEM 4.11), and so on. There are two features of hypostatic abstraction in mathematics of which Peirce made particular note. We shall wish to also, because they mark both the difference and the analogy between hypostatic abstraction in mathematics and in empirical science. They are that in mathematics: 1. hypostatic abstraction "is a necessary inference whose conclusion refers to a subject not referred to in the premiss; ... " (4.463), and 2. " ... the new individual spoken of is an ens rationis; that is, its being consists in some other fact" (4.463). "For what is an abstraction but an object whose being consists in facts about other things?" (NEM 4.11). Consider (2) first. The existence of a height consists in the fact that an object may be measured. The existence of a class, XPX, consists in

Weiss, and Burks, editors, Harvard University Press, 1931-1958. Similar references preceded by 'NEM' are to page m of volume n of The New Elements of Mathematics of Charles S. Peirce, Carolyn Eisele, editor, The Hague, 1976.