N. Goodman and W. V. Quine, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic, 12 (1947).
1. Renunciation of Abstract Entities
We do not believe in abstract entities. No one supposes that abstract entities -- classes, relations, properties, etc. -- exist in space-time; but we mean more than this. We renounce them altogether. We shall not forego all use of predicates and other words that are often taken to name abstract objects. We may still write "x is a dog", or "x is between y and z"; for here "is a dog" and "is between . . . and" can be construed as syncategorematic: significant in context but naming nothing. But we cannot use variables that call for abstract objects as values.1 In "x is a dog", only concrete objects are appropriate values of the variable. In contrast, the variable in "x is a zoological species" calls for abstract objects as values (unless of course, we can somehow identify the various zoological species with certain concrete objects). Any system that countenances abstract entities we deem unsatisfactory as a final philosophy. Renunciation of abstract objects may leave us with a world composed of physical objects or events, or of units of sense experience, depending upon decisions that need not be made here. Moreover, even when a brand of empiricism is maintained which acknowledges repeatable sensory qualities as well as sensory events,2 the philosophy of mathematics still faces essentially the same problem that it does when all universals are repudiated. Mere sensory qualities afford no adequate basis for the unlimited universe of numbers, functions, and other classes claimed as values of the variables of classical mathematics. Why do we refuse to admit the abstract objects that mathematics needs? Fundamentally this refusal is based on a philosophical intuition that cannot be justified by appeal to anything more ultimate. It is fortified, however, by certain a posteriori considerations. What seems to be the most natural principle for abstracting classes or properties leads to paradoxes. Escape from the paradoxes can apparently be effected only by recourse to alternative rules whose artificiality and arbitrariness arouse suspicion that we are lost in a world of make-believe.3 Notes 1. That it is in the values of the variables, and not in the supposed designata of constant terms, that the ontology of a theory is to be sought, has been urged by W. V. Quine in "Notes on Existence and Necessity", The Journal of Philosophy, Vol. 40 (1943), pp. 113-127; also in "Designation and Existence", ibid., Vol. 36 (1939), pp. 701-709. 2 As for example in SQ [N. Goodman, A Study of Qualities (Ph. D. dissertation, Harvard University, 1941)]. Qualitative ('abstract') particles of experience and spatio-temporally bounded ('concrete') particles are there regarded as equally acceptable basic elements for a system. Devices described in the present paper will probably make it possible so to revise that study that no construction will depend upon the existence of classes. [This has since been carried out in SA [N. Goodman, The Structure of Appearance (1st ed., 1951; 2nd ed., Bobbs-Merrill, 1966] 3 The simple principle of class abstraction which leads to Russell's paradox and others, is this: Given any formula containing the variable "x", there is a class whose members are all and only the objects x for which that formula holds. See W. V. Quine, Mathematical Logic, pp. 128-130. For a brief survey of systems designed to exclude the paradoxes, see pp. 163-166, op. cit.; also "Element and Number", Journal of Symbolic Logic, Vol. 6 (1941), pp. 135-149. Contents -- Go to §2