N. Goodman and W. V. Quine, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic, 12 (1947). 2. Renunciation of Infinity
We decline to assume that there are infinitely many objects. Not only is our own experience finite, but there is no general agreement among physicists that there are more than finitely many physical objects in all space-time.4 If in fact the concrete world is finite, acceptance of any theory that presupposes infinity would require us to assume that in addition to the concrete objects, finite in number, there are also abstract entities. Classical arithmetic presupposes an infinite realm of numbers. Hence if, in an effort to reconcile arithmetic with our renunciation of abstract entities, we were to undertake to identify numbers arbitrarily with certain things in the concrete world, we should thereby drastically curtail classical arithmetic; for, we cannot assume there are infinitely many such things. Classical syntax, like classical arithmetic, presupposes an infinite realm of objects; for it assumes that the expressions it treats of admit concatenation to form longer expressions without end. But if expressions must, like everything else, be found within the concrete world, then a limitless realm of expressions cannot be assumed. Indeed, expressions construed in the customary way as abstract typographical shapes do not exist at all in the concrete world; the language elements in the concrete world are rather inscriptions or marks, the shaped objects rather than the shapes.5 The stock of available inscriptions can be vastly increased if we include, not only those that have colors or sounds contrasting with the surroundings, but all appropriately shaped spatio-temporal regions even though they be indistinguishable from their surroundings in color, sound, texture, etc. But the number and length of inscriptions will still be limited insofar as the spatio-temporal world itself is limited. Consequently we cannot say that in general, given any two inscriptions, there is an inscription long enough to be the concatenation of the two. Furthermore, there can be at most only as many inscriptions as concrete objects. Hence, if concrete objects are finite in number, there are bound to be some for which there are no names or descriptions whatever. Otherwise every concrete object would have to be the name or description of a unique and distinct concrete object; and we should thus be deprived of all predicates and connectives, to say nothing of synonyms, duplicate inscriptions, and non-inscriptions. Notes 4 According to quantum physics, each physical object consists of a finite number of spatio-temporally scattered quanta of action. For there to be infinitely many physical objects, then, the world would have to have infinite extent along at least one of its spatio-temporal dimensions. Whether it has is a question upon which the current speculation of physicists seems to be divided. 5 A nominalistic syntax language may of course, still contain shape-predicates, enabling us to say that a given prescription is, for example, dot-shaped, dotted-line-shaped, Odyssey-shaped. See 5 and 10 below. Contents -- Go to §3