N. Goodman and W. V. Quine, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic, 12 (1947).
5. Elements of Nominalistic Syntax
It may naturally be asked how, if we regard the sentences of mathematics merely as strings of marks without meaning, we can account for the fact that mathematicians can proceed with such remarkable agreement as to methods and results. Our answer is that such intelligibility as mathematics possesses derives from the syntactical or metamathematical rules governing those marks. Accordingly we shall try to develop a syntax language that will treat mathematical expressions as concrete objects -- as actual strings of physical marks.11 Since one mark is as concrete as another, we can deal with such marks and strings as "epsilon" and "(v)(v epsilon v | v epsilon v)" quite as well as with ones like "(" or "Eiffel Tower". But our syntax language must itself be purely nominalistic; it must make no use of terms or devices that involve commitment to abstract entities. It might seem that this program could be carried out without any difficulty once we have specified that we are dealing with concrete marks; but actually classical syntax has depended so heavily upon platonistic devices in constructing its definitions that the nominalist is faced with the necessity of finding new means of definition at almost every step. Not only subsidiary terms, but such key terms as "formula", "substitution", and "theorem" have to be defined by quite new routes.12 The platonistic object language that our nominalistic syntax is to treat of must contain notations for truth-functions, quantification, and membership. All we need for these purposes are parentheses, variables, the stroke ''|'' of alternative denial, and the sign "epsilon" of membership. Parentheses will serve both for enclosing alternative denials to indicate groupings and for enclosing variables to form universal quantifiers. To simplify our syntactical treatment, let us require that each alternative denial be enclosed in parentheses even when it stands apart from any broader context. As variables we may use "v", "v' ", "v'' ", etc., so that the simple typographical shapes of the object language reduce to six: "v", " ' ", "(", ")", "|", and "epsilon". As already mentioned, the characters of our language are not these abstract shapes -- which we, as nominalists, cannot countenance -- but rather concrete marks or inscriptions. We can, however, apply shape-predicates to such individuals; thus "Vee x" will mean that the object x is a vee (i.e., a "v"-shaped inscription), and "Ac x" will mean that x is an accent (i.e., a " ' "-shaped inscription), and "LPar x" will mean that x is a left parenthesis, and "RPar x" will mean that x is a right parenthesis, and "Str x" will mean that x is a stroke (a "|"-shaped inscription), and "Ep x" will mean that x is an epsilon. But it happens actually that left parentheses and right parentheses are alike in shape, and distinguishable only by their orientation in broader contexts. It would appear therefore that instead of writing "LPar z", to mean that x is intrinsically a left parenthesis, we should write "LPar xy", meaning that x is a left parenthesis from the point of view of its orientation within the longer inscription y; and correspondingly for "RPar". Since however this exceptional treatment is made necessary solely by a typographical idiosyncrasy, we may disregard it. The reader may, if he likes, restore an intrinsic distinction between left and right parentheses by thinking of each left parenthesis as comprising within itself the straight uninked line joining its tips. Our nominalistic syntax must contain, besides the six shape-predicates, some means of expressing the concatenation of expressions. We shall write "Cxyz" to mean that x and y and z are composed of whole characters of the language, in normal orientation to one another, and contain neither split-off fragments of characters nor anything extraneous, and that the inscription x consists of y followed by z. The characters comprising y and z may be irregularly spaced; furthermore the inscription x will be considered to consist of y followed by z no matter what the spatial interval between y and z, provided that x contains no character that occurs in that interval. The two remaining primitives of our syntax language are abbreviations of the familiar predicates "is part of" and "is bigger than". "Part xy" means that x, whether or not it is identical with y, is contained entirely within y. "Bgr xy" means that x is spatially bigger than y. Our syntax language, then, contains the nine predicates "Vee", "Ac", "LPar", "RPar", "Str", "Ep", "C", "Part", and "Bgr", together with variables, quantifiers and the usual truth-functional notations "V", "&", etc. The variables take as values any concrete objects. Notes 11 We might, equally consistently with nominalism, construe marks phenomenally, as events in the visual (or in the auditory or tactual) field. Moreover, although we shall regard an appropriate object during its entire existence as a single mark, we could equally well -- and even advantageously if we want to increase the supply of marks -- construe a mark as comprising the object in question during only a single moment of time. 12 The idea of dealing with the language of classical mathematics in terms of a nuclear syntax language that would meet nominalistic demands was suggested in 1940 by Tarski. In the course of that year the project was discussed among Tarski, Carnap, and the present writers, but solutions were not found at that time for the technical problems involved. Contents -- Go to §6