Cornell University Press has done philosophy of mathematics a service by reprinting this 1983 collection of the main papers of Charles Parsons from the 1960s and 1970s in that area. The reprint is unchanged except for a short preface on p.14, noting a correction. (Correction to the correction: Read “S4.2” and “S4.3” for “54.2” and “54.3.”) Again and again one is surprised to see how points that appear in the later literature are already to be found, clearly but unemphatically stated, in these early papers.
Papers on Ontology. “Ontology and Mathematics” is the most sensible piece ever written on its subject. If one limits one's attention to theories formulated using classical logic, concerns oneself solely with questions “ontology” as contrasted with “ideology,” and solely with the distinction between the “abstract” and the “concrete,” there is nothing interesting to be said: Any mathematics involves ontological commitment to abstract entities. Interesting questions arise when one considers either: (i) non-classical logics (modal, intutionistic, substitutional); (ii) distinctions among abstract entities (those that are equivalence types of perceptible or linguistic entities with respect to perceptible or linguistic equivalence relations); or (iii) questions of “ideology” as opposed to “ontology” (inductive definitions, truth). All of (i)–(iii) are touched on, and topics taken up in the later papers given a first exposure in this paper. The final remark at the end of the paper is worth quoting: “The ontology of a theory is, roughly, what the theory says that there is. But this provides only a minimal characterization of what someone commits oneself to who accepts the theory, since he might be able to justify or even understand the propositions of the theory only by relating them to statements which involve additional entities.”
“A Plea for Substitutional Quantification” opposes Quine, for whom the introduction of Lesniewskian operators does not reduce a theory's ontology, but simply makes the notion of ontology inapplicable. Parsons considers a theory in which an existential quantification over classes is explained in the metalanguage as being true if and only if a certain condition is fulfilled whose formulation involves the existence of expression types and a notion of truth. (He notes the seldom-noted complications that arise if the original domain contains nameless objects.) Contra Quine he writes (66): “The fact that the substitutional interpretation yields truth-conditions for quantified sentences means that everything necessary for speaking of these classes as entities is present, and the request for some more absolute verification of their existence seems senseless.” But note that for someone fixated on the crude distinction between the abstract and the concrete there is no gain, since expression types are required in the metalanguage and are just as lacking in causal powers as are classes.
“Informal Axiomatization, Formalization, and the Concept of Truth” considers the simplest argument for the consistency of first-order arithmetic: that its axioms are all true, while any set of truths is consistent. In what sense if any are second-order notions required to explain the notion of truth? The usual explicit definition involves impredicative second-order quantification, but the most natural definition is an inductive one (where delicate questions about the distinction between universally quantified formulas and free-variable schemata arise). The discussion is sophisticated and in places technical. Briefer consideration is given to the situation in axiomatic set theory.
Historical Papers. Next come two papers on Kant, and one each on Frege and Quine. I am not qualified to comment on Kant exegesis, though much material in the Kant papers, such as the claim that the phenomenology of “thoroughly finite” beings like ourselves provides no basis for the thought that space is potentially infinite in extent or divisibility, is of independent interest.
“Frege's Theory of Number” is famous among specialists today for its observation that Frege's development of arithmetic in his logic uses his inconsistent basic law V only to establish what is now called “Hume's Principle.” But there is more in the paper. For instance, the last two sections of the paper and a postscript provide a sensitive discussion of “Poincaré-style” objections that mathematical induction is required either at the outset of the logicist project, in the explanation of the notion of formula, or at the end of it, to connect the natural numbers as intuitively understood with the natural numbers as defined by logicists.
“Quine on the Philosophy of Mathematics” does not contain Parsons’ best-known objection against Quine (that his holist conception of the justification of mathematics fails to account for the obviousness of elementary arithmetic), which came later, but there are many other cogent points. Parsons undertakes the unusual project of trying to relate Quine's views to contemporary views on necessity, a notion anathema to Quine himself.
Papers on Set Theory. “Sets and Classes” concerns the role of classes in set theory, and the rough equivalence of the admission of classes and the admission of a satisfaction predicate, part of Parsons’ motivation for taking up semantic paradoxes in “The Liar Paradox.” That paper antedates the work of Kripke and the Gupta-Herzberger school, which the 1983 postscript hardly provides adequate space to discuss. Though this makes the paper a bit dated, the discussion of “systematic” or “typical” ambiguity is still worth studying. The paper is also unusual for being willing to discuss (Chihara's version of) Tarski's view that all one can look for is a useful repair of a flawed intuitive notion, a view most treat as unmentionable.
“What Is the Iterative Conception of Set?” begins with criticism of the appeal to a picture of sets as constructed in temporal stages that can be found in standard expositions of the iterative conceptions of set, and ends by proposing the substitution of modal (actual vs. possible) for temporal (present vs. future) language. (In between, Parsons revisits the notion of proper class.)
The volume culminates with “Sets and Modality,” in which Parsons puts forward a formal proposal for a modal set theory of the kind towards which his discussion in some of the earlier papers has been pointing. The technicalities cannot be done justice in a short review. The main philosophical problem is perhaps that of the characterizing the notion of “necessity” involved, labeled “mathematical” since both logical and metaphysical seem ruled out.
There remains vastly more in the book that will repay careful reading than there has been space here even to mention.
