Title: Constructive Logicism and the Rationals
The Constructive Logicist’s account of the natural numbers satisfies an adequacy condition employing Schema N. This is to the effect that the number of Fs is n if and only if there are exactly n Fs. The right-hand side carries no ontological commitment at all to natural numbers as entities.
This study smoothly generalizes this kind of account to the rational numbers. It examines the conceptual and procedural mastery required for a grasp of the rational numbers as abstract, dimensionless entities. It furnishes introduction and elimination rules for the important notions involved, and shows that they satisfy a new and appropriate adequacy condition: that one should be able to derive every instance of Schema Q. This is to the effect that that fraction of a whole F that t constitutes is p:q if and only if a certain condition obtains concerning t and the natural numbers p and q—a condition whose expression carries no ontological commitment at all to fractions as entities.
Our treatment avoids all the problems that beset both type-raising and double-barreled abstraction principles. It reveals that the conceptual resources needed for a grasp of fractions are very modest indeed. One needs no grasp of addition or multiplication in order to attain to a grasp of, and to incur commitment to, rational numbers. Our treatment reveals the naturals, qua rationals, to be the original naturals, and not just some ‘counterparts’ of them in an altogether different and newly created structure. (The rational number n is the natural number n.) Our treatment also explains, via Schema Q, how it is that rational numbers are applicable in our thought and talk about the wider world. Free Core Logic regiments all the reasoning involved.