Quantity and Number
Kant argues that no metaphysical principle can be verified save by showing what must hold of whatever exists in order for it to be possible to secure knowledge of it. In Chapter 4 we have noticed how Kant brings out that any possible objects of knowledge must be quantitative. We cannot secure any knowledge without thinking. With what we think we do not obtain any knowledge of objects unless we refer it to some objects or other. But we cannot refer to any objects what is in our thoughts without somehow representing to ourselves the objects to which we refer it and thinking it to pertain to them. We also cannot refer to them what is in our thoughts without thinking whether it pertains to all of them, to some of them, or to just one object. Kant not only points out that we cannot secure any knowledge of objects without uniting at least one concept with another and regarding what is thought by one to pertain to what is represented by the other; he also points out that no judgments can be formed without quantifying concepts. Quantifying concepts are not merely connective concepts, required for uniting any concepts into judgments.2 Kant furthermore points out that quantifying concepts are objective concepts, holding of objects of judgments generally; for we cannot form a judgment without thinking how much of its subject the predicate pertains to. 3 However, these considerations no more enable it to be concluded that whatever exists must be quantitative than the consideration that we cannot form a judgment without thinking what pertains to what shows that whatever exists must be a thing or a property of a thing. Take, for example, the judgment that the points at which the sides of a square intersect are at the ends of the sides. We cannot make this judgment
withoutthinkingwhetherthepredicateholdsofallofthepointsorof someofthem.Wecanmakeitwithoutthinkingthateachpointhasa magnitude.