Russell acknowledges that we cannot prove there are qualities, the universals represented by adjectives and substantives, whereas we can prove the existence of relations. If we believe in whiteness, a universal, we say that white things are so because they share an abstract quality of whiteness. The empiricists Berkeley and Hume denied the existence of "bstract ideas."They suggested that what really happens when we choose to think of whiteness is that we isolate an image of a particular white thing and reason from it, "taking care not to deduce anything concerning it which we cannot see to be equally true of any other white thing." Russell compares this example to reasoning about a triangle in geometry. We reason as Berkeley and Hume prescribe. However, as soon as we wonder how we know that something we have chosen is white or a triangle, then we must choose something else white or triangular and use it as a criterion, saying that one "must have the right sort of resemblance to our chosen" other particular. Thus, resemblance is presupposed in the choosing of one particular and resemblance is a universal. It is a relation that holds for all pairs that are white; the empiricist alternative indirectly appeals to the abstract universal.
The relation of resemblance is a true universal. It transcends expression in any single particular. Having admitted this, it is faulty reasoning to deny universals about qualities, like whiteness and triangularity. Having proved that universals do exist, next Russell sets the discussion towards proving that universals are not mental in nature. He holds that they are independent of thought or apprehension.
He considers the proposition "Edinburgh is north of London." This relation between two places seems clearly independent of our knowing it. Our knowing it does not make it true; rather, we just "apprehend a fact" that existed prior to us. Even if no one existed, the place where Edinburgh is would still be in a relation north of the place where London is. It seems true, then, that nothing mental is assumed in the fact about these two places. Yet, this fact involves the relation called "north of," a universal. Since the proposition above does not involve anything mental, "north of" must be non-mental as well. We can therefore state that the relation is like the "terms it relates," independent of thought. "North of" is independent of thought and yet it cannot be said to exist exactly in the way that London and Edinburgh exist. There is no place or time in which the relation exists. Since "everything that can be apprehended by the sense or by introspection exists at some particular time," Russell concludes that the relation "north of" is something radically different from these other things; it is not physical or mental.
Russell concludes this chapter with a note on terminology. "Thoughts and feelings, minds and physical objects" exist "in time," in our normal sense. However, universals do not; they "subsist or have being. Being is timeless. Russell divides the world of existence from the world of being. The latter, he writes, is delightfully precise and exact, "unchangeable, rigid," to "all those who love perfection more than life." Comparatively, the world of existence is "fleeting, vague," a jumble of physical and mental things, and contains "everything that makes any difference to the value and life of the world."
This idiosyncratic type of being, not apparently physical or mental, has led to previous philosophical confusion that the universal is really mental in nature. Thinking about universals exists in the mind of course, and in that sense it may be mental. This argument depends on an equivocation similar to Berkeley's ambiguous "idea" discussed in chapter four. Think of whiteness. If we call the action of thought "whiteness," we can say that it is mental by this argument. However, the sense of the universal that we think of is the sense that denotes an object, the idea of whiteness. Russell claims that thought is necessarily differentiable from the universal because taking the universal to be identical with thought robs it of its essential universality. Since "one man's act of thought is necessarily a different thing from another man's," the idea of whiteness cannot be identical between them. What is common between their thought is an abstract object called "whiteness."
Russell's philosophy is most conspicuously neo-Platonic with respect to universals. The metaphysical dichotomy of the world is in complete harmony with Platonic metaphysics. The concrete world we know directly is like a collection of imperfect shadows of the world of ideals or universals. Since Russell also holds that we cannot directly access most of the world, but can only be acquainted with sense-data, universals neatly contain the counterpart essences after which we grope blindly. Neo-Platonic realism is essentially indirect realism, the theory that our understanding of reality is mediated by the veil of our senses. We access reality indirectly, gathering a muddy picture from what we can perceive and our reasoning about it.