In this early chapter, Russell addresses one major issue—matter. He sets out to decide whether we can be sure that matter exists or if we must admit that matter is something imagined, as real as a dream might be said to be real. The criterion for our certainty is the independent existence of physical objects, for we have identified matter with physical objects in the preceding chapter. The object now is to establish what many philosophers suspect, that the table exists independent of our perception of it, that if we turn away from it the table is still there. Initially, Russell reminds us that while we are doubting the physical existence of an object, "we are not doubting the sense-data, which made us think there was a table," the immediate experiences of sensation.

If the table is real, then our confidence in our senses has been well-placed, and we might be said to have reasonably inferred reality from its appearance. If we find, with Russell, that the table is not real, then the "whole outer world is a dream." It is vital to grasp the difference between these two hypotheses. One affirms our common-sense view of reality, and the other holds that "we alone exist" and nothing we experience is real in our ordinary sense. Russell will contend that it cannot be proved that we are not dreaming "alone in a desert," but also argues that there is no reason for supposing that this is the case.

Here, Russell refers to Descartes' Meditations. Descartes believed in nothing that was not clearly and distinctly true. He imagined the possibility of a disordered, deceptive reality. Descartes considered the deceitful demon possible because he could not prove that it wasn't the case. However, Descartes found that it could not be the case that he himself did not exist; it was impossible because if he did not exist, then he could not be deceived by a demon. Since he doubted, he necessarily existed. Russell highlights Descartes' service to philosophy as that of illustrating that "subjective things are the most certain."

A formal statement of the problem goes: "Granted that we are certain of our own sense-data, have we any reason for regarding them as signs of the existence of something else, which we can call the physical object." The first reason that Russell examines involves the idea of public experience versus private experience. If a group of people is sitting together at a dinner party around a table, then it is reasonable to assume that they see the same forks and knives, the same tablecloth, the same glasses. Since the sense-data is private to each person, "what is immediately present to the sight of one is not immediately present to the sight of another," and it is reasonable to infer that "they all see things from slightly different points of view, and therefore see them slightly differently." Common experience suggests that we believe in such "public neutral objects," as Russell calls them. And if these objects are to exist, objects which more than one person can know, then it seems as though there must be something that transcends the private experience of sense-data. Further examining this reason for believing in the independent existence of physical objects, Russell next inquires as to why we should believe in public neutral objects.

It is true that though people's experiences may vary a little, they can be remarkably similar. It is also true that the variations between their descriptions may vary according to scientific principles having to do with perspective and reflection. However, at this moment Russell retreats from the height of this inquiry to point out that to the extent that we have admitted the experiences of other people, we have made a mistake. Supposing that other people exist begs the question at stake, since the existence of other people is predicated on the assumption that physical objects exist independently. In this stage of the argument, other people are only represented by sense-data. Russell's conclusion from this example is that we must make no appeal to sense- data outside our own private experience.

Here Russell acknowledges that, strictly speaking, we could never truly know that the whole outer world is not a dream. It is always a logical possibility that we are deceived about the true nature of reality and that it is hidden from us. It is possible because "no logical absurdity results from the hypothesis that the world consists of myself and my thoughts and feelings and sensations." However, Russell's argument is that though there may be no way to disprove this "uncomfortable" possibility, there is no reason in support of it either. What is simpler and more plausible is the hypothesis that independent physical objects exist "whose action on us causes our sensations." The advantage of this hypothesis is in its simplicity.