Sense-Data and the Physical World
Russell's chief innovation with The Problems of Philosophy was the concept of sense-data. Sense-data are the impressions that the appearance of reality offers our senses. We have a sensation of a piece of sense-data. Sense-data is an important concept distinguished from the physical world full of physical objects because it is the only part of the world with which we have direct acquaintance. The physical world is an external conception, in Russell's philosophy, which reaches us only through an interface with our senses.
Ideas and Universals
Russell's neo-Platonism is palpable in his substantial defense of the reality of universals. His theory of universals is remarkably similar to the Platonic "theory of ideas" or "forms." In The Republic, justice is the ideal form, which Plato discerns as essential to all just acts. Russell's universal example "whiteness" is analogous as an ideal in which particulars, white things, participate for their being.
A Theory of Knowledge
The primary concern of The Problems of Philosophy is the establishment of a practicable theory of knowledge. Russell's famous innovations for his theory are knowledge by acquaintance and knowledge by description. We have knowledge by acquaintance when we are directly aware of our sense-data. Knowledge by description is based on our acquaintance and some knowledge of truths. Our acquaintance is the most direct access we have to things in reality. Description allows us, from a distance, to infer knowledge about the actual world. It is thus that direct and indirect realism is a dichotomy conspicuous in Russell's thought, though not made explicit.
Russell's style of enquiry in this work appropriates Cartesian radical doubt as a tool to do analytic philosophy and conceive of new possibilities. Descartes refused to accept anything as true unless it struck him as clearly and distinctly true, in itself. Russell revives this method in his first chapter's account of the dubious nature of our so-called "knowledge." Throughout the work, he prompts us to doubt and reconsider our everyday conception of reality, knowledge, and truth.
The Problems of Other Philosophers
Russell's work is revisionary in its capacity as introduction. When offering his account of other philosophical arguments, he first gives an overview, then points out the strengths or weaknesses of the reasoning. In fact, much of Russell's career was built on applying his new methods of logic to old philosophical problems. He resolves tensions between empiricists and rationalists, exploring arguments made by Hume and Kant. He critiques the popular program of idealists like Berkeley and Leibniz. The beginning of modern philosophy finds expression through his synopsis of Descartes and late contemporary thought through Hegel.