Part of the problem is to find a refution of Berkeleyan idealism, of showing how the existence and real nature of matter can be proved. In Problems Russell argues that matter is a well supported hypothesis that explains our experiences. This is the notion of hypothesis which Russell contrasts with construction in the passage above. Russell saw an analogy between the case of simply hypothesizing the existence of numbers with certain properties, those described by axioms, and hypothesizing the existence of matter. Examples include the most general spatiotemporal properties of objects, that no two can occupy the same place at the same time, and so on.
Material objects are now to be seen as collections of sense data.
Russell saw some support for this in the theory of relativity, and the fundumental importance of frames of reference in the new physics. These prominent examples are not the only use of the notion of construction in Russell's thought. Like the construction of numbers, this construction abstracts out what a number of occurrences of a belief have in common, a believer and various objects in a certain order. Propositional functions are abstracted from their values, propositions. Viewing propositional functions as constructions from propositions which are in turn constructions by the multiple relation theory helps to make sense of the theory of types of propositional functions in Principia Mathematica .