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. Russell's multiple relation theory, that he held from 1910 to 1919 or so, argued that the constitituents of propositions, say 'Desdemona loves Cassio', which is false, are unified in a way that does not make it the case that they constitute a fact by themselves. Othello judges that Desdemona loves Cassio'. 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.