The Principles of Mathematics

„The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works.”

„logic aims at independence of empirical fact, and the existence of the universe is an empirical fact. It is true that if the world did not exist, logic-books would not exist; but the existence of logic-books is not one of the premisses of logic, nor can it be inferred from any proposition that has a right to be in a logic-book.”

„To define logic, or mathematics, is therefore by no means easy except in relation to some given set of premisses. A logical premiss must have certain characteristics which can be defined: it must have complete generality, in the sense that it mentions no particular thing or quality; and it must be true in virtue of its form. Given a definite set of logical premisses, we can define logic, in relation to them, as whatever they enable us to demonstrate. But (1) it is hard to say what makes a proposition true in virtue of its form; (2) it is difficult to see any way of proving that the system resulting from a given set of premisses is complete, in the sense of embracing everything that we should wish to include among logical propositions. As regards this second point, it has been customary to accept current logic and mathematics as a datum, and seek the fewest premisses from which this datum can be reconstructed. But when doubts arise — as they have arisen — concerning the validity of certain parts of mathematics, this method leaves us in the lurch.”

„the mind, in fact, is as purely receptive in inference as common sense supposes it to be in perception of sensible objects.”

„A proper name, when it occurs in a proposition, is always, at least according to one of the possible ways of analysis (where there are several), the subject that the proposition or some subordinate constituent proposition is about, and not what is said about the subject.”