The approach to the philosophy of mathematics pioneered by Frege and Russell. According to logicism the truths of mathematics are logical truths, deducible by logical laws from basic logical axioms. The programme of showing this started with Frege's brilliant demonstration that elementary truths of counting (e.g. ‘there are four apples here’) can be formalized using only the quantifiers and identity. No irreducible mention of number is demanded. The greatest achievement of logicism was Principia Mathematica (1912) by Russell and Whitehead. The problem for the programme was that the complexity necessary to avoid paradoxes led to a mapping of mathematics onto set theory, with its own structures and axioms, rather than to anything recognizable as ‘purely’ logical.