Mathematics as Logical Tautologies

Logicism tried to show that the foundation of mathematics is in logic. The logicist programme was launched by Frege, and elaborated in technical details by Whitehead and Russell. Logicism confronts the idea that mathematical notions and theorems are grounded in empirical observations and personal experiences; logicism confronts any such form of psychologism. A key point in the logicist programme is Frege’s definition of number in terms of set theoretical notions. According to logicism all mathematical concepts can be defined by logical concepts, and all mathematical theorems can be derived from logical theorems. Consequently, logicism sees mathematics as logic. Since logical theorems can be shown to be tautologies, all mathematical theorems consequently become tautologies. Logicism inspires the idea that the proper language of science is a formal mathematical language, where the meanings of concepts are defined in terms of sets, and the meanings of propositions are defined in terms of their truth values. A similar theory of meaning has found its way into mathematics education dressed up like the Modern Mathematics Movements.