Logicist Notions in Philosophy of Mathematics

Abstract

The point of view according to which logic has priority over mathematics has been maintained by some philosophers and mathematicians in two senses: a strong view and a weak view. Both of them are logicist: The first one regards mathematics as reducible to logic. The second one considers that the essence of mathematics can be known by researching the logical consequences of a certain system of postulates or axioms; thus, the underlying logic (whatever the mathematical theory might be) is crucial for this sort of studies.

Some logicist concepts are interesting in philosophy of mathematics. So it is necessary to study the state of being in use of each point of view. In fact, the weak view has prevailed. To show that, we settle down how logicist statements have been influenced by Gödel’s theorem, though that goes against formalist philosophy. Afterwards, we present a formal system for second order logic; treatment of nominalization is enclosed, and every Frege’s law is proved to be a theorem. Finally, a short balance is made.