---
tag: TT2ASTQ0WE
kind: definition
name: Logicism
msc: [03A05]
defines: 5PGSQYB5Y1
uses: [EFFRJFW67Y, 086QKPF0A6]
sources:
  - url: https://en.wikipedia.org/wiki/Logicism
---

## Definition

*Logicism* is the position in the [[foundations of mathematics|EFFRJFW67Y]] that consists of two theses:

1. every mathematical concept, or every concept of arithmetic in the weaker form, can be defined from purely logical concepts;
2. every mathematical truth, or every truth of arithmetic in the weaker form, can be derived from purely logical laws by purely logical rules of inference.

## Remarks

Frege's definition of number follows the first thesis: the number of a concept $F$ is the class of all concepts that are in one-to-one correspondence with $F$.
The axiom of infinity of Principia Mathematica does not fit the second thesis easily.
It asserts that infinitely many objects exist, and it is not clear that logic alone can assert this.
The underlying logic of logicism is a strong logic, such as higher-order logic or a theory of types, not only [[predicate logic|086QKPF0A6]].