---
tag: 5PGSQYB5Y1
kind: concept
name: Logicism
msc: [03A05]
sources:
  - url: https://en.wikipedia.org/wiki/Logicism
same_as:
  wikidata: Q845691
  wikipedia: Logicism
---

## Summary

Logicism is the view that mathematics, or at least arithmetic, is a part of logic.
Its claim is that mathematical notions can be defined in purely logical terms and that mathematical truths can be proved from logical laws alone.
Gottlob Frege, and later Bertrand Russell and Alfred North Whitehead in Principia Mathematica, tried to carry out this program.
It is one of the classical answers in the [[foundations of mathematics|EFFRJFW67Y]].

## Explanations

### L1 human

A logicist says that numbers are not new things beyond logic.
For example, the number $2$ can be described as what all two-element collections have in common.
And "two-element" can be said in logic: "there are $x$ and $y$ with $x \ne y$, and nothing else".
Then a fact such as $1 + 1 = 2$ would be a truth of logic.
Frege's system turned out to be inconsistent because of Russell's paradox.
Later systems avoided the paradox, but they needed axioms, such as an axiom of infinity, that are hard to call purely logical.