---
tag: N1N5N2QJ86
kind: definition
name: Formalism
msc: [03A05]
defines: 46JCQ6KDA4
uses: [EFFRJFW67Y, 0B94TKFQJX]
sources:
  - url: https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)
---

## Definition

*Formalism* is a position in the [[foundations of mathematics|EFFRJFW67Y]].
It holds that mathematical statements are strings of symbols in a formal system.
A statement is established when it is derived from the axioms of the system by its rules of inference.
The derivation is the whole content of a [[proof|0B94TKFQJX]], and a statement need not refer to abstract objects.

## Remarks

In a formal system for arithmetic, the string $S(0) + S(0) = S(S(0))$ is established by a derivation from the axioms; a formalist accepts it for that reason alone.
Some statements are true in the standard natural numbers but have no derivation in the system.
An example is a Gödel sentence of the [[incompleteness theorems|X9WYNYE29C]], and formalism does not count it as established.
Hilbert's version of formalism kept a contentful core of finitary mathematics.
It treated the rest as a formal system whose consistency should be proved by finitary means; this is [[Hilbert's program|P1RW6JWX73]].