Math 2.0

Mathematical logic and foundations Definition

Formalism

Builds on Foundations of mathematicsMathematical proof

Definition

Formalism is a position in the foundations of mathematics. 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, 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))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, 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.

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To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Foundations of mathematics, concept, no evidence levelFoundations of mathem…conceptProposition, concept, no evidence levelPropositionconceptImplication, definition, no evidence levelImplicationdefinitionModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptFormalism, definition, no evidence levelFormalismdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Foundations of mathematics concept
  2. Proposition concept
  3. Implication definition
  4. Modus ponens concept
  5. Mathematical proof concept
  6. Formalism definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)
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MSC03A05
DefinesFormalism
UsesFoundations of mathematics, Mathematical proof
Changed2 November 2026

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