Math 2.0

Mathematical logic and foundations Definition

Logicism

Builds on Foundations of mathematicsPredicate logic

Definition

Logicism is the position in the foundations of mathematics 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 FF is the class of all concepts that are in one-to-one correspondence with FF. 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.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Foundations of mathematics, concept, no evidence levelFoundations of mathem…conceptProposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptLogicism, definition, no evidence levelLogicismdefinition
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. Conjunction definition
  4. Disjunction definition
  5. Implication definition
  6. Negation definition
  7. Predicate definition
  8. Biconditional definition
  9. Propositional logic concept
  10. Predicate logic concept
  11. Logicism definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Logicism
About this page
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Kinddefinition
WritingW0 stub
MSC03A05
DefinesLogicism
UsesFoundations of mathematics, Predicate logic
Changed2 November 2026

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