Mathematics & Knowledge (DP IB Theory of Knowledge): Revision Note

Mathematics & knowledge

• Mathematics produces knowledge through abstract reasoning and proof rather than observation alone

• This raises questions about whether mathematical knowledge is discovered, invented, or shaped by human choices

Mathematics as discovery or invention

• Mathematics can be seen as discovered when patterns and relationships seem to exist independently of humans

  • E.g. once you define prime numbers, their properties constrain what is possible

    • Proof uncovers truths you cannot change

• Mathematics can be seen as invented when it depends on human choices about definitions, symbols and axioms

  • E.g. changing axioms in geometry

    • There are different but internally consistent systems

    • Multiple “true” models within different rules

• The discovery vs invention debate affects how mathematical certainty is interpreted

  • The discovery view says that certainty comes from uncovering objective structure

    • Knowledge feels inevitable

  • The invention view says that certainty comes from consistency within chosen rules

    • Knowledge depends on human starting points

The universality of Mathematics

• The universality of mathematics supports the idea that mathematical knowledge applies across cultures and languages, e.g.: 

  • Independent communities develop counting and geometry

  • Shared structures emerge

  • Results can be translated across contexts

• Universality is strengthened because mathematical justification relies on proof, not authority or tradition

  • Proof can be checked by any competent knower

  • This reduces dependence on local belief systems

• Universality can be questioned because humans choose what to formalise and what problems to prioritise

  • Different priorities shape which concepts are developed

  • This influences what is treated as important knowledge

• Universality does not mean mathematical knowledge is value-free

  • Values influence research goals

  • This affects what mathematics is funded, taught, and applied

Mathematical objects and abstraction

• Mathematical objects are abstract because they are defined by properties rather than physical features

  • A triangle in mathematics is an idealised object with exact properties

    • Physical drawings are approximations, not the object itself

• Abstraction allows mathematics to produce knowledge that applies to unlimited cases at once

  • Generalisation means that one proof covers many situations

    • Knowledge extends beyond individual examples

• Abstraction can increase certainty within mathematics, but can seem to distance it from reality

  • Defined objects are perfectly consistent

    • So there is certainty about claims inside the system

  • But real-world objects are messy and variable

    • They are harder to match to ideal mathematical forms

• Interpretation matters when linking abstract mathematics to the world

  • Choosing what counts as “the same” structure in reality affects which results are treated as applicable

Purity vs application

• Pure mathematics develops knowledge for internal coherence rather than immediate practical use

  • Focus on proof and structure strengthens justification

  • Truth depends on the axioms and logic used

• Applied mathematics uses mathematical structures to model and explain real-world phenomena

  • Modelling choices introduce assumptions

  • This affects how reliable conclusions are about reality

• The purity vs application distinction shows two ways mathematical knowledge can be justified

  • Pure mathematics is justified by logical proof from axioms

    • Certainty comes from deduction

  • Applied mathematics is justified by predictive success and fit with evidence

    • Its usefulness depends on model accuracy

• Disagreement in applied mathematics often concerns how well a model represents reality

  • Different assumptions produce different outcomes

  • This changes what counts as strong justification