TOK knowledge questions about mathematics

Culture, history and language: TOK knowledge questions

Mathematics is often called universal, but it is made by people in particular places and times. These questions look at who gets the credit, how notation shapes thought, and the mathematics in everyday practices.

Questions on this page

Each question has claims, counterclaims, real-life situations from mathematics and pointers for your essay and exhibition. Check dates and figures in a reliable source before you use them, and cite that source. Link a question to the prescribed title or exhibition prompt you are working on, in your own words: see using maths examples in your essay and choosing exhibition objects. The study headings (scope, perspectives, methods and tools, ethics) are our own.

Whose contributions are recognised in the history of mathematics, and why does it matter?

Knowledge and politics Knowledge and the knower Perspectives

Textbooks name theorems after a few famous people, mostly European men. Historians have shown that many results were found earlier or elsewhere, and that many contributors were overlooked. This question asks how recognition affects what and how we know.

Claims

  • Mathematical truth does not depend on who discovered it, so recognition affects history but not mathematical knowledge itself.
  • Correcting the historical record improves our knowledge of how mathematics developed, which is valuable in itself.

Counterclaims

  • Who is recognised shapes who feels mathematics is 'for them', which affects who becomes a mathematician and so which questions are asked.
  • Naming conventions can hide the international, collaborative nature of mathematics.

Real-life situations from mathematics

Pascal's triangle. The triangle of binomial coefficients appears in the work of Jia Xian in eleventh-century China and of Yang Hui (1261), and in Persian and Indian mathematics, centuries before Blaise Pascal's treatise. In China it is often called Yang Hui's triangle.

Recognition and women in mathematics. Emmy Noether proved fundamental results in algebra and physics but was initially not allowed a paid post at Göttingen. Katherine Johnson's calculations supported early US space flights, including John Glenn's orbit in 1962. In 2014 Maryam Mirzakhani became the first woman to win a Fields Medal.

Use this in your TOK work

Essay. Fits titles about the history of knowledge, perspectives or the knower's identity.

Exhibition. A textbook page showing 'Pascal's triangle', with a note on its earlier history, is a thoughtful object.

The mathematics behind it

Is mathematics the same in every culture?

Knowledge and indigenous societies Knowledge and language Perspectives

Different civilisations counted in different bases, used different notations and valued different problems, yet often reached the same results. This question asks whether mathematics is universal, cultural, or both at different levels.

Claims

  • The results are universal: a proof of Pythagoras' theorem works the same in any language or culture.
  • Independent discovery of the same results in different places suggests mathematics is not merely cultural.

Counterclaims

  • What counts as a problem worth solving, what counts as a proof and how results are written down have varied widely between cultures.
  • Calling one tradition 'mathematics' and others 'folk practice' reflects historical power rather than the nature of knowledge.

Real-life situations from mathematics

Base 60 and base 20. Babylonian mathematicians used a base-60 system, which survives in 60 minutes in an hour and 360 degrees in a full turn. The Maya used a base-20 system with a symbol for zero.

The Kerala school. Mathematicians in Kerala, India, beginning with Madhava (about 1340–1425), found infinite series for π and for trigonometric functions about two centuries before similar series were published in Europe by Gregory and Leibniz.

Use this in your TOK work

Essay. Fits titles about culture, universality and perspectives.

Exhibition. A clock face is a simple, clever object for this question: it carries Babylonian mathematics into every classroom.

The mathematics behind it

Does it matter who gets credit for a mathematical discovery?

Knowledge and politics Ethics

Mathematicians care deeply about priority: being first to prove something. Disputes over credit have been bitter and sometimes national. This question asks whether credit affects knowledge itself or only the people involved.

Claims

  • Credit rewards the effort of proof and encourages people to publish openly, which spreads knowledge.
  • Accurate credit lets us trace where ideas came from, which helps us understand them.

Counterclaims

  • Priority disputes can divide communities and slow progress, as happened between British and continental mathematicians.
  • Many discoveries are collective or simultaneous, and naming them after one person distorts how knowledge is made.

Real-life situations from mathematics

Newton and Leibniz. Isaac Newton and Gottfried Leibniz developed calculus independently. In 1712 a Royal Society committee found in Newton's favour; its report was largely written by Newton himself, who was the Society's president.

Secret discoveries. In 1973 Clifford Cocks, working at the UK's intelligence agency GCHQ, devised a public-key encryption method essentially the same as RSA. It stayed secret until 1997, so the public credit went to Rivest, Shamir and Adleman, who published in 1977.

Use this in your TOK work

Essay. Fits titles about the social side of knowledge, individuals and communities, or ethics.

Exhibition. A banknote or stamp honouring a mathematician is an object that invites questions about who is remembered.

The mathematics behind it

Can knowledge built into practices such as weaving, building or navigation count as mathematics?

Knowledge and indigenous societies Knowledge and language Scope

Many communities use sophisticated patterns, measurements and calculations in crafts and navigation without writing them as theorems. Ethnomathematics, a field associated with Ubiratan D'Ambrosio, studies such practices. This question asks where the boundary of mathematics lies.

Claims

  • If a practice reliably uses number, pattern, symmetry or measurement, it contains mathematical knowledge, whether or not it is written as a theorem.
  • Recognising such knowledge widens our understanding of what mathematics is and who does it.

Counterclaims

  • Mathematics as an area of knowledge is defined by proof and generality; skilful practice without justification is valuable knowledge, but of a different kind.
  • Outsiders may 'find' mathematics in a practice that its practitioners would describe quite differently.

Real-life situations from mathematics

Inca khipu. The Inca recorded numbers using knotted cords called khipu, with a place-value system in base 10. Scholars are still working out whether some khipu also recorded words.

Polynesian wayfinding. In 1976 the voyaging canoe Hōkūleʻa sailed from Hawaiʻi to Tahiti navigated by Mau Piailug using stars, swells and birds, without instruments, showing the accuracy of traditional navigation knowledge.

Use this in your TOK work

Essay. Fits titles about indigenous knowledge, the boundaries of areas of knowledge or practical versus theoretical knowledge.

Exhibition. A woven textile with a repeating pattern, or a photo of a traditional navigation chart, suits the indigenous societies theme.

The mathematics behind it

Does mathematical notation shape what mathematicians are able to think?

Knowledge and language Knowledge and technology Methods and tools

Doing long multiplication in Roman numerals is very hard; in Hindu–Arabic numerals it is routine. Notation is more than a container for ideas. This question asks whether the symbols we use limit or extend mathematical knowledge.

Claims

  • Good notation makes patterns visible and calculations manageable, so it opens up new knowledge that was hard to reach before.
  • The spread of a notation, such as place value, changed whole economies and sciences, which shows its power.

Counterclaims

  • The underlying mathematics is the same in any notation; notation affects convenience, not truth.
  • Mathematicians adapt notation to their thinking, so thought may shape notation more than the other way round.

Real-life situations from mathematics

Hindu–Arabic numerals in Europe. Fibonacci's Liber Abaci (1202) introduced Hindu–Arabic numerals and place value to many European merchants. Adoption was gradual; some cities restricted their use in official records for a time.

Napier's logarithms. John Napier published logarithms in 1614, turning multiplication into addition. Tables of logarithms transformed astronomy and navigation calculations for over three centuries.

Use this in your TOK work

Essay. Fits titles about language, tools and how knowledge is represented.

Exhibition. A slide rule or a book of log tables is a lovely historical object for a prompt about tools and knowledge.

The mathematics behind it

Other areas: Proof · Nature of maths · Infinity · Models · Statistics · Technology · The knower · Ethics · all 49 knowledge questions