TOK knowledge questions about mathematics

Models and the real world: TOK knowledge questions

Every model leaves something out. These questions ask what a model, a map or a measurement can tell us about the world, and what is lost or assumed along the way.

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.

If every mathematical model leaves something out, how can models produce knowledge?

Knowledge and politics Knowledge and technology Methods and tools

A model keeps some features of a situation and ignores others so the mathematics can be done. The statistician George Box made the point that all models are wrong, but some are useful. This question asks how a deliberately incomplete picture can still give knowledge, and how we tell when a model has left out too much.

Claims

  • Simplifying is what makes knowledge possible: a model that included everything would be as complicated as the world and explain nothing.
  • Models can be tested against data, and a model that predicts well, within stated limits, gives reliable knowledge about those limits.

Counterclaims

  • Users of a model often forget its assumptions, so its conclusions can be trusted beyond the range where they hold.
  • Choosing what to leave out is a judgement, and different modellers can make different choices that lead to different conclusions.

Real-life situations from mathematics

COVID-19 epidemic models. In March 2020 epidemic models from Imperial College London projected deaths under different policies and influenced the UK's decision to introduce a lockdown. The models were praised and criticised for their assumptions about how people would behave.

SIR models in the classroom. The simple SIR model splits a population into susceptible, infected and recovered groups. It captures the rise and fall of an epidemic but ignores ages, places and behaviour, which is both its power and its limit.

Use this in your TOK work

Essay. Fits titles about models, simplification or prediction across the natural and human sciences. Your IA modelling experience is a strong personal example.

Exhibition. A screenshot of an epidemic curve from 2020 is a powerful object for a prompt about the reliability or consequences of knowledge.

The mathematics behind it

Can a representation be accurate and misleading at the same time?

Knowledge and politics Knowledge and language Perspectives

Every flat map of the round Earth distorts something: areas, angles, distances or shapes. A map can be perfectly accurate for its purpose and still give a false impression. This question asks how mathematics helps us understand, and choose between, such trade-offs.

Claims

  • Mathematics shows exactly what each projection preserves and distorts, so a well-chosen map is accurate knowledge for its purpose.
  • Being misleading is a fault of the user, not the representation; knowing the method removes the problem.

Counterclaims

  • Most people never learn how a map was made, so the choice of representation shapes their beliefs, for example about the size of countries.
  • Choosing a representation can reflect political or cultural perspectives, so mathematics does not make the choice neutral.

Real-life situations from mathematics

The Mercator projection. Gerardus Mercator's 1569 map keeps compass directions as straight lines, which made it ideal for navigation, but it greatly enlarges areas far from the equator. Greenland appears similar in size to Africa, which is about fourteen times larger.

The Gall–Peters debate. In 1973 Arno Peters promoted an equal-area projection as fairer to countries near the equator. Cartographers pointed out that equal-area maps were already known and that his map distorted shapes, and the debate became as much about politics as mathematics.

Use this in your TOK work

Essay. Excellent for titles about representation, perspective or whether knowledge can be neutral.

Exhibition. A Mercator wall map is one of the most popular and effective maths-linked objects; make your commentary specific about what it preserves and distorts.

The mathematics behind it

Does chaos theory set a limit on what mathematics can know about the future?

Knowledge and technology Scope

Some systems follow exact rules yet are practically unpredictable, because tiny differences in starting values grow quickly. This question asks whether that is a limit on knowledge itself or only on our measurements and computers.

Claims

  • Chaos theory is itself mathematical knowledge: it tells us precisely how fast errors grow and therefore how far ahead prediction is possible.
  • In chaotic systems we can still know long-term patterns and probabilities, even when exact prediction fails.

Counterclaims

  • If the future of a deterministic system cannot be predicted in practice, then knowing the rules gives less knowledge than we might hope.
  • Measurements are always finite, so chaos guarantees a horizon beyond which no improvement in technology helps.

Real-life situations from mathematics

Lorenz's rounding. In 1961 Edward Lorenz restarted a weather simulation by typing in a value rounded to three decimal places instead of six. The new run soon diverged completely from the old one. His 1963 paper became a founding work of chaos theory.

Weather forecasts. Forecasters now run many simulations from slightly different starting values and report probabilities, an approach shaped by the discovery that the atmosphere is chaotic.

Use this in your TOK work

Essay. Fits titles about prediction, certainty or the limits of knowledge in the natural sciences.

Exhibition. A weather forecast showing a percentage chance of rain is an ordinary object that opens up chaos and probability.

The mathematics behind it

Are units of measurement discovered in nature or decided by agreement?

Knowledge and language Knowledge and politics Methods and tools

A metre and a kilogram feel like natural facts, but they are defined by international agreement, and the definitions have changed. This question asks how much of quantitative knowledge depends on human conventions.

Claims

  • The choice of unit is conventional, but the quantity measured is real, so measurement gives objective knowledge expressed in an agreed language.
  • Modern definitions are tied to constants of nature, which makes them as objective as possible.

Counterclaims

  • What we choose to measure, and how precisely, reflects human purposes, so measurement is never purely a reading of nature.
  • Agreements need institutions; the authority of a measurement depends partly on trust in those institutions.

Real-life situations from mathematics

Redefining the kilogram. Until May 2019 the kilogram was defined by a metal cylinder kept near Paris. It was then redefined using the Planck constant, a constant of nature.

Defining the metre. Since 1983 the metre has been defined by the distance light travels in a vacuum in a set fraction of a second, so the speed of light is now exact by definition.

Use this in your TOK work

Essay. Fits titles about conventions, objectivity or the role of institutions in knowledge.

Exhibition. A ruler or a kitchen scale makes a simple but rich object for a prompt about who decides what counts as knowledge.

The mathematics behind it

How do the assumptions built into a model shape the conclusions it can reach?

Knowledge and politics Methods and tools

Whether a population is modelled as growing exponentially or levelling off, or whether people are assumed to act independently, can decide what a model concludes. This question asks how far a model's results are discovered from data and how far they are built in by its assumptions.

Claims

  • Assumptions are stated openly and can be tested, so a good model makes its own limitations visible and its conclusions conditional.
  • Comparing models with different assumptions shows which conclusions are robust and which depend on one choice.

Counterclaims

  • Once results are reported, the assumptions are often lost, so readers take conditional conclusions as unconditional facts.
  • Modellers' own expectations can guide which assumptions seem reasonable, so a model may confirm what its makers already believed.

Real-life situations from mathematics

Malthus and population. In 1798 Thomas Malthus argued that population tends to grow geometrically while food supply grows arithmetically, predicting recurring famine. Changes in farming technology and birth rates meant his assumptions did not hold as he expected.

Logistic versus exponential growth. Fitting early epidemic or population data with an exponential model and with a logistic model can give similar fits at first but very different forecasts later, which is why the choice of model must be justified.

Use this in your TOK work

Essay. Fits titles about assumptions, bias or the reliability of predictions in the human and natural sciences.

Exhibition. Two graphs from the same data, one fitted exponentially and one logistically, show how assumptions decide conclusions.

The mathematics behind it

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