TOK knowledge questions about mathematics

Ethics and responsibility: TOK knowledge questions

Mathematics is used to count people, convict them, elect leaders and price risk. These questions ask what responsibility comes with that power, for mathematicians and for everyone who uses their results.

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.

How can mistakes in mathematical reasoning lead to injustice?

Knowledge and politics Knowledge and the knower Ethics

Numbers carry authority in courtrooms and public decisions. When a calculation is presented confidently by an expert, few people are in a position to question it. This question asks who is responsible for making sure mathematical evidence is understood correctly.

Claims

  • Statistical reasoning can protect justice by replacing vague impressions with clear evidence, as long as it is done correctly.
  • Professional bodies of statisticians can and do correct misuse, as in the case below.

Counterclaims

  • Juries and judges often cannot evaluate mathematical evidence, so they trust the expert, and a single error can decide a case.
  • Presenting a number gives a false sense of precision about uncertain situations, which can be unfair to the accused.

Real-life situations from mathematics

Sally Clark. In 1999 Sally Clark was convicted in England of murdering her two baby sons. An expert witness told the court that the chance of two sudden infant deaths in such a family was about 1 in 73 million, a figure obtained by squaring the chance for one death as if the two deaths were independent. The Royal Statistical Society publicly criticised the reasoning, and her conviction was overturned in 2003.

The prosecutor's fallacy. Confusing the probability of the evidence given innocence with the probability of innocence given the evidence is a known error, called the prosecutor's fallacy. Conditional probability shows why the two can be very different.

Use this in your TOK work

Essay. Fits titles about the responsibilities of experts, ethics of knowledge, or trust.

Exhibition. A newspaper report of a court case involving statistics is a serious object for a prompt about the consequences of knowledge.

The mathematics behind it

Do mathematicians have ethical responsibilities for how their work is used?

Knowledge and politics Knowledge and technology Ethics

Mathematics is often described as neutral: theorems are neither good nor bad. Yet mathematical work has been used in weapons, surveillance, finance and medicine. This question asks whether the people who create mathematical knowledge share responsibility for its uses.

Claims

  • A theorem can be used for good or harm, so responsibility lies with those who apply it, not with those who prove it.
  • Mathematicians cannot predict the uses of their work; pure results have found applications decades later.

Counterclaims

  • Many mathematicians work directly on applications, such as algorithms or models, where uses are foreseeable, so they share responsibility.
  • Projects such as the University of Cambridge's Ethics in Mathematics Project argue that mathematicians need ethical training, as doctors and engineers do.

Real-life situations from mathematics

Codebreaking in the Second World War. Mathematicians at Bletchley Park, including Alan Turing, broke German codes. Their work is widely thought to have shortened the war. It also shows mathematics serving national security in secret.

Weapons of Math Destruction. In her 2016 book, the mathematician Cathy O'Neil argued that opaque mathematical models used in hiring, policing and lending can harm people at scale, and called for accountability.

Use this in your TOK work

Essay. Fits titles about ethics and knowledge, or responsibility of knowers.

Exhibition. A model of an Enigma rotor or a printout of a cipher can anchor a prompt about ethics.

The mathematics behind it

Can mathematics tell us what a fair voting system is?

Knowledge and politics Ethics

Every voting system turns individual preferences into a group decision, and mathematics shows that different systems can choose different winners from the same votes. This question asks whether mathematics can settle questions of fairness or only clarify them.

Claims

  • Mathematics can state fairness conditions precisely and prove which systems satisfy them, giving real knowledge about voting.
  • Without mathematics, debates about fairness rely on slogans; with it, trade-offs become clear.

Counterclaims

  • Mathematics can show that no system satisfies all desirable conditions, so the final choice depends on values, not proof.
  • Boundaries and districts can be drawn to favour one side while every vote is counted correctly, so arithmetic accuracy is not the same as fairness.

Real-life situations from mathematics

Arrow's impossibility theorem. In 1951 Kenneth Arrow proved that, with three or more options, no ranked voting system can satisfy a short list of reasonable fairness conditions all at once.

Gerrymandering. The word comes from an 1812 electoral district in Massachusetts, approved by Governor Elbridge Gerry, whose shape was likened to a salamander. Mathematicians now propose measures to detect unfair district maps.

Use this in your TOK work

Essay. Fits titles about politics and knowledge, or the limits of mathematical answers.

Exhibition. A ballot paper (a specimen or a class vote) is an excellent object for a politics prompt.

The mathematics behind it

Should numbers decide questions about health and life?

Knowledge and politics Ethics

Health services must decide which treatments to fund, and they often use calculations of cost and benefit. This question asks whether turning lives into numbers makes such decisions fairer or hides moral choices behind mathematics.

Claims

  • When resources are limited, a transparent calculation treats everyone by the same rule and makes decisions open to challenge.
  • Without numbers, decisions depend on who shouts loudest, which is less fair.

Counterclaims

  • Any formula embeds value judgements, such as how much a year of life in poor health is worth, which mathematics cannot justify.
  • A precise number can make an uncertain, moral decision look objective and settled.

Real-life situations from mathematics

QALYs in the UK. The UK's National Institute for Health and Care Excellence (NICE) uses quality-adjusted life years (QALYs) to compare treatments, and has generally treated a cost of about £20,000 to £30,000 per QALY as a guide to value for money.

Triage during the pandemic. During 2020 some hospitals drew up scoring systems to decide which patients would receive intensive care if beds ran out, prompting public debate about fairness to older and disabled people.

Use this in your TOK work

Essay. Fits titles about ethics, objectivity and the human sciences.

Exhibition. A medicine leaflet with dosage tables, or a health service leaflet, is an object that connects numbers and lives.

The mathematics behind it

What is gained and lost when a state counts its people?

Knowledge and politics Knowledge and language Ethics

A census turns a population into numbers that guide funding, representation and planning. It also defines categories people must fit into. This question asks how counting shapes knowledge of a society and the people being counted.

Claims

  • Without accurate counts, governments cannot plan schools, hospitals or fair representation, so counting is essential knowledge.
  • Statistical methods let agencies protect individual privacy while still publishing useful totals.

Counterclaims

  • The categories on a form, such as ethnicity, are human choices, and they change; counting creates the groups it claims to measure.
  • Data collected for one purpose can be used for another, sometimes harmfully.

Real-life situations from mathematics

The ethnicity question. The census in Great Britain first asked about ethnic group in 1991. The categories have been revised in later censuses, showing that what is counted reflects changing ideas about identity.

Privacy and the 2020 US census. The US Census Bureau applied a mathematical technique called differential privacy to its 2020 results, adding carefully calibrated noise to protect individuals. Some users of the data argued that it reduced accuracy for small communities.

Use this in your TOK work

Essay. Fits titles about classification, politics and the human sciences.

Exhibition. A census form is a classic, powerful object; comment on one specific question it asks.

The mathematics behind it

Who is responsible when a trusted mathematical model fails?

Knowledge and politics Knowledge and technology Ethics

Banks and investors use mathematical models to price risk. When those models fail, the consequences can be global. This question asks how responsibility should be shared between the model, its makers and its users.

Claims

  • Models are tools; responsibility lies with the people who decide to rely on them beyond their limits.
  • Model makers who state their assumptions clearly have met their responsibility.

Counterclaims

  • The authority of mathematics can discourage users from questioning a model, so its makers share responsibility for how confidently it is presented.
  • Some risks, such as rare extreme events, are systematically underestimated by common models, so the mathematics itself can mislead.

Real-life situations from mathematics

Long-Term Capital Management. This hedge fund, whose partners included the Nobel-winning economists Robert Merton and Myron Scholes, relied heavily on mathematical models. In 1998 it suffered huge losses, and the US Federal Reserve organised a rescue by banks to avoid wider damage.

The 2008 financial crisis. Many analyses of the 2008 crisis argued that models used to price mortgage-linked securities underestimated how likely it was that many borrowers would default at the same time.

Use this in your TOK work

Essay. Fits titles about trust in experts, models and the ethics of knowledge.

Exhibition. A bank statement or a loan agreement showing an interest rate is an everyday object for a prompt about consequences of knowledge.

The mathematics behind it

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