Area of knowledge - Mathematics

Area of knowledge - Mathematics

Mathematics is sometimes seen to have a degree of certainty that is unmatched by other areas of knowledge or is seen to be founded on a set of more or less universally accepted definitions and basic assumptions. This makes mathematics an excellent source of material for TOK discussions.

One interesting focus for discussions could be the status of mathematics as an area of knowledge. Students could consider why disciplines in the human sciences are often keen to cast their conclusions in mathematical terms, or why mathematical treatments of a topic are often taken by many to be a sign of intellectual rigour. They could also consider why mathematics is often given a privileged position in many education systems.

Another rich source of material for TOK discussions can be the role of creativity, imagination, beauty and elegance in mathematics. Despite, or perhaps because of, the strict confines of mathematical logic, mathematics can be an enormously creative subject, asking its practitioners to make great leaps of imagination. This could lead to discussion of whether, or why, elegance and beauty should be relevant to mathematical value.

Another interesting focus could be the relationship between mathematics and the world around us. Mathematics is often used to model real-world processes. Yet, in some ways, mathematics can also seem quite abstract and detached from the real world, strongly focused on the application of reason rather than relying on experience and observation of the world.

Students could also consider the role and significance of proof in mathematics, and how this relates to concepts such as truth. They could reflect on whether the term “proof” is used differently in mathematics compared to how it is used in our everyday lives or in other areas of knowledge.

Examples of knowledge questions arising from this area of knowledge are suggested below.

Examples of knowledge questions

Scope

• Why is mathematics so important in other areas of knowledge, particularly the natural sciences?

• How have technological innovations, such as developments in computing, affected the scope and nature of mathematics as an area of knowledge?

• Is absolute certainty attainable in mathematics?

• Is there a distinction between truth and certainty in mathematics?

• Should mathematics be defined as a language?

• Is mathematics better defined by its subject matter or its method?

• Does mathematics only yield knowledge about the real world when it is combined with other areas of knowledge?

• Is there a hierarchy of areas of knowledge in terms of their usefulness in solving problems?

Perspectives

• What is it about mathematics that enables mathematical results to remain unchanged over time?

• How significant have notable individuals been in shaping the nature and development of mathematics as an area of knowledge?

• What is the role of the mathematical community in determining the validity of a mathematical proof?

• Is mathematical knowledge embedded in particular cultures or traditions?

• Does personal experience play any role in the formation of claims in mathematics?

• Is progress harder to make in mathematics than in other areas of knowledge?

• If mathematics is created by humans, is it still possible to accept mathematical truths as objective facts about the world?

• Are all of the areas of knowledge in the TOK course themselves embedded in a particular tradition or bound to a particular culture?

Methods and tools

• Is mathematical reasoning different from scientific reasoning or reasoning in other areas of knowledge?

• What is meant by the term “proof” in mathematics, and how is this similar to, or different from what is meant by this term in other areas of knowledge?

• How do mathematicians reconcile the fact that some conclusions seem to conflict with our intuitions?

• What does it mean to say that mathematics is an axiomatic system?

• How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?

• Do mathematical symbols have meaning in the same way that words have meaning? Is personal experience more important or less important in mathematics compared to other areas of knowledge?

Ethics

• If mathematical knowledge is highly valued, does this place special ethical responsibilities on mathematicians when they are making claims?

• On what criteria could we decide whether mathematicians should be held responsible for unethical applications of their work?

• How are unethical practices, such as “data dredging”, used by statisticians to deliberately manipulate and mislead people?

• Is it ethically justifiable for academic mathematicians to spend time doing research that does not have immediate useful applications?

• Do mathematical judgments and ethical judgments face similar challenges in terms of the evidence available to support them?

• Are mathematicians the people best placed to create codes of ethics for professional mathematicians?

Making connections to the core theme

• Why do you think mathematics enjoys a privileged status in many education systems? (scope)

• Who judges the validity of a proof? (perspectives)

• What steps can we take to help ourselves avoid being misled by statistics used in unclear or disingenuous ways in the media? (methods and tools)

• To what extent do you agree with the claim that mathematics “serves as a training that shapes thinking in an ethics-free and amoral way” (Paul Ernest)? (ethics)