Mathematics

Description

Key Vocabulary

Proof
Theorem
Axiom

Complexity
Logic
Algorithm
Intuition

Deductive reasoning
Number theory
Incompleteness

Knowledge Framework

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 & 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?

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)

Quotes about Math

"As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality." (Albert Einstein)

"What is mathematics? It is only a systematic effort of solving puzzles posed by nature." (Shakuntala Devi)

"Mathematics is the music of reason." (James Joseph Sylvester, English mathematician)

"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country." (David Hilbert, German mathematician)

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." (William Paul Thurston, American mathematician)

"Nature is written in mathematical language." (Galileo Galilei, Italian astronomer, physicist and engineer)

"It is impossible to be a mathematician without being a poet in soul." (Sofia Kovalevskaya, Russian mathematician)

"Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigor should be a signal to the historians that the maps have been made, and the real explorers have gone elsewhere." (W.S. Anglin, Mathematics author)

"Logic and mathematics are nothing but specialised linguistic structures." (Jean Piaget).

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