talking about, nor whether what we are saying is true.
Bertrand Russell (1917)
From a TOK point of view mathematics is a rather special area of knowledge. On the one hand it seems
to supply a certainty often missing in other disciplines. On the other, its methods—for example, the
application of strict logical procedures to supposedly self-evident first principles—suggest a subject
matter that is removed from the real world. It is hardly surprising then to find a variety of responses to
mathematical knowledge, from astonishment at the beauty of some mathematical argument, to wonder
at the power of mathematics to solve problems in the sciences or engineering, to frustration in the face
of apparently meaningless symbols manipulated as part of a pointless game.
What is unarguable is the ability of mathematics to yield important knowledge about the world, often
in conjunction with other areas of knowledge. Why mathematics should be so successful in this regard
rests upon a number of questions concerning the nature of mathematics itself and its relation to the
world and to human intelligence. Some mathematicians argue that their subject is a language, that it is,
in some sense, universal or that there is great beauty to be found in it. What is clear, in any case, is that
mathematics is a rich area of exploration for the TOK student.
sense “already there” to be discovered?
• What does it mean to say that mathematics can be regarded as a formal game devoid of intrinsic
meaning? If this is the case, how can there be such a wealth of applications in the real world?
• What does it mean to say that mathematics is an axiomatic system?
• Some educational systems make a distinction between pure mathematics and applied mathematics.
Does this reflect a fundamental difference in approach to mathematical knowledge?
• It is sometimes said that mathematical reasoning is a process of logical deduction. If this is true, and
if the conclusion of a proof must always be implied by (contained in) its premises, how can there
ever be new mathematical knowledge?
Mathematics and the world
• We can use mathematics successfully to model real-world processes. Is this because we create
mathematics to mirror the world or because the world is intrinsically mathematical?
• Some major advances in physics, for example, discoveries of elementary particles, have come about
through arguments involving the beauty, elegance or symmetry of the underlying mathematics.
What does this tell us about the relationship between the natural sciences, mathematics and the
• Is mathematics better defined by its method or by its subject matter?
• In the light of the questions above, is mathematics invented or discovered?
• Mathematicians marvel at some of the deep connections between disparate parts of their subject.
Is this evidence for a simple underlying mathematical reality?
other areas of knowledge?
• What are the roles of empirical evidence and inductive reasoning in establishing a mathematical
• Can a mathematical statement be true before it has been proven?
• In hypothesis testing, a statistician could state that a result was true at the 5% significance level.
What does this mean?
• It has been argued that we come to know the number 3 through examples such as three oranges or
three cups. Does this support the independent existence of the number 3 and, by extension, numbers
in general? If so, what of numbers such as 0, -1, i (the square root of -1) and a trillion? If not, in what
sense do numbers exist?
• In the light of the question above, why might it be said that mathematics makes true claims about
• In what sense might chaos (non-linear dynamical systems) theory suggest a limit to the applicability
of mathematics to the real world?
• Can mathematics be characterized as a universal language?
• To what extent is mathematics a product of human social interaction?
• What is the role of the mathematical community in determining the validity of a mathematical proof?
• Why is it that mathematics is considered to be of different value in different cultures?
• How would you account for the following features that seem to belong particularly to mathematics:
some people learn it very easily and outperform their peers by years; some people find it almost
impossible to learn, however hard they try; most outstanding mathematicians supposedly achieve
their best work before they reach the age of 30?
• What counts as understanding in mathematics? Is it sufficient to get the right answer to a mathematical
problem to say that one understands the relevant mathematics?
• Are there aspects of mathematics that one can choose whether or not to believe?
• How do we choose the axioms underlying mathematics? Is this an act of faith?
• Do the terms “beauty” or “elegance” have a role in mathematical thought?
• Is there a correlation between mathematical ability and intelligence?
• Is there a clear-cut distinction between being good or bad at mathematics?