TOK - Mathematics
The big picture
Scope
Perspectives
Mathematics is based on
a series of axioms
Axiom: A statement that is taken
to be true and serves as a starting
point for further argumentation.
Three philosophical
approaches to mathematics
Mathematical nominalists
Mathematical fictionalists
Mathematical platonists
Bertrand Russell &
North Whitehead
Goal: to establish a foundation
of mathematics based on proof and
evidence instead of assumptions.
Believed that knowledge created within
the area of knowledge of mathematics is
objectively only if the knowledge is based
on previously established truths.
Mathematics relies on:
Axioms which are taken to be
self-evident, and thus do not
require proof
Conjectures which have not
been proved, but are
assumed to be true.
Branches of
mathematics
Pure mathematics: is abstract
and not directly applied to in,
another area of knowledge
Applied mathematics:
is applied to or utilized in
real-world contexts
The platonist perspective
The formalist perspective
The role of experts
in mathematics
Mathematics exists a priori,
outside of our experience
Evidence for the Platonist perspective being valid includes the experiential certainty that two objects placed in a group with two other objects creates a group of four independent objects
Mathematics is a human-invented method of description akin to language (both describe the world)
Evidence for this claim is that mathematical objects such as numbers do not exist outside of the minds.
Responsibility and power to explain knowledge within that AOK to non-experts. This explanatory role includes both interpretation and explanation of knowledge.
Mathematics is an AOK in which individual contributions are significant to the advancement of knowledge