, PLEASE USE THIS DOCUMENT AS A GUIDE TO ANSWER YOUR ASSIGNMENT
Question 1: Mathematics in Society
1.1. Define mathematics using the three views presented in your study guide
(instrumentalist/toolbox, Platonist, and system view). Provide one real-life or classroom example
to illustrate each view.
The Toolbox/Instrumentalist View
The instrumentalist or toolbox view of mathematics sees it as a collection of tools, facts, and rules
that are used to achieve external objectives. This perspective focuses on the practical application of
mathematical knowledge, often emphasizing performance and the correct use of techniques rather
than deep understanding. Mathematics is seen as a means to an end, where the main goal is solving
specific problems through computation and formula application.
Example: In a classroom, a teacher might instruct students on how to calculate the area of a
1
triangle using the formula x base x height . The students’ task is to apply this formula to solve
2
various exercises, concentrating solely on correctly substituting values into the equation and
arriving at the right numerical answer. The focus is on completing the task efficiently rather
than exploring the deeper conceptual meaning of why the formula works.
The Platonist View
According to the Platonist view, mathematics is seen as a timeless and objective reality that exists
independently of human thought. In this view, mathematical truths are not created by people but are
discovered, much like uncovering pre-existing truths within an abstract mathematical universe. The
belief is that mathematical objects and structures exist in a separate realm, independent of human
perception or understanding.
Example: A teacher might explain the concept of prime numbers as fundamental, indivisible
numbers that exist within an immutable mathematical system. Students would learn that prime
numbers are not human inventions but rather objective truths that exist regardless of whether
we have named or discovered them. The teacher may frame the process of discovering new
primes as uncovering these timeless truths, which already exist in the mathematical realm.
The System View
The system view of mathematics emphasizes the logical and deductive nature of mathematical
knowledge. It sees mathematics as a structured, axiomatic system where results are derived from a
set of foundational principles. The focus is on the importance of proofs and the interconnection of
mathematical concepts, with an emphasis on consistency, rigor, and the formalization of
mathematical theories.
Example: In geometry, a teacher using the system view would encourage students to
understand the relationships between geometric shapes by proving key theorems, such as the
property that opposite angles of a parallelogram are equal. Students would begin with basic
axioms, such as the definition of parallel lines, and use deductive reasoning to build a logical
argument. This method reinforces the idea that mathematics is a coherent and unified system
where each new result is connected to and supported by earlier truths.
Question 1: Mathematics in Society
1.1. Define mathematics using the three views presented in your study guide
(instrumentalist/toolbox, Platonist, and system view). Provide one real-life or classroom example
to illustrate each view.
The Toolbox/Instrumentalist View
The instrumentalist or toolbox view of mathematics sees it as a collection of tools, facts, and rules
that are used to achieve external objectives. This perspective focuses on the practical application of
mathematical knowledge, often emphasizing performance and the correct use of techniques rather
than deep understanding. Mathematics is seen as a means to an end, where the main goal is solving
specific problems through computation and formula application.
Example: In a classroom, a teacher might instruct students on how to calculate the area of a
1
triangle using the formula x base x height . The students’ task is to apply this formula to solve
2
various exercises, concentrating solely on correctly substituting values into the equation and
arriving at the right numerical answer. The focus is on completing the task efficiently rather
than exploring the deeper conceptual meaning of why the formula works.
The Platonist View
According to the Platonist view, mathematics is seen as a timeless and objective reality that exists
independently of human thought. In this view, mathematical truths are not created by people but are
discovered, much like uncovering pre-existing truths within an abstract mathematical universe. The
belief is that mathematical objects and structures exist in a separate realm, independent of human
perception or understanding.
Example: A teacher might explain the concept of prime numbers as fundamental, indivisible
numbers that exist within an immutable mathematical system. Students would learn that prime
numbers are not human inventions but rather objective truths that exist regardless of whether
we have named or discovered them. The teacher may frame the process of discovering new
primes as uncovering these timeless truths, which already exist in the mathematical realm.
The System View
The system view of mathematics emphasizes the logical and deductive nature of mathematical
knowledge. It sees mathematics as a structured, axiomatic system where results are derived from a
set of foundational principles. The focus is on the importance of proofs and the interconnection of
mathematical concepts, with an emphasis on consistency, rigor, and the formalization of
mathematical theories.
Example: In geometry, a teacher using the system view would encourage students to
understand the relationships between geometric shapes by proving key theorems, such as the
property that opposite angles of a parallelogram are equal. Students would begin with basic
axioms, such as the definition of parallel lines, and use deductive reasoning to build a logical
argument. This method reinforces the idea that mathematics is a coherent and unified system
where each new result is connected to and supported by earlier truths.
