MATH223 - History and Philosophy of Mathematics

Year

Credit points

Campus offering

Prerequisites

MATH107 Introduction to Logic and Algebra

Incompatible

MATH313 History and Philosophy of Mathematics

Unit rationale, description and aim

This unit provides an opportunity for students to learn of the beginnings of some important topics in mathematics. This will reinforce the fact that most topics began with the need to solve a problem.

Typically, mathematics is taught as a series of, often, disconnected topics and considerable care must be taken to inform students of the interconnections between those topics. As well as providing the names of those who first developed various areas of mathematics, study of the history of mathematics reinforces the connections between those areas. The history of mathematics also replaces those topics within their context, motivates their study and explains what lead to their development and what discoveries arose from them.

This unit focuses on the ides of and need for proof. Building on previous Mathematics units and by looking at the development of counting and number systems, the evolution of mathematical concepts in response to need and the development of proof and its various forms is studied. That evolution is traced both from ancient cultures to the present day and across a variety of cultures and civilisations.

The aim of this unit is to place the development of modern ideas of number systems and mathematics within the context in which they were first developed. This unit will also study the need for the sophisticated definitions that are currently used, by considering the problems that needed to be resolved.

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Describe the role that use played in the evolution of counting. Including Aboriginal, Torres Strait Islander and other multicultural aspects (GA1, GA5)

LO2 - Explain the development of numbers and number forms including natural numbers, integers, irrational numbers and complex numbers (GA5, GA9)

LO3 - Identify the links with Geometry and the notion of proof (GA4, GA5)

LO4 - Prove some major results in mathematics including infinity of primes, irrationality of √2, countability of rational and uncountability of the reals (GA4, GA5, GA8)

LO5 - Describe the importance of conjectures in evolving mathematical thinking and identify some of these key problems in the history of mathematics (GA5, GA7, GA9)

Graduate attributes

GA1 - demonstrate respect for the dignity of each individual and for human diversity

GA4 - think critically and reflectively 

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA7 - work both autonomously and collaboratively 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA9 - demonstrate effective communication in oral and written English language and visual media 

Content

Topics will include:

1. History of Counting from pre-historical to historical times: the natural numbers.
2. History of operating with numbers and notation used.
3. Extending number systems.
4. Geometry and the development of proof.
5. Different methods of proof.
6. Cardinality and notions of infinity. Countability of sets.
7. Axiomatic systems and Gödel’s Incompleteness Theorem
8. Intuitionism
9. Famous problems in the evolution of mathematics
10. Computing and the future of mathematics

Learning and teaching strategy and rationale

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology.

This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial.

150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study.

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit.

This unit provides an opportunity for students to focus on a period in history that particularly appeals to them, or that they think is important. By combining a written task on a set topic, together with a presentation to the class with associated report on a topic of their choice (with guidance), students will be able concentrate upon either a subject of their choice or consider the origin, across cultures, of an aspect of mathematics.

Overview of assessments

Representative texts and references

Cameron, M. (1983). Heritage mathematics. North Melbourne: Hargreen.

Crossley, J. N. (1987). The emergence of number (2nd ed.). World Scientific.

Hofstadter, D. (2011). Gödel Escher Bach: The eternal golden braid. (20th ed.). New York: Basic Books.

Kline, M. (2001). Mathematical thought from ancient to modern times (v1–3) New Edition. Cambridge University Press.

Lakatos, I. (2018). Proofs and refutations: The logic of mathematical discovery. Cambridge; New York: Cambridge University Press.

Pottage, J. (1982). Geometrical investigations: Illustrating the art of discovery in the mathematical field. Reading, MA: Addison Wesley.

Smullyan, R. (2014). A beginner’s guide to mathematical logic. Dover.

Stillwell, J. (2020). Mathematics and Its History: A Concise Edition. Cham: Springer Nature

Struik, D. (1987). A concise history of mathematics (4th rev. ed.). New York: Dover.