Foundations of Abstract Mathematics Online Course

Department of Mathematical Sciences, Stellenbosch University

This is a preliminary demonstration version of the site with limited functionality

Founding authors of the project:

Dr Howell, Dr Janelidze, and Professor Rewitzky

Project supported by the FINLO grant of Stellenbosch University

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A message from a student (Thomas Weighill), 2011

It has been a privilege to take this course this year, as I rarely get an opportunity to do in-depth rigorous mathematics in my Engineering degree. I think that this course has greatly developed my skills in logic and mathematics as well as increased my interest in mathematics and in particular, abstract mathematics.

Perhaps one frustrating aspect was that a few of the students and I found the pace a bit slow. But the slow pace is definitely necessary to include those classmates who had little experience in real mathematical thinking.

I think it is important that this course be presented to people who do not have a strong mathematical background or who may not use the information they get in this course directly in their careers, in addition to those who excel in and enjoy mathematics. In every field and every faculty, people are needed who have a strong sense of both creativity as well as exactness, and this is what this course fosters in its students.

I think I would recommend this course to two types of people: firstly, people who are very interested in mathematics and maybe even want to pursue a research career or use mathematics in their career one day. This course will give them a solid mathematical foundation which is sorely lacking in maths education and also teach them how to think beyond the obvious and the straight-forward.

I would also recommend it to anyone who wants to broaden their mind and improve their thinking skills, or who are merely curious about mathematics. Often people end up in their careers without a real desire for perfection or rigour. Or they end up using maths in their job without understanding what the formulae really mean and where they come from. I think that there is a lot to learn in this course besides mathematical theory.

Dr Janelidze is enthusiastic and has a love for maths. He is also patient with those who are not so strong in mathematics and very accessible to students.

Areas for improvement in the course are perhaps a bit more historical information. Maybe it is just me, but I am interested to know when people discovered things and how. It doesn’t need to be examinable work just a mention every now and then.

Lastly, maybe an extra class to accommodate more advanced students. Advanced students should, however, not be accelerated to the point that they don’t have a solid understanding of the concepts, so I would be cautious. One of the best things about this course is the time and effort put into laying a solid foundation and understanding of the work.

A message from a student (Pillip-Jan van Zyl), 2011

Foundation of Mathematics II is an introductory course to precise axiomatic logic through a set theoretic basis. This module should appeal to students who's exposure to mathematics have left them seeking a more satisfactory and unambiguous explanation of both familiar conclusions and the more fundamental inner workings of mathematics alike.

Emphasis is placed on a deeper understanding of mathematical concepts and structure: the critical importance of the definition, the synthesis of assumptions into axioms, its logical progression to sometimes surprising theorems, the "naturalness" of some phenomena such as dualities and the conduciveness of contradiction into new mathematical structures.

Numbers and cardinality are treated later in the course when earlier concepts such as partitions, relations and functions are fully understood and the student should be have adequate knowledge to be exposed to the somewhat counterintuitive nature of infinite sets.

I would highly recommend the course to students that value clarity and completeness in arguments, for at its essence mathematics provides certainty by means of assumption in an otherwise rather uncertain reality.

Foundations II is a succinct and fitting means to explore this beauty of mathematics, an art that encourages knowledge and experience to be approached with the inquisitiveness and originality of a child.

A message from a student (Vareesé van Tonder), 2011

The course Foundations of Abstract Mathematics has provided me with the ability to read and write in the mathematical language. The course also gave me a deeper understanding of set theory and insight into proving theorems. Having these skills at hand, my problem solving and analysis techniques greatly improved and this was reflected in my marks. Above all, the course gave me an appreciation for the unambiguity and preciseness of mathematics.