## FAM seminars

Foundations of Abstract Mathematics I and II are second and third year courses delivered by the Mathematics Division of Stellenbosch University since 2011-2012.

The official Stellenbosch University codes for these courses are 21539 278 and 21539 378.

These courses are essentially collections of seminars, which vary from year to year. The aim of these courses is to develop mathematial culture in students who have special love for the subject.

# What do students say

## Charlotte Rabie, 2013

Foundations is the reason I love mathematics once again. It opened my eyes to a whole new mathematics I didn’t realize existed. It not only changed the way I see the mathematical world but it changed the way I approach obstacles in life. I would recommend this course not only mathematicians but for all.

## Mark Chimes, 2012

The coursework in Foundations in Abstract Mathematics is varied, and will interest those who enjoy mathematics. It is certainly a change from the constant stream of calculus and linear algebra taught in the traditional mathematics courses offered in second year at Stellenbosch University, and gives a much more rounded view of mathematics. It is also a lot more difficult. In the traditional courses, the student is taught a set of procedures. The difficulty in solving a problem lies in remembering the procedures and deciding which ones to use in each situation. In the Foundations course, learning the material is just the beginning; the student must then write or complete proofs for different theorems, or find non-obvious mathematical properties and examples. A lot of the time in the Foundations course will be spent simply thinking, or otherwise trying out different approaches, most of which will fail, until one works. This can be frustrating, since the results are less quickly visible than when simply applying known procedures. Nevertheless, it teaches skills which are essential for anyone wanting to pursue pure mathematics and discover new results of their own, and grants an understanding of mathematics which can help a great deal for those entering mathematically technical fields.

## 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.

## Pillip-Jan van Zyl, 2011

Foundation of Abstract 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.

## 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.

# 2013

## Seminars

### First order language and axiomatic set theory

by Dr Janelidze.

### Language, proofs, and introduction to structures

by Dr Janelidze.

### Real number system

by Professor Breuer.

### Sets and arithmetic

by Dr Janelidze.

### Problems from classical mathematics

by Professor Wagner.

## Students

Mr BERNDT,

Mr BREYTENBACH completed with distinction,

Ms CARSTENS,

Mr CHIMES complete with distinction,

Mr COETZEE,

Ms DU TOIT,

Ms FILLIES,

Ms FOURIE,

Mr HOEFNAGEL completed with distinction,

Mr LANG,

Ms LAUBSCHER completed with distinction,

Ms LE ROUX completed with distinction,

Ms MARX,

Ms MEKANARISHVILI,

Ms NIKOLOV completed with distinction,

Ms RABIE completed with distinction,

Ms VAN SCHALKWYK completed with distinction.

Mr VAN NIEKERK completed with distinction.

# 2012

## Seminars

### Discrete Structures

by Professor Wagner: I will aim to give an introduction to discrete (typically finite) structures such as partially ordered sets, lattices, permutations, tree structures, graphs, etc. We will be mostly dealing with structural and combinatorial properties and possibly touch on algorithmic aspects (depending on students’ interest). Potential topics include: inversions in permutations and the precise definition of a determinant, chains and antichains in partially ordered sets, stable marriages and related combinatorial problems, enumeration by bijections, …

### Elements of Mathematical Reasoning

by Professor Rewitzky: We will explore basic principles of mathematical reasoning and study the structure of the formal mathematical language. This is essential not only in the study of advanced topics in abstract mathematics, but also for a better understanding of some of the topics in more elementary mathematics, where students struggle usually because of not being able to `speak in the mathematical language.

### Introduction to Number Theory

by Professor Breuer: “Number theory is an area of mathematics which originates with the study of arithmetic of natural numbers. One of the charms of the subject is that it becomes possible to formulate seemingly simple problems which may turn out to be extremely difficult to solve. Attempts to solve these problems often lead to a beautiful synthesis of advanced algebra and geometry. We will touch on some of these aspects of modern number theory, as well as more classical topics from elementary number theory.

### Set Theory

by Dr Ouwehand: Even though the notion of set is very elementary, in some sense even more elementary than that of number, almost all of modern mathematics can be formulated in the language of set theory. In this course we will quickly review, at an intuitive level, the basic notions that form an essential part of every mathematician’s toolkit. It is well-known, however, that an intuitive approach to set theory can lead to contradiction. We will therefore proceed to study set theory from an axiomatic point of view, starting from the Zermelo-Fraenkel axioms.

### First Steps in Relative Incidence Geometry

by Dr Janelidze

### Categorical Algebra

by Dr Janelidze

## Students

Ms BLEKER,

Mr CHIMES completed with distinction,

Ms CILLIÉ completed with distinction,

Mr DE BEER,

Mr DREYER completed with distinction,

Mr HARRISON,

Mr HUGHES,

Ms KEYTER completed with distinction,

Mr LOUW completed with special distinction (100%),

Ms LE ROUX,

Ms MARX,

Mr OUKO,

Mr PATERSON,

Mr PRETORIUS,

Mr PRINS,

Ms RHODA,

Mr TLALI,

Mr VAN NIEKERK,

Mr VAN VUUREN.

# 2011

## Seminars

### Informal constructive set theory

by Dr Janelidze

### Introduction to group theory

by Professor Fransman, Dr Howell, and Dr Janelidze

## Students

Ms J R CHIPFAKACHA (rating: 2448) completed with distinction,

Ms M CUNLIFFE (rating: 195),

Mr J ENGELBRECHT (rating: 912),

Mr GOWER (rating: 150),

Ms D A GROBLER (rating: 26100) completed with special distinction (100%),

Ms M KEYTER (rating: 9702) completed with distinction,

Ms J KOTZEE (rating: 732),

Ms N MEKANARISHVILI (rating: 440),

Ms R T MYOLI (rating: 2422) completed with distinction,

Mr D A POPOV (rating: 156),

Mr R PRENTER (rating: 464),

Mr T TLALI (rating: 210),

Mr C N A TROCH (rating: 182),

Ms V VAN TONDER (rating: 1260) completed with distinction,

Mr P J VAN ZYL (rating: 9184) completed with distinction,

Mr Z N VILJOEN (rating: 825),

Mr T WEIGHILL (rating: 8320) completed with distinction.

Students who attended the lectures but were not officially registered for the course:

Mr CHIMES

Ms A ROBERTSON

Mr FK VAN NIEKERK