FAM at Stellenbosch University

What is FAM?

"FAM" is an acronym for an undergraduate course, "Foundations of Abstract Mathematics", offered by the Department of Mathematical Sciences of Stellenbosch University (FAM I and FAM II, respectively) since 2012/2011. The course consists of two year-long modules, FAM I (Mathematics 278) and FAM II (Mathematics 378), offered at the second and the third year, respectively. It is possible to enroll for only one of the two modules. Neither of the modules has any prerequisites, although note that admission to the third-year module is subject to approval by the Department of Mathematical Sciences. The course aims to let the students experience mathematical research, at the level corresponding to students' mathematical skills, and in this process, to uplift those skills.

A bit of history

When teaching a course in calculus at the University of Cape Town in 2008, Zurab Janelidze was approached by a student, Pieter du Toit, with a request to show him mathematics beyond what was taught in the course. This led to a Seminar in Pure Mathematics, which was run by Zurab and attended by Pieter and some of his friends. When Zurab was appointed as a lecturer at Stellenbosch University in 2009, he offered Ingrid Rewitzky and Karin Howell to jointly run a seminar, based on the model of the Seminar in Pure Mathematics, for undergraduate students. The enthusiasm of students attending this seminar led to the conception of the Foundations of Abstract Mathematics modules, based on a framework proposed by Ingrid and Zurab. Thanks to the support of the Head of the Mathematics Division at the time, David Holgate, the third-year module was introduced in 2011, and the second-year module in 2012. The Faculty approved this initiative under the understanding that lecturers involved in teaching these modules would do so in their free time instead of it being part of their official teaching workload. Teaching of these modules became part of the official teaching load by the initiative of the new Division Head, Florian Breuer, a few years after the introduction of the modules. He, Zurab, and Stephan Wagner were the main lecturers in the module for several years, until the resignation of Florian and Stephan (who took up academic positions in Australia and Sweden, respectively). During these years, the module was convened by Zurab. In 2023, the module is jointly convened by Zurab and Sophie Marques. A majority of postgraduate students in Mathematics at Stellenbosch University, who have completed their undergraduate studies at SU, would have followed one or both of the FAM modules during their undergraduate studies. For some of them, these modules played a crucial role in inspiring them to switch their existing career choice (e.g., engineering, computer science, theoretical physics) to mathematics. There are examples where ideas conceived in FAM I and FAM II have led to research topics at the Honors, Masters and PhD levels. 

FAM assignments

Assessment in FAM is mainly based on assignments, where students need to apply their creativity to solve problems not discussed in class (or tutorial), compose proofs of theorems, sometimes in a symbolic language (in a formal proof system), come up with examples or counterexamples to a concept/hypothesis, or come up with their own theorems (and their proofs). Occasionally, some of these assignments involve coding proofs using a proof assistant software. Students who show readiness for engaging with a mathematical exploration project (a longer version of the usual assignment), which often comes close to an honors project in terms of its quality and content, are given the opportunity to engage with a longer project in the place of smaller assignments. 

The following is a fragment of a conversation among a group of students who were required to append this recording, discussing their semi-joint assignment, to the latter (rewind to play the full video - but its long!). Note that the recording is shared with the permission of all students in the video.

How do students find the experience?

The video above ends with the group members discussing how FAM has impacted them. Here is what some of the other students have said when asked what they have learned in the first quarter of the FAM module in 2021 (once again, shared under permission):

• Reading and understanding equations. We knew about logical operators, but now we know how to use them more effectively to get actual results. This equips one with a toolbox to use in other math modules. Reading and interpreting equations in other modules specifically.
• A philosophical look at mathematics: instead of being given a problem and asked to solve it, now we look at the mechanics of how we can solve the problem and what really encompasses mathematical activity. Comparison with language is fascinating. It is a good life skill to understand logic, which this term contributed to.
• Originally, I thought of this like every other math course: numbers and calculations. Now I view this course more as a course in logic which teaches you how to think. This was very cool, very unlike to what I have done before. Excellent pacing: it was important not to go fast to get a good understanding of what we are working on.
• This term gave me a deeper understanding of mathematics - it was not just about learning a method and solving problems. It was nice that in the beginning more emphasis was placed on effort rather than accuracy. Instead of trying to get it right, one had the opportunity to engage deeper and learn more about the subject, than in other modules where the emphasis is to learn something to get it right. In this module, you learn to understand. The focus was more on understanding concepts rather than grasping the language used to interpret the concept.
• Usually, the student is on the receiving end - now it is the student who was expected to produce a precise mathematical statement that others would be able to interpret correctly.
• The concept of breaking things down and unpacking in proofs. A cool skill to learn. Mathematics is neither invented or discovered. Mathematics is rather something that is within every human being.
• How anything can be turned into math. Mathematics can be made from a normal conversation. How to write down logical reasoning through mathematical steps.
• How mathematics is really so broad around us. I kind of new this, but I did not realise the actual broad extent if this.
• The seminar does not force you to parrot learn - it is much more understanding based. It is a nice thing that the focus is on understanding the work.
• Instead of repetitive information, the lecturer gives us information and lets us build on it while learning from each other. I wish other modules were like that too.
• This is probably the only course that brings thought into it. After the lecture, instead of being happy that the lecture is done, you are still thinking about the lecture. Assessments reflect this too. Putting in extra thought and creativity gives you marks. So assessments allow thought input. You also have the freedom to interpret things in your own way.
• This module teaches you how to formulate your thoughts and structure them in terms of assumptions and conclusions. You must think carefully and understand the process, rather than go through everything step by step or parrot learning, as is often the case in other math modules.

See https://foabma.blogspot.com/ for further feedback.

The Abstract Mathematics stream

A few years ago, FAM has inspired creation of a new focal area (aka stream) within the BSc Mathematical Sciences programme. It features FAM I and II as compulsory modules, among other mathematics modules at the second and third year levels. This stream gives students flexibility to combine a mathematics major with majors in biochemistry, chemistry, physics, genetics, computer science, applied mathematics, and mathematical statistics (only four major combinations are possible in all other focal areas of the programme).  

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The Kindest Ending (Chapter 1)

The Kindest Ending

A novel by 

Natia Kuparadze and Zurab Janelidze

 Chapter 1. Small Wooden House

The morning was pleasantly cold and fragrant. The slightly painted balcony door gently let the spring sunlight into the bedroom, which slowly crept towards the middle of the room. Just a little, and the glittering rays would gently touch and caress the sleeping Noah's face. Noah loved his morning naps, when half-awake he convinced himself he was dreaming of a colorful world of his own making. These were magical moments that became more and more beautiful the closer they got to being given up. The rays of the sun flashed on his face. 

Noah shook a little, then remained motionless, until finally he stretched his hands and opened his eyes. For a while he looked at the space of the room, as if he had not returned from the world of dreams. 

He got up. 

What would today be like? The mind's eye went over the day’s possible developments. He sorted them in his mind, then changed them over and over a few times... he returned to some thoughts several times. While in his thoughts, he walked barefoot in the room, went out to the balcony and greedily inhaled the scent of the spring morning. He then felt a rush of energy, which seems to have helped him to decide firmly to do today what he had been preparing for for a very long time. Indeed, for a very long time… 

Noah rushed to put on his favorite blue jeans and a blue T-shirt, grabbed his backpack and ran down the stairs before you could find him walking quickly along the street that sparked with sunlight, until Noah’s figure dissolved in the morning silence.

Noah was a young writer. It can be said that he spent most of his childhood free time writing. Although he had many good friends and was the best student in school, he did not enjoy spending time with friends or studying as much as putting his fictional stories on paper. In his early childhood, before he knew how to write, he used to tell his invented stories to his family and friends.

"Great storyteller", that's how his beloved grandmother called him. 

She listened attentively to his stories and showed more interest in his grandson's passion than anyone else. She used to ask him many questions about his stories. These questions gave rise to new and new ideas in Noah, and often several different endings appeared for the same story. 

"Noah, now tell me, which of these endings is the true ending of your story?" Grandma used to ask. Noah used to answer: 

"Grandma, whichever you like more". Then grandma would say, 

"I prefer the kindest ending…" and then make her choice. 

The title of Noah's first published collection of short stories was "The Kindest Ending". It consisted of several stories that had the same beginning but different endings. In the preface, the author warned the reader that he himself should choose the kindest ending. Readers seemed to have learned to enjoy this unexpected freedom very much. The collection of Noah's stories soon gained great recognition. Unfortunately, the grandmother did not live to witness this recognition. However, in the depths of his heart, Noah firmly felt that his grandmother was following his successful career from a place that was not far at all, a world that was much more interesting and magical than the world in which Noah had still a very long time to spend before he would get a chance to meet his grandmother again… but of course, during the course of this time, he did meet with her many times, in his dreams while asleep… and there, she continued to be a source of inspiration for many of the ideas in his stories.

Often, when a new idea came to him while in a discussion with his grandmother, he would say "Thank you, grandma!". It was these words that were running through his head when today, he walked quickly down the street that sparkled with sunlight and dissolved into the morning silence. Nobody else but him and his grandmother knew what Noah was up to on this day, but you and I will find out too, soon enough, dear reader.

The quiet day was slowly turning into a boisterous one, preparing Noah for a day… full of miracles?!

A small wooden house stood on the outskirts of the city, in solemnity and silence. At first glance, it might not even have caught your eye, so much so that it was embedded in the picture of nature. It was as if it had grown into a row of trees and a colorful garden that surrounded it. 

But an observant eye, especially one that had seen a lot of the bright colors of the Magical World, would surely be attracted to a small wooden house with a beautiful porch, old windows, wooden stairs, a wonderfully manicured garden, where many colorful flowers bloomed and indicated the flawless love and care of their caretaker. The observant eye would definitely notice the path paved with small stones that led to the house and would look at the beautiful lanterns overhead. The observant eye would notice too that in the middle of the garden, there was an old swing that was swayed by the wind.

Every detail of the house and its garden emphasized the refined, deep and tasteful nature of its owner. It was a kind of fairy tale world that summoned you… it would call you and make you think, as if it were telling you an old good tale, where you were brought back to your childhood and got to be reminded of the mostly forgotten, warmest and dearest things to you. These were the feelings Noah had when he approached the house. He stood motionless for a long time, almost out of breath and excited. He felt and read every nook and cranny of the house with his eyes. But he did not yet dare to enter and only looked into its depths through the old cracked fence.

He stood like this for a while… then, thoughts involuntarily brought him back to his grandmother. He clearly imagined that bright and sunny day when he would sit down with her on the staircase of the house. Grandmother was humming a beautiful melody in a low voice while stroking Noah's head. How vivid this memory was in his imagination! It was as if he could still feel the warmth of her beloved grandmother's hands on his head and a pleasant tingle went through the back of his head. Well... it was that very day, who can forget it?! That very day when grandma confided in Noah her unbelievable story…

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Lesson 5 in Perceptive Mathematics: union of sets

If you love collecting shells from a beach, you surely have at your home shells collected from several different beaches. 

Let's say you have shells from three different beaches. Let B1 be the set of shells collected from the first beach, B2 the set of shells collected from the second beach, and B3 the set of shells collected from the third beach. Altogether, your collecting activity can be recorded by the set A={B1, B2, B3}. However, when you bring those shells at home, you will most likely mix them all up in one heap of shells. So if for instance, B1={S1, S2, S3}, B2={S4, S5} and B3={S6, S7, S8, S9}, where the symbols S1, ...., S9 represent the collected shells, then your collection at home, after the shells have been placed all together, is given by the set C={S1, S2, ..., S9}. Now, more generally, the union of a set A is the set C of all elements of individual elements of A. We write the union of A as UA, thus C=UA.

Exploration 5.1. Compute the following unions (i.e., write down what set they equal to by explicitly listing the elements enclosed between braces).  

• U{}=?
• U{{}}=?
• U{{{}}}=?
• U{{},{A}}=?
• U{{A, B}, {A, {}}, {C}}=?
• U{{A, B}, {A, C}, {{A, C, {}}}}=?

The union of a set can be seen as a reverse construction both to the singleton construction as well as the quotient construction. If we begin with a set A, then take its singleton, {A}, and then take the union, U{A}, we get back the same set A. Indeed, the elements of U{A} must be elements of elements of {A}. But {A} has only one element, it is A. So elements of U{A} are the same as elements of A, which means that the two sets are the same set. Similarly, if Q is a quotient of the set A, then elements of UQ are all elements of classes formed during partitioning. We know by the first axiom of partitioning that these elements must be elements of A. Moreover, by the second axiom of partitioning, no element of A will be missed (every element of A is an element of some class, i.e., of some element of Q). So the sets UQ and A are once again the same set. Intuitively, if partitioning separates a set into pieces, the union puts these pieces back together.

Exploration 5.2. 

• Draw several different tree diagrams for sets, ensuring in each case that every element of the set being considered is itself a set. Then, for each of these drawings, draw the tree diagram for the union of the corresponding set. 
• While busy with the task above, did you notice that the union construction shrinks the height of the tree? In contrast, the singleton and the quotient constructions grow the trees (make a few sketches to get convinced of this). 

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Lesson 4 in Perceptive Mathematics: tree diagrams for sets

 A set can be pictured as a tree. 

We describe this by examples. Consider for instance the set {A, B, {A, C, D, E}}. It has three elements, A, B, and the set {A, C, D, E}, which in turn has four elements, A, C, D, and E. Let us call this set X, so X={A, B, {A, C, D, E}}. Note that by coincidence, X and one of its elements, namely, {A, C, D, E}, share a common element A. This is permissible and it does not mean that there are two copies of A -- the A that is an element of X is the same A that is an element of X. Of course, in notation, we are forced to write out the set with two copies of A, but the two are supposed to represent the same object A. Now, to represent the set X as a tree, we start X and branch it out in as many elements as X has. So there will be three branches going out of X. The first two branches terminate with labels A and B, respectively. The third branch branches out further into four branches, which terminate with labels A, C, D, and E. As a result, we get the following "tree":

If one of A, B, C, D, E in this example is itself a set, we can expand the diagram further. For instance, if A={C, D}, then we get the tree

Exploration 4.1. How would you represent a singleton and the empty set as a tree? Using this insight, represent the following sets as trees:

• {{}}
• {{{}}}
• {{},{{}}}
• {{A, {{}, B}}}

Note that representing a set as a tree introduces even more redundant information than in the symbolic notation of sets. First, let's list some redundancies in the symbolic notation:

• Repetition of elements: the set {A} is the same set as the set {A, A} since both sets have exactly one element, the object A (recall that just because we mention a symbol twice does not mean it has two separate meanings).
• Order of elements: the set {A, B} is the same set as the set {B, A} because both sets have the same elements.

With the tree notation, in addition to the redundancies inherited from the symbolic notation, there is also a redundancy of the shape of the drawing, such as the length of branches, for instance. For instance, the last diagram above represents the same set as the following one:

Exploration 4.2. Represent as trees each of the quotients of the set X from the example above.

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Lesson 3 in Perceptive Mathematics: singletons

A singleton is a set having exactly one element. For instance, {R} is a singleton, where R is a rose. Now, we do want to think of a rose R as a different object to the singleton {R}. These objects even have different types. The first is a plant, and the second is a set. Here is an illustrative analogy: a rose is not the same thing as as a vase with a rose in it.

The empty set {} is not a singleton, since it does not have any elements. Neither is a set having two or more different elements a singleton. We can turn any object B into a singleton, by considering the set having B as its unique element, i.e., the set {B}. What happens when B={}? In notation, we get {B}={{}}. Now, what is the set {{}}? Is it still the empty set? Let us see. The empty set does not have any elements. What about the set {{}}? Does it have any elements? Well, this set is a singleton (remember, it is equal to {B}, where B={}), so yes, it has exactly one element, that element being the empty set B={}. So, since {} has no elements and {{}} has an element, these two sets cannot be the same set (if they were the same set, they would have the same elements).

Exploration 3.1. Decide whether the set {{{}}} is the same set as the set {{}}.

Consider again the set {R}. Is there a way to partition this set? Recall that during partitioning we arrange elements of a set in classes. Since a class cannot be empty (the third axiom for partitioning), there is only one class that can be formed, the class consisting of the unique element of the given set, so it is the class {R}. What is the corresponding quotient? Well, a quotient is the set of all classes formed during partitioning. Since there is only one class that can be formed, we get that the quotient is the singleton {{R}} (elements of the partition must be the classes). So if V={R} then the singleton of V is its quotient {V}={{R}}, which happens to be the only quotient that V={R} has.

Note that the singleton {R} and its quotient {{R}} are different sets, since, they do have different elements. The element of the first singleton is R, the rose, while the element of the second singleton is the set {R}, and we already agreed that R and {R} are different from each other. 

Exploration 3.2. How many quotients does the empty set have? Is the quotient of the empty set {} the singleton {{}}? Revisit the axioms for partitioning to get help with answering this question.

Here is another way of understanding singletons and sets in general. The thought of a rose is different from the actual rose. The thought of the thought of a rose is different from both. Every time we add "the thought of" we can interpret this as a process of taking a singleton:

• R is the actual rose
• {R} is the thought of a rose
• {{R}} is the thought of the thought of a rose, etc. 

In this analogy, the empty set can be interpreted as the thought of "nothing". Nothing does not exist, but the thought of nothing does.

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Lesson 2 in Perceptive Mathematics: quotient of a set

Elements in a set do not have to have any similar features. For instance, we can have a set {A1, A2, M1, M2, M3} where A1, A2 are apples and M1, M2, M3 are mountains. When we have a set whose elements are of different types, we often like to sort these elements by grouping elements of the same type together. Thus, for instance, in the set {A1, A2, M1, M2, M3} we may wish to group the apples together into one set {A1, A2} and group the mountains together into another set {M1, M2, M3}. This process is called a partitioning of a set, and the groupings we get as a result of partitioning are called classes. Note that the classes are themselves sets, such as the class {A1, A2} and the class {M1, M2, M3} for the partitioning just discussed.

Exploration 2.1. Come up with real-life examples of partitioning and in each case, describe what the classes are. Here is a picture for inspiration:

In the example we discussed, can we partition {A1, A2, M1, M2, M3} into the classes {A1, A2, M1} and {M2, M3}? From the first look, it does not seem like apples and the mountain are of the same type, right? But of course, it very much depends on the rule of sorting. For instance, the apples A1, A2, and M1 can come from one country, while M2 and M3 can come from another country. Then if we sort according to which country these entities come from, we will get indeed the classes {A1, A2, M1} and {M2, M3}. In general, partitioning does not restrict the rule of sorting in any way, as long as the classes fulfill the following axioms ("axiom" is a scientific word for a "rule"):

• Every element of each class must be an element in the set that is being partitioned.
• Every element in the set that is being partitioned must belong to exactly one class.
• A class of a partition cannot be the empty set.

We can organize the classes created during partitioning into one set: the elements of this new set are all classes created during the partitioning of the old set. This new set is called a quotient (or a partition) of the old set. For instance, for the partitioning of {A1, A2, M1, M2, M3} where the classes are {A1, A2} and {M1, M2, M3}, the quotient is {{A1, A2}, {M1, M2, M3}}. In our second example of partitioning, where the classes were {A1, A2, M1} and {M2, M3}, the corresponding quotient is {{A1, A2, M1}, {M2, M3}}. Thus, both {{A1, A2}, {M1, M2, M3}} and {{A1, A2, M1}, {M2, M3}} are quotients of the set {A1, A2, M1, M2, M3}.

Exploration 2.2. List all possible quotients of the set {A, B, C}. Make sure that the partitioning used for each quotient satisfies the three axioms stated above. Furthermore, to ensure that your list does contain all possible quotients of {A, B, C}, verify that every possible outcome of partitioning that satisfies these three axioms has been considered. How many quotients does the set {A, B} have?

Here are some notes on terminology. The terms "partition" and "quotient" are interchangeable. Note that in the English language, "partition" is both a verb and a noun, while "quotient" is only a noun (so we will never say "quotienting"). The quotient of a set is often called a quotient set, to distinguish it from the quotient of numbers.

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Lesson 1 in Perceptive Mathematics: the concept of a set

 Look at this photo of eight apples.

How many apples are there in the photo? Eight. How many photos are there? One. Several objects or entities seen as one thing is called a set. The objects that make up the set are called its elements. Unlike in a photo, the arrangement of these objects among each other, or any information about the elements of the set apart from knowing what these elements are, is not considered to be part of the information about the set. Thus, for instance, the set of the eight apples that we see in the photo would be the same as the set of the same apples shown in a different photo, where these apples have been rearranged.

If A1, A2, ..., A8 are symbolic representations of the apples shown in the photo, then {A1, A2, A3, A4, A5, A6, A7, A8} is how we symbolically represent the set of those apples. 

Exploration 1.1. According to your interpretation of the definition of a set given above, which of the following sets should be the same set as the set {A1, A2, A3, A4, A5, A6, A7, A8}?

• {A1, A3, A2, A4, A5, A6, A7, A8}
• {A1, A2, A3, A4, A5, A6, A7}
• {A1, A2, A3, A4, A4, A6, A7, A8}
• {A1, A2, A3, A4, A4, A5, A6, A7, A8}

Just like we could add apples to a photo, or take some away, we can add elements to a set, or take them away. For instance, let us start with the set {A1, A2, A3}. If we take A3 away from this set, we get the set {A1, A2}. What happens if we take away A1 and A2 as well? Can we do that? Well, taking away A2 gives us {A1}, and taking away A1 from this set should perhaps give us {}? Such a set, that is, a set having no elements, is also allowed to be a set. It is called the empty set.

Exploration 1.2. Start with a set of three apples and a set of three mountains (represent the mountains by M1, M2, M3). Take away one element from each set, and do this three times. Will the resulting sets be the same set or will they be different sets? More generally, is there only one empty set, or are there many empty sets?

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On the structure of information

A well-known metaphor: a house is not a mere collection of bricks. It is, rather, a collection of bricks that has been organized in a certain structure. Organizing information into structure seems to be something our brains are good at. What if this organization is the means by which our brain comprehends, stores, and transmits information? In other words, the structure of information is all there is. "Meaning" may simply be certain types of structure of information that our brain distinguishes from others. When this distinction occurs, the brain provides us with emotional impulses, which creates this sensation of "aha, that is quite deep". If something like this is true, then the brain ought to be programmed to recognize structure in a way that is synchronized with the structure of the universe surrounding us, since those "aha" moments led us to a point where we can make predictions about nature, communicate over a large distance, etc. 

It is difficult to test out such a theory since perceived information is usually matched with existing knowledge in our subconscious... so it is difficult to isolate a describable portion of self-contained information, which would be necessary for a rigorous study of how the structure of information determines the meaning. Except perhaps in an art form, where the "meaning" is least dependent on existing knowledge, such as music.

Here is my (almost) first attempt at the study of how a meaning of a musical piece could be interpreted via the structure of the organization of its sounds. I improvised this short piece: 

Zurab's Music Channel · 1805

And then isolated various layers of its musical structure:

Each colored dot in the bottom layer corresponds to a half bar. Here is the score, for reference (bars are numbered):

The colors encode musical similarities. For instance, notice that the fourth and the fifth bars both have a C chord in an extended half note. This is marked by the fact that the seventh and the ninth dots (from left) both have the same red color. The higher layers are combinations of half-bars again categorized according to the similarity of their musical structure. Now, my hypothesis is that when listening to this musical piece our brain generates (partially subconsciously) the structure displayed in the image above. The mere possibility of, and the easiness by which the brain generates this structure gives us the illusion of "meaning".

Mathematically, the structure we are talking about here can be seen as a collection of subsets of a partially ordered set (poset) of "pieces" of given information. In the example above, this would be the poset of intervals of the musical piece. In the picture, these intervals are continuous bars. The set of bars of the same color constitutes one subset. What is extraordinary in this example is the organization of these subsets into partitions of the entire piece (one partition for each line of bars in the picture, so four altogether). Another interesting phenomenon is that bars of the same color always occur in the same line. I do not know to what extent these rules are universal to musical compositions (of a certain type?). 

As for applications of the study of the structure of information, well, if "meaning" can be reduced to "structure", then by embedding structure into artificial intelligence, we should be able to produce a machine that is more human-like. This feels scary, I know, but I hope that my theory has a sufficient amount of flaws that it will not bring us closer to the terminator judgment day anytime soon. 

Another possible application is in education: by identifying and emphasizing the structure in learning, the brain of a learner may be able to acquire the skill/knowledge more efficiently.

And perhaps, there can be applications in psychology too, where structure can be a key in helping a brain make sense of life experiences...

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