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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Escape (opus 1801)

When the world around us gets too difficult to bear, the mind escapes to a different world where a rainbow is all it takes to brighten up the day. And yes there will be a river of tears, but also a bridge will rise over it. Beyond the bridge, beyond the rainbow, there are bright yellow planes. But you decide to stay where you are since the tears have made the ground on which you stand fertile.

NFT

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2022 Academic Activities

Summary

• Initiated seven new collaborative research projects within the Mathematical Structures research programme, that includes researchers and postgraduate students from various universities in South Africa: operator semigroups, measure structures, metric frames, canonical extensions, ranked monoids, sum structures, lower topology.
• Supervised and co-supervised nine postgraduate students (two honors, two masters, and five phd).
• Represented South Africa at the General Assembly of the International Mathematical Union along with a colleague in Mathematics Education.
• In collaboration with colleagues and students, developed and delivered a successful math-music theatrical production for the celebration of the International Year of Basic Sciences for Sustainable Development. The production was supported by NITheCS, ASSAf and DSI.
• Developed and delivered four national postgraduate courses online: SOFiA on python, mathematical structures (in collaboration), introductory set theory (in collaboration), category theory.
• Executed presidential duties for SAMS: chairing of SAMS council meetings, of the AGM, opening and closure of the SAMS congress, etc. Prepared and delivered presidential address at the AGM (in consultation with the SAMS Council) to give a direction to SAMS activities in the coming years.
• Elected as NITheCS associate co-representative, and in this role, served on the NITheCS management committee monthly meetings.
• Ran the national research programme in mathematical structures under NITheCS along with three other principal investigators in the research programme.
• Two co-authored papers published, one in Journal of Symbolic Logic. Co-authored paper in Order accepted for publication.
• Served on the programme committee of the international conference "Topology, algebra and categories in logic" held in Coimbra, Portugal.
• Gave two interviews (radio and youtube).
• Taught and co-taught and/or convened six modules at Stellenbosch University, including two engineering mathematics modules, one honors module and two third-year modules.
• Progress made on existing and new research projects and delivered talks on those.
• Carried out duties in the role as mathematical sciences programme coordinator and member of a university research committee.
• Carried out refereeing and editorial duties (not listed below).

November-December 2022

• Research discussion (9 December) with Dr. Christian Budde: started research project on the category theory of operator semigroups.
• Chaired the Annual General Meeting of the South African Mathematical Society (8 December).
• Gave a SAMS Congress talk on the noetherian form of sets.
• Gave opening and closing speeches at the 65th Congress of the South African Mathematical Society (6-8 December), held at Stellenbosch University. 
• Gave an opening speech at the special meeting of the Mathematics section of National Graduate Academy (5 December).
• Chaired the fourth Council Meeting of the South African Mathematical Society (2 December).
• Conducted weekly 6-hour tutorial sessions in November for students in Foundations of Abstract Mathematics I for additional assessment opportunity.
• The paper on ordinal number systems fully published in the Journal of Symbolic Logic.
• Submitted author comments on the journal proofs of the paper on stack combinatorics (joint work with Helmut Prodinger and Francois van Niekerk). The paper is being published by Springer Order.
• Hosted research visit (18-20 November) of Dr. Cerene Rathilal. Started joint work on measure structures.
• Submitted a report on the Mathematical Structures Research Programme at NITheCS and delivered a talk at the NITheCS Associates Workshop on the progress of the research programme.
• Made progress with Kishan Dayaram on diagram lemmas in the context of noetherian forms.
• Fundamano production (4 November) was a success -- full house attendance and well received. See: videos, press release.

September-October 2022

• Gave a talk on at the "Topology, Algebra, and Category Theory" international conference (19-22 September) dedicated to the 65th birthday of Themba Dube. The subject of the talk was metric frames.
• Supervised original honors projects of Gregor Feierabend and Gideo Joubert. 
• Gave a semester honors course on Logic.
• Taught the English group of Engineering Mathematics 242 in the second semester of 2022.
• Chaired the third Council Meeting of the South African Mathematical Society (7 October).
• Hosted research visit (20 September - 8 October) of my PhD student, Noluntu Baart, to work on deductive reasoning in intermediate-phase mathematics education.
• Hosted research visit (9 October - 9 December) of my PhD student, Kishan Dayaram, to make progress on three joint papers.
• Hosted research visit (9-26 October) of Dr. Partha Pratim Ghosh. Joint work on canonical extensions started.
• Rehearsed and prepared for the Fundamano production in a team of students. This is a theatrical production bringing mathematics on stage, celebrating the international year for basic sciences.
• Drafted a paper based on the research on the category of near-vector spaces (co-authored with my MSc student, Daniella Moore, and the co-supervisor, Dr. Sophie Marques).
• Gave a National Graduate Academy course on category theory. Click here for videos and lecture notes.
• Gave a South African Theory School course on mathematical structures (jointly with Dr. Cerene Rathilal and Dr. Partha Pratim Ghosh). Click here for videos and lecture notes.
• Spoke on "Is Maths Trauma a real thing?" at the radio show Weekend Breakfast with Refiloe Mpakanyane. Click here for the podcast.

July-August 2022

• Organised a Research Workshop (5 July) on the occasion of visit (5 July) of Dr. Francois Schulz. Collaboration started on ranked monoids.
• Organised a Research Workshop (14 July) on the occasion of the research visit of Prof. Dharmanand Baboolal and Dr. Cerene Rathilal. Collaboration started on metric frames.
• Represented South Africa at the General Assembly of the International Mathematical Union (July 3-4, the report of the meeting is available here). 
• Gave the August NITheCS mini-school on Elementary Introduction to Set Theory together with Dr. Amartya Goswami.
• Gave a Foundations of Abstract Mathematics I seminar on arithmetic and proof composition.
• Started research on the category of near-vector spaces (joint work with Dr. Sophie Marques and Daniella Moore).
• Leading programme renewal discussions in Mathematics in the second semester of 2022.

May-June 2022

• The paper on matrix taxonomy was published in Theory and Applications of Categories.
• Hosted research visit (1-4 June) of Dr. Charles Msipha to advance progress on sum structures.
• Continued research on a noetherian form of sets -- see the updated paper.
• Chaired the second Council Meeting of the South African Mathematical Society (26 May).
• Prepared an International Year for Basic Sciences for Sustainable Development project, which would later be called Fundamano. The project is listed on the official website of this international initiative. Dr. Charles Msipha and Dr. Sophie Marques are co-founders of the project.
• Elected as a NITheCS Associate Representative. Duties include serving on the NITheCS Management Committee (meetings are held monthly).
• Served on the programme committee of the international conference "Topology, algebra and categories in logic" held in Coimbra, Portugal.

March-April 2022

• Revisited research on a noetherian form of sets (joint work with Dr. Francois van Niekerk).
• Organised a Research Workshop on Monoidal Sum Structures at Stellenbosch University (20-25 March) and hosted the visit of Dr. Charles Msipha (Tshwane University of Technology). See the Mathematical Structures Research Programme website for further information. Two research projects dealing with sum structures were initiated at this workshop.
• Organised a Research Workshop on Lower Topology at Stellenbosch University (3-10 April) and hosted the visit of Dr. Amartya Goswami and Ms. Micheala Hoenselaar (University of Johannesburg). A research project on lower topology was initiated at this workshop.
• Gave an interview at the Meet a Mathematician series (see https://youtu.be/lOLIc8Jnja4).
• Supervised a 3rd year research project by Jean du Plessis (under Foundations of Abstract Mathematics II).

January-February 2022

• Serving on the Subcommittee B of the Research Committee of Stellenbosch University for 2022.
• Serving on the Programme Committee of the Faculty of Science of Stellenbosch University for 2022.
• Setting up Mathematical Structures Research Programme at the National Institute for Theoretical and Computational Sciences, along with Prof. Yorick Hardy, Dr. Partha Pratim Ghosh, and Dr. Cerene Rathilal.
• Delivered online lecture series Python-Based Introduction to Mathematical Proofs for the The 12th CHPC Introductory Programming School and The 4th NITheCS Summer School on the Foundations of Theoretical and Computational Science.
• Teaching Engineering Mathematics 214 (together with Dr. Liam Baker, Dr. Ronalda Benjamin, and Dr. Michael Hoefnagel) in the first semester and giving a Foundations of Abstract Mathematics I seminar in Mathematical Reasoning in the first term. Also teaching a third-year module, Topology, in the first semester. 
• Convening Foundations of Abstract Mathematics I & II (year modules) and Topology (semester module) in 2022.
• Started/resumed (co-)supervision of the following postgraduate students: Noluntu Baart (PhD), Roy Ferguson (MSc), Kishan Dayaram (PhD), Paul Hugo (PhD), Brandon Laing (PhD), Daniella Moore (MSc), Ineke van der Berg (PhD).
• The paper on ordinal number systems appeared online in the Journal of Symbolic Logic (joint work with Ineke van der Berg).
• Assumed the role of the President of the South African Mathematical Society for the term 2022-2023. Chaired the first Council meeting (11 Feb).
• Under the research assistantship of Gregor Feierabend, the first prototype of a Haskell implementation of the SOFiA proof assistant was produced. See source code on GitHub or the live software.

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Fundamano

 

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Resilience (Opus 1015)

This piece is to remind you of resilience, or toughness. Life is not a straight path and there come moments when the best you can do is endure. This requires bring out the fighter within you. It may also require you to stay focused. 

Look out for the following objects in the video, which carry the symbolism as described below:

• Light sources (light bulbs, lanterns, etc.): ideas that could help you get through the difficult times
• Switched off TV screen with headphones over it: the feeling of emptiness
• Guns, glasses and the helmet: self-defense mechanisms 
• Male and Female characters: your body (male character) and your soul (female character)

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The Proof Course: Lecture 2

Many real-life situations lead us to considering a mathematical problem dealing with finding all possible numbers \(x\) satisfying a certain formula. In most primitive cases, this formula is an equation involving basic arithmetic operations (like the one we considered in Lecture 1). As an example of a formula that does not fall in this category, consider the following one:

\(x<y^2\) for every value of \(y\) (Formula A)

In other words, the formula expresses the property that no matter what value of \(y\) we pick, we will always have \(x<y^2\). Let us write this purely symbolically as follows (so that it looks more like a formula!):

\(y\Rightarrow x<y^2\) (symbolic form of Formula A)

In general, the symbol "\(\Rightarrow\)" describes logical implication of statements. Here the implication is: if \(y\) has a specific value then \(x<y^2\). In the symbolic form above, the assumption that \(y\) has a specific value is expressed by just writing \(y\) on the LHS (left-hand-side) of the implication symbol "\(\Rightarrow\)". Since we are not giving any further detail as to which specific value does \(y\) have, the implication must not be dependent on such detail, and hence the RHS (right-hand-side), \(x<y^2\), must hold for all values of \(y\). Note however that this type of symbolic forms, where variables are allowed to be written on their own like in the LHS of the implication symbol above, is not a standard practice. We will nevertheless stick to it, as it makes understanding proofs easier. 

So, what is the solution of Formula A? If \(x<y^2\) needs to hold for every value of \(y\), then in particular, it must hold for \(y=0\), giving us \(x<0^2=0\). This can be written out purely symbolically, as a proof:

1. \(y\Rightarrow x<y^2\)
2. \(x<0^2\)
3. \(x<0\)

However, as we know from Lecture 1 already, this proof only proves that if Formula A is true then \(x<0\). In order for \(x<0\) to be the solution of Formula A, we also need to prove that if \(x<0\) then Formula A is true. Well, since \(0\leqslant y^2\) is true for every \(y\), combining \(x<0\) with \(0\leqslant y^2\) we will get \(x<y^2\), as required in Formula A. So the proof is:

1. \(x<0\)
2. \(y\Rightarrow 0\leqslant y^2\)
3. \(y\Rightarrow x<y^2\)

Note that it seems as if this proof violates our requirement that in a basic proof, every line except the first one must be a logical conclusion of the previous one or several lines. Line 2 does not necessarily seem to be a conclusion of Line 1. Instead, it is simply a general true fact that does not seem to logically depend on Line 1 at all: it says that the square of every number is greater or equal to \(0\). We can account for such situations by agreeing that "several" in "one or several lines" includes the case of "\(0\) many". So in a basic proof we can also include lines that recall facts we know. If we had not done that in the above proof, we would have to skip from Line 1 directly to Line 3, and it may not have been so clear how does one logically conclude Line 3 from Line 1. So we allow inclusion of known facts as lines in a basic proof for the sake of clarity. Knowing this, we might want to make the first proof clearer by inserting one such line:

1. \(y\Rightarrow x<y^2\)
2. \(x<0^2\)
3. \(0^2=0\)
4. \(x<0\)

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The Proof Course: Lecture 1

In this blog-based lecture course we will learn how to build mathematical proofs.

Let us begin with something simple. You are most likely familiar with "solving an equation". You are given an "equation", say \[x+2=2x-3\] with an "unknown" number \(x\) and you need to find all possible values of \(x\), so that the equation holds true. You then follow a certain process of creating new equations from the given one until you reach the solution: \[2+3=2x-x\] \[5=x\] This computation is in fact an example of a proof. To be more precise, there are two proofs here: one for proving that

if \(x+2=2x-3\) then \(x=5\) (Proposition A),

and the other proving that

if \(x=5\) then \(x+2=2x-3\) (Proposition B).

The first proof is the same as the series of equations above. The second proof is still the same series, but in reverse direction. The two Propositions A and B together guarantee that not only \(x=5\) fulfills the original equation (Proposition B), but that there is no other value of \(x\) that would fulfill the same equation (Proposition A). It is because of the presence of these two proofs in our computation that we can be sure that \(x=5\) is indeed the solution of the equation \(x+2=2x-3\).

In general, a proof is a series of mathematical formulas, like the equations above. However, in addition to a "vertical" structure of a proof, where each line displays a formula that has been derived from one or more previous lines, there is also a "horizontal" structure, where each line of a proof has a certain horizontal offset. This is, at least, according to a certain proof calculus formulated by someone by the name of Fitch. There are other ways of defining/describing proofs; in fact, there is an entire subject of proof theory, which studies these other ways. We will care little about those other ways and stick to the one we started describing, as it is closest to how mathematicians actually compose proofs in their everyday job.

So where were we? We were talking about "vertical" and "horizontal" structure of a proof. Not to complicate things too much at once, let us first get a handle on the vertical structure of proofs, illustrating it on various example proofs that have most primitive possible horizontal structure. We will then, slowly, complexify the horizontal structure as well.

For Proposition A, the proof goes like this:

1. \(x+2=2x-3\)
2. \(2+3=2x-x\)
3. \(5=x\)

The numbers at the start of each line are just for our reference purposes, they do not form part of the proof. Line 2 is a logical conclusion of Line 1: if \(x+2=2x-3\) then it must be so that \(2+3=2x-x\), since we could add \(3\) to both sides of the equality and subtract \(x\) as well – a process under which the equality will remain true if it were true at the start.

Line 3 is (again) a logical conclusion of Line 2: since \(5=2+3\) and \(2x-x=x\), so if the equality in Line 2 were true then the equality in Line 3 must be true as well.

A series of lines of mathematical formulas where every next line is a logical conclusion of the previous one or more lines, is a mathematical proof with simplest possible horizontal structure. We will call such proofs "basic".

Proposition B also has a basic proof:

1. \(5=x\)
2. \(2+3=2x-x\)
3. \(x+2=2x-3\)

Just as before, every next line is a logical conclusion of the previous one.

What about the first line (in each proof)? If the first line were to also satisfy the requirement that it is a logical conclusion of the previous lines, then, since there are no lines before the first line, it would appear that the first line is true on its own, without a need for justification. If course, in both proofs this is false: in the first proof, we cannot claim that Line 1 is true. Truth of Line 1 in the first proof depends on the value of \(x\). Without knowing anything about the value of \(x\), we cannot claim that \(x+2=2x-3\), since if, say, \(x=0\), then \(x+2=2x-3\) is clearly false. The same for the second proof - we cannot claim that Line 1 is true. Instead, the role of the first line in each of the proofs is to "assume" they are true, and then see what conclusions can be drawn from such assumption. Recall that Proposition B, for instance, states that if \(x=5\) then \(x+2=2x-3\). It does not state that 

\(x=5\) and \(x+2=2x-3\), 

or that 

\(x=5\) or \(x+2=2x-3\), 

and so on. So in a basic proof the first line will always be an assumption, unlike the rest of the lines, which are conclusions from the previous one or several lines.

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Pure Mathematics: Job Description

What does a "pure mathematician" do? A shoemaker makes shoes, a musician makes music, an applied mathematician uses mathematics to solve some real-life problems... Each of these job descriptions have some sort of measurable output. What is such output for a pure mathematician? 

Some will say that a pure mathematician solves problems in mathematics, i.e., mathematical problems that are not necessarily related to "real life". This does not do justice to the efforts of a pure mathematician: if you are keen to solve problems, rather solve real-life problems! The problem is that the language in which these "pure" mathematical problems are solved is such that it cannot (always) be used to solve the "real-life" problems. A pure mathematician wants to solve only those problems whose solutions are expressed in a pure mathematical language. This does not do justice to the efforts of a pure mathematician either: what a picky attitude! Besides, solve-a-problem style job description applies to every other job. Indeed, any job for which you expect to get paid requires some sort of problem-solving.

The job description of a pure mathematician is actually quite straightforward. A pure mathematician builds "proofs". A proof is a discussion that reaches a certain conclusion with a life-time guarantee of truthfulness of this conclusion. In no other discipline are you able to establish proofs with such a guarantee. Surely having a certainty in a certain fact is a useful thing in any area of life. Unfortunately though, as soon as your conclusions come close to describing how something in "real life" works, their certainty can no longer be guaranteed, i.e., they step out of the reach of pure mathematics. Still, pure mathematics is extremely useful in establishing the real-life-like close-to-certain conclusions, otherwise the disciplines such as applied mathematics, physics, chemistry, and many others, would hardly make any progress (for those who may not be aware of this, these disciplines, as well as many others, rely a lot on conclusions proved in pure mathematics).

The conclusions that a proof proves are called "theorems". Then there are "definitions", which are essentially shortcuts for building complex proofs. Now a proof starts with certain assumptions (always, in fact, for those who may have been deceived that unlike religion, science does not rely on unproved assumptions, but this is a topic for an entirely different discussion...). The universal assumptions, i.e., those that are used over and over in many different proofs, are called "axioms". Part of the task of a pure mathematician is coming up with appropriate definitions and axioms. In the end, they are to be used in a proof, otherwise, they are useless. Solving a pure mathematical problem is all about finding a proof: of a theorem, its negation, or if the theorem has not been precisely stated, finding a precise statement and then its proof. So fair and square, a pure mathematician is someone who builds proofs!

 

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The SOFiA Proof Assistant Project

Background

The goal of this project is to build a proof assistant based on the SOFiA proof system, where the capital letters in SOFiA stands for Synaptic First Order Assembler (the purpose of the lower-case "i" will be explained further below). The use of terms "synapsis" and "assembler" is a suggestion of Brandon Laing, who wrote an MSc Thesis, "Sketching SOFiA" (2020), where the notion of an assembler was introduced: an assembler is the monoid of words in a given alphabet, seen as a monoidal category. The main result of his MSc Thesis was a characterization of assemblers using intrinsic properties of a monoidal category. An assembler gives a robust theoretical framework which guides the syntactical structure of the SOFiA proof system. The latter has been refined through a series of discussions with Louise Beyers and Gregor Feierabend in 2021, after which the first computer implementation of the SOFiA proof system was produced, based on the Python programming language. You can learn about it here. In January 2021, Gregor Feierabend developed a self-contained Haskell implementation, with user interface and documentation, which can be accessed here.

Overview of the SOFiA Proof System

The SOFiA proof system is an adaptation of the Fitch notation for natural deduction. The main novelty of the SOFiA proof system is the use of variables as statements, which leads to reducing quantified statements to implications. This allows unification of deduction rules for implication with those for the universal and existential quantifiers. The basic deduction rules for the proof system then are:

• Making an assumption (no restrictions except that the assumption must be a valid SOFiA expression).
• Restating an already stated SOFiA expression.
• Recalling a theorem or an axiom, external to the proof.
• Equating a stated SOFiA expression with itself.
• Synapsis: stepping out of an assumption block (this allows to conclude quantified statements, as well as implications).
• Application a SOFiA expression (this allows to conclude from quantified statements as a generalization of the modus ponens rule).
• Substitution: substituting SOFiA expressions within each other based on already stated equalities.

These deduction rules do not include rules for disjunction or fallacy. The latter can be implemented as axiom schemes. So at its base, the SOFiA proof system embodies a bit less than intuitionistic logic. This is marked by the appearance of lower-case "i" in "SOFiA". Note however that because in the SOFiA syntax there is no distinction between "objects" and "statements about objects", the SOFiA proof system is not quite the same as the usual proof system of a first-order logic, although in a loose sense SOFiA does have the structure of a first-order language. One of the key differences with standard first-order languages is that in SOFiA one does not introduce additional relational or functional symbols. Instead, one may write any sequence of allowed characters in SOFiA which can be given the intended meaning of a relational or a functional symbol by means of axioms. Possibility for a sound and complete embedding of any first-order logic in SOFiA still needs to be proved and is currently one of the founding themes of PhD research by Brandon Laing.

Developing the Proof Assistant

The current version(s) of the SOFiA proof assistant have the following shortcomings, which are to be addressed in the near future:

• The proofs can only be built line-by-line, it is currently not possible for the computer to fill the missing lines. This applies to both the Python and Haskell implementations. 
• The Python implementation source code is messy and there is currently no documentation.
• The Haskell implementation contains bugs.
• There Python implementation does not have a user interface. 
• Python and Haskell implementations come with modules for Boolean Logic and Peano Arithmetic, but they do not yet come with a module for Set Theory.

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Bracket Notation for Mathematical Proofs

The bracket notion for mathematical proofs is an adaptation of the Fitch notation for Gentzen's natural deduction proof system. It has led to the development of the SOFiA proof assistant. This post brings together some videos explaining the bracket notation and the first-order formal language for mathematics in the context of the bracket notation.

1. General Overview

~ 20 min

2. Building Blocks for Statements

~ 1 hour

3. Examples of Forming Statements

~ 40 min

4. Examples of Forming Statements (Continued)

~ 35 min

5. Concluding Quantified Statements

~ 35 min

6. Concluding from Quantified Statements

~ 1 hour

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Category Theory Course 2022

Here you will find the content for the Category Theory course given under the National Graduate Academy NGA-Coursework of the CoE-MaSS. The lectures are on Saturdays 9:00-11:00. 

Register here to receive the Zoom link for joining the lectures

There is also a Discord channel for this course, which you can find on the Discord server of the NGA-Coursework project. 

This is a video-based course aimed at post-graduate students and as well academics interested to learn about category theory, with live participation of the audience shaping the content of the course. For a reading course at the South African honors level, see:

• Category Theory: A First Course by George Janelidze

For an introduction to category theory for non-mathematicians and undergraduate students, see:

• Elementary Introduction to Category Theory by Amartya Goswami and Zurab Janelidze 

Lecture 1: Categories

Lecture 2: Functors

Lecture 3: Natural Transformations

Lecture 4: Adjunctions

Lecture 5: Limits

Lecture 6: Duality

Lecture 7: Yoneda Embedding

Lecture 8: Equivalence of Categories

9. Exponentiation

10. Universal constructions

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Mathematical Structures Course 2022

This page contains resources for a SATACS course at NITheCS that runs over the second semester of 2022. The lectures take place on Zoom on Fridays 17:00-19:00. If you would like to join them, register here.

Lecture 1: Magmas

Lecture 2: Join Semi-Lattices

Lecture 3: Relations

Lecture 4: Universes

Lecture 5: Posets

Lecture 6: Groups

Lecture 7: Topological Spaces

Lecture 8: Posets II

Lecture 9: Posets III

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