Zurab Janelidze
December 14, 2022
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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.

- 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).

- 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.

- 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.

- 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.

- 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.

- 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).

- 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.

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)

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:

- \(y\Rightarrow x<y^2\)
- \(x<0^2\)
- \(x<0\)

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:

- \(x<0\)
- \(y\Rightarrow 0\leqslant y^2\)
- \(y\Rightarrow x<y^2\)

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:

- \(y\Rightarrow x<y^2\)
- \(x<0^2\)
- \(0^2=0\)
- \(x<0\)

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:

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

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:

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

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.

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!

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.

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.

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.

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.

~ 20 min

~ 1 hour

~ 40 min

~ 35 min

~ 35 min

~ 1 hour

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

This is the blog post of the 2022 August NITheCS Mini-School. Let us begin with some useful links:

- Lecture notes on universes of sets (introductory)
- Some videos explaining the concepts from the lecture notes above

This picture symbolizes a human state when one is working on a routine task, while one's mind looks into the bigger picture of things. While guitarists hands are busy playing on the guitar, his eyes are looking into the open space from a balcony. The fence of the balcony symbolizes the restrictions imposed on us by the necessity of a routine task.

This picture shows the back of a woman with yellow hair, in a stylish red dress, gazing at the white moon. The hair is blowing in a light wind. Mountains covered in snow are in the background. Her outfit is certainly not a match for the cold weather, but contemplation will keep her body warm. This picture symbolizes that deeper things in life can give us physical strength.

This drawing shows two sides of Camille Vasquez, one of the lawyers defending Johnny Depp in the livestreamed trial of defamation of Johnny Depp in 2022. The picture shows a head and a body side-by-side. The head, with a thoughtful expression of the face, represents the successful lawyer. The body, in a formal dress, represents the celebrity status that she obtained during the trial. The white color of the outfit is a reminder of the white suit that she wore during her closing argument. The body and the head are flipped relative to each other to emphasize the contrast between what they represent. This artwork is a symbolism for the duality of being good at what you do and being famous.

This freehand digital artwork represents the idea that significant change requires restructuring of foundations. The solid ground on the right middle side of the picture (the foundations) dissolves into an uproar of waves that illustrate the process of restructuring, which may appear to be chaotic. Going back from the left to the right side of the picture the waves subside into (new) foundations.

The inspiration for this composition was a discussion with the Composer Hans Roosenschoon, during which I presented to him some of my musical works, and also had a chance to listen to some of his unpublished works. I composed the piece the next day after the discussion. It was originally intended to be only the first sketch of the composition.

This piece signifies beauty of creativity. The video shows two contrasting characters, which represent the mind and the soul in a creative process. The roses are the creation. They appear in different color, form and contexts, to signify diversity of creation. Thus, the video provides a symbolic interpretation of the process and product of creativity.

Elaboration of some of the symbolism in the video:

- The bud with the sun in the background shown at the start of the video represents an idea that starts the creative process.
- The rose opening up, which is repeated three times in the video, represents the anticipation of the fulfillment, the fulfillment, and the reflection on the fulfillment of the creative process.
- The first character, dressed in conservative clothes, symbolizes the mind. The second character symbolizes the soul. The first character is reserved in her display of emotions as well as in her interaction with the roses. The second character is spontaneous and emotional, who interacts more intimately with the roses and displays enjoyment in such interaction. These represent the rational approach of the mind and the contrasting intuitive approach of the soul in a creative process.
- The first character wears black top throughout the video. The second character wears brighter tops. The first represents the critical approach of the mind and the struggles of the creative process, while the second represents the positive approach of the soul and the joy of the creative process. The positive/critical disposition of the soul versus the mind is symbolized also in the brighter lighting background for the second character versus the first character.
- For the most part of the video the character representing the mind has roses separated in bottles in the foreground. This represents the attitude of the mind to concentrate on the details in isolation from each other. The character representing the soul is, in contrast, shown with a bucket of flowers. This symbolizes the holistic approach of the soul in the creative process. The single flower that the second character appears to have isolated from the bucket symbolizes the driving idea behind the creative process.
- At the end of the video, the flowers in front of the character representing the mind are no longer separated in their bottles. Instead they appear lying in a heap in front of her, with one flower from the heap in her hands. This represents the conclusion of the creative process, when the mind dismisses the details and brings them all together, leading to the emergence of the contour of the bigger picture as a detail of its own.
- Just before the last scene, the character representing the soul passes the single rose she is holding towards the screen. This symbolizes disengagement of the soul at the end of the creative process. In the final scene, however, the other character remains with the roses. For the first time here, she smiles, but momentarily, while smelling the flower she is holding. This symbolizes that what remains after conclusion of the creative process is just mental image of what has been created. The excitement has subsided and there is only one emotion left, the unique positive experience of the mind in the process, which lasts only for one moment, making that moment worth the creative process: the feeling of accomplishment.

The story begins with an observation made to me by my father, on 22 February 2022, that it is a special date. Afterwards, I found out he was representing this date as 20220222 (year first, month and then day). In my original interpretation, it was 22022022. I was not satisfied just with the fact that this is a palindrome (a number which, when reversed, gives back the same number). So I looked into its prime factors: 22022022 = 2 x 3 x 11 x 2 x 3 x 11 x 333667. After a while of staring at this, I checked what would happen if I reversed 333667. That number (766333) turned out to be a prime as well! I learned afterwards that prime numbers whose reverse (in decimal notation) are prime, are called *reverse primes*. So then 22022022 is a *composite reverse prime*, i.e., a natural number who all prime factors are reverse primes. The question was: how special does this make the number 22022022? *To be continued.*

Read More
*These are notes in progress for a talk given at the online user group conference of the advanced programme mathematics organized by ieb (19 February 2022)*

In my experience, what makes transition from school mathematics to university mathematics hard is the depth of engagement with mathematics that university mathematics requires of students, compared to the depth of engagement that school mathematics requires. Do you agree or disagree with the following thesis:

If you do not agree and think that a university student should understand university mathematics more deeply than a school learner understands school mathematics, this means that you expect a learner transitioning to university not only having to learn more advanced mathematics, but to understand it more deeply than they understand its foundation, the school mathematics. Is that possible?!

What hinders a school learner to understand school mathematics deeply? Learning is driven by assessment: eventually, the task of a school teachers is to bring a pupil to the point of passing the final exam paper. Learning happens through textbooks which engages learners in a learning process that is based on answering questions that resemble those in the final exam. So then, it is natural to conclude that the mathematical questions students have to work on, whether from the textbook or the exam, paint the image of the mathematics they are learning. What if these questions mislead the learner and provide them with a wrong image of mathematics?

Here we give some examples of questions that have a great potential of misleading the learners.

The school expects the answer to be 8. In actual mathematical reality, it can be any other number as well, since we have not given any restriction on the sequence apart from its first eight digits. There are 1669 entries on these eight terms in the Encyclopedia of Integer Sequences.

This question misleads a pupil to thinking that the first idea that comes to mind (which would be indeed 8 for most humans) is the right answer to a mathematical question, and hence, if no idea comes at first, then there is no way to answer a question.

The school expects the answer to be entire real line. In actual mathematical reality, a formula does not define a function uniquely. To have a function, you need to first name a domain (and a "codomain", which is not necessarily the same as the "range") and then, state the formula which is to be interpreted for the named domain and codomain. Otherwise, why the entire real line -- the domain could have been the complex plane just as well!

This question misleads a pupil to thinking that mathematical equations contain complete information about mathematical objects. This eventually results in them dismissing words in a complex logical statement, focusing on a flattened picture comprising of just the equations in the statement. For example, they read "if 1=2 then 3=4" as "1=2 and 3=4". The latter is a false statement. The former is not: from 1=2 we can indeed derive 3=4 (simply add 2 to both sides).

The school expects the students to perform a sequence of manipulations eventually leading to the "answer" *x=7*. In actual mathematical reality, each step needs to be confirmed to be reversible to be certain that *7* is indeed a solution of the equation (that is it the only solution, is confirmed by the derivation).

This question misleads a pupil to thinking that solving an equation means applying some procedures in one direction. With this, not only the pupils do not understand the concept of a "solution", but they tend to think that any mathematical problem can be solved by a sequence of manipulations which need not be logically justified. This also results in them not understanding the difference between "implies" and "is equivalent to".

These are just some examples of misleading questions. There are many more!

Note:

- "sketch" instead of "graph" (or "sketch of the graph").
- Wilson says "-3 is not included" (it is rather the paint (-3,-1) that is not included in the graph) but "4 is included" (similarly, 4 is merely the x-coordinate of the point included in the graph).
- Wilson says that the domain is "where your graph is on the x-axis", and "range is where the graph is on the y-axis".
- Wilson says "if it is not defined, we put a round bracket, if it is defined, we put a square bracket".

In each of these examples we see a simplification of the language for pointing out something to the pupil. These simplifications may cost knowledge: a pupil who is not well familiar with the material receives logically incomplete information and hence enters into a mode of memorization where certain phrases are paired up with certain settings in a mathematical question. Instead of learning to actually comprehend the question, the learners become accustomed to break up the question in several keywords which in their memory relate (without any insight as to why) to some procedures that the learning needs to perform.

With this method of learning, which is present at university too, the learner/student comprehends mathematics as a set of memorized principles whose only application that they can experience is writing an exam. Mathematics for them is not something in which they can look for a story, an insight, or meaning.

Factual teaching vs insightful teaching compares well with learning music based on reading music notation vs based on listening to the instrument:

It is customary to think that pupils lose passion for mathematics when they encounter challenges in it. I do not believe this to be the case. To the contrary, I believe that people get inspired by challenge, as long as it stimulates intellect. Nobody has passion for something that is easy to get! My thesis is:

Sure, it is important for a learner to develop the skill of working hard. However, mathematics is not the right subject for that. Mathematics is supposed to be the subject that awakens a genius in a human being. There is even scientific evidence to back something that true mathematicians know very well: to be smart, you need to be lazy. The reason for this seemingly paradoxical relation is that if you are not lazy, you are likely to involve yourself with various activities that makes use of the rest of your body more than your brain. As a mathematician, the more mathematical research I do, the less I want to do anything else, including going to a shop to buy bread -- I become too lazy to even do that, not to mention doing admin at the university! On the other hand, when I force myself to do hard work with the admin (or physical exercise, which for the sake of winning time involves bicycling to the bread shop), I am no longer able to do mathematics effectively. My brain sort of flattens out -- I am not able to think sufficiently deeply any more.

Doing mathematics properly and working hard are in direct contradiction. Part of mathematical ingenuity is about finding a solution to a problem that saves your time. Laziness actually drives mathematical enquiry. But as soon as you do mathematics the right way, your brain releases chemicals which enable you to work for long hours, without having a feeling that you worked hard. You get exhausted, and yet you feel you did not work hard enough. That is the truth about mathematics.

When mathematics is taught through hard work, it kills passion simply because hard work kills mathematical creativity in a professional mathematics researcher too. If you are doing mathematics and you feel you are working hard, as far as I can tell, you are not doing actual mathematics.

A mathematical challenge is supposed to ignite passion and not kill it. If it kills passion, it may not be a challenge worth pursuing. In fact, research mathematicians use this principle to navigate their way in research: selection of which problem to pursue and which not to pursue is very much determined whether the problem ignites or kills passion.

Thus, when mathematics is done the right way, the diagram is:

challenge => passion ignites => hard work

When it is done the wrong way, the diagram becomes:

hard work => passion dies => challenge

In general, school mathematics is much about learning procedures to solve computational problems, without understanding why do those procedures work, not to mention a chance to self-discover the procedures. Intellectual effort of the student is reduced to writing in one's memory bank these procedures and practicing their application as far as answering exam questions is concerned.

Instead, university mathematics is more about understanding concepts intuitively, allowing a student to apply the understanding to solve a problem by self-discovering a procedure. Not only students are expected to explain why a procedure works, but they are also expected to come up with a procedure (a proof) which would confirm validity of a mathematical statement.

Those who have a talent for memorizing procedures usually lack the talent of creativity in mathematics, and vice versa. In other words, school mathematics favors learners with a certain intellectual profile, which is likely to exclude those who are capable of taking their mathematics studies at an advanced university level.

The issues discussed here do not only apply to transition from school mathematics to university mathematics. Similar issues arise in transition from undergraduate mathematics to postgraduate mathematics, and from postgraduate mathematics to research mathematics. In all cases, incorrect approach to mathematics leads to lack of sufficient foundation to advance to the next step.

Mathematical proof is a method of discourse which allows a human being to:

- discover new mathematical knowledge,
- analyze existing mathematical knowledge,
- verify truthfulness of a piece of mathematical knowledge.

Mathematical knowledge is knowledge of abstract principles about our universe. As such, it requires use of symbols to represent entities
that are inherently abstract. For example, the symbol *2* may represent *2 apples* or *2 pears*. The number *2* is an abstract entity, since it is not confined to any
of these concrete representations.

Mathematics functions at different levels of
abstraction too. For instance, we may write a symbol, such as *n*, to represent
any number. In one case we could have *n = 2*, and in another case we could have *n = 3*.
This is a second layer of abstraction compared to the layer of each specific number,
such as number *2*. Symbols representing abstract entities form basic ingredients of mathematical proof. The most complex parts of mathematical proofs deal with manipulations of these symbols, which sometimes may take an extremely long time. To optimize a proof, it is important to understand its most fundamental components. The aim of these lectures is to provide an exposition of these fundamental components.

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