Showing posts with label talk. Show all posts
Showing posts with label talk. Show all posts
The Mind, Mathematics and Sustainable Development
This post is about the talk I gave on the International Day of Mathematics, March 14, 2023.
Abstract:
In this talk we will first reflect on how mathematics emerges from the ability of the mind to "zoom in" on the logical structure of the universe. Guided by this point of view, we will engage in a discussion about challenges in teaching and learning mathematics and the prospects for the bigger role that mathematics could play towards sustainable development.
About the speaker:
Zurab Janelidze is a professor of mathematics at Stellenbosch University. He is an associate of NITheCS and a principal investigator in one of the research programmes at NITheCS. He serves on the editorial boards of two international journals in his field of expertise, category theory, and serves as the president of the South African Mathematical Society. He is passionate about discovering and teaching mathematics, as well as looking for mathematical structures in other art forms.
Slides:
The talk:Feel free to leave your comments below. By the way, one "real-life" application of the abstract theorem from Example 2 is that it gives a proof why 2 to the power n is always strictly greater than n, for any natural number n. Let me know if you do not see how.
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.
Elementary Introduction to Set Theory
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
Lecture 1
Lecture 2
Lecture 3
Lecture 4
The Transition from High School Mathematics to University Mathematics
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)
1. Introduction
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:
A school learner must understand school mathematics at the same depth that a university student is expected to understands university mathematics.
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?!
2. Misleading Questions
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.
What is the next number in the sequence 0,1,2,3,4,5,6,7,...?
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.
What is the domain of the function f(x)=x^2?
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).
Solve the equation (x+3)/2=5.
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!
3. Factual Teaching vs Insightful Teaching
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:
4. Contribution from Dr. Cerene Rathilal
5. Challenge, Passion and Hard Work
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:
What kills mathematical passion in learners is realization that to overcome a mathematical challenge all it takes is routine hard word (e.g., memorization), instead of intellectual enquiry.
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
6. Final Note ♪
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.
7. Some Feedback from Students
The content of the first-year math courses is a lot of new work which builds onto the foundation that high school laid, therefore it was ideal to enter the first year with my mind still fresh from the matric exam the previous year. I personally found the way of learning maths in high school - listening in class, practising and using textbooks - quite similar to learning 1st-year math, only the workload increased and the pace at which it was presented. One big difference from high school is that you are given the free will to attend lectures and use your time wisely for practising problems. You are given examples to work through weekly, but it is basically up to the student if they want to make it their "homework" and that is what will separate the cream of the crop when they get tested. Thus if one did not practice self-discipline in high school, University is where they learn to. Another difference from high school is different lecturers for different courses with their own unique teaching styles. Compared to one high-school teacher, students now face the reality that they must adapt to new lecturers for every course each promoting new environments of learning, but most important growth. The biggest difference for me from high school is the inquisitiveness and enthusiasm. I remember in high-school everyone always complained about this assignment and that test, and actually lost the marvel of learning. In University however you are surrounded by a sea of individuals with the same interests as you, further igniting your passion for math. I just started my second year, and I just keep discovering new benefits of studying math. It depends on what module one takes, but the math still continues to enlighten sides of myself that I forgot I had, for example thinking outside of the box, or getting excited if a math problem sounds crazy or makes absolutely no sense! My advice would be to any scholar coming to university to think about math creatively, observe it around you, talk about it, be brave, ask many questions and most importantly have fun! - Nina Smit
The transition from school Mathematics to University Mathematics is quite challenging. For one, you do not know what to expect and I feel the NSC system fails students who take a deep interest in the STEM field. The mathematics you learn in school is pretty straight forward and answers to questions often follow an obvious algorithm that you can use without even understanding the fundamentals of what you were taught. Mathematics forms an integral part of STEM and I feel schools (or schools in my town) did not emphasize this enough. I know some schools offer Ad Math but the schools in my town didn't which was sad. I feel all schools should offer students the opportunity to challenge themselves with "difficult Math". This would lay a good foundation for students transitioning into university. - Cole Tymothy Paulse
Transitioning from school mathematics to university mathematics can be tricky. The main difference is the way it is taught and examined. In a school mathematics test/exam, you will be expected to answer questions that are similar to the examples you have been shown in class or your homework questions. This is very different in a university setting. Many questions don't look the same as the given examples so you will have to apply the knowledge acquired from the "homework" to solve a problem that is presented in a format that you probably haven't seen. You can't only use the fact that you know the formula and format to solve a university mathematics problem, you need to develop your logical and critical thinking skills and get creative to understand and answer the question. - Refentse Makweya
School mathematics relies heavily on memorization while university mathematics relies on more problem-solving. At school, we are taught a method to solve a specific problem. At university, we are taught certain tools that can be applied to solve various problems in interesting and creative ways. University mathematics courses contain far more work than high school mathematics and learning happens fast. University mathematics assessments often feel more like mathematics olympiads in the sense that the questions are often testing problem-solving abilities over simple memorization. - Anonymous
For my transition...I was really blown out of my mind at how much math in university depended on trigonometric work as well as graphs. I wish the teacher in school could emphasize more on the importance of those topics. It also on the other hand showed me just how much i did not know about maths and to what extent it can be applied. I really do look forward to my journey in maths and all the components I will still learn in the years to come. - Ancois Huysamen
I enjoyed my primary School very much. It should not be surprising that I excelled at mathematics and I was given enough space to attempt other problem-solving methods and also try some harder problems in competitions. In high school, I was given similar freedom to do those things, even if teachers did not know much other mathematics outside the syllabus. I stayed in a traditional high school for two years before being homeschooled. I did the Cambridge syllabus as opposed to CAPS. Cambridge syllabus was significantly more robotic. Every past paper has very similar questions that you can learn how to solve. Also, the memos of these past papers gave marks for some unnecessary steps so memorising solutions on past paper memos was even more vital. In university, I feel a bit more freedom in the methods I can use and I can be confident that as long as I'm clear about what I'm doing, I'll get all the marks. I can also be confident that my lecturer can answer any math-related question I have. - Jayden Thomas Dickson
In short, it was and still is one of the toughest but most rewarding challenges I've ever faced. In my experience, there was a big shift in what we were trying to achieve with mathematics. In high school, I was almost always finding a single answer or a number. The goal was mostly just to find the value of something and use Mathematics as the tool to do so. When I first started University Mathematics I saw quite quickly that most of our time was spent doing something much more useful and amazing. We were using Mathematics to understand and describe things. I started to see and understand how Mathematics was not only a method of problem-solving but also an amazing language of description. A very clear example I could give is my completely different experience with Probability and Statistics. Throughout high school, Statistics was a section I really disliked. It felt boring, repetitive, and pointless. However, when I started with my first-semester module on Probability Theory and Statistics (which thankfully was a compulsory module or I never would have chosen it), it quickly become one of the subjects I most enjoyed. It really challenged how I perceived so many everyday events and interactions. The necessity of going from axioms to proof made the well-established formulas I was shown in high school come to life because I could see with more intuition every term, constant and co-efficient was meaningful and not just conveniently helpful.
All this did come with a much higher requirement for practice, repetition, and the need for failure which was a new experience for me in mathematics. Getting things wrong for the process of learning became a common theme in anything I encountered at university. I realized extremely quickly that battling for hours on a topic and having assessments go terribly on a somewhat frequent basis was a reality I need to accept when it happened and learn from those mistakes because feeling disheartened and frustrated was going to only hurt the process of doing better in future. What made this process less demotivating is also a change in the culture around getting things wrong. In High School getting something wrong felt like a worse fate than trying. Success often seemed the only acceptable outcome. What changed dramatically was that failure became something celebrated as a learning opportunity and was met with countless resources to capitalize on it and the focus from my lecturers wasn't getting me to pass the module but to understand the content and become a better mathematician.
To summarize the transition was a massive one. It has challenged me more than I ever thought it would, but it has given me a viewpoint and appreciation for mathematics that I have never had before. I do wonder how many more of my friends in high school would have a changed perception of mathematics if they could get a taste of what the subject can be in a different environment. Looking back I am also amazed by how much I have progressed throughout the last year and hope that in the future more high school students will be inspired to consider a career in Science and Mathematics. - Chad Robert Davies (see also https://foabma.blogspot.com/2022/02/school-vs-university-mathematics.html)
Noetherian information systems
These are notes for a colloquium talk to be given at NITheCS.
The Snake Lemma from this fragment of a 1980 film ("It's My Turn", starring Jill Clayburgh and Michael Douglas), along with many other similar theorems in abstract algebra, known to be true for a variety different algebraic settings, can all be established in a unified setting of noetherian forms. This post attempts to give a preliminary step towards a possibly ambitious goal of applying noetherian forms outside abstract mathematics. In this light we propose a variation of this notion, a "noetherian information system", which is intended to be more agile in terms of identifying applications.
General Information Systems
By a network we mean a web of devices and directed binary channels between them ( = a graph in the sense of category theory). An information system over a network consisting of just one device and no channels, is a collection of information clusters, where some clusters may be parts of others.
The picture above describes an example of an information system. The device is a cellphone. A cluster is an approximate GPS location of the cellphone. Six clusters are displayed above, marked by A, B, C, D, E, F. The blue clusters represent GPS locations measured at different times on a given day. The orange ones represent GPS locations measured on another day. In general, the idea is that information clusters in an information system are all possible data that the device is theoretically able to capture/process.
There are two ways to interpret the meaning for a cluster E to be part of a cluster C:
- If we think of E as the range of possible locations of the cellphone, then E gives more information about the location of the cellphone than C does. We call this the classical interpretation. In this interpretation, E being part of C gets interpreted as E "implying" C in the sense of classical mathematical logic.
- If we think of each possible location as an attribute of the cellphone, then we can interpret E to be a state of the cellphone in which the cellphone has less attributes than in the state C. We call this the quantum interpretation, since with this interpretation, the information cluster E is seen as a state where the cellphone is simultaneously in all locations within the region E.
The general notion of an information cluster is an abstract one, as is the notion of one cluster being part of another. We only require that "is part of" relation between clusters is reflexive, antisymmetric and transitive.
A transmission from one information system to another maps each cluster in one system to a cluster in another system, so that if among two clusters, one is part of another, then a similar relation will hold for the mapped clusters (this property is called monotonicity of transmission). An example of transmission is calculation of distance range to a pinned location, based on the approximate GPS location of the cellphone.
In the example above, note that while D is not part of C, it gets mapped to a part of where C maps to. So, in general, transmission does not reflect the "part of" relation between clusters (while it preserves it, by monotonicity).
An information system over a general network consists of information systems over individual devices of the network, where each channel f from a device A to a device B determines a transmission of information from the information system over A to the information system over B. Note that a single transmission itself can be seen as an information system, where the network consists of two devices and one channel between them.
In an information system, given a finite chain of channels
Henceforth, we we speak of a "transmission" we refer to a "composite transmission", i.e., a morphism in the transmission category.
Inputs and Outputs
Define a stash of a transmission to be any cluster that maps to the smallest cluster in the target of the transmission (provided such exists), and define a reach to be any cluster such that there is a cluster in the source mapping to it.
In quantum interpretation, the smallest cluster (when it exists) is given by the least possible attribute set (it must be part of any cluster). Then a stash can be seen as a piece of information that gets concealed (as much as possible) in a transmission. If the smallest cluster is the empty set of attributes, then a stash is a piece of information that does not get transmitted at all.
In classical interpretation, the smallest cluster is a piece of information that logically implies any other piece of information: so it is the logical contradiction (the falsity). In this interpretation, a stash is a piece of information whose transmission results in contradiction. So once again, we may think of this as information that will not get transmitted.
In both classical and quantum interpretations, reach is a piece of received information.
An input is a channel that has the following properties:
- Any transmission with the same target as that of the input, whose every reach is a reach of the input, arises as a composite of the input transmission with a transmission to the source of the input.
- The input transmission maps clusters injectively (i.e., different clusters do not transmit to the same cluster).
- Any cluster that is part of a reach of the input transmission is itself a reach of the input transmission.
An output is a channel that has the following property:
- Any transmission with the same source as the output, whose every non-stash is a non-stash of the output, arises as a composite of the output with a transmission from the target of the output.
- If the output transmission maps a cluster B to part of a clusters A, and all stashes are part of A, then B is part of A as well.
- Any cluster in the target of the output is a reach of the output transmission.
An information system is said to be prenoetherian when the following three conditions hold:
- Any transmission decomposes as an output transmission followed by an isotransmission and followed by an input transmission.
- For any two inputs there is a third input whose reaches are precisely those clusters which are reaches of both initial inputs.
- For any two outputs there is a third output whose stashes are precisely those clusters which are parts of every single cluster containing all stashes of both initial outputs.
Mathematical Examples
Consider vector spaces as devices. Define a channel from a vector space V to a vector space W to be a linear map from V to W. Declare an information cluster to be a subspace of a vector space and declare one cluster to be part of another when the first subspace lies inside the second one. Transmission of subspaces is given by direct image of a subspace under a linear map. The corresponding category of transmissions is equivalent to the category of projective spaces (the quotient of the category of vector spaces, which identifies two linear maps when they act the same way on subspaces). This information system is prenoetherian thanks to the fact that the category of vector spaces is an abelian category. The required decomposition of a transmission is given by decomposition of a linear map as a quotient map, followed by an isomorphism and followed by subspace inclusion. The isomorphism in this decomposition is guaranteed by Noether's First Isomorphism Theorem.
Vector spaces here can be replaced by many other group-like structures. More generally, semi-abelian categories give rise to a prenoetherian information systems similarly to how abelian categories do -- once again, relying on Noether's First Isomorphism Theorem, as well as a few other important "exactness properties".
It has been recently shown that we can even consider a prenoetherian information system where the transmission category is equivalent to the category of sets. Devices in this example are sets. Channels are functions between sets. An information cluster partitions the set into equivalence classes and either distinguishes one of the classes or not. One cluster is part of another, if each equivalence class in the first is a subset of an equivalence class in the second and the distinguished equivalence class (if there is one) in the first cluster is a subset of the distinguished one in the second cluster.
A channel transmission then acts as suggested in the following example.
In these examples, every composite transmission is a channel transmission (which is because in each case our starting network was already a category, and transmissions were chosen functorially). When this happens, the decomposition required can by simplified to a decomposition into an output following by an input. This is thanks to the fact that inputs and outputs are stable under composition with channel isotransmissions.
In order to be able to recover all Noether isomorphism theorems in the transmission category of a prenoetherian information system (as well as many other homorphism theorems, such as homological diagram lemmas of homological algebra, for instance), we need to assume that the clusters admit finite suprema and finite infima (i.e., that information systems above each device are lattices in the sense of order theory) and that each transmission is a left adjoint in a Galois connection -- we call such information system a noetherian information system. We consider below a special case of this scenario, where clusters have arbitrary suprema and transmissions preserve them (these correspond to topological functors, whose applications in general topology have been a center of attention for many years in the research group of Professor Guillaume Brümmer from the University of Cape Town).
Topological Information Systems
An information system over a network consisting of just one device and no channels is complete when any number of clusters can be, in the terminology suggested by the quantum interpretation, superposed to form a new cluster. What we mean by "superposition" is nothing other than "join" in the terminology of order theory (so, "disjunction" in the classical interpretation). Superposition of no clusters should also be possible. In this case, the result is the smallest cluster.
None of the information systems considered in the first section of this post are topological.
- The first information system is not topological since clusters there are always circular regions of a plane. It is impossible to superpose two circular regions into another circular region. Note that a superposition of a set S of clusters is formally defined as a cluster J such that every member of S is part of J and moreover, J is part of any other cluster K that has the same property (i.e., that every member of S is part of K). So superposition of two disks should be a disk which contains both, but which is contained in any other disk containing both. Such disk does not exist unless one of the two disks contains the other: on the illustration below, an attempt to superpose two blue disks must produce a disk that wholly lies both in the yellow disk and the red disk, i.e., that lies in the orange region, while at the same time contains both blue disks -- this is not possible.
- Although non-zero finitely many closed intervals can be superposed, infinitely many, in general, cannot be superposed.
- In both cases, empty superposition is not possible.
A transmission is said to be continuous when transmission of information preserves superposition of clusters. In particular, when:
- the smallest cluster is transmitted to the smallest cluster, and
- superposing clusters in the source information system and then transmitting the resulting cluster, is the same as first transmitting the initial clusters and then superposing them in the target information system.
A general information system is topological when all information systems over individual devices are complete and all transmissions are continuous.
It is not difficult to amend our example(s) considered in the first section of this post to get topological information systems. For approximate GPS locations, we can allow any planar region to be a cluster and not just a circular one. For distance ranges, we could allow any set of numbers to be a cluster. The transmission of measuring distance ranges from a pinned location will then be continuous too for general mathematical reasons.
In a topological information system, every transmission can be reversed: the reverse of a transmission f is given by mapping a cluster D in the target of f to the superposition of all clusters that by f are mapped to parts of D. Writing f S for the result of mapping S by f, and writing Df for the result of mapping D by the referse of f, we get the following impressive law:
In our example, reverse transmission of a distance range will result in the following region, which is the region of all locations that are in that distance range from the given location.
The seemingly simple law above has a number of useful mathematical consequences. We describe some of these below:
- Reverse transmission is monotone and it preserves meets of clusters (a meet of a set of clusters is defined as the largest cluster that is part of each cluster).
- Transmission followed by reverse transmission results in expansion of the cluster.
- Reverse transmission followed by transmission results in shrinking of the cluster.
- Transmission, then reverse transmission, and then transmission again, results in the same cluster as by initial transmission. There is a similar property starting with reverse transmission in the place of transmission.
Concluding Remarks
The notion of a noetherian information system from this post is an adaptation of the notion of a noetherian form from
A. Goswami and Z. Janelidze, Duality in non-abelian algebra IV. Duality for groups and a universal isomorphism theorem, Advances in Mathematics 349, 2019, 781–812.
Related references, and especially those which led to the development of this concept, can be found in the article above. In particular, it contains the following reference to the paper of Emmy Noether where her isomorphism theorems were first established:
E. Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern,
Mathematische Annalen 96 (1927), 26–61.
In the following paper, Saunders Mac Lane, suggested that there should be a unified categorical approached to isomorphism and other such theorems based on duality:
S. Mac Lane, Duality for groups, Bull. Am. Math. Soc. 56, 1950, 485–516.
Noetherian forms provide such approach. The example dealing with the category of sets can be generalized to any topos (joint work in progress with Francois van Niekerk). Keeping in mind that the notion of a topos is a notion of a mathematical universe, we see how noetherian information systems have a wide reach in abstract mathematics. This makes one wonder whether they can be found outside of mathematics as well? In particular:
- Is there a useful real-life interpretation of a noetherian information system?
- If yes, does it lead to the ability to usefully model real-life information systems as noetherian information systems?
- In particular, are there any applications in machine learning or data science?
- Or, is it perhaps possible to use noetherian information systems to usefully model function of a living organism, or maybe, cognitive function of a human being?
- Does the category of Hilbert spaces, which is neither an abelian nor a semi-abelian category, but which plays an important role in quantum mechanics, have a noetherian form?
- Can the physical universe be modelled as a noetherian information system?
The poset of matrix properties
Below are the notes for the talk above, given at the Algebra, Geometry, Topology & Applications seminar.
1. Bird's-Eye View of Exactness Properties
One of the active areas of research in Categorical Algebra is the study of various properties of categories expressed using limits and colimits. Such properties are usually referred to as exactness properties. This terminology comes from the fact that, historically, the first such properties emerged in the study of exact sequences in the sense of Homological Algebra. The matrix properties in the title of this post are particular types of exactness properties, which can be encoded using integer matrices. Before explaining what they are, let us first recall the notions of limit and colimit.
Given a diagram of objects and arrows (objects are certain mathematical structures and arrows are morphisms between them), a limit (of the diagram) is a way to encode the information about the diagram in a single object; it is a terminal (commutative) cone over the diagram. The notion of a colimit is defined by "reversing arrows" in the definition of a limit.
Examples of limits and colimits abound in mathematics:
- Various complex topological spaces are colimits of diagrams of simpler spaces.
- Higher dimensional vector spaces are limits of lower dimensional spaces.
- Cartesian products of mathematical structures (e.g., sets, groups, topological spaces with the Tykhonov topology) are instances of limits.
- Every set is a colimit of singleton sets. A singleton set itself is a limit of the empty diagram.
- Starting with a unit interval, combining limits and colimits we can build spaces such as the sphere, the torus and the Klein bottle (among many others).
- The monoid of words decomposes as a colimit of multiple copies of the additive monoid of natural numbers.
- Taking the quotient of a mathematical structure (e.g., of a set by an equivalence relation, or of a group by a normal subgroup), is an example of a colimit.
- Intersection of subsets of a set and inverse image of a subset along a function are examples of limits.
- Profinite structures (e.g., profinite groups) are special types of limits of finite structures.
- Addition and multiplication of natural numbers are examples of colimits and limits.
- etc.
Each of these examples require a suitable category, in which it is specified what objects are, what morphisms are between objects (arrows), and how we compose them (which we need for expressing commutativity). The following two points are worth noting:
- For a given diagram, its limit (as well as colimit), when the latter exists, is unique, but only up to isomorphism.
- Every limit is a colimit in the dual category (a category obtained from a given one by reversing the direction of arrows).
From a concrete perspective, which looks at mathematical structures through the lenses of their elements, limits and colimits are tools for constructing local objects in a given universe of mathematical structures. From a more abstract perspective, limits and colimits are tools for analyzing global properties of universes of structures. Here is a striking analogy: in real life, addition and multiplication of natural numbers are means for accumulating quantity; in number theory, they are means for studying the natural number system. This is not a mere analogy, as addition and multiplication of natural numbers are instances of colimits and limits in the category where objects are natural numbers and a morphism from, e.g., 2 to 5, is a function from the set {0,1} to the set {0,1,2,3,4}.
Picking up on the previous analogy, as much as the study of the natural number system employs analysis of properties of multiplication and addition, the study of a category employs analysis of properties of limits and colimits. It turns out that properties which involve limits or colimits only, are much easier to study, than properties which involve both limits and colimits. This is in fact true in number theory as well, where purely from the additive point of view, the natural number system is nothing other than the monoid of words of a singleton alphabet, i.e., it is a free monoid over a singleton, and from the multiplicative point of view, it is a free commutative monoid over a countable set (namely, the set of prime numbers via the prime number decomposition). However, unlike in number theory, we are not studying here a single category, but all of them, which brings in properties that are not necessarily true in every category. Determining implications between those properties can be a nontrivial task, even if the properties are expressed in terms of limits or colimits alone.
2. Matrix Properties
In an n-dimensional cube, mark some vertices with a blue marker, and mark one of them with red. Then, consider a property of a mathematical structure X which states that for every representation of the cube in the n-th power of X, and for every substructure S of the n-th power of X, if the blue vertices fall in S, then so must the red vertices.
Points: C D F B H I P
Axis AI: 0 0 0 0 0 1 1
Axis AB: 0 0 0 1 1 0 1
Axis AD: 0 1 1 0 1 0 1
Axis AC: 1 0 1 0 1 0 1
Now, representing the cube in the n-th power of X amounts to choosing elements of X that will be called 0 and 1. In this case n = 4 and the property states that for arbitrary choice of 0 and 1 in X, if all columns of the matrix above belong to some substructure of X, then so does the column of zeros. We give here two examples of categories having this specific matrix property:
- The category of groups. If X is a group, then we can express A as e.g. A = C - F + D.
- The category of lattices. If X is a lattice, then we can express A as e.g. A = (C /\ D) \/ (D /\ B) \/ (B /\ C).
In the figure above, if we delete the points B, H, I, P, we will obtain a matrix property that defines Mal'tsev categories, among which are many categories of group-like structures as well as the dual of any topos. On the other hand, if we delete the points F, H, I, P, then we will obtain a matrix property that defines majority categories. Majority categories play a similar role for lattice-like structures as Mal'tsev categories do for group-like structures. Many theorems for groups and other similar structures can be proved in Mal'tsev categories. Similarly, many theorems for lattice-like structures can be proved in majority categories. The dual of the category of topological spaces happens to be a majority category, but not a Mal'tsev one.
The matrix property defining Mal'tsev categories implies the property given by the matrix above, simply because one matrix can be obtain from the other by deleting some columns. This is intuitively clear: if fewer blue vertices guarantee the red, then a surplus of blue vertices will still do the same. The majority property implies the property above for the same reason. However, in general, implications of matrix properties are not given by merely addition of columns. These implications are context sensitive (for which class of categories do we want the implication to hold?) and in the context of algebraic categories, where matrix properties become a special type of linear Mal'tsev conditions, we do not know yet an algorithm for deciding them. Very recently, we found an algorithm for deciding implications of matrix properties in the context of categories having finite limits. This context is the context for definitions of Mal'tsev category, majority category, as well as many others that have been studied in Categorical Algebra since 1990's, and hence it is a natural habitat of matrix properties. The algorithm settles a problem that has been open since a general notion of a matrix property was first proposed in my MSc Thesis in 2004.
3. The Algorithm
The algorithm for deciding implications of matrix properties has a geometric description. To explain it, we first need to define a notion of a reduction of a matrix. We will say that a matrix L is a reduction of a matrix M when L can be obtained from M by a succession of the following transformations:
- Expansion: when a new dimension is added to the cube (in any position), but the colored vertices do not change their position.
- Collapse: when one of the dimensions of the cube is collapsed and the colored vertices are projected along the collapsing dimension.
- Tilt: when given two dimensions in the cube, such that all colored vertices are on the same end of the first dimension, colored vertices on one side of the second dimension slide along the edges of the first dimension.
- Inside-out turn: when all colored vertices change their position on the edge of a given dimension.
If the structure of blue vertices of a reduction L of M can be embedded in N, then we can add to N a blue vertex at the relative position of the red vertex in L. The algorithm says that M will imply N provided after a series of increments of blue vertices in N using reductions of M, the red vertex of N will get colored blue.
The only time this algorithm will fail to detect matrix implication is when M has the following reduction
1 0
0 1
and N has the following reduction
1
In these two cases, the matrices M and N do imply each other and moreover, they imply all other (nonempty) matrix properties. Categories having these properties are preorders -- this is the trivial case.
A similar algorithm works if we want to establish that a conjunction of a family of matrix properties implies another matrix property, except that then the reductions L which are used to increment N can each time be reductions of a different matrix in the family.
4. Computer-Generated Results
The algorithm allowed us to use computer programming for computing various fragments of the poset of matrix properties of categories having finite limits. The image below displays the poset of non-trivial 3-dimensional matrix properties (excluding the matrix property that holds for all categories -- the anti-trivial case). In this image, shaded cells represent the entry 1 in the matrix, while unshaded ones represent the entry 0.
All of the implications above can actually follow from deletion of columns except the very bottom one. We leave it as an exercise to establish the bottom implication using the algorithm described above. The poset of (non-trivial and non-anti-trivial) 4-dimensional matrix properties is much larger. There are altogether 442 non-equivalent 4-dimensional matrix properties, with the following distribution, where the numbers on the horizontal axis count blue vertices on the 4-dimensional cube:
It is difficult to display all 442 of the 4-dimensional matrix properties on one picture, so will limit the number of columns (blue vertices) to 6:
- Mal'tsev + majority ("arithmetical"),
- Mal'tsev + 4-near-unanimity,
- majority,
- 4-minority,
- refinement of Mal'tsev with directly decomposable congruences,
- minority,
- Mal'tsev,
- 3-edge,
- 4-ary near unanimity,
- 4-edge,
- refinement of directly decomposable congruence classes,
- refinement of the egg-box property,
- refinement of normal local projections.
5. Some Theorems and Concluding Remarks
The algorithm is useful not only for using the computer to generate fragments of the infinite poset of exactness properties, but also for proving theorems about exactness properties that help one understand the structure of this poset. Some of these theorems were inspired by the computer-generated pictures. Here we briefly review some of the theorems that we managed to establish thus far.
Theorem. The infinite poset of matrix properties given by diagonal matrices is as described in the figure below.
Diagonal matrices are those where the only non-zero entries lie on the main diagonal, but the red vertex need not have all coordinates equal to 0. In the figure below, m represents the dimension of the diagonal matrix and k represents the number of non-zero coordinates of the red vertex.
Theorem. The conjunction of Mal'tsev and majority properties (i.e., the "arithmetical" case) implies any other non-trivial matrix property.
Theorem. Each matrix property is either implied by the majority matrix or implies the Mal'tsev property (but not both).
Theorem. Among matrix properties that are not anti-trivial, there are no maximal matrix properties.
We conclude with a few important remarks:
- In this post, we limited our attention to binary matrix properties, i.e., when the entries in the matrix are either 0 or 1. The theorems above are valid only in the binary case. The geometric interpretation of matrix properties using n-dimensional cubes is also only valid in the binary case.
- The algorithm for deciding implications of matrix properties was established in [HJJ]. The pictures of some fragments of the poset of matrix properties are also from [HJJ].
- The theorems above are from [HJJvdW].
- Matrix properties were first introduced in [J] (see [HJJ] for additional references).
- Majority categories were first introduced in [H], by an application of the "matrix method" for translating universal-algebraic properties into exactness properties described in [J].
- The notion of a Mal'tsev category, defined in the context of categories having finite limits, first appeared in [CPP].
- This post is an extension of another post, which gives a shorter and a complementary account of the results discussed here.
- The geometric interpretation of the algorithm presented in this post is an original contribution of the post. A paper will be written up on it, hopefully some time soon.
- Moving diagrams were designed on Geogebra, recorded using Wondershare UniConverter, and turned into gif files using ezgif.
- Python implementation of the algorithm can be found here. The software is called "mclex".
- The fourth author of [HJJvdW] was a first-year student at Stellenbosch University, when he joined the research project.
References
[CPP] A. Carboni, M.C. Pedicchio, and N. Pirovano, Internal graphs and internal groupoids in Mal’cev categories, Canadian Math. Soc. Conference proceedings 13, 1992, 97-109.
[H] M. Hoefnagel, Majority categories, Theory and Applications of Categories 34, 249-268, 2019.
[HJJvdW] M. Hoefnagel, P.-A. Jacqmin, Z. Janelidze, and E. van der Walt, On binary matrix properties, submitted for publication, 2021.
[J] Z. Janelidze, Matrices of terms and relations in categories, MSc Thesis, Tbilisi State University, 2004.