Sticky Post – Read this first. Categories and Links in Mathemafrica

The navigability of Mathemafrica isn’t ideal, so I have created this post which might guide you to what you are looking for. Here are a number of different categories of post which you might like to take a look at:

Always write in a comment if there is anything you would like to see us write about, or you would like to write about.…

By | January 17th, 2018|Uncategorized|0 Comments

Patterns, Predictions and Actions: Foundations of Machine Learning, by Hardt and Recht – a review

NB. I was sent this book as a review copy.

From Princeton University Press

I’ve just taught a course on mathematics for data science. Sadly it was only ten hours long, so there was only so much that I could cover. However, I feel that was taught was sufficient to get my students to the point that they would feel both comfortable with, and highly motivated to read Patterns, Predictions and Actions.

The balance between theory, application and narrative in the book is, I think, just right, making it a genuinely pleasurable book to read cover to cover, or to dip into a given topic to find the mathematical details (or at least what you need to get started). As with any foundational book, each topic could be covered in massively more detail, but that would simply make it a book, different from the authors’ intentions. The jump between the ideas and mathematical principle of Support Vector Machines, as given on one page, the optimisation methods of linear programming, and the practical aspect of coding up of such an algorithm are missing, but given the aims of the book, this doesn’t feel like a loss.…

By | January 29th, 2023|Book reviews, Reviews|1 Comment

The Story of Proof: Logic and the History of Mathematics, by John Stillwell – a review

NB. I was sent this book as a review copy.

The last book of Stillwell’s that I reviewed was Reverse Mathematics which was utterly fascinating, and truly mind-bending. I was very much looking forward to another of his books, and this one did not disappoint. It is a much less alternative perspective on mathematics than the previous, but no less beautifully written or compelling.

I teach pure mathematics to first year undergraduates (amongst others), and so often find that the very concept of a mathematical proof is something that is so hard to grasp. What is sufficient to concretely prove something? What can be assumed? What sort of proof is appropriate within a given context? High school maths generally sets students up very badly in this realm.

Stillwell’s book on the Story of Proof is perhaps a little beyond what could be grasped easily by most first year students, though very keen ones, with patience could certainly make their way through it, and would benefit enormously from doing so.…

By | January 29th, 2023|Book reviews, Reviews|1 Comment

THE BIG BANG OF NUMBERS. How to build the universe using only maths, by Manil Suri (Bloomsbury, 2022) – a review by Henri Laurie

Goodreads link.

Oh no. Not another overview of mathematics, for “everyone”. Set theory, numbers from natural to complex, geometry, algebra. Axiomatics. Gödel. Infinity. Applications. Philosophy??

Isn’t this all a big yawn? Hasn’t this been done again and again? For example by Lancelot Hogben, Eric Temple Bell, Reuben Hersch (and that’s just off the top of my head)?

Not at all, as it turns out. Suri has a marvellous new angle, one that allows him to bypass almost everything those authors wrote about. There is nary a formula or a figure or a proof (except for the endnotes), nor much about big names or history or the various  fields of mathematics.

Instead, there is an emphasis on *ideas* and on pursuing them wherever they may lead. Indeed, Suri gives us mathematics as the passionate pursuit of meaning, as engrossing as physics or music.

The conceit he uses is that one can design a universe very like ours starting from nothing – that is, from the empty set.…

By | November 7th, 2022|Book reviews, Reviews|0 Comments

The best writing on Mathematics, 2021, Edited by Mircea Pitici – a review

NB. I was sent this book as a review copy.

I’ve been reading this series every year now for the last five years or so, and it never disappoints. Mircea does an amazing job each time at collecting such a diverse ideas, voices, and areas of mathematics, that I usually find the vast majority of them to be exceptional. This year is no different.

The book came out during Covid, and rather aptly starts off talking about the effects on mathematicians of involuntary confinement of one form or another. In fact the very first chapter talks of the work of Poncelet, who was involved in Napoleon’s failed campaign, and subsequently imprisoned in Russia, and of his teacher’s Monge, who studied aspects of projective geometry. It just so happened that the diagram of Monge’s published in the essay was precisely what I had needed for a particular problem that I was working on (though in higher dimensions).…

By | October 6th, 2022|Book reviews, Reviews|0 Comments

On useful study habits

I’ve been teaching MAM1000W for around 9 years now, and I am learning all the time. I learn both about new ways to think about old subjects (and how to try and best explain them), and I learn about the way students study, about what works and what doesn’t, and what are some of the habits of students who succeed. Not all of these ideas will be perfect for everyone, but I hope that they will help.

Passive versus active learning

Trying to teach as clearly as possible is a double-edged sword. Of course I want students to come away feeling like they have understood the subject, but if they come away with too much confidence, then they won’t do the one thing which they have to do to actually understand it…and that is practice, but practice of a very particular kind. There is a balance that we should all be thinking about when trying to improve on something (be it sports, music, languages, or maths), and that is finding the right questions to practice on which are hard enough to make us have to sweat a little, but not so hard so as to make us give up completely.…

By | May 13th, 2022|Courses, First year, MAM1000, Undergraduate|2 Comments

In pursuit of Zeta-3 – The World’s Most Mysterious Unsolved Math Problem, by Paul Nahin – a review

NB. I was sent this book as a review copy.

I have to admit that I felt very skeptical when I started reading this book. In the prologue it is stated that the book is aimed at enthusiastic readers of mathematics with an AP level of high school maths. Then, diving into the book one sees what looks at first sight like a pure maths textbook at graduate level. But Paul Nahin isn’t one to pull a fast one like that, so I read further. In fact, I raced through it, hugely enjoyed it, and in the end agree with Nahin that someone with a US AP level of high school maths, or here in South Africa a confident first year undergraduate could actually understand everything in the book.

The book is not written as a textbook on mathematics, much as it might look like one, but rather it is taking an historical path through the investigations into the mysteries of zeta(3).…

By | April 23rd, 2022|Book reviews, Reviews|3 Comments

When least is best, by Paul Nahin – a review

NB. I was sent this book as a review copy.

For my review of Nahin’s superb book “How to fall slower than gravity”, see here.

While not often taught as a topic with such wide-ranging uses in maths classes, finding the maxima or minima of functions is one of the most important areas in all of applied mathematics. I say this as a practitioner of machine learning, where most of what we do is trying to find the minimum of a loss function, and as a physicist where in quantum field theory, the dynamical equations come from trying to extremise an action. While these areas aren’t discussed in the book (the closest it gets is looking at the classical Euler-Lagrange problem), to get students to think about how useful it is to find the maxima and minima of a function is really a powerful thing.

Nahin takes on this challenge and succeeds in the same way that he succeeded in making the problems in the previous book of his that I reviewed both fascinating and easy to follow.…

By | April 23rd, 2022|Book reviews, Reviews|1 Comment

A course in Complex Analysis, by Saeed Zakeri – a review

NB. I was sent this book as a review copy.

This is a no-nonsense, clearly written graduate level textbook on complex analysis, and while it is written for a graduate audience, I think that the way it is laid out, with clear examples throughout, a keen undergraduate with a background in analysis and topology. As such it is far more approachable than many other books on complex analysis and I would say that it would be perfectly suited for physics students wanting to go into areas like quantum field theory, particularly string theorists where the sections on conformal metrics and the modular group would be very helpful.

One thing to look out for in a book like this is the clarity of the proofs, and the number of intermediate lines which are included, and in this case I think that there is just the right amount to make everything easy to follow, but not overwhelming the material.…

By | April 23rd, 2022|Book reviews, Reviews|0 Comments

Visual Differential Geometry and Forms – a mathematical drama in five acts, by Tristan Needham – a review

NB. I was sent this book as a review copy.

Studying physics, some two decades ago at The University of Bristol, I found the majority of what we covered relatively intuitive. Even the arcane world of quantum mechanics, while impossible to truly visualise, is, paradoxically, often relatively simple to calculate, and the objects that you use are directly from the world of complex numbers, differential equations and linear algebra. What stumped me however were tensors. I found it so hard to really picture what was going on with these objects. Vectors were ok, and the metric tensor I could handle, but as soon as you got onto differential forms, all my intuition went out the window. The world of differential geometry, while I could plug and chug, felt like putting together sentences in a foreign language where all I had were rules for using the syntax and grammar, without a deep understanding of what the objects were

This book would have answered all of my prayers back then.…

By | December 11th, 2021|Book reviews, Reviews|0 Comments

CRL Task 5: Learning Causal Models

We’ve now come to one of the most vital aspects of this theory – how can we learn causal models? Learning models is often an exceptionally computationally intensive process, so getting this right is crucial. We now develop some mathematical results which guarantee bounds on our learning. We’ll start by discussing the current state of this field in relation to causal inference and reinforcement learning.

This Series

  1. Causal Reinforcement Learning
  2. Preliminaries for CRL
  3. CRL Task 1: Generalised Policy Learning
  4. CRL Task 2: Interventions – When and Where?
  5. CRL Task 3: Counterfactual Decision Making
  6. CRL Task 4: Generalisability and Robustness
  7. Task 5: Learning Causal Models
  8. (Coming soon) Task 6: Causal Imitation Learning
  9. (Coming soon) Wrapping Up: Where To From Here?

Learning Causal Models

Perhaps one of the most computationally difficult processes in the field of causal inference is that of learning underlying causal structure by algorithmically identifying cause-effect relationships. In recent years there has been a surge of interest in learning such relationships in the fields of machine learning and artificial intelligence, though it has been relatively prevalent in the social sciences for many years now (e.g.…

By | September 19th, 2021|Uncategorized|0 Comments