Book Review – Mathemafrica

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

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

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 Jonathan Shock| 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.

From Princeton University Press

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 Jonathan Shock| 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 Jonathan Shock| November 7th, 2022|Book reviews, Reviews|0 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.

From Princeton University Press

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 Jonathan Shock| 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.

From Princeton University Press

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 Jonathan Shock| 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.

From Princeton University Press

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 Jonathan Shock| 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.

From Princeton University Press

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 Jonathan Shock| December 11th, 2021|Book reviews, Reviews|0 Comments

Curves for the Mathematically Curious – an anthology of the unpredictable, historical, beautiful and romantic, by Julian Havil – a review

NB I was sent this book as a review copy.

From Princeton University Press.

What a beautiful idea. What a beautiful book! In studying mathematics, one comes across various different curves while studying calculus, or number theory, or geometry in various forms and they are asides of the particular subject. The idea however of flipping the script and looking at curves themselves and from them gaining insight into: statistics, combinatorics, number theory, analysis, cryptography, fractals, Fourier series, axiomatic set theory and so much more is just wonderful.

This book looks at ten carefully chosen curves and from them shows how much insight one can get into vast swathes of mathematics and mathematical history. The curves chosen are:

The Euler Spiral – an elegant spiral which leads to many other interesting parametrically defined curves
The Weierstrass Curve – an everywhere continuous but nowhere differentiable function
Bezier Curves – which show up in computer graphics and beyond
The Rectangular Hyperbola – which leads to the investigation of logarithms and exponentials
The Quadratrix of Hippies – which are tightly linked to the impossible problems of antiquity
Peano’s Function and Hilbert’s Curve – space filling curves which lead to a completely flipped understanding of the possibilities of infinitely thin lines
Curves of Constant Width – curves which can perfectly fit down a hallway as they rotate.

…

By Jonathan Shock| March 15th, 2020|Book reviews, Reviews, Uncategorized|1 Comment

Prime Suspects – The anatomy of integers and permutations, by Andrew Granville and Jennifer Granville, illustrated by Robert Lewis – a review

NB I was sent this book as a review copy.

From Princeton University Press.

What a spectacular book! I am rather blown away by it. This is a graphic novel written about two bodies discovered by cops in an American city some time around the present day, and the forensic investigation which goes into solving the case, and somehow the authors have managed to make the whole book about number theory and combinatorics.

I have to admit that when I started reading the book I was worried that it was going to have the all-too-common flaw of starting off very simple and then suddenly getting way too complicated for the average reader, but they have managed to somehow avoid that remarkably well.

It is however a book that should be read with pen and paper, or preferably computer by one’s side. As I read through and mathematical claims were made, about prime factors of the integers and about cycle groups of permutations, I coded up each one to see if I was following along, and I would recommend this to be a good way to really follow the book.…

By Jonathan Shock| July 9th, 2019|Book reviews, Reviews|1 Comment

How to Fall Slower Than Gravity And Other Everyday (and Not So Everyday) Uses of Mathematics and Physical Reasoning – by Paul J. Nahin, a review

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

From Princeton University Press.

This book is without a doubt the most enjoyable, stimulating book of mathematical physics (and occasionally more pure branches of maths) puzzles that I have ever read. It’s essentially a series of cleverly, and occasionally fiendishly put-together mathematics and physics challenge questions, each of which gets you thinking in a new and fascinating way.

The level of mathematics needed is generally only up to relatively basic calculus, though there is the occasional diversion into a slightly more complex area, though anyone with basic first year university mathematics, or even a keen high school student who has done a little reading ahead, would be able to get a lot from the questions.

I found that there were a number of ways of going through the questions. Some of them are enjoyable to read, and simply ponder. For me, occasionally figuring out what should be done, without writing anything down, was enough to be pretty confident that I saw the ingenuity in the puzzle and the solution and I was happy to leave it at that.…

By Jonathan Shock| January 10th, 2019|Book reviews, Reviews, Uncategorized|2 Comments

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