The most recommended math books

Nagel’s book is the most understandable explanation I’ve found about one of the most cosmically seminal math proofs: Godel’s incompleteness theorem.

It takes the idea of recursion and self-reference to the ultimate conclusion about truth, understanding, and boundaries of existence. This may sound hootie-tootie, but think of the sequence of abstraction in math: arithmetic, algebra, calculus, etc.

Once you add in recursion (self-reference), you’ve got the most important concepts underlying true understanding and productivity in software – something never discussed in computer science, and certainly not in practical programming.

Think about the so-called Von Neumann computer architecture, which underlies all modern computers – instructions are data, stored in the same place as ordinary data, and some instructions write data that are instructions. This is recursion taken to the next level. 

When people think of financial planning, their first thought is investing. Their second thought is retirement.

Kaplans explain risk succinctly: “Everything is possible, yet only one thing happens.” People understand risk but don’t really understand how to apply it rationally to investing (market risks) or to retirement (longevity risk).

But first, having an understanding of what risk is and isn’t, and where it comes from is important before you can apply it to what fuels your plans – markets and longevity.

This book helped me formulate the basic planning concepts I use in my book since personal finance is all about taking risks – as are any other decisions and actions you take in life.

Great Circles is a unique tale of the life and works of mathematicians, scientists, philosophers, poets, and other literary figures. It is collections of circles of thoughts and implications that return on themselves as if they are gravitationally attached to some core red dwarf of universal meaning.  

I loved reading this book. One moment I was into the math, and in the next, I was immersed in a relevant poem or was personality attached to some math or a philosophical thought about a connection of a poem with the math. It was a ride more than a read. It is a calming cognitive exercise on tour through and between chapters – mind wandering not permitted-- with a smooth comfort of thought as if Grosholz is in the room (or perhaps in your brain) reading and guiding.  

The poetry is gripping and wonderfully placed between the appropriate background materials. 

This book goes beyond the hype of data science, the details of machine learning methods, and the coding so closely associated with data science. Rather, it emphasizes the real types of problems for which data science may help, and explains the practical issues (“the real work”) that often lead to failure in data science projects.

These issues tend to be overlooked in more technical presentations of data science. They include such critical considerations as defining the right problem to begin with, understanding the “pedigree” (background and quality) of any data used, and ensuring that the right people are involved from the start.

The first time I came across this wonderful book, I did the inevitable double take: I had happened to study infinity in its many possible renditions, and the title of this superb book seems to be a mistake: an obvious, silly contradiction in terms, because we can always add one to any purported 'biggest number in the world' and thereby immediately produce an even bigger number. Yet...This book describes the weird, weird world of huge numbers, the race amongst a rarefied coterie of aficionados to find ever bigger specific, provable numbers, and much else besides. 

Mind-blowing.

Scholarship, regardless of the particular field, is always a creative process. Craft and method, the fruits of a rigorous education and hard work, are essential prerequisites, but genuine breakthroughs often seem to result from some mysterious incubation: a wild dance of ideas that emerge from the subconscious, stirred up by the keen will to understand something. Only a few of them eventually make it, by way of sensual representations, to the surface of consciousness, where they are formed and articulated by logic and language. In this book, the great French mathematician Jacques Hadamard captivatingly describes his investigation into the psychological underpinnings of creativity. He stresses the role of images and emotions in thought processes. I have always liked his conclusion that every significant invention requires at least some poetic feel.

This book is one of the nuttiest I’ve read in its imaginings of a modern world on steroids. The part that teases most is that its predictions might come true.

I know our smartphones and laptops have changed our world, but cures that would extend longevity ad infinitum? Neural interfaces that would connect us directly to the internet? Nanobots that would reduce the cost of goods to zero?

Because the topic of modernity has come to engross me so, the big question always is, “How far will it carry us?” The answer to which (according to Kurzweil) is much farther than you can possibly imagine and much sooner than you think. One should have one’s mind blown every couple of years, and Kurzweil does exactly that.

When I introduce measurements to my students, I talk about “an exciting adventure across centuries.” I love this book because it tells the long history of measurement, starting from prehistory and arriving at our days as if it were a true, fantastic adventure humankind has lived since the beginning of its existence.

This is not a technical book, nor a book of history, in a strict sense. It is written by a journalist as an adventure novel, telling the story of a fantastic journey across generations to build our present knowledge.

I read it as a novel, eager to discover the next adventure, always discovering players and facts I ignored. 

For those intending to use R with an eye on the popular 'Tidyverse' suite of packages – which facilitate the handling, manipulation, and visualisation of data sets – it's hard to go past this book. From the founding contributors of the RStudio/Tidyverse worlds, this is a great way to learn about this dialect of R against the overarching backdrop of statistical data analysis and data science.

In the mood for a graphic novel starring Bertrand Russell and a supporting cast of famous thinkers like Whitehead, Frege, Gödel, and Wittgenstein? Logicomix is for you!

Flip to Chapter Six, “Incompleteness,” for a peek of Vienna in the 1930s. The logic and philosophy illustrated throughout provide a great context for the work of Vienna’s famous philosophical circle led by Moritz Schlick, whose 1936 murder provides a chilling contrast to the intellectual pursuits of that time.