100 books like Wonders Beyond Numbers

This book profoundly influenced my thinking process, combining the worlds of mathematics, art, and music. I was captivated by how the book explores the deep connections between Gödel’s incompleteness theorems, Escher’s art, and Bach’s art of counterpoint.

The book’s puzzles and thought experiments pushed me to think more abstractly and critically. Despite being dense, I found it incredibly rewarding and eye-opening. I recommend this book to anyone interested in logic, creativity, and the nature of human thought. It’s a masterpiece!

Mathematicians are constantly baffled by the public’s lack of awareness, not just of what mathematics does, but what it is. Today’s technological society functions only because of a vast range of mathematical concepts, techniques, and discoveries, which go far beyond elementary arithmetic and algebra. This was one of the first books to tackle these misunderstandings head on. It does so by examining not just the math and what it’s used for, but the social structures, the ‘conditions of civilization’ that have brought us to this curious state: utterly dependent on math, almost universally unaware that we are. 

I was given this book when I was about 15, and devoured it. It is an eclectic collection of mathematical paradoxes, fallacies, and curiosities so strange that they seem impossible. Mathematical magic tricks, a proof that all numbers are equal, a proof that all triangles are isosceles, a curve whose length is infinite but whose area is finite, a curve that crosses itself at every point, a curve that fills the interior of a square. Infinities that are bigger than other infinities. The Saint Petersburg Paradox in probability, a calculation that you should pay the bank an infinite amount of money to play one fair coin-tossing game. The smallest number that cannot be named in fewer than thirteen words (which I’ve just named in twelve words).

The Four Color Problem was one of the most baffling questions in mathematics for over 120 years. First posed in 1852, it asks whether every map can be colored with four colors, or fewer, so that regions adjacent along a boundary have different colors. The answer (yes) was finally obtained in 1976, with massive computer assistance. This method was initially controversial, but the result is now firmly established. This highly readable account, with full-color illustrations, opens up the history and the personalities who tackled this topological enigma, as well as making the mathematics comprehensible. The path to the final solution is littered with blunders and mistakes, but also illustrates how mathematicians can join forces across the generations to chip away at a problem until it cracks wide open. 

A renowned historian of science, Clagett carries the story of Greek science forward all the way to the sixth century CE—a span of 1200 years. From that point in time, Greek science passed into the hands of Islamic scholars who advanced it further, especially the mathematical sciences.

This book is not, like ours, organized chronologically and developmentally but according to modern scientific domains—biology and medicine, mathematics, physics, and astronomy. And it focuses on specific scientific inquiries, while we focus on more general and fundamental things like ontology (what exists), cosmology (the overall structure of the universe), the laws of nature, and the drivers of change and motion.

This book is thus a complement to ours in its wide historical sweep and in what it highlights.

Discover Condorcet!

Most people, when asked to name a philosopher who wrote about inequality, would think of Rousseau. Condorcet was the last of the Encyclopédistes, young enough to experience the revolution in 1789—sadly, also one of its victims.

Unlike his philosopher colleagues, he participated actively in public policymaking, first in the Ministry of Finance, later as an elected member of the Legislative Assembly after the revolution. He chaired an organization working for the abolition of slavery. He argued for equal rights for women before Olympe de Gouges and Mary Wollstonecraft had published their more well-known pamphlets. He co-authored the Declaration of the Rights of Man and of the Citizen and also wrote a proposal for new constitution for France.

Most importantly, he realized the fundamental role of education as a means to reduce inequality and liberate mankind, and he even developed curricula for the various stages of a general… show more.

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.

Mathematics requires accurate calculation, and students sometimes think that getting the right answer is enough. But mathematics is also about valid logical arguments, and the demand for clear communication increases through an undergraduate degree. Students, therefore, need to learn to write professionally, with attention to general issues like good grammar, and mathematics-specific issues like accuracy in notation, precision in logical language, and structure in extended arguments. Vivaldi’s book has a great many examples and exercises, and students could benefit from studying it systematically or from dipping into it occasionally and reflecting on small ways to improve.

Mesopotamian mathematics is a fascinating subject; their numerical system was based on 60, and the ancient thinkers were adept at many types of calculations and word problems. Hundreds of clay tablets reflect their advanced understanding of mathematical principles. Eleanor Robson explains clearly in this book how historians and mathematicians have interpreted the evidence, and she discusses not just specific mathematical texts, how they are understood, and the way ideas were expressed, but she also introduces the scribes who developed and learned it all, and even the buildings in which they worked. The book is a “social history,” as the subtitle notes, and also an intellectual adventure.

Many undergraduate mathematics books – even those aimed at new students – are dense, technical, and difficult to read at any sort of speed. This is a natural feature of books in a deductive science, but it can be very discouraging, even for dedicated students. Houston’s book covers many ideas useful at the transition to proof-based mathematics, and he has worked extensively and attentively with students at that stage. Consequently, his book maintains high mathematical integrity and has lots of useful exercises while also being an unusually friendly read.