Taming the Infinite – Ian Stewart

Chapter 1. Tokens, Tallies, and Tablets (The Birth of Numbers):

The first chapter is purely history of mathematics and how the different civilizations around the world came with their different symbols and mathematical notations. This really shows the importance of numbers to humanity, since it proves the need of many completely unrelated cultures each coming up with their own language to express math.

We can trace back the concept of numbers to around 10,000 years ago, when tokens where used to keep records or track of financial information.

Numbers and mathematics remained fairly simple throughout a long period of time, and still now most of the operations we do everyday are of little to no mathematical complexity. However, it’s interesting to think about the relationship between cultural changes and mathematical evolution throughout our history.

Chapter 2. The Logic of Shape (First steps in Geometry):

Geometry is my favourite type(?) of math. I was very impressed to learn it was one of the first fields pursued, and with all logical sense, I learned from this chapter about how geometry is actually the base for much of the mathematical progress that came after. Geometry rarely dealt with numbers, it was all about relations and abstractions.

This chapter mainly covers Pythagoras, Euclid, Archimedes. Hypatia, Exodus, and Parthenon are tangentially covered. This are all people I greatly admire and undoubtedly incredible geniuses , its really almost above comprehension how they could achieve such radical innovations that have shaped history ever since.

It was cool to know that Euclid made a few mistakes in his Elements, with a couple of them being very big and well-known by now. I did not want to look much into this because I’m now really excited to find them during our Euclid classes. (Except I took note of this one to check out later: Euclid’s Parallel Axiom is non intuitive 35)

The greeks contributed two principal things to human development. The first is a systematic understanding of geometry, using it as a tool to understand our universe. The second was the systematic use of logical deduction to make sure that what was being asserted could be justified.

Chapter 3, Notations and Numbers: Where Our Number Symbols Come From

Before we settled on the notation 1 2 3 4 5 6 7 8 9 0, the romans, the greeks, the arabs, the hindus, the mayans, used different systems with different notations and without the use of 10 decimal digits. This chapter offers a brief hindsight of this history before concluding on the importance of learning arithmetics contrary to just using a calculator and learning to read the notation. The response is rather obvious.

Chapter 4, Lure of the unknown: X marks the spot

This chapter consists mainly of the birth of algebra (which means adding equal amounts to both sides of the equation), from linear to quadratic, cubic, conic, and even quartic equations in the 16th century. Algebra began as a way to systematize problems in arithmetic. Highlights were the method used by the babylonians to solve quadratic equations and the dispute between Cardano and Tartaglia for the discovery of the solution to conic equations, in a time where mathematicians used to duel each other on the streets for money and fame. Interesting as well is that the symbols + and – appeared around the 16th century as well.

Chapter 5, Eternal Triangles: Trigonometry and Logarithms

Every polygon can be built from triangles, and all the other interesting figures (circles and ellipses) can be approximated by polygons. Trigonometry, which basically means ‘measuring triangles’ is the practical use of the many formulas that have to do with the metric of triangles (angles, length of sides, total area). Astronomy was one of the earliest applications of trigonometry, and it remarkably used a fairly complicated one, and because of the link to astronomy almost all trigonometry was spherical until 1450. Nowadays this is very used from surveying to navigation and GPS location.

The second big theme of this chapter is logarithms, which were very useful because logarithms could be used to convert complicated multiplications into more simple additions. This was very important for a long time but since the 1960s and the introduction of computers logarithms have lost their groove, because computers can now easily and very rapidly solve complex multiplications.

Chapter 6, Curves and Coordinates: Geometry is Algebra is Geometry

This chapters starts by saying there is no real difference between algebra and geometry, and actually one could be represented using only the other. However, it is very useful to see things from a geometrical perspective in certain situations, as from an algebraic in others, being this the reason why we maintain both. Most of the innovation in mathematics (and everywhere actually) comes from when someone realizes there’s a connection between to seemingly unrelated subjects, we see examples of this through the chapter in the lives of Fermat and Descartes.

“Mathematics is the ultimate technology transfer. And it is those cross-connections, revealed to us over the past 4000 years, that make mathematics a single, unified subject.”

Chapter 7, Patterns in Numbers: The Origins of Number Theory

Starting with Euclid and going through Fermat, Lagrange, and Gauss, number theory studies the relations between whole numbers and had been a useless science for the last 2500 years, until computers came along, since they work with electronic representations of whole numbers. Most of the work in number theory is based on prime numbers. This was a very technical and interesting chapter, my favorite so far.

Chapter 8, The System of the World: The Invention of Calculus

So we got to 1680, Leibniz published first but Newton was more well-known, accused him of plagiarism, and turned calculus into a central technique of mathematical physics. Calculus is the mathematics of instantaneous rates of change, how rapidly is some quantity changing at this very instant?

As the physical astronomical background that led up to calculus, we go through the lives of Copernicus, Kepler, and Galileo, and their explanations of the movement of the planets in our solar system. Then, to study the history of the creation of Calculus we go through Leibniz and Newton.

“The Laws of nature are written in the language of calculus; what matters are not the values of physical variables, but the rates at which they change. It was a profound insight, and it created a revolution, leading more or less directly to modern science, and changing our planet forever.”

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