How to read this book
Mathematics follows a definition-theorem-proof structure that might be difficult to follow at first. If you are unfamiliar with such a flow, don’t worry. I’ll give a gentle introduction right now.
In essence, mathematics is the study of abstract objects (such as functions) through their fundamental properties. Instead of empirical observations, mathematics is based on logic, making it universal. If we want to use the powerful tool of logic, the mathematical objects need to be precisely defined. Definitions are presented in boxes like this below.
Definition 1. (An example definition)
Definitions appear like this.
Given a definition, results are formulated as if A, then B statements, where A is the premise, and B is the conclusion. Such results are called theorems. For instance, if a function is differentiable, then it is also continuous. If a function is convex, then it has global minima. If we have a function, then we can approximate it with arbitrary precision using a single-layer neural network. You get the pattern. Theorems are the core of mathematics.
We must provide a sound logical argument to accept the validity of a proposition, one that deduces the conclusion from the premise. This is called a proof, responsible for the steep learning curve of mathematics. Contrary to other scientific disciplines, proofs in mathematics are indisputable statements, set in stone forever. On a practical note, look out for these boxes.
Theorem 1. (An example theorem)
Let x be a fancy mathematical object. The following two statements hold.
(a If A, then B.
(b) If C and D, then E.
Proof. This is where the proof goes.
To enhance the learning experience, I’ll often make good-to-know but not absolutely essential information into remarks.
Remark 1. (An exciting remark)
Mathematics is awesome. You’ll be a better engineer because of it.
The most effective way of learning is building things and putting theory into practice. In mathematics, this is the only way to learn. What this means is that you need to read through the text carefully. Don’t take anything for granted just because it is written down. Think through every sentence. Take every argument and calculation apart. Try to prove theorems by yourself before reading the proofs.
With that in mind, let’s get to it! Buckle up for the ride; the road is long and full of twists and turns.
