Principia Mathematica is modern and insightful

Principia Mathematica, authored by Alfred North Whitehead and Bertrand Russell, is renowned for its extensive nature and meticulous proofs. The book demonstrates that only a handful of fundamental concepts are sufficient to encompass the entirety of mathematics. In Chapter 1, the authors introduce definitions that they consider crucial, which they view as more informative than the propositions in which they are used. These definitions help establish the subjects and judgments deemed most important.

The concept of variables is explored in depth, with the authors distinguishing between 'apparent variables' (bound variables in modern terminology) and 'real variables' (free variables). The authors demonstrate that asserting something containing a real variable, such as x = x, is equivalent to asserting that the formula holds for all possible values of x. This assertion allows for the proof of existence theorems, where a particular instance of truth can be found.

The authors also introduce the notion of 'types' on page 21, which is considered a pioneering usage of the word 'type' in the context of programming. On page 26, the authors explain that the symbol for set membership is derived from the Greek epsilon, the first letter of the word ἐστί, meaning 'to be.' Therefore, x ∈ man means 'x is a man.' Lastly, on page 33, the authors provide a modern definition of a function as a particular form of a binary relation, marking an early instance of this concept.