Principia Mathematica

Principia Mathematica (1910) is a work of philosophy in the domains of logic and mathematics by Bertrand Russell, with contributions from a contemporary mathematician, Alfred North Whitehead. It was intended to extend an earlier work by Russell called Principles of Mathematics, but ended up being a longer, more nebulous endeavor, undergoing several editions. Principia Mathematica endorses a thesis introduced by modern logicians arguing that mathematical language can be broken down into a more fundamental logical language. Russell holds that logic, by nature, is the most accurate general language with which to describe reality. The book is considered a seminal contribution to that which we now know about mathematics and its relation to the describing activities of human behavior, as well as to the reality outside the bounds of human consciousness.

Russell, the principal writer, begins with the argument that it would be easier to “prove” the validity of mathematics if we could show its logical soundness. He elevates logical truth above all other truths that humans can create, arguing that it has its own form, unlike other truth-statements, which primarily state a semantic content. In addition, logic is a natural human faculty, which seems to extend out of the rational mind. He takes a few examples of logical laws we have recognized to exist in basic human reasoning that existed even before we had a logical language, or even an alphabet, for describing them. Russell traces logical form back to Aristotle, the first great example of an effort to give linguistic shape to these natural logical forms. Russell states the project of his book is to show that this effort can be extended into the domain of math.

Russell’s first effort along these lines is to describe propositional logic and attach it to a formal system, the former built on a finite number of logical axioms. He thinks of propositions and other logical operators as things we can represent using simple symbols. From the expressions we can make by linking these symbols together, Russell posits, we can deduce other logical expressions that are true because they stem from earlier, logically true statements. In general, he argues that a formal system must start with a finite set of axioms, which he also calls assumptions. For his own system, he makes sure that his axioms are ones that humankind considers self-evident when applied to sets and classes. He qualifies that they need not necessarily be true when applied to real objects.



Next, Russell and Whitehead develop their theory of mathematics as an extension of logic. They first spend time creating a “theory of types” defined by their axiomatic language. Then, they provide a formal definition of number in a non-circular manner. To do this, they borrow from the thinking of Gottlob Frege, a German mathematical philosopher, and define the number as an object we reach via counting, rather than via abstraction. Counting consists of identifying numbers with objects, such as one’s fingers, and building up a sequence through correspondence. They extrapolate this idea to set theory, showing that two sets are equivalent in cardinality, or “size,” if this one-to-one correspondence exhausts itself without any leftover elements in any set. They claim that numbers do not exist in the abstract form at all, but rather are sets in their own right, obtained via this equivalence relation.

The remainder of Principia Mathematica is dedicated to more complex derivations in number theory and arithmetic. The authors add two final axioms to their formal system: the axiom of infinity, which states that numbers are positioned in an infinite sequence; and the axiom of reducibility, which prevents a possible paradox which Russell points out. With these axioms, the authors eventually develop a logical basis to pure mathematics.

Many mathematicians have disputed the thinking or claims of Russell and Whitehead in later literature, but few dispute that Principia Mathematica is rigorous and compelling. The book has thus made it into the canon of mathematical argumentation and has been found useful in a diverse range of scientific domains both in and outside of math and philosophy.