Principia Mathematica is one of the seminal works of mathematical logic. Russell coauthored it with the mathematician Alfred North Whitehead over a ten-year period beginning in 1903. Originally conceived as an elaboration of Russell’s earlier Principles of Mathematics, the Principia’s three volumes eventually grew to eclipse Principles in scope and depth.
The goal of the Principia is to defend the logicist thesis that mathematics can be reduced to logic. Russell believed that logical knowledge enjoys a privileged status in comparison with other types of knowledge about the world. If we could know that mathematics is derived purely from logic, we could be more certain that mathematics was true. Russell and other philosophers believed that logical truths are special for several reasons. First, they have the distinguishing characteristic that they are true in virtue of their form rather than their content. Second, we have knowledge of them a priori, meaning without experience. Take, for example, the statement “Penguins either do or do not live in Antarctica.” This is a logical truth, an example of what logicians call the Law of Excluded Middle. Regardless of whether we know anything about penguins or frogs or X, we can say with certainty that this statement is true. On the other hand, we cannot know whether penguins are good swimmers without having observed some penguins (or at least looking in a book). Logicians, beginning with Aristotle, have studied statements and arguments that have the quality of certainty and tried to distill what in their form makes them certain. The Principia is in some sense an extension of this project from general logical arguments to mathematical ones. It aims to show that mathematical truths like “two plus two equals four” are true for the same reasons as our first statement about penguins.
The Principia’s three massive volumes are divided into six sections. Like most modern logic texts, the Principia begins by laying out a formal system of propositional logic and then proceeds to develop the theorems (or consequences) of the system. The basic idea is to use symbols to stand for propositions. A proposition is a statement that can be deemed either true or false. For example, P could stand for the proposition that penguins live in Antarctica and ¬P (read “not P”) for the proposition that penguins do not live in Antarctica. Russell and Whitehead introduce symbols like these and then add rules for combining them into complex statements using logical connectors, the English language equivalents of which are and, or, not, and if . . . then. Our original penguin statement would then read “P or ¬P.” In addition to this vocabulary for formalizing propositions, there is also a set of rules for making deductions. A deduction is simply a way to express a valid argument using symbols. (Recall that an argument is valid if the truth of its premises or assumptions guarantees the truth of its conclusion.) A simple deduction rule used in Principia is called modus ponens. It goes:
If P, then Q.
As in the penguin example, P and Q can stand for any propositions, so the following is a valid use of modus ponens:
If it rains, then the ground will be wet.
It has rained.
Therefore the ground is wet.
Typically, a formal system also contains a set of axioms or assumptions that form the starting point for applying deduction rules. In the case of Principia, the axioms are a select group of self-evident logical truths of the penguin type, except that they are about classes and sets instead of concrete physical objects.
After specifying these axioms and rules, Russell and Whitehead spend the bulk of Principia methodically developing their consequences. First, they develop their theory of types within the formal language. Next, they define the concept of number. Defining the concept of number is quite difficult to do without being circular. For example, it is hard to imagine how one would explain what the number 2 is without having to refer to the concept of 2. The key insight into this problem, which was originally conceived by the German philosopher Gottlob Frege and adopted by Russell and Whitehead, is to think of numbers in terms of concrete counting, not in terms of abstract numbers. When we first learn to count, we use our fingers to mark off the items as we count them. Each finger corresponds to one item. One can do the same thing to see if two sets are the same size by marking off items two at a time, one from each set. If there are no items left over in either set after pairing everything, the sets are the same size. The technical expression of this operation is somewhat complicated, but the basic idea is that the “number” of a set is the set of all sets that are the same size, as measured by our counting procedure. Russell and Whitehead were able to prove that this procedure produces objects that behave just like numbers. In fact, Russell and Whitehead go even further and make the claim that numbers simply are these sets. The number 2 is a shorthand way of referring to “the set of all sets of couples,” the number 3 is a shorthand for “the set of all sets of trios,” and so on.