### Summary

*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.

P.

Therefore 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.