With the definition of number settled, Russell and Whitehead spend
the rest of *Principia* deriving more complicated
math, including arithmetic and number theory. However, to do this,
Russell and Whitehead were forced to add two additional axioms to
their system. The first is the axiom of infinity, which postulates
that there is an infinity of numbers. This axion is necessary to
derive real numbers. The second is the axiom of reducibility, which
is necessary to avoid Russell’s paradox. Using these two new axioms
in combination with the original logical axioms and *modus
ponens*, Russell and Whitehead spend the second and third
volumes of *Principia* deriving much of pure mathematics
in their system of formal logic.

### Analysis

Russell and Whitehead’s *Principia*, like
Newton’s similarly titled book two centuries earlier, was truly
groundbreaking. Just as Newton’s *Principia* revolutionized
physics, Russell and Whitehead’s treatise forever changed mathematics
and philosophy. The *Principia* has produced at least
three lasting, important effects. First, the *Principia* brought
mathematical logic to the forefront as a philosophical discipline.
It inspired much follow-up work in logic and led directly to the
development of *metalogic*, or the study of what
properties different logical systems have. Obscure as this may sound, many,
if not most, of the interesting results in logic in the twentieth century
are actually in metalogic, and these results have had profound implications
for epistemology and metaphysics. Second, the methods of mathematical
logic have had a great effect on the practice of *analytic
philosophy*. Analytic philosophy refers to a method of doing
philosophy by making arguments, the assumptions and structure of
which are as explicit and clear as possible. This idea is directly
parallel to the use of axioms and inference rules in formal systems.
From metaphysics to the philosophy of science to ethics, modern
philosophers in the Anglo-American tradition try to justify each
step of their arguments by some clear assumption or principle. Third,
both the technical apparatus of mathematical logic and its principles
of rigorous, step-by-step reasoning have found application in fields
ranging from computer science to psychology to linguistics. Computer
scientists, for example, have used logic to prove the limits of
what computers can do, and linguists have used it to model the structure
of natural language. None of these advances would have been possible
without Russell and Whitehead’s pioneering work.

However, the modern *Principia* also resembles
Newton’s work in a less flattering respect. Just as Einstein’s theory
of relativity overthrew Newton’s ideas about force, mass, and energy,
the work of later logicians and philosophers such as Kurt Gödel
and W. V. O. Quine has cast the results of *Principia* and
the logicist project into doubt. Recall that the aim of *Principia* was
to show that all mathematical knowledge could be derived from purely
logical principles. It was with this goal in mind that Russell and
Whitehead carefully selected logical axioms and rules of inference
that appeared to be a priori logical truths. However, two of these
axioms—the axiom of infinity and the axiom of reducibility—arguably
do not fit the bill. Consider our statement about penguins: there
either are or are not penguins in Antarctica. This statement seems
impossible to deny. Now consider the assertion that there is an
infinity of numbers. What makes this logically necessary? Is there
an infinite number of atoms? How can we have any knowledge of infinites?
Some critics have argued that the axiom of infinity is not a priori
in nature but is an empirical question whose answer depends on experience.
If this is so, any mathematical results derived from it must also
depend on experience, and the logicist program is in peril. Critics
have also focused on the axiom of reducibility. This axiom is necessary
to avoid Russell’s Paradox, but apart from that it does not seem
to have a purely logical justification. Critics have assailed it
as ad hoc, or assumed just to get a desired result. If this is the
case and it does not have a more fundamental nature, all of the
results derived from it are in doubt or at least not logically self-evident,
as Russell and Whitehead hoped to show.

The work of the logician Kurt Gödel has raised special
doubts about the *Principia*’s supposed proof of
the logicist program. Recall that one goal of the *Principia* was
to show that all of mathematics could be captured in a formal system.
This should be distinguished from the central logicist thesis that
mathematics was reducible to logic, but it was still crucial to
Russell and Whitehead’s method of proving this thesis. Gödel, in
a famous 1931 response to the *Principia*, showed
that this goal was unachievable, that no formal system could capture
all mathematical truths. This famous result is known as Gödel’s
Incompleteness Theorem. Its significance was in establishing that
there are some mathematical truths that cannot be deduced in any
formal system. This proved a major obstacle to logicists like Russell
who hoped to show formally that mathematics was just logic. However,
the logicist program is not yet completely dead, and the substantial
contributions of the *Principia* are still being
felt throughout math, philosophy, and beyond.