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.
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.