What did Whitehead and Russell's "Principia Mathematica" achieve?

In philosophical contexts, the Principia Mathematica is sometimes held in high regard as a demonstration of a logical system.

But what did Whitehead and Russell's Principia Mathematica achieve for mathematics?

Solution 1:

I'll try to answer referring to the Introduction to the 1st edition of W&R's Principia (3 vols, 1910-13); see :

• Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (2nd ed,1927), page 1:

THE mathematical logic which occupies Part I of the present work has been constructed under the guidance of three different purposes. In the first place, it aims at effecting the greatest possible analysis of the ideas with which it deals and of the processes by which it conducts demonstrations, and at diminishing to the utmost the number of the undefined ideas and undemonstrated propositions (called respectively primitive ideas and primitive propositions) from which it starts. In the second place, it is framed with a view to the perfectly precise expression, in its symbols, of mathematical propositions: to secure such expression, and to secure it in the simplest and most convenient notation possible, is the chief motive in the choice of topics. In the third place, the system is specially framed to solve the paradoxes which, in recent years, have troubled students of symbolic logic and the theory of aggregates; it is believed that the theory of types, as set forth in what follows, leads both to the avoidance of contradictions, and to the detection of the precise fallacy which has given rise to them. [emphasis added.]

Simplifying a little bit, the three purposes of the work are :

• the foundation of mathematical logic

• the formalization of mathematics

• the development of the philosophical project called Logicism.

I'll not touch the third point.

Regarding the first one, PM are unquestionably the basic building block of modern mathemtical logic.

Unfortunately, its cumbersome notation and the "intermingling" of technical aspects and philosophical ones prevent for using it (at least the initial chapters) as a textbook.

Compare with the first "fully modern" textbook of mathematical logic :

• David Hilbert & Wilhelm Ackermann Principles of Mathematical Logic, the 1950 translation of the 1938 second edition of text Grundzüge der theoretischen Logik, firstly published in 1928.

See the Introduction [page 2] :

symbolic logic received a new impetus from the need of mathematics for an exact foundation and strict axiomatic treatment. G.Frege published his Begriffsschrift in 1879 and his Grundgesetze der Arithmetik in 1893-1903. G.Peano and his co-workers began in 1894 the publication of the Formulaire de Mathématiques, in which all the mathematical disciplines were to be presented in terms of the logical calculus. A high point of this development is the appearance of the Principia Mathematica (1910-1913) by A.N. Whitehead and B. Russell.

H&A's work is a "textbook" because - in spite of Hilbert's deep involvement with is foundational project - it is devoted to a plain exposition of "technical" issues, without philosophical discussions.

Now for my tentative answer to the question :

what did Whitehead and Russell's Principia Mathematica achieve for mathematics?

The first (and unprecedented) fully-flegded formalization of a huge part of mathematics, mainly the Cantorian mathematics of the infinite.

Unfotunately again, we have a cumbersome symbolism, as well as an axiomatization based on the theory of classes (and not : sets) that has been subsequently "surpassed" by Zermelo's axiomatization.

But we can find there "perfectly precise expression of mathematical propositions [and concepts]", starting from the elementary ones.

Some examples regarding operations on classes:

*22.01. $\alpha \subset \beta \ . =_{\text {Df}} . \ x \in \alpha \supset_x x \in \beta$

[in modern notation : $\forall x \ (x \in \alpha \to x \in \beta)$]

This defines "the class $\alpha$ is contained in the class $\beta$," or "all $\alpha$'s are $\beta$'s."

and the definition of singleton:

[the] function $\iota 'x$, meaning "the class of terms which are identical with $x$" which is the same thing as "the class whose only member is $x$." We are thus to have

$$\iota'x = \hat y(y = x).$$

[in modern notation : $\{ x \} = \{ y \mid y=x \}]$

[...] The distinction between $x$ and $\iota'x$ is one of the merits of Peano's symbolic logic, as well as of Frege's. [....] Let $\alpha$ be a class; then the class whose only member is $\alpha$ has only one member, namely $\alpha$, while $\alpha$ may have many members. Hence the class whose only member is a cannot be identical with $\alpha$*. [...]

*51.15. $y \in \iota'x \ . \equiv . \ y = x$

[in modern notation : $y \in \{ x \} \leftrightarrow y=x$].

Solution 2:

Whilst Whitehead and Russell failed in their main aim of deriving mathematics from logic, the logic that they developed is the mathematical logic that is used universaly today.