principia mathematica

The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–27, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A that replaced ✸9 and all-new Appendix B and Appendix C.

PM was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy,^[1] being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.

One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.

PM is not to be confused with Russell's 1903 The Principles of Mathematics. PM states: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."

PM has long been known for its typographical complexity. Famously, several hundred pages are required in PM to prove the validity of the proposition 1+1=2. The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.^[2]

I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated....

Hardy, G. H. (2004) [1940]. A Mathematician's Apology. Cambridge: University Press. p. 83. ISBN 978-0-521-42706-7.

He [Russell] said once, after some contact with the Chinese language, that he was horrified to find that the language of Principia Mathematica was an Indo-European one

Littlewood, J. E. (1985). A Mathematician's Miscellany. Cambridge: University Press. p. 130.

Theoretical basis

As noted in the criticism of the theory by Kurt Gödel (below), unlike a formalist theory, the "logicistic" theory of PM has no "precise statement of the syntax of the formalism". Another observation is that almost immediately in the theory, interpretations (in the sense of model theory) are presented in terms of truth-values for the behaviour of the symbols "⊢" (assertion of truth), "~" (logical not), and "V" (logical inclusive OR).

Truth-values: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw (pure) formalist theory would not provide the meaning of the symbols that form a "primitive proposition"—the symbols themselves could be absolutely arbitrary and unfamiliar. The theory would specify only how the symbols behave based on the grammar of the theory. Then later, by assignment of "values", a model would specify an interpretation of what the formulas are saying. Thus in the formal Kleene symbol set below, the "interpretation" of what the symbols commonly mean, and by implication how they end up being used, is given in parentheses, e.g., "¬ (not)". But this is not a pure Formalist theory.

Contemporary construction of a formal theory

The following formalist theory is offered as contrast to the logicistic theory of PM. A contemporary formal system would be constructed as follows:

1. Symbols used: This set is the starting set, and other symbols can appear but only by definition from these beginning symbols. A starting set might be the following set derived from Kleene 1952: logical symbols "→" (implies, IF-THEN, "⊃"), "&" (and), "V" (or), "¬" (not), "∀" (for all), "∃" (there exists); predicate symbol "=" (equals); function symbols "+" (arithmetic addition), "∙" (arithmetic multiplication), "'" (successor); individual symbol "0" (zero); variables "a", "b", "c", etc.; and parentheses "(" and ")".^[3]
2. Symbol strings: The theory will build "strings" of these symbols by concatenation (juxtaposition).^[4]
3. Formation rules: The theory specifies the rules of syntax (rules of grammar) usually as a recursive definition that starts with "0" and specifies how to build acceptable strings or "well-formed formulas" (wffs).^[5] This includes a rule for "substitution".^[6] of strings for the symbols called "variables" (as opposed to the other symbol-types).
4. Transformation rule(s): The axioms that specify the behaviours of the symbols and symbol sequences.
5. Rule of inference, detachment, modus ponens : The rule that allows the theory to "detach" a "conclusion" from the "premises" that led up to it, and thereafter to discard the "premises" (symbols to the left of the line │, or symbols above the line if horizontal). If this were not the case, then substitution would result in longer and longer strings that have to be carried forward. Indeed, after the application of modus ponens, nothing is left but the conclusion, the rest disappears forever.

Contemporary theories often specify as their first axiom the classical or modus ponens or "the rule of detachment":

A, A ⊃ B │ B

The symbol "│" is usually written as a horizontal line, here "⊃" means "implies". The symbols A and B are "stand-ins" for strings; this form of notation is called an "axiom schema" (i.e., there is a countable number of specific forms the notation could take). This can be read in a manner similar to IF-THEN but with a difference: given symbol string IF A and A implies B THEN B (and retain only B for further use). But the symbols have no "interpretation" (e.g., no "truth table" or "truth values" or "truth functions") and modus ponens proceeds mechanistically, by grammar alone.

Construction

The theory of PM has both significant similarities, and similar differences, to a contemporary formal theory. Kleene states that "this deduction of mathematics from logic was offered as intuitive axiomatics. The axioms were intended to be believed, or at least to be accepted as plausible hypotheses concerning the world".^[7] Indeed, unlike a Formalist theory that manipulates symbols according to rules of grammar, PM introduces the notion of "truth-values", i.e., truth and falsity in the real-world sense, and the "assertion of truth" almost immediately as the fifth and sixth elements in the structure of the theory (PM 1962:4–36):

1. Variables
2. Uses of various letters
3. The fundamental functions of propositions: "the Contradictory Function" symbolised by "~" and the "Logical Sum or Disjunctive Function" symbolised by "∨" being taken as primitive and logical implication defined (the following example also used to illustrate 9. Definition below) as

p ⊃ q .=. ~ p ∨ q Df. (PM 1962:11)

and logical product defined as

p . q .=. ~(~p ∨ ~q) Df. (PM 1962:12)

1. Equivalence: Logical equivalence, not arithmetic equivalence: "≡" given as a demonstration of how the symbols are used, i.e., "Thus ' p ≡ q ' stands for '( p ⊃ q ) . ( q ⊃ p )'." (PM 1962:7). Notice that to discuss a notation PM identifies a "meta"-notation with "[space] ... [space]":^[8]

Logical equivalence appears again as a definition:

p ≡ q .=. ( p ⊃ q ) . ( q ⊃ p ) (PM 1962:12),

Notice the appearance of parentheses. This grammatical usage is not specified and appears sporadically; parentheses do play an important role in symbol strings, however, e.g., the notation "(x)" for the contemporary "∀x".

1. Truth-values: "The 'Truth-value' of a proposition is truth if it is true, and falsehood if it is false" (this phrase is due to Frege) (PM 1962:7).
2. Assertion-sign: "'⊦'. p may be read 'it is true that' ... thus '⊦: p .⊃. q ' means 'it is true that p implies q ', whereas '⊦. p .⊃⊦. q ' means ' p is true; therefore q is true'. The first of these does not necessarily involve the truth either of p or of q, while the second involves the truth of both" (PM 1962:92).
3. Inference: PM 's version of modus ponens. "[If] '⊦. p ' and '⊦ (p ⊃ q)' have occurred, then '⊦ . q ' will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of '⊦. q ' [in other words, the symbols on the left disappear or can be erased]" (PM 1962:9).
4. The Use of Dots
5. Definitions: These use the "=" sign with "Df" at the right end.
6. Summary of preceding statements: brief discussion of the primitive ideas "~ p" and "p ∨ q" and "⊦" prefixed to a proposition.
7. Primitive propositions: the axioms or postulates. This was significantly modified in the 2nd edition.
8. Propositional functions: The notion of "proposition" was significantly modified in the 2nd edition, including the introduction of "atomic" propositions linked by logical signs to form "molecular" propositions, and the use of substitution of molecular propositions into atomic or molecular propositions to create new ｅxpressions.
9. The range of values and total variation
10. Ambiguous assertion and the real variable: This and the next two sections were modified or abandoned in the 2nd edition. In particular, the distinction between the concepts defined in sections 15. Definition and the real variable and 16 Propositions connecting real and apparent variables was abandoned in the second edition.
11. Formal implication and formal equivalence
12. Identity
13. Classes and relations
14. Various descriptive functions of relations
15. Plural descriptive functions
16. Unit classes