Principia Mathematica
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The Principia Mathematica is a 3-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910–1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. One of the main inspirations and motivations for the Principia was Frege's earlier work on logic, which had led to paradoxes discovered by Russell. These were avoided in the Principia by building an elaborate system of types: a set has a higher type than its elements and one can not speak of the "set of all sets" and similar constructs which lead to paradoxes (see Russell's paradox).
The Principia is widely considered by specialists in the subject to be one of the most important and seminal works in mathematical logic and philosophy. According to the Modern Library, it was the 23rd most important book of the twentieth century.[1]
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[edit] Scope of foundations laid
The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be.
A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.
[edit] Consistency issue
The questions remained
- whether a contradiction could be derived from the Principia's axioms (the question of inconsistency), and
- whether there exists a mathematical statement which could neither be proven nor disproven in the system (the question of completeness).
Propositional logic itself was known to be both consistent and complete, but the same had not been established for Principia's axioms of set theory. (See Hilbert's second problem.)
Gödel's incompleteness theorems cast unexpected light on these two related questions.
Gödel's first incompleteness theorem showed that Principia could not be both consistent and complete. According to the theorem, for every sufficiently powerful logical system (such as Principia), there exists a statement G that essentially reads, "The statement G cannot be proved." Such a statement is a sort of Catch-22: if G is provable, then it's false, and the system is therefore inconsistent; and if G is not provable, then it's true, and the system is therefore incomplete.
Gödel's second incompleteness theorem shows that basic arithmetic cannot be used to prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger. In other words, the statement "there are no contradictions in the Principia system" cannot be proven true or false in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).[citation needed] (Note: Actually, though the last sentence may be true, it does not quite follow from the one before it. For example, there is a proof of the consistency of basic arithmetic. This does not run afoul of the second incompleteness theorem because the proof uses a logical system that is strictly different to basic arithmetic itself.)
[edit] Quotation
- "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." – Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version).
- The proof is actually completed in *110.643 (Volume II, 1st edition, page 86), accompanied by the comment, "The above proposition is occasionally useful."
[edit] See also
[edit] References
- Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962.
[edit] External links
- Stanford Encyclopedia of Philosophy:
- Principia Mathematica -- by A. D. Irvine.
- The Notation in Principia Mathematica -- by Bernard Linsky.
- Principia Mathematica online (University of Michigan Historical Math Collection):
