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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.
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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 of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot 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 since Aristotle's Organon. The Modern Library placed it 23rd in a list of the top 100 nonfiction books of the twentieth century.
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.
Consistency and criticisms
The questions remained:
- whether a contradiction could be derived from the Principias 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 Principias 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 is false, and the system is therefore inconsistent; and if G is not provable, then it is true, and the system is therefore incomplete.
Gödel's second incompleteness theorem shows that no formal system extending basic arithmetic can be used to prove its own consistency. Thus, the statement "there are no contradictions in the Principia system" cannot be proven in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).
Wittgenstein (e.g. in his Lectures on the Foundations of Mathematics, Cambridge 1939) criticised Principia on various grounds, such as:
- It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and Principia, this would be treated as evidence of an error in Principia (e.g. that Principia did not characterize numbers or addition correctly), not as evidence of an error in everyday counting.
- The calculating methods in Principia can only be used in practice with very small numbers. To calculate using large numbers (e.g. billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental - and hence questionable - methods such as induction). So again Principia depends on everyday techniques, not vice versa.
However Wittgenstein did concede that Principia may nonetheless make some aspects of everyday arithmetic clearer.
Quotations
- "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." – Volume I, 1st edition, (page 362 in 2nd edition; page 360 in abridged version).
- The proof is actually completed in (Volume II, 1st edition, page 86), accompanied by the comment, "The above proposition is occasionally useful."
See also
External links
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