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Principia Mathematica

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For Isaac Newton's book containing basic laws of physics, see Philosophiæ Naturalis Principia Mathematica.
The title page of the shortened version of the Principia Mathematica to *56.

The Principia Mathematica 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 1927 it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced *9 and an all-new Appendix C.

PM, as it is often abbreviated (not to be confused with Russell's 1903 Principles of Mathematics), 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 PM was Frege's earlier work on logic, which had led to paradoxes discovered by Russell. These were avoided in PM 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).

PM 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.[1] The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.[2]

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

According to Carnap's "Logicist Foundations of Mathematics", Russell wanted a theory that could plausibly be said to derive all of mathematics from purely logical axioms. However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility. Since the first two were existential axioms, Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be sure that the formal statements even properly express statements of real analysis, so that statements depending on it could not be reformulated as conditionals. Frank P. Ramsey tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.

Beyond the status of the axioms as logical truths, 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 consistent, but the same had not been established for Principia's axioms of set theory. (See Hilbert's second problem.)

In 1930, Gödel's completeness theorem showed that propositional logic itself was complete in a much weaker sense - that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms. However, this is not the stronger sense of completeness desired for Principia Mathematica, since a given system of axioms (such as those of Principia Mathematica) may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.

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 379 (page 362 in 2nd edition; page 360 in abridged version).
File:Principia Mathematica theorem 54-43.png
*54.43: From this proposition it will follow, ... that 1+1=2
  • The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful."

See also

References

  1. ^ Irvine, Andrew D. (2003-05-01). "Principia Mathematica (Stanford Encyclopedia of Philosophy)". Metaphysics Research Lab, CSLI, Stanford University. Retrieved 2009-08-05.
  2. ^ "The Modern Library's Top 100 Nonfiction Books of the Century". The New York Times Company. 1999-04-30. Retrieved 2009-08-05.

Primary:

  • 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.

Secondary:

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