Talk:Principia Mathematica
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The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published from 1910 to 1913. It is (is/was?) 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 (is/was?) Frege's earlier work on logic, which had led to some contradictions discovered by Russell in 1901 (see Russell's paradox). These contradictions were avoided in the Principia Mathematica by building an elaborate system of types. A set has a higher type than its elements so that one cannot speak of the "set of all sets" and similar constructs which lead to paradoxes.
The Principia covered/covers 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 (is/was?) clear that all known mathematics could in principle be developed in the adopted formalism.
After the publication of Principia Mathematica, questions remained whether a contradiction could be derived from its axioms, and whether there exists/existed a mathematical statement which could neither be proven nor disproven in the system. These questions were settled by Gödel's incompleteness theorem in 1931. Gödel's second incompleteness (is this the same 1931 theorem?) 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).
Yet, as Douglas Hofstadter (has) pointed out, there may be additional levels of potential contradiction in the Principia. A central principle of the "system of types" mentioned above is that statements that are self-referential are forbidden, to avoid Russell's paradox. Loops of statements that are self-referential (circular definitions) are also forbidden. However, the statement "We do not allow self-referential statements in Principia Mathematica" is a seeming violation of the rule against self-referential statements, an apparent contradiction at the heart of the philosophy, although it may be interpreted as meaning that none of the following statements in the formal system itself would be self-referential. That is, this statement may mean "in the following formal axiomatic system self-referential statements are not allowed," which clearly is not self-referential.
A fourth volume on the foundations of geometry had been planned (by Whitehead and Russell?), but the authors admitted to intellectual exhaustion upon completion of the third volume. A fourth volume did not appear.
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.
[edit] 1+1=2
In my edition, proposition *54·43 (from which “will follow, when arithmetical addition has been defined, that 1+1=2”) occurs on page 360, not 362 (see fac simile). Should I correct the article, or is there some other edition in which it occurs on page 362? --Gro-Tsen 22:46, 5 February 2006 (UTC)
- I think I put that in there, and I got the 362 from The Mathematical Experience', by Davis and Hersh, page 334. So if 360 is the correct page, go ahead and correct it. Bubba73 (talk), 23:16, 5 February 2006 (UTC)
- In the edition in the library at Texas A&M-Commerce, that proposition occurs on page 362. I happened to scan the page several years ago, and in fact still have a copy: Principia page 362. According to the university's website, this is the 2nd edition, Cambridge [Eng.] University Press, 1925-1927. Agarvin 19:29, 8 February 2006 (UTC)
[edit] Wikisource it!
Given that Principia Mathematica is public domain by now, I think it would be a good idea to make it available at Wikisource. Would anyone else be interested in contributing to such a thing? (The full text is available online anyway; it's just a matter of transferring and wikifying it.) --Ian Maxwell 00:24, 27 March 2006 (UTC)
- I would imagine Principia Mathematica being quite painful to wikify. :) (see some online version like http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAT3201.0001.001) Mathematical notation is exact, but OCR:ing it correctly would need some heavy customizationing. Quick analyzing and error-proofing program would probably be nice too -- and output in LaTeX format... Talamus 19:23, 5 May 2006 (UTC)
[edit] Pronunciation
I don't really know IPA well enough to use it, but I think it'd be helpful to add the pronunciation, specifically that in Principia the 'c' is hard; I always thought it was a soft c until I heard it said aloud. When I went online to check it out, I had to search for quite a while before I found a definitive reference.
- For future reference, all cs in Latin are "hard" (i.e. sound like English ks) - "Caesar" for example, should be pronounced much more like the german word "Kaiser" than the modern English pronunciation of "Ceasar". -- Tyler 07:41, 9 May 2006 (UTC)
A story here: When I was young I refered to it as Prin-cip-ia Mathematica and my father corrected me to Prin-kip-ia Mathematica. So I've always used the hard k. I agree with your finding: but ... we need to find a definitive source that corroborates this and here's why: My Merriam-Websters New Collegiate Dictionary 1990 doesn't have "Principia Mathematica" as an entry but it does have "principium" [L. beginning, basis, a fundamental principle] and it offers two alternate pronunciations, (the first the preferred): prin-sip-e-em, prin-kip-e-em. We need a bona fide Latin expert here. ("weenie weedie weekie" comes to mind). But the "sip" form may be more a matter of common usage in the English-speaking community, or not? Now I am confused. wvbaileyWvbailey 14:26, 6 June 2006 (UTC)
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The original Classical Latin pronunciation (as spoken by the Romans) is with the hard k, prinkipia. Later, every country adopted its own pronunciation (because Latin was taught as a dead language), so in English-speaking countries it was pronounced prinsipia. So they are both correct, in a sense. See Latin spelling and pronunciation and Latin regional pronunciation.
[edit] 1+1=2
This article explains the Principia Mathematica 1+1=2 proof, and discusses other related matters. Is it worth listing it in the "External links" section? -- Dominus 11:08, 20 June 2006 (UTC) Bold text
[edit] To do
I don't know much about the subject but, from what I do, the article should refer to Peano axioms and identify Russell as the main orchestrator of the project. --Ghirla -трёп- 09:44, 10 October 2006 (UTC)
