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.

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)

The 1910 Cambridge University Press edition has it on page 379. [1] Talamus 19:08, 5 May 2006 (UTC)

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)

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.

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

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)

Category: Seminal Works removal

I am curious why BetacommandBot removed "Category:Seminal works" on 25 September 2006 on line 37. Is this not a seminal work? Malangthon 02:47, 27 January 2007 (UTC)

Influence of Frege

The article currently states: "One of the main inspirations and motivations for the Principia was Frege's earlier work on logic." This needs to be analyzed and not stated as a truism. Russell had never even heard of Frege until 1903 at which point he had nearly completed his earlier work Principles of Mathematics which arguably is the main inspiration and motivation for the larger Principia. DJProFusion 21:54, 20 October 2007 (UTC)

Russell was Whitehead's student (in this time beween circa 1898-1903). Russell discovered his paradox in June 1901 and sent a letter to Frege dated 16 June 1902. In this he writes: "For a year and a half I have been acquainted with your Grundgesetze der Arithmetik [1893], but it is only now that I have been able to find the time for the thorough study I intended to make of your work..." (Letter to Frege in van Heijenoort 1967, 1976 3rd edition:125). Russell states that "Especially so far as function is concerned (§9 of your Begriffsschrift) [1879], I have been led on my own to views that are the same even in the details." (ibid) Frege answered promptly (22 June 1902), acknowledging the defect in his (Frege's) work. van Heijenoort notes that Frege received the letter from Russell when his (Frege's) 2nd volume was at the printers, and he barely had time to add the necessary emendation. In turn, Russell's Principles of Mathematics (1903) was at the printshop when Russell got a cc of Frege's 2nd volume with its emendation, and he too (Russell) added an appendix endorsing Frege's emendation. (cf van Heijenoort:126). He failed to resolve the paradox and proposed his "theory of types" (1908) to escape it (cf van Heijenoort:150). All this happened long before the PM was completed (cf van Heijenoort:216).

That's all the info I have. From this it's very unclear as to what Russell knew and didn't know about Frege's work before 1903. After then, it's obvious the whole house of cards had to be built on the sticks and straw of the "theory of types". Bill Wvbailey 23:30, 20 October 2007 (UTC)

External links

The links below "Principia Mathematica online (University of Michigan Historical Math Collection)" do not work (as of 28 February 2009). Can anyone provide alternative links?

Ignacio González (talk) 19:35, 28 February 2009 (UTC)

completeness theorem

The "consistency and criticisms" section says:

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...

The completeness theorem is actually about predicate logic, not propositional logic. I'm a bit reluctant to change the first sentence to "Predicate logic itself was known to be consistent" without knowing if that was the case. It's also not stated what kind of logic is used in Principia (in conjunction with its type system). Anyway, I think the paragraph needs repair, but I'm not sure what to do, so I'll leave it alone. 69.228.171.150 (talk) 20:49, 22 October 2009 (UTC)

Make it First-order predicate logic and you're good. --Ancheta Wis (talk) 21:13, 22 October 2009 (UTC)

I'm not conversant enough with the history of the subject to claim that first-order predicate logic was known in 1910 to be consistent, but if you want to make that edit, I won't argue with it. I really don't have any clear idea of what kind of deductive system Principia used. 69.228.171.150 (talk) 08:16, 23 October 2009 (UTC)

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Some factual history: It was only in 1930 that Kurt Goedel, for his doctoral dissertation at the University of Vienna --

"proved that the predicate calculus of first order is complete, in the sense that every valid formula is provable . . . The statement that the pure predicate calculus of first order is complete, that is, that every valid formula is provable, is equivalent to the statement that every formula is either refutable [Goedel's footnote 10: "A is refutable" is to mean "NOT-A is provable"] or satisfiable. Goedel actually proves a stronger statement, namely, that every formula is either refutable or ℵ₀[aleph0] satisfiable." (boldface added; van Heijenoort's introduction to Goedel 1930 van Heijenoort 1967:582-3)

Bernays 1926 proved "that every correct formula of the propositional calculus does indeed follow from the axioms given in Principia Mathematica (Goedel 1930 van H. 1967:583 and footnote 2).

Admittedly, all of this is terribly confusing, especially the difference between "complete" and "consistent". Here's what wikipedia says (confusingly -- a thicket of too many undefined words too fast):

"The semantic definition [of consistency] states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.

"If these semantic and syntactic definitions are equivalent for a particular logic, the logic is complete."

"In theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula φ in its language, at least one of φ or ¬ φ is a logical consequence of the theory.

This wiki-thicket is not too good. So we wonder what Goedel -- probably the clearest-thinking mathematician ever to live -- meant when he used his words -- consistent, complete, valid, satisfiable (satisfies). Goedel begins by defining hthe notion of "restricted functional calculus of logic" [modern first order predicate calculus] and valid (aka "tautological") this way:

"³In terminology and symbolism this paper follows Hilbert and Acermann 1928. According to that work, the restricted functional calculus contains the logical expressions that are constructed from propositional variables, X, Y, Z, . . ., and functional variables (that is, variables for properties and relations) of type 1, F(x), G(x,y), H(x, y, z), . . ., by means of the operations V (or), ~(not), ∀x (for all), ∃x (there exists), with the variable in the prefixes ∀x and ∃x ranging over individuals only, not over functions. A formula of this kind is said to be valid (tautological) if a true proposition results from every substitution of specific propositons and functions for X, Y, Z, ... and F(x), G(x, y), ... respectively (for example)∀x[ F(x) V ~F(x)]. (footnote 3 of Goedel's 1930, boldface added. I substituted the modern ∀x for Goedel's (x), the modern ∃x for Goedel's (Ex) and the symbol ~ for Goedel's overscore)

In the body of the paper he defines the notion of "complete". At the start of his paper he invokes "Whitehead and Russell" but wonders if

"when such a procedure is followed the question at once arises whether the initially postulated system of axioms and principles of inference is complete, that is, whether if it actually suffices for the derivation of every logic-mathematical proposition, or whether it is conceivable that there are true propositions (which may even be provable by means of other principles) that cannot be derived in the system under consideration." (ibid, boldface added)

In other words, if the theory is complete then every single possible valid formula can be derived within the system. In footnote 4 he defines his use of "valid", and then he defines (his use of) "satisfiable" in terms of his "valid":

"For a formula with free individual variables, A(x, y, z, . . ., w), "valid" means that ∀x∀y...∀w: A(x, y, . . ., w) is valid and "satisfiable" that ∃x∃y...∃w: A(x, y, z, . . ., w) is satisfiable, so that the following holds without exception: "A is valid" is equivalent to "~A is not satisfiable". (ibid, used modern symbols)

I'm going to leave this and go away and do more research. Whether or not the article is correct remains to be seen. Bill Wvbailey (talk) 18:11, 23 October 2009 (UTC)