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This is an old revision of this page, as edited by 37.60.109.133 (talk) at 15:36, 28 January 2024 (→Isn't the article too praiseworthy to be neutral: new section). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.


Possible error in notation section

The explanation for Example 1 in section "An introduction to the notation of "Section A Mathematical Logic"" looks incorrect. The example line "✸3.12. ⊢ : ~p . v . ~q . v . p . q" has five single dots, but the explanation that follows it only talks about four of them. There doesn't seem to be any explanation for the parentheses around (~p) v (~q). It seems like those parenthesis must come from the third single dot (acting as a right parenthesis) in the original line, but the rules given would have that dot terminated by the second dot rather than going all the way to the starting colon.

Untitled

Please append new sections to the foot of the page, to retain the time order. --Ancheta Wis (talk) 02:30, 21 February 2011 (UTC)[reply]

Helpful edits?

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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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.

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Perhaps the question should not be how the Romans would have pronounced it, but rather: how would Newton have pronounced it? — Preceding unsigned comment added by Dradler (talk • contribs) 03:27, 24 August 2012 (UTC)[reply]

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Russell pronounced "Principia" with a soft 'c' (and all the 'i'-s in the usual Romance-language way, as long-'e'-s).12.20.236.2 (talk) 22:13, 17 September 2012 (UTC)[reply]

---

In classical Latin it's "Prin-chee-pia", not "Prin-kee-pia". That pronunciation has been inherited by Italian, where we say "principi", pronounced "prin-chee-pi" (not to be confused with "princes" which is spelled the same).

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I found an introduction to an interview with Russell, and the BBC reporter used a hard /k/: https://www.youtube.com/watch?v=1bZv3pSaLtY&t=0s As far as I can tell the original classical Latin pronunciation of "C" used a hard /k/.

The article Traditional English pronunciation of Latin seems to explain why there's dissent on this point:

"The traditional English pronunciation of Latin, and Classical Greek words borrowed through Latin, is the way the Latin language was traditionally pronounced by speakers of English until the early 20th century. In the Middle Ages speakers of English, from Middle English onward, pronounced Latin not as the ancient Romans did, but in the way that had developed among speakers of French. This traditional pronunciation then became closely linked to the pronunciation of English, and as the pronunciation of English changed with time, the English pronunciation of Latin changed as well. Until the beginning of the 19th century all English speakers used this pronunciation, including Roman Catholics for liturgical purposes. Following Catholic emancipation in Britain in 1829 and the subsequent Oxford Movement, newly converted Catholics preferred the Italianate pronunciation which became the norm for the Catholic liturgy. Meanwhile, scholarly proposals were made for a reconstructed Classical pronunciation, close to the pronunciation used in the late Roman Republic and early Empire, and with a more transparent relationship between spelling and pronunciation.

One immediately audible difference between the pronunciations was in the treatment of stressed vowels, in which the English version followed the sound changes that had affected English itself, the stressed vowels being quite different from their unstressed counterparts, whereas in the other two versions they remained the same. Among the consonants, treatment of the letter c followed by a front vowel was an obvious distinction. Thus the name Cicero was spoken in the English version as Sisero, in the Italianate as Chichero and in the restored classical as Kikero. (Similarly with et cetera, etc.)

The competition between the three pronunciations grew towards the end of the 19th century. By the beginning of the 20th century, however, a consensus for change had developed. The Classical Association, shortly after its foundation in 1903, put forward a detailed proposal for a reconstructed classical pronunciation. This was supported by other professional and learned bodies. Finally in February 1907 their proposal was officially recommended by the Board of Education for use in schools throughout the UK. Adoption of the "new pronunciation" was a long drawn out process,[4] but by the mid-20th century, classroom use of the traditional pronunciation had ceased.

As I understand it, in French the C became an /s/ (and thus in words later inherited by English), in Italian the C became /ch/, and in Spain it became /ts/. But since the mid-1900s English speakers who are trying to speak a Latin phrase have been taught to use the original classical Latin version.

I'm currently using a hard /k/ for the 'c' in Principia Mathematica, simply because I understand that's how it would have been done in classical Latin, the title is intentionally in Latin, and speakers who learned Latin in the UK have been taught to do the same. That at least seems justifiable to me.

Dwheeler (talk) 15:07, 8 August 2016 (UTC)[reply]

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[reply]

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)[reply]


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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Make it First-order predicate logic and you're good. --Ancheta Wis (talk) 21:13, 22 October 2009 (UTC)[reply]
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)[reply]
PM uses a ramified type theory with all finite types and no transfinite one. It was Russell's own invention. 86.178.149.192 (talk) 16:13, 26 June 2020 (UTC)[reply]

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)[reply]

Yes, it's a standard source of confusion that "complete" as in "completeness theorem" and "complete" as in "incompleteness theorem" have completely different meanings. But that's not what's at issue here. The Hilbert school made a lot of progress on consistency questions in the 1920's. But the article seems to say something about the state of knowledge in 1910 when Principia was written, and I don't know 1) whether the consistency of predicate logic was known at that time; and 2) whether Principia even used predicate logic. The article really needs some expansion to explain Principia's deductive system and axioms, but that too is a separate matter. 69.228.171.150 (talk) 13:43, 24 October 2009 (UTC)[reply]

To your point this article is in dire need of work. The short answer to your 1) is NO, and to 2) YES. As far as I'm concerned the following nails it, unless someone knows of a proof that existed before Post 1921.

Not only was the consistency of predicate logic unknown, but the consistency of propositional logic was unknown until Post 1922 (and maybe H. M. Sheffer's Total determinations of deductive systems with special reference to the algebrea of Logic 1921? -- see references in 2nd edition of Principia p. xivi 1927). Because propositional logic is just predicate logic without use of the quantifier(s), you need the proof for propositional logic before you can have one for predicate logic, and that one (I believe) falls out of Goedel 1930 completeness proof. Only because of Hilbert was good predicate logic (that used by Goedel 1930, for instance) available in 1925-1927 (cf Hilbert's 1925 On the Infinite in van H. pp. 381ff, and especially Hilbert's 1927 The foundations of mathematics address (cf van H. pp. 464ff).

Until Post 1921, it seems that the consistency of the Laws of Thought and Boolean algebra were taken for granted -- with a couple prominent exceptions (e.g. the Ladd-Franklin and Huntington references in Couturat 1914, see below). In Post's 1921 Introduction to a general theory of elementary propositions (van Heijenoort 1967:264ff) Post introduces the problem as follows:

"Our most important theory gives a uniform method for testing the truth of any proposition of the system . . . these relations definitely show that the postulates of Principia are capable of developing the complete system of the logic of propositions without ever introducing results extraneous to that system -- a conclusion that could hardly have been arrived at by the particular processes used in that book" (Post's Introduction, van H 1967:265).

In particular is Post's theorem and corollary:

"THEOREM. The system of elementary propositions of Principia is consistent. (p. 272)
"THEOREM: Every function of the system either can be asserted by means of the postulates or is inconsistent with them. (p. 272)
"COROLLARY. A function is either asserted as a result of the postulates or else its assertion will bring about the assertion of every possible elementary proposition. (p. 272)

There's more to it than this, however. Even Post 1921 invokes Nicod's use of Sheffer's "stroke"(cf. p. 275) and he invokes Schroeder 1891, but dismisses him in his footnote 7. And he invokes Whitehead 1898 for a part of his proofs (cf footnote 11 p. 271). So it's not clear at all that a "proof" of some sort did not exist before Post. Until Russell and Frege we know that logic was Boolean in nature; Russell introduced the world to Frege via Russell's 1903 Prinicples of Mathematics (cf Grattain-Guinness 2000). So I go to my texts: I have a cc of Frege 1887; he defines the notion of "judgement", Conditonality, (modern implication), and Negation. re Peano 1889 the commentary by van Heijenoort treats his logic unkindly: "his logical laws should perhaps be taken as rules of inference, not as formulas in a logical language" (p. 84 in van Heijenoort).

The introduction to Couturat 1914 by Jourdain specifically calls out Frege and Russell (all these texts can be gotten at googlebooks). But the logic was still Boolean, e.g. Couturat lists the two laws of thought -- Contradiction and the Excluded Middle -- this way:

"By definition, a term and its negative verify the two formulas aa' = 0, a + a' = 1, which represent respectively the principle of contradiction and the principle of excluded middle.1
1 As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Tbought"), the principle of

contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of discourse).

I haven't looked at Ladd-Franklin yet but I suspect this will be a very good reference. In Couturat we also find this very useful tidbit on pages 4-5:

1 See HUNTINGTON, "Sets of Independent Postulates for the Algebra of Logic", Transactions of the Am. Math. Soc., Vol. V. 1904. pp. 288-309. [Here he says: "Any set of consistent postulates would give rise to a corresponding algebra, viz., the totality of propositions which follow from these postulates by logical deductions. Every set of postulates should be free from redundances, in other words, the postulates of each set should be independent, no one of them deducible from the rest." (bracket in original: Louis Couturat The algebra of Logic 1914:4-5 footnote 1)

Whether he proves anything, and how he does it, can only be answered by examining the paper. Russell 1903 has limited references to "consistent" or "consistency". Here is one re material implication: "It is plain that true and false propositions alike are entities of a kind, but that true propositions have a quality not belonging to false ones, a quality which, in a non-psychological sense, may be called being asserted. Yet there are grave difficulties in forming a consistent theory on this point . . .."(p.35).

I don't have Schroeder, nor Pierce.

Jevons 1880 (ELEMENTARY LESSONS IN LOGIC: DEDUCTIVE AND INDUCTIVE,WITH COPIOUS QUESTIONS AND EXAMPLES, AND A VOCABULARY OF LOGICAL TERMS. BY W. STANLEY JEVONS, M.A)

"There is in short something in which all sciences must be similar; to which they must conform so long as they maintain what is true and self-consistent; and the work of logic is to explain this common basis of all science" (p. 6)
"Provided it is consistent with the laws of thought there is nothing that we may not have to accept as a probable hypothesis, however difficult it may be to conceive and understand.( p. 271)

Etc. In other words, "consistency" was a well-known issue that presumably goes back to Plato and Aristotle. From Boole's 1854 Laws of Thought we find "a logical inconsistency, or contradiction in the idea itself" but that's about it.

Bill Wvbailey (talk) 17:13, 24 October 2009 (UTC)[reply]

True to form, in a book 1883 Studies in Logic by Peirce, Ladd (before she was Ladd-Franklin), Mitchell, et. al. produce charts all 16 forms of 2 Boolean variables -- anyone who knows what this means will instantly recognize the tables on pages 62 (Ladd's contribution) and 75-76 (Mitchell's contribution). Amazingly the notion of minterm by this time was well developed, as was the notion of "universe of discourse" (cf page 19: "The symbol ∞ represents the universe of discourse. (Wundt, Peirce.)"). Also amazingly, she references Frege. Bill Wvbailey (talk) 22:09, 24 October 2009 (UTC)[reply]

Summary of Proofs:

  • Axioms of propositional calculus [of Principia] are independent -- Bernays 1918 (published 1926) (van H. 1967:264)
  • propositional calculus [of Principia] is consistent -- Post 1921 (cf Post 1921 in van H. 1967:265ff)
  • propositional calculus [of Principia] is complete -- Post 1921(cf Post 1921 in van H. 1967:265ff)
  • Minimal base of propositional calculus -- Zylinski (1925), Post (1941), Wernick (1942) (cf van H. 1967:265)
  • Restricted functional calculus [cf definition in Goedel 1930] is consistent -- Bernays 1926 (cf footnote 2 in Goedel 1930 van H. 1967:582)
  • Restricted functional calculus [cf definition in Goedel 1930] is complete -- Goedel 1930 (cf van H. 1967:582ff)
  • Every valid formula is provable i.e. is either refutable or satisfiable

Wvbailey (talk) 15:41, 25 October 2009 (UTC)[reply]

Cantor

I have removed the sentence about whether there will be an independent statement in the system formulated by Whitehead and Russell -- yes, there is one, the Continuum Hypothesis discovered by Cantor in 1877. —Preceding unsigned comment added by 207.6.250.155 (talk • contribs) 05:40, 7 September 2010 (UTC)[reply]


No Classes

The very first paragraph is very misleading. Type theory is not a theory of types of sets. Russell held a no classes theory. As a theory of sets, types are unmotivated. The trouble is that it's intuitively easy for people familiar with set theory to think of types that way, but it is simply historically inaccurate. Russell speaks of types of propositions and types of propositional functions. PM is not an attempt to reduce all of mathematics to set theory (dodging paradox by types). It rejects an ontology of sets. If anything, it is a theory of propositional functions, however even this ontology is disputed because there is a great deal of historical evidence that Russell envisioned a nominalist semantics for propositional functions through a substitional interpretation of the higher-order quantifiers. To be sure, you have to start somewhere and I don't think the wiki should involve all scholarly disputes, but should the very first paragraph be so misleading?~~ —Preceding unsigned comment added by 173.19.227.52 (talk) 01:42, 20 February 2011 (UTC)[reply]

I'm not sure that what you wrote is correct. But I'm not sure it's entirely incorrect, either. Therein hidden in the tall grass is a difficulty. After his discovery of "the paradoxes" and his publishing of his Principles of Mathematics in 1903 in particular the appendix where he proposed a solution to the paradox he discovered in Frege, Russell went through 4 or 5 iterations of possible "solutions" including his "no-class theory", his "zig-zag theory", his "theory of limitation of size" followed by his "theory of types" and then his "ramified types". (I've not read the papers -- they're obscure--but I have read the relevant letters betweeen Russel and Frege (see van Heijenoort 1967) and the appendix to Russell's 1903 which you can get from "googlebooks"). In the prefaces to the second edition of Principia Mathematica (PM) (ca 1927) he addressed the advances and criticisms between the 1st and 2nd editions of PM with his "fully extensional" notion of "matrix". For a nuanced analysis see the 1944 paper by Kurt Goedel, and an analysis of Goedel's analysis written by Charles Parsons ca 1990, re Kurt Goedel's 1944 paper "Russell's mathematical logic", a reprint of which appears in Feferman et. al. 1990 Kurt Goedel Collected Works Volume II, Oxford University Press, New York NY, ISBN-13 978-0-19-514721-6(v.2.pbk.) pages 102ff (including the paper itself from pages 119ff). Without a close examination of Goedel's analysis (even the commentator admits to difficulties) I'd be hesitant to assert that Russell didn't acknowledge the existence of "classes" that derive from descriptive functions. Bill Wvbailey (talk) 02:17, 21 February 2011 (UTC)[reply]

Do we really need an image for 1+1=2?

The image is simply scanned from a book and has no context from which it follows from Principia - that is, the text has no meaning that ascribes it to Russell. It is a fascinating illustration of the exhaustiveness of the text, but the illustration (.png) itself, if relevant, should just be TeXified and briefly explained (simply how the proposition itself works should be described).

Does everyone understand my argument, because I think it's fairly straightforward - using a .png image for this proof is like scanning some bit of text in a later print of Catcher in the Rye instead of simply using blockquotes. I'll give this a week before TeXifying, unless someone wants to do it earlier. SamuelRiv (talk) 03:43, 22 February 2011 (UTC)[reply]

I think there's some historical interest and some value in the scan itself. A TeX version would not preserve the original typography, which is quite distinctive. —Mark Dominus (talk) 04:26, 22 February 2011 (UTC)[reply]
We can reproduce the typography set exactly with TeX, and the choice of symbols is a standard set for some logicians. Furthermore, can you be sure this scan is from a first edition? If it's not, then I can see no historical relevance. SamuelRiv (talk) 04:46, 22 February 2011 (UTC)[reply]
Can you please elaborate your argument? I do not understand it. Your analogy with "a later print of Catcher in the Rye" seems false to me, because the typography and layout of Principia Mathematica is quite unusual, and that typography and layout is represented accurately in the scan. I am skeptical that you can reproduce the typography with TeX, but I will withhold judgment until I see your attempt.
And yes, it is definitely a scan of the first edition, although I am mystified as to why you think that should matter. —Mark Dominus (talk) 04:59, 22 February 2011 (UTC)[reply]
"The above proposition is occasionally useful" (Russell and Whitehead 1912:86).
Mark I'm with you. When I encountered the image I was fascinated, especially the quote. I don't understand why someone would want to Bowdlerize the original when the original, in the original typography, is still available for sale in the 2nd edition and available as a public-domain pdf through www.books.google.com . Observe also that the frontice-page is also reproduced at the top; per SamuelRiv's argument this be converted to LaTeX as well. But to do so is probably illegal, ditto for any other images derived from the 2nd edition. Volume II (dated 1912) is in the public domain (or at least googlebooks cc'd it and says so, and that's how I got the image directly above) but Volume I is either not in the public domain, or has not been cc'd. Thus I'm considering the fronticepage image of Volume II dated 1912 to be a free-use image, but I'm uncertain as to whether or not the image at the top of the page (the abridged 1927 cc) is covered by fair use. Ditto for the image derived from it that is in the article now. Thoughts?. Bill Wvbailey (talk) 16:36, 22 February 2011 (UTC)[reply]
The fact that there is a proof is interesting, but there is absolutely zero explanation in the article of how it actually works. That said, how it works requires a definition of arithmetic - the rest is a rather simple formalism of how sets combine, so the details of the proof displayed are too muddled in notation (that is defined elsewhere in the book) to illustrate anything other than "this looks like a really complicated way to say 1+1=2, therefore it must be profound." Now that I say that out loud, it actually makes the image seem misleading, putting, as you say, typography ahead of science.
That said, I fail to understand how the typography is interesting: the font appears to be Times, and again the logic notation conventions he uses are still used today.
The quotation in the current image is pretty generic. The quotation "occasionally useful" is interesting and funny, but I think it is done plenty of justice in the article text, since until now there hasn't been an image of that part of the text.
"Bowdlerize"? Are you sure you meant to use that word? If so, please explain how my proposal to TeXify does anything even close to the sort. Illegal to TeXify? It's first a quotation, and since it's factual it doesn't even need to be attributed, and since it's public domain it doesn't even need to be cited!
Why to actually remove the image: it requires additional bandwidth for slower browsers to get a simple text-only image, and more importantly, it has no accessibility to vision-impaired users.
Also, without explanation or, by your assertion, non-"distinctive" typography (that is, layperson-legible longhand), it simply serves to confuse and bewilder the reader into thinking logic is some kind of mysticism, when again I must state that both the proposition and proof are rather simple. SamuelRiv (talk) 02:05, 23 February 2011 (UTC)[reply]

M. SamuelRiv is very unhappy with me, as a cc of the following post shows. M. SamuelRiv seems unaware that often this is the process here in wikipediaville, where we parse-and-respond by paragraph. Thus, the order of what follows is not in the original order, as I parsed by paragraph and responded. Cc'd from my talk page here's M. SamuelRiv's explanation for why he altered the order:

Talk page butchery (Principia Mathematica)
Please don't chop up my comments on Principia Mathematica Talk, as it destroyed the original flow (granted, there wasn't much flow originally) and also made the signature near-impossible to follow. That was really not cool, and strongly discouraged in the future. SamuelRiv (talk) 00:40, 24 February 2011 (UTC)[reply]

As I have not too much emotional stake in this discussion I'll just leave the following as M. SamuelRiv has left it, unless someone wants to revert to what to where we were responding to his various points via the process of indentation-response. BillWvbailey (talk) 01:07, 24 February 2011 (UTC)[reply]

  • Again: it's just an illustrative example of what the contents of the book looks like, should the reader be curious. It's not meant to be the topic of technical discussion. BillWvbailey (talk) 14:29, 23 February 2011 (UTC)[reply]
  • It's what is called an "illustrative example". Its content is not meant to be a topic of a long-winded technical discussion. As in "this is an illustration/example of the type/style/appearance of the topic in discusion". For instance, someone might put in an image from one of Da Vinci's notebooks, "an example" of a sketch, say of a flying machine, without going into detail about its aerodynamic properties were the sketch constructed, nor the physical/mechanical/aerodynamic properties of airplanes or helicopters in general. It's just there for those just curious enough about what the contents of the book (or the authors' works) might appear without going out and buying the relevant text(s), or downloading off www.books.google.com the books in the public domain.
Thanks for explaining this, Bill. I was not sure how to put it, but you have expressed my thoughts as well. —Mark Dominus (talk) 18:42, 23 February 2011 (UTC)[reply]
  • Bowdlerize is the correct description of what you want to do: you want to destroy an image, for some ulterior (ethical? moral?) purpose we cannot imagine. The above image is an image. It is not a quotation. There is a difference with respect to copyright law. Because it is an image, and a faithful one at that, it is susceptible to copyright law, whereas a brief quotation is usually okay for scholarly purposes. The one in the article is apparently in copyright and shouldn't be there, and it should not be Bowdlerized unless the Bowdlerization results in pure text (again, why you want to do this is beyond our comprehension). The above image is not in copyright, so it could go into the article as is an image or be Bowdlerized.
  • I'll give you real-life example of how copyright works in wikipedia: For the article Teapot Dome scandal I wanted to insert a few black and white photographs that I took as an undergraduate student (in the 1960's) from journals such as The Oil Weekly (e.g. a photo of the Teapot Rock that gave the oilfield its name; an image of the camp that set up in the oil fields, etc). At the time developed these and inserted them in a paper I took for a course (this was okay: there was no commercial gain, and it was a "scholarly" effort). Move forward to 2009: with a digital camera I took photographs of the original photographic prints and then enhanced them -- sharpened them, tried to remove some of the blur and the sepia tint. Then I called the magazine (still around after 75 years or so) and asked the editor for permission to publish as public-domain images. The problem was, I missed the year that they fall into the public domain by 1 year, and the editor just gave me verbal permission, not written. I should have had him send me a written, signed permission slip, and then I could have uploaded the permission form as well. So some sharp tool deleted my images. (They are now in the public domain). I think you'll see the blank spots on the article's discussion page.
I don't think there are any copyright problems with this image. Any facsimiles of the content of the first edition of PM, such as File:Principia_Mathematica_theorem_54-43.png, are in the public domain, as the original work was published in the United States prior to 1923, so is itself in the public domain. The cover image File:Pmdsgdbhxdfgb2.jpg at the top of the page is the cover of the work being described ("Cover art from various items, for identification only in the context of critical commentary of that item"), and complies with Wikipedia's non-free content policy under the Fair Use exemption. —Mark Dominus (talk) 18:42, 23 February 2011 (UTC)[reply]
I'm not having a problem with the frontice-page image, myself. But I am uncertain about the 54-43 image. Here's why: I found the image 54-43 in my 2nd edition (1927) on page 360. This is the "abridged version" that as far as I know, is still in print. (my cc was printed in 1962, is in tatters. 1962 was the first time it was printed in paperback: "Paperback Edition to *56". My question is: given that 54-43 appeared in Volume I (ca 1911-1912) [DID IT?] does the fact that it was republished with new intros and etc, in an "abridged" format, move the copyright up to 1927 (2nd edition). Or even up to 1962 when it was published in the abridged version (there is a warning on page xi to this effect --"All cross-referrences in the text, including references to definitions and propositions, relate to the Second Edition (1927) and may not necessarily be found in this abridged version." This strikes me as the sort of the trick used by Disney to keep the copyright alive on Goofy and Mickey, et. al. Or does the clock start at "first use"? (Or does anybody really care?) BillWvbailey (talk) 23:04, 23 February 2011 (UTC)[reply]
  • I think you're grabbing at straws here. Reducing your argument to absurdity: you are proposing that every image in wikipedia must be converted to LaTex or deleted? There must be millions of images. Good luck.
  • BTW I've seen people put banners up requesting no TeX in articles. Here's one from Euclidean algorithm (it's hidden and right at the top): FOR REASONS OF ACCESSIBILITY TO VISUALLY-IMPAIRED READERS (see WP:ACCESS), THIS ARTICLE AVOIDS MATH MODE, UNLESS IT'S NECESSARY. PLEASE DO NOT ADD MATH-MODE FORMULAE, UNLESS YOU ALSO ADD THE CORRESPONDING ALT TEXT AS WELL, E.G., math alt="description". EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION. If you look through the article almost all the math is in "plain text", and as the banner promises, the Matrix method section violates this.

I think that the point of the image, somewhat, is to show just how cryptic the notation was. The article at [2] could be used as a source for a quote like "The notation in that work has been superseded by the subsequent development of logic during the 20th century, to the extent that the beginner has trouble reading PM at all." or "the very notation of the work has become alien to contemporary students of logic, and that has become a barrier to the study of Principia Mathematica." — Carl (CBM · talk) 20:22, 23 February 2011 (UTC)[reply]

My comment was previously sliced up, so I think I need to reiterate that the image is an image of text only. There is no content in the image other than raw text. The point of TeXifying (I should say Unicodifying, since it doesn't need TeX at all) is that it is more user-accessible than an image, because those who are interested in the actual notation can actually follow the code or Unicode conversion if necessary. Let me say this again: the image is of text exclusively, in a non-distinctive typography with not-unusual symbols (that would be faithfully reproduced anyway), and I think is properly served as a block quotation faithful to notation and quotations have nothing to do with copyright so let's please drop that line of discussion. SamuelRiv (talk) 00:58, 24 February 2011 (UTC)[reply]
To me it appears that what you really want is a TeX version of the technical content of the 1+1=2 proof, but that you mistakenly framed this as wanting to remove the historical image of the actual PM.
What if you were to create your TeX version of the text without also removing the image? If you were interested, you could even start a new article, explaining the technical content of the 1+1=2 proof, with your TeX translation as the centerpiece, and link to it from here. This might achieve your goals of making the content more user-accessible. It might address your worry about presenting the logic as something confusing, bewildering, or mystical. I quite agree that the translation could be accomplished without any loss of mathematical content.
I agree that the copyright issues are peripheral, and I also would like to omit them from this discussion. —Mark Dominus (talk) 05:40, 24 February 2011 (UTC)[reply]
RE a new page: Good grief Mark -- you've trod this ground already! I found this in a section above. But no one answered, and I scanned the article and didn't find a link:
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)[reply]
RE demystifying the notation: (This helped me, when I realized that Russell was using Peano as a source; also see his comments in the preface to his Principles of Mathematics 1903). Russell spells out how he adopted his notation -- "The notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modelled on those which he prefixes to his Formulario Mathematico [i.e. Peano 1889]. His use of dots as brackets is adopted, and so are many of his symbols" (Russell-Whitehead 1927:4). For comparison, see the translated portion of Peano 1889 in van Heijenoort 1967:81ff. About the only major change I can see is the substitution of ⊃ for Ɔ as used by Peano.
RE a proof reduced to more-modern symbols: see the link on the page. Bill Wvbailey (talk) 23:32, 24 February 2011 (UTC)[reply]

Why don't you start by reproducing the text here on the talk page so that people can see what it looks like? I need to see whether the alignment is good before I can say whether I think a text version would be appropriate. — Carl (CBM · talk) 22:35, 24 February 2011 (UTC)[reply]

My blog article is not a reliable source in the Wikipedia sense. But there is a detailed discussion of PM's proof of 1+1=2 in Nagel, Ernest (1956). "Symbolic Notation, Haddocks' Eyes and the Dog-Walking Ordinance". In Newman, James R. (ed.). The World of Mathematics. Vol. 3. New York: Simon and Schuster. p. 1894–1900.. —Mark Dominus (talk) 16:51, 25 February 2011 (UTC)[reply]

Section on notation

∙⋁ ∼▪ ■∗∗ ✸ ∾ ⊃ ≡ ⊦ ⊢ ︰

RE typography -- dots: I'm using Arial unicode MS, and having problems. The sharp-eyed reader of the original will see PM 's "big square dots" being used in a different way than the commonplace period. The "big square dots" are indeed square, and they're "bigger". Here is an experiment using big square dots:

" ▪ ", " ■ "

Either would work, IMHO. Unfortunately there are no stacked versions of "big square dots" except maybe this one (it's not a colon) "︰ ". So three dots would be " ▪︰ "

Ergo from page 10 (1927): " ⊦︰▪ p ⊃ q ▪ ⊃ ︰ q ⊃ r ▪ ⊃ ▪ p ⊃ r. " [Note the period which is bringing this example to an end and is not part of the main formula.

RE typography -- assertion ⊦: looks lousy on my browser. The other symbol is ⊢ (right tack). It looks lousy too (bottom vertical is shorter than the top vertical).

RE typography -- star: *1.1 vs ✸1.1. On my browser the 2nd is more faithful to the original.

RE modus ponens: PM's treatment of modus ponens is wonderfully written. But I'm not comfortable with my summary of it. PM writes on page 9 (1927 edition, and more at p. 94):

The trust in inference is the belief that if the two former assertions are not in error, the final assertion is not in error. Accordingly whenever, in symbols, where p and q have of course special determinations,
" ⊦p " and " ⊦(p ⊃ q)
have occurred, the " ⊦q will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of " ⊦q ". It is of course convenient, even at the reisk of repetition, to write " ⊦p " and " ⊦(p ⊃ q) in close justaposition before proceeding to " ⊦q " as the result of an inference. When this is to be done, for the sake of drawing attention to the inference which is being made, we shall write instead:
" ⊦p ⊃ ⊦q "
which is to be considered as a mere abbreviation of the threefold statement
" ⊦p " and " ⊦(p ⊃ q) and " ⊦q ".
Thus " ⊦p " may be read "p, therefore q," being in fact the same abbreviation, essentially, as this is; for " p, therefore q " does not explicitely state, what is part of its meaning, that p implies q. An inference is the dropping of a true premiss; it is the dissolution of an implication". (p. 94).

But then on pages 98ff we see symbolism (sort of) similar to the contemporary . . . so I'm going to strike this from the article until I (or someone) can figure it out. BillWvbailey (talk) 18:02, 25 February 2011 (UTC)[reply]

Mr.M - The big square blocks are absolutely horrendous. A bold colin or period is much closer typographically to the PM. For the sake of the reader's eyes, and not spreading even more fear of the notation than is already present, I'm changing it. — Preceding unsigned comment added by 199.89.180.254 (talk) 03:36, 12 April 2012 (UTC)[reply]

Mr.M again - Please please please please do not revert the dots. Bold periods and colons become sufficiently square as to become as close to identical with the original typography as possible(. : :. ::)! Also, ⊦ and v are far closer than ⊢ V, and I have tried to replace them respectively (especially with the disjunction operator, as otherwise it could be mistaken for the universal class symbol) Lastly, spacing. No space between operators and dots, and space between variables and dots seems to the the closest to the text and most visually pleasing, so I have adopted it. — Preceding unsigned comment added by 199.89.180.254 (talk) 06:19, 12 April 2012 (UTC)[reply]

Article in need of help from experts

In my last edit, I added the Expert-subject template to the top of the article, because even to my non-mathematician's eye, the article appears as a bit of a grab-bag. A great deal of it is devoted to analyzing the notation of the work. While that is useful, a lot more could probably be said about its contents, especially given how important the work is. --Teemu Leisti (talk) 00:54, 10 August 2011 (UTC)[reply]

I was the editor who added most of the content that you accurately describe as a "grab bag". What happened was: I started into it and got daunted by the magnitude of the project. Lately I've been exploring the literature re Logicism (see also User:Wvbailey/Logicism); that's why the most recent additions to the article came about.
What is generally the problem here is that PM is a mess (see the criticism of Goedel 1944). The reasons are two-fold. First, I'm not sure that Russell-the-mathematician had the concept of the editorial "red pencil", i.e. of cutting anything. Everything mathematical that Russell writes is highly repetitive, waffly, unstructured, unlike his philosophy. It's as if he wrote down every twist and turn of his thought-processes and didn't bother to cut away the useless dead-ends. The best way to see how bad his mathematical writing is is to compare it with that of the Formalist writing of Hilbert and Bernays -- what takes Russell 1000 pages to write (1903 PM and 1913 PM) Hilbert can do in 20 pages (you can see examples of Hilbert's writings in van Heijenoort:1967, for example). Second, as a mathematical theory, PM is rotten at the core. I've found criticism in Grattan-Guinness that points out at least one reason why: Russell used language to talk about language (see Bernstein's criticsm in Grattan-Guinness page 454 and subsequent discussion "8.5.4 Groping towards metalogic; he also cites Sheffer's similar criticism). There are other reasons too (pointed out by Goedel 1944, and a close reading of Russell 1903: the refusal to accept the notion of "class" as an object in its own right, the absence of an axiom of pairing and thus no ordered pairs (Grattan-Guinness points this defect out on page 442).
The problem starts with his 1903 Principle of Mathematics, all 500 or so sloppy pages of it. This was supposed to be "Volume I" of two parts, volume I being a verbal "explication" to be followed by the circa 1912-1913 Principia Mathematica with all the symbolism in a formal presentation. But then at the last second Russell discovered the paradox in Frege (Russell paradox), so he had to go back and add a new appendix, modify his 1903 text, and rewrite his introduction. During the years between 1903 and 1912 he failed to resolve the matter of "impredicative definitions" in his own work, and he concocted the "axiom of reducibility" (which Goedel 1944 flatly states as demonstrably false, given Russell's refusal to accept "classess" [sets] as constructed objects in their own right, cf Goedel 1944:140). [This is a serious problem with Russell's logicism, really peculiar; it first appears in his 1903]. But the undaunted Russell published it anyway. In the meanwhile he wrote his 1912 The Problems of Philosophy where the more philosophic issues of his epistemology appear. By 1919 sales were falling off (and the war intervened with his spending time in jail) so he wrote the non-mathematical "Introduction to Mathematical Philosophy", a more precise and coherent version of his 1903 at about 170 pages. In this work he condemns, and virtually gives up on, his "axiom of reducibility", but he had nothing to replace it with. Then along came Wittgenstein (he'd spent his war years in a prisoner-of-war camp writing) with his Tractatus. While at Cambridge Russell and Wittgenstein had been teacher and student + friends, so Russell paid close attention to Wittgenstein's criticism. This he incorporated in to the second edition of PM (see the Intro to the Second Edition and new Appendix C), but essentially this destroyed the effort; from that point on, Russell gave it (mathematics and logicism) up.
A very useful discussion/summary appears in Grattan-Guiness 2000:556ff, i.e. Chapter 10 in The Search for Mathematical Roots, Princeton University Press, Princeton, NJ ISBN 0-691-05858-X.
Bill Wvbailey (talk) 15:06, 10 August 2011 (UTC)[reply]
Teemu Leisti, I think the editors you seek are already working (albeit in solitude) on the PM article. --Ancheta Wis (talk) 15:17, 10 August 2011 (UTC)[reply]
Wvbailey: thanks for the explanation. Perhaps you could add some of your criticism to the article itself? That is, sourced from elsewhere, as Wikipedia allows no "original research". And you didn't delete my signature; instead, I'd forgotten to add it.
Ancheta Wis: OK, cool. Teemu Leisti (talk) 01:25, 16 August 2011 (UTC)[reply]
After I wrote the above, I noticed that you had already followed my advice, Wvbailey. Teemu Leisti (talk) 01:43, 16 August 2011 (UTC)[reply]

Alright, a couple points. One, Wvbailey, your comments about the PM strike me as odd. Considering most of what was written for the Principia by Russell never saw the light of day (he constructed a predecessor to the lambda calculus in his correspondence with Frege, and carried it to quite reasonable level of development before abandoning it for philosophical reasons, as well as several variants of his 'substitutional calculus', and the method of 'quadratic forms' outlined in Principles; see http://people.umass.edu/klement/lambda.pdf), the comment that he had no notion of the editors pen seems a-historical at best, and insulting at worst. This is particularly odd considering his tendency to the exactl opposite vice: he had a fetish for trying to compress a point into as few words as possible, making his philosophical as well as his mathematical writing cryptic and unhelpful without recourse to past works or his private notes. Comparing the work to that of Hilbert is comparing apples and oranges, considering the incredibly different aims and philosophies of the two men.

Two, again Wvbailey, your history is simply incorrect. The history of the axiom of reducibility, in brief is as follows: In Russell's 1905 paper "On Some Difficulties in the Theory of Transfinite Numbers and Order Types" Russell provides the first tentative suggestion of his substitutional theory. In the theory, propositions are reified in a manner suggested in Principles, while denoting concepts are now capable of being done away with thanks to the theory of definite descriptions. As a result, propositional functions are proxied through the operation of substitution of entities in propositions. The calculus proxies a type stratified functional calculus by treating propositional functions as incomplete symbols, while preserving a single style of variable. Finding the system philosophically satisfying, he starts attempting to construct a proxy for set theory in the system. Due to difficulties in dealing with the system, Whitehead objects, and simply typed functional calculus is taken as the official system, with the substitutional theory expected to be set forth in an appendex as a philosophical explanation for types. After his 1906 "On the Substitutional Theory of Classes and Relations", Russell discovers a contradiction which has been named the p0/a0 paradox, and is forced to modify the substitutional theory by adding orders to propositions. In particular, in "Mathematical Logic as Based on the Theory of Types" which you will find in (van Heijenoort), you will see the ramified heirachy briefly explained in terms of substitutional theory, before passing on to the functional heirarchy as more convenient in practice. Ramification cripples the system however, so Russell adds the axiom of reducibility to restore the power of the system, while preventing paradoxes from forming in the substitutional base of the theory. For more, see Landini's Russell's Hidden Substitutional Theory, and Hylton's Propositional Functions and Analysis.

Three, in general. The Principia Mathematica is a difficult work. It's hard. It was written by two of the greatest philosophers of their age, and was written before the use of arithmetization emerged as a standard method for dealing with syntax, and Tarskian semantics was adopted as a lingua franca for the field. As more work is being done outside those strictures, it is increasingly being viewed as likely that substitutional quantification and truth-value semantics is what gives the clearest reading of the PM in terms of the introduction, and with this in mind, the denunciation of the system as "rotten at the core" is beginning to seem premature. It may very well be either that is the case, but the point is that simply denouncing the entire thing as muddled and confused is not a good way to begin an attempt to understand the work itself. — Preceding unsigned comment added by Monadologiser (talk • contribs) 00:20, 2 November 2012 (UTC)[reply]

Reorganizing the Article

I think much of this article needs to be rewritten and organized into new sections. The article has no coherent overal structure, and I'm afraid that if I were to try to add one in piecemeal, what is currently there would seem out of place and incoherent. Since the edits I have in mind are pretty sweeping, I thought I would just propose a format, and open discussion about how to best make it fit the pattern, rather than take a chainsaw to the whole thing.

Alright, as I see it, the following points need to be touched on: The prehistory of the PM and its development. The former should mention the algebraic logicians, Peano, and Frege. The second part of that should try to show the development from the Principles of Mathematics to publication: discussing the paradoxes, the ontological shift from the theory of descriptions, and Russell's various proposed and attempted resolutions during that period. I mention this in particular because of his substitutional calculus, which offers the clearest account of the notion of a propositional function, and why he treats the symbols the way he does, as well as supplying his original reasons for ramification. The theory treats propositional functions as incomplete symbols, with the entities quantified over being propositions (which he saw as mind independent entities akin to facts or states of affairs), and a primitive 4 place predicate (p/a);x!q, reading "q is like p except containing x wherever p contains a". After that, there should be a brief description of the philosophy of mathematics underlying the system. Next a summary of each volume. If anyone is fammilar with the third volume, I could use some help writing on the construction of real numbers, which is highly original, and deserves to be described. After that, should come the aftermath. The impact on Carnap and Quine, Wittgenstein's and Ramsey's criticisms, comparison of the construction with Tarskian semantics, etc. Honestly though, I'm inclined to funnell any further criticism passed that to its own page, as the book is big and difficult enough as it is. Maybe a page comparing the system of the Principia with various others in the literature (Quine, Church, Tarski, Godel, Rosser) might even be called for. Also, changes to the second edition. Finally, I thought it might be fun to end with some of the humorous quotes alluded to by G.H.Hardy in his review. The following sources I think will be useful: G. Landini - Russell's Hidden Substitutional Theory ; P. Hylton - Propositional Functions and Analysis; Russell - Essays in Analysis (compiled by D. Lackey), Principles, and obviously PM; van Heijenoort - From Frege to Godel; Quine - Set Theory and It's Logic, Mathematical Logic, System of Logistic; in addition, there is the Stanford Encyclopedia of Philosophy, and the website of a Russell scholar with a lot of his papers http://people.umass.edu/klement/works.html (Note the "please do not cite without permission" for works in progress")

So, the proposed skeleton: I.Introduction II. History a. Contemporaries and the state of logic (summary, linking to ...) b. The development of the system of the PM. III. Philosophy of Mathematics (summary, linking to logicism) IV. Vol 1 a. Propositions, and propositional functions b. Quantification, *9 and *10 c. Reducibility d. Incomplete Symbols, definite descriptions, scope, contextual definition of classes, and typical ambiguity. e. Theory of classes, relations, functions, etc. V. Vol 2 etc. VI. Vol 3 etc. VII. Changes to the Second Edition VIII. Reception and Criticisms IX. Quotes and Humor

If this were to be done, it should probably be stored elsewhere until finished, but I'd appreciate feedback, suggestions, and offers to exploit people for free intellectual labor. — Preceding unsigned comment added by Monadologiser (talk • contribs) 02:34, 2 November 2012 (UTC)[reply]

It's a brave proposal. There could be a sandbox or a working page which could be used to hold the text. I'm a °little worried about hindsight bias. For example the article's notation might be better served if side-by-side versions PM vs other were employed. §--Ancheta Wis   (talk | contribs) 02:56, 2 November 2012 (UTC)[reply]
Sandboxing is a good idea. Also: limit your scope a bit, it is too broad. For example, there's an almost ten-year lead-up to the writing of PM, and PM itself. I'd concentrate on PM itself, in particular how to present the arcane symbolism. Then if you still have the energy, work on the "lead-up". With regards to PM's critics: If you can get your hands on Goedel's 1944:119 Russell's mathematical logic, and Parson's note to this paper on pages 102ff, these might change your thinking about the quality of Russell and Whitehead's work (Kurt Goedel Collected Works Volume II); in fact Russell writes to the editor about Goedel's criticisms and says "[I] think it highly probable that many of his criticsms of me are justified. The writing of Principia Mathematica was completed thiry-three years ago, and obviously, in view of subsequent advances in the subject, it needs amending in various ways. If I had the leisure, I should be glad to attempt a revision of its introductory portions, but external circumstances make this impossible. (p. 102). Also the biography, the big one by Ronald Clark, discusses the quality problem.
RE your edit: the symbolism of a --> b --> c is utterly different than a < b < c. In the case of the logical symbols, their syntax is critical: to evaluate the expression the symbols are assigned truth values, and the outcome of a --> b is a "true, false" that can be chained. Unfortunately, the order of the chaining needs to be controlled by a syntactic (i.e. a formation-) rule. In the case of (a < b) < c, this has no meaning because it mixes a truth-functional outcome (a < b) might yield a "true" or a "false", but you cannot chain it, "truth" < c is meaningless. Because his formal symbols and their formation rules were insufficient Russell was forced, to define the symbol-string a --> b --> c as logically equivalent to (a --> b) & (b --> c) (§3.02); this should have "fallen out" of pre-established formation rules (e.g. see Kleene 1952:69-71). BillWvbailey (talk) 15:36, 2 November 2012 (UTC)[reply]

On the chaining of implications: First, I cannot find the Kleene article you are talking about; giving an actual name is helpful. I checked my copies of metamath and mathematical logic, but neither seemed to have something relevant on the cited pages, so I am at a loss as to what you are referring to. For want of this information, I am forced to delay my conclusion that you are saying nonsense. The analogy between A -> B -> C and x < y < z is just the parsing of either into (A->B)^(B->C) and (x < y) ^ (y < z) respectively, which was a shared notation among the Peanese disciples. Yes, the expression was defined, but there is no ambiguity in reducing that to primitive notation, so truth does not enter into it. The actual process of 'chaining' is an abbreviation for proofs relying on mp. Whether definite descriptions or class expression involve such an ambiguity may be up in the air for now, but I am utterly at a loss to understand what you are talking about. Arguing though that Kleene set up a system in 1952 where the abbreviation would not work is not an effective means of arguing that it is actually ambiguous in the PM.

Kleene 1952 is a book, Introduction to Metamathematics, North-Holland Publishers (see Ancheta Wis entry below). Kleene's formal system was derived mainly by the Formalists including Hilbert and Bernays(as a reaction against Russell's logicism) and adopted by Goedel among others, starting in 1904; you can find Hilbert's first paper and the evolution of his thinking in van Heijenoort. Another useful reference (because it has papers by Bernays in it) is Mancosu 1998 From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, Oxford Universtiy Press, ISBN 0-19-509632-0 pb. The point is: Russell didn't distinguish between the theory and the metatheory (see more below). E.g. you adopt a system that consists of a collection of signs that include parentheses and make composite symbols by means of formal syntax and two formation rules, modus ponens and substitution (there's a lot about this problem in Grattain-Guiness, ref below). such a system will not allow ambiguous expressions to be created: thus your system can produce ((a --> b) --> c) or (a --> (b --> c)) but never a --> b --> c, nor (a --> b --> c) etc. Russell adopted Peano as his mentor (see Russell 1903); whereas Frege's work has a formal syntax, Peano does not, and PM adopts the intuitive Peano symbolism with all its failings (see quotes below). Similarly, the logic of PM does not have a formal syntax. This is what van Heijenoort and Quine have to say about this point. Re Peano:
From van Heijenoort 1967:84 preface to Peano's The Principles of Arithmetic:
"There is, however, a grave defect. The formulas are simply listed, not derived; and they could not be derived, because no rules of inference are given. Peano introduces a notation for substitution (V 4, p. 91), but does not state any rule. What is far more important, he does not have any rule that would play the role of the rule of detachment. The result is that, for all his meticulousness in the writing of formulas, he has no logic that he can use . . . In the proof [van H. gives an example] (and it is typical of Peano’s proofs, the passage from formulas (1) and (2) to formula (3) cannot be carried out by a ‘’formal’’ procedure; it requires some intuitive logical argument, which the reader has to supply. The proof brings out the whole difference between an axiomatization, even written in symbols and however careful it may be, and a formalization [etc. van Heijenoort goes on for another column about this failing]." (p. 84).
This sort of “intuitive” logic is contrasted, by van Heijenoort, with that of Frege (cf page 4 in van Heijenoort).
And then there is the problem in PM mentioned by Wittgenstein in his Tractatus -- the distinction between a "theory" -- a formalized system -- and the "metatheory":
”3.3 In logical syntax the meaning of a sign ought never to play a role, it must admit of being established without mention being thereby made of the meaning of a sign; it out to presuppose only the descriptions of the expressions.
”3.331 From this observation we get a further view – into Russell’s theory of Types. Russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak of the meaning of the signs. [etc]
”5.4 Here it becomes clear that there are no such things as “logical objects” or “logical constants” (in the sense of Frege and Russell).
”5.452 The introduction of a new expedient in the symbolism of logic must always be an event full of consequences. No new symbol may be introduced in logic in brackets or in the margin – with, so to speak, an entirely innocent face.
”(Thus in the “Principia Mathematica” of Russell and Whitehead there occur definitions and primitive propositions in words. Why suddenly words here? This would need a justification. There was none, and be none for the process is actually not allowed).
From this we can understand, in van Heijenoort 1967:151, Quine's preface to Russell's 1908 Mathematical logic as based on the theory of types:
"Failure to distinguish thus between open sentences on the one hand and attributes and relations on the other had grave consequences for this paper and equally for Principia mathematica, for which this paper sets the style . . . Russell's failure [re the ramification of types] is due to his failure to distinguish between propositional functions as notations and propositional functions as attributes and relations. [etc. for another half column, where Quine also implicates Ramsey for a similar lack of understanding.] (p. 153)
With regards to a critical examination of and response to PM there’s a very useful chapter to be found in Grattain-Guinness 2000 The Search for Mathematical Roots 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Goedel, Princeton University Press, ISBN 0-691-05857-1 (alk. paper). This chapter is titled “Chapter 8 Influence of Logicism, 1910-1930’’ and runs from pages 410-505. With regards to Wittgenstein’s Tractatus, and Russell’s response:
”From their earlier discussion Wittgenstein must have realized that Russell had mixed logic and logicism together . . . Russell [countered] “that every language has, as Mr. Wittgenstein says, a structure concerning which, ‘’in the language’’, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit” (1922a,xxxii).
”To Russell this proposal extended his propositional hierarchy of types (1940a, 62). But to us a feature of capital importance to philosophy was proposed here: a general and fundamental distinction of language from metalanguage, and by implication (as it were) of logic from metalogic, and of theory from metatheory.” (p. 437)
So I repeat: read the 1944 article by Goedel. If you're going to craft a really exceptional article about PM I suggest you adopt a balanced-but-critical and skeptical outlook. Also, a technical wiki-issue: we're supposed to use secondary sources, e.g. commentary by van Heijenoort or Quine or Goedel or Grattain-Guiness or Mancosu, not the primary sources. You're not supposed to interpret the primary sources -- this is "original research" (OR) and this transgression is punishable by wiki-death. But sometimes this becomes difficult avoid; I suggest quotations from the originals to hammer home a point. Bill Wvbailey (talk) 22:52, 7 November 2012 (UTC)[reply]

Currently I am digitizing several of Russell's early articles on substitution and propositional functions for wikisource, and will respond more later. Monadologiser (talk)

74, the article lists Stephen Kleene 1952 Introduction to Meta-Mathematics, 6th Reprint, North-Holland Publishing Company, Amsterdam NY, ISBN 0-7204-2103-9. I personally use the wikipedia page for ISBNs to get the locations of print copies. --Ancheta Wis   (talk | contribs) 00:51, 7 November 2012 (UTC)[reply]

I have read the article by Godel, and I have read the Graham Guiness. I have been, and continue to be skeptical about the Principia, as a former member of the Church of WVO Quine, who in my opinion does a much better job of making the PM look like a pile of muddled confusion. On the other hand, I have read the PM, which you don't seem to regard as very much important to the job at hand, prefering hermaneutics of a Godel article instead. Personally, the points which have convinced me to the contrary are Landini's work cited above, some papers by Klemmet also cited, and Kripke's 'Is There a Problem With Substitutional Quantification?', for arriving at a construction very simmilar to the rammified hierarchy with substitutional quantifiers, and suggesting that the objectual reading of the quantifiers is misplaced.

Now, your points, one at a time. On the subject of p -> q -> r. You are mistaken. The expression is not a primitive expression of the language of the principia, in any reasonable interpretation. For WFFs A, B, (A v B) is a WFF. (A v B v C) is simply ill-formed. As such, since it is not an expression which can occur naturally, it is the perogative of the author to use it as a defined expression. There is nothing contradictory in rendering (A v B v C) as ( ~( ( (A<->B) <-> C ) v B) . A ), and if he or she is feeling sufficiently malicious, it is a perfectly valid definition schema. However, ( A -> B -> C ) can never be rendered as (A -> B) -> C in the Principia, since as you well know, the definition supplies the rule that it be rendered (A -> B).( B -> C). There is nothing here which speaks to the meaning of the signs. The comparison to < is only insofar as the superficial simmilarity of notational conveniences for chaining. I do not know how to say this forcefully enough: you are wrong. If you actually read the book whose page you have deigned to grace with your presence, you would realize this.

As to the distinction between object language and meta-language in the Principia, while I recognize that this is the interpretation of van Heijenoort, I personally disagree with it, although I recognize that it is a popular reading which needs to be recognized. While it is obvious that the system is not set forth as an application of the first order thory of concatenation, it does not seem fair to judge the work at this venture on those grounds alone. Instead, I agree with Landini in interpreting most of the awkwardness of the Principia as stemming from the fact that most of his previous work had been on a calculus of objectual propositions, and subsequently being still influenced by the force of habit. The nominalistic semantics there suggested has been increasingly favored as an interpretation due to the fact that it actually makes sense of the introduction, which most interpretations do not. Still, this is a technical matter, and if you want to see the arguments for it presented in greater detail, see Landini's Russell's Hidden Substitutional Theory. For a conflicting modern interpretation, see Linsky's Evolution of the Principia Mathematica [he also did the SEP article on the notation].

With regard to Quine's criticisms, see my historical marks at the end of the previous section [i.e. article in need of help from experts]. Russell's work on substitution explains a great deal about what sort of thing a propositional function was supposed to be, as referenced even in Mathematical Logic as Based on the Theory of Types, which you cite. This is not original work on my part. But this strongly opposes Quine's interpretation of the quantifiers of PM as ranging over attributes in the sense which he uses the word. In addition, it provides syntactic reasons for his simultaneous introduction of the axiom of reducibility and the ramified hierarchy. While I agree in general with Quine's criticisms of propositions as entities, there is nothing syntactically shady about the theory any more than there is with modal logic, which he similarly despised semantically, which respecting its syntactic viability.

To conclude, the page called for an expert with big letters on the top of the page, and I thought it would be a nice thing to help out, since I have benefited so much from Wikipedia throughout my education. I am a graduate student in logic with a special interest in the history and philosophy of the subject, I am one of the few people who has actually read the book in question, and I am caught up with, and involved with, the contemporary scholarship on the subject. I am also familiar with the history of its interpretation, ranging from Church to Godel to Quine, etc. As such, I am in a decent position to present the multiple conflicting readings of the subject which are now common aside one another for the purpose of the article. While I appreciate the need for communication, as a scholar of the history of analytic philosophy, please respect the fact that I often have multiple opposing sources saying things that disagree with one another. I congratulate you on having read a few books and articles on the subject, but my not agreeing with them is not evidence of not having read them. Please keep that in mind. What you cite as fact are the interpretation of individual scholars, which I have spent some time weighing against one another with regard to merit, and which I personally find less convincing than others.

— Preceding unsigned comment added by Monadologiser (talk • contribs) 07:38, 8 November 2012 (UTC)[reply]

You seem a bit defensive. Nobody's challenging your expertise. Nobody's standing in your way to be an editor. It would be good to register as a full-fledged wikipedian, though. You are a newbie and us old folk are supposed to treat newbies gently, a challenge to do in your case :-). As a really old guy who's been at this since Jan 2006 I've learned a few things. To survive as a wikipedia editor you go through a painful phase of growing a very thick skin. This isn't the same as writing a paper or a thesis: others are going to challenge and change your writings (just as you want to change this article), they're going to revert you on occasion. In a few years some young whipper-snappers's going to come along and tear up your work, and you're going to have to sit and take it, and sometimes you're going to end up arguing with them on the talk pages. I had to develop some skills with respect to this peculiar type of collaboration -- it's more like forming alliances in politics (I've been there, done it) -- e.g. not insulting other editors or making assumptions about their expertise, but arguing only from the facts (e.g. quotes from sources). (In that regard: actually I've read all of the first volume of PM up to about section D, plus the Appendix A and C and the 2nd edition intro, plus hunks of his 1903 esp null and unit class stuff and the addendum where he tries and fails to patch up the paradox, his 1912 (a decent piece of writing) and Dedekind's Nature and Theory of Numbers, and Peano and Frege and Hilbert etc (in van H), and this is just a fraction).
I like the different take on PM that you're suggesting. I was a fanboy of Russell's (especially his writing style and some of his epistemology) until I encountered his 1903 and his failure to stop right there and fix it. (Again, the biography talks about Russell's leaving loose threads hanging and plowing ahead with the threads still hanging out, to get caught and trip him up later). So instead of fixing the problem or abandoning his "logicising" he farts around for 8 or so years (while falling in love with Whitehead's wife and maybe having an affair, nice guy) and eventually ends up in his no-class theory. And as Goedel 1944:132 notes, "The result has been in this case essentially negative; i.e., the classes and concepts introduced in this way do not have all the properties required for their use in mathematics . . ..". Whether you agree or not isn't the point: this is an encyclopedia that needs to present as many scholarly points of view as there are legit published sources. It's a huge topic and a huge undertaking. So have at it. Good luck. Bill Wvbailey (talk) 16:11, 8 November 2012 (UTC)[reply]

I actually am coming from the opposite perspective. I was a fanboy of Tarski and Quine, and thought the PM was a muddled pile of tissue until a certain professor showed me some things I hadn't noticed. With the recent publication of his manuscripts from the period during the writing of the Principia though, it has become increasingly apparent that Russell's thoughts on logic during that time were far different than supposed. In brief, Russell appears to have constructed, in short succession, a wide variety of systems which never saw the light of day. Please see http://people.umass.edu/klement/lambda.pdf for one notable example. The one which I personally find the most fascinating is the substitutional calculus, as aside from Quine's System of Logistic, I know of no other noteworthy system with reified propositions. But each seems to have left an indelible mark on the system of the Principia, including the ramified hierarchy. With these in view, it is finally possible to give an account of the syntax and semantics of the Principia which doesn't conflict horribly with the introduction. I don't plan on introducing all of this into the article of course, but there are certain aspects which are relevant specifically to the controversy over what propositional functions are, and I think a few paragraphs should be sufficient. But the issue of the nature of propositional functions is one of the key points of both difficulty and intense criticism, so providing background on that is rather important to understanding the syntax of the system. I really suggest reading the Landini book.

What I think though is that the article itself should not explicitly take the stance that the content of the PM is nonsense. In all honesty, I consider the task to be more akin to that of Biblical interpretation, or of interpreting a lousy work of fiction charitably in assuming for the sake of argument that it has a plot (the twilight saga springs to mind). The process of doing so requires the capacity to stomach the writing of the person in question, and to attempt to present a decent narrative when the whole thing is over. That being said, since the work is so large and complicated as it is, and considering the sheer volume of criticism, I thought a separate page for criticisms of the Principia might be called for. I've got quite a lot of material from Quine, Wittgenstein, Church, Curry, Putnam, Kripke, etc. which could easily fill such a page, and you probably have a lot more. — Preceding unsigned comment added by Monadologiser (talk • contribs) 20:34, 8 November 2012 (UTC)[reply]

I'm printing off the Klement paper as I write this, will puruse the Landini if it doesn't cost too much. I agree with everything you've written in the last paragraph. In particular the idea of splitting off a page for the criticism. A similar thought: to split off the symbolism discussion into a sub-article. In that regard: it is here and the next point where I see the show-stoppers: First, how do you present PM to a student/reader who is not familiar with the work and will be thoroughly intimidated by it when (and if) they develop the courage to open the cover and don't slam it shut in a panic. And rightfully so: I don't know which presentation is more offputting, Frege's presentation or PM's. [Frege's, probably, is worse. To Russell's credit, though, the ~, V, ⊃ are perfectly fine, as are the ( ) to indicate "for all" and ∃. But the dots are just awful, and the mixing of dots and logical AND is a capital offense. And the hats over the symbols and the horseshoes with dots and etc etc, just nasty.] It may seem strange to a modern logician, but rather than the contemporary symbols (, ), &, V, ~ or "bent-bar" , we engineers adopted and were taught the Boolean forms e.g. xy =def (x & y), x+y =def (x V y) and a line over a symbol or an expression means "not", as does an apostrophe as in x'(x+y)'. And we get along just fine with this, sending men to the moon, designing computers etc etc. I just scanned my cc Peirce and Ladd-Franklin 1883, and I see the exact same "Boolean" symbolism that we learned in college. So it appears that an entire profession rejected the chicken-scratchings that Russell presented them with. (This is actually pretty interesting, this split in symbolism; you see it in some very important early enginnering papers such as Shannon, Veitch and Karnaugh and the symbolism is still with us e.g. Karnaugh map). So maybe something can be done to help with the symbolism (what really needs to be done by a scholar is convert PM to modern symbolism(s), and republish it; see the next comment in a similar regard).
Another problem is what to do about the 2nd edition. Really, the second edition should have been a stand-alone text, and not a reprinting of the first edition together with Appendices, so that you have to go through and expurgate this and that and then add that and this. The fact that a student needs to (1) be aware that you have to do this, and (2) then actually do this [which means they have to read the "Intro to the Second Edition" very carefully] just adds to the angst (anger, frustration). I think that an upfront admission that PM's presentation is really offputting would help; I've seen writers call it "hen scratchings" or something similar but I'll be damned if I can remember where. Bill Wvbailey (talk) 16:59, 9 November 2012 (UTC)[reply]

I must confess right off the bat that I disagree with your assessment of the symbolism. But first, a historic point. Almost all of the notation for the PM is either directly adapted from Peano, whose notation was beginning to be quite popular, or due to Whitehead, who was almost as fetishistic about his notation as Peano. And the '.' for 'and' was actually supposed to be reminiscent of the usual dot for multiplication, while staying distinct [it was constantly referred to throughout the book as 'logical product']. To blame the PM on this count is unfair, as to provide a completely new notation that not even a Peanese disciple (I'm tempted to write Peanist) could read would be to cast themselves off from close to the only friends and supporters they had. I think though most people, especially engineers, (and suprisingly mathematicians) are deathly afraid of any new notation. The fact that a lot of people find the notation of the principia frightening and frustrating speaks to this irrational intolerance. As such, people tend to look on any old notation which does not agree with the current as horribly complicated and abtruse. But this it shares with cultures and languages generally, and I don't think the page should begin with a disclaimer any more than I think the page on Shakespear should open with a recognition of how frustrating the language is for almost all school children and most adults to make sense of, or a page on Aristotle should open with a statement of how abtruse and unnatural his terminology is (even if both are true).

I will submit G.H. Hardy's review of it [ http://www.cl.cam.ac.uk/~ns441/files/hardy-principia.pdf ], which sums up my stance pretty well. The notation is not difficult to learn, and if you actually write out the proofs and formulae by hand, I think you will be pleasantly suprised at how easy the dot notation becomes with a little practice. I agree that a page for the notation might be called for, but that depends on one point. The article on the notation for the SEP is exceptional, and I would be tempted to just include a link to it in the text and say that this is perhaps the best resource to learn the notation. I'm not sure what the policy is though on doing such things, so perhaps making a page for it would be the next best thing.

On the second edition: I am not quite sure what to make of it. When I read what is written, what I see is an admittedly incomplete sketch about how certain potential revisions to main body of the text (suggested by Wittgenstein) might be carried out if one were so inclined (as Russell was increasingly sympathetic to the extensional view). This includes a detailed appendex on how the theory of quantification might be reformed. What I don't see is a rewriting of the entire body, or even a claim of having rewritten it, so I tend to view it as just that. It ends with a somber note that most of the upper reaches of the system would fail (just as he notes in his introduction to the Tractatis), and that it remains unknown whether the system could be so adapted without adverse consequences. Yet there are those who claim it intended more, but they are so at odds with what I see written that I am unable to comprehend them. To me it seems like an atheist accepting the ad-hoc Christian interpretation of the old testament as predicting the coming of Jesus, only for the purpose of criticising the old testament since he obviously doesn't believe the predictions of the coming of Jesus were correct. There are far fairer methods of criticism than this I believe. — Preceding unsigned comment added by Monadologiser (talk • contribs) 20:17, 9 November 2012 (UTC)[reply]

How many pages?

Whats the total page number of each edition? -194.138.39.54 (talk) 11:16, 7 October 2013 (UTC)[reply]

Edit Request

In the section "The construction of the theory of PM", I do not understand what is meant by the sentence starting with:"Another observation is that almost immediately in the theory...". Either this sentence was butchered or the translation (from the German??) failed. My point is that observation .. in the theory will have no clear meaning to the average reader. Could someone rewrite this to make it intelligible? Its NOT a technical issue, its about using correct English. Thx.173.189.78.173 (talk) 16:02, 26 April 2015 (UTC)[reply]

I too was disturbed by this phrasing when I read this sentence, and I also thought that the awkward formulation may have come from the deletion of part of the phrase, but after giving it some thought I think that the sentence is grammatically correct but just very awkwardly put. I believe that it is meant to say "Furthermore in the theory, it is almost immediately observable that interpretations...". Do you (or anyone who may come across this thread) think my proposition would be an adequate change? Given that this thread was started years ago, I'm not so sure the original poster will ever see this response, but given that I believe my proposition to be clearer and more legible than the current form of the sentence, I will implement my change now.Tommpouce (talk) 20:50, 30 May 2021 (UTC)[reply]

Missing comment?

In the past, I've seen the quote (I think it was Russell but it could have been Whitehead's) about what an enormous waste of time their 10 year effort to write this had been. It's important enough, I think, to be included here. I've not been able to find it after a brief search and besides I don't know if he maintained that POV or it was possibly just part of the "intellectual exhaustion" alluded to in the article. Either way its important, imho. Oh, while I'm at it, the fact that the first printing was published at a loss, subsidized by the authors 50 each, the Royal Society 200 and 300 (pounds) by Cambridge University Press (see Wikipedia article on Whitehead) should also be included here, certainly (in fact that whole section should be included).173.189.78.173 (talk) 16:30, 26 April 2015 (UTC)[reply]

I should also add that http://plato.stanford.edu/entries/principia-mathematica/ (which is an external link (I think) of the article) is imho a far more clear exposition about Principia than this article. You'll profit much more from reading that than this, although this goes into the nuts and bolts (so deeply, in fact, that this looses sight of the forest through the trees).173.189.78.173 (talk) 17:18, 26 April 2015 (UTC)[reply]
Was Gottlob Frege's work a waste of time as well? Remember, it was invalid. The work Russell & Whitehead did was exhausting for them, and fit for machines. Herbert A. Simon records Russell's thankful reaction to Simon's program, that there must be a better way. --Ancheta Wis   (talk | contribs) 19:22, 26 April 2015 (UTC)[reply]

✸

What is with the overuse of what Wiktionary calls the "heavy eight-pointed rectilinear black star"? What does it even mean? I've never come across it in an article before and can't find anything online, which worries me regarding the average user! UaMaol (talk) 20:00, 2 April 2017 (UTC)[reply]

It's a specific character of PM. Millbug talk 22:56, 5 April 2019 (UTC)[reply]
It would be good to mention this in the article and explain it's meaning. --Meillo (talk) 12:23, 5 December 2021 (UTC)[reply]
A better asterisk is ✳ (U-2733, bolded). It's 8-pointed but doesn't overkill with the black blob in the center. Milt (talk) 20:09, 6 April 2019 (UTC)[reply]
However, when it is inserted on an article page, it is not bolded, too light and slanted. The closest ✳ to PM is in the 2nd example at

https://tex.stackexchange.com/questions/157416/typesetting-famous-54-43-proof-11-2-in-principia-mathematica. --Milt (talk) 11:25, 10 April 2019 (UTC)[reply]

catch 22

Is there really a need to treat contradictions as being "some sort of catch 22"? it's strange, to say the least

are people unaware that contradiction exists outside of that popular novel? — Preceding unsigned comment added by 188.26.59.167 (talk) 13:50, 27 August 2018 (UTC)[reply]

Possibly incorrect translation of a PM formula into modern notation

IMHO the original translation (as of 2020-04-05 10:37:51 UTC) for the formula: 

   p⊃q:q⊃r.⊃.p⊃r

as 

   ((p⊃q) ∧ (q⊃r))⊃(p⊃r)

was incorrect.

The rule, as stated in PM vol. I (2nd ed., 1963, https://ia800602.us.archive.org/35/items/PrincipiaMathematicaVolumeI/WhiteheadRussell-PrincipiaMathematicaVolumeI_text.pdf [viewed 2020-04-04]) reads:

"The scope of the bracket indicated by any collection of dots extends backwards or forwards beyond any smaller number of dots, or any equal number from a group of less force, until we reach either the end of the asserted proposition or a greater number of dots or an equal number belonging to a group of equal or superior force." (p. 9)

Thus, the scope of ":", even though if is in group III which has lower priority as group I single dots in the formula, still extends to the end of the formula over the smaller number of dots (single dots round the '⊃' symbol).

Also, the example on page 10 of PM is very thorough:

"“p⊃q:q⊃r.⊃.p⊃r“ will mean “if p implies q; and if q implies r then p implies r.“ (This is not true in general). Here the two dots indicate a logical product; since two dots do not occur anywhere else, _the scope of these two dots_ extends backwards to the beginning of the proposition, and _forwards to the end_." (emphasis mine)

I've fixed the formula. The text should probably also be amended, "having the same priority" -> "having higher priority".

Please double-check. Pterodaktilis (talk) 16:50, 5 April 2020 (UTC)[reply]

Lead section is too obscure

The lead is full of trivia, obscurantism, and is not written for the general reader. On the plus side, it does seem to contain all the relevant points buried in this muddled mess. Viriditas (talk) 00:09, 22 October 2021 (UTC)[reply]

The problem is with the work itself. The obscurity of the article is a result of trying to describe the work in admiring terms, rather than giving an honest critique from the perspective of Godel, Wittgenstein, set theory, and mathematics as actually practiced, which all ought to be promoted to the lead to explain why this book is not worth studying.
To call Russell's principia trivial would be undeserved praise: it is merely unsound. basically unsound. Jaredscribe (talk) 22:43, 13 November 2022 (UTC)[reply]

"contemporary"

Section 2 uses the word "comtemporary" a lot. However, it is not clear whether it is using the word in the sense of "contemporary with PM", or in the sense "present-day". I suspect the latter, in which case it is confusing, and should be replaced by "modern". Since I'm unsure, I haven't made the change. JCBradfield (talk) 09:58, 15 March 2023 (UTC)[reply]

Isn't the article too praiseworthy to be neutral

Such as "There can be no doubt that PM is(...)" 37.60.109.133 (talk) 15:36, 28 January 2024 (UTC)[reply]

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