Editing Principia Mathematica (section)

{{Short description|3-volume treatise on mathematics, 1910–1913}}
{{For|Isaac Newton's 1687 book|Philosophiæ Naturalis Principia Mathematica}}
{{distinguish|text=[[The Principles of Mathematics]] – another book by Russell, published in 1903}}
{{Italic title}}
{{Use British English|date=September 2013}}
{{Use dmy dates|date=December 2024}}
[[Image:Russell, Whitehead - Principia Mathematica to 56.jpg|200px|right|thumb|The title page of the shortened ''Principia Mathematica to'' ✱56]]
[[Image:Principia Mathematica 54-43.png|thumb|right|500px|'''✱54.43''':
"From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2."  – Volume I, 1st edition, [http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath&cc=umhistmath&idno=aat3201.0001.001&frm=frameset&view=image&seq=401 p. 379] (p.&nbsp;362 in 2nd edition; p.&nbsp;360 in abridged version). (The proof is actually completed in Volume II, 1st edition, [http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath&cc=umhistmath&idno=aat3201.0002.001&frm=frameset&view=image&seq=126 page 86], accompanied by the comment, "The above proposition is occasionally useful." They go on to say "It is used at least three times, in ✱113.66 and ✱120.123.472.")]]

{{quote box|width=35%|right|quote=I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of ''Principia Mathematica''. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated....
|author=[[G. H. Hardy]]
|source=''[[A Mathematician's Apology]]'' (1940){{sfn|Hardy|2004|p=83}}
|salign=right
}}
{{quote box
| width = 35%|right
| quote = He [Russell] said once, after some contact with the Chinese language, that he was horrified to find that the language of ''Principia Mathematica'' was an Indo-European one.
| author = [[John Edensor Littlewood]]
| source = ''[[Littlewood's Miscellany]]'' (1986){{sfn|Littlewood|1986|p=130}}
|salign=right
}}

The '''''Principia Mathematica''''' (often abbreviated '''''PM''''') is a three-volume work on the [[foundations of mathematics]] written by the mathematician–philosophers [[Alfred North Whitehead]] and [[Bertrand Russell]] and published in 1910, 1912, and 1913. In 1925–1927, it appeared in a second edition with an important ''Introduction to the Second Edition'', an ''Appendix A'' that replaced '''✱9''' with a new ''Appendix B'' and ''Appendix C''. ''PM'' was conceived as a sequel to Russell's 1903 ''[[The Principles of Mathematics]]'', but as ''PM'' states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was originally intended by us to be comprised in a second volume of ''Principles of Mathematics''... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."

''PM'', according to its introduction, had three aims: (1) to analyze to the greatest possible extent the ideas and methods of mathematical logic and to minimize the number of [[primitive notion]]s, [[axiom]]s, and [[inference rule]]s; (2) to precisely express mathematical propositions in [[Mathematical logic|symbolic logic]] using the most convenient notation that precise expression allows; (3) to solve the paradoxes that plagued logic and [[set theory]] at the turn of the 20th century, like [[Russell's paradox]].<ref>{{Cite book|url=https://archive.org/details/PrincipiaMathematicaVolumeI|title=Principia Mathematica|last1=Whitehead|first1= Alfred North |first2=Bertrand |last2=Russell|publisher=Cambridge University Press|year=1963|location=Cambridge|pages=[https://archive.org/details/PrincipiaMathematicaVolumeI/page/n46 1]}}</ref>

This third aim motivated the adoption of the theory of [[system of types|types]] in ''PM''. The theory of types adopts grammatical restrictions on formulas that rule out the unrestricted comprehension of classes, properties, and functions. The effect of this is that formulas such as would allow the comprehension of objects like the Russell set turn out to be ill-formed: they violate the grammatical restrictions of the system of ''PM''.

''PM'' sparked interest in symbolic logic and advanced the subject, popularizing it and demonstrating its power.<ref name="SEP">{{cite web
|url=http://plato.stanford.edu/entries/principia-mathematica/#SOPM
|title=Principia Mathematica (Stanford Encyclopedia of Philosophy)
|last=Irvine
|first=Andrew D.
|author-link=Andrew David Irvine
|date=1 May 2003
|publisher=Metaphysics Research Lab, CSLI, Stanford University
|access-date=5 August 2009}}</ref> The [[Modern Library]] placed ''PM'' 23rd in their list of the top 100 English-language nonfiction books of the twentieth century.<ref name="ML">{{cite web
|url=https://www.nytimes.com/library/books/042999best-nonfiction-list.html
|title=The Modern Library's Top 100 Nonfiction Books of the Century
|date=30 April 1999
|publisher=The New York Times Company
|access-date=5 August 2009
}}</ref>