Editing Natural number (section)

==History==
<!--This section is linked from [[Numeral system]]-->

===Ancient roots===
{{further|Prehistoric counting}}

[[File:Ishango bone (cropped).jpg|thumb|The [[Ishango bone]] (on exhibition at the [[Royal Belgian Institute of Natural Sciences]])<ref name=RBINS_intro>{{cite web |title=Introduction |series=[[Ishango bone]] |publisher=[[Royal Belgian Institute of Natural Sciences]] |location=Brussels, Belgium |url=https://www.naturalsciences.be/expo/old_ishango/en/ishango/introduction.html |archive-url=https://web.archive.org/web/20160304051733/https://www.naturalsciences.be/expo/old_ishango/en/ishango/introduction.html |archive-date=4 March 2016}}</ref><ref name=RBINS_flash>{{cite web |title=Flash presentation |series=[[Ishango bone]] |publisher=[[Royal Belgian Institute of Natural Sciences]] |place=Brussels, Belgium |url=http://ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html |archive-url=https://web.archive.org/web/20160527164619/http://ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html |archive-date=27 May 2016}}</ref><ref name=UNESCO>{{cite web |title=The Ishango Bone, Democratic Republic of the Congo |website=[[UNESCO]]'s Portal to the Heritage of Astronomy |url=http://www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1 |archive-url=https://web.archive.org/web/20141110195426/http://www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1 |archive-date=10 November 2014}}, on permanent display at the [[Royal Belgian Institute of Natural Sciences]], Brussels, Belgium.</ref> is believed to have been used 20,000&nbsp;years ago for natural number arithmetic.]]

The most primitive method of representing a natural number is to use one's fingers, as in [[finger counting]]. Putting down a [[tally mark]] for each object is another primitive method. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set.

The first major advance in abstraction was the use of [[numeral system|numerals]] to represent numbers. This allowed systems to be developed for recording large numbers. The ancient [[History of Ancient Egypt|Egyptians]] developed a powerful system of numerals with distinct [[Egyptian hieroglyphs|hieroglyphs]] for&nbsp;1, 10, and all powers of&nbsp;10 up to over 1&nbsp;million. A stone carving from [[Karnak]], dating back from around 1500&nbsp;BCE and now at the [[Louvre]] in Paris, depicts&nbsp;276 as 2&nbsp;hundreds, 7&nbsp;tens, and 6&nbsp;ones; and similarly for the number&nbsp;4,622. The [[Babylonia]]ns had a [[Positional notation|place-value]] system based essentially on the numerals for&nbsp;1 and&nbsp;10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context.<ref>{{cite book |first=Georges |last=Ifrah |year=2000 |title=The Universal History of Numbers |publisher=Wiley |isbn=0-471-37568-3}}</ref>

A much later advance was the development of the idea that&nbsp;{{num|0}} can be considered as a number, with its own numeral. The use of a&nbsp;0 [[numerical digit|digit]] in place-value notation (within other numbers) dates back as early as 700&nbsp;BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number.{{efn| A tablet found at Kish ... thought to date from around 700&nbsp;BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place.<ref>{{cite web |title=A history of Zero |website=MacTutor History of Mathematics |url=http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html |url-status=live |access-date=23 January 2013 |archive-url=https://web.archive.org/web/20130119083234/http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html |archive-date=19 January 2013}}</ref>}} The [[Olmec]] and [[Maya civilization]]s used&nbsp;0 as a separate number as early as the {{nowrap|1st century BCE}}, but this usage did not spread beyond [[Mesoamerica]].<ref>{{cite book |first=Charles C. |last=Mann |year=2005 |title=1491: New Revelations of the Americas before Columbus |page=19 |publisher=Knopf |isbn=978-1-4000-4006-3 |url=https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19 |url-status=live |via=Google Books |access-date=3 February 2015 |archive-url=https://web.archive.org/web/20150514105855/https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19 |archive-date=14 May 2015}}</ref><ref>{{cite book |first=Brian |last=Evans |year=2014 |title=The Development of Mathematics Throughout the Centuries: A brief history in a cultural context |publisher=John Wiley & Sons |isbn=978-1-118-85397-9 |chapter=Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations |via=Google Books |chapter-url=https://books.google.com/books?id=3CPwAgAAQBAJ&pg=PT73}}</ref> The use of a numeral&nbsp;0 in modern times originated with the Indian mathematician [[Brahmagupta]] in 628&nbsp;CE. However, 0 had been used as a number in the medieval [[computus]] (the calculation of the date of Easter), beginning with [[Dionysius Exiguus]] in 525&nbsp;CE, without being denoted by a numeral. Standard [[Roman numerals]] do not have a symbol for&nbsp;0; instead, ''nulla'' (or the genitive form ''nullae'') from {{Lang|la|nullus}}, the Latin word for "none", was employed to denote a&nbsp;0 value.<ref>{{cite web |first=Michael |last=Deckers |title=Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius |url=http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |publisher=Hbar.phys.msu.ru |date=25 August 2003 |access-date=13 February 2012 |archive-url=https://web.archive.org/web/20190115083618/http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |archive-date=15 January 2019 |url-status=live }}</ref>

The first systematic study of numbers as [[abstraction]]s is usually credited to the [[ancient Greece|Greek]] philosophers [[Pythagoras]] and [[Archimedes]]. Some Greek mathematicians treated the number&nbsp;1 differently than larger numbers, sometimes even not as a number at all.{{efn|This convention is used, for example, in [[Euclid's Elements]], see D.&nbsp;Joyce's web edition of Book VII.<ref name=EuclidVIIJoyce>{{cite book |author=Euclid |author-link=Euclid |editor-first=D. |editor-last=Joyce|editor-link=David E. Joyce (mathematician) |chapter=Book VII, definitions 1 and 2 |title=[[Euclid's Elements|Elements]] |publisher=Clark University |chapter-url=http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html }}</ref>}} [[Euclid]], for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).<ref name="Mueller 2006 p. 58">{{cite book |last=Mueller |first=Ian |year=2006 |title=Philosophy of mathematics and deductive structure in [[Euclid's Elements]] |page=58 |publisher=Dover Publications |location=Mineola, New York |isbn=978-0-486-45300-2 |oclc=69792712}}</ref> However, in the definition of [[perfect number]] which comes shortly afterward, Euclid treats 1 as a number like any other.<ref>{{cite book |author=Euclid |author-link=Euclid |editor-first=D. |editor-last=Joyce |chapter=Book VII, definition 22 |title=[[Euclid's Elements|Elements]] |publisher=Clark University |chapter-url=http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII22.html |quote=A perfect number is that which is equal to the sum of its own parts. }} In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example {{math|1=6 = 1 + 2 + 3}} is a perfect number.</ref>

Independent studies on numbers also occurred at around the same time in [[India]], China, and [[Mesoamerica]].<ref>{{cite book |first=Morris |last=Kline |year=1990 |orig-year=1972 |title=Mathematical Thought from Ancient to Modern Times |publisher=Oxford University Press |isbn=0-19-506135-7}}</ref>

===Emergence as a term===
[[Nicolas Chuquet]] used the term ''progression naturelle'' (natural progression) in 1484.<ref>{{cite book |last1=Chuquet |first1=Nicolas|author-link=Nicolas Chuquet |title=Le Triparty en la science des nombres|date=1881 |orig-date=1484 |url=https://gallica.bnf.fr/ark:/12148/bpt6k62599266/f75.image |language=fr}}</ref> The earliest known use of "natural number" as a complete English phrase is in 1763.<ref>{{cite book |last1=Emerson |first1=William |title=The method of increments|date=1763 |page=113 |url=https://archive.org/details/bim_eighteenth-century_the-method-of-increments_emerson-william_1763/page/112/mode/2up}}</ref><ref name="MacTutor"/> The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.<ref name="MacTutor">{{cite web |title=Earliest Known Uses of Some of the Words of Mathematics (N) |url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/n/ |website=Maths History |language=en}}</ref>

Starting at 0 or 1 has long been a matter of definition. In 1727, [[Bernard Le Bovier de Fontenelle]] wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.<ref>{{cite book |last1=Fontenelle |first1=Bernard de |title=Eléments de la géométrie de l'infini |date=1727 |page=3 |url=https://gallica.bnf.fr/ark:/12148/bpt6k64762n/f31.item |language=fr}}</ref> In 1889, [[Giuseppe Peano]] used N for the positive integers and started at 1,<ref>{{cite book |title=Arithmetices principia: nova methodo |date=1889 |publisher=Fratres Bocca |url=https://archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up|page=12 |language=Latin}}</ref> but he later changed to using N<sub>0</sub> and N<sub>1</sub>.<ref>{{cite book |last1=Peano |first1=Giuseppe |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up|language=fr}}</ref> Historically, most definitions have excluded 0,<ref name="MacTutor"/><ref>{{cite book |last1=Fine |first1=Henry Burchard |title=A College Algebra |date=1904 |publisher=Ginn |page=6 |url=https://books.google.com/books?id=RR4PAAAAIAAJ&dq=%22natural%20number%22&pg=PA6 |language=en}}</ref><ref>{{cite book |title=Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166 |date=1958 |publisher=United States Armed Forces Institute |page=12 |url=https://books.google.com/books?id=184i06Py1ZYC&dq=%22natural%20number%22%201&pg=PA12 |language=en}}</ref> but many mathematicians such as [[George A. Wentworth]], [[Bertrand Russell]], [[Nicolas Bourbaki]], [[Paul Halmos]], [[Stephen Cole Kleene]], and [[John Horton Conway]] have preferred to include 0.<ref>{{cite web |title=Natural Number |url=https://archive.lib.msu.edu/crcmath/math/math/n/n035.htm |website=archive.lib.msu.edu}}</ref><ref name="MacTutor"/>

Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0,<ref name="MacTutor"/>{{efn|name=MacLaneBirkhoff1999p15|{{harvtxt|Mac Lane|Birkhoff|1999|page=15}} include zero in the natural numbers: 'Intuitively, the set <math>\N=\{0,1,2,\ldots\}</math> of all ''natural numbers'' may be described as follows: <math>\N</math> contains an "initial" number {{math|0}}; ...'. They follow that with their version of the [[Peano's axioms]].}} number theory and analysis texts excluding 0,<ref name="MacTutor"/><ref name="Křížek">{{cite book |last1=Křížek |first1=Michal |last2=Somer |first2=Lawrence |last3=Šolcová |first3=Alena |title=From Great Discoveries in Number Theory to Applications |date=21 September 2021 |publisher=Springer Nature |isbn=978-3-030-83899-7 |page=6 |url=https://books.google.com/books?id=tklEEAAAQBAJ&dq=natural%20numbers%20zero&pg=PA6 |language=en}}</ref><ref>See, for example, {{harvtxt|Carothers|2000|p=3}} or {{harvtxt|Thomson|Bruckner|Bruckner|2008|p=2}}</ref> logic and set theory texts including 0,<ref>{{cite book |last1=Gowers |first1=Timothy |title=The Princeton companion to mathematics |date=2008 |publisher=Princeton university press |location=Princeton |isbn=978-0-691-11880-2 |page=17}}</ref><ref>{{cite book |last1=Bagaria |first1=Joan |title=Set Theory |url=http://plato.stanford.edu/entries/set-theory/ |publisher=The Stanford Encyclopedia of Philosophy |edition=Winter 2014 |year=2017 |access-date=13 February 2015 |archive-url=https://web.archive.org/web/20150314173026/http://plato.stanford.edu/entries/set-theory/ |archive-date=14 March 2015 |url-status=live}}</ref><ref>{{cite book |last1=Goldrei |first1=Derek |title=Classic Set Theory: A guided independent study |url=https://archive.org/details/classicsettheory00gold |url-access=limited |date=1998 |publisher=Chapman & Hall/CRC |location=Boca Raton, Fla. [u.a.] |isbn=978-0-412-60610-6 |page=[https://archive.org/details/classicsettheory00gold/page/n39 33] |edition=1. ed., 1. print|chapter=3}}</ref> dictionaries excluding 0,<ref name="MacTutor"/><ref>{{cite dictionary|url=http://www.merriam-webster.com/dictionary/natural%20number|title=natural number|dictionary=Merriam-Webster.com|publisher=[[Merriam-Webster]]|access-date=4 October 2014|archive-url=https://web.archive.org/web/20191213133201/https://www.merriam-webster.com/dictionary/natural%20number| archive-date=13 December 2019|url-status=live}}</ref> school books (through high-school level) excluding 0, and upper-division college-level books including 0.<ref name="Enderton">{{cite book |last1=Enderton |first1=Herbert B. |title=Elements of set theory |date=1977 |publisher=Academic Press |location=New York |isbn=0122384407 |page=66}}</ref> There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include [[division by zero]]<ref name="Křížek"/> and the size of the [[empty set]]. [[Computer language]]s often [[Zero-based numbering|start from zero]] when enumerating items like [[For loop|loop counters]] and [[String (computer science)|string-]] or [[Array data structure|array-elements]].<ref>{{cite journal |last1=Brown |first1=Jim |title=In defense of index origin 0 |journal=ACM SIGAPL APL Quote Quad |date=1978 |volume=9 |issue=2 |page=7 |doi=10.1145/586050.586053|s2cid=40187000 }}</ref><ref>{{cite web |last1=Hui |first1=Roger |title=Is index origin 0 a hindrance? |url=http://www.jsoftware.com/papers/indexorigin.htm |website=jsoftware.com |access-date=19 January 2015 |archive-url=https://web.archive.org/web/20151020195547/http://www.jsoftware.com/papers/indexorigin.htm |archive-date=20 October 2015 |url-status=live}}</ref> Including 0 began to rise in popularity in the 1960s.<ref name="MacTutor"/> The [[ISO 31-11]] standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as [[ISO/IEC 80000|ISO 80000-2]].<ref name=ISO80000/>

===Formal construction===

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. [[Henri Poincaré]] stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.<ref>{{cite book |last1=Poincaré |first1=Henri|translator1-first=William John|translator1-last= Greenstreet |title=La Science et l'hypothèse|trans-title=Science and Hypothesis|orig-date=1902|date=1905|chapter=On the nature of mathematical reasoning|chapter-url=https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_1|at=VI}}</ref> [[Leopold Kronecker]] summarized his belief as "God made the integers, all else is the work of man".{{efn|The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."<ref>{{cite book
 |last=Gray |first=Jeremy |author-link=Jeremy Gray
 |year=2008
 |title=Plato's Ghost: The modernist transformation of mathematics
 |page=153
 |publisher=Princeton University Press
 |isbn=978-1-4008-2904-0
 |via=Google Books
 |url=https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22
 |url-status=live
 |archive-url=https://web.archive.org/web/20170329150904/https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22#v=snippet&q=%22God%20made%20the%20integers%2C%20all%20else%20is%20the%20work%20of%20man.%22&f=false
 |archive-date=29 March 2017
}}</ref><ref>{{cite book
 |last=Weber |first=Heinrich L.
 |year=1891–1892
 |chapter=Kronecker
 |chapter-url=http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6 
 |archive-url=https://web.archive.org/web/20180809110042/http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6
 |archive-date=9 August 2018
 |title=''Jahresbericht der Deutschen Mathematiker-Vereinigung''
 |trans-title=Annual report of the German Mathematicians Association
 |pages=2:5–23. (The quote is on p.&nbsp;19)
 |postscript=;
}} {{cite web
 |title=access to ''Jahresbericht der Deutschen Mathematiker-Vereinigung''
 |url=http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
 |archive-url=https://web.archive.org/web/20170820201100/http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
 |archive-date=20 August 2017 
}}</ref>}}

The [[Constructivism (mathematics)|constructivists]] saw a need to improve upon the logical rigor in the [[foundations of mathematics]].{{efn|"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." {{harv|Eves|1990|p=606}} }} In the 1860s, [[Hermann Grassmann]] suggested a [[recursive definition]] for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications.

[[Set-theoretical definitions of natural numbers]] were initiated by [[Gottlob Frege|Frege]]. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including [[Russell's paradox]]. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.<ref>{{harvnb|Eves|1990|loc=Chapter 15}}</ref>

In 1881, [[Charles Sanders Peirce]] provided the first [[Axiomatic system#Axiomatization|axiomatization]] of natural-number arithmetic.<ref>{{cite journal|last= Peirce|first= C.&nbsp;S.|author-link= Charles Sanders Peirce|year= 1881|title= On the Logic of Number|url= https://archive.org/details/jstor-2369151|journal= American Journal of Mathematics|volume= 4|issue= 1|pages= 85–95|doi= 10.2307/2369151|mr= 1507856|jstor= 2369151}}</ref><ref>{{cite book|last= Shields|first= Paul|year= 1997|title= Studies in the Logic of Charles Sanders Peirce|url= https://archive.org/details/studiesinlogicof00nath|url-access= registration|chapter= 3. Peirce's Axiomatization of Arithmetic|chapter-url= https://books.google.com/books?id=pWjOg-zbtMAC&pg=PA43|editor1-last= Houser
|editor1-first= Nathan|editor2-last= Roberts|editor2-first= Don D.|editor3-last= Van Evra|editor3-first= James|publisher= Indiana University Press|isbn= 0-253-33020-3|pages= 43–52}}</ref> In 1888, [[Richard Dedekind]] proposed another axiomatization of natural-number arithmetic,<ref>{{cite book |title=Was sind und was sollen die Zahlen? |date=1893 |publisher=F. Vieweg |url=https://archive.org/details/wassindundwasso00dedegoog/page/n42/mode/2up |language=German|at=71–73}}</ref> and in&nbsp;1889, Peano published a simplified version of Dedekind's axioms in his book ''The principles of arithmetic presented by a new method'' ({{langx|la|[[Arithmetices principia, nova methodo exposita]]}}). This approach is now called [[Peano arithmetic]]. It is based on an [[axiomatization]] of the properties of [[ordinal number]]s: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is [[equiconsistent]] with several weak systems of [[set theory]]. One such system is [[ZFC]] with the [[axiom of infinity]] replaced by its negation.<ref>{{cite journal
 | last1 = Baratella | first1 = Stefano
 | last2 = Ferro | first2 = Ruggero
 | doi = 10.1002/malq.19930390138
 | issue = 3
 | journal = Mathematical Logic Quarterly
 | mr = 1270381
 | pages = 338–352
 | title = A theory of sets with the negation of the axiom of infinity
 | volume = 39
 | year = 1993}}</ref> Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include [[Goodstein's theorem]].<ref>{{cite journal | last1=Kirby | first1=Laurie | last2=Paris | first2=Jeff | title=Accessible Independence Results for Peano Arithmetic | journal=Bulletin of the London Mathematical Society | publisher=Wiley | volume=14 | issue=4 | year=1982 | issn=0024-6093 | doi=10.1112/blms/14.4.285 | pages=285–293}}</ref>