Editing Foundations of mathematics (section)

=== Natural numbers ===
{{Main|Peano arithmetic}}

The work of [[#Real analysis|making rigorous real analysis and the definition of real numbers]], consisted of reducing everything to [[rational number]]s and thus to [[natural number]]s, since positive rational numbers are fractions of natural numbers. There was therefore a need of a formal definition of natural numbers, which imply as [[axiomatic theory]] of [[arithmetic]]. This was started with [[Charles Sanders Peirce]] in 1881 and [[Richard Dedekind]] in 1888, who defined a natural numbers as the [[cardinality]] of a [[finite set]].<ref>{{cite book |last1=Dedekind |first1=Richard |title=What Are and What Should the Numbers Be? Continuity and Irrational Numbers |publisher=Springer |isbn=978-3-662-70059-4}}</ref> However, this involves [[set theory]], which was not formalized at this time.

[[Giuseppe Peano]] provided in 1888 a complete axiomatisation based on the [[ordinal numeral|ordinal]] property of the natural numbers. The last Peano's axiom is the only one that induces logical difficulties, as it begin with either "if {{mvar|S}} is a set then" or "if <math>\varphi</math> is a [[predicate (mathematical logic)|predicate]] then". So, Peano's axioms induce a [[quantification (logic)|quantification]] on infinite sets, and this means that Peano arithmetic is what is presently called a [[Second-order logic]].

This was not well understood at that times, but the fact that [[infinity]] occurred in the definition of the natural numbers was a problem for many mathematicians of this time. For example, [[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> This applies in particular to the use of the last Peano axiom for showing that the [[successor function]] generates all natural numbers. Also, [[Leopold Kronecker]] said "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>}} This may be interpreted as "the integers cannot be mathematically defined".