Mathematics

===Reality===
The connection between mathematics and material reality has led to philosophical debates since at least the time of [[Pythagoras]]. The ancient philosopher [[Plato]] argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as [[Mathematical Platonism|Platonism]]. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.<ref name=SEP-Platonism>{{cite encyclopedia |title=Platonism in Metaphysics |encyclopedia=The Stanford Encyclopedia of Philosophy |last=Balaguer |first=Mark |editor-last=Zalta |editor-first=Edward N. |year=2016 |edition=Spring 2016 |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/archives/spr2016/entries/platonism |access-date=April 2, 2022 |archive-date=January 30, 2022 |archive-url=https://web.archive.org/web/20220130174043/https://plato.stanford.edu/archives/spr2016/entries/platonism/ |url-status=live }}</ref>

[[Armand Borel]] summarized this view of mathematics reality as follows, and provided quotations of [[G. H. Hardy]], [[Charles Hermite]], [[Henri Poincaré]] and Albert Einstein that support his views.<ref name=borel />
{{blockquote| Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.<ref>See {{cite journal
 | first=L. | last=White | year=1947
 | title=The locus of mathematical reality: An anthropological footnote
 | journal=[[Philosophy of Science (journal)|Philosophy of Science]]
 | volume=14|issue=4 | pages=289–303
 | doi=10.1086/286957 | s2cid=119887253
 | id=189303 | postscript=;
}} also in {{cite book
 | first=J. R. | last=Newman | year=1956
 | title=The World of Mathematics
 | publisher=Simon and Schuster | location=New York
 | volume=4 | pages=2348–2364
}}</ref> Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a {{em|feeling}} of an objective existence, of a reality of mathematics ...}}

Nevertheless, Platonism and the concurrent views on abstraction do not explain the [[#Unreasonable effectiveness|unreasonable effectiveness]] of mathematics.<ref>{{cite book
 | title=The Software of the Universe, An Introduction to the History and Philosophy of Laws of Nature
 | first=Mauro
 | last=Dorato
 | year=2005
 | chapter=Why are laws mathematical?
 | pages=31–66
 | isbn=978-0-7546-3994-7
 | publisher=Ashgate
 | chapter-url=https://www.academia.edu/download/52076815/2ch.pdf
 | access-date=December 5, 2022
 | archive-url=https://web.archive.org/web/20230817111932/https://d1wqtxts1xzle7.cloudfront.net/52076815/2ch-libre.pdf?1488997736=&response-content-disposition=inline%3B+filename%3DChapter_2_of_the_book_the_software_of_th.pdf&Expires=1692274771&Signature=PXpNLBsmWMkz9YUs6~LUOfXNkmkCAmDfxQUoWOkGJKP4YqPGQUFMuP1I0xFycLZkL0dyfGwdGQ7mPk44nvmpM3YpKBSeVCZRXtDMiwgqs1JhEWrJovAhrchPLM1mGn3pw5P6LPo0sDZsl7uaPoZHMyCyJpayHvFtpyj1oUMIdmGuYM5P3euy1R87g6xlKyNAp~~BR5I4gVpopzLoeZn7d3oEnOOua0GjsqsZ6H9mEgcZMpH-qF8w9iFa9aSXFpqxagQwcVVkg7DXkOjVV5jyzctBUKQtOQQ~-9EN1y-c9pFV-Xu-NNuoN3Ij6K4SwvjYv0a8DMs8T5SVj1Kz9i4CEQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
 | archive-date=August 17, 2023
 | url-status=live
 }}</ref>