Science

=== Formal science ===
[[Formal science]] is an area of study that generates knowledge using [[formal system]]s.<ref name="wluniversity2021">{{cite web | url = https://my.wlu.edu/the-sciences-at-wandl/formal-sciences | title = Formal Sciences: Washington and Lee University | website = Washington and Lee University | access-date = May 14, 2021 | quote = A "formal science" is an area of study that uses formal systems to generate knowledge such as in Mathematics and Computer Science.  Formal sciences are important subjects because all of quantitative science depends on them. | archive-date = May 14, 2021 | archive-url = https://web.archive.org/web/20210514125428/https://my.wlu.edu/the-sciences-at-wandl/formal-sciences | url-status = live }}</ref><ref name="löwe2002"/><ref name="rucker2019"/> A formal system is an [[abstract structure]] used for inferring [[theorem]]s from [[axiom]]s according to a set of rules.<ref>{{Cite web |title=formal system |url=https://www.britannica.com/topic/formal-system |access-date=May 30, 2022 |website=[[Encyclopædia Britannica]] |language=en |archive-date=April 29, 2008 |archive-url=https://web.archive.org/web/20080429174130/http://www.britannica.com/eb/article-9034889/formal-system |url-status=live }}</ref> It includes mathematics,<ref>{{cite book |last=Tomalin |first=Marcus |year=2006 |title=Linguistics and the Formal Sciences }}</ref><ref>{{Cite journal|title=The Formal Sciences: Their Scope, Their Foundations, and Their Unity|journal=Synthese|volume=133 |pages=5–11| doi=10.1023/a:1020887832028|year=2002|last1=Löwe|first1=Benedikt|s2cid=9272212}}</ref> [[systems theory]], and [[theoretical computer science]]. The formal sciences share similarities with the other two branches by relying on objective, careful, and systematic study of an area of knowledge. They are, however, different from the empirical sciences as they rely exclusively on deductive reasoning, without the need for empirical evidence, to verify their abstract concepts.<ref name="Fetzer2013"/><ref name="Bill2007">{{cite book |first=Thompson | last = Bill |title=The Nature of Statistical Evidence |chapter=2.4 Formal Science and Applied Mathematics |publisher=Springer |series=Lecture Notes in Statistics |volume=189 |year=2007 |page=15}}</ref><ref name="popper2002a" /> The formal sciences are therefore [[A priori and a posteriori|''a priori'']] disciplines and because of this, there is disagreement on whether they constitute a science.<ref name="Bishop1991"/><ref name="Bunge 1998">{{cite book |last1=Bunge |first1=Mario |title=Philosophy of Science: Volume 1, From Problem to Theory |date=1998 |publisher=Routledge |isbn=978-0-7658-0413-6 |edition=revised |volume=1 |location=New York |pages=3–50 |chapter=The Scientific Approach}}</ref> Nevertheless, the formal sciences play an important role in the empirical sciences. [[Calculus]], for example, was initially invented to understand [[Motion (physics)|motion]] in physics.<ref name="MujumdarandTejinder2016">{{Cite book | last1 = Mujumdar | first1 = Anshu Gupta | last2 = Singh | first2 = Tejinder | year = 2016 | chapter = Cognitive science and the connection between physics and mathematics | title = Trick or Truth?: The Mysterious Connection Between Physics and Mathematics | editor-first1 = Anthony | editor-last1 = Aguirre | editor-first2 = Brendan | editor-last2 = Foster | edition =  | series = The Frontiers Collection | pages = 201–218 | location = Switzerland | publisher = SpringerNature | isbn = 978-3-319-27494-2}}</ref> Natural and social sciences that rely heavily on mathematical applications include [[mathematical physics]],<ref>{{cite web |title=About the Journal |url=http://jmp.aip.org/jmp/staff.jsp |url-status=dead |archive-url=https://web.archive.org/web/20061003233339/http://jmp.aip.org/jmp/staff.jsp |archive-date=October 3, 2006 |access-date=October 3, 2006 |website=[[Journal of Mathematical Physics]]}}</ref> [[mathematical chemistry|chemistry]],<ref>{{cite book|last=Restrepo|first=G.|chapter=Mathematical chemistry, a new discipline|title=Essays in the philosophy of chemistry|editor-last1=Scerri|editor-first1=E.|editor-last2=Fisher|editor-first2=G.|publisher=Oxford University Press|location=New York, UK|year=2016|pages=332–351|isbn=978-0-19-049459-9 |url=https://global.oup.com/academic/product/essays-in-the-philosophy-of-chemistry-9780190494599?cc=de&lang=en&|archive-url=https://web.archive.org/web/20210610130352/https://global.oup.com/academic/product/essays-in-the-philosophy-of-chemistry-9780190494599?cc=de&lang=en& |archive-date=June 10, 2021 }}</ref> [[mathematical biology|biology]],<ref>{{Cite web |title=What is mathematical biology|publisher=Centre for Mathematical Biology, University of Bath|url=http://www.bath.ac.uk/cmb/mathBiology/ |url-status=dead |archive-url=https://web.archive.org/web/20180923070442/http://www.bath.ac.uk/cmb/mathBiology/ |archive-date=September 23, 2018 |access-date=June 7, 2018}}</ref> [[mathematical finance|finance]],<ref>{{cite web |last=Johnson |first=Tim |date=September 1, 2009 |title=What is financial mathematics? |url=https://plus.maths.org/content/what-financial-mathematics |access-date=March 1, 2021 |work=+Plus Magazine |archive-date=April 8, 2022 |archive-url=https://web.archive.org/web/20220408231344/https://plus.maths.org/content/what-financial-mathematics |url-status=live }}</ref> and [[mathematical economics|economics]].<ref>{{cite book|last=Varian|first=Hal|author-link=Hal Varian|year=1997|chapter=What Use Is Economic Theory?|editor-last1=D'Autume|editor-first1=A.|editor-last2=Cartelier|editor-first2=J.|title=Is Economics Becoming a Hard Science?|publisher=Edward Elgar}} [http://www.sims.berkeley.edu/~hal/Papers/theory.pdf Pre-publication]. {{Webarchive|url=https://web.archive.org/web/20060625062619/http://www.sims.berkeley.edu/~hal/Papers/theory.pdf |date=June 25, 2006 }}. Retrieved April 1, 2008.</ref>