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Relationship between mathematics and physics
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<h2 id="historical-interplay">Historical interplay</h2> <p>Before giving a <a href="/facts/Mathematical_proof/7ECDrU80">mathematical proof</a> for the formula for the <a href="/facts/Volume/aeAXCBUd">volume</a> of a <a href="/facts/Sphere/4Ivb1cTj">sphere</a>, <a href="/facts/Archimedes/sIeMhSVF">Archimedes</a> used physical reasoning to discover the solution (imagining the balancing of bodies on a scale).<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> <a href="/facts/Aristotle/U2bDlBFW">Aristotle</a> classified physics and mathematics as theoretical sciences, in contrast to practical sciences (like <a href="/facts/Ethics/wNVNgL5q">ethics</a> or <a href="/facts/Politics/1gDNy3RP">politics</a>) and to productive sciences (like <a href="/facts/Medicine/2fnp90Je">medicine</a> or <a href="/facts/Botany/PaIRoxnb">botany</a>).<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p><p>From the seventeenth century, many of the most important advances in mathematics appeared motivated by the study of physics, and this continued in the following centuries (although in the nineteenth century mathematics started to become increasingly independent from physics).<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> The creation and development of <a href="/facts/Calculus/MEmUTYoz">calculus</a> were strongly linked to the needs of physics:<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> There was a need for a new mathematical language to deal with the new <a href="/facts/Dynamics_(mechanics)/JLk81H71">dynamics</a> that had arisen from the work of scholars such as Galileo Galilei and <a href="/facts/Isaac_Newton/4L7gTncN">Isaac Newton</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> The concept of <a href="/facts/Derivative/7Ito70Dh">derivative</a> was needed, Newton did not have the modern concept of <a href="/facts/Limit_of_a_function/HFbQ69e5">limits</a>, and instead employed <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimals</a>, which lacked a rigorous foundation at that time.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> During this period there was little distinction between physics and mathematics;<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> as an example, Newton regarded <a href="/facts/Geometry/wTQf5ffQ">geometry</a> as a branch of <a href="/facts/Mechanics/A3yHLQA5">mechanics</a>.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p><p><a href="/facts/Non-Euclidean_geometry/efKa7De2">Non-Euclidean geometry</a>, as formulated by <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a>, <a href="/facts/J%25C3%25A1nos_Bolyai/JeXBranr">János Bolyai</a>, <a href="/facts/Nikolai_Lobachevsky/0fe8z6jY">Nikolai Lobachevsky</a>, and <a href="/facts/Bernhard_Riemann/oV2fL0k8">Bernhard Riemann</a>, freed physics from the limitation of a single Euclidean geometry.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> A version of non-Euclidean geometry, called <a href="/facts/Riemannian_geometry/pPEzjyQC">Riemannian geometry</a>, enabled <a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a> to develop <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> by providing the key mathematical framework on which he fit his physical ideas of gravity.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p><p>In the 19th century <a href="/facts/Auguste_Comte/Jr0EkS1q">Auguste Comte</a> in his <a href="/facts/Hierarchy_of_the_sciences/z1wTmhT8">hierarchy of the sciences</a>, placed physics and astronomy as less general and more complex than mathematics, as both depend on it.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> In 1900, <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a> in his <a href="/facts/Hilbert%2527s_problems/HaQWIxvV">23 problems</a> for the advancement of mathematical science, considered the axiomatization of physics as his <a href="/facts/Hilbert%2527s_sixth_problem/HFapMXG0">sixth problem</a>. The problem remains open.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> </p><p>In 1930, <a href="/facts/Paul_Dirac/6bt5yRdk">Paul Dirac</a> invented the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> which produced a single value when used in an integral. The mathematical rigor of this function was in doubt until the mathematician <a href="/facts/Laurent_Schwartz/Kxelk3ju">Laurent Schwartz</a> developed on the theory of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> </p><p>Connections between the two fields sometimes only require identifying similar concepts by different names, as shown in the 1975 <a href="/facts/Wu%25E2%2580%2593Yang_dictionary/fSPXTfQS">Wu–Yang dictionary</a>,<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> that related concepts of <a href="/facts/Gauge_theory/fSW3gggv">gauge theory</a> with <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a>: 332 </p> <h2 id="physics-is-not-mathematics">Physics is not mathematics</h2> <p class="note">See also: <a href="/facts/Deductive_reasoning/HadA1zWS">Deductive reasoning</a> and <a href="/facts/Inductive_reasoning/EJ992JJ9">Inductive reasoning</a></p> <p>Despite the close relationship between math and physics, they are not synonyms. In mathematics objects can be defined exactly and logically related, but the object need have no relationship to experimental <a href="/facts/Measurement/rsLLbn1s">measurements</a>. In physics, definitions are abstractions or idealizations, approximations adequate when compared to the natural world. In 1960, <a href="/facts/Georg_Rasch/XWxhXvPF">Georg Rasch</a> noted that <a href="/facts/All_models_are_wrong/equVtJBv">no models are ever true</a>, not even <a href="/facts/Newton%2527s_laws_of_motion/XKzGTXK1">Newton's laws</a>, emphasizing that models should not be evaluated based on truth but on their applicability for a given purpose.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> For example, Newton built a physical model around definitions like his <a href="/facts/Newton%2527s_laws_of_motion/XKzGTXK1">second law of motion</a> F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } based on observations, leading to the development of calculus and highly accurate planetary mechanics, but later this definition was superseded by improved models of mechanics.<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> Mathematics deals with entities whose properties can be known with <a href="/facts/Certainty/rKcz7Upd">certainty</a>.<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> According to <a href="/facts/David_Hume/aTwlJ80V">David Hume</a>, only statements that deal solely with ideas themselves—such as those encountered in mathematics—can be demonstrated to be true with certainty, while any conclusions pertaining to experiences of the real world can only be achieved via "probable reasoning".<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> This leads to a situation that was put by <a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a> as "No number of experiments can prove me right; a single experiment can prove me wrong."<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> The ultimate goal in research in <a href="/facts/Pure_mathematics/V29nQC1C">pure mathematics</a> are rigorous <a href="/facts/Mathematical_proof/7ECDrU80">proofs</a>, while in physics <a href="/facts/Heuristic_argument/v5fdQvTX">heuristic arguments</a> may sometimes suffice in leading-edge research.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> In short, the methods and goals of physicists and mathematicians are different.<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a> Nonetheless, according to <a href="/facts/Roland_Omn%25C3%25A8s/dHbNfVdd">Roland Omnès</a>, the <a href="/facts/Axiom/mLjP14Mc">axioms</a> of mathematics are not mere conventions, but have physical origins.<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> </p> <h3>Mathematics is physics</h3> <p>A well-known dictum of the Russian and Soviet mathematician <a href="/facts/Vladimir_Arnold/IMBhyfMS">Vladimir Arnold</a> is "Mathematics is the part of physics where experiments are cheap".<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a><a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> While the phrase generated controversy and even parodies, Arnold has defended it.<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> Mathematicians <a href="/facts/Arthur_Jaffe/0pF6APjR">Arthur Jaffe</a> and <a href="/facts/Frank_Quinn_(mathematician)/Q6aLxgF2">Frank Quinn</a> have noted trends in mathematics towards more focus on intuition even at the cost of rigor and suggest this trend is due to interactions between math and physics.<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> In the framework of <a href="/facts/Willard_van_Orman_Quine/2RrhATxa">Quine</a>'s <a href="/facts/Epistemological_holism/MTEc5HEJ">epistemological holism</a>, our <a href="/facts/Belief/HbPvkGT6">beliefs</a>, even in mathematics, are subjected to the "tribunal of experience", just like in physics.<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> </p> <h2 id="role-of-rigor-in-physics">Role of rigor in physics</h2> <p class="note">See also: <a href="/facts/Theoretical_physics/Cs2Y12rQ">Theoretical physics</a> and <a href="/facts/Mathematical_physics/sLKg8T1G">Mathematical physics</a></p> <p>Rigor is indispensable in pure mathematics.<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a> But many definitions and arguments found in the physics literature involve concepts and ideas that are not up to the standards of <a href="/facts/Rigour/v23FKfhw">rigor in mathematics</a>.<a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a><a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a><a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a><a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a> </p><p>For example, <a href="/facts/Freeman_Dyson/kSRxTh2r">Freeman Dyson</a> characterized <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> as having two "faces". The outward face looked at nature and there the predictions of quantum field theory are exceptionally successful. The inward face looked at mathematical foundations and found inconsistency and mystery. The success of the physical theory comes despite its lack of rigorous mathematical backing.<a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a>: ix <a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a>: 2 </p><p>Some mathematicians, such as <a href="/facts/Arthur_Jaffe/0pF6APjR">Arthur Jaffe</a> and <a href="/facts/Frank_Quinn_(mathematician)/Q6aLxgF2">Frank Quinn</a>, argue that non-rigorous mathematical work can sometimes bring benefits too.<a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> </p> <h2 id="philosophical-problems">Philosophical problems</h2> <p>Some of the problems considered in the <a href="/facts/Philosophy_of_mathematics/z7xBcdWA">philosophy of mathematics</a> are the following: </p> <ul><li>Explain the effectiveness of mathematics in the study of the physical world: "At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" —<a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a>, in <i>Geometry and Experience</i> (1921).<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a></li> <li>Clearly delineate mathematics and physics: For some results or discoveries, it is difficult to say to which area they belong: to the mathematics or to physics.<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a></li> <li>What is the geometry of physical space?<a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a></li> <li>What is the origin of the axioms of mathematics?<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a></li> <li>How does the already existing mathematics influence in the creation and development of <a href="/facts/Theoretical_physics/Cs2Y12rQ">physical theories</a>?<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a></li> <li>Is arithmetic <i>analytic</i> or synthetic? (from <a href="/facts/Immanuel_Kant/vIxjdy6I">Kant</a>, see <a href="/facts/Analytic%25E2%2580%2593synthetic_distinction/J1nXtGzd">Analytic–synthetic distinction</a>)<a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a></li> <li>What is essentially different between doing a physical experiment to see the result and making a mathematical calculation to see the result? (from the <a href="/facts/Alan_Turing/klov9b6g">Turing</a>–<a href="/facts/Ludwig_Wittgenstein/wN9bAgMz">Wittgenstein</a> debate)<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a></li> <li>Do <a href="/facts/G%25C3%25B6del%2527s_incompleteness_theorems/du8PRu8g">Gödel's incompleteness theorems</a> imply that physical theories will always be incomplete? (from <a href="/facts/Stephen_Hawking/b7dcce7n">Stephen Hawking</a>)<a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a><a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a></li> <li>Is mathematics invented or discovered? (millennia-old question, raised among others by <a href="/facts/Mario_Livio/RGuRt6zK">Mario Livio</a>)<a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a></li></ul> <h2 id="education">Education</h2> <p>In recent times the two disciplines have most often been taught separately, despite all the interrelations between physics and mathematics.<a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a> This led some professional mathematicians who were also interested in <a href="/facts/Mathematics_education/6hXltmmF">mathematics education</a>, such as <a href="/facts/Felix_Klein/XMILhEfy">Felix Klein</a>, <a href="/facts/Richard_Courant/QDToNfWl">Richard Courant</a>, <a href="/facts/Vladimir_Arnold/IMBhyfMS">Vladimir Arnold</a> and <a href="/facts/Morris_Kline/kVyuJIh2">Morris Kline</a>, to strongly advocate teaching mathematics in a way more closely related to the physical sciences.<a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a><a class="footnote-ref" id="fnref:59" href="#fn:59"><sup>59</sup></a> The initial courses of mathematics for college students of physics are often taught by mathematicians, despite the differences in "ways of thinking" of physicists and mathematicians about those traditional courses and how they are used in the physics courses classes thereafter.<a class="footnote-ref" id="fnref:60" href="#fn:60"><sup>60</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Non-Euclidean_geometry/efKa7De2">Non-Euclidean geometry</a></li> <li><a href="/facts/Kepler%2527s_laws_of_planetary_motion/jYQoIBjM">Kepler's laws of planetary motion</a></li> <li><a href="/facts/Saving_the_phenomena/okVy87CW">Saving the phenomena</a></li> <li><a href="/facts/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences/n6U6Rdte">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a></li> <li><a href="/facts/Mathematical_universe_hypothesis/b7JIz2kF">Mathematical universe hypothesis</a></li> <li><a href="/facts/Zeno%2527s_paradoxes/sxw7Bp4p">Zeno's paradoxes</a></li> <li><a href="/facts/Mathematical_model/CGhyuxgz">Mathematical model</a></li> <li><a href="/facts/Empiricism/f68IpCeE">Empiricism</a></li> <li><a href="/facts/Mathematics_of_general_relativity/5RktLkom">Mathematics of general relativity</a></li> <li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki</a></li> <li><a href="/facts/Experimental_mathematics/NfuxNMIz">Experimental mathematics</a></li> <li><a href="/facts/History_of_Maxwell%2527s_equations/48kLar3T">History of Maxwell's equations</a></li> <li><a href="/facts/Philosophy_of_mathematics/z7xBcdWA">Philosophy of mathematics § Platonism</a></li> <li><a href="/facts/History_of_astronomy/js3QcRnX">History of astronomy</a></li> <li><i><a href="/facts/Why_Johnny_Can%2527t_Add/Wtt24FYL">Why Johnny Can't Add</a></i></li> <li><a href="/facts/Mathematical_formulation_of_quantum_mechanics/fRLw4sFU">Mathematical formulation of quantum mechanics</a></li> <li><a href="/facts/Scientific_modelling/aNyvIyUQ">Scientific modelling</a></li> <li><a href="/facts/All_models_are_wrong/equVtJBv">All models are wrong</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Vladimir_Arnold/IMBhyfMS">Arnold, V. I.</a> (1999). "Mathematics and physics: mother and daughter or sisters?". <i>Physics-Uspekhi</i>. 42 (12): 1205–1217. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1999PhyU...42.1205A">1999PhyU...42.1205A</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1070%2Fpu1999v042n12abeh000673">10.1070/pu1999v042n12abeh000673</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:250835608">250835608</a>.</li> <li><a href="/facts/Vladimir_Arnold/IMBhyfMS">Arnold, V. I.</a> (1998). <a href="https://web.archive.org/web/20170428233041/http://pauli.uni-muenster.de/~munsteg/arnold.html">"On teaching mathematics"</a>. <i>Russian Mathematical Surveys</i>. 53 (1). Translated by A. V. Goryunov: 229–236. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1998RuMaS..53..229A">1998RuMaS..53..229A</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1070%2FRM1998v053n01ABEH000005">10.1070/RM1998v053n01ABEH000005</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:250833432">250833432</a>. Archived from <a href="http://pauli.uni-muenster.de/~munsteg/arnold.html">the original</a> on 28 April 2017. Retrieved 29 May 2014.</li> <li><a href="/facts/Michael_Atiyah/RQVJ9ATE">Atiyah, M.</a>; Dijkgraaf, R.; Hitchin, N. (1 February 2010). <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3263806">"Geometry and physics"</a>. <i>Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. 368 (1914): 913–926. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2010RSPTA.368..913A">2010RSPTA.368..913A</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1098%2Frsta.2009.0227">10.1098/rsta.2009.0227</a>. <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3263806">3263806</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/20123740">20123740</a>.</li> <li>Boniolo, Giovanni; Budinich, Paolo; Trobok, Majda, eds. (2005). <i>The Role of Mathematics in Physical Sciences: Interdisciplinary and Philosophical Aspects</i>. Dordrecht: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781402031069.</li> <li>Colyvan, Mark (2001). <a href="http://www.colyvan.com/papers/miracle.pdf">"The Miracle of Applied Mathematics"</a> (PDF). <i>Synthese</i>. 127 (3): 265–277. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1010309227321">10.1023/A:1010309227321</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:40819230">40819230</a>. Retrieved 30 May 2014.</li> <li><a href="/facts/Paul_Dirac/6bt5yRdk">Dirac, Paul</a> (1938–1939). <a href="http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html">"The Relation between Mathematics and Physics"</a>. <i>Proceedings of the Royal Society of Edinburgh</i>. 59 Part II: 122–129. Retrieved 30 March 2014.</li> <li><a href="/facts/Richard_Feynman/32vGFrtB">Feynman, Richard P.</a> (1992). "The Relation of Mathematics to Physics". <i>The Character of Physical Law</i> (Reprint ed.). London: Penguin Books. pp. 35–58. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0140175059.</li> <li><a href="/facts/G._H._Hardy/zBkDLb05">Hardy, G. H.</a> (2005). <a href="https://web.archive.org/web/20211009101447/https://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%20Apology.pdf"><i>A Mathematician's Apology</i></a> (PDF) (First electronic ed.). University of Alberta Mathematical Sciences Society. Archived from <a href="http://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%20Apology.pdf">the original</a> (PDF) on 9 October 2021. Retrieved 30 May 2014.</li> <li><a href="/facts/Nigel_Hitchin/HY5u59I5">Hitchin, Nigel</a> (2007). <a href="http://arbor.revistas.csic.es/index.php/arbor/article/viewFile/115/116">"Interaction between mathematics and physics"</a>. <i>ARBOR Ciencia, Pensamiento y Cultura</i>. 725. Retrieved 31 May 2014.</li> <li>Harvey, Alex (2012). "The Reasonable Effectiveness of Mathematics in the Physical Sciences". <i>General Relativity and Gravitation</i>. 43 (2011): 3057–3064. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1212.5854">1212.5854</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2011GReGr..43.3657H">2011GReGr..43.3657H</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs10714-011-1248-9">10.1007/s10714-011-1248-9</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121985996">121985996</a>.</li> <li><a href="/facts/John_von_Neumann/aUk6urof">Neumann, John von</a> (1947). "The Mathematician". <i>Works of the Mind</i>. 1 (1): 180–196. 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Retrieved 2014-05-27.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf">Gregory W. Moore – Physical Mathematics and the Future (July 4, 2014)</a></li> <li><a href="http://www.iop.org/publications/iop/2014/file_64122.pdf">IOP Institute of Physics – Mathematical Physics: What is it and why do we need it? (September 2014)</a></li> <li><a href="https://www.youtube.com/watch?v=B-eh2SD54fM">Feynman explaining the differences between mathematics and physics in a video available on YouTube</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Uhden, Olaf; Karam, Ricardo; Pietrocola, Maurício; Pospiech, Gesche (20 October 2011). "Modelling Mathematical Reasoning in Physics Education". Science & Education. 21 (4): 485–506. Bibcode:2012Sc&Ed..21..485U. doi:10.1007/s11191-011-9396-6. 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