View source for The Unreasonable Effectiveness of Mathematics in the Natural Sciences

{{short description|1960 article by Eugene Wigner}}

"'''The Unreasonable Effectiveness of Mathematics in the Natural Sciences'''" is a 1960 article written by the [[physicist]] [[Eugene Wigner]], published in ''[[Communications on Pure and Applied Mathematics|Communication in Pure and Applied Mathematics]]''.<ref>{{Cite journal|last1=Wigner|first1=E. P.|author-link=Eugene Wigner|year=1960|title=The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959|url=http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html|journal=Communications on Pure and Applied Mathematics|volume=13|issue=1|pages=1–14|bibcode=1960CPAM...13....1W|doi=10.1002/cpa.3160130102|s2cid=6112252|archive-url=https://web.archive.org/web/20210212111540/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html|archive-date=2021-02-12}}</ref><ref>Note: Wigner's mention of Kellner and Hilleraas "... Jordan felt that we would have been, at least temporarily, helpless had an unexpected disagreement occurred in the theory of the helium atom. This was, at that time, developed by Kellner and by Hilleraas ..." refers to Georg W. Kellner ({{cite journal|doi=10.1007/BF01391720|title=Die Ionisierungsspannung des Heliums nach der Schrödingerschen Theorie|year=1927|last1=Kellner|first1=Georg W.|journal=Zeitschrift für Physik|volume=44|issue=1–2|pages=91–109|bibcode=1927ZPhy...44...91K|s2cid=122213875}}) and to [[Egil Hylleraas]].</ref> In it, Wigner observes that a [[Theoretical physics|theoretical physics's]] [[Mathematics|mathematical]] structure often points the way to further advances in that theory and to [[empirical]] predictions. Mathematical theories often have [[predictive power]] in describing nature.

==Observations and arguments==
Wigner argues that mathematical concepts have applicability far beyond the context in which they were originally developed. He writes: "It is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena."<ref name="Wigner60p8">{{harvnb|Wigner|1960|loc=§Is the Success of Physical Theories Truly Surprising? p. 8}}</ref> He adds that the observation "the laws of nature are written in the [[language of mathematics]]," properly made by [[Galileo Galilei|Galileo]] three hundred years ago, "is now truer than ever before."

Wigner's first example is the [[gravity|law of gravitation]] formulated by [[Isaac Newton]]. Originally used to model freely falling bodies on the surface of the Earth, this law was extended based on what Wigner terms "very scanty observations"<ref name="Wigner60p8" /> to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."<ref>{{harvnb|Wigner|1960|p=9}}</ref> Wigner says that "Newton{{Nbsp}}... noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence."

Wigner's second example comes from [[quantum mechanics]]: [[Max Born]] "noticed that some rules of computation, given by [[Werner Heisenberg|Heisenberg]], were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, [[Pascual Jordan|Jordan]], and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions." But [[Wolfgang Pauli]] found their work accurately described the [[hydrogen atom]]: "This application gave results in agreement with experience." The [[helium atom]], with two electrons, is more complex, but "nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by [[Toichiro Kinoshita|Kinoshita]] at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we 'got something out' of the equations that we did not put in." The same is true of the [[Emission spectrum|atomic spectra]] of heavier elements.

Wigner's last example comes from [[quantum electrodynamics]]: "Whereas Newton's theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg's prescriptions. The quantum theory of the [[Lamb shift]], as conceived by [[Hans Bethe|Bethe]] and established by [[Julian Schwinger|Schwinger]], is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand."

There are examples beyond the ones mentioned by Wigner. Another often cited example is [[Maxwell's equations]], derived to model the elementary electrical and magnetic phenomena known in the mid-19th century. The equations also describe radio waves, discovered by [[David Edward Hughes]] in 1879, around the time of [[James Clerk Maxwell]]'s death.

==Responses==
The responses the thesis received include:
* [[Richard Hamming]] in computer science, "The Unreasonable Effectiveness of Mathematics".<ref name="hamming">{{Cite journal |last1=Hamming |first1=R. W. |author-link=Richard Hamming |year=1980 |title=The Unreasonable Effectiveness of Mathematics |url=https://math.dartmouth.edu/~matc/MathDrama/reading/Hamming.html |url-status=live |journal=The American Mathematical Monthly |volume=87 |issue=2 |pages=81–90 |doi=10.2307/2321982 |jstor=2321982 |archive-url=https://web.archive.org/web/20220622104638/https://math.dartmouth.edu/~matc/MathDrama/reading/Hamming.html |archive-date=2022-06-22 |access-date=2021-07-30 |hdl=10945/55827|hdl-access=free }}</ref>
* [[Arthur M. Lesk|Arthur Lesk]] in molecular biology, "The Unreasonable Effectiveness of Mathematics in Molecular Biology".<ref name="lesk">{{Cite journal|last1=Lesk|first1=A. M.|title=The unreasonable effectiveness of mathematics in molecular biology|doi=10.1007/BF03025372|journal=The Mathematical Intelligencer|volume=22|issue=2|pages=28–37|year=2000|s2cid=120102813}}</ref>
* [[Peter Norvig]] in artificial intelligence,  "The Unreasonable Effectiveness of Data"<ref name="norvig">{{Cite journal|last1=Halevy|first1=A.|author-link1=Alon Y. Halevy|last2=Norvig|first2=P.|author-link2=Peter Norvig|last3=Pereira|first3=F.|doi=10.1109/MIS.2009.36|title=The Unreasonable Effectiveness of Data|journal=IEEE Intelligent Systems|volume=24|issue=2|pages=8–12|year=2009|s2cid=14300215|url=http://static.googleusercontent.com/media/research.google.com/en//pubs/archive/35179.pdf|access-date=2015-09-04|archive-date=2022-08-09|archive-url=https://web.archive.org/web/20220809171235/https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/35179.pdf|url-status=live}}</ref>
* [[Max Tegmark]] in physics, "The Mathematical Universe".<ref name="tegmark" />
* [[Ivor Grattan-Guinness]] in mathematics, "Solving Wigner's mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences".<ref name="ivor">{{Cite journal|last1=Grattan-Guinness|first1=I.|title=Solving Wigner's mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences|doi=10.1007/BF02985373|journal=The Mathematical Intelligencer|volume=30|issue=3|pages=7–17|year=2008|s2cid=123174309}}</ref>
* [[Vela Velupillai]] in economics, "The Unreasonable Ineffectiveness of Mathematics in Economics".<ref name="velupillai">{{Cite journal|last1=Velupillai|first1=K. V.|title=The unreasonable ineffectiveness of mathematics in economics|doi=10.1093/cje/bei084|journal=Cambridge Journal of Economics|volume=29|issue=6|pages=849–872|year=2005|citeseerx=10.1.1.194.6586}}</ref>
* [[Terry Sejnowski|Terrence Joseph Sejnowski]] in Artificial Intelligence: The Unreasonable Effectiveness of Deep Learning in Artificial Intelligence".<ref>"The Unreasonable Effectiveness of Deep Learning in Artificial Intelligence".{{cite journal
 | last = Sejnowski
 | first = Terrence J.
 | title = The unreasonable effectiveness of deep learning in artificial intelligence
 | journal = Proc. Natl. Acad. Sci. U.S.A.
 | volume = 117
 | issue = 48
 | pages = 30033–30038
 | date = 2020
 | doi = 10.1073/pnas.1907373117
 | doi-access = free
 | pmid = 32457138
 | pmc = 7293599
 | arxiv = 2002.04806
 }}</ref> 

===Richard Hamming===
{{more citations needed|section|date=April 2022}}
Mathematician and [[Turing Award]] laureate [[Richard Hamming]] reflected on and extended Wigner's ''Unreasonable Effectiveness'' in 1980, discussing four "partial explanations" for it,<ref name="hamming"/> and concluding that they were unsatisfactory. They were:

1. ''Humans see what they look for''. The belief that science is experimentally grounded is only partially true. Hamming gives four examples of nontrivial physical phenomena he believes arose from the mathematical tools employed and not from the intrinsic properties of physical reality.
* Hamming proposes that [[Galileo]] discovered the [[Galileo Galilei#Falling bodies|law of falling bodies]] not by experimenting, but by simple, though careful, thinking.<ref>{{Cite web |title=The Unreasonable Effectiveness of Mathematics - R.W. Hamming - Some Partial Explanations |url=https://ned.ipac.caltech.edu/level5/March02/Hamming/Hamming.html |access-date=2024-01-06 |website=ned.ipac.caltech.edu}}</ref> Hamming imagines Galileo as having engaged in the following [[thought experiment]] (the experiment, which Hamming calls "scholastic reasoning", is described in Galileo's book ''On Motion''.):

: <blockquote>Suppose that a falling body broke into two pieces. Of course, the two pieces would immediately slow down to their appropriate speeds. But suppose further that one piece happened to touch the other one. Would they now be one piece and both speed up? Suppose I tie the two pieces together. How tightly must I do it to make them one piece? A light string? A rope? Glue? When are two pieces one?<ref>{{cite web|first=Albert|last=Van Helden|title=On Motion|work=The Galileo Project|url=http://galileo.rice.edu/sci/theories/on_motion.html|date=1995|access-date=16 October 2013|archive-date=21 December 2017|archive-url=https://web.archive.org/web/20171221120150/http://galileo.rice.edu/sci/theories/on_motion.html|url-status=live}}</ref></blockquote>

: There is simply no way a falling body can "answer" such hypothetical "questions." Hence Galileo would have concluded that "falling bodies need not know anything if they all fall with the same velocity, unless interfered with by another force." After coming up with this argument, Hamming found a related discussion in Pólya (1963: 83-85).<ref>{{cite book|author-link=George Pólya|first1=George|last1=Pólya|author-link2=Leon Bowden|first2=Leon|last2=Bowden|author3=School Mathematics Study Group|title=Mathematical methods in science; a course of lectures|publisher=Stanford: School Mathematics Study Group|year=1963|oclc=227871299|series=Studies in mathematics|volume=11}}</ref> Hamming's account does not reveal an awareness of the 20th-century scholarly debate over just what Galileo did.{{clarify|date=March 2018}}

*The inverse square [[law of universal gravitation]] necessarily follows from the [[conservation of energy]] and of space having [[spacetime|three dimensions]]. Measuring the exponent in the law of universal gravitation is more a test of whether space is [[Euclidean space|Euclidean]] than a test of the properties of the [[gravitational field]].
*The inequality at the heart of the [[uncertainty principle]] of [[quantum mechanics]] follows from the properties of [[Fourier integral]]s and from assuming [[time invariant|time invariance]].<ref>{{cite journal|first1=Gerald B.|last1=Folland|first2=Alladi|last2=Sitaram|title=The Uncertainty Principle: A Mathematical Survey|journal=Journal of Fourier Analysis and Applications|volume=3|issue=3|pages=207–238|year=1997|doi=10.1007/BF02649110|s2cid=121355943}}<!-- Needs expert confirmation PDF available at Citeseer --></ref>
*Hamming argues that [[Albert Einstein]]'s pioneering work on [[special relativity]] was largely "scholastic" in its approach. He knew from the outset what the theory should look like (although he only knew this because of the [[Michelson–Morley experiment]]), and explored candidate theories with mathematical tools, not actual experiments. Hamming alleges that Einstein was so confident that his relativity theories were correct that the outcomes of observations designed to test them did not much interest him. If the observations were inconsistent with his theories, it would be the observations that were at fault.

2. ''Humans create and select the mathematics that fit a situation''. The mathematics at hand does not always work. For example, when mere [[scalar (mathematics)|scalar]]s proved awkward for understanding forces, first [[vector (geometry)|vector]]s, then [[tensor]]s, were invented.

3. ''Mathematics addresses only a part of human experience''. Much of human experience does not fall under science or mathematics but under the [[value theory|philosophy of value]], including [[ethics]], [[aesthetics]], and [[political philosophy]]. To assert that the world can be explained via mathematics amounts to an act of faith.

4. ''[[Evolution]] has primed humans to think mathematically''. The earliest lifeforms must have contained the seeds of the human ability to create and follow long chains of close reasoning.

===Max Tegmark===
Physicist [[Max Tegmark]] argued that the effectiveness of mathematics in describing external physical reality is because the physical world is an abstract mathematical structure.<ref name="tegmark">{{cite journal|title=The Mathematical Universe|year=2008|first=Max|last=Tegmark|author-link=Max Tegmark|doi=10.1007/s10701-007-9186-9|journal=Foundations of Physics|volume=38|issue=2|pages=101–150|arxiv=0704.0646|bibcode=2008FoPh...38..101T|s2cid=9890455}}</ref><ref>{{cite book|first=Max|last=Tegmark|title=Our Mathematical Universe|publisher=Knopf|year=2014|isbn=978-0-307-59980-3|title-link=Our Mathematical Universe}}</ref> This theory, referred to as the [[mathematical universe hypothesis]], mirrors ideas previously advanced by [[Peter Atkins]].<ref>{{cite book|first=Peter|last=Atkins|author-link=Peter Atkins|title=Creation Revisited|publisher=W.H.Freeman|year=1992|isbn=978-0-7167-4500-6}}</ref> However, Tegmark explicitly states that "the true mathematical structure isomorphic to our world, if it exists, has not yet been found." Rather, mathematical theories in physics are successful because they approximate more complex and predictive mathematics. According to Tegmark, "Our successful theories are not mathematics approximating physics, but simple mathematics approximating more complex mathematics."

===Ivor Grattan-Guinness===
[[Ivor Grattan-Guinness]] found the effectiveness in question eminently reasonable and explicable in terms of concepts such as analogy, generalization, and metaphor. He emphasizes that Wigner largely ignores "the effectiveness of the natural sciences in mathematics, in that much mathematics has been motivated by interpretations in the sciences".<ref name="ivor"/>

===Michael Atiyah===
The tables were turned by [[Michael Atiyah]] with his essay "The unreasonable effectiveness of physics in mathematics." He argued that the toolbox of physics enables a practitioner like [[Edward Witten]] to go beyond standard mathematics, in particular the geometry of [[4-manifold]]s. The tools of a physicist are cited as [[quantum field theory]], [[special relativity]], non-abelian [[gauge theory]], [[Spin (physics)|spin]], [[chirality (physics)|chirality]], [[supersymmetry]], and the [[Duality (electricity and magnetism)|electromagnetic duality]].<ref>{{cite book|editor-first=A.S.|editor-last=Fokas|author-link=Michael Atiyah|first=Michael|last=Atiyah|date=2002|chapter=The unreasonable effectiveness of physics in mathematics|title=Highlights of Mathematical Physics|pages=25–38|publisher=[[American Mathematical Society]] |isbn=0-8218-3223-9|oclc=50164838}}</ref>

== Prior versions of the argument ==
German scholar [[Moritz Wilhelm Drobisch|Moritz Drobisch]] was known to have revered the "mathematical fundament" of most sciences, as it put students in the position to "awe at the teleological coherence" and "recognise a superhuman, ordering wisdom whose purposes [...] [they] will gradually understand".<ref>Drobisch, Philologie 1832, pp. 24f.</ref> Relevantly,<ref>{{Cite web |title=Between "Bildung" and "Wissenschaft": The 19th-Century German Ideal of Scientific Education — EGO |url=https://ieg-ego.eu/en/threads/models-and-stereotypes/germanophilia-and-germanophobia/bas-van-bommel-between-bildung-and-wissenschaft-the-19th-century-german-ideal-of-scientific-education |access-date=2025-05-10 |website=EGO {{!}} Europäische Geschichte Online |language=de}}</ref> he stated of [[astronomy]]: <blockquote>[T]he harmonious order, in which the celestial bodies describe their orbits, the eternally consistent regularity, touches a deep sounding string within us and elevates us – far from just letting the dead mechanism of chance unwind before us – to the notion of the supreme wise being.<ref>Drobisch, Philologie 1832, p. 22.</ref></blockquote>

==See also==
{{Columns-list|colwidth=30em|
*[[Cosmology]]
*[[Foundations of mathematics]]
*[[Mark Steiner]]
*[[Mathematical universe hypothesis]]
*[[Philosophy of science]]
*[[Quasi-empiricism in mathematics]]
*[[Relationship between mathematics and physics]]
*[[Scientific structuralism]]
*[[Unreasonable ineffectiveness of mathematics]]
*''[[Where Mathematics Comes From]]''
}}

==References==

{{reflist|30em}}

==Further reading==
{{refbegin}}
*{{cite journal|author=Sundar Sarukkai|title=Revisiting the 'unreasonable effectiveness' of mathematics|journal=Current Science|volume=88|issue=3|date=10 February 2005|pages=415–423|jstor=24110208}}
*{{cite journal|first=Alex|last=Kasman|title=Unreasonable Effectiveness|journal=Math Horizons Magazine|pages=29–31|date=April 2003|volume=10|issue=4|doi=10.1080/10724117.2003.12023669|s2cid=218542870|url=http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf334|access-date=2010-09-30|archive-date=2010-06-26|archive-url=https://web.archive.org/web/20100626053603/http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf334|url-status=live}}, a piece of "mathematical fiction."
*{{cite book|first=Mark|last=Colyvan|chapter=Indispensability Arguments in the Philosophy of Mathematics|date=Spring 2015|title=Stanford Encyclopedia of Philosophy|publisher=Metaphysics Research Lab, Stanford University|editor-first=Edward N.|editor-last=Zalta|chapter-url=http://plato.stanford.edu/entries/mathphil-indis/|access-date=2011-02-17|archive-date=2011-05-24|archive-url=https://web.archive.org/web/20110524214649/http://plato.stanford.edu/entries/mathphil-indis/|url-status=live}}
*{{cite book|first=Sorin|last=Bangu|title=[[The Applicability of Mathematics in Science: Indispensability and Ontology]]|publisher=Plagrave MacMillan|location=London|year=2012|isbn=978-0230285200|series=New Directions in the Philosophy of Science}}
* {{cite magazine|last=Wolchover|first=Natalie|date=9 December 2019|title=Why the Laws of Physics Are Inevitable|magazine=Quanta Magazine|url=https://www.quantamagazine.org/how-simple-rules-bootstrap-the-laws-of-physics-20191209/|access-date=12 December 2019|archive-date=12 December 2019|archive-url=https://web.archive.org/web/20191212201218/https://www.quantamagazine.org/how-simple-rules-bootstrap-the-laws-of-physics-20191209/|url-status=live}}
* [https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html Mathematical drama] from [[Stanford University]]
{{refend}}

{{Industrial and applied mathematics}}

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