The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Last updated August 28, 2024

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article written by the physicist Eugene Wigner, published in Communication in Pure and Applied Mathematics .^[1]^[2] In it, Wigner observes that a theoretical physics's 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."^[3] He adds that the observation "the laws of nature are written in the language of mathematics," properly made by Galileo three hundred years ago, "is now truer than ever before."

Wigner's first example is the 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"^[3] to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."^[4] Wigner says that "Newton ... 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 Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, 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 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 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 Bethe and established by 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

Amongst the responses the thesis received include:

Richard Hamming in computer science, "The Unreasonable Effectiveness of Mathematics".^[5]
Arthur Lesk in molecular biology, "The Unreasonable Effectiveness of Mathematics in Molecular Biology".^[6]
Peter Norvig in artificial intelligence, "The Unreasonable Effectiveness of Data"^[7]
Max Tegmark in physics, "The Mathematical Universe".^[8]
Ivor Grattan-Guinness in mathematics, "Solving Wigner's mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences".^[9]
Vela Velupillai in economics, "The Unreasonable Ineffectiveness of Mathematics in Economics".^[10]

Richard Hamming

Mathematician and Turing Award laureate Richard Hamming reflected on and extended Wigner's Unreasonable Effectiveness in 1980, discussing four "partial explanations" for it,^[5] 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 law of falling bodies not by experimenting, but by simple, though careful, thinking.^[11] 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.):

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?^[12]

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).^[13] Hamming's account does not reveal an awareness of the 20th-century scholarly debate over just what Galileo did.^{[ clarification needed ]}

The inverse square law of universal gravitation necessarily follows from the conservation of energy and of space having three dimensions. Measuring the exponent in the law of universal gravitation is more a test of whether space is 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 integrals and from assuming time invariance.^[14]
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 scalars proved awkward for understanding forces, first vectors, then tensors, 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 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.^[8]^[15] This theory, referred to as the mathematical universe hypothesis, mirrors ideas previously advanced by Peter Atkins.^[16] 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".^[9]^{[ clarification needed ]}

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-manifolds. The tools of a physicist are cited as quantum field theory, special relativity, non-abelian gauge theory, spin, chirality, supersymmetry, and the electromagnetic duality.^[17]

Related Research Articles

The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, stemming from the work of Niels Bohr, Werner Heisenberg, Max Born, and others. While "Copenhagen" refers to the Danish city, the use as an "interpretation" was apparently coined by Heisenberg during the 1950s to refer to ideas developed in the 1925–1927 period, glossing over his disagreements with Bohr. Consequently, there is no definitive historical statement of what the interpretation entails.

The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wave function collapse. This implies that all possible outcomes of quantum measurements are physically realized in some "world" or universe. The evolution of reality as a whole in MWI is rigidly deterministic and local. Many-worlds is also called the relative state formulation or the Everett interpretation, after physicist Hugh Everett, who first proposed it in 1957. Bryce DeWitt popularized the formulation and named it many-worlds in the 1970s.

<span class="mw-page-title-main">Eugene Wigner</span> Hungarian-American physicist and mathematician (1902–1995)

Eugene Paul Wigner was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".

Physics is a branch of science whose primary objects of study are matter and energy. Discoveries of physics find applications throughout the natural sciences and in technology. Historically, physics emerged from the scientific revolution of the 17th century, grew rapidly in the 19th century, then was transformed by a series of discoveries in the 20th century. Physics today may be divided loosely into classical physics and modern physics.

The multiverse is the hypothetical set of all universes. Together, these universes are presumed to comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and constants that describe them. The different universes within the multiverse are called "parallel universes", "flat universes", "other universes", "alternate universes", "multiple universes", "plane universes", "parent and child universes", "many universes", or "many worlds". One common assumption is that the multiverse is a "patchwork quilt of separate universes all bound by the same laws of physics."

Physics is the natural science of matter, involving the study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. Physics is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics is called a physicist.

Paul Adrien Maurice Dirac was an English mathematical and theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for both quantum electrodynamics and quantum field theory. He was the Lucasian Professor of Mathematics at the University of Cambridge, a professor of physics at Florida State University, and a 1933 Nobel Prize in Physics recipient.

A theory of everything (TOE), final theory, ultimate theory, unified field theory, or master theory is a hypothetical, singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all aspects of the universe. Finding a theory of everything is one of the major unsolved problems in physics.

In physics, gravity (from Latin gravitas 'weight') is a fundamental interaction primarily observed as mutual attraction between all things that have mass. Gravity is, by far, the weakest of the four fundamental interactions, approximately 10³⁸ times weaker than the strong interaction, 10³⁶ times weaker than the electromagnetic force and 10²⁹ times weaker than the weak interaction. As a result, it has no significant influence at the level of subatomic particles. However, gravity is the most significant interaction between objects at the macroscopic scale, and it determines the motion of planets, stars, galaxies, and even light.

Reality is the sum or aggregate of all that is real or existent within the universe, as opposed to that which is only imaginary, nonexistent or nonactual. The term is also used to refer to the ontological status of things, indicating their existence. In physical terms, reality is the totality of a system, known and unknown.

An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments. However, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, local or non-local, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters.

In philosophy, the philosophy of physics deals with conceptual and interpretational issues in physics, many of which overlap with research done by certain kinds of theoretical physicists. Historically, philosophers of physics have engaged with questions such as the nature of space, time, matter and the laws that govern their interactions, as well as the epistemological and ontological basis of the theories used by practicing physicists. The discipline draws upon insights from various areas of philosophy, including metaphysics, epistemology, and philosophy of science, while also engaging with the latest developments in theoretical and experimental physics.

Max Erik Tegmark is a Swedish-American physicist, machine learning researcher and author. He is best known for his book Life 3.0 about what the world might look like as artificial intelligence continues to improve. Tegmark is a professor at the Massachusetts Institute of Technology and the president of the Future of Life Institute.

In physics, action at a distance is the concept that an object's motion can be affected by another object without being in physical contact with it; that is, the non-local interaction of objects that are separated in space. Coulomb's law and Newton's law of universal gravitation are based on action at a distance.

In physics, mechanics is the study of objects, their interaction, and motion; classical mechanics is mechanics limited to non-relativistic and non-quantum approximations. Most of the techniques of classical mechanics were developed before 1900 so the term classical mechanics refers to that historical era as well as the approximations. Other fields of physics that were developed in the same era, that use the same approximations, and are also considered "classical" include thermodynamics and electromagnetism.

This timeline lists significant discoveries in physics and the laws of nature, including experimental discoveries, theoretical proposals that were confirmed experimentally, and theories that have significantly influenced current thinking in modern physics. Such discoveries are often a multi-step, multi-person process. Multiple discovery sometimes occurs when multiple research groups discover the same phenomenon at about the same time, and scientific priority is often disputed. The listings below include some of the most significant people and ideas by date of publication or experiment.

There is a diversity of views that propose interpretations of quantum mechanics. They vary in how many physicists accept or reject them. An interpretation of quantum mechanics is a conceptual scheme that proposes to relate the mathematical formalism to the physical phenomena of interest. The present article is about those interpretations which, independently of their intrinsic value, remain today less known, or are simply less debated by the scientific community, for different reasons.

<span class="mw-page-title-main">Theoretical physics</span> Branch of physics

Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena.

The von Neumann–Wigner interpretation, also described as "consciousness causes collapse", is an interpretation of quantum mechanics in which consciousness is postulated to be necessary for the completion of the process of quantum measurement.

<span class="mw-page-title-main">Relationship between mathematics and physics</span>

The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since antiquity, and more recently also by historians and educators. Generally considered a relationship of great intimacy, mathematics has been described as "an essential tool for physics" and physics has been described as "a rich source of inspiration and insight in mathematics".

References

↑ Wigner, E. P. (1960). "The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. S2CID 6112252. Archived from the original on 2021-02-12.
↑ 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 (Kellner, Georg W. (1927). "Die Ionisierungsspannung des Heliums nach der Schrödingerschen Theorie". Zeitschrift für Physik. 44 (1–2): 91–109. Bibcode:1927ZPhy...44...91K. doi:10.1007/BF01391720. S2CID 122213875.) and to Egil Hylleraas.
1 2 Wigner 1960 , §Is the Success of Physical Theories Truly Surprising? p. 8
↑ Wigner 1960 , p. 9
1 2 Hamming, R. W. (1980). "The Unreasonable Effectiveness of Mathematics". The American Mathematical Monthly. 87 (2): 81–90. doi:10.2307/2321982. hdl: 10945/55827 . JSTOR 2321982. Archived from the original on 2022-06-22. Retrieved 2021-07-30.
↑ Lesk, A. M. (2000). "The unreasonable effectiveness of mathematics in molecular biology". The Mathematical Intelligencer. 22 (2): 28–37. doi:10.1007/BF03025372. S2CID 120102813.
↑ Halevy, A.; Norvig, P.; Pereira, F. (2009). "The Unreasonable Effectiveness of Data" (PDF). IEEE Intelligent Systems. 24 (2): 8–12. doi:10.1109/MIS.2009.36. S2CID 14300215. Archived (PDF) from the original on 2022-08-09. Retrieved 2015-09-04.
1 2 Tegmark, Max (2008). "The Mathematical Universe". Foundations of Physics. 38 (2): 101–150. arXiv: 0704.0646 . Bibcode:2008FoPh...38..101T. doi:10.1007/s10701-007-9186-9. S2CID 9890455.
1 2 Grattan-Guinness, I. (2008). "Solving Wigner's mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences". The Mathematical Intelligencer. 30 (3): 7–17. doi:10.1007/BF02985373. S2CID 123174309.
↑ Velupillai, K. V. (2005). "The unreasonable ineffectiveness of mathematics in economics". Cambridge Journal of Economics. 29 (6): 849–872. CiteSeerX 10.1.1.194.6586 . doi:10.1093/cje/bei084.
↑ "The Unreasonable Effectiveness of Mathematics - R.W. Hamming - Some Partial Explanations". ned.ipac.caltech.edu. Retrieved 2024-01-06.
↑ Van Helden, Albert (1995). "On Motion". The Galileo Project. Archived from the original on 21 December 2017. Retrieved 16 October 2013.
↑ Pólya, George; Bowden, Leon; School Mathematics Study Group (1963). Mathematical methods in science; a course of lectures. Studies in mathematics. Vol. 11. Stanford: School Mathematics Study Group. OCLC 227871299.
↑ Folland, Gerald B.; Sitaram, Alladi (1997). "The Uncertainty Principle: A Mathematical Survey". Journal of Fourier Analysis and Applications. 3 (3): 207–238. doi:10.1007/BF02649110. S2CID 121355943.
↑ Tegmark, Max (2014). Our Mathematical Universe. Knopf. ISBN 978-0-307-59980-3.
↑ Atkins, Peter (1992). Creation Revisited. W.H.Freeman. ISBN 978-0-7167-4500-6.
↑ Atiyah, Michael (2002). "The unreasonable effectiveness of physics in mathematics". In Fokas, A.S. (ed.). Highlights of Mathematical Physics. American Mathematical Society. pp. 25–38. ISBN 0-8218-3223-9. OCLC 50164838.