The relationship between mathematics and physics has been a subject of study of

philosopher Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

s and

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...

s since antiquity, and more recently also by

historian A historian is a person who studies and writes about the past and is regarded as an authority on it. Historians are concerned with the continuous, methodical narrative and research of past events as relating to the human species; as well as the ...

s and

educator A teacher, also called a schoolteacher or formally an educator, is a person who helps students to acquire knowledge, competence, or virtue, via the practice of teaching. ''Informally'' the role of teacher may be taken on by anyone (e.g. w ...

s. Generally considered a relationship of great intimacy,

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

has been described as "an essential tool for physics" and

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

has been described as "a rich source of inspiration and insight in mathematics". Some of the oldest and most discussed themes are about the main differences between the two subjects, their mutual influence, the role of

mathematical rigor Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mat ...

in physics, and the problem of explaining the effectiveness of mathematics in physics. In his work ''

Physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

'', one of the topics treated by

Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...

is about how the study carried out by mathematicians differs from that carried out by physicists. Considerations about mathematics being the language of

nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...

can be found in the ideas of the

Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...

: the convictions that "Numbers rule the world" and "All is number", and two millennia later were also expressed by

Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...

: "The book of nature is written in the language of mathematics".

Historical interplay

Before giving a

mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...

for the formula for the

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

of a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

used physical reasoning to discover the solution (imagining the balancing of bodies on a scale).

classified physics and mathematics as theoretical sciences, in contrast to practical sciences (like

ethics Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...

politics Politics () is the set of activities that are associated with decision-making, making decisions in social group, groups, or other forms of power (social and political), power relations among individuals, such as the distribution of Social sta ...

) and to productive sciences (like

medicine Medicine is the science and Praxis (process), practice of caring for patients, managing the Medical diagnosis, diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment, Palliative care, palliation of their injury or disease, ...

botany Botany, also called plant science, is the branch of natural science and biology studying plants, especially Plant anatomy, their anatomy, Plant taxonomy, taxonomy, and Plant ecology, ecology. A botanist or plant scientist is a scientist who s ...

). 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).E. J. Post
A History of Physics as an Exercise in Philosophy, p. 76.
/ref> The creation and development of

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

were strongly linked to the needs of physics: There was a need for a new mathematical language to deal with the new dynamics that had arisen from the work of scholars such as Galileo Galilei and

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

. The concept of

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

was needed, Newton did not have the modern concept of limits, and instead employed

infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

s, which lacked a rigorous foundation at that time. During this period there was little distinction between physics and mathematics; as an example, Newton regarded

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

as a branch of

mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

. Non-Euclidean geometry, as formulated by

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

, János Bolyai,

Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky (; , ; – ) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, kno ...

, and

Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

, freed physics from the limitation of a single Euclidean geometry. A version of non-Euclidean geometry, called

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

, enabled

Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

to develop

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

by providing the key mathematical framework on which he fit his physical ideas of gravity. In the 19th century

Auguste Comte Isidore Auguste Marie François Xavier Comte (; ; 19 January 1798 – 5 September 1857) was a French philosopher, mathematician and writer who formulated the doctrine of positivism. He is often regarded as the first philosopher of science in the ...

in his hierarchy of the sciences, placed physics and astronomy as less general and more complex than mathematics, as both depend on it. In 1900,

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

in his 23 problems for the advancement of mathematical science, considered the axiomatization of physics as his sixth problem. The problem remains open. In 1930,

Paul Dirac Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...

invented the

Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

which produced a single value when used in an integral. The mathematical rigor of this function was in doubt until the mathematician Laurent Schwartz developed on the theory of distributions. Connections between the two fields sometimes only require identifying similar concepts by different names, as shown in the 1975 Wu–Yang dictionary, that related concepts of

gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

with

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

Physics is not mathematics

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

measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to ...

s. In physics, definitions are abstractions or idealizations, approximations adequate when compared to the natural world. In 1960, Georg Rasch noted that no models are ever true, not even Newton's laws, emphasizing that models should not be evaluated based on truth but on their applicability for a given purpose. For example, Newton built a physical model around definitions like his second law of motion

\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. Mathematics deals with entities whose properties can be known with

certainty Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and ...

. According to

David Hume David Hume (; born David Home; – 25 August 1776) was a Scottish philosopher, historian, economist, and essayist who was best known for his highly influential system of empiricism, philosophical scepticism and metaphysical naturalism. Beg ...

, 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". This leads to a situation that was put by

as "No number of experiments can prove me right; a single experiment can prove me wrong." The ultimate goal in research in

pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...

are rigorous proofs, while in physics heuristic arguments may sometimes suffice in leading-edge research.MICHAEL ATIYAH ET AL. "RESPONSES TO ''THEORETICAL MATHEMATICS: TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS'', BY A. JAFFE AND F. QUINN. https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00503-8/S0273-0979-1994-00503-8.pdf" In short, the methods and goals of physicists and mathematicians are different. Nonetheless, according to Roland Omnès, the

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s of mathematics are not mere conventions, but have physical origins.

Role of rigor in physics

Rigor is indispensable in pure mathematics. But many definitions and arguments found in the physics literature involve concepts and ideas that are not up to the standards of rigor in mathematics. For example,

Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ...

characterized

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

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.

Philosophical problems

Some of the problems considered in the

philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...

are the following: *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?" —

, in ''Geometry and Experience'' (1921). *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. *What is the geometry of physical space? *What is the origin of the axioms of mathematics? *How does the already existing mathematics influence in the creation and development of physical theories? *Is arithmetic ''analytic'' or synthetic? (from Kant, see Analytic–synthetic distinction) *What is essentially different between doing a physical experiment to see the result and making a mathematical calculation to see the result? (from the Turing–

Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...

debate) *Do

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...

imply that physical theories will always be incomplete? (from

Stephen Hawking Stephen William Hawking (8January 194214March 2018) was an English theoretical physics, theoretical physicist, cosmologist, and author who was director of research at the Centre for Theoretical Cosmology at the University of Cambridge. Between ...

) *Is mathematics invented or discovered? (millennia-old question, raised among others by

Mario Livio Mario Livio (born June 19, 1945) is an astrophysics, astrophysicist and an author of works that popularize science and mathematics. For 24 years (1991–2015) he was an astrophysicist at the Space Telescope Science Institute, which operates the H ...

)

Education

In recent times the two disciplines have most often been taught separately, despite all the interrelations between physics and mathematics. This led some professional mathematicians who were also interested in

mathematics education In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out Scholarly method, scholarly research into the transfer of mathematical know ...

, such as

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

, Richard Courant, Vladimir Arnold and Morris Kline, to strongly advocate teaching mathematics in a way more closely related to the physical sciences. 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.https://bridge.math.oregonstate.edu/papers/ampere.pdf

References

External links

Gregory W. Moore – Physical Mathematics and the Future (July 4, 2014)

IOP Institute of Physics – Mathematical Physics: What is it and why do we need it? (September 2014)

Feynman explaining the differences between mathematics and physics in a video available on YouTube
Philosophy of physics Philosophy of mathematics History of science Mathematics education Physics education Foundations of mathematics History of mathematics History of physics