The subject of physical mathematics is concerned with

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 ...

that is motivated by

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 ...

and is considered by some as a subfield of

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

Overview

Physically motivated mathematics existed within a tradition of mathematical analysis of nature that goes back to the ancient Greeks. A good example is Archimedes' ''Method of Mechanical Theorems'', where the principle of the balance is used to find results in pure

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 ...

. This tradition, elaborated further by Islamic and Byzantine scholars, was reintroduced to the West in the

12th century The 12th century is the period from 1101 to 1200 in accordance with the Julian calendar. In the history of European culture, this period is considered part of the High Middle Ages and overlaps with what is often called the Golden Age' of the ...

and during the

Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...

. It became known as "mixed mathematics" and was a major contributor to the emergence of modern mathematical physics in the

17th century The 17th century lasted from January 1, 1601 (represented by the Roman numerals MDCI), to December 31, 1700 (MDCC). It falls into the early modern period of Europe and in that continent (whose impact on the world was increasing) was characterized ...

. The details of

physical unit A unit of measurement, or unit of measure, is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...

s and their manipulation were addressed by

Alexander Macfarlane Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician. Life Macfarlane was born in Blairgowrie, Scotland, to Daniel MacFarlane (Shoemaker, Blairgowrie) and Ann Small. He s ...

in ''Physical Arithmetic'' in 1885. The science of

kinematics In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with s ...

created a need for mathematical representation of

motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...

and has found expression with

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

s, and

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

. At the

University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

the

Mathematical Tripos The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics, University of Cambridge, Faculty of Mathematics at the University of Cambridge. Origin In its classical nineteenth-century form, the tripos was a di ...

tested students on their knowledge of "mixed mathematics". "... w books which appeared in the mid-eighteenth century offered a systematic introduction to the fundamental operations of the fluxional calculus and showed how it could be applied to a wide range of mathematical and physical problems. ... The strongly problem-oriented presentation in the treatises ... made it much easier for university students to master the fluxional calculus and its applications ndhelped define a new field of mixed mathematical studies..." An adventurous expression of physical mathematics is found in

Maxwell Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (disambiguation) * Maxwell baronets, in the Baronetage of N ...

's ''

A Treatise on Electricity and Magnetism ''A Treatise on Electricity and Magnetism'' is a two-volume treatise on electromagnetism written by James Clerk Maxwell in 1873. Maxwell was revising the ''Treatise'' for a second edition when he died in 1879. The revision was completed by Wil ...

'', which used

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s. The text aspired to describe phenomena in four dimensions, but the foundation for this physical world,

Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...

, trailed by forty years. String theorist Greg Moore said this about physical mathematics in his vision talk at Strings 2014.

References

Eric Zaslow Eric Zaslow is an American mathematical physicist at Northwestern University. Biography Zaslow attended Harvard University, earning his Ph.D. in physics in 1995, with thesis "Kinks, twists, and folds : exploring the geometric musculature of qu ...

, Physmatics, *

Arthur Jaffe Arthur Michael Jaffe (; born December 22, 1937) is an American mathematical physicist at Harvard University, where in 1985 he succeeded George Mackey as the Landon T. Clay Professor of Mathematics and Theoretical Science. Education and career ...

, Frank Quinn, "Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics",

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

30: 178-207, 1994, *

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...

et al., "Responses to Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics, by A. Jaffe and F. Quinn", Bull. Am. Math. Soc. 30: 178-207, 1994, * Michael Stöltzner, "Theoretical Mathematics: On the Philosophical Significance of the Jaffe-Quinn Debate", in: ''The Role of Mathematics in Physical Sciences'', pages 197-222, * Kevin Hartnett (November 30, 2017
"Secret link discovered between pure math and physics"

Quanta Magazine ''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science ...

Applied mathematics Mathematical physics {{math-physics-stub

Overview

See also

References