Before Newton There is a tradition of mathematical analysis of nature that goes back to the ancient Greeks; examples include
Euclid (
Optics),
Archimedes (
On the Equilibrium of Planes,
On Floating Bodies), and
Ptolemy (
Optics,
Harmonics). Later,
Islamic and
Byzantine scholars built on these works, and these ultimately were reintroduced or became available to the West in the
12th century and during the
Renaissance. In the first decade of the 16th century, amateur astronomer
Nicolaus Copernicus proposed
heliocentrism, and published a treatise on it in 1543. He retained the
Ptolemaic idea of
epicycles, and merely sought to simplify astronomy by constructing simpler sets of epicyclic orbits. Epicycles consist of circles upon circles. According to
Aristotelian physics, the circle was the perfect form of motion, and was the intrinsic motion of Aristotle's
fifth element—the quintessence or universal essence known in Greek as
aether for the English
pure air—that was the pure substance beyond the
sublunary sphere, and thus was celestial entities' pure composition. The German
Johannes Kepler [1571–1630],
Tycho Brahe's assistant, modified Copernican orbits to
ellipses, formalized in the equations of Kepler's
laws of planetary motion. An enthusiastic atomist,
Galileo Galilei in his 1623 book
The Assayer asserted that the "book of nature is written in mathematics". His 1632 book, about his telescopic observations, supported heliocentrism. Having made use of experimentation, Galileo then refuted geocentric
cosmology by refuting Aristotelian physics itself. Galileo's 1638 book
Discourse on Two New Sciences established the law of equal free fall as well as the principles of inertial motion, two central concepts of what today is known as
classical mechanics.
Christiaan Huygens, a talented mathematician and physicist and older contemporary of Newton, was the first to successfully idealize a physical problem by a set of mathematical parameters in
Horologium Oscillatorum (1673), and the first to fully mathematize a mechanistic explanation of an unobservable physical phenomenon in
Traité de la Lumière (1690). He is thus considered a forerunner of
theoretical physics and one of the founders of modern mathematical physics.
Newtonian and post-Newtonian physics The prevailing framework for science in the 16th and early 17th centuries was one borrowed from
ancient Greek mathematics, where geometrical shapes formed the building blocks to describe and think about space, and time was often thought as a separate entity. With the introduction of algebra into geometry, and with it the idea of a coordinate system, time and space could now be thought as axes belonging to the same plane. This essential mathematical framework is at the base of all modern physics and used in all further mathematical frameworks developed in next centuries. By the middle of the 17th century, important concepts such as the
fundamental theorem of calculus (proved in 1668 by Scottish mathematician
James Gregory) and finding extrema and minima of functions via differentiation using Fermat's theorem (by French mathematician
Pierre de Fermat) were already known before Leibniz and Newton.
Isaac Newton (1642–1727) developed
calculus (although
Gottfried Wilhelm Leibniz developed similar concepts outside the context of physics) and
Newton's method to solve problems in mathematics and physics. He was extremely successful in his application of
calculus and other methods to the study of motion. Newton's theory of motion, culminating in his
Philosophiæ Naturalis Principia Mathematica (
Mathematical Principles of Natural Philosophy) in 1687, modeled three Galilean laws of motion along with Newton's
law of universal gravitation on a framework of
absolute space—hypothesized by Newton as a physically real entity of Euclidean geometric structure extending infinitely in all directions—while presuming
absolute time, supposedly justifying knowledge of absolute motion, the object's motion with respect to absolute space. The principle of Galilean invariance/relativity was merely implicit in Newton's theory of motion. Having ostensibly reduced the
Kepler's laws as well as Galileo's terrestrial laws of motion to a unifying force, Newton achieved great mathematical rigor, but with theoretical laxity. In the 18th century, the Swiss
Daniel Bernoulli (1700–1782) made contributions to
fluid dynamics, and
vibrating strings. The Swiss
Leonhard Euler (1707–1783) did special work in
variational calculus, dynamics, fluid dynamics, and other areas. Also notable was the Italian
Joseph-Louis Lagrange (1736–1813), for work in
analytical mechanics: he formulated
Lagrangian mechanics and variational methods. A major contribution to the formulation of Analytical Dynamics called
Hamiltonian dynamics was also made by the Irish physicist, astronomer and mathematician,
William Rowan Hamilton (1805–1865). Hamiltonian dynamics had played an important role in the formulation of modern theories in physics, including field theory and quantum mechanics. The French mathematical physicist
Joseph Fourier (1768 – 1830) introduced the notion of
Fourier series to solve the
heat equation, giving rise to a new approach to solving partial differential equations by means of
integral transforms. Into the early 19th century, following mathematicians in France, Germany and England had contributed to mathematical physics. The French
Pierre-Simon Laplace (1749–1827) made paramount contributions to mathematical
astronomy,
potential theory.
Siméon Denis Poisson (1781–1840) worked in
analytical mechanics and
potential theory. In Germany,
Carl Friedrich Gauss (1777–1855) made key contributions to the theoretical foundations of
electricity,
magnetism,
mechanics, and
fluid dynamics. In England,
George Green (1793–1841) published
An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in 1828, which in addition to its significant contributions to mathematics made early progress towards laying down the mathematical foundations of electricity and magnetism. A couple of decades ahead of Newton's publication of a particle theory of light, the Dutch
Christiaan Huygens (1629–1695) developed the wave theory of light, published in 1690. By 1804,
Thomas Young's double-slit experiment revealed an interference pattern, as though light were a wave, and thus Huygens's wave theory of light, as well as Huygens's inference that light waves were vibrations of the
luminiferous aether, was accepted.
Jean-Augustin Fresnel modeled hypothetical behavior of the aether. The English physicist
Michael Faraday introduced the theoretical concept of a field—not action at a distance. Mid-19th century, the Scottish
James Clerk Maxwell (1831–1879) reduced electricity and magnetism to Maxwell's electromagnetic field theory, whittled down by others to the four
Maxwell's equations. Initially, optics was found consequent of Maxwell's field. Later, radiation and then today's known
electromagnetic spectrum were found also consequent of this electromagnetic field. in the context of electrodynamics The English physicist
Lord Rayleigh [1842–1919] worked on
sound. The Irishmen
William Rowan Hamilton (1805–1865),
George Gabriel Stokes (1819–1903) and
Lord Kelvin (1824–1907) produced several major works: Stokes was a leader in
optics and fluid dynamics; Kelvin made substantial discoveries in
thermodynamics; Hamilton did notable work on
analytical mechanics, discovering a new and powerful approach nowadays known as
Hamiltonian mechanics. Very relevant contributions to this approach are due to his German colleague mathematician
Carl Gustav Jacobi (1804–1851) in particular referring to
canonical transformations. The German
Hermann von Helmholtz (1821–1894) made substantial contributions in the fields of
electromagnetism, waves,
fluids, and sound. In the United States, the pioneering work of
Josiah Willard Gibbs (1839–1903) became the basis for
statistical mechanics. Fundamental theoretical results in this area were achieved by the German
Ludwig Boltzmann (1844–1906). Together, these individuals laid the foundations of electromagnetic theory, fluid dynamics, and statistical mechanics.
Relativistic By the 1880s, there was a prominent paradox that an observer within Maxwell's electromagnetic field measured it at approximately constant speed, regardless of the observer's speed relative to other objects within the electromagnetic field. Thus, although the observer's speed was continually lost relative to the electromagnetic field, it was preserved relative to other objects
in the electromagnetic field. And yet no violation of
Galilean invariance within physical interactions among objects was detected. As Maxwell's electromagnetic field was modeled as oscillations of the
aether, physicists inferred that motion within the aether resulted in
aether drift, shifting the electromagnetic field, explaining the observer's missing speed relative to it. The
Galilean transformation had been the mathematical process used to translate the positions in one reference frame to predictions of positions in another reference frame, all plotted on
Cartesian coordinates, but this process was replaced by
Lorentz transformation, modeled by the Dutch
Hendrik Lorentz. In 1908, Einstein's former mathematics professor
Hermann Minkowski, applied
linear algebra to model 3D space together with the 1D axis of time by treating the temporal axis like a fourth spatial dimension—altogether 4D spacetime—and declared the imminent demise of the separation of space and time. In 1887, experimentalists Michelson and Morley failed to detect aether drift, however. It was hypothesized that motion
into the aether prompted aether's shortening, too, as modeled in the
Lorentz contraction. It was hypothesized that the aether thus kept Maxwell's electromagnetic field aligned with the principle of Galilean invariance across all
inertial frames of reference, while Newton's theory of motion was spared. Austrian theoretical physicist and philosopher
Ernst Mach criticized Newton's postulated absolute space. Mathematician
Jules-Henri Poincaré (1854–1912) questioned even absolute time. In 1905,
Pierre Duhem published a devastating criticism of the foundation of Newton's theory of motion. extending invariance to all reference frames—whether perceived as inertial or as accelerated—and credited this to Minkowski, by then deceased. General relativity replaces Cartesian coordinates with
Gaussian coordinates, and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton's
vector of hypothetical gravitational force—an instant
action at a distance—with a gravitational
field. The gravitational field is
Minkowski spacetime itself, the 4D
topology of Einstein aether modeled on a
Lorentzian manifold that "curves" geometrically, according to the
Riemann curvature tensor. The concept of Newton's gravity: "two masses attract each other" replaced by the geometrical argument: "mass transform curvatures of
spacetime and free falling particles with mass move along a geodesic curve in the spacetime". The innovations in geometry were foreshadowed by
non-Euclidean geometry and the explorations of
curvature by
C. F. Gauss. Even massless energy exerts gravitational effect by its
mass equivalence locally "curving" the geometry of the four, unified dimensions of space and time.
Quantum mechanics Another revolutionary development of the 20th century was
quantum theory, which emerged from the seminal contributions of
Max Planck (1856–1947) (on
black-body radiation) and Einstein's work on the
photoelectric effect. In 1912, mathematician
Henri Poincaré published
Sur la théorie des quanta. He introduced the first non-naïve definition of quantization in this paper. The development of early quantum physics followed by a heuristic framework devised by
Arnold Sommerfeld (1868–1951) and
Niels Bohr (1885–1962), but this was soon replaced by the
quantum mechanics developed by
Max Born (1882–1970),
Louis de Broglie (1892–1987),
Werner Heisenberg (1901–1976),
Paul Dirac (1902–1984),
Erwin Schrödinger (1887–1961),
Satyendra Nath Bose (1894–1974), and
Wolfgang Pauli (1900–1958). This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of
self-adjoint operators on an infinite-dimensional vector space. That is called
Hilbert space (introduced by mathematicians
David Hilbert (1862–1943),
Erhard Schmidt (1876–1959) and
Frigyes Riesz (1880–1956) in search of generalization of Euclidean space and study of integral equations), and rigorously defined within the axiomatic modern version by
John von Neumann in his celebrated book
Mathematical Foundations of Quantum Mechanics, where he built up a relevant part of modern functional analysis on Hilbert spaces, the
spectral theory (introduced by
David Hilbert who investigated
quadratic forms with infinitely many variables. Many years later, it had been revealed that his spectral theory is associated with the spectrum of the hydrogen atom. He was surprised by this application.) in particular. Paul Dirac used algebraic constructions to produce a relativistic model for the
electron, predicting its
magnetic moment and the existence of its antiparticle, the
positron.
List of prominent contributors to mathematical physics in the 20th century Prominent contributors to the 20th century's mathematical physics include (ordered by birth date): •
William Thomson (Lord Kelvin) (1824–1907) •
Oliver Heaviside (1850–1925) •
Jules Henri Poincaré (1854–1912) •
David Hilbert (1862–1943) •
Arnold Sommerfeld (1868–1951) •
Constantin Carathéodory (1873–1950) •
Albert Einstein (1879–1955) •
Emmy Noether (1882–1935) •
Max Born (1882–1970) •
George David Birkhoff (1884–1944) •
Hermann Weyl (1885–1955) •
Satyendra Nath Bose (1894–1974) •
Louis de Broglie (1892–1987) •
Norbert Wiener (1894–1964) •
John Lighton Synge (1897–1995) •
Mário Schenberg (1914–1990) •
Wolfgang Pauli (1900–1958) •
Paul Dirac (1902–1984) •
Eugene Wigner (1902–1995) •
Andrey Kolmogorov (1903–1987) •
Lars Onsager (1903–1976) •
John von Neumann (1903–1957) •
Sin-Itiro Tomonaga (1906–1979) •
Hideki Yukawa (1907–1981) •
Nikolay Nikolayevich Bogolyubov (1909–1992) •
Subrahmanyan Chandrasekhar (1910–1995) •
Mark Kac (1914–1984) •
Julian Schwinger (1918–1994) •
Richard Phillips Feynman (1918–1988) •
Irving Ezra Segal (1918–1998) •
Ryogo Kubo (1920–1995) •
Arthur Strong Wightman (1922–2013) •
Chen-Ning Yang (1922–2025) •
Rudolf Haag (1922–2016) •
Freeman John Dyson (1923–2020) •
Martin Gutzwiller (1925–2014) •
Abdus Salam (1926–1996) •
Jürgen Moser (1928–1999) •
Michael Francis Atiyah (1929–2019) •
Joel Louis Lebowitz (1930–) •
Roger Penrose (1931–) •
Elliott Hershel Lieb (1932–) •
Yakir Aharonov (1932–) •
Sheldon Glashow (1932–) •
Steven Weinberg (1933–2021) •
Ludvig Dmitrievich Faddeev (1934–2017) •
David Ruelle (1935–) •
Yakov Grigorevich Sinai (1935–) •
Vladimir Igorevich Arnold (1937–2010) •
Arthur Michael Jaffe (1937–) •
Roman Wladimir Jackiw (1939–) •
Leonard Susskind (1940–) •
Rodney James Baxter (1940–) •
Michael Victor Berry (1941–) •
Giovanni Gallavotti (1941–) •
Stephen William Hawking (1942–2018) •
Jerrold Eldon Marsden (1942–2010) •
Michael C. Reed (1942–) •
John Michael Kosterlitz (1943–) •
Israel Michael Sigal (1945–) •
Alexander Markovich Polyakov (1945–) •
Barry Simon (1946–) •
Herbert Spohn (1946–) •
John Lawrence Cardy (1947–) •
Giorgio Parisi (1948-) •
Abhay Ashtekar (1949-) •
Edward Witten (1951–) •
F. Duncan Haldane (1951–) •
Ashoke Sen (1956–) •
Juan Martín Maldacena (1968–) == See also ==