“The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration ofjkbni semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form.”

— Leonid Kantorovich

"On one class of functional equations" (1936), as cited in: O'Connor, John J.; Robertson, Edmund F., " Leonid Kantorovich http://www-history.mcs.st-andrews.ac.uk/Biographies/Kantorovich.html", MacTutor History of Mathematics archive, University of St Andrews

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Leonid Kantorovich8

Russian mathematician 1912–1986

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“Ideas and techniques known in quantum electrodynamics have been applied to the Bardeen-Cooper-Schrieffer theory of superconductivity. In an approximation which corresponds to a generalization of the Hartree-Fock fields, one can write down an integral equation defining the self-energy of an electron in an electron gas with phonon and Coulomb interaction. The form of the equation implies the existence of a particular solution which does not follow from perturbation theory, and which leads to the energy gap equation and the quasi-particle picture analogous to Bogolyubov's.”

Yoichiro Nambu (1921–2015) American physicist

[Quasi-particles and gauge invariance in the theory of superconductivity, Physical Review, 117, 3, February 1960, 648–663, 10.1103/PhysRev.117.648]

“The field equation may… be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium—in complete analogy with… the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Footnote
Geometry as a Branch of Physics (1949)

“A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side.
Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.”

Rudolf E. Kálmán (1930–2016) Hungarian-born American electrical engineer

Source: New results in linear filtering and prediction theory (1961), p. 95 Article summary; cited in: " Rudolf E. Kálmán http://www-history.mcs.st-andrews.ac.uk/Biographies/Kalman.html", MacTutor History of Mathematics archive, 2010

“Almost all of fluid dynamics follows from a differential equation called the Navier-Stokes equation. But this general equation has not, in practice, led to solutions of real problems of any complexity. In this sense, the curve of a baseball is not understood; the Navier-Stokes equation applied to a base ball has not been solved.”

Robert Adair (physicist) (1924) Physicist and author

Source: The Physics Of Baseball (Second Edition - Revised), Chapter 2, The Flight Of The baseball, p. 22

“Any progress in the theory of partial differential equations must also bring about a progress in Mechanics.”

Carl Gustav Jacob Jacobi (1804–1851) German mathematician

Vorlesungen über Dynamik http://archive.org/details/cgjjacobisvorle00lottgoog [Lectures on Dynamics] (1842/3; publ. 1884).

“If the variables are continuous, this definition [Ashby’s fundamental concept of machine] corresponds to the description of a dynamic system by a set of ordinary differential equations with time as the independent variable. However, such representation by differential equations is too restricted for a theory to include biological systems and calculating machines where discontinuities are ubiquitous.”

Ludwig von Bertalanffy (1901–1972) austrian biologist and philosopher

Source: General System Theory (1968), 4. Advances in General Systems Theory, p. 96, as cited in: Vincent Vesterby (2013) From Bertalanffy to Discipline-Independent-Transdisciplinarity http://journals.isss.org/index.php/proceedings56th/article/viewFile/1886/672

“Science is a differential equation. Religion is a boundary condition.”

Alan Turing (1912–1954) British mathematician, logician, cryptanalyst, and computer scientist

Epigram to Robin Gandy (1954); reprinted in Andrew Hodges, Alan Turing: the Enigma (Vintage edition 1992), p. 513.

“Elegance requires that the number of defining equations be small. Five is better than ten, and one is better than five. On this score, one might facetiously say that String Theory is the ultimate epitome of elegance. With all the years that String Theory has been studied, no one has found even a single defining equation! The number at present count is zero. We know neither what the fundamental equations of the theory are nor even if it has any.”

Leonard Susskind book The Cosmic Landscape

[The Cosmic Landscape: String Theory and the Illusion of Intelligent Design, 2008, Little, Brown, https://books.google.com/books?id=RIW9E1sOyxUC&q=epitome#v=snippet&q=epitome&f=false]

“The same equations have the same solutions”

Richard Feynman (1918–1988) American theoretical physicist

volume II; lecture 12, "Electrostatic Analogs"; p. 12-1
The Feynman Lectures on Physics (1964)

“The most fatal trap into which thinking may fall is the equation of existence and expediency.”

Abraham Joshua Heschel (1907–1972) Polish-American Conservative Judaism Rabbi

Source: Who Is Man? (1965), Ch. 5
Context: Man is naturally self-centered and he is inclined to regard expediency as the supreme standard for what is right and wrong. However, we must not convert an inclination into an axiom that just as man's perceptions cannot operate outside time and space, so his motivations cannot operate outside expediency; that man can never transcend his own self. The most fatal trap into which thinking may fall is the equation of existence and expediency.

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Leonid Kantorovich8

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