“We completely agree that description by differential equations is not only a clumsy but, in principle, inadequate way to deal with many problems of organization.”

—  Ludwig von Bertalanffy

Source: General System Theory (1968), 4. Advances in General Systems Theory, p. 97

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Ludwig von Bertalanffy 65
austrian biologist and philosopher 1901–1972

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“General systems theory deals with the most fundamental concepts and aspects of systems. Many theories dealing with more specific types of systems (e. g., dynamical systems, automata, control systems, game-theoretic systems, among many others) have been under development for quite some time. General systems theory is concerned with the basic issues common to all these specialized treatments. Also, for truly complex phenomena, such as those found predominantly in the social and biological sciences, the specialized descriptions used in classical theories (which are based on special mathematical structures such as differential or difference equations, numerical or abstract algebras, etc.) do not adequately and properly represent the actual events. Either because of this inadequate match between the events and types of descriptions available or because of the pure lack of knowledge, for many truly complex problems one can give only the most general statements, which are qualitative and too often even only verbal. General systems theory is aimed at providing a description and explanation for such complex phenomena.”

Mihajlo D. Mesarovic (1928) Serbian academic

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“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

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“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

“By the term functionalism, considered as a principle of organization, we mean the differentiation or distinction between kind of duties. Thus it is clearly distinguished from the scalar principle, in which there is also differentiation, but of quite another kind. The scalar differentiation refers simply to degrees or gradations of authority.”

James D. Mooney (1884–1957) American businessman

Source: Onward Industry!, 1931, p. 45

“Concepts like those of organization, wholeness, directiveness, teleology, control, self-regulation, differentiation, and the like are alien to conventional science. However, they pop up everywhere in the biological, behavioral, and social sciences and are, in fact, indispensable for dealing with living organisms or social groups. Thus, a basic problem posed to modem science is a general theory of organization. General Systems Theory is, in principle, capable of giving exact definitions for such concepts.”

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

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“All this takes many clumsy and inexact word-descriptions to describe, but if we read paintings like we read books, it would not be such a hidden language for painting can effortlessly produce such elegant solutions.”

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“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 (1912–1986) Russian mathematician

"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

“Today our main problem is that of organized complexity. Concepts like those of organization, wholeness, directiveness, teleology, control, self-regulation, differentiation and the like are alien to conventional physics. However, they pop up everywhere in the biological, behavioural and social sciences, and are, in fact, indispensable for dealing with living organisms or social groups. Thus, a basic problem posed to modern science is a general theory of organization.”

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“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.”

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“Ours, according to Leibnitz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations.”

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[Lectures on Celestial Mechanics, https://books.google.com/books?id=mZtrCQAAQBAJ&pg=PA1] (p. 1)

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