“We have already various treatises on Mechanics, but the plan of this one is entirely new. I intend to reduce the theory of this Science, and the art of solving problems relating to it, to general formulae, the simple development of which provides all the equations necessary for the solution of each problem. I hope that the manner in which I have tried to attain this object will leave nothing to be desired. No diagrams will be found in this work. The methods that I explain require neither geometrical, nor mechanical, constructions or reasoning, but only algebraical operations in accordance with regular and uniform procedure. Those who love Analysis will see with pleasure that Mechanics has become a branch of it, and will be grateful to me for having thus extended its domain.”

— Joseph Louis Lagrange, book Mécanique analytique

Mécanique analytique (1788) as quoted by E. W. Hobson, Mathematics, from the points of view of the Mathematician and of the Physicist (1912) an address delivered to the Mathematical and Physical Society of University College London, p.13. https://books.google.com/books?id=H7Y_AQAAIAAJ&pg=PA13

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Joseph Louis Lagrange6

Italian mathematician and mathematical physicist 1736–1813

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“I discovered that a whole range of problems of the most diverse character relating to the scientiﬁc organization of production (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i. e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion.
I have succeeded in ﬁnding a comparatively simple general method of solving this group of problems which is applicable to all the problems I have mentioned, and is sufﬁciently simple and effective for their solution to be made completely achievable under practical conditions.”

Leonid Kantorovich (1912–1986) Russian mathematician

Kantorovich (1960) "Mathematical Methods of Organizing and Planning Production." Management Science, 6(4):366–422, 1960, p. 368); As cited in: Cockshott, W. Paul. " Mises, Kantorovich and economic computation http://www.dcs.gla.ac.uk/publications/PAPERS/8707/standalonearticle.pdf." (2007).

“The solution of the higher indeterminates depends almost entirely on very favourable numerical conditions and his methods are defective. But the extraordinary ability of Diophantus appears rather in the other department of his art, namely the ingenuity with which he reduces every problem to an equation which he is competent to solve.”

James Gow (scholar) (1854–1923) scholar

p, 125
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“Whenever a theory appears to you as the only possible one, take this as a sign that you have neither understood the theory nor the problem which it was intended to solve.”

Karl Popper (1902–1994) Austrian-British philosopher of science

Objective Knowledge: An Evolutionary Approach (1972)

“First of all I want to remind you of the essential features of models. In my opinion they are: (i) drawing up a list of the variables to be considered; (ii) drawing up a list of the equations or relations the variables have to obey and (iii) testing the validity of the equations, which implies the estimation of their coefficients, if any. As a consequence of especially (iii) we may have to revise (i) and (ii) so as to arrive at a satisfactory degree of realism of the theory embodied in the model. Then, the model may be used for various purposes, that is, for the solution of various problems. The advantages of models are, on one hand, that they force us to present a "complete" theory by which I mean a theory taking into account all relevant phenomena and relations and, on the other hand, the confrontation with observation, that is, reality. Of course these remarks are far from new.”

Jan Tinbergen (1903–1994) Dutch economist

"The Use of Models: Experience," 1969

“This work… was designed in the first instance to be a second edition of a Treatise on Algebra, published in 1830, and which has been long out of print; but I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iii
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“Ugly problems often require ugly solutions. Solving an ugly problem in a pure manner is bloody hard.”

Rasmus Lerdorf (1968) Danish programmer and creator of PHP

php.devel Mailing List, October 2002 http://osdir.com/ml/php.devel/2002-10/msg00704.html

“Each problem that I solved became a rule which served afterwards to solve other problems”

René Descartes (1596–1650) French philosopher, mathematician, and scientist

“Art is this: art is the solution of a problem which cannot be expressed explicitly until it is solved.”

Piet Hein (1905–1996) Danish puzzle designer, mathematician, author, poet

As quoted in Man Creates Art Creates Man (1973) by Duane Preble, p. 14
Variant translation: Art is solving problems that cannot be formulated before they have been solved. The shaping of the question is part of the answer.
As quoted in Architecture: form, space, and order (2007) by Francis D.K. Ching, p. ix
Context: After all, what is art? Art is the creative process and it goes through all fields. Einstein’s theory of relativity — now that is a work of art! Einstein was more of an artist in physics than on his violin.
Art is this: art is the solution of a problem which cannot be expressed explicitly until it is solved.

“In order to this, I found it absolutely necessary to consider this Subject entirely anew, as if it had never been treated of before; the Principles of the old Perspective being so narrow, and so confined, that they could be of no use in my Design: And I was forced to invent new Terms of Art, those already in use being so peculiarly adapted to the imperfect Notions that have hitherto been had of this Art, that I could make no use of them in explaining those general Principles I intended to establish.”

Brook Taylor (1685–1731) English mathematician

New Principles of Linear Perspective (1715, 1749)

“The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.”

Niels Henrik Abel (1802–1829) Norwegian mathematician

A Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree (1824) Tr. W. H. Langdon, as quote in A Source Book in Mathematics (1929) ed. David Eugene Smith

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Joseph Louis Lagrange6

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