# „It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. …Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.“

—  Thomas Little Heath, Diophantos of Alexandria: A Study in the History of Greek Algebra (1885)
##### Thomas Little Heath46
British civil servant and academic 1861 - 1940

### „When an equation…clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots. …Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions.“

—  Morris Kline American mathematician 1908 - 1992
Mathematical Thought from Ancient to Modern Times (1972), p. 143.

### „Find a fraction which, multiplied by itself, shall give 6, or… find the square root of 6. This can be shown to be an impossible problem; for it can be shown that no fraction whatsoever multiplied by itself, can give a whole number, unless it be itself a whole number disguised in a fractional form, such as 4&frasl;2 or 21&frasl;3. To this problem, then, there is but one answer, that it is self-contradictory. But if we propose the following problem,—to find a fraction which, multiplied by itself, shall give a product lying between 6 and 6 + a; we find that this problem admits of solution in every case. It therefore admits of solution however small a may be… as small as you please.“

—  Augustus De Morgan British mathematician, philosopher and university teacher (1806-1871) 1806 - 1871
The Differential and Integral Calculus (1836), ...there is such a thing as the square root of 6, and it is denoted by &radic;<span style="text-decoration: overline">6</span>. But we do not say we actually find this, but that we approximate to it.

### „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) scholar 1854 - 1923
A Short History of Greek Mathematics (1884), p, 125

### „Sometimes… Diophantus solves a problem wholly or in part by synthesis…. Although… Diophantus does not treat his problems generally and is usually content with finding any particular numbers which happen to satisfy the conditions of his problems,… he does occasionally attempt such general solutions as were possible to him. But these solutions are not often exhaustive because he had no symbol for a general coefficient.“

—  James Gow (scholar) scholar 1854 - 1923
A Short History of Greek Mathematics (1884), p, 125

### „Causality applies only to a system which is left undisturbed. If a system is small, we cannot observe it without producing a serious disturbance and hence we cannot expect to find any causal connexion between the results of our observations. Causality will still be assumed to apply to undisturbed systems and the equations which will be set up to describe an undisturbed system will be differential equations expressing a causal connexion between conditions at one time and conditions at a later time. These equations will be in close correspondence with the equations of classical mechanics, but they will be connected only indirectly with the results of observations.“

—  Paul Dirac, livro Principles of Quantum Mechanics
The Principles of Quantum Mechanics (4th ed. 1958), I. The Principle of Superposition - 1. The Need for a Quantum Theory

### „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 Hungarian-born American electrical engineer 1930 - 2016
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

### „The spinor genus is due to Eichler. His early results (1952) established the theory over the rational field and also, in certain special cases, over a number field. Kneser (1956) extended this to number fields in general. At about the same time Watson obtained Eichler's results by elementary methods over the rational field.“

—  O. Timothy O'Meara American mathematician 1928 - 2018
[Review: Integral quadratic forms by G. L. Watson, Bull. Amer. Math. Soc., 67, 1961, 536–538, 10.1090/S0002-9904-1961-10673-3] (quote from p. 537)

### „The Hindus introduced negative numbers… The first known use is Brahmagupta about 628; he also states the rules for the four operations with negative numbers. Bhāskara points out that the square root of a positive number is twofold, positive and negative. He brings up the matter of the square root of a negative number but says that there is no square root because a negative number is not a square. No definitions, axioms, or theorems are given.The Hindus did not unreservedly accept negative numbers. Even Bhāskara, while giving 50 and -5 as two solutions of a problem, says, "The second value is in this case not to be taken, for it is inadequate; people do not approve of negative solutions."“

—  Morris Kline American mathematician 1908 - 1992
Mathematical Thought from Ancient to Modern Times (1972), However, negative numbers gained acceptance slowly. p. 185.

### „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) Physicist and author 1924
The Physics Of Baseball (Second Edition - Revised), Chapter 2, The Flight Of The baseball, p. 22

### „Since the 1950s, the key equation of quantum gravity has been called the Wheeler-DeWitt equation. Bryce DeWitt and John Wheeler wrote it down, but in all the time since then, no one had been able to solve it. We found we could solve it exactly, and in fact we found an infinite number of exact solutions.“

—  Lee Smolin American cosmologist 1955
"Loop Quantum Gravity," The New Humanists: Science at the Edge (2003), Describing work with Ted Jacobson

### „Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers).“

—  Henry Burchard Fine American academic 1858 - 1928
The Number-System of Algebra, (1890), p. 86; Reported in Moritz (1914, 282)

### „Diophantos lived in a period when the Greek mathematicians of great original power had been succeeded by a number of learned commentators, who confined their investigations within the limits already reached, without attempting to further the development of the science. To this general rule there are two most striking exceptions, in different branches of mathematics, Diophantos and Pappos. These two mathematicians, who would have been an ornament to any age, were destined by fate to live and labour at a time when their work could not check the decay of mathematical learning. There is scarcely a passage in any Greek writer where either of the two is so much as mentioned. The neglect of their works by their countrymen and contemporaries can be explained only by the fact that they were not appreciated or understood. The reason why Diophantos was the earliest of the Greek mathematicians to be forgotten is also probably the reason why he was the last to be re-discovered after the Revival of Learning. The oblivion, in fact, into which his writings and methods fell is due to the circumstance that they were not understood. That being so, we are able to understand why there is so much obscurity concerning his personality and the time at which he lived. Indeed, when we consider how little he was understood, and in consequence how little esteemed, we can only congratulate ourselves that so much of his work has survived to the present day.“

—  Thomas Little Heath British civil servant and academic 1861 - 1940
Diophantos of Alexandria: A Study in the History of Greek Algebra (1885), Historical Introduction, p.17

### „The systems approach to problems does not mean that the most generally formulated problem must be solved in one research project. However desirable this may be, it is seldom possible to realize it in practice. In practice, parts of the total problem are usually solved in sequence. In many cases the total problem cannot be formulated in advance but the solution of one phase of it helps define the next phase. For example, a production control project may require determination of the most economic production quantities of different items. Once these are found it may turn out that these quantities cannot be produced on the available equipment in the available time. This, then, gives rise to a new problem whose solution will affect the solution obtained in the first phase.“

—  C. West Churchman American philosopher and systems scientist 1913 - 2004
1940s - 1950s, Introduction to Operations Research (1957), p. 7

### „The principle of bounded rationality [is] the capacity of the human mind for formulating and solving complex problems is very small compared with the size of the problems whose solution is required for objectively rational behavior in the real world — or even for a reasonable approximation to such objective rationality.“

—  Herbert A. Simon, livro Administrative Behavior
1940s-1950s, Administrative Behavior, 1947, p. 198.

### „The problems for which I could find no solution in fact had no solution.“

—  Michael Moorcock, livro The Eternal Champion
The Eternal Champion (1970), Chapter 23 “In Loos Ptokai” (p. 137)

### „In 1922, Friedmann… broke new ground by investigating non-static solutions to Einstein's field equations, in which the radius of curvature of space varies with time. This Possibility had already been envisaged, in a general sense, by Clifford in the eighties.“

—  Gerald James Whitrow British mathematician 1912 - 2000
The Structure of the Universe: An Introduction to Cosmology (1949), Context: The models of Einstein and de Sitter are static solutions of Einstein's modified gravitational equations for a world-wide homogeneous system. They both involve a positive cosmological constant &lambda;, determining the curvature of space. If this constant is zero, we obtain a third model in classical infinite Euclidean space. This model is empty, the space-time being that of Special Relativity. It has been shown that these are the only possible static world models based on Einstein's theory. In 1922, Friedmann... broke new ground by investigating non-static solutions to Einstein's field equations, in which the radius of curvature of space varies with time. This Possibility had already been envisaged, in a general sense, by Clifford in the eighties.<!--p.82

### „The great problem of man, how to live in conscioues harmony with himself, with his neighbor, and with the whole to which he belongs, admits of as many solutions as there are provinces in our Father's kingdom; and it is in this, and not in the material sphere, that individuals and nations display their divergences of character.“

—  Theodor Mommsen German classical scholar, historian, jurist, journalist, politician, archaeologist and writer 1817 - 1903
The History of Rome - Volume 1, Context: The great problem of man, how to live in conscioues harmony with himself, with his neighbor, and with the whole to which he belongs, admits of as many solutions as there are provinces in our Father's kingdom; and it is in this, and not in the material sphere, that individuals and nations display their divergences of character. The exciting causes which gave rise to this intrinsic contrast must have been in the Græco-Italian period as yet wanting; it was not until the Hellenes and Italians separated that deep-seated diversity of mental character became manifest, the effects of which contiue to the present day. The family and the state, religion and art, received in Italy and in Greece respectively a development so peculiar and so thoroughly national, that the common basis, on which in these respects also the two peoples rested, has been so overgrown as to be almost concealed from our view. That Hellenic character, which sacrificed the whole to its individual elements, the nation to the single state, and the single state to the citizen; whose ideal of life was the beautiful and the good; and, only too often, the pleasure of idleness; whose political development consisted in intensifying the original individualism of the several cantons, and subsequently led to the internal dissolution of the authority of the state; whose view of religion first invested its gods with human attributes, and then denied their existence; which gave full play to the limbs in the sports of the naked youth, and gave free scope to thought in all its grandeur and in all its awefulness;- and taht Roman character, which solemnly bound the son to reverence the father, the citizen to reverence the ruler, and all to reverence the gods; which required nothing; and honoured nothing, but the useful act, and compelled every citizen to fill up every moment of his brief life with unceasing work; which made it a duty even in the boy to modestly to cover the body; which deemed every one a bad citizen who wished to be different from his fellows; which viewed the states as all in all, and a desire for the state's extension as the only aspiration not liable to censure,- who can in thought trace back these sharply-marked contrasts to that original unity which embraced them both, prepared the way for their development, and at length produced them? Vol. 1, pt. 1, Chapter 2: "Into Italy" Translated by W.P.Dickson.

### „It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning.“

—  David Hilbert, Mathematical Problems
Mathematical Problems (1900)

### „I've been in the inventor business for 33 years and my experience is that for every problem the Lord has made, He has also made a solution. If you and I can't find the solution, the let's honestly admit that you and I are damn fools but why blame it on the Lord and say He created something 'impossible.“

—  Thomas Edison American inventor and businessman 1847 - 1931

### „The type of rationality we assume in economics — perfect, logical, deductive rationality — is extremely useful in generating solutions to theoretical problems. But it demands much of human behavior — much more in fact than it can usually deliver. If we were to imagine the vast collection of decision problems economic agents might conceivably deal with as a sea or an ocean, with the easier problems on top and more complicated ones at increasing depth, then deductive rationality would describe human behavior accurately only within a few feet of the surface. For example, the game Tic-Tac-Toe is simple, and we can readily find a perfectly rational, minimax solution to it. But we do not find rational “solutions” at the depth of Checkers; and certainly not at the still modest depths of Chess and Go.“

—  W. Brian Arthur American economist 1946
Inductive Reasoning and Bounded Rationality (The El Farol Problem) (1994), p. 1

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