“1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.
2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.
3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.
Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.”

— Nicholas Murray Butler

Editor's Introduction, The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906) by David Eugene Smith

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Nicholas Murray Butler 17

American philosopher, diplomat, and educator 1862–1947

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“1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.
2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.
3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.
Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.”

David Eugene Smith (1860–1944) American mathematician

David Eugene Smith, "Editor's Introduction," in: The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906)

“In Foundations of the Theory of Signs (p. 6), the three terms in question were defined follows: pragmatics as the study of "the relation of signs to interpreters", semantics as the study of "the relations of signs to the objects to which the signs are applicable", syntactics as the study of "the formal relations of signs to one another."”

Charles W. Morris (1903–1979) American philosopher

Source: Writings on the General Theory of Signs, 1971, p. 301

“We can never wholly separate the human from the mechanical side… But you all see every day that the study of human relations in business and the study of operating are bound up together.”

Mary Parker Follett (1868–1933) American academic

Attributed to Follett in: Richard C. Wallace, ‎David E. Engel, ‎Dr. James E. Mooney (1997). The learning school: a guide to vision-based leadership. p. ix
Attributed from postum publications

“The study of human relations in business and the study of the technology of operating are bound up together.”

Mary Parker Follett (1868–1933) American academic

Attributed to Mary Parker Follett in: Business: The Ultimate Resource, 2001. p. 904.
Attributed from postum publications

“If we wished to relate the space of the [Cubist] painters to geometry, we should have to refer it to the non-Euclidean mathematicians; we should have to study, at some length, certain of Riemann's theorems.”

Jean Metzinger (1883–1956) French painter

Du Cubisme (1912)

Albert Gleizes (1881–1953) French painter

1910s, Du Cubisme (1912)

“Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone.”

Henri Poincaré book Science and Hypothesis

Les mathématiciens n'étudient pas des objets, mais des relations entre les objets ; il leur est donc indifférent de remplacer ces objets par d'autres, pourvu que les relations ne changent pas. La matière ne leur importe pas, la forme seule les intéresse.
Source: Science and Hypothesis (1901), Ch. II: Dover abridged edition (1952), p. 20

“To investigate the essence of a natural phenomenon, three conditions are necessary: We must first study and know the phenomenon itself, from all sides; we must then determine in what relation it stands to other natural phenomena; and lastly, when we have ascertained all these relations, we have to solve the problem of measuring these relations and the laws of mutual dependence—that is, of expressing them in numbers. In the first period of chemistry, all the powers of men's minds were devoted to acquiring a knowledge of the properties of bodies; it was necessary to discover, observe, and ascertain their peculiarities. This is the alchemistical period. The second period embraces the determination of the mutual relations or connections of these properties; this is the period of phlogistic chemistry. In the third period, in which we now are, we ascertain by weight and measure and express in numbers the degree in which the properties of bodies are mutually dependent. The inductive sciences begin with the substance itself, then come just ideas, and lastly, mathematics are called in, and with the aid of numbers, complete the work.”

Justus von Liebig (1803–1873) German chemist

Familiar Letters on Chemistry, Tr. Blythe, 4th ed., London, 1859, p. 60 as quoted by John Theodore Merz, A History of European Thought in the Nineteenth Century Vol.1 http://books.google.com/books?id=xqwQAAAAYAAJ (1903).

“In order to improve your game you must study the endgame before anything else; for, whereas the endings can be studied and mastered by themselves, the middlegame and the opening must be studied in relation to the endgame.”

José Raúl Capablanca (1888–1942) Cuban chess player

Last lectures (1966), p. 19

“Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression.”

Antoine Augustin Cournot Researches into the Mathematical Principles of the Theory of Wealth

Source: Researches into the Mathematical Principles of the Theory of Wealth, 1897, p. 3 ; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/198/mode/2up, (1914) p. 33: About the nature of mathematics

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Nicholas Murray Butler 17

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