“The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things.
Considered from this point of view, mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.
Its chief attribute is clearness; it has no marks to express confused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them.”

— Joseph Fourier , book The Analytical Theory of Heat

Preliminary Discourse, p.7 Note: often quoted as Mathematics [or mathematical analysis] compares the most diverse phenomena and discovers the secret analogies that unite them.
The Analytical Theory of Heat (1878)

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Joseph Fourier 4

French mathematician and physicist 1768–1830

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“I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely.”

René Descartes book La Géométrie

Second Book
La Géométrie (1637)

“The surfaces of three-dimensional space are distinguished from each other not only by their curvature but also by certain more general properties. A spherical surface, for instance, differs from a plane not only by its roundness but also by its finiteness. Finiteness is a holistic property.”

Hans Reichenbach (1891–1953) American philosopher

The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes. ...but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.
The Philosophy of Space and Time (1928, tr. 1957)

“There is no more open door by which you can enter into the study of natural philosophy than by considering the physical phenomena of a candle.”

Michael Faraday book The Chemical History of a Candle

The Chemical History of a Candle (1860)

“The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical mechanics all the manual arts belong, from which mechanics took its name.”

Isaac Newton book Philosophiae Naturalis Principia Mathematica

Preface (8 May 1686)
Philosophiae Naturalis Principia Mathematica (1687)
Context: The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical; what is less so is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic: and if any could work with perfect accuracy, he would be the most perfect mechanic of all; for the description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved.

“Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Source: On the Study and Difficulties of Mathematics (1831), Ch. I.

“The natural way of doing this [seeking scientific knowledge or explanation of fact] is to start from the things which are more knowable and obvious to us and proceed towards those which are clearer and more knowable by nature; for the same things are not 'knowable relatively to us' and 'knowable' without qualification. So in the present inquiry we must follow this method and advance from what is more obscure by nature, but clearer to us, towards what is more clear and more knowable by nature. Now what is to us plain and obvious at first is rather confused masses, the elements and principles of which became known to us by later analysis…”

Aristotle book Physics

A.1, 184a.16 sqq, source: http://classics.mit.edu//Aristotle/physics.html, Book I, Part 1, Tr. R. P. Hardie, R. K. Gaye.
Physics

“In a logically perfect language, there will be one word and no more for every simple object, and everything that is not simple will be expressed by a combination of words, by a combination derived, of course, from the words for the simple things that enter in, one word for each simple component.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

1910s, The Philosophy of Logical Atomism (1918)

“The errors which arise from the absence of facts are far more numerous and more durable than those which result from unsound reasoning respecting true data.”

Charles Babbage On the Economy of Machinery and Manufactures

Source: On the Economy of Machinery and Manufactures, 1832/1841, p. 156. Ch. 17 "Of Price as Measured by Money"

“Suppose we consider, not the rationality of holism, but its spirituality. Holism traditionally says that a collection of beings may have a collective property that cannot be inferred from the properties of its members.”

C. West Churchman (1913–2004) American philosopher and systems scientist

Source: 1960s - 1970s, The Systems Approach and Its Enemies (1979), p. 212; cited in Janet Judy McIntyre-Mills (2003) Critical Systemic Praxis for Social and Environmental Justice. p. 65

“There is nothing more universal than the individual, for what is the property of each is the property of all.”

Miguel de Unamuno (1864–1936) 19th-20th century Spanish writer and philosopher

The Tragic Sense of Life (1913), III : The Hunger of Immortality
Context: Egoism you say? There is nothing more universal than the individual, for what is the property of each is the property of all. Each man is worth more than the whole of humanity, nor will it do to sacrifice each to all save in so far as all sacrifice themselves to each. That which we call egoism is the principle of psychic gravity, the necessary postulate. "Love thy neighbor as thyself," we are told, the presupposition being that each man loves himself; and it is not said "Love thyself." And nevertheless, we do not know how to love ourselves.

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Joseph Fourier 4

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