Charles Sanders Peirce quote: “The one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habit…”

“Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy.
Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics.”

Joseph Fourier book The Analytical Theory of Heat

Source: The Analytical Theory of Heat (1878), Ch. 1, p. 1

“The object of pure Physic[s] is the unfolding of the laws of the intelligible world; the object of pure Mathematic[s] that of unfolding the laws of human intelligence.”

James Joseph Sylvester (1814–1897) English mathematician

Reported in: Memorabilia Mathematica by Robert Edouard Moritz, quote #129.

“A scientific theory is a concise and coherent set of concepts, claims, and laws (frequently expressed mathematically) that can be used to precisely and accurately explain and predict natural phenomena.
A theory should include a mechanism that explains how its concepts, claims, and laws arise from lower-level theories.”

Mordechai Ben-Ari (1948) Israeli computer scientist

Source: Just a Theory: Exploring the Nature of Science (2005), Chapter 2, “Just a Theory: What Scientists Do” (p. 24)

“Galileo had provided the methodology for the analysis of motions on and near the earth and had applied it successfully. Copernicus and Kepler had previously obtained the laws of motion of the planets and their satellites. …But Galileo had succeeded in deriving numerous laws from a few physical principles and… the axioms and theorems of mathematics. …The Keplerian laws …were not logically related to each other. Each was an independent inference from observations. …They seemed to be suspended in the same vacuum in which the planets moved.
Galileo's laws had the additional advantage of supplying physical insight. The first law of motion and the law that the force of graviation gives… a downward acceleration of 32 ft/sec2… explain the vertrical rise and fall of bodies, motion on slopes, and projectile motion. Kepler's laws… had no physical basis. …Kepler tried to introduce the idea of a magnetic force which the sun exerted… But he failed to related the behavior of the planets to the precise laws of planetary motion. …
The new astronomical theory was completely isolated from the theory of motion on earth. …it bothered mathematicians and scientists who believed that all the phenomena of the universe were governed by one master plan instituted by the master planner—God.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), pp. 224-225

“Harmony is the characteristic of the intellectual system of the universe; and immutable laws of moral existence must pervade all time and all space, all ages and all worlds.”

George Bancroft (1800–1891) American historian and statesman

Literary and Historical Miscellanies (1855), The Necessity, the Reality, and the Promise of the Progress of the Human Race (1854)
Context: The unchanging character of law is the only basis on which continuous action can rest. Without it man would be but as the traveller over endless morasses; the builder on quicksands; the mariner without compass or rudder, driven successively whithersoever changing winds may blow. The universe is the reflex and image of its Creator. "The true work of art," says Michael Angelo, "is but a shadow of the Divine perfections." We may say in a more general manner, that Beauty Itself Is But The Sensible Image Of The Infinite; that all creation is a manifestation of the Almighty; not the result of caprice, but the glorious display of his perfection; and as the universe thus produced, is always in the course of change, so its regulating mind is a living Providence, perpetually exerting itself anew. If his designs could be thwarted, we should lose the great evidence of his unity, as well as the anchor of our own hope.
Harmony is the characteristic of the intellectual system of the universe; and immutable laws of moral existence must pervade all time and all space, all ages and all worlds.

“By his theory of relativity Albert Einstein has provoked a revolution of thought in physical science. ...Physical space and time are found to be closely bound... with... motion of the observer; and only an amorphous combination of the two is left... It is my aim to give an account of this work without introducing anything very technical in... mathematics, physics, or philosophy. ...[T]he task is one of interpreting a clear-cut theory... although perhaps not everyone would accept the author's views of its meaning.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

Space, Time and Gravitation (1920)

“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 (1862–1947) American philosopher, diplomat, and educator

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

“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)

“Money, having freed itself from the physical universe, has become number itself, and finance a strange form of mathematical alchemy.”

David Orrell (1962) Canadian mathematician

Source: The Other Side Of The Coin (2008), Chapter 3, One Versus Plurality, p. 89

“The first question we should face is: What is the aim of a physical theory? To this question diverse answers have been made, but all of them may be reduced to two main principles:
"A physical theory," certain logicians have replied, "has for its object the explanation of a group of laws experimentally established."
"A physical theory," other thinkers have said, "is an abstract system whose aim is to summarize and classify logically a group of experimental laws without claiming to explain these laws…
Now these two questions — Does there exist a material reality distinct from sensible appearances? and What is the nature of reality? — do not have their source in experimental method, which is acquainted only with sensible appearances and can discover nothing beyond them. The resolution of these questions transcends the methods used by physics; it is the object of metaphysics.
Therefore, if the aim of physical theories is to explain experimental laws, theoretical physics is not an autonomous science; it is subordinate to metaphysics…
Now, to make physical theories depend on metaphysics is surely not the way to let them enjoy the privilege of universal consent.”

Pierre Duhem (1861–1916) French physicist, historian of science

Notice sur les Titres et Travaux scientifiques de Pierre Duhem rédigée par lui-même lors de sa candidature à l'Académie des sciences (mai 1913), The Aim and Structure of Physical Theory (1906)

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