On Invariance (-1899)

"The calculation of probabilities is of the utmost value, […] but in statistical inquiries there is need not so much of mathematical subtlety as of a precise statement of all the circumstances. The possible contingencies are too numerous to be covered by a finite number of experiments, and exact calculation is, therefore, out of the question. Although nature has her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical calculation, although it is inexact, may be adequate in affairs of practice." (Gottfried W Leibniz [letter to Bernoulli], 1703)

"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." (Joseph Fourier, "The Analytical Theory of Heat", 1822)

"There cannot be a language more universal and more simple, more free from errors and obscurities [...] more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes." (Joseph Fourier, "The Analytical Theory of Heat", 1822)

"Every scientific task aims at the determination of equivalences and at the discovery of their invariants and for it the following line of poetry holds: 'the wise one seeks a stable pole amid the flight of phenomena'." (Leopold Kronecker,"Zur Theorie der elliptischen Functionen", 1889-1890)

"With the happy expression 'Invariants' chosen by Mr. Sylvester, and quite appropriate to the meaning of the matter, one originally denotes only rational functions of the coefficients of forms that remain unchanged under certain linear transformations of the variables of the forms. But the same expression has since then also been extended to some other entities [Bildungen] that remain unchanged under transformation. This multiple applicability of the concept of invariants rests upon the fact that it belongs to a much more general and abstract realm of ideas. In fact, when the concept of invariants is separated from the direct formal relation to a process of transformation and it is tied rather to the general concept of equivalence, then the concept of invariants reaches the most general realm of thought. For, every abstraction, - say an abstraction from certain differences that are presented by a number of objects, - states an equivalence and the concept originating from the abstraction, for instance the concept of a species, represents the 'invariant of the equivalence'." (Leopold Kronecker,"Zur Theorie der elliptischen Functionen", 1889-1890)

"In our century the conceptions substitution and substitution group, transformation and transformation group, operation and operation group, invariant, differential invariant and differential parameter, appear more and more clearly as the most important conceptions of mathematics." (Sophus Lie, Leipziger Berichte No. 47, 1896)

"In mathematics there is no understanding. In mathematics there are only necessities, laws of existence, invariant relationships. Thus any mathematico-mechanistic outlook must, in the last analysis, waive all understanding. For, we only understand when we know the motives; where there are no motives, all understanding ceases." (Friedrich W Nietzsche) [attributed]