"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." (Paul Painlevé, "Analyse des travaux scientifiques", 1900)
"The constructive process inheres in all forms of synergy, and the cooperation of antithetical forces in nature always results in making, that is, in creating something that did not exist before. But in the organic world this character of structure becomes the leading feature, and we have synthetic products consisting of tissues and organs serving definite purposes, which we call functions." (Lester F Ward, "Pure Sociology", 1903)
"Allow us now a hypothesis that is arbitrary but not self-contradictory. One might encounter instances where using a function without a derivative would be simpler than using one that can be differentiated. When this happens, the mathematical study of irregular continua will prove its practical value." (Jean-Baptiste Perrin, 1906)"
"Consider, for instance, one of the white flakes that are obtained by salting a solution of soap. At a distance its contour may appear sharply defined, but as we draw nearer its sharpness disappears. The eye can no longer draw a tangent at any point. A line that at first sight would seem to be satisfactory appears on close scrutiny to be perpendicular or oblique. The use of a magnifying glass or microscope leaves us just as uncertain, for fresh irregularities appear every time we increase the magnification, and we never succeed in getting a sharp, smooth impression, as given, for example, by a steel ball. So, if we accept the latter as illustrating the classical form of continuity, our flake could just as logically suggest the more general notion of a continuous function without a derivative." (Jean-Baptiste Perrin, 1906)
"It must be borne in mind that, although closer observation of any object generally leads to the discovery of a highly irregular structure, we often can with advantage approximate its properties by continuous functions. Although wood may be indefinitely porous, it is useful to speak of a beam that has been sawed and planed as having a finite area. In other words, at certain scales and for certain methods of investigation, many phenomena may be represented by regular continuous functions, somewhat in the same way that a sheet of tinfoil may be wrapped round a sponge without following accurately the latter's complicated contour." (Jean-Baptiste Perrin, 1906)
"Mathematicians, however, are well aware that it is childish to try to show by drawing curves that every continuous function has a derivative. Though differentiable functions are the simplest and the easiest to deal with, they are exceptional. Using geometrical language, curves that have no tangents are the rule, and regular curves, such as the circle, are interesting but quite special." (Jean-Baptiste Perrin, 1906)
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions. […] A curve is the totality of points, whose coordinates are functions of a parameter which may be differentiated as often as may be required." (Felix Klein, "Elementar Mathematik vom hoheren Standpunkte aus" Vol. 2, 1909)
"I do not say that the notion of infinity should be banished; I only call attention to its exceptional nature, and this so far as I can see, is due to the part which zero plays in it, and we must never forget that like the irrational it represents a function which possesses a definite character but can never be executed to the finish If we bear in mind the imaginary nature of these functions, their oddities will not disturb us, but if we misunderstand their origin and significance we are confronted by impossibilities." (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)
"It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations." (Felix Klein, "Lectures on Mathematics", 1911)
"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)
"Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions." (George H Howison, Journal of Speculative Philosophy Vol. 5, 1914)
"The combinatory analysis as considered in this work occupies the ground between algebra, properly so called, and the higher arithmetic. The methods employed are distinctly algebraical and not arithmetical. The essential connecting link between algebra and arithmetic is found in the circumstance that a particular case of algebraic multiplication involves arithmetical addition. [...] This link was forged by Euler for use in the theory of partitions of numbers. It is used here for the most general theory of combinations of which the partition of numbers is a particular case. [...] The theory of the partition of numbers belongs partly to algebra and partly to the higher arithmetic. The former aspect is treated here. It is remarkable that in the international organization of the subject-matter of mathematics ‘Partitions’ is considered to be a part of the Theory of Numbers which is an alternative name for the Higher Arithmetic, whereas it is essentially a subdivision of Combinatory Analysis which is not considered to be within the purview of the Theory of Numbers. The fact is that up to the point of determining the real and enumerating Generating Functions the theory is essentially algebraical [..]" (Percy A MacMahon, "Combinatory Analysis", Vol. 1, 1915)
"In the realm of physics it is perhaps only the theory of relativity which has made it quite clear that the two essences, space and time, entering into our intuition, have no place in the world constructed by mathematical physics. Colours are thus 'really' not even æther-vibrations, but merely a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time." (Hermann Weyl, "Space, Time, Matter", 1922)
"One says that y is a function of x if to a value of x corresponds a value of y. One indicates this correspondence by the equation y=f(x)." (Édouard Goursat, 1923)
Resources: [1] MacTutor (2005) The function concept [link]
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