On Functions (1875-1899)

"The concept of power, which includes as a special case the concept of whole number, that foundation of the theory of number, and which ought to be considered as the most general genuine origin of sets [Moment bei Mannigfaltigkeiten], is by no means restricted to linear point sets, but can be regarded as an attribute of any well-defined collection, whatever may be the character of its elements. [...] Set theory in the conception used here, if we only consider mathematics for now and forget other applications, includes the areas of arithmetic, function theory and geometry. It contains them in terms of the concept of power and brings them all together in a higher unity. Discontinuity and continuity are similarly considered from the same point of view and are thus measured with the same measure." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"The probable error, the mean error, the mean square of error, are forms divined to resemble in an essential feature the real object of which they are the imperfect symbols - the quantity of evil, the diminution of pleasure, incurred by error. The proper symbol, it is submitted, for the quantity of evil incurred by a simple error is not any power of the error, nor any definite function at all, but an almost arbitrary function, restricted only by the conditions that it should vanish when the independent variable, the error, vanishes, and continually increase with the increase of the error." (Francis Y Edgeworth, "The method of least squares", 1883)

"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)

"Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skillful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures."(Johann F Herbart, 1890)

"We are led naturally to extend the language of geometry to the case of any number of variables, still using the word point to designate any system of values of n variables (the coördinates of the point), the word space (of n dimensions) to designate the totality of all these points or systems of values, curves or surface to designate the spread composed of points whose coördinates are given functions (with the proper restrictions) of one or two parameters (the straight line or plane, when they are linear fractional functions with the same denominator), etc. Such an extension has come to be a necessity in a large number of investigations, in order as well to give them the greatest generality as to preserve in them the intuitive character of geometry. But it has been noted that in such use of geometric language we are no longer constructing truly a geometry, for the forms that we have been considering are essentially analytic, and that, for example, the general projective geometry constructed in this way is in substance nothing more than the algebra of linear transformations." (Corradi Segre, "Rivista di Matematica" Vol. I, 1891)

"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)

"The atomic theory plays a part in physics similar to that of certain auxiliary concepts in mathematics: it is a mathematical model for facilitating the mental reproduction of facts. Although we represent vibrations by the harmonic formula, the phenomena of cooling by exponentials, falls by squares of time, etc, no one would fancy that vibrations in themselves have anything to do with circular functions, or the motion of falling bodies with squares." (Ernst Mach, "The Science of Mechanic", 1893)

"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,"Mathematical Theory of the Principles of Wealth", 1897)

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)