On Numbers (1870-1879)

"The Mathematician deals with two properties of objects only, number and extension, and all the inductions he wants have been formed and finished ages ago. He is now occupied with nothing but deductions and verification." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870)

"Mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number." (George Holmes Howison, "The Departments of Mathematics, and their Mutual Relations", Journal of Speculative Philosophy Vol. 5, No. 2, 1871)

"Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this." (Richard Dedekind, "On Continuity and Irrational Numbers", 1872)

"Music is like geometric figures and numbers, which are the universal forms of all possible objects of experience." (Friedrich Nietzsche, "Birth of Tragedy", 1872)

"That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers." (Richard Dedekind, "Stetigkeit und irrationale Zahlen", 1872)

"The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains." (Richard Dedekind,"Stetigkeit und irrationale Zahle", 1872)

"The Mathematician deals with two properties of objects only, number and extension, and all the inductions he wants have been formed and finished ages ago. He is now occupied with nothing but deductions and verification." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1872)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision. " (George H Lewes "Problems of Life and Mind", 1873)

"We produce these representations in and from ourselves with the same necessity with which the spider spins. If we are forced to comprehend all things only under these forms, then it ceases to be amazing that in all things we actually comprehend nothing but these forms. For they must all bear within themselves the laws of number, and it is precisely number which is most astonishing in things. All that conformity to law, which impresses us so much in the movement of the stars and in chemical processes, coincides at bottom with those properties which we bring to things. Thus it is we who impress ourselves in this way." (Friedrich Nietzsche, "On Truth and Lie in an Extra-Moral Sense", 1873)

"[…] with few exceptions all the operations and concepts that occur in the case of real numbers can indeed be carried over unchanged to complex ones. However, the concept of being greater cannot very well be applied to complex numbers. In the case of integration, too, there appear differences which rest on the multplicity of possible paths of integration when we are dealing with complex variables. Nevertheless, the large extent to which imaginary forms conform to the same laws as real ones justifies the introduction of imaginary forms into geometry." (Gottlob Frege,"On a Geometrical Representation of Imaginary forms in the Plane", 1873)

"Nature eludes calculation. Number is a grim pullulation. Nature is the thing that cannot be numbered." (Victor Hugo, "The Toilers of the Sea", 1874)

"One microscopic glittering point; then another; and another, and still another; they are scarcely perceptible, yet they are enormous. This light is a focus; this focus, a star; this star, a sun; this sun, a universe; this universe, nothing. Every number is zero in the presence of the infinite." (Victor Hugo, "The Toilers of the Sea", 1874)

"When we consider complex numbers and their geometrical representation, we leave the field of the original concept of quantity, as contained especially in the quantities of Euclidean geometry: its lines, surfaces and volumes. According to the old conception, length appears as something material which fills the straight line between its end points and at the same time prevents another thing from penetrating into its space by its rigidity. In adding quantities, we are therefore forced to place one quantity against another. Something similar holds for surfaces and solid contents. The introduction of negative quantities made a dent in this conception, and imaginary quantities made it completely impossible. Now all that matters is the point of origin and the end point; whether there is a continuous line between them, and if so which, appears to make no difference whatsoever; the idea of filling space has been completely lost. All that has remained is certain general properties of addition, which now emerge as the essential characteristic marks of quantity. The concept has thus gradually freed itself from intuition and made itself independent. This is quite unobjectionable, especially since its earlier intuitive character was at bottom mere appearance. Bounded straight lines and planes enclosed by curves can certainly be intuited, but what is quantitative about them, what is common to lengths and surfaces, escapes our intuition." (Gottlob Frege, "Methods of Calculation based on an Extension of the Concept of Quantity", 1874)

"In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction." (Carl Friedrich Gauss, Werke, 1876

"If statistical graphics, although born just yesterday, extends its reach every day, it is because it replaces long tables of numbers and it allows one not only to embrace at glance the series of phenomena, but also to signal the correspondences or anomalies, to find the causes, to identify the laws." (Émile Cheysson, cca. 1877)

"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)