Budget of Paradoxes (London, 1872), p. 377.
[1950]. We have adroitly defined the infinite in arithmetic by a loveknot, in this manner ∞; but we possess not therefore the clearer notion of it.—Voltaire.
A Philosophical Dictionary; Article “Infinity.” (Boston, 1881).
[1951]. I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.—Gauss.
Brief an Schumacher (1831); Werke, Bd. 8 p. 216.
[1952]. In spite of the essential difference between the conceptions of the potential and the actual infinite, the former signifying a variable finite magnitude increasing beyond all finite limits, while the latter is a fixed, constant quantity lying beyond all finite magnitudes, it happens only too often that the one is mistaken for the other.... Owing to a justifiable aversion to such illegitimate actual infinities and the influence of the modern epicuric-materialistic tendency, a certain horror infiniti has grown up in extended scientific circles, which finds its classic expression and support in the letter of Gauss [see 1951], yet it seems to me that the consequent uncritical rejection of the legitimate actual infinite is no lesser violation of the nature of things, which must be taken as they are.—Cantor, G.
Zum Problem des actualen Unendlichen; Natur und Offenbarung, Bd. 32 (1886), p. 226.
[1953]. The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits.—Lewes, G. H.
Problems of Life and Mind (Boston, 1875), Vol. 2, p. 384.