[2034]. According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name Bœotians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically.—Sartorius, W. v. Waltershausen.
Gauss zum Gedächtniss (Leipzig, 1856), p. 81.
[2035]. There is many a rational logos, and the mathematician has high delight in the contemplation of inconsistent systems of consistent relationships. There are, for example, a Euclidean geometry and more than one species of non-Euclidean. As theories of a given space, these are not compatible. If our universe be, as Plato thought, and nature-science takes for granted, a space-conditioned, geometrised affair, one of these geometries may be, none of them may be, not all of them can be, valid in it. But in the vaster world of thought, all of them are valid, there they co-exist, and interlace among themselves and others, as differing component strains of a higher, strictly supernatural, hypercosmic, harmony.—Keyser, C. J.
The Universe and Beyond; Hibbert Journal, Vol. 3 (1904-1905), p. 313.
[2036]. The introduction into geometrical work of conceptions such as the infinite, the imaginary, and the relations of hyperspace, none of which can be directly imagined, has a psychological significance well worthy of examination. It gives a deep insight into the resources and working of the human mind. We arrive at the borderland of mathematics and psychology.—Merz, J. T.
History of European Thought in the Nineteenth Century (Edinburgh and London, 1903), p. 716.
[2037]. Among the splendid generalizations effected by modern mathematics, there is none more brilliant or more inspiring or more fruitful, and none more commensurate with the limitless immensity of being itself, than that which produced the great concept designated ... hyperspace or multidimensional space.—Keyser, C. J.
Mathematical Emancipations; Monist, Vol. 16 (1906), p. 65.
[2038]. The great generalization [of hyperspace] has made it possible to enrich, quicken and beautify analysis with the terse, sensuous, artistic, stimulating language of geometry. On the other hand, the hyperspaces are in themselves immeasurably interesting and inexhaustibly rich fields of research. Not only does the geometrician find light in them for the illumination of otherwise dark and undiscovered properties of ordinary spaces of intuition, but he also discovers there wondrous structures quite unknown to ordinary space.... It is by creation of hyperspaces that the rational spirit secures release from limitation. In them it lives ever joyously, sustained by an unfailing sense of infinite freedom.—Keyser, C. J.
Mathematical Emancipations; Monist, Vol. 16 (1906), p. 83.