"With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it ‘mind-mass’. All thinking is, accordingly, formation of new mind masses." (Bernhard Riemann, "Gesammelte Mathematische Werke", 1876)
"If two well-defined manifolds M and N can be coordinated with each other univocally and completely, element by element (which, if possible in one way, can always happen in many others), we shall employ in the sequel the expression, that those manifolds have the same power or, also, that they are equivalent." (Georg Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre", 1878)
"The analysis of Nature into its individual parts, the grouping of the different natural processes and natural objects in definite classes, the study of the internal anatomy of organic bodies in their manifold forms - these were the fundamental conditions of the gigantic strides in our knowledge of Nature which have been made during the last four hundred years. But this method of investigation has also left us as a legacy the habit of observing natural objects and natural processes in their isolation, detached from the whole vast interconnection of things; and therefore not in their motion, but in their repose; not as essentially changing, but fixed constants; not in their life, but in their death." (Friedrich Engels, "Herr Eugen Dühring's Revolution in Science", 1878)
"I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined, when on the basis of its definition and as a consequence of the logical principle of excluded middle it must be regarded as internally determined, both whether an object pertaining to the same conceptual sphere belongs or not as an element to the manifold, and whether two objects belonging to the set are equal to each other or not, despite formal differences in the ways of determination." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)
"By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic form or idea." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)
"The foregoing account of my researches in the theory of manifolds has reached a point where further progress depends on extending the concept of true integral number beyond the previous boundaries; this extension lies in a direction which, to my knowledge, no one has yet attempted to explore. My dependence on this extension of number concept is so great, that without it I should be unable to take freely the smallest step further in the theory of sets." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)
"Instead of the points of a line, plane, space, or any manifold under investigation, we may use instead any figure contained within the manifold: a group of points, curve, surface, etc. As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these figures should depend, the number of dimensions of the line, plane, space, etc. is likewise arbitrary and depends only on the choice of space element. But so long as we base our geometrical investigation on the same group of transformations, the geometrical content remains unchanged. That is, every theorem resulting from one choice of space element will also be a theorem under any other choice; only the arrangement and correlation of the theorems will be changed. The essential thing is thus the group of transformations; the number of dimensions to be assigned to a manifold is only of secondary importance." (Felix Klein, "A comparative review of recent researches in geometry", Bulletin of the American Mathematoical Society 2(10), 1893)
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