On Topology (1900-1949)

"That branch of mathematics which deals with the continuity properties of two- (and more) dimensional manifolds is called analysis situs or topology. […] Two manifolds must be regarded as equivalent in the topological sense if they can be mapped point for point in a reversibly neighborhood-true (topological) fashion on each other." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"The disadvantage of differential geometry, as compared with Euclidean or projective geometry and also topology, is that we are not in a position to found it on invariant basic concepts (fundamental relations) and axioms therefor. The situation is no different for conformal geometry on a Riemann surface." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"Two Riemann surfaces which can be mapped conformally onto each other are (conformally) equivalent and are to be regarded as different representations of one and the same ideal Riemann surface. The intrinsic properties of a Riemann surface will include only those properties which are invariant under conformal maps; that is, those properties which, if possessed by one Riemann surface are possessed by every equivalent surface. Obviously all topological properties are intrinsic properties of a Riemann surface; similarly with those properties belonging to the surface by virtue of its smoothness." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"Imagine any sort of model and a copy of it done by an awkward artist: the proportions are altered, lines drawn by a trembling hand are subject to excessive deviation and go off in unexpected directions. From the point of view of metric or even projective geometry these figures are not equivalent, but they appear as such from the point of view of geometry of position [that is, topology]." (Henri Poincaré, "Dernières pensées", 1920)

"The young mathematical disciple 'topology' might be of some help in making psychology a real science." (Kurt Lewin, Principles of topological psychology, 1936)

"In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain." (HermannWeyl. "Invariants", Duke Math. J. 5, 1939)

"It is possible that analysis in the large may eventually reduce to topology, but not until topology has been greatly broadened. It is equally conceivable that the apparently less general situations which arise with such frequency in problems in analysis in the large may form the canonical cases about which the topology of the future can be built." (Marston Morse, "What is Analysis in the Large?", The American Mathematical Monthly Vol. 49 (6), 1942) 

"The first attempts to consider the behavior of so-called ‘random neural nets’ in a systematic way have led to a series of problems concerned with relations between the 'structure' and the ‘function’ of such nets. The ‘structure’ of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of "this" neuron synapsing on ‘that’ one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport. "Cycle distributions in random nets." The Bulletin of Mathematical Biophysics 10 (3), 1948)

"A definition is topological if it makes no use of mathematical elements other than those defined in terms of continuous deformations or transformations. Such deformations or transformations take the straightness out of planes and alter lengths and areas." (Marston Morse, "Equilibria in Nature: Stable and Unstable", Proceedings of the American Philosophical Society Vol. 93 (3), 1949)