"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)
"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)
"Theorems valid 'in the small' are those which affirm a statement about a certain neighborhood of a point without making any statement about the size of that neighborhood." (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)
"In the realm of physics it is perhaps only the theory of relativity which has made it quite clear that the two essences, space and time, entering into our intuition, have no place in the world constructed by mathematical physics. Colours are thus 'really' not even æther-vibrations, but merely a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time." (Hermann Weyl, "Space, Time, Matter", 1922)
"Matter [...] could be measured as a quantity and [...] its characteristic expression as a substance was the Law of Conservation of Matter [...] This, which has hitherto represented our knowledge of space and matter, and which was in many quarters claimed by philosophers as a priori knowledge, absolutely general and necessary, stands to-day a tottering structure." (Hermann Weyl, "Space, Time, Matter", 1922)
"Space and time are commonly regarded as the forms of existence of the real world, matter as its substance. A definite portion of matter occupies a definite part of space at a definite moment of time. It is in the composite idea of motion that these three fundamental conceptions enter into intimate relationship." (Hermann Weyl, "Space, Time, Matter", 1922)
"It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought to be taken as a basis for the analysis of a continuum." (Hermann Weyl, "Riemanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie", 1925)
"Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory." (Hermann Weyl, "Levels of Infinity", cca. 1930)
"A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)
"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means." ( Hermann Weyl, "The Open World: Three Lectures In the Metaphysical Implications of Science", 1932)
"We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for […] The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for." (Hermann Weyl, "The Mathematical Way of Thinking", 1940)
"May the God who watches over the right use of mathematical symbols, in manuscript, print, and on the blackboard, forgive me [my sins]." (Hermann Weyl, "The Classical Groups", 1946)
"The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]" (Hermann Weyl, "Mathematics and Logic", 1946)
"The mystery that clings to numbers, the magic of numbers, may spring from this very fact, that the intellect, in the form of the number series, creates an infinite manifold of well-distinguished individuals. Even we enlightened scientists can still feel it, e.g., in the impenetrable law of the distribution of prime numbers." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1949)
"The constructions of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow-mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity." (Hermann Weyl, "A Half-Century of Mathematics", The American Mathematical Monthly, 1951)
"A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before." (Hermann Weyl, "Symmetry", 1952)
"As far as I can see, all a priori statements in physics have their origin in symmetry." (Hermann Weyl, "Symmetry", 1952)
"In the one sense symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole. Beauty is bound up with symmetry." (Hermann Weyl, "Symmetry", 1952)
"Rut seldom is asymmetry merely the absence of symmetry. Even in asymmetric designs one feels symmetry as the norm from which one deviates under the influence of forces of non-formal character." (Hermann Weyl, "Symmetry", 1952)
"Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection." (Hermann Weyl, Symmetry, 1952)
"Numbers have neither substance, nor meaning, nor qualities. They are nothing but marks, and all that is in them we have put into them by the simple rule of straight succession." (Hermann Weyl, "Mathematics and the Laws of Nature", 1959)
"Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory." (Hermann Weyl, "Axiomatic versus constructive procedures in mathematics", The Mathematical Intelligencer, 1985)
"Besides language and music, mathematics is one of the primary manifestations of the free creative power of the human mind." (Hermann Weyl)
"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl)
"Mathematical inquiry lifts the human mind into closer proximity with the divine than is attainable through any other medium." (Hermann Weyl)
"Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect." (Hermann Weyl)
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