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affine connection analytic applied arbitrary Backlund called Cartan Cauchy characteristic classical coefficients Collected Papers vol complete system condition consider constant contains coordinate cross-section curvature curves defined Definition denote determined developed differential forms differential geometry differential operators dimensional elementary particle elements Estabrook example exterior differential system finite transformations foliation formula Gaussian geometric given Hamiltonian Heisenberg Hence Hermann Hilbert space holomorphic ideas independent infinite continuous group infinite group infinitely small transformation infinitesimal transformations inner product Interdisciplinary Mathematics internally similar intuition inverses isomorphic Lie algebra Lie group linear map linear subspace linear system manifold Math Sci Press mathematicians one-forms one-term group orbits parameters partial differential equations physicists physics Poisson bracket quantum mechanics real-valued functions relations Riemannian metric satisfied Sine-Gordon solitons solutions structure submanifold Suppose symplectic system theory tangent tensor Theorem tion transformation group variables vector bundle vector field vector space Wahlquist
Page 17 - We seek reality, but what is reality ? The physiologists tell us that organisms are formed of cells; the chemists add that cells themselves are formed of atoms. Does this mean that these atoms or these cells constitute reality, or rather the sole reality? The way in which these cells are arranged and from which results the unity of the individual, is not it also a reality much more interesting than that of the isolated elements, and should a naturalist who had never studied the elephant except by...
Page 16 - ... finally attained absolute rigor? At each stage of the evolution our fathers also thought they had reached it. If they deceived themselves, do we not likewise cheat ourselves? We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new ; not from it alone can any science issue.
Page 18 - Reflections number of elementary operations; when we have examined these operations one after the other and ascertained that each is correct, are we to think we have grasped the real meaning of the demonstration? Shall we have understood it even when, by an effort of memory, we have become able to repeat this proof by reproducing all these elementary operations in just the order in which the inventor had arranged them? Evidently not; we shall not yet possess the entire reality; that I know not what...
Page 18 - Thus logic and intuition have each their necessary role. Each is indispensable. Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention.
Page 15 - ... straight line, we see it under the form of a rectilinear band of a certain breadth. We well know these lines have no width; we try to imagine them narrower and narrower and thus to approach the limit; so we do in a certain measure, but we shall never attain this limit. And then it is clear we can always picture these two narrow bands, one straight, one curved, in a position such that they encroach slightly one upon the other without crossing. We shall thus be led, unless warned by a rigorous...
Page 18 - Yet this is what the reader of a book on mathematics would do if he were a logician only. To understand the game is wholly another matter; it is to know why the player moves this piece rather than that other which he could have moved without breaking the rules of the game. It is to perceive the inward reason which makes of this series of successive moves a sort of organized whole. This faculty is still more necessary for the player himself, that is, for the inventor. Let us drop this comparison and...
Page 17 - ... physiologists tell us that organisms are formed of cells; the chemists add that cells themselves are formed of atoms. Does this mean that these atoms or these cells constitute reality, or rather the sole reality? The way in which these cells are arranged and from which results the unity of the individual, is not it also a reality much more interesting than that of the isolated elements, and should a naturalist who had never studied the elephant except by means of the microscope think himself...
Page 14 - Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.
Page 19 - VI. But at the moment of formulating this conclusion I am seized with scruples. At the outset I distinguished two kinds of mathematical minds, the one sort logicians and analysts, the others intuitionalists and geometers. Well, the analysts also have been inventors. The names I have just cited make my insistence on this unnecessary. Here is a contradiction, at least apparently, which needs explanation. And first, do you think these logicians have always proceeded from the general to the particular,...
Page 15 - ... perplexed. The two sorts of minds are equally necessary for the progress of science; both the logicians and the intuitionalists have achieved great things that others could not have done. Who would venture to say whether he preferred that Weierstrass had never written or that there had never been a Riemann? Analysis and synthesis have then both their legitimate roles. But it is interesting to study more closely in the history of science the part which belongs to each.