- differentiable manifolds being replaced by Hilbert space - exterior derivatives being replaced by inner products - affine connexions being replaced by rotating differentials - tensor analysis being replaced by Pseudo-Differential Operator theory - affine (only) infinitesimal small geometry (affine connexions) being replaced by truly infinitesimal small (Hilbert space) geometry - Lie's contact transform becoming obsolete
We emphasis that the (Poisson formula) series representations of the cot(x) and the Dirac functions are convergent in H(a) with a<.-1/2.
The singular (Calderon-Zygmund Pseudo Differential) operator with domain of (Cartan's) differential forms is proposed to be the non-standard alternative to the "standard", non-bounded (momentum) differential operator. It is basically an isomorphism from H(a+1) --> H(a) with a real. The requirements from physics determines the setting of the scale factor a: it is proposed to put a:=-1 in order to ensure that the range of S is isomorph to L(2)=H(0).
In today's world this number line is perceived as a simple term. But this is not the case. A". "point" on the number line is a whole universe, if one realizes that such a "point" is a whole universe, which is about a Dedekind cut of the infinite number of rational numbers(WeH), p. 1, "Preface", 1917: " At the center of my reflections stands the conceptual problem posed by the continuum - a problem which ought to bear the name of Pythagoras and which we currently attempt to solve by means of the arithmetical theory of irrational numbers".(WeH1), p. 86: " While topology has succeeded fairly well in mastering continuity, we do not yet understand the ...inner meaning of the restriction to differential manifolds. Perhaps one day physics will be able to discard it. At present it seems indispensable since the laws of transformation of most physical quantities are intimately connected with that of the differentials dx(i)."... " As the true lawfulness of nature, according to Leibniz's continuity principle, finds ist expression in laws of nearby action, connecting only the values of physical quantities at space-time points in the immediate vicinity of one another, so the basic relations of geometry should concern only infinitely closely adjacent points ('near-geometry' as opposed to far-geometry'). Only in the infinitely small may we expect to encounter the elementary and uniform laws, hence the world must be comprehended through its behavior in the infinitely small".(WiL), Preface: "....The book deals with the problems of philosophy and shows, as I believe, that the method of formulating these problems rests on the miss understanding of the logic of our language. Its whole meaning could be summed up somewhat as follows: What can be said at all can be said clearly; and whereof one cannot speak thereof one must be silent. .... The book will, therefore draw a limit to thinking, or rather - not to thinking, but to the expression of thoughts; for, in order to draw a limit to thinking we should have to be able to think both sides of this limit (we should therefore have to be able to think what cannot be thought). ... The limit can, therefore, only be drawn in language and what lies on the other side of the limit will be simply nonsense."(ScE), p. 90".... beide Paradoxa lösen wird (...), indem man dem Bau unsrer westlichen Naturwissenschaft, die östliche Identitätslehre einverleibt. ... Ich wage, den Geist unzerstörbar zu nennen, denn er hat sein eigenes und besonderes Zeitmaß; nämlich er ist jederzeit j e t z t. Für ihn gibt es in Wahrheit weder früher noch später, sondern nur Jetzt, in das die Erinnerungen und die Erwartungen einbeschlossen sind."(ToA), chapter I: " Time and numbers are different terms, which seem to be independent. This is valid as long as one do not wants to measure. The question, if "time" is measurable or not can be answered positively or negatively. ... Bergson defined the duration as the "real" time."(ScE): “Bohr's standpoint, that a space-time description is impossible, I reject a limine. Physics does not consist only of atomic research, science does not consist only of physics, and life does not consist only of science. The aim of atomic research is to fit our empirical knowledge concerning it into our other thinking. All of this other thinking, so far as it concerns the outer world, is active in space and time. If it cannot be fitted into space and time, then it fails in its whole aim and one does not know what purpose it really serves.”The
Some early thought about the topics above:
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