This preface has been edited at the end of a long journey, which started in 2010 until today. During this period of time this homepage has been developed. The journey is still going on, but the main pillars and their underlying mathematical challenges and delivered solutions are established yet. there are the three views on the considered problems, which are the physical, the mathematical and the philosophical views. Kant's critique of pure reason gives the rational for the interface and boundaries of those three areas, which is governed by the term "transcendence". We emphasis that the term "transcendental" in mathematics (number theory) is even beyond Kant's definition of the term "transcendental": the transcendental numbers are a subset of the set of the irrational numbers (from a mathematical (definition) point of view), but already the irrational numbers are transcendental in the sense of Kant. The mathematical terms "continuity" and "Riemann integral" are building on the concept of irrational numbers, i.e. they are also transcendental terms. The Lebesgue integral is defined as a generalization of the Riemann integral. In the framework of the Lebesgue integral concept the set of rational numbers is a so-called zero-set, only (!), i.e. the probability to pick a rational number out of the set of the real numbers is zero. Our proposed mathematical model is building on the (Leibniz) mathematical transcendental term " we "just" propose and show evidence of a consistent mathematical language (definitions, axioms) in an unusual distributional Hilbert space framework, which is less regular than the L(2)-test space, but still more regular than the domain of the Dirac function, while still applying standard functional analysis/spectral analysis/variational theory. There are multiple handicaps regarding the usage of the Dirac "function" as a central concept in the quantum theory: let e denote an arbitrarily small positive real number and n denote the space dimension. The Dirac "function" is a distribution which is not an element of the quantum state Hilbert space L(2)=H(0). Its regularity depends from the space dimension n, i.e. the Dirac "function" is an element of the Hilbert space H(-n/2-e). Our approach builds on an alternative quantum state Hilbert space H(-1/2). Its definition is enabled by the Riesz and Calderon-Zygmund integrodifferential operators. We note that in case of space dimension n=1 the Riesz operators are identical to the Hilbert transform operator. The considered (distributional) Hilbert space framework enables a truly infinitesimal geometry (WeH); as one first consequence the manifold concept of Einstein's field equations with its handicap of A common Hilbert space framework for PDE field equations and quantum dynamics enables an integrated mathematical quantum and gravity field theory model, including a gravitational collapse and space-time singularity theory (R. Penrose). The Berry-Keating (Hilbert-Polya refinement) conjecture is verified by a convolution representation of the Zeta function, enabled by the distributional Fourier series representation of the cot(x)-function (S. Ramanujan). This provides an answer to Derbyshine's question (in "Prime Obsession"): The common distributional Hilbert space framework of classical field (PD) equations and quantum field equations and its corresponding classical and variational (weak) mathematical models require a change of a current paradigm: now the classical models become the mathematical approximations to the weak (Pseudo-) Differential Equations models and not the other way around. Another consequence is that the term "force" is only valid for classical PDE, when the Lagrange formalism is equivalent to the Hamiltonian formalism due to a defined Legendre transform. Another consequence is the fact that the energy inequality (with respect to the newly proposed H(1/2) energy space) of the non-linear, non-stationary NSE now also anticipates a contribution of the non-linear term, while, at the same time, enabling a global bounded energy inequality for the non-linear, non-stationary NSE in case of space dimension n=3. Leibniz's monad concept is an extension of the real numbers to ideal/hyper-real numbers. Those are nothing more than another set of "transcendental numbers" in the sense of Kant (whereby the term "real" for the real numbers is already miss-leading); the properties of the set of the ideal numbers are identical to those of the real numbers (which are (in a physical sense) not "real" at all with 100% probability), except only one missing valid axiom, the The fascination, motivation and energy to walk through this journey was and is primarily to contribute as much as possible to all those subject areas at that moment in time, when the one or the other idea popped up. The main drivers are “amazement” and “pursuit of new”, and not to follow academicals career paths. In this sense "prosit" (lat. "may it be useful") : From a philosophical perspective the relationship of current and newly proposed “ideal” (transcendental) mathematical objects to describe very large and very small physical phenomena (R. Penrose) is still affecting open, valid philosophical questions, as e.g. addressed in (RuB). I would summarize my general hypothesis thus: consciousness is associated with the learning of living substance; its knowing how (Können) is unconscious.Mind has erected the objective outside world of the natural philosopher out of its own stuff. Mind could note cope with this gigantic task otherwise than by the simplifying device of excluding itself - withdrawing from its conceptual creation. Hence the latter does not contain its creator.Physical science ... faces us with the impasse that mind per se cannot play piano - mind per se cannot move a finger of a hand. Then the impasse meets us, the blank of the "how" of mind's leverage on matter. The inconsequence staggers us. It is a misunderstanding?Neither can the body determine the mind to think, nor the mind determine the body to motion or rest or anything else (if such there be)."(ScE1) 'THE VEDANTIC VISION': "F or philosophy, then, the real difficulty lies in the spatial and temporal multiplicity of observing and thinking individuals. If all events took place in one consciousness, the whole situation would be extremely simple. There would then be something given, a simple datum, and this, however otherwise constituted, could scarily present us with a difficulty of such magnitude as the one we do in fact have on our hands."I do not think that this difficulty can be logically resolved, by consistent thought, within intellects. But it is quite easy to express the solution in words, thus: the plurality that we perceive is only appearance; it is not real. Vedantic philosophy, ..... (ScE2) 'FORM, NOT SUBSTANCE, THE FUNDAMENTAL CONCEPT': " It is clearly the peculiar form or shape (German: Gestalt) that raises the identity beyond doubt, not the material content. Had the material been melted and cast into the shape of a man, the identity would be much more difficult to establish. And what is more: even if the material identity were established beyond doubt, it would be of very restricted interest. I should probably not care very much about identity or not of that mass of iron, and should declare that my souvenir had been destroyed."(ScE2) 'THE NATURE OF OUR MODELS': "In this we must, of course, take shape (or Gestalt) in a much wider sense than as geometrical shape. Indeed there is no observation concerned with the geometrical shape of a particle or even of an atom. It is true that in thinking about the atom, in drafting theories to meet the observed facts, we do very often draw geometrical pictures on the black-board, or on a piece of paper, or more often just only in our mind, the details of the picture being given by a mathematical formula with much greater precision and in a much handier fashion then pencil or pen could ever give. That is true. ...."(ScE2) 'THE ALLEGED BREAK-DOWN OF THE BARRIER BETWEEN SUBJECT AND OBJECT': "For physical action always is inter-action, it always is mutual. What remains doubtful to me is only just this: whether it is adequate to term one of the two physically interacting systems the 'subject'. For the observing mind is not a physical system, it cannot interact with any physical system. And it might be better to reserve the term 'subject' for the observing mind."The OVERVIEW page of this homepage is frequently updated. A pdf version with status Jan 2018 is provided in
(HoJ) Hold J., Why does the World Exists? An Existential Detective Story, Liveright Publishing Corporation, New York, 2012 (RuB) Russel B., The Problems of Philosophy, Oxford university Press, Oxford, 1912 (ScE) Schrödinger E., Mind and Matter, Cambridge University Press, 1958 (ScE1) Schrödinger E., My View of the world, Cambridge University Press, 1964 (WeH) Weyl H. The Continuum, A Critical Examination of the Foundation of Analysis, Dover Publications, Inc. New York, 1994 (WeH1) Weyl H. Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, 1949, 2009 | |||||||||||||||||