This preface has been edited at the end of a long journey, which started in 2010 until today (2017). During this period of time this homepage has been developped. 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 beyong 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 trancendental 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.
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 (wheryby 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 not at all "real" with 100% probability), except only one missing valid axiom, the Archimedean axiom, which is related to physical measurement capabilites of a lenght by a given standard measurement length (!). The set of real numbers provides the baseline for standard analysis with the concepts of the Riemann and the Lebesgue integrals. The latter one is the fundamental concept to define the inner product of the test (Hilbert) space L(2) resp. the Dirichlet (energy) integral with its underlying domain, the (Sobolev) Hilbert space H(1), which is a subset of the test space L(2). The set of Leibniz's ideal numbers provides the baseline of non-standard analysis. In this framework the Stieltjes integral can be interpreted as the counterpat of the Lebesgue integral.
Our proposed mathematical model integrating all considered problem areas is building the the latter (Leibniz) mathematical transcendental terms (differentials, 1-forms). This means that there is no additional transcendence level added (which would be anyway a contradiction by itself), but the mathematical model is applicable to all considered problem Areas. The physical-mathematical modelling requirements (measurement/ observation/ test results validation) is still building on the test space L(2).
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 academical career paths. In this sense
"prosit" (lat. "may it be useful") :
We "just" propose and show evidence of a consistent mathematical language (definitions, axioms) in a unusual distributional Hilbert space framework less regular than the L(2)-test space, but still more regular than the domain of the Dirac function), while applying standard functional analysis/spectral analysis/variational theory.
The considered (distributional) Hilbert space framework enables a truly infinitesimal geometry (WeH), which enables by the same time an integrated mathematical quantum and gravity field theory model. The key message related to the physical problem areas (Navier-Stokes, Maxwell, Einstein field equations) and its related physical classical and variational (weak) mathematical models is, that the classical models are the approximations to the weak (Pseudo-) Differential Equations models and not the other way around. As a consequence, the term "force" is only valid for classical PDE, when the Lagrange formalism is equivalent to the Hamiltonean formalism due to a defined Legendre transform and the energy inequality estimate of the non-linear, non-stationary NSE includes also the contribution of the non-linear term and leading to global bounded energy estimates for space dimension n=3.
The approach omits concepts like exterior tensor & exterior algebras and exterior differential forms, as well as corresponding gauge theories. It provides a mathemtical truly infinitesimal geometry e.g. resulting in a new gravitational collapse and space-time singularity theory (R. Penrose).
However, the relationship of those “ideal” mathematical objects (even if they describe only "weak", distributional conceptual elements/concepts) to observed physical phenomena is still affecting open, valid philosophical questions, as e.g. addressed in (RuB).
(KnA) Kneser A., Das Prinzip der kleinsten Wirkung von Leibniz bis zur Gegenwart, B. G. Teubner Verlag, Leipzig, Berlin, 1928
(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