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.
The term “journey” already indicates, why the results are "published" in this form and not following standard processes in academic world per each of the identified problem areas, respectively per underlying specific problems within those areas.
The considered problem areas are currently seen as either purely mathematical (e.g. RH) or physical problems (quantum field vs. gravity field theory and its union) or, in case of the RH, there is also a mix between mathematical and physical problems (e.g. Berry conjecture). At the end of this journey we claim that all those problems are purely mathematical modelling problems:
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 "differential". It is interesting to be mentioned that also Schrödinger in (ScE1) uses the term "differential" (in German version) to explain "perception process between "subconscious" and "awareness" of human mind. The Leibniz transcendental concept of "differential = monad" means that there is no additional transcendence level added (which would be anyway a contradiction by itself), but the mathematical model becomes now 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):
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 differentiable manifolds (which is a purely mathematical requirement without any physical meaning/justification) can be omitted. The approach also omits concepts like exterior tensor & exterior algebras and exterior differential forms, as well as corresponding gauge theories. It leads to a modified Einstein-Hilbert action functional newly based on a Stieltjes integral representation replacing the Lebesgue measure dx(4) by the corresponding Stieltjes integral measure dg(x(4)). Complementary variational principles can be derived from this, whereby the corresponding (classical) PDEs representation could be well defined without any boundary conditions (A. M. Arthurs, L. B. Rall).
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 non-trivial zeros of Riemann's zeta function arise from inquiries into the distribution of prime numbers. The eigenvalues of a random Hermitian matrix arise from inquiries into the behavior of systems of subatomic particles under the laws of quantum mechanics. What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?"
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 Archimedean axiom: this is related to physical measurement capabilities of a length 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 the non-standard analysis. In this framework the Stieltjes integral can be interpreted as the counterpart of the Lebesgue integral going along with a reduced regularity requirements of the corresponding domain, which becomes the newly proposed energy Hilbert space H(1/2). The corresponding quantum state Hilbert space framework changes from the current test space L(2)=H(0) to the distributions Hilbert space H(-1/2) whereby the test space L(2) is compactly embedded. The complementary subspace H(-1/2)-H(0) is closed enabling the definition of an orthogonal projection operator from H(-1/2) onto the test space H(0). It therefore provides the framework to model also an additional continuous spectrum of the (energy) Hamiltonian quantum operator (Berry conjecture), as well as superconductivity, super-fluids and condensates.
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") :
there is a common baseline of all considered conceptual problem areas, which can be overcome by one single common mathematical model. From a mathematical-philosophical point of view this is not surprising at all ((KnA), (ScE), (ScE1), (WeH), (WeH1)): already an irrational number is an "universe" by itself, i.e. an "object" defined as an "existing" (i.e. mathematical defined) limit of an infinite sequence of "rational" "objects". In other words, the existence of any "irrational number" "object" "exists" per (mathematical) completeness axiom, only. The mathematical concepts of "continuity" and "differentiability" are then building on the concept of "real" numbers (the extension of the set of rational numbers). The (physical) body-contact problem, (mathematically, not physically required) differentiable (not continuous) manifolds concepts to model space-time structure in Einstein's field theory (WeH1), a well-defined, but still mysterious Dirac function with space-dimension depending distributional Hilbert space domain (!) and other concepts are built on top of it.
From a mathematical perspective there are two axioms essentially:
- the completeness axiom to enable the "building" of "real" numbers, including irrational numbers
- the Archimedean axiom originally formulated for segments, which states that if the smaller one of two given segments is marked off sufficient number of times, it will always produce a segment larger then the larger one of the original two segments or, in simple words, to enable the measurement of a given (potentially very large) number per n times a given "unit" of measure length.
Putting the journey as the objective and not the
gathering and “owning” a list of deliverables/papers is a purely
personal thing. Nevertheless, there is a famous book related to this
different way looking at it, which might be interested to the one or
other visitor of this homepage (FrE).
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).
J. Gaarder addresses some of those philosophical questions in relation to a depiction of life in "Maya" (GaJ). It is about the universe, its related evolutionary theory and corresponding questions that tangle us here on earth: one protagonist's states that "the universe seeks to understand itself and the eye that looks into the universe is the eye of the universe itself". Anyway, and in any case reading the book of J. Gaarder is just fun; ... another related book, again just for fun, is ((HoJ). The underlying conceptual idea is going back to (NaT). .... or from another perspective: Yoda: "the power ('the vacuum energy') be ('is') with you ('with us')" ...:).
There are two basic assumptions to Th. Nagel's conceptial thoughts (NaT):
- awareness ("Bewusstsein") is an essential element/part of the evolution of the cosmos - values are lens ("objektiv") and independent from the point of view of a judgmental (evaluating) person.
Based on those assumptions he concluded that "the Materialist Neo-Darwinian Conception of Nature is Almost Certainly False". From a mathematical point of view the most easiest counter argument to his thesis is just to not accept at least one of those assumptions. All other kind of reasoning which are not based on "assumption - conclusion" format is anyway out of scope of the descriptive science, mathematics. And here's the overall challenge to all of those kind of philosophical discussions, whenever the word "cosmos" is part of it: Mathematics is THE only language of theoretical physics to describe the "cosmos" on the one hand side, while on the other hand side mathematics is a descriptive science, only. If it is possible that mathematics provides a "field/mind" description which can be consistently embedded into existing mathematical-physical models (including "initial value" "functions"), already this kind of formal descriptions, only (based on common sense-mathematical logic principles) go beyond human observation's horizon (and Kant's "critique of pure reason" boundaries of human understanding, as well). Already this would be another wonderful thing, what human beings are able to build. With respect to Nagel's second assumption then it would be only a small next step to its acceptance, as well.
There is a similar view possible on Th. Nagel's "View from Nowhere" (NaT1): human beings only observe parts of a "total", which shows several (or even infinite) aspects of all kinds of "infinities". (This is already the case when the baselines for all mathematical concepts are defined, which is about the "real" numbers. Already each irrational number is a very simple example of this kind of infinity (it is already an universe by itself) and the set of all irrational number, as well). Any human being observer "realizes" just from his individual perspective. There is no (objective) observer perspective from somewhere possible. Again, the descriptive science, the mathematics, is able to deal with different kinds and cardinalities of infiniteness. The concept of "continuous" "function" and "differentiable" "functions" is already a next dimension on infiniteness on top of "real" numbers or "ideal" numbers.
E. Schrödinger was also concerned with "Mind and Matter" questions (ScE): "The objective world has only been constructed at the prize of taking the self, that is, mind, out of it remaking it; mind is not part of it; obviously, therefore, it can neither act on it nor be acted on by any of its parts. If this problem of the action of mind on matter cannot be solved within the framework of our scientific representation of the objective world, where and how can it be solved?" ... "No single man can make a distinction between the realm of his perceptions and the realm of things that cause it, since however detailed the knowledge he may have acquired about the whole world, the story is occurring only once and not twice. The duplication is an allegory suggested mainly by communication with other beings."
(ScE) A TENTATIVE ANSWER:'"A single experience that is never to repeat itself is biologically irrelevant. Biologic value lies only in learning the suitable reaction to a situation that offers itself again and again, in many cases periodically, and always requires the same response if the organism is to hold its ground." ... But whenever the situation exhibits a relevant differential - let us say the road is up at the place where we used to cross it, so that we have to make a detour - this differential and our response to it intrude into consciousness, from which, however, they soon fade below the threshold, if the differential becomes a constantly repeating feature. .... Now in those fashion differentials, variants of response, bifurcations, etc., are piled up one upon the other in unsurvey able abundance, but only the mose recent ones remain in the domain of consciousness, only those with regard to which the living substance is still in the stage of learning or practicing.
... 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': "For 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."
(FrE) Fromm E., To Have or to Be, Harper & Row, New York, 1976