- 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

Manifolds/Tensor Analysis & Hilbert space/ PDO

Generalized functions on Hilbert spaces

The proposed mathematical Hilbert space framework in combination with singular integral operators has been successfully applied also for aerodynamics and electrodynamics:

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.

Calderon-Zygmund Pseudo Differential Operator

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).

@ http://www.navier-stokes-equations.com/Author-s-papers

Riesz Operators

The Riesz operators fulfill the following crucially property with respect to the rotation group SO(n), ((StE), (BrK) p.13):

Let m be the Mikhlin multipliers of the Riesz operators and r an element of SO(n), then m(r(x))=r(m(x)).

As a consequence there is a corresponding change from a Riemann manifold (with the related concept of “extension quantities” (Grassmann)) to a Hilbert space framework for differentials (see also (ScE) 1.1.3, to model "extended quantities" in a "continuum", whereby differentiable manifolds are required in case of a Riemannian manifold).  We note some other properties of the Riesz resp. the Hilbert operators: The Hilbert transform (as well as the Riesz operators) are “symplectic-like” in the sense, that it holds (Hu,v)=-(u,Hv), H*H=-I. The Riesz operators commutes with translations and homothesis (PeB) example 9.9.-9.11).

The constant Fourier coefficient of a Hilbert transformed function vanishes. This property plays a key role in the two proofs of the Riemann Hypothesis. At the same time there is a similarity to the "cusp form" with its vanishing zero mode in the context of spectral theory in hyperbolic surfaces. This fact indicates a relationship to the proposed vector (domain) field (see e. g. Borthwick D., Introduction to Spectral theory in Hyperbolic Surfaces).

Riemann's continuous manifolds (which ends up to be necessarily differentiable (!!))

it is proposed to replaced manifolds by distributional Hilbert space, which allows a truly modelling of geometry.

The terminology of "multiple extended quantities" was used by B. Riemann synonym to a "continuous manifold", conceptually based on two essential attributes: "continuity" and "multiple extensions". Since Helmholtz, Riemann, Poincare and Lie the history of manifolds  are the attempt to build a mathematical structure to model the whole (the continuum) and the particular (the part) to put its combination then into relationship to describe motion, action etc. From the paper from E. Scholz below we recall the two conceptual design strategies:

Strategy I: design of an "atomistic" theory of the continuum:  to H. Weyls's opinion this contradicts to the essence of the continuum by itself

Strategy II: develop a mathematical framework, which symbolically explores the "relationship between the part and the whole" for the case of the continuum.

The later one leads to the concept of affine connexion, based on the concept of a manifold, which were developed during a time period of about 100 years.

The concept of manifolds leads to the concept of co-variant derivatives, affine connexion and Lie algebra to enable analysis and differential geometry, but (according to H. Weyl in E. Scholz 1) ...a .." truly infinitesimal geometry ... should know a transfer principle for length measurements between infinitely close points only."

Inner products and first fundamental form

We note, that the 1st fundamental form is related to (inner) geometry concepts like lengths, angles, Christoffel symbols & the Levi-Civita derivative. The corresponding mathematical model concepts are inner products and (dual) Hilbert spaces. The 2nd fundamental form addresses the (parallel/affine) displacement of tangential (vector) spaces, i.e. it leaves the (inner geometry) Hilbert space framework. The additionally required mathematical concepts are about "hyper areas" and related distance functions. Therefore, not only the terminology changes to "exterior geometry". The gauge theory framework is a consequence to re-build again necessary vector space properties.

Lie transforms

We refer to

Lie S., Ueber die Grundlagen der Geometrie (1890), Wissenschaftliche Buchgesellschaft, Band CXX, Darmstadt, Sonderausgabe MCMLXVII

Quote (p.2): "Für den dreifach ausgedehnten Raum können die betreffenden Eigenschaften folgendermassen zusammengefasst werden:

die Bewegungen des dreifach ausgedehnten Raumes bilden eine Gruppe von reellen Transformationen, welche die folgende Eigenschaft besitzt: Wird ein reeller Punkt und ein reelles hindurchgehendes Linienelement festgehalten, so ist immer noch continuierliche Bewegung möglich; wird jedoch ausserdem ein durch das Linienelement gehendes reelles Flächenstück festgehalten, so bleiben alle Punkte des Raumes in Ruhe.

Diese Eigenschaft kommt der Gruppe der Euclidischen und der Gruppe der Nichteuclidischen Bewegungen, aber keiner anderen Gruppe zu. .....

In einem Raum mit mehr als drei Dimensionen lassen sich die beiden betreffenden Gruppen in ganz entsprechender Weise charakterisieren. Dagegen stellt sich die Sache wesentlich anders in einem zweifach ausgedehnten Raume; in der Ebene giebt es noch weitere Gruppen, welche die genannten Eigenschaften besitzen."

Lie's theory of "contact transformation", which builds the foundation of the Lie theory in the context of the manifolds:

http://openlibrary.org/books/OL7045943M/Geometrie_der_beruÌˆhrungstransformationen

Lie S., "Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen". Bearb. und hrsg. von Georg Scheffers (1893)

Lie S., "Ueber die Grundlagen der Geometrie", Wissenschaftliche Buchgesellschaft Darmstadt, MCMLXVII

Bell J. L., "Hermann Weyl on intuition and the continuum"

Scholz E., "H. Weyl’s and E. Cartan’s proposals for infinitesimal geometry in the early 1920s"

Legendre transform

We note that also the Legendre transform is a contact transformation. If the Legendre transform is applicable (ensured by (!) sufficiently high regularity assumptions) it is applied to prove the equivalence of the Lagrange and the Hamiltonian formalisms. We emphasis, that it would be sufficient to have a Hamilton (energy minimization functional) formalism, only, to define existing physical laws in the framework of variational theory. In case of PDO of negative order (in opposite to PDO of postive order, as model of PDE with corresponding regularityx assumption to its domain) the induced Hilbert space with respect to the  energy norm is a compactly embedded sub set of the induced Hilbert space with respect to the operator (graph) norm.

Therefore, minimization representations with respect to operator norm are defined, w/o the need that the "standard" minimzation representation with respect to the energy norm (which defined classical resp. weal PDE representation) are not neccessarily defined.

Therefore, it’s not a necessary (from a mathematical modelling perspective) that today's PDEs (representing physical models, e.g. the Maxwell equations) need to be valid for both representations, the integral form and in the differential form.

Cartan’s differential forms

Probably interesting to mention that today physicists calculate with differentials as "number" objects, but they neglect its physical existence as "particle" objects, while mathematicians calculate with differentials only as "functionals" or within the Cartan differential form calculus, but accept those "objects" as well defined existing "objects" of an e.g. Hilbert space  (which is the today´s mathematical standard framework for quantum mechanics modeling quantum "objects", ending up with quotes like the following one from N. Bohr: "If people are not scared about the quantum theory, they haven´t understood it").

Berkeley described Leibniz' differentials as "ghosts of departed quantities":

Dray T., Manogue C. A., Putting Differentials Back into Calculus

The alternative normal derivative definition of J. Plemelj

A new mathematical concept to define the normal derivative on the boundary with only "continuous" regularity assumption (only using interior domain values) was given by

J. Plemelj, Potentialtheoretische Untersuchungen, B. G. Teubner Verlag Leipzig, 1911

(pdf-copy: see www.navier-stokes-equations.com)

In J. Plemelj´s mathematical concept there exists a massless particle in the form of a differential connected to potentials defined by Stieltjes integrals“; in section I, §8 he states: "bisher war es ueblich fuer das Potential V(p) die Form (...) vorauszusetzen, wobei dann (...) die Massendichtigkeit der Belegung genannt wurde. Eine solche Annahme erweist sich aber als eine derart folgenschwere Einschraenkung, dass dadurch dem Potentials V(p) der groesste Teil seiner Leistungsfaehigkeit hinweg genommen wird."

Archimedian non-ordered fields

Preliminary Notes

(StJ), p.27: "...the set of real numbers is seen as a model for the number line. 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 Archimedean axiom (axiom of Eudoxos, (WeH) p. 41, 45):

From every positive number a one can obtain a number greater than 1 by repeating addition

a+a+a....+a (n times) > 1

where n is a positive integer number.

The field of Non-Standard numbers *R is an Archimedean non-ordered field, while the field of real numbers R is an Archimedean ordered field.

This is curious information, whereby in gravitation theory a black whole is seems and accepted as a "real" "objects" by  human's mind (still sophisticated phenomena, but consistently described in mathematical language).

A. Robinson The metaphysics of calculus II.pdf

We note that the "real number" field without the axiom of Eudoxus expands to the "Non Standard number" field, with same cardinality (A. Robinson) and same all other properties. Needless to mention, that experimental physics is anyway only requiring rational numbers, while theoretical physics models are calculating with differentials in same manner as with irrational numbers.

The Archimedean axiom is "just" about "distance" measurements of the real x-axis by an integer multiple of a given length standards. Now the delta of non-standard to standard is not about the way, how to measure, but about "ordered field" versus "non-ordered field". Human beings might need this specific type of "order", but does Nature need this as well?

The most probably strongest principles in Nature is "entropy", which is the opposite of "ordering".

How our current understanding and interpretation of the physical/ measurable world would look like, if our children would learn right from the beginning mathematical analysis as described in the language of "ideal" points? Analysis, as teached in school, become "standard", because it's part of the standard education program; if it would be replaced by "Non-Standard Analysis" this would be perceived as "standard".The current "Non-Standard-Analysis" would be a standard one and the other way around. This would mean that our universe would be realized and interpreted by mind as "Non-Standard", as we all were learned at school, but "standard" in the way, how Leibniz would have been defined/interpreted the term "differential" and ist actions in the universe. Singularities would become "natural" and consistent to the corresponding physical-mathematical models; big bang would require no t=0perception of R. Penrose versus S. Hawking (mathematicians vs physicists) of what matter are finally goes back to Newton´s miss understanding/ interpretation of Leibniz´s concept of the infinitesimals (monads). Imagine that the development of the infinitesimal calculus would have been built on the original thoughts of Leibniz instead? This would have been lead to the fact, that Robinson´s Non-Standard Analysis would be teach at school as "standard" and an "ideal" point would be a natural "object", as it is a "real" objects/numbers (even if it is a sophisticated transcendental number) today.

Fundamental theorem of set theory (K. Gödel, P. Cohen):

The Cantor continuum hypothesis is neither provably, nor refutably.

(RiM): The system *R of hyper-real numbers (nonstandard reals) is a way of treating infinite and infinitesimal small quantities. The cardinality of the real and hyper-real number fields is the same. The Archimedean axiom, which is related to measure distances on the real number axis with a finite measuring stick of finite length, is valid only for the real number field:

*R is a Archimedean non-ordered field, while the field R of the real numbers is an Archimedean ordered field.

Some early thought about the topics above: