4.4 'Solutions' proposed

4.4 'Solutions' proposed 149

Classical Dynamics reconstructed

CONCLUSIONS

4.4 'Solutions' proposed

"1937 Philos. Rev. 46 439: outswell v.: In one of those 5

terminal notes which together outswell the essay, he inti-mates that writers on the philosophy of science suffer from ignorance of mechanics*." * In the reference: chemistry.

Oxford English Dictionary: Word of the Day, 10

30.03.2009.

PROBLEMS

Not only mechanicians have been pondering the foundations of mechanics and the concept of force in particular, but also philosophers, and the latter have even proposed solutions for the problems, they believe to have identified. Most of these 15

expositions are neither enlightening nor useful; often the problems claimed to be identified are pseudo-problems.

MODELS/GOALS/PLANS

The detailed analysis of a recent philosophical treatise permits to wind up the preceding sections, substantiating the claim that the expositions of the fundamen-20

tals of classical mechanics and the corresponding historical and philosophical lit-eratures are in a deplorable state. For impatient readers the analysis will at the same time provide a detailed outlook on the problems and solutions subsequently to be elaborated on in extenso in the present treatise.

4.4.1 'Philosophical' approach 25

"Philosophen versuchen häufig im Hase-und-Igel-Spiel die Igel zu spielen und sind doch immer nur die Hasen."

Jürgen Mittelstraß (2008).

4.4.1.1 PROBLEM/MODEL/GOAL

A recent climax of the confusion and of the continuing, self-perpetuating 30

interest of philosophers in the problems of mechanics is the treatise 'Zur Konzeption der Kraft in [der Geschichte] der Mechanik [bis 1915]' by Lopes Coelho (2000, 2001). As indicated by the author's additions […], already the title is felt to be incomplete and thus to be grossly misleading.

150 4 Opening operations

Schmiechen 10.09.2009

The extended discussion of that treatise permits to wind up the preceding sections and, at the same time, the discussion will provide, for impatient readers, an outlook on abstract elementary mechanics, on its operational in-terpretation and on elementary physics, details to be discussed in great depth in the following more than thousand pages. 5

The publication of the habilitation treatise is advertised with the sentences (2001):

"Was ist die Kraft? Die Konzeption der Kraft der Mechanik stellt ein bisher ungelöstes Problem der physikalischen Theorie dar. Durch einen Rekurs auf die Philosophie und Wissenschaftstheorie wird hier ein Lö-10

sungsvorschlag unterbreitet."

The problem and the proposal for its solution in terms of philosophy and theory of science are only apparently the same as those of the present trea-tise. The details and the results, in fact the whole conceptual frame work are altogether totally different from those of the present exercise aiming to un-15

derstand classical mechanics in Goethe's sense and in Einstein's spirit.

As has been stated over and over again there is a problem and it cannot be solved 'inside' mechanics, but only from 'outside', by meta-physics, cau-tiously called meta-mechanics in the present treatise in order to avoid any further confusion. 20

This is not one option among others as stated in the first sentences of the preface (2000/III). The opinion that philosophy and theory of science are only helping maids is felt to misjudge their roles in constructing and under-standing theories and in using them as power tools in the daily work of sci-entists and engineers (2000/IV). 25

As will be demonstrated in the present exposition the problem of force can be understood and thus can be solved concerning not only the concep-tual framework but also its interpretation as well as its physics. Solutions in the spirit of more or less obscure 'meta-physics' are no longer meeting the current standards of rationality and are no longer serving the current pur-30

poses.

4.4.1.2 AXIOMATIC APPROACH

The problem stated, the circularity in 'defining' mass and force in terms of the 'basic equation: mass times acceleration equals force' is called circulus vitiosus without noticing that all 'mythical' foundations, all initial problems 35

are exhibiting exactly the same 'problems' (2000/1 f).

By their very nature, being axiomatic systems, they are provisional, step-ping stones, roots in the swamp to help ourselves out of the morass. Thus the problem as stated is a pseudo-problem and not surprisingly, this funda-mental problem is not further addressed, not adequately discussed and not 40

solved in the treatise scrutinised.



The epistemology underlying the exposition is simply too vague, suffer-ing from the lack of explicit models of theories etc, ignoring even the sim-plest such models. How else is the sentence (2000/9):

"Die …Principia … sind gemäß der Euklidischen Form dargestellt, … "

to be understood? This is not just elegant jargon, but clearly indicating that 5

the essence and the usage of axiomatic systems and Newton's axiomatic sys-tem in particular have not been understood, finally resulting in the 'banal' central arguments and results of the treatise.

The fact that Newton's axiomatic system already meets the current stan-dards, ultimately established by Hilbert, has escaped the attention of most 10

philosophers and physicists. Else many discussions found in the literature and in the treatise under scrutiny would not have been possible. Contrary to Euclid's empty 'Definitions' of basic concepts Newton's first three Defini-tions are correct formal definitions of derived concepts.

A pragmatic decision would have been to stop any further discussion of 15

the habilitation thesis at this stage, but the following notes are provided to elucidate some essential points already at this stage.

4.4.1.3 HERMENEUTICS

"Lange Überlegungen zeigen gewöhnlich, daß man den Punkt nicht im Auge hat, von dem die Rede ist, … " 20

Johann Wolfgang Goethe: Wilhelm Meisters Lehrjahre (BA 10/451).

In the first chapter twelve prominent works of mechanics, selected from more than 250 references listed, are 'examined' in detail. They are claimed to be interpreted in a hermeneutical fashion adapted to the purpose at hand, 25

'each on its own, considered as a system' (Italics: MS) and stressing the con-cept of force; results to be used in the second chapter (2000/7 f).

The author has inspected the 150 pages and found the 'interpretations' of the expositions lacking any interpretations worth that name, hopelessly in-adequate, not meeting the current standards set by Lorraine Daston and her 30

'consorts' at the Max-Planck-Institut for the History of Science in Berlin. This fundamental work has not even been mentioned.

And even worse, the coverage is without any useful results, without any critical appraisals of the essentials. The approach is clearly different from the one followed in the present treatise, trying to analyse and to understand, 35

mostly the same famous works, to trace the evolution and point out the dis-coveries and the fallacies.

All works are 'interpreted' by quoting the originals in extenso, but without adequate exegesis. Just as an example, Newton's epistemology is not ad-dressed; it is not mentioned that Newton, in accordance with the dual model 40



of theories, expressis verbis clearly distinguished between the abstract, 'ab-solute' theory and its operational, 'relative' interpretation using 'instruments', the latter subject to the laws of mechanics themselves, a 'banal fact' forgot-ten for a long time, re-discovered by Einstein two hundred years later.

The real problem is to set up the simplest meta-mechanics and meta-5

principles adequate for the purposes at hand and if that has been done the concepts and 'natures' of forces will be evident, in that explicitly constructed representation space.

And they can be operationally interpreted and identified as a matter of professional craftsmanship as clearly described by representatives of the Er-10

langen school, Janich in particular. As has clearly been pointed out in a let-ter by Borzeszkowski and Wahsner to Lorenzen there are no proto-mechanics and no measurements before abstract mechanics has been set up (1995/38-41).

All this is well-known and well understood since long and in the present 15

treatise it will be shown how powerful the axiomatic, 'conventional', 'con-structive' approach is in solving problems not solved so far, among them the philosophical problems in question, and the physical problems of inertia and gravity.

4.4.1.4 THESE 1 20

" … übereilte Handlungen [zeigen gewöhnlich], daß man ihn [, den Punkt, von dem die Rede ist,] gar nicht kennt."

Johann Wolfgang Goethe: Wilhelm Meisters Lehrjahre (BA 10/451). 25

The second chapter, devoted to the 'solution proposed', starts 'correctly' with the section headed 'Bestimmung des Problems', what ever 'Bestim-mung' may imply, maybe 'definition'? But the surprise follows instantly with the sub-section THESE 1 (2000/157):

"These 1: Wenn das Trägheitsgesetz angenommen wird, dann folgt als 30

logische Konsequenz, daß eine außerhalb des Körpers liegende Ursache be-stehen muß, um dessen beschleunigte Bewegung zu rechtfertigen."

Whatever 'rechtfertigen' may imply.

And this proposition is formally 'proved' and it is substantiated by refer-ring to all the famous expositions of the inertia principle, ending with the 35

glorious 'conclusion' (2000/177):

"Indem die aus dem Trägheitsgesetz gezogene Konsequenz eine logische und darüber hinaus eine Übereinstimmung, wie gezeigt wurde, mit der Ge-schichte der Mechanik wie auch deren Gegenwart besteht, scheint, daß die



These als eine gültige betrachtet werden kann, und daher gehen wir zur nächsten über."

This is felt to be an incredible corruption of Newton's very clear statement of his first law, to be 'polite' jargon missing all the essential points, fashion-able 'non-sense' of the type pinpointed by Sokal, though not quite as ex-5

travagant.

4.4.1.5 THESE 2

So, if this was not a problem to be solved, what then is the problem to be solved? And again the surprise follows instantly in sub-section THESE 2 (2000/177): 10

"These 2: Der Ursache der Veränderung der eigenen Bewegung des Körpers wird in der Theorie Ausdruck gegeben, sie wird Kraft genannt. So wird die Theorie logisch kohärent, sie versagt aber in bezug auf die Phä-nomene, weil es keinen natürlichen Ausdruck für die von ihr bedingte Kon-junktion gibt. 15

Nachzuweisen ist also:

a) Die Ursache der Veränderung wird Kraft genannt;

b) Die Theorie wird kohärent;

c) Die Theorie steht nicht im Einklang mit den Phänomenen."

And again these cannot possibly be the problems to be solved! 20

It will be hard to match or even surpass the sheer beauty of the hollow phrases 'es scheint, dass die These als eine gültige betrachtet werden kann', 'der in der Theorie Ausdruck gegeben wird' and 'nachzuweisen ist also: Die Ursache der Veränderung wird Kraft genannt'.

Ad a): What does the proof of the usage of the term 'force' prove? To 25

'confuse' the two mechanisms causing changes of momentum, a quantity proper (Menge), the surface forces and the body forces formally into the one 'concept of force', forgetting that this is a sum of essentially different physi-cal processes, not quantities (Grössen, not Mengen) is going back to pre-Newtonian mechanics. Even in the stone ages human beings 'must' have dis-30

tinguished surface and body forces.

Ad b): That 'the theory becomes coherent', that the momentum balance remains correct under this formal 'confusion' is trivial, 'banal' Einstein would have said.

Ad c): The present author admits that he does not understand the meaning 35

of the remaining sentence:

" … sie versagt aber in bezug auf die Phänomene, weil es keinen natürli-chen Ausdruck für die von ihr bedingte Konjunktion gibt."

What is a natural expression? How does a conditional conjunction come in?



Forgetting about these 'philosophical' reservations the author does not un-derstand why the theory, which has perfectly and on celestial scale with ex-treme precision served many purposes for more than three hundred years 'fails concerning the phenomena'.

The only reason detectable in the context of the treatise and in line with 5

its style of arguments is again a trivial one: the 'confusion' of surface forces and body forces, clearly distinguished by Newton in the first law, into the one 'concept of force' is indeed 'useless', not only for the theory but also for the practice of mechanics as well, and definitely for the discussion of the fundamentals. 10

4.4.1.6 EVALUATION/ASSESSMENT

" 'Ich weiß nicht, was ich meinen soll', entgegnete Sancho, ich bin nicht so belesen wie Ihr in den metaphy-sischen* Schriften. Dennoch wage ich zu sagen, ja zu be-schwören, dass die Gaukelbilder da bestimmt keine Un-15

schuldsengel sind.' " * In the reference: fahrenden.

Miguel de Cervantes Saavedra: Don Quijote (2008/I, 531).

Even the first impression of the treatise scrutinised is rather disappointing, to say it politely. The coverage of the classical literature is completely in-20

adequate. In a language and methodology hopelessly inadequate for the pur-poses at hand THESE 1 and THESE 2 provide solutions for problems, which turn out to be none. The central problems, among others the problems of logical analysis and of operational interpretation, mentioned on the first page are not treated any further. 25

Similar 'meta-physical' ideas have been expressed by Wahsner and Bor-zeszkowski in their edition of Mach's 'Mechanik in ihrer Entwicklung'. Of particular interest are their Notes 17 thru 22 concerning Mach's discussion of the law of inertia (1988/533-536), with the crucial paragraph (1988/535):

" … Als 'natürliche' Bewegung wird in der ausgearbeiteten klassischen 30

Mechanik dann die Trägheitsbewegung unterstellt. Sie fungiert als Be-wegungsetalon, an dem die Bewegung gemessen wird. (Ausführlicher da-zu: H.-H. v. Borzeszkowski/R. Wahsner, Physikalischer Dualismus und dialektischer Widerspruch. Studien zum physikalischen Bewegungsbegriff, Darmstadt 1988). 35

21. Diese berühmte Kritik Machs beruht auf der auch heute noch stark verbreiteten Meinung, daß das zweite Newtonsche Axiom, das den Kraft-begriff definiert; das erste Axiom schon enthalte. Diese Interpretation ist jedoch falsch. Denn das erste Axiom hält fest, daß ein Körper auf sich selbst keine Kraft ausübt, d. h. daß die Kraft gleich Null ist, wenn nur ein 40

einzelner Körper vorhanden ist. Damit eine Kraft vorhanden sein kann,



muß noch etwas physikalisch Zweites, ein zweiter 'Körper' da sein. Diese Feststellung folgt nun aber nicht schlechthin aus dem zweiten Axiom, son-dern muß gesondert gefordert werden."

'Natural motion' and an 'etalon of motion' are concepts not in accordance with the relativity of the intensity and extensity of motion, both depending 5

on the observation space 'chosen' for convenience. Thus 'eigene Bewegung' is a non-sensical notion.

For ready reference Mach's discussion is quoted in extenso (Mach, 1988/161 f):

11. Wir haben schon erwähnt, daß Galilei ganz nebenher das sogenannte 10

Gesetz der Trägheit gefunden hat. Ein Körper, auf welchen, wie man zu sa-gen pflegt, keine Kraft wirkt, behält seine Richtung und Geschwindigkeit unverändert bei. Mit diesem Gesetz der Trägheit ist es sonderbar zugegan-gen. Bei Galilei scheint es nie eine besondere Rolle gespielt zu haben. Die Nachfolger aber, namentlich Huyghens und Newton, haben es als ein be-15

sonderes Gesetz formuliert. Ja letzterer hat sogar aus der Trägheit eine all-gemeine Eigenschaft der Materie gemacht. Man erkennt aber leicht, daß das Trägheitsgesetz gar kein besonderes Gesetz ist, sondern in der Galilei-schen Anschauung, daß alle bewegungsbestimmenden Umstände (Kräfte) Beschleunigungen setzen, schon mit enthalten ist. 20

In der Tat, wenn eine Kraft keine Lage und keine Geschwindigkeit, son-dern eine Beschleunigung, eine Geschwindigkeitsänderung bestimmt, so versteht es sich, daß wo keine Kraft ist, auch keine Änderung der Ge-schwindigkeit stattfindet. Man hat nicht nötig, das besonders auszuspre-chen. Nur die Befangenheit des Anfängers, die sich auch der großen For-25

scher der Fülle des neuen Stoffes gegenüber bemächtigte, konnte bewirken, daß sie sich dieselbe Tatsache als zwei verschiedene Tatsachen vorstellten und dieselbe zweimal formulierten.21

Die Trägheit als selbstverständlich darzustellen oder sie aus dem allge-meinen Satz 'die Wirkung einer Ursache verharrt' abzuleiten, ist jedenfalls 30

durchaus verfehlt. Nur ein falsches Streben nach Strenge kann auf solche Abwege führen. Mit scholastischen Sätzen, wie mit dem angeführten, ist auf diesem Gebiete nichts zu verrichten." Final italics: MS.

Wahsner's and Borzeszkowski's extended Note 21 ends with the following paragraph succeeded by Note 22 with a remarkable 'insight' (1988/536): 35

"21.cont'd. Eine Diskussion mit Mach würde vermutlich zu keinem Er-gebnis führen, da er die Notwendigkeit des ersten Axioms auch nicht im Sinne der hier gegebenen Interpretation einsehen würde. Sein Argument: Eine Kraft ist selbstverständlich eine Wirkung zwischen zwei Körpern.

22. Das physikalische Trägheitsprinzip kann aus einem metaphysischen 40

Satz tatsächlich nicht als unmittelbar einleuchtend abgeleitet werden. Diese Feststellung impliziert aber nicht die Ansicht, das Trägheitsgesetz sei nichts weiter als die Folge des zweiten Axioms." Italics: MS.



Finally the authors admit that 'metaphysics' is not helpful to understand the problem, unless understood pragmatically as a coherent hierarchy of meta-theories, as meta-mechanics in particular.

Of interest are not 'opinions implied' but are coherent axiomatic systems of conventions capable of operational interpretations. The point here is that 5

various expositions are not competing, are not more or less true or false, but may be equivalent and more or less adequate for the purposes at hand in any particular case, anything else is additional, superfluous 'superstition'.

4.4.2 'Rational' approach

4.4.2.1 FORMAL ANALYSIS 10

" … Denn die Bücher ohne Formeln Haben meistens keinen Sinn … "

From an apocryphal paraphrase of Bertold Brecht's 'Drei Groschen Oper' (Beggars' Opera) for Max Born's fiftieth birthday (TellerE, 1991/1). 15

Any attempt to clarify the 'fundamental' problems of classical mechanics in an informal fashion is doomed to fail (Suppes, 1983.f). Although the axiomatic system in question is very simple as, e. g., that of Euclidean ge-ometry and that of the rational theory of propulsion, the structure of the rep-resentation space spanned is extremely rich and intricate. 20

As other neural networks, human brains just do not (like to) 'walk' these ways and easily loose their ways. As has been described and will further be explained in great detail the history of mechanics provides ample evidence to that effect. Most striking is that so far no classical theory of general rela-tivity and of gravity has been developed. 25

Thus, in order to provide for the transparency necessary for scrutiny tabu-lar formats are adopted for Newton's axiomatic system and for the following axiomatics of rational mechanics promoted. The former will be embedded into the latter to elucidate its formal and the factual implications.

30

Newton's axiomatic system

symbolic explanation Principia

Basic concepts

density ρ basic unit: density of water

Def. 1

volume V Def. 1



velocity, speed v i translational mo-tion,

'only relatively to be distinguished'

Def. 2

Def. 3

inertial force F inert Def. 3

surface force F surf i Law 1.2

surface force acting at the 'environment'

F surf env i Law 3

body force F body i Law 1.2

Derived concepts formally defined

quantity of matter, mass

m ≡ ρ V capacity of transla-tional motion

Def. 1

momentum M i ≡ m v i quantity of motion, quantitas motus

Def. 2 Law 2

inertial 'force' F inert ≡ − m d t v i 'resistance to change of motion'

Def. 3

Basic propositions

Axioms

Axiom 2 m d t v i = F surf i + F body

i balance of transla-tory momentum in

any observation space given

'classical principle of general relativ-

ity'

Law 2.1

Def. 4

Axiom 3 F surf i + F surf env

i = 0 i continuity of mo-mentum flow

across the surface, implicit dynamical

definition of the surface

Law 3

Der'd propositions

Theorems

formally deduced

Theorem 1

'rational' rendering

if [F surf i = 0 i

and F body i = 0 i ]

then d t v i = 0 i

body moving freely

in an inertial space:

'principle of inertia'

Law 1.2

Law 1.2

Law 1.1

Theorem 1

Newton's rendering

d t v i = 0 i

if not [F surf i not = 0 i

and F body i not = 0 i ]

'principle of inertia'

body moving freely

in an inertial space

Law 1.1

Law 1.2

Law 1.2

Def. 3



Theorem 1.1

if F surf i = 0 i then

m d t v i = F body i

body moving freely, in vacuo

Theorem 1.2

if F body i = 0 i then

m d t v i = F surf i

body moving in in-ertial spaces

Theorem 2 if d t v i = 0 i then

F surf i + F body

i = 0 i

steady states in the observation space

given

Theorem 3 if v i = 0 i then

F surf i + F body mat

i = 0 i

balance of diffu-sive momentum in-flow and material

momentum produc-tion

Def. 4

This sketch of an axiomatic system of elementary dynamics in Goethe's spirit may appear naïve compared with axiomatisations for Newton's parti-cle mechanics, e. g., by SimonHA (1983/239-244) and ScheibeE (1983 /180 ff). But as Maxwell requested it is carefully phrased in the language of 5

dynamics contrary to the paper of SimonHA, which is hopelessly shrouded in mathematical language, although specifically written to satisfy the criteria of logicians and practising scientists at the same time.

4.4.2.2 EXPLANATORY NOTES.

„Man muss den Heiligen des Evangelii solange inter-10

pretieren, bis etwas Vernünftiges herauskömmt.“

Immanuel Kant (Quotation following Nordhoven, 2009).

GENERAL. Already this very short table leaves no doubts concerning a number of fundamental issues subjects of endless, mostly inadequate and fu-15

tile discussions. The following additional notes are sketches of detailed dis-cussions to follow in due course.

A problem in the analysis is that Newton's explanations of the Definitions sometimes appear to include his Laws. In the table it is explicitly stated that Newton's first and second laws consist of two 'parts' each, the second parts 20

often overlooked and/or 'forgotten' and/or not understood.

The concepts introduced are meaningful only in the context of the few axioms stated. Thus since publication of the Principia the concepts of forces are correctly defined only implicitly in the given frame work of the momen-tum balance. Pre-Newtonian, 'independent' definitions of basic concepts, 25

similar to the Euclid's empty 'definitions', are still to be found even in recent textbooks.



QUANTITY OF MATTER. The first observation is that Newton's first defini-tion is by no means circular as is being ritually, but falsely repeated. His ba-sic concept is that of density and the density of water is the basic unit as Ca-jori has clearly pointed out in his note on Definition 1 (PM/638 f).

Newton's definition has still been the basis of the original definition of the 5

mass etalon, the unit of mass measurement: 1 kg is the mass of one 1 litre = 1 dm^3 of distilled water at the temperature of 4°C and the pressure of 760 mm Hg (SchmidtErn, 1953/1-17). Modern attempts to replace the 'Ur'-kilogramme, the Platinum-Iridium mass etalon at Sevres by a Silicon sphere are essentially still adhering to the exactly the same definition of mass. 10

'CAPACITY' OF MOTION. Though the basic unit of density is that of water densities are in fact compared by their specific weights in any given obser-vation space. According to the notes following the individual Definitions Newton conceived 'mass' from beginning as 'inertia', as a mechanical prop-erty of a 'body', a 'quantity (Menge) of matter' (PM/1): 15

"It is this quantity that I mean hereafter everywhere under the name body or mass. And the same (the quantity of matter) is known by the weight of each body, for it is proportional to the weight, as I have found by experi-ments on pendulums, very accurately made, which shall be shown hereaf-ter." 20

Accordingly CohenIB in his 'Guide' explicitly states that for Newton in the Principia mass is not of interest as quantity of matter, as the derived concept formally defined before, but as a basic concept implicitly defined by his second law, measured by the weight of the body of matter under investiga-tion in the body fixed observation space (1999/95). 25

Consequently Newton's second 'Definition' is introduced as first 'Axiom' in the rational axiomatics to follow. In view of the fact that this axiom in-cludes a phenomenological parameter, the inertia, the capacity of motion, its more cautious qualification as 'constitutive law' is felt to be more appropri-ate and will be used in cases. 30

'PRINCIPLE OF INERTIA'. In Newton's own exposition the first law, the so-called 'principle of inertia' is expressis verbis introduced as a theorem, with a small 'difference', the double negation: 'unless it is', 'if it is not' compelled … by forces [not vanishing] impressed'. Contrary to THESE 1 already at this stage reference to forces is necessary. If only one condition holds, further 35

derived sentences, theorems may be deduced as shown in the table.

Though Newton correctly stated a theorem he called it an axiom. Even the uninitiated, or only those, will argue that such a general statement cannot possibly be meaningful without specification of the conditions under which it is 'valid'. 'Experts' as Mach, contrary to 'philosophers', have always 40

pointed out that the first law is not an axiom but a theorem.



Thus the first law does not imply that Newtonian dynamics is limited to freely moving bodies and/or inertial spaces. As a matter of fact, most obser-vation spaces of interest and considered by Newton in celestial and terres-trial mechanics do not belong to this very limited class of observation spaces. 5

But the theorem, mistakenly called an axiom, has led not only non-dyna-micists ritually to repeat the mistake, that Newton's dynamics is valid only in inertial spaces. In other words, Newton's lex prima is not an axiomatic convention restricting the choice of observation spaces, so-called 'inertial spaces', in which freely moving bodies proceed on straight paths. The 10

widely held instinctive believe, that the theory of classical, Newtonian me-chanics holds only in these spaces, is plainly wrong.

The consequence may be phrased as a 'rule': Only if [an author has (not read and/or not understood the Principia)] then [that author will claim that (Newton's first law is an axiom) and/or (Newtonian and classical mechanics 15

apply only in inertial spaces)].

DISCUSSION. CohenIB in discussing the first axiom claims that Newton 'needs' the first law as an axiom, in view of the fact that he phrases the sec-ond law in terms of impulsive forces (1999.g/109-111). But this point of view suffers from two serious mistakes. Firstly Newton did not 'need' the 20

first 'axiom', but 'used' the first 'theorem' to demonstrate and illustrate his second law in terms of impulsive momentum changes due to impulsive forces. Secondly the same theorem is useful in case of continuously acting forces.

Concerning Newton's concepts of force an observation by Brackenridge is 25

pertinent here (1996/VIII f):

" … the claim often made that Newton's early work reveals a confusion concerning force that was later eliminated, specifically that he attempted to combine two or more different force concepts. It is my contention that an analysis of the details of Newton's solution reveals no such confusion. One 30

must understand the details in order to make an informed decision. … "

4.4.2.3 'CONSEQUENCES'

THESE 1 reads:

"Wenn das Trägheitsgesetz angenommen wird, dann folgt als logische Konsequenz, daß eine außerhalb des Körpers liegende Ursache bestehen 35

muß, um dessen beschleunigte Bewegung zu rechtfertigen." Italics: MS.

What 'außerhalb des Körpers liegend' implies will be discussed immedi-ately. The point here is that THESE 1 states nothing else than Newton's first law:



"Every body continues in its state of rest, or of uniform motion in a right line, unless [i. e. if not] it is compelled to change that state by [non-vanishing] forces impressed upon it." Italics: MS.

Further, already the first statement in THESE 2 cannot be defended:

"These 2: Die Ursache der Veränderung der eigenen Bewegung des Kör-5

pers wird in der Theorie Ausdruck gegeben, sie wird Kraft genannt." Ital-ics: MS.

'Die eigene Bewegung' is supposedly the essential point of the whole the-sis, the reference to 'ontology' as also expressed by Wahsner and Borzesz-kowski. The 'pity' is that any motion is relative as Newton states explicitly at 10

the end of Definition 3, thus it is felt to be hardly useful as an ontological basis (PM/2).

And all the authors adhering to the ontological doctrine miss to note that in single body 'universes' the concept of motion is 'meaningless'. There 'are' no different observation spaces; they cannot even be conceived, forgetting 15

about their operational interpretations. 'Bewegung an sich' is an inherently meaningless, a non-sensical notion.

Traditionally 'causes' of momentum changes are, despite the differences in their physical mechanisms, 'uniformly' called 'forces'. Further both, 'cause of momentum change' and 'force', are usually considered as essentially iden-20

tical concepts. But 'unfortunately' it is a 'fact', known 'in der Geschichte der Mechanik, wie auch deren Gegenwart', that both concepts are essentially different, wesensverschieden.

In case of the surface forces the causes, adjacent other bodies lie indeed visibly outside the body, while the resulting forces 'lie in the surface'; mo-25

mentum is diffusively flowing across the surface into the body. In case of the body forces the causes, distant other bodies lie again visibly outside the body, while the forces lie inside the body, momentum is produced inside the body of matter.

In the ill-defined 'conclusion' it is claimed, 'dass THESE 1 in Überein-30

stimmung mit der Geschichte der Mechanik, wie auch deren Gegenwart, als eine gültige betrachtet werden kann' (2000/177). The only consequence to be drawn is that Lopes Coelho in accordance with Wahsner and Borzesz-kowski introduces another axiom, subject of a literature of its own: Bodies themselves cannot generate causes of momentum changes and subsequently 35

'forces'. In global or aggregate mechanics a related axiom is phrased in en-gineering jargon: internal forces in a system of bodies of matter cannot move the system as a whole.

Conceiving momentum as quantity proper in a body of matter, as will be done throughout the present treatise, implies that momentum changes ex ni-40

hilo, other than through surface forces, momentum diffusion across the body surface into the body, and through body forces, momentum productions in-



side the body of matter, are excluded explicitly by the second law; there is no need for a separate axiom.

4.4.2.4 CONCLUSIONS

The author is firmly convinced that analyses similar to the preceding may be found in the literature. But he admits that he gave up his search after 5

finding mostly inadequate discussions, often even plain non-sense. In order to escape this deadlock, hopefully avoiding all pitfalls he performed the pre-sent exercise in elementary axiomatics and logics during the study of perti-nent paragraphs in CohenIB 's 'Guide to the Principia'.

All the problems mentioned in the 'Prologue' and 'Opening operations' are 10

indeed culminating in the treatise scrutinised. Texts have not been read and/or have not been analysed with the care necessary, as has been the case over the centuries. The sad story is that scholars and philosophers of New-tonian mechanics have not been helpful in escaping the misconceptions, the mistakes and the superstition handed down from generation to generation in 15

the academic game of 'Stille Post'.

CohenIB refers expressis verbis to the very great responsibility of the translators, but in his and Whitman's own translation and in his 'Guide' ad-heres to traditional (mis-)conceptions as Schüller does in his translation, both translations based on the same Latin original of the third edition of the 20

Principia, as reconstructed by CohenIB and Whitman.

To substantiate the above criticism in detail there is no other way than to reconstruct classical mechanics in a rather formal fashion as the preceding analysis of Newton's axiomatics has shown. Accordingly an outline of clas-sical mechanics in the spirit of Newton, Euler, d'Alembert and Einstein will 25

be sketched.

Not only uninitiated readers, provided they happen to stumble into this analysis, will wonder how cumbersome the reconstruction of the most ele-mentary concepts is, even forgetting about historical texts, and that it will take another more than thousand pages to unfold the details. 30

4.4.3 Elementary mechanics

"As far as possible, one ought to let things speak for themselves; one must first labor to find the appropriate Anschauung, onlooking, so that one might develop an adequate terminology and method of presentation that 35

expresses tolerably well the full gamut [entire range] of the phenomena."

Dennis Sepper, summing up the rationale of Goethe's method (1988/45).



4.4.3.1 ELEMENTARY DYNAMICS

Elementary mechanics, conceived here as the common root of classical mechanics, is concerned with the translational motions of 'small' solid bod-ies of ponderable matter in three-dimensional Euclidean space. Rotational motions of the bodies are excluded from the outset, to be treated in subse-5

quent sections of the treatise.

The same motions may be observed from spaces in relative translational motion to each other. In elementary mechanics Cartesian frames at rest in the (observation) spaces are adopted throughout as (reference) frames. In global or aggregate mechanics reference frames at rest in the spaces of gen-10

eralised motions, e. g., in moving solid bodies will be more convenient.

Following Galilei, Newton and Euler in rational mechanics the extensities of matter and of motion of material bodies provide instances of the abstract concept of quantities contained in boundaries. Thus the quantity of motion in a given body of ponderable matter may change due to surface 'forces' 15

and/or due to body 'forces'.

The balance of the quantity of motion, of momentum in the traditional format

m d t v i = F surf i + F body

i

already implies the isotropy of the translational inertia of solid bodies and 20

the invariance of its invariant, the mass.

To introduce the 'concept of force'

F i ≡ Σ j F j

i ≡ F surf i + F body

i ,

as in the treatise scrutinised, forgetting that it is the sum of at least two es-sentially different physical mechanisms, wesensverschiedener Mechanis-25

men, explicitly mentioned in the first law of the Principia, indicates lack of the most elementary 'Anschauung', plain ignorance of the essentials and fundamentals.

The following table of an adequate rational axiomatics explicitly states the various mechanisms of momentum flow and production to be distin-30

guished in meaningful talk about 'forces'. Further, at this stage the artificial numbering of axioms has been dropped in favour of names proper in order to prevent misleading connotations as far as possible.

'Rational' elementary dynamics

symbolic explanation

Concepts

velocity, speed v i ≡ mot I intensity of motion



momentum M i ≡ mot E extensity of motion, quantitas motus

inertia I i j ≡ mot C capacity of motion, inertia

mass m capacity of transla-tional motion

momentum Flows M F i , M

C i , M

D i into the body: Con-

vective and Diffusive

Diffusive momentum flows

M D i , M

L i , M

M i into the body:

macro- (Large), mi-cro- (Molecular)

scale

momentum Production M P i , M

G i , M

K i inside the body of

matter: dynamical (Gravitational), Kin-ematical (apparent)

Gravitational momen-tum production

M G i = M N

i + M O i inside the body of

matter: actual (mate-rial, Nuclear), pOten-

tial (immaterial)

U i j unit tensor

Axioms

inertia I i j = m U i j

momentum, quantity of translational motion

M i = m v i

local balance of trans-lational momentum

∂ t M i = M F i + M P

i in a space fixed con-trol volume

momentum flow M F i = M C

i + M D i

diffusive momentum flow

M D i = M L

i + M M i

momentum production M P i = M G

i + M K i

gravitational momen-tum production

M G i = M N

i + M O i

Theorems

'substantial' balance of translational momen-

tum

d t M i ≡ ∂ t M i − M C i

= M D i + M P

i

in the body of matter

material component of the momentum bal-

ance

0 i = M M i + M N

i momentum balance in the body fixed ob-

servation space

immaterial component of the momentum bal-

ance

m d t v i = M O i + M K

i additional in any other observation

space



In addition the interconnections of the various 'components' of momen-tum production are shown in the following table of terminology, to be read top-down.

Momentum production: terminology

space invariant space variant

physical apparent

dynamical kinematical

actual potential 'relative'

material immaterial

5

4.4.3.2 ELEMENTARY PHYSICS

According to the meta-model surface forces in the balance (Bilanz) of momentum are identified as momentum diffusion

F surf i = M D

i = M L i + M M

i ,

into the body, due to material processes, and the body forces are identified 10

as momentum production in the body, with the actual and the potential gravitational components and the kinematical component

F body i = M P

i = M N i + M O

i + M K i

and thus the balance of momentum in elementary mechanics is rendered in the format 15

m d t v i = M D i + M P

i = M D i + M N

i + M O i + M K

i

as used in the present treatise. In case of solid bodies considered the diffu-sive term reduces to the micro-scale diffusion, in the limit molecular diffu-sion.

4.4.3.3 ANTI-COPERNICAN TURN 20

This balance may be split into the material component

0 i = M M i + M N

i ,

the momentum balance in the body fixed space, and the additional immate-rial component

m d t v a b

i = M O i + M K a b

i 25

in any other space in arbitrary relative translatory motion to the former.

Already at this stage it is evident that the body fixed observation space is the fundamental observation space. This insight will be referred to in catch-ing jargon as the 'anti-Copernican turn'.



The separation of the balance of momentum into its material and immate-rial components is nothing else but d'Alembert's principle, applied to a sin-gle body. And the material balance of momentum is nothing else but New-ton's fourth 'definition', while the immaterial balance of momentum is noth-ing else but the balance of force free motions of the reference mollusc con-5

sidered in celestial mechanics, all of them to be discussed in detail.

Concerning the terminology the following considerations are worth not-ing. In case of the term 'material balance of momentum' the 'alternative' term 'balance of material momentum' is not an alternative, but a non-sensical no-tion. There is no 'material momentum', so it cannot, it needs not be bal-10

anced. But in case of the term 'immaterial balance of momentum' the 'alter-native' term 'balance of immaterial momentum' might be an alternative. All momentum is immaterial, apparent.

4.4.3.4 INTENSITIES OF PRODUCTION

In terms of mass specific magnitudes, for short intensities, the momentum 15

balance becomes

d t v i = M M i

/ m + f P i = M M

i / m + f N

i + f O i + f K

i

with v i the mass specific momentum or the intensity of motion, and with

f i ≡ f X i ≡ M X

i / m ,

the (mean) mass specific momentum productions or intensities of momen-20

tum production in the body.

And thus the corresponding material and immaterial momentum balances in the body fixed space in terms of intensities are

0 i = M M i

/ m + f N i

and 25

d t v a b

i = f O i + f K a b

i ,

respectively.

The qualification 'immaterial' for the second equation has cautiously been chosen, although 'apparent' would have been quite appropriate, while Ein-stein did not subscribe to his. Two of the terms in the equation are evidently 30

apparent, while in case of the potential gravity this is not so evident. But in view of the anti-Copernican turn, referring to body fixed observation space, the intensity of the potential gravity is indeed 'apparent'.

The term 'mass specific momentum productions' implies that the intensi-ties of momentum production are the same for all bodies, solely depending 35

on their masses, their inertias, and not on other properties, e. g., the material of the body of matter under consideration or any other properties.



The terms 'mass specific momentum diffusion' or 'intensity momentum diffusion' cannot be introduced in the same way, as the diffusion is not re-lated to the mass of the body of matter. Defined that way both notions would be inappropriate and misleading as will be discussed in more detail in due course. 5

4.4.3.5 ELEMENTARY KINEMATICS

According to the momentum balance, the dynamic state equation, the (time) rate of change of the velocity, the acceleration, equals the sum of the mass specific forces as the rate of change of the location along the path equals the velocity according to the kinematical state equation 10

d t s i = v i .

Introducing the unit tangent

t i ≡ d s s i

and the modulus of velocity, often referred to as speed in the narrow sense

v ≡ d s / d t , 15

the velocity is obtained in the format

v i ≡ v t i

and thus the acceleration in the format

d t v i ≡ ∂ t v t i + v 2 c i

with the curvature of the path 20

c i ≡ d s 2

s i .

Introducing the unit normal with local radius

n i ≡ r c i

the acceleration is finally obtained in the format

d t v i ≡ ∂ t v t i + v 2 / r n i . 25

Accordingly the acceleration consists of two components: the acceleration tangential to the path and the acceleration normal to the path. In general the acceleration referred to a rotating reference frame is only a 'partial' accelera-tion to be complemented by the changes due to the changes of the reference frame 30

d t v i = ∂ t v i + ε i j k ω j v k .

If, and only if, the reference frame is defined by the tangent and the curva-ture the relations

∂ t v i = ∂ t v t i ,



ε i j k ω j v k = v 2 / r n i .

hold.

For convenience the equation of motion

∂ t v t i + v 2 / r n i = M M i

/ m + f N i + f O

i + f K i

may be split up into the two scalar component balances 5

∂ t v = (M M i

/ m + f N i + f O

i + f K i) t i

v 2 / r = (M M i

/ m + f N i + f O

i + f K i) n i .

All the implications of the abstract model up to this point, inadequately touched, if at all, in the treatise scrutinised, will be developed in detail in the present treatise. In the following further concepts are introduced necessary 10

for the discussion of elementary dynamics.

4.4.3.6 INVARIANCE PRINCIPLES

According to the classical principle of general relativity the balance of momentum can be set up in any observation space, e. g., in two spaces in arbitrary relative translatory motion to each other 15

d t v b

i = M M i

/ m + f N b i + f O b

i + f K b i ,

d t v a

i = M M i

/ m + f N a i + f O a

i + f K a i .

Due to the space invariance of the momentum diffusion, of the compo-nents of the gravitational momentum production and of the mass the differ-ence of the balances is 20

d t v b a

i = f K b a i = − d t v

a b i .

This kinematical 'balance' is not to be confused with the principle of local equivalence to be discussed. It is nothing else but the equation of kinemati-cal relativity

v b a i = − v a b

i . 25

The spatially uniform intensity of the apparent momentum production is not only equivalent, but is even identical with the acceleration of the body in the observation space.

Being strictly a 'matter' of kinematics this relation tells us nothing what-soever about dynamics, the physics of body forces, which 'cancelled' out of 30

consideration. But the rule of transformation

f O b i = f O a

i + f K b a i

implies, that the potential production of momentum is at least locally equi-valent to the apparent field of acceleration due to the acceleration of the ob-servation space relative to the body under consideration. 35



Although freely moving bodies are not of primary interest in the present treatise free motions will be considered next. As Newton convincingly dem-onstrated in the first book of the Principia dealing nearly exclusively with free motions, this can be done 'before' a theory of momentum production, of gravity in particular is being developed. 5

As a matter of fact no satisfactory theory has been proposed in the past three hundred years. How such a theory in the context of classical dynamics might look like will be discussed in detail.

4.4.4 Freely moving bodies

4.4.4.1 PRINCIPLE OF LOCAL EQUIVALENCE 10

" … der glücklichste Gedanke meines Lebens."

Albert Einstein (Pais, 1983/177 ff).

If the motions of the small body under consideration are not constrained, if the momentum diffusion vanishes

M M i = 0 i , 15

the material momentum balance reduces to

0 i = 0 i + M N i

in the body and the additionally immaterial balance in any other space re-duces to

m d t v a b i = M O

i + M K a b i , 20

or, in terms of intensities, for short

d t v i = f O

i + f K i ,

i. e. in components

∂ t v = (f O i + f K

i) t i

v 2 / r = (f O i + f K

i) n i . 25

Thus in the body fixed observation space the intensities of potential gravi-tational and kinematical momentum productions balance each other

0 i = f O i + f K

i .

This theorem is called 'principle of local equivalence'.

Thus unconstrained motions of bodies of ponderable matter take place 30

• without any material momentum diffusion and production into and in the body, respectively,



and according to the above theorem

• on curved paths, in general,

• with the total immaterial intensity of momentum production vanish-ing in the body fixed observation space.

Long before Einstein enjoyed 'den glücklichsten Gedanken seines Lebens' 5

classical general relativity has been subject of amusing party talk at Oxford (Carroll, 1988/311-313; complete quotation to follow).

4.4.4.2 REFERENCE MOLLUSC

The equation of motion

d t v i = f O

i 10

in the observation space defined by the condition

d t v i = − f K

i

is thus the equation of force free motions of small bodies of matter as parts of the reference mollusc. This 'equation of general relativity' is not a strictly kinematical relationship, although no forces take place at and in the body. 15

The reference mollusc represents the inertia of the whole universe and the implication of limiting considerations to small bodies is that their inertia does not change the motions of the reference mollusc.

Accordingly the equation of general relativity may be said to be the equa-tion motions of the reference mollusc. The author has not seen any exposi-20

tion clearly stating this essential content of the principle of local equiva-lence. But a caveat is in place here in order to protect readers from miscon-ceptions.

The dynamical state equation is a first order differential equation and the kinematical state equation is another first order differential equation, Thus 25

the 'motion' of a body, its path in the observation space depends on two 'ad-ditional', arbitrary parameters, the initial conditions of the motion under consideration.

Usually the 'equivalence' of 'inertial' and 'heavy' mass is being ritually in-voked, although it remains obscure where these obsolete concepts originated 30

and how they may be interpreted operationally, details to be discussed in considerable detail for historical interest.

4.4.4.3 INERTIAL SPACES

Observation spaces in which freely moving bodies experience no momen-tum production and consequently no acceleration are traditionally called 'in-35

ertial' spaces. Thus any observation space moving at constant translational speed relative to the body is locally an inertial space as well.



In this case the equations of motion reduce to

∂ t v = 0 ,

v 2 / r = 0 .

Thus the bodies of matter move at constant speed on straight paths, on paths with vanishing curvature. 5

Today everybody has seen TV coverages of material objects freely float-ing in satellites, being themselves small bodies providing inertial observa-tion spaces. The freely floating objects provide ideal cases for philosophical studies into the 'inertia principle', the small curvatures of their paths and other 'disturbing' effects being practically negligible. 10

Motions on straight paths may be considered as limiting cases of circular motions

∂ t v = 0 ,

v 2 / r = const .

Ideally circular motions have played a prominent role in the history of me-15

chanics, often strictly speculative, later in Copernicus' harmonia mundis and finally implied as special cases by Kepler's laws and by Newton's theory of gravitation. The more general case of 'central' motions on conic sections tak-ing place in potentials of single masses will be discussed in due course.

4.4.4.4 BODIES FALLING JOINTLY 20

If the intensity of momentum production in one body of matter is locally uniform in a more or less extended region of an observation space 'given' or chosen, the theorem on freely moving bodies formally implies that all other (small, non-interacting) bodies, at rest relative to the 'first' body in the ob-servation space fixed in the 'first' body, move or 'fall' in exactly parallel 25

ways in such regions of uniform intensity of momentum production

d t v i = f O

i + f K i ,

not necessarily invariant in time. In these cases of practically uniform inten-sity of momentum production the effects of the latter cannot be distin-guished from those observed in an accelerated observation space. 30

Careful experiments, already performed by Newton with 'pendulous bod-ies' at the surface of the Earth, have confirmed this implication of his axio-matic theory and the implication of the term 'mass specific momentum pro-duction' suggested earlier and to be considered again. These experiments thus confirm the adequacy of the axiomatic theory for the purposes at hand. 35

As in case of kinematical relativity this relation tells us nothing what-soever about dynamics, the physics of body forces, gravity forces in particu-



lar to be considered next. The magnitudes (Grössen) on both sides of the immaterial balance in terms of intensities are locally, in the small body 'equivalent' and the theorem is accordingly called principle of local equiva-lence.

The first and the last magnitudes in the balance are apparent, spatially 5

uniform kinematical magnitudes depending on the observation space cho-sen. The second magnitude is at least locally approximately uniform de-pending, but this does not imply that the intensity of potential momentum production is apparent (Einstein, 1997/46).

In the general case of non-uniform intensity of momentum production, in 10

the presence of 'tidal' forces, the paths of bodies falling freely in vacuo will not be parallel any more, but the distances of the bodies will be changing as the bodies fall. These effects permit to distinguish intensities of momentum production from translatory accelerations even in closed 'elevator' cabins (Pössel, 2005/104 ff). 15

Falsely attributing these changes in distance between falling test bodies to gravitational interactions between the test bodies leads directly to the wrong 'consequence' that gravity is a phenomenon due to tidal forces and thus due to the curvature of Einstein's four-dimensional space-time. The 'relativistic' phenomena, accounted for in Einstein's field equations, are secondary ef-20

fects not subject of classical mechanics.

4.4.5 Constrained motions

4.4.5.1 MATERIAL MOMENTUM PRODUCTION

In case of constrained motions the complete balance of momentum in a body fixed observation space is 25

0 i = M M i

/ m + f N i .

explicitly showing that the momentum production

M N i

≡ M P b i

and its intensity in the body fixed space

f N i ≡ f P b

i 30

are material as are the momentum diffusion and the inertia.

The material forces balancing each other in the fundamental space, body fixed observation space

0 i = M M i + M N

i



in accordance with Newton's 'Definition' IV and d'Alembert's principle are called nearly universally 'lost forces'. This name is felt to be completely mis-leading, indicating that the nature of these forces has not been understood.

These forces are the only material forces 'taking place' at and in the body. Thus the case of vanishing forces, of unconstrained motions in an inertial 5

space, in the treatise scrutinised claimed to be ontologically particularly in-teresting, is the least interesting of all.

Traditionally the material momentum production in the body is called its gravity and the intensity of momentum production in the body is called gravity field strength in the body. The balancing momentum diffusion into 10

the body is called the supporting force. If the support system is calibrated it is called a dynamometer permitting to measure the material momentum pro-duction

M N i = − M M

i = M M dyn i .

according to Newton's third Law and fourth Definition. 15

In the observation space fixed at the surface of the Earth the gravity of the body is called its 'weight' and denoted by

W i ≡ M N i ≡ M P b

i .

while the intensity of gravity in the body, denoted by

g i ≡ W i / m , 20

is falsely called the 'acceleration of free fall' due to the immaterial mass spe-cific balance

d t v i = g i

in case of unconstrained motion.

4.4.5.2 STEADY MOTIONS 25

If the surface forces acting on the body, the momentum diffusion across its surface, and of body forces acting in the body, the momentum production in the body in the observation space we happen to be in, balance each other, the 'net force'

M M i + M P

i = 0 i 30

vanishes and the momentum balance reduces to

d t v i = ∂ t v t i + v 2 c i = 0 i .

the body moves at constant, maybe vanishing speed. As has been noted over and over again by many scholars, Euler is being quoted in a footnote (2000/300), this is a theorem, the theorem on steady motions in any given 35

observation space.



In this case the condition of vanishing tangential acceleration implies vanishing curvature of its path. Newton's first law may also be read in this sense (PM/13):

"Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon 5

it." Italics: MS.

In this sense the theorem does not need a derivation and certainly not a pseudo-scientific 'proof'; it is evident, as even the uninitiated will 'see'. Ex-amples are airplanes taking off.

4.4.5.3 'TRÄGHEITSGESETZ' 10

The motions are not the 'eigen-motions' the body performs 'für sich' (2000/157), the 'Einheitsbewegung' the philosopher and others are singling out (2000/301). They claim that the special case of surface forces and body forces vanishing separately is 'ontologically' different and refer to Newton's lex prima as 'inertia principle'. 15

And, most importantly, they further claim that only after Newton's first law has been 'firmly' established the concept of forces, causing changes of 'eigen-motions', can be introduced in a satisfactory fashion (2000/301-302). As has already been demonstrated this point of view is based on obsolete 'metaphysical' conceptions, forgetting Hilbert's lessons. The notion of 'mo-20

tion in itself', 'Bewegung an sich' is non-sensical.

In axiomatic systems the basic concepts are jointly 'defined' by the axi-oms, 'implicitly' as Hilbert said or 'coherently' as the author prefers to say, in the present case essentially by the momentum balance. The step-by-step in-troduction of concepts, 'preferably' by separate, incoherent definitions in 25

Euclid's fashion, 'gemäß der Euklidischen Form', also found elsewhere in various guises, is obscuring the essential issues.

Not only according to the principle of relativity, but already in accordance with d'Alembert's principle the body fixed observation space is the preferred space. In this space the momentum balance is simply 30

0 i = M M i + M N

i

and as Newton states in his fourth 'definition', which is in fact a theorem, a consequence of the principle of general relativity, in agreement with obser-vations, the material body forces vanish with the (material) surface forces, the constraints. This fact will be the basis of a model of matter and of the 35

theory of inertia and of gravity to be developed here and in detail in due course.

In the detailed discussion the separation of the momentum balance into its material and immaterial components will be shown to be a consequence of



the principles of relativity, objectivity and materiality relating the abstract theory with reality.

4.4.5.4 EVALUATIONS/ASSESSMENTS

In a nutshell this sketch of abstract elementary mechanics provides a con-ceptual framework and a wealth of insights of considerable interest and im-5

portance in the following operational and physical 'interpretations', which again will only be sketched at this stage.

4.4.6 Operational interpretation


Only after the preceding elaborations the original problem stated, the op-10

erational interpretation of the concepts (Lopes Coelho, 2000/1), can be dis-cussed in an intellectually satisfactory fashion.

Instead of the very involved traditional, 'static' procedure of setting up weight 'scales', calibrating dynamometers etc the most elegant way to cali-brate the dynamometer, measure the mass of bodies etc is a 'kinetic' proce-15

dure, also avoiding the intricacies of collisions of bodies, the tests often considered in philosophical treatises.

4.4.6.2 ABSTRACT MODEL

For the present purpose, for the exposition of the essentials the dyna-mometer is assumed to be an ideal, mass-less, isotropic spring system with 20

linear stiffness.

In an observation space with time and space invariant momentum pro-duction in the body the material momentum balance of the system

0 i = − c x 0 i + W i

results in an equilibrium elongation. Superimposed are oscillations accord-25

ing to the immaterial momentum balance

m d t 2 ∆ x i = − c ∆ x i .

Transformation of the latter into the frequency domain

(− m ω 2 + c) ∆ x F i = 0 i ,

results in the eigen-value equation 30

− m ω 2 + c = 0

with the circular eigen-frequency or 'natural' frequency of the spring-mass system

ω ≡ 2 π / T



in terms of its eigen-period.

The eigen-value equation is the basis of extremely simple rules for the following fundamental operations.

4.4.6.3 PROTO-MECHANICS

For the purpose at hand the following units of measurement can be estab-5

lished. Linking them up with the legal units is not subject of this exposition, as it does not contribute to the clarification of the fundamentals.

Taking the mass of a given body of matter as reference mass, its mass m 0 adopted as mass etalon, as unit of mass measurement, and a given spring as reference dynamometer the spring-mass system can be used as reference 10

clock

ω 0 2 = c / m 0 .

the eigen-period T 0 adopted as time etalon, as unit of time measurement.

Due to the fact that the phenomenological parameters, the mass and the so far unknown stiffness of the dynamometer are space invariant the eigen-15

period of the system is space invariant as well, independent of the observa-tion space we happen to be in. Thus, different from pendulum clocks, these clocks even work in inertial spaces.

This procedure is in accordance with the principle of objectivity requiring that the whole theory and its interpretation have to be independent of the 20

choice of the units of measurement. And this requirement is met, if the the-ory is unit-free, the necessary condition being stated in Buckingham's Π-theorem.

4.4.6.4 INTERPRETATION

Having established the units of mass and of time measurement the stiff-25

ness of the ideal reference dynamometer is obtained

c 0 = m 0 ω 0 2 = m 0 (2

π / T 0) 2 .

In practice the dynamometer will be calibrated by using the legal mass and time scales; but, to repeat, this is not the problem under discussion.

A second dynamometer with the same stiffness can now be used to com-30

pare masses, to 'measure' other masses

m = c 0 / ω

2 = m 0 T 2 / T 0

2 .

The ratio of masses equals the ratio of the squares of the eigen-periods.

Finally the reference system can be used to measure the mass specific weight or momentum production in the body in the given observation space 35

g i ≡ W i / m = x 0

i ω 2 .



Taking any linear extension of a given solid body of matter as reference length, its length L 0 adopted as length etalon, as unit of length measure-ment, the static elongation

e i = x 0 i

/ L 0

becomes and thus the local mass specific momentum production, falsely 5

called 'acceleration of free fall'

g i = e i L0 (2 π / T 0)

2 .

So it is not necessary to measure the acceleration of a freely falling body ac-cording to the equation

d t v i = g i ≡ W i / m . 10

The acceleration equals the mass specific weight in value, but both are es-sentially different concepts, wesensverschieden.


Thus the problem posed at the beginning of the treatise scrutinised has been solved, at least in principle as required for the present purpose. We are 15

not stuck in a circulus vitiosus; we have pulled ourselves out of the morass as elegantly as Münchhausen did.

The purpose of this detailed exposition is to show by way of an up-to-date professional approach the intricacy of the problem and the underlying hier-archy of theories required for its solution. Suppes' remarks concerning this 20

problem have already been quoted (1983.d).

The exposition has shown that any attempts to solve the problem 'inside' mechanics or to find a solution by way of 'philosophical hermeneutics', even if that would meet decent standards, are doomed to fail.

4.4.7 Universes, reference molluscs 25

"Napoleon remarked that Laplace had not mentioned God in his work [the 'Méchanique Celeste' dedicated to Napoleon] and to which Laplace replied: 'Sire, I had no need for the hypothesis.' Lagrange on hearing Laplace's reply observed: 'But it is a beautiful hypothesis with 30

which many things can be explained away.' "

Jean Louis Lagrange (1997/XXIX).


The equation of unconstrained motions is not particularly interesting on terrestrial scale, where motions are governed by surface forces, i. e., by sur-35



rounding bodies, may be fluids, but the equation governs celestial mechan-ics with 'phantastic' accuracies compared to those attainable in terrestrial mechanics. Still, Newton carefully mentions resistance on celestial scale in the explanation to the first law (PM/13):

"The greater bodies of the planets and comets, meeting with less resis-5

tance in freer spaces, preserve their motions both progressive and circular for a much longer time."

And again there are no universes with only one body; the motions of many bodies have to be considered, the motions of Einstein's 'reference mol-luscs' (Bezugsmollusken), the most prominent example in the history of as-10

tronomy being our planetary system. It is noted expressis verbis that the catching name does not imply that molluscs proper are moving freely in the sea, without surface forces acting at their skins.

The free motions of reference molluscs are 'neutral', forceless in accor-dance with the conceptions of Aristotle, Galilei and Einstein and with the 15

instinctive belief still shared by most people not exposed to classical me-chanics distorted by 'Stille Post' over three centuries. Einstein's return to the pre-Copernican point of view has been named the anti-Copernican turn.

Referring to reference molluscs is not particularly convenient, thus in practice observation spaces at rest in dominant bodies are chosen, on planet-20

ary scale, e. g., the Sun, in satellite navigation, e. g., the Earth. The 'disad-vantage' of the freedom of choosing arbitrary observation spaces is that 'ap-parent' intensities of momentum production have to be accounted for, in the case of interest the immaterial potential gravity.

4.4.7.2 NEWTON'S LAW OF GRAVITATION 25

According to Newton's law of gravitation 'small' bodies of reference mol-luscs 'follow' the gradients of the mass potential

d t v i = f O i

= − G ∂ i u m .

Gradients of the mass potential are causing intensities of momentum pro-duction and thus momentum production in bodies of matter. 30

That bodies 'attract each other' is 'standard' jargon, though grossly mis-leading generations of scholars and students. Newton himself carefully avoided this terminology (CohenIB, 1999/61 f). Further, insiders will notice that the obsolete concepts of 'inertial mass', 'heavy mass' and 'gravitational mass' are not being referred to. 35

Thus in general bodies of reference molluscs move on curved paths

c i = f O i

/ v 2 ,



the curvature being proportional to the momentum production caused by the gradient of the mass potential and inversely proportional to the square of the speed along the path.

Contrary to the opinion widely held mass specific momentum production, body force fields outside bodies of ponderable matter do not exist, they 5

'cannot' exist, neither conceptually, momentum production in 'empty' space is non-sensical, nor physically, there is no way to prove their existence, any probe is another body of matter.

Accordingly Einstein states very cautiously, explicitly referring to the 'force of attraction' (Anziehungskraft) acting at a probe in the vicinity of a 10

sphere, maybe the Sun, (1950/152 f):

"Die Strahlen in unserer Skizze sind die Kraft l inien des Schwere-feldes. Die Kraftlinien können nämlich im leeren Raum konstruiert wer-den, ohne daß Materie vorhanden zu sein braucht, und vorläufig zeigen sie alle – oder, wie man auch sagen kann -, zeigt das Feld lediglich an, wie 15

sich ein Prüfkörper verhalten würde, den man in die Nähe der Kugel [, des Körpers] brächte, deren Feld wir konstruieren." Italics: MS.

Most surprisingly Einstein does not further elaborate on this fundamental observation and does not exploit its full potential, but abruptly changes the micro-universe of discourse (1950/154): 20

"Auf das Gravitationsproblem wollen wir nun allerdings noch nicht ein-gehen. Es sollte uns nur als Einführung dienen und das Verständnis ande-rer, ähnlicher Überlegungen aus der Elektrizitätslehre erleichtern."

4.4.7.3 MACH'S PRINCIPLE

According to the instinctive belief of human beings since times unknown, 25

all bodies in the universe are parts of 'a whole', 'the whole'. This belief has in recent time been called Mach's principle. Einstein referred to all bodies of the universe as the reference mollusc.

The model of 'the whole' adopted in the context of this treatise, the micro-universe of discourse being classical dynamics, is the mass potential due to 30

the mass distribution in space

U m = − ∫ V ρ dV / r = − ∫ m dm / r ,

which is free of rotation and obeys Poisson's partial differential equation

∂ i ∂ i u m = − 4 π ρ .

Due to the motions of the reference mollusc the mass potential is not sta-35

tionary, even if the observation space is chosen in the 'dominant' body. As a consequence the kinetic energies of the individual bodies are not invariant and as one consequence the planets and any other satellites are not moving on perfectly elliptical paths in 'central' mass potentials.



The prominent example is the path of Mercury. The total observed pre-cession of its perihelion, 5600" per century, is accounted for by Newton's theory of gravitation to 99.23 % . The unexplained 'rest' of 43'' per century has been explained by Einstein's theory of general relativity taking into ac-count the curvature of the four dimensional space-time. 5

Thus Newton's theory perfectly serves the purposes of planetary astron-omy and satellite 'navigation', provided all relevant bodies of the reference mollusc are taken into account, as is standard practice in astronomy and in satellite tracking systems.

The mass potential may conveniently be considered as the 'aether' postu-10

lated by Einstein in his lectures at Leyden in the 1920th, later abandoned by him (1920/15):

"Gemäß der allgemeinen Relativitätstheorie ist ein Raum ohne Äther un-denkbar; denn in einem solchen gäbe es nicht nur keine Lichtfortpflanzung, sondern auch keine Existenzmöglichkeit von Maßstäben und Uhren, also 15

auch keine räumlich-zeitlichen Entfernungen im Sinne der Physik."

Recently the physics of the 'vacuum has become an important topic of re-search' (Pais, 1983/288). In the treatise scrutinised only Einstein's Lecture Notes of 1909-1910 and a Draft of the theory of general relativity of 1913 are listed (2000/324). More recently Laughlin has referred to the physics of 20

the mass potential, the aether, alias 'vacuum', in his 'Abschied von der Welt-formel' (2007).

While in Einstein's theory of general relativity the mass distribution 'con-stitutes' space, in classical mechanics the mass potential together with its singularities and the geometrical space are conveniently distinguished. Even 25

on planetary scale the curvature of Einstein's four dimensional space-time is negligible for most practical purposes; the universal time and plane tangent space and Euclidean stereometry are serving most purposes.

The curvature of Einstein's four dimensional space-time should not be confused with the curvature of the mass potential, although Einstein's ideas 30

have reportedly been inspired by the latter (Laughlin, 2007/185, quotation to follow).

4.4.7.4 PHENOMENOLOGY

According to the preceding exposition the local intensity of gravitational momentum production and the local gradient of the mass potential are the 35

same for all bodies. Consequently the phenomenological parameter in New-ton's law, called 'constant of gravitation', is a 'universal' property of ponder-able matter. Its value can be determined in accordance with the equation of unconstrained motion only in laboratories by simulating astronomical situa-tions (Pohl, 1947/43). 40



The simple equation for the mass specific weight at distance r from the centre of a homogeneous spherical central body of mass m, and thus for the acceleration is

d t v = − G m / r 2 .

Based on this equation Pohl has derived the order of magnitude of the con-5

stant of gravitation in one of his famous lecture hall experiments (1947/44).

Measurements of Cavendish and followers have confirmed that the con-stant of gravitation is a 'universal' property of ponderable matter independ-ent of the sizes of the bodies and the materials they consist of. In due course this subject will be further discussed. 10

4.4.7.5 PROPOSED CONVENTION

The proposal to adopt the convention

G = 1

and obtain the mass

m = − r 2 d t v 15

in units L 0 3 / T 0

2 without measuring forces, is not particularly interesting and convenient (Pohl, 1947/44 f). As has been shown masses can be com-pared without measuring forces much more easily.

But the value of the constant of gravitation may be standardised, as has the value of the speed of light in vacuo. According to the latter standard the 20

speed of light can no longer be measured, it is the measure. Standardising the constant of gravitation might resolve the problem of replacing the Ur-kilogramme as presently attempted following two different approaches, de-tails to be discussed.

4.4.7.6 EVALUATIONS/ASSESSMENTS 25

According to Newton's law of gravitation momentum production taking place in bodies of ponderable matter due to the presence of distant bodies is caused by gradients of the mass potential at the locations of the bodies. The universal phenomenological constant of gravitation 'describes' how ponder-able matter 'reacts' with momentum production on gradients of the mass po-30

tential.

4.4.8 Theory of gravitation

"Raffiniert ist der Herr Gott, aber boshaft ist er nicht. Subtle is the Lord, but malicious He is not." Italics: translation A. Pais. 35

Albert Einstein (Pais, 1982/113).




Any theory of gravitation worth that name has to be a theory of the mac-roscopic constant of gravitation. It has to elucidate the mechanism of inertia and gravity and finally to explain why it is a 'universal' property of ponder-able matter in terms of models of matter. As has been shown anything else 5

is understood in classical mechanics in the spirits of Newton, Euler, d'Alembert and Einstein.

4.4.8.2 GLOBAL MODEL OF MATTER

So far Newton's theory of gravitation has been linked up with the theory of general relativity. Further, his fourth 'definition' suggests a simple model 10

of matter permitting to link it up with the standard model of particle physics as well.

Newton's fundamental observation of material momentum diffusion and production vanishing together permits to conceive a solid body of ponder-able matter as a massless isotropic spring system of very large stiffness c agg 15

with the aggregate mass

m agg = m ,

suspended at the centre of mass, the mass 'condensed', concentrated with very high mass density ρ agg in a very small volume.

The aggregate mass is a singularity of the mass potential 20

∂ i ∂ i u m = − 4 π ρ agg .

and thus the balance of momentum

0 i = M M i + M N

i = − c agg x i − m agg G ∂ i u m

after elimination of the deflection of the spring system reduces to the equa-tion 25

0 = − ω agg 2 + G 4 π ρ agg

for the constant of gravitation

G = ω agg 2 / (4 π ρ agg)

with the circular eigen-frequency of the system

ω agg 2 = c agg / m agg . 30

This global model of matter does 'of course' not permit to identify the constant of gravitation of ponderable matter. But already at this stage every-body with the slightest bit of curiosity and imagination will guess how this might be done. But before that is being sketched the implications of the global model are exploited. 35



4.4.8.3 WEIGHT, INERTIA EXPLAINED

The aggregate model of matter described permits to 'explain' the phenom-ena of weight and inertia. If the body moves unconstrained the singularity moves as part of the reference mollusc, the spring system is not deflected, no material forces 'take place'. 5

But if the body is constrained, not permitted to follow its free motion as part of the reference mollusc, the spring system is deflected. According to the momentum balance in the body fixed observation space

m d t v b

i = 0 i = M M i + M N

i = − c agg x i + W i

the weight of the aggregate mass is balanced by the supporting spring sys-10

tem assumed to be linear in the range of interest.

Observing the same set-up in the space accelerated relative to the body so that the apparent momentum production balances the weight

f O a b i = d t v

a b i = − W i

/ m

the momentum balance is 15

m d t v a

i = − c agg x i + 0 i ,

inertial resistance, inertia is experienced, the material deflection of the spring system remaining exactly the same as before.

The important observations are that

• the phenomena of weight and inertia only occur in bodies con-20

strained in motion,

• they thus occur only if bodies are in contact with each other,

• the 'physics' of the phenomena are identically the same,

• the 'difference' in phenomena is a matter of the observation space chosen. 25

4.4.8.4 EINSTEIN'S EXPLANATIONS

" …, mit denen man sich auch lange zufrieden gegeben hat."

Rudolf Carnap: Der Raum (1922/35).

The fact that a given 'property' has the same value for all bodies may be 30

explained in two ways. One way has been discussed by Carnap using as an example the changes in the distance of benchmarks with the temperature of the bench (1922/35 f).

Instead of accounting for the temperature changes of the bench one could talk of changes in the lengths of the objects measured depending on the dif-35

ference in temperature between bench and object:



"… also eine Fernwirkung, grundsätzlich nicht widersinniger als die elektrostatische und die der Schwerkraft, mit denen man sich auch lange zufrieden gegeben hat." Italics: MS.

What is 'widersinnig' and has been felt satisfactory for a long time will be seen immediately. 5

The second half of the paragraph reads:

"In unserem Beispiel liegt die Sache so: bei der Messung verschiedener Körper, die im Temperaturgleichgewicht sind mit dem erwärmten Maßstab, zeigt sich, noch bevor man überhaupt daran denken kann, jene Wirkung in ein Naturgesetz zusammenzufassen, der auffällige Umstand, daß nicht nur 10

bei allen Körpern jene Fernwirkung eintritt, sondern daß sie sogar zahlen-mäßig bei allen die gleiche ist, gleichgültig aus welchem Stoff sie bestehen. Hier kommt folgender Grundsatz des wissenschaftlichen Verfahrens zur Geltung: Zeigen inbezug auf einen Vergleichskörper die andern Körper bei aller sonstigen Verschiedenheit in irgend einer Beziehung ein zahlenmäßig 15

gleiches Verhalten, so ist zur Vereinfachung der gesetzmäßigen Darstellung zu versuchen, diese Übereinstimmung als nur scheinbar hinzustellen, da-durch daß dem Vergleichskörper das entgegengesetzte Verhalten beigelegt wird. Dieser Grundsatz, ein Sonderfall des Machschen Grundsatzes der wissenschaftlichen Sparsamkeit, ist es, der den Auffassungen der Erddre-20

hung, der Erdbewegung um die Sonne, der Bewegung der Sonne inbezug auf die Fixsterne gegenüber den älteren, entgegengesetzten Auffassungen den Vorzug gibt. Derselbe Grundsatz in andrer Wendung hat auch, ange-sichts der Tatsache der gleichen Fallbeschleunigung für alle Körper, zum Einsteinschen Äquivalenzprinzip der Schwerkraft geführt. Dieser Grundsatz 25

nun veranlaßt uns, die [eine] Maßsetzung …, der [anderen] … vorzuziehen. Aber, und darauf liegt hier unser Augenmerk, die Erfahrungstatsachen können uns nicht dazu zwingen. In diesem Sinne ist die Wahl der Maßset-zung frei und unabhängig von der Erfahrung; nicht aber ist die Wahl will-kürlich, sondern sie wird durch Grundsätze, ähnlich dem angeführten ge-30

leitet und kann dabei die Erfahrungstatsachen berücksichtigen." Italics: MS.

As it happens, Carnap's opinion, as far as gravity is concerned, is felt to be based on fundamental misconceptions, invalid reasoning and ignoring parti-cle physics. 35

Material gravity is not apparent, it is not an action at a distance (Fern-wirkung) without mediation, but due to gradients of the mass potential, the physics of which after all is subject of intense current research. 'Zur Verein-fachung der gesetzmäßigen Darstellung zu versuchen, diese Übereinstim-mung als nur scheinbar hinzustellen' is not in accordance with Newton's 40

fourth 'definition', the most fundamental observation concerning material forces, resulting in bodies of ponderable matter.



4.4.8.5 PARTICLE MODEL OF MATTER

Consequently the empirical fact that the phenomenological, material con-stant of gravitation is a 'universal' macroscopic property of ponderable mat-ter must have another explanation. The simplest proposal is to base it on the universal structure of ponderable matter. 5

The starting point of this approach is the global model of bodies described along with its implications. Evidently the 'ersatz' model applies to any mac-roscopic part of the bodies of ponderable matter. Still, such 'more local' models are as insufficient as the global model.

Further 'down' in scale, 'through' the molecules and the atoms and their 10

nuclei, the search for the singularities of the mass potential ends in the nu-cleons 'containing' the quarks, the mass singularities, suspended in gluons, massless spring systems.

This standard particle model of the structure of nucleons, not a mechanis-tic model, is a very close, not so say a perfect analogue of the global model, 15

first described by the author right before he stumbled over a pictorial sketch of the structure of protons (Klanner, 2001). The details and the minute dif-ferences between protons and neutrons will become of interest to physicist as soon as they can measure the constant of gravity with the precision nec-essary. 20

4.4.8.6 'GRAND UNIFICATION'

Everybody having followed the preceding exposition will immediately start speculating about the 'grand unification' of forces, since Einstein's un-successful attempts still one of the unsolved problems of physics. The ques-tion is of course what did Einstein try and why? 25

The model of matter developed in accordance with Newton's axiomatic theory and Einstein's theory of the force free motion of the reference mol-lusc suggests that there is no 'force' of gravitation on the macroscopic level of ponderable matter and further down, which could be united with other forces of physics. 'Are physicists barking up the wrong tree' concerning 30

gravitation, as Greene has suspected? In view of the preceding analysis this is no longer a vague, unqualified suspicion.

In order to 'derive' the constant of gravitation according to the rule stated for the global model the low frequency behaviour of the particle systems in the nucleons have to be studied. Destroying protons in colliders of ever in-35

creasing power can safely be discarded as the wrong approach, as is killing living creatures in search of their souls. 'Further down' the search for souls on the quantum level has recently been revived.

Most of the particles identified by physicists are unstable. Only free pro-tons are stable, even 'free neutrons are unstable and decay after a lifetime of 40



about 15 minutes into a proton, an electron, and an anti-neutrino' (NIST Neutron Lifetime Experiment). Only if neutrons are part of nuclei of atoms they are stable. More recent accounts of the research on the structure of nu-cleons have been published by BassSD and Samulat (2008), by DürrS et alii (2008) and by Spillner (2009). 5

Concerning the standard model of matter, which is far beyond the horizon of the present treatise, a recent paper discussing 'Die mathematische Zäh-mung des Standard-Modells' is of interest (Bär, 2009). Its abstract reads as follows:

"Die moderne Theorie der Elementarteilchen, die Quanten-Yang-Mills-10

Theorie, vermag die experimentellen Befunde mit unerhörter Genauigkeit wiederzugeben, es fehlt ihr jedoch bisher ein solides mathematisches Fun-dament. Über dessen Gestalt gibt es nur sehr nebelhafte Vorstellungen." Italics: MS.

This abstract invokes the layman's question: Are not the 'nebulous concep-15

tions' concerning physics rather than the mathematics of the model?


Classical mechanics does not only link gravitation to the whole universe according to Mach's principle and Newton's law of gravitation, but also to the standard model of matter in accordance with Newton's fourth 'definition' 20

and d'Alembert's principle, the fundamental observation concerning the ma-terial momentum diffusion and production balancing each other in bodies constrained in their free motions as part of the 'reference mollusc'.

According to the model proposed gravity of ponderable matter is due to the natural frequency of the quark-gluon systems in the nucleons 'governed' 25

by the strong forces on that scale. As will be outlined in due course this model, in getting rid of the 'unbelievable' concept of a gravity field offers dramatic advantages in understanding the world we live in.

4.4.9 'Zu Ende gekommen'

4.4.9.1 PROBLEM/MODEL/GOAL 30

In order to demonstrate in detail the non-sensical nature of the exposition of Lopes Coelho a sketch of classical mechanics has been provided based on the works of Newton, Euler and d'Alembert, which have also been referred to in the treatise scrutinised.

4.4.9.2 'BILANZ' 35

In concluding this 'appraisal' the last page of the third, the final chapter 'Bilanz' is quoted just for ready reference without further explicit analysis



for those wanting to use the solution proposed for problems, which are none (2000/318):

" … Der Begriff stimmt mit den Meßverfahren überein und wirft sogar ein Licht auf sie hinsichtlich ihrer konzeptuellen Anpassung an die Erfah-rung. Vom Begriff leitet man auf eine klare und logische Weise die Grund-5

gleichung der Mechanik ab, über welche Einstimmigkeit unter den Physi-kern herrscht.

Aus der Geschichte des Kraftbegriffs werden im ersten Kapitel verschie-dene Auffassungen, Schwierigkeiten und Kritiken dargestellt. Derjenige Teil der Begriffe, der die Systematisierung der Bewegungen in der Theorie 10

oder die Behandlung der mechanischen Probleme betrifft, wird durch die vorgeschlagene Konzeption subsumiert; der andere Teil, der die Rechtferti-gung der Phänomene betrifft, z. B. zu sagen, es gebe innewohnende Kräfte, die Kraft sei die Ursache der Veränderung der eigenen Bewegung des Kör-pers usw., wird dagegen aus der Konzeption ausgeschlossen. Diesen letzten 15

Teil beiseite zu lassen und jenen anderen einzuschließen, scheint positiv zu sein, und erweist sich auch so, denn die Schwierigkeiten zur Kraft und die-jenigen, die mit dem Trägheitsgesetz zusammenhängen, verschwinden, und damit auch die auf ihnen beruhenden Kritiken. All dies scheint darauf hin-zuweisen, daß die von Poincaré ausgesprochene Schwierigkeit, eine befrie-20

digende Idee der Kraft zu geben, überwunden wird. Das Kraft-Problem, mithin die Frage nach der Konzeption der Kraft, scheint so zu Ende ge-kommen zu sein, und damit auch die vorliegende Arbeit." Italics: MS.

What has come to an end is hopefully this type of non-sensical 'philosophy'.

How is it possible, 'von einem Begriff auf eine klare und logische Weise 25

die Grundgleichung der Mechanik abzuleiten'? In conclusion the whole ex-ercise scrutinised is felt to be quite 'incredible' even by the most elementary standards and even if one forgets, that it has been published about hundred years after Einstein published his theory of general relativity, as has been shown in essential parts to be explain in terms of classical mechanics. 30

In large parts the work is not meeting the state of the art neither in ele-mentary philosophy and hermeneutics, nor in elementary axiomatics and lo-gics, nor in elementary mechanics. It 'simply' contributes to the bad reputa-tion of this sort of 'philosophy', as the example explicitly shows not just a cliché among those practising mechanics, provided they care to take notice. 35

4.4.9.3 CLASSICAL MECHANICS

Already the crude sketch of the 'picture' (Hertz) of classical mechanics shows that many 'theoretical' and 'philosophical' expositions concerning this matter are ridiculous caricatures based on plain superstition and/or on lack of comprehension of elementary mechanics and the classical literature. 40

'Up to' the planetary scale the theory of general relativity is nothing else but classical mechanics as developed by Newton, Euler and d'Alembert, re-



discovered by Einstein without always referring to the classical works and their implications.

The ritual repetitions of Einstein's formulations clearly indicate that they have not been understood. As Popper has noted, the frequently held view that the fundamental models are incomprehensible, in particular those of 5

general relativity, indicates that the problems and their solutions have not been understood.

The theory of gravitation in accordance with Newton's law of gravitation, with his observation concerning the balance of the material forces and with the standard model of particle physics may be of interest to physicists, may 10

be a stepping stone for understanding Einstein's achievements even beyond classical mechanics, the micro-universe to which the present discourse is limited.


The exercise has shown once again that people who have not worked in 15

mechanics should better refrain from talking about mechanics. Or as Boethius said: 'O si tacuisses, philosophus mansisses.'

What has been done in this sketch and will be done in the following much more detailed exposition is to establish a coherent model of classical me-chanics that philosophers can and, hopefully, will scrutinise instead of fur-20

ther 'analysing' caricatures of what they believe to be classical mechanics.

In developing that model the author explicitly responds to the request stated by Mittelstraß (1981/93):

"Der Wissenschaftstheoretiker stellt die existierenden Wissenschaften nicht in Frage. Vielmehr versucht er deren Rekonstruktion unter der Vor-25

aussetzung, daß eine rationale Rekonstruktion möglich ist. Stellt sich her-aus, daß diese Voraussetzung nicht gegeben ist, entfällt, so darf man Steg-müller verstehen, bis auf weiteres eine wissenschaftstheoretische Aufgabe; die Wissenschaften selbst sind aufgerufen, hier erst einmal für wissen-schaftstheoriedienliche Verhältnisse zu sorgen." Final italics: MS. 30

Exactly this task will be undertaken. The detailed painting of the 'picture', of the foundations and unfolding the applications will be subject of the remain-ing thousand pages.

EVALUATIONS/ASSESSMENTS

The analysis of a philosophical treatise on the concept of force has shown that 35

the problems identified are none and consequently the solution proposed is trivial, not serving any theoretical and/or practical purposes.

CONCLUSIONS

Having thus demonstrated in great detail the need for reconstruction the follow-ing section has to cover the remaining opening operations of the treatise. 40

4.5 Model, Goal, Plan 189


4.5 Model, Goal, Plan

"Goethische Behandlung der Wissenschaft – mein Pro-jekt."

Novalis: Allgemeines Brouillon, 1798-99 (1981/494). 5

"Die letzte Steigerung des Komplizierten ist das Einfa-che."

Oscar Wilde (Tesar, 1909/191).

PROBLEMS

The problem identified in the foregoing sections is the need for reconstruction of 10

classical mechanics.

MODELS/GOALS

According to the model of the problem solving process the goal of this section is to provide the model for the reconstruction of classical mechanics and to define, in terms of this model, the goal and the plan of the treatise. 15

PLANS

Following the detailed statements concerning the instinctive beliefs and the problems underlying the treatise the statements of the model and the goal will be extremely short, while the plan will be elaborated on in some detail in a number of separate sub-sections. 20

4.5.1 Model of treatise

"Alice thought to herself 'Then there is no use in speaking'. The voices did not join in, this time, as she hadn't spoken, but, to her great surprise, they all thought in chorus (I hope you understand what thinking in chorus 25

means − for I must confess that I don't). 'Better say nothing at all. Language is worth a thou-

sand pounds a word!' "

Lewis Carroll: Through the Looking-Glass (1988/155). 30

According to the conviction of the author most of the confusion in phi-losophy and mechanics can be avoided, if the very many levels of discourse are clearly separated and on each level explicit models are being introduced, openly accessible and intelligible for each participant in the 'game'. Prob-lems are not solved by confusing the issues but by carefully separating 35

them, though without destroying their inherent, essential interrelations!



Consequently the problem of disentangling our fundamental beliefs in classical mechanics will be solved by embedding mechanics into adequate working theories of knowledge, of theories, of proto-mechanics, of meta-mechanics in general, of meta-mechanics ad hoc, of elementary mechanics global and local. 5

Novalis has specified exactly what he felt to be Goethe's way of treating science (Novalis, 1981.a /302):

"Er abstrahiert mit einer seltnen Genauigkeit, aber nie ohne das Objekt zugleich zu konstruieren, dem die Abstraktion entspricht. Dies ist nichts als angewandte Philosophie – ... ". 10

This dictum refers to the fact that the model of a hierarchy of theories un-derlying the present treatise is not a static model but implies a continuous oscillations between abstractions and constructions on each level, or as Goethe himself said in his 'Buch des Sängers' in 'West-östlicher Divan' (BA 03/12): 15

"Im Atemholen sind zweierlei Gnaden, die Luft einziehen, sich ihrer entladen."

At an earlier stage the feedback loops in the problem solving process have already been mentioned.

Goethe's scientific approach has been summarised by Sepper (1988/45): 20

"We may sum up the rationale of Goethe's method on the eve of his opti-cal studies as follows: ... As far as possible one ought to let things speak for themselves; one must first labor to find the appropriate Anschauung, onlooking, so that one might develop an adequate terminology and method of presentation that expresses tolerably well the full gamut [entire range] of 25

the phenomena."

and further (1988/64):

"If Goethe feared abstraction, he feared it when it was applied abstractly, rather than concretely."

and described in detail in chapters on 'Factuality, certainty, and the organisa-30

tion of science' and on 'Goethe and the ethos of science'.

Feyerabend has pointed out the playful character of the process (1983/24):

"Daß Interessen, Macht, Propaganda und Gehirnwäschemethoden in der Entwicklung der Erkenntnis und der Wissenschaft eine viel größere Rolle 35

spielen, als allgemein angenommen wird, das läßt sich auch an einer Ana-lyse des Verhältnisses von Denken und Handeln erkennen. Es wird oft für selbstverständlich gehalten, daß ein klares und deutliches Verständnis neuer Ideen ihrer Formulierung und Institutionalisierung vorangeht und vorange-hen sollte. (Eine Untersuchung beginnt mit einem Problem, sagt Popper.) 40

Zuerst hat man einen Gedanken oder ein Problem, dann handelt man, d. h. redet, baut oder zerstört. Doch so entwickeln sich gewiß nicht kleine Kin-



der. Sie gebrauchen Wörter, verbinden sie, spielen mit ihnen, bis sie eine Bedeutung erfassen, die ihnen bisher unzugänglich war. Und die anfängli-che spielerische Tätigkeit ist eine wesentliche Voraussetzung für das schließliche Verstehen. Es gibt keinen Grund, warum dieser Mechanismus beim Erwachsenen nicht mehr arbeiten sollte. … Die Schaffung eines Ge-5

genstands und die Schaffung und das vollständige Verständnis einer richti-gen Vorstellung von dem Gegenstand gehören sehr oft zu ein und demsel-ben unteilbaren Vorgang und lassen sich nicht trennen, ohne diesen zu un-terbrechen. Der Vorgang selbst wird von keinem wohldefinierten Pro-gramm geleitet, denn er enthält die Bedingungen für die Verwirklichung 10

möglicher Programme. Er wird vielmehr von einem unbestimmten Drang geleitet, einer 'Leidenschaft' (Kierkegaard). Aus ihr entspringt ein bestimm-tes Verhalten, das wiederum die Umstände und die Ideen hervorbringt, die für die Analyse und Erklärung des Vorgangs nötig sind, also nötig sind, um ihn 'rational' zu machen." 15

But Sepper warns (1988/21):

"In reading Goethe as scientist, as philosopher of science, and as histo-rian of science, we shall occasionally find ways in which he anticipated later work. But it would be a mistake to try to rehabilitate Goethe merely by turning him into a twentieth-century scientist or philosopher or historian 20

before his time."

This warning certainly applies to many similar cases. In his study on Goethe's scientific approach Bortoft referred to the same problem (1995/9):

"Es ist eine oberflächliche Denkgewohnheit, sich die Vergangenheit so zu 'erfinden', daß sie zur Gegenwart paßt." 25

In view of the multilevel and multi-stage nature of the decision processes human beings are permanently engaged in educated guesses (Vermutungen) and convictions (Überzeugungen). Both are absolutely necessary and have proved to be extremely efficient in the game of survival. So Nietzsche's ver-dict: 'Die Feinde der Wahrheit sind nicht Lügen, sondern Überzeugungen' is 30

too harsh, but pointing out a serious problem on the way to disentangle 'facts and fictions' in our knowledge, 'Dichtung und Wahrheit' as Goethe called his autobiography in 'reverse' order, reportedly for melodious reasons.

The stipulation of an 'opposition' of 'rationality' and 'imagination' is as simple-minded as the stipulation of the banal 'opposition' of 'facts' and 'fic-35

tions', alias theories. As the opposition of objectivity and subjectivity it came about only during the first decade of the nineteenth century and has since undergone a very diverse development (Daston, 2001, 2007).

In a lecture on 'Images of Objectivity and Subjectivity' Peter Galison, of the Harvard University at the Humboldt-Universität zu Berlin, 20.01.2005 40

has provided a survey of the 'pictorial' history from Goethe's time of ingen-ious insights (1730-1830), through the time of photographically correct im-ages (1830-1920) to the time of images interpreted by trained, experienced



experts. In the meantime the ideas have been published in the book 'Objek-tivität' together with Lorraine Daston (2008).

All these types of pictorial representation are coexisting today and with the advent of simulations to study of implications of theories, the results presented in pictorial format, a new type of images and a new style of ex-5

planation have evolved.

The 'opposition' of 'practice' and 'theory' is much older (Wikipedia):

"Banause wurde von der im antiken Griechenland üblichen Bezeichnung bánausos (… ursprünglich 'der am Ofen Arbeitende', im weiteren Sinne '(Kunst-)Handwerker', schließlich 'vulgär') abgeleitet, mit der diejenigen 10

abwertend benannt wurden, die nicht frei geboren waren und ihren Lebens-unterhalt durch körperliche Arbeit verdienen mussten. Dazu gehörten neben den Handwerkern auch diejenigen Künstler, die die praktischen Künste (ar-tes mechanicae) ausübten. Der Zugang zu den freien Künsten (artes libera-les) blieb diesen Schichten verwehrt. In der antiken Gesellschaft waren nur 15

diejenigen Tätigkeiten gesellschaftlich akzeptiert, die ohne Erwerbsabsich-ten und fast ausschließlich auf geistigen Fähigkeiten beruhten. Neben Handwerk und Kunst wurden auch Handel und Landarbeit als banausisch angesehen."

Accordingly the author uses 'Banausen' for 'craftsmen and artisans, who 20

know how to do it, but do not know why to do it'. According to dictionaries consulted the common translation of 'Banausen' today is 'Philistines' imply-ing a completely different meaning.

4.5.2 Goal of the treatise

"Wer wird sich dazu versteigen, über himmlische Din-25

ge zu urteilen, obwohl ihm die irdischen noch rätselhaft sind?"

Eusebius HieronymusE: Über Isaia VI, 1-7. Aus dem Lateinischen übersetzt von Ludwig Schade.

It is not physics in general which will be dealt with, but only classical 30

mechanics, again not in general but mostly dynamics and it is not some 'new' dynamics, as Gummert suspected in a discussion, but classical dynam-ics in the sense of Boltzmann (1897/V f):

"Man sprach in neuerer Zeit viel über die Dunkelheiten in den Principien der Mechanik und suchte sie dadurch zu beseitigen, dass man der Mechanik 35

ein ganz neues, fremdartiges Gewand gab. Ich habe hier den entgegen ge-setzten Weg eingeschlagen und versucht, ob sich nicht bei möglichst treuer Darstellung der Mechanik in ihrer classischen Form die Dunkelheiten eben-falls vermeiden liessen, theils indem ich gewisse Dinge, die man früher überging oder als selbstverständlich nur obenhin berührte, ausführlich be-40

handelte, theils indem ich jede berechtigte Kritik sorgfältig berücksichtigte.



Besonders kann ich da den Bemerkungen Hertz' über die einschlägige Schriften Mach's nur aufs Wärmste zustimmen, wenn ich auch keineswegs überall derselben Meinung bin, wie Mach."

And this will be done step by step, fully exploiting the implications of every model. The underlying model of mechanics shows two totally differ-5

ent interpretations of the term 'dynamics', both being used in the literature. Here dynamics, the theory of forces is in focus. 'Dynamics', the theory of moving systems, does not 'explain' motions.

Mechanics: 'dual model'

Kinetics

Theory of motions due to forces:

Momentum balance in bodies of ponderable matter

at rest: Statics, in motion: 'Dynamics'

Kinematics

Theory of motions

Dynamics

Theory of forces

10

The dual usage of the term 'dynamics' has a long tradition. At the begin-ning of his discussion of the 'Begründung der Dynamik durch Galilei' Dühring notes (1887/17):

"Man hat daher ein Recht, die Dynamik ganz allgemein als eine Lehre von den Ursachen und Gesetzen der Bewegung aufzufassen, gleichviel ob 15

es sich um die Combination bloßer Beharrungsbewegungen oder um Orts-veränderungen unter dem Einfluss stets von Neuem wirkender Kräfte hand-le."

The disadvantages of not copying from other textbooks have been vividly described by JamesW. The disadvantage of the stepwise procedure is that the 20

author and his readers become impatient. They already know 'everything' and they know many cross-connections. So part of the effort will have to be catching the interest of the readers by opening unexpected avenues at any stage of the development.

In order to support this endeavour the author does not avoid expletive 25

phrases as described in the Oxford English Dictionary: Word of the Day' (12.06.2009):

"expletive, a. and n.

A. adj.

1. Serving to fill out; introduced merely to occupy space, or to make up a 30

required quantity or number: a. gen. 1656-81 in BLOUNT Glossogr. 1666 TILLOTSON Rule of Faith I. §3 Those expletive topicks which popish writers … do generally make use of to help out a book. …

B. n.



1. An expletive word or phrase, one used for filling up a sentence, eking out a metrical line, etc. without adding anything to the sense. … 1607 HIERON Defence I. 160 To be put in expletively and by way of explica-tion. … "

The goal is to show the readers, that they often do not 'know', but instinc-5

tively belief, and that often their beliefs are misleading, incoherent or plainly falsely held. The attempt will be made to convince readers to depart from old beliefs and superstition, dating back to their school and university training, maybe just 'conditioning' as described by Pavlov.

The author's oscillations between abstractions and constructions, his cum-10

bersome departure from professional superstition will not be subject of the treatise. According to the account given of the state of affairs the goal of the treatise is to provide an adequate coherent foundation for practical teaching and applications, non-traditional and non-routine applications in particular, and not to add another useless paper to the pile. 15

The goal of revisiting Newton's Principia and embedding classical me-chanics into sound meta-mechanics is very different from Chandrasekhar's 'Newton's Principia for the Common Reader' (2003), which is an incom-plete, annotated version of Motte's translation, using current mathematical notation for the reconstruction of Newton's proofs only as far as they relate 20

to celestial mechanics.

The annotations, proudly claimed to be 'the first analysis of the Principia without recourse to secondary sources', sometimes just underlining parts of quotations, are very superficial and not enlightening at all. Three examples will suffice to prove this verdict. 25

Definition III on 'the vis insita or the innate force of matter ...' is com-mented in the sentence (2003/19):

"There is hardly anything that one can usefully add to Newton's careful explanation of the concept of inertia. But note particularly the statements that are underlined." 30

Underlined among others is 'a body only exerts this force when another force, impressed upon it, endeavours to change its condition; and the exer-cise of this force may be considered as both resistance and impulse'. Even expert readers would appreciate some 'useful addition' as this is one of the absolutely crucial statements as will be seen. 35

The chapter on 'Basic concepts: definitions' ends with the remark (2003/22):

" ... concerning the notions of 'absolute time' and 'absolute space' on which Newton bases his dynamics, it will suffice to say that, in current ter-minology, the space-time manifold that is assumed is the Cartesian product 40

t x Euclidean 3-space, where t is Newton's 'equable time'."



What is 'the Common Reader' supposed to conclude from this empty formal statement? It does neither explain the success of Newton's proto-mechanics nor its limits.

Following the quotation of Newton's statement of Law II Chandrasekhar notes (2003/23): 5

"The statement of the Law is self-explanatory."

As shown in the present treatise this is not true, in no sense. A statement cannot be self-explanatory, but can only be explained by embedding it into an adequate meta-theory. Further, history shows that the implications of the basic axiom have been and still are widely not understood. And to be sure 10

'the Common Reader' is totally lost already at this early stage, nearly six hundred pages to follow, not a ponderous German 'Einführung', but the body of the text.

By any standard, 'common', advanced or expert, this is not the analysis promised in the title and expected by 'the Common Reader', as documented 15

in the comments on the Amazon website. It is only mentioned that on the whole the Principia are somehow coherent (2003/17):

"The concepts are complex and interrelated. Newton's Definitions and Axioms should be read in their totality and in their context keeping in mind, that they '... shall be explained more at large in the following trea-20

tise. For to this end it was that I composed it.' " Italics: Quotation accord-ing to Newton, PM/12.

This is very much loose talk! No attempt has been made to explain to 'the Common Reader' how 'intricate and interrelated the concepts' are, how to 'read in totality' and what the context might be. Instead, the details of proofs 25

are elaborated, which are certainly not of interest in the present context and definitely not to the 'Common Reader'.

4.5.3 'Engineering' philosophy

" 'Thinking again?' the Duchess asked, with another dig of her sharp chin. 30

'I've a right to think,' said Alice sharply, for she was beginning to feel a little worried.

'Just about as much right', said the Duchess, 'as pigs have to fly; ...' "

Lewis Carroll: Alice's Adventures in Wonderland 35

(1988/89).

Before the treatise proper starts various aspects and ideas will be dis-cussed which are underlying the following exposition. Readers only inter-ested in 'solid results' may find these to be unnecessary diversions. But they



definitely underestimate the practical importance of being explicitly aware of the 'environment' into which mechanics is embedded.

'Philosophy' in most textbooks of mechanics is 'limited to prefaces and in-troductions' (Raatzsch, 2000/39), maybe to 'metaphysical prologues' (Tesar, 1909). Even in the 'New Foundations for Classical Mechanics' by Hestenes 5

meta-mechanics is treated, though inadequately, not in the opening, but in the closing chapter on the 'Foundations of Mechanics' already mentioned (1986/574-602).

In the present treatise philosophy will not only be 'the label on the bottle, not only providing the flavour of the wine inside' but will be constitutive for 10

the whole treatise, providing the framework, setting the scene. It is directly, naïvely in the spirit of the age of enlightenment (Kant, 1916/1):

"Aufklärung ist der Ausgang des Menschen aus seiner selbstverschulde-ten Unmündigkeit. Unmündigkeit ist das Unvermögen, sich seines Vers-tandes ohne Leitung eines anderen zu bedienen. Selbstverschuldet ist diese 15

Unmündigkeit, wenn die Ursache derselben nicht am Mangel des Verstan-des, sondern der Entschließung und des Mutes liegt, sich seiner ohne Lei-tung eines anderen zu bedienen! Sapere aude! Habe Mut, dich deines eige-nen Verstandes zu bedienen! ist also der Wahlspruch der Aufklärung.

Faulheit und Feigheit sind die Ursachen, warum ein so großer Teil der 20

Menschen, nachdem sie die Natur längst von fremder Leitung freigespro-chen dennoch gern zeitlebens unmündig bleiben; und warum es anderen so leicht wird, sich zu deren Vormündern aufzuwerfen. Es ist so bequem, un-mündig zu sein. ... "

The author dares to transcend boundaries of academic disciplines, know-25

ing about the dangers, but being convinced of the necessity for the purpose at hand, the adequate reconstruction of classical mechanics. As has been mentioned this reconstruction cannot possibly be achieved inside mechanics or by the history and philosophy of mechanics in their present states.

The specific way in which real and ideal things are being looked at de-30

pends on the specific subject looked at (Raatzsch, 2000/96). This is in ac-cordance with the earlier remark that insights and arguments depend cru-cially on our convictions and instinctive beliefs, which themselves do not depend on insights and arguments, but are at best 'in accord' with those.

But the idea to reconstruct philosophy from a meta-philosophy is not fea-35

sible as has been explained by Raatzsch (2000/26 f):

"Hegel sieht also das Dilemma. Sein Versuch der Auflösung ist so ein-fach wie grandios. Er besagt: Der Maßstab, den wir in der Kritik verwen-den, ist 'von dem ewigen und unwandelbaren Urbild der Sache [der Philo-sophie] selbst hergenommen ...' Ein solches Urbild kann aber nicht etwas 40

vor der Philosophie Bestehendes sein, denn mit ihm ist zugleich die Philo-sophie schon gegeben. Die Philosophie muß nur nicht stets schon in ihrer vollen Form, dem System gegeben sein. Das Urbild muß es nur deshalb ge-



ben, weil kein grundsätzlicher Unterschied besteht zwischen einem Beurtei-len der Wahrheit und einem Verstehen des Inhaltes dessen was Philoso-phen als solche sagen. ...Bis ins Detail hinein gleiche Gedanken findet man bei – Frege."

In the present context the preceding quotation can be paraphrased by re-5

placing 'philosophy' with 'mechanics'. And the ideal of mechanics, SchillerF 's 'dunkle Totalidee', the author has in his mind was the reference used in the criticism of the present state of mechanics and will be the refer-ence of the system of classical dynamics to be developed.

Under the title 'Glückliches Ereignis' Goethe remembered a discussion 10

with SchillerF on the subject (Neunzig, 2002/11-12). The story 'Der ersten Bekanntschaft mit SchillerF ' following a lecture in a series at Jena on July 20, 1794, documented many years later (Damm, 2004/198) is of fundamen-tal interest in the context of the present work (Goethe, 1949/V 433 f; GA 16/867): 15

" …; einstmals fand ich Schillern daselbst, wir gingen zufällig beide zu-gleich heraus, ein Gespräch knüpfte sich an, er schien an dem Vorgetrage-nen teilzunehmen, bemerkte aber verständig und einsichtig und mir sehr willkommen, wie eine so zerstückelte Art, die Natur zu behandeln, den Laien, der sich gerne darauf einließe, keineswegs anmuten könne. 20

Ich erwiderte darauf, daß sie den Eingeweihten selbst vielleicht unheim-lich bleibe, und daß es doch wohl noch eine andere Weise geben könne, die Natur nicht gesondert und vereinzelt vorzunehmen, sondern sie wirkend und lebendig, aus dem Ganzen in die Teile strebend darzustellen. Er wünschte darüber aufgeklärt zu sein, verbarg aber seine Zweifel nicht; er 25

konnte nicht eingestehen, daß ein solches, wie ich behauptete, schon aus der Erfahrung hervorgehe.

Wir gelangten zu seinem Hause, das Gespräch lockte mich hinein; da trug ich die Metamorphose der Pflanzen lebhaft vor und ließ, mit manchen charakteristischen Federstrichen, eine symbolische Pflanze vor seinen Au-30

gen entstehen. Er vernahm und schaut das alles mit grosser Teilnahme, mit entschiedener Fassungskraft; als ich aber geredet hatte, schüttelte er den Kopf und sagte: Das ist keine Erfahrung, das ist eine Idee. Ich stutzte, ver-drießlich einigermaßen: denn der Punkt, der uns trennte, war dadurch auf das strengste bezeichnet. Die Behauptung aus Anmut und Würde fiel mir 35

wieder ein, der alte Groll wollte sich regen, ich nahm mich aber zusammen und versetzte: das kann mir sehr lieb sein, daß ich Ideen habe, ohne es zu wissen, und sie sogar mit Augen sehe." Italics: MS.

Mach in his 'Principles of Thermodynamics' has referred to the genesis of theories (Heller, 1964/80): 40

"Wer sich mit der Forschung beschäftigt hat, wird schwerlich glauben, daß die Entdeckungen nach dem Aristotelischen oder Bacon'schen Schema der Induktion ... zustande kommen. Da wäre ja das Entdecken ein behagli-



ches Handwerk. Die Tatsachen, deren Erkenntnis eine Entdeckung vor-stellt, werden vielmehr erschaut."

The literature on this topic alone is inexhaustible, but is not of much help in practical research situations. Consequently it is not being considered here. Of course Leibniz' ars inveniendi (Popp, 2000/25) comes to mind here, 5

which is followed pragmatically. Knowledge will be generated by organis-ing a large body of diverse knowledge in an operational fashion, though here only in the very limited micro-universe of classical dynamics.

The 'theory of theories' developed, maybe providing a sound foundation for most scientific and engineering work, has been the central part of the 10

lectures the author has held over thirty years. The subject has been profes-sional problem solving in hydro-mechanical systems engineering conceived as multi-level and multi-stage control and decision processes of interlinked macro-operations. This model is also underlying the doctoral dissertation and the paper 'Zur Logik ingenieurwissenschaftlicher Arbeit' by Blank 15

(1980).

4.5.4 Proto- and meta-mechanics

"Die gerade Linie wird eher in sich selbst wieder zu-rückkehren, als ich von meiner Richtung abweichen; sage mir einen Weg der noch näher ist als der geradeste und 20

ich will den jetzigen fahren lassen und deiner Anweisung folgen."

Georg Christoph Lichtenberg: Gedankenbücher, B (1768-71) (1963/44).

Before mechanics can be discussed adequately proto-mechanics, the theo-25

ries of time, space, motion and matter will have to be outlined. This will be the first instance not only to introduce the principle of invariance, but to show its power in constituting Newton's 'absolute, mathematical' chronome-try and stereometry, homogeneous in the present context of classical me-chanics, in a rational fashion free of superstition avoiding the terms 'time' 30

and 'space' as far a possible. By the way Newton's usage of 'absolute' and 'relative' time and space will be shown to be in perfect accordance with the dual model of theories.

The author arrived at the proto-theories following Janich's exposition in 'Kleine Philosophie der Naturwissenschaften' (1990) being impressed with 35

the constructive program of Lorenzen (1974) and the Erlangen School, re-constructing current scientific practice from an operational point of view, much in line with his own guiding principles, practice and teaching. Among others, important sources of inspiration have been Klein's 'Geometry',



Weyl's 'Space – Time – Matter' and Friedmann's 'Die Welt als Raum und Zeit', studied repeatedly.

The original intention was to cover only this topic, the theories of time and space, which the author before had neither understood and nor time to consider. But in the process of studying and understanding the subject a 5

more comprehensive presentation became necessary to provide the various levels of instinctive beliefs, into which this theory is embedded and make the ideas proposed comprehensible.

During the work on the exposition the goals and the corresponding ar-rangement of the material had to be changed a number of times in order to 10

meet the requirements of the step by step procedure.

Following the 'Proto-mechanics abstract' the first theory to be introduced in the hierarchy of generic theories will be 'Meta-mechanics abstract', the general theory of systems in terms of the black box model and correspond-ing output equivalent state space models. This theory may be considered as 15

a perfect meta-mechanics, to avoid the term meta-physics, an operational model of Plato's parable of the prisoners in the cave and of the 'engineering' philosophy developed.

As the concept of force the concept of state is often used only vaguely in very many different 'interpretations'. In order to avoid the resulting confu-20

sion the 'interpretation' used in the present context will be abstract, mathe-matically defined, and free of all the usual 'metaphysical' or 'thermodynami-cal' connotations and implications.

In view of the purpose of the treatise and in view of interpreting abstract proto-mechanics 'Meta-mechanics: ad hoc' will be developed as an intuitive, 25

in fact the simplest instantiation of the theory of systems the balance of quantities in the original sense of extensities (Mengen), not just magnitudes (Grössen), and definitely not substances. It provides a meta-theory, the 'higher standpoint' Mach has been looking for, embracing all 'quantitative' sciences, mechanics in particular as will shown in detail. 30

The following chapter on 'Elementary mechanics: abstract' may be con-sidered as the cornerstone of the whole treatise. It will be introduced as an instance of the meta-theory of quantities, the quantity under investigation being Newton's quantitas motus, momentum of translational motions of ex-tended bodies, not of point masses, alias mass points, as usual. This is 35

strictly in the sense of Newton and even more so in the sense of Euler. The chapter will end with a section on 'Elementary kinematics'.

Only at this stage an 'abstract interpretation' of abstract proto-mechanics becomes possible. The chapter on 'Proto-mechanics: time, space' will be fol-lowed by an extended interlude, a chapter on 'perspective relativity'. The lat-40

ter will be discussed in the spirit of Augustinus, in terms of 'macroscopic'



messengers of limited speed, maybe runners on deck or bullets in vacuo, and will provide a model of Einstein's theory of special relativity.

4.5.5 Elementary and local dynamics

"Jeder Paragraph in der neuen Physik sollte so behan-delt werden, daß man sähe, daß man ihn nicht abge-5

schrieben, sondern selbst dabei gedacht hat."

Georg Christoph Lichtenberg (Joost, 2000/85).

Elementary mechanics, the common root of mechanics, will be developed according to the dual model of theories. The chapter on interpretation, 'Ele-mentary dynamics', will be followed by an additional chapter on 'Elemen-10

tary physics', on gravitation. Readers will hardly imagine how painful the departure from long cherished jargon and associated beliefs turned out to be and how much work went into the clarification of elementary mechanics and its basic applications.

Local mechanics will be embedded into the meta-theory of continua. As 15

first instance of particular interest the meta-theory of motions of ideal con-tinua will be developed, providing a meta-theory of Schrödinger's equation for wave mechanics in general, non-relativistic, 'classical' quantum mechan-ics in particular. As second instance of more interest in the present treatise the meta-theory of material continua will be developed. 20

Cauchy's universal equation for the motion of material continua will be introduced efficiently as an instance of the latter, thus providing insights into all its logical and factual implications. This equation is sometimes 'pos-tulated' axiomatically (Truesdell, 1966/3), but it will be noticed how much needs to be said and known about its implications in view of the fact that 25

continuum dynamics is the common trunk of all branches of classical me-chanics. The kinematics of continua will be dealt with at some length, due to the fact that even 'standard' textbooks leave many questions open (Eringen, 1962).

Important subjects will be the equation of and the constitutive equations 30

for classical materials. In the context of the momentum balance the theory of materials will be treated as the theory of diffusive fluxes of momentum. The principle of invariance with respect to certain groups of transforma-tions, constituting specific classes of materials in the narrow sense, will only be touched. 35

In the present context the relevant structures and scales are not necessarily those of the molecular structures of matter, but rather the macroscopic struc-tures and scales governing the diffusive momentum flow. Typical examples are the scales of the anisotropic structures of wood, of the polycristalline



structures of metals, of the granular structures of grain or ore in bulk carri-ers, or of the turbulent structures in boundary layers.

After having settled these fundamental questions the following develop-ments will be of a more technical nature. But again it will be shown that 'many beliefs have, by habit and association become entangled with other 5

beliefs, not really instinctive, but falsely supposed to be part of what is be-lieved instinctively'.

4.5.6 Global dynamics, 'principles'

"In the elder days of Art, Builders wrought with greatest care 10

Each minute and unseen part, For the gods are everywhere."

Henry Wadsworth Longfellow, quoted by Ludwig Wittgenstein in his Manuscript 120 (Volume XVI), 20.04.1938 with the addition: (könnte mir [person-15

ally] als Motto dienen).

In dealing with global balances of momentum, spin and energy the limit-ing cases of control boundaries at rest and moving with the body of matter, respectively, will be discussed in detail. For classical materials the global balance of spin reduces to the balance of the moment of momentum, usually 20

introduced as Euler's law or as Newton's 'second' law, as it is called in China, both balances together more recently in Germany being called New-ton-Euler equations.

In addition to these basic principles of mechanics partial energy balances will provide the explicit Euler-Lagrange equations of motions in terms of 25

generalised non-holonomic velocities. They reduce to the generalised Kane's equations for systems with additive partial velocities.

The variational principles of mechanics will be shown to be a subset of classical mechanics for systems with 'degenerate', holonomic kinematics and forces derived from energy functions. Typically terrestrial systems of inter-30

est, fluid-mechanical systems in particular, do not belong to this degenerate type.

The necessary generalisation leading back to the state space models will not be followed in detail. These models permit to account for the past his-tory of the global motions in terms of 'hidden' states not to be confused with 35

the 'hidden motions' Hertz was concerned with in his attempt to 'understand' gravity.

Rigid body dynamics will be discussed in terms of 'stacked' (Mathcad) equations of momentum (Wucht) and spin (Drall), the generalised Euler



equations in body fixed coordinates, already used by von Mises in his 'mo-tor' calculus. They are invariant under changes of the reference point and of the observation space and, not surprisingly, they can be shown to be identi-cal with the Euler-Lagrange equations of rigid body motions.

The kinematics of rigid bodies will be discussed in terms of Euler angles. 5

As an example the identification of the location of the reference point in the body and of the orientation of the body by moving platform systems based on strap-down field meters, alias accelerometers, will be treated.

4.5.7 Rigid bodies, in fluids

" ..., meine Arbeit war eigentlich wieder die des Klä-10

rens. Ich glaube, das Wesentliche ist, daß die Tätigkeit des Klärens mit Mut betrieben werden muß: fehlt der, so wird sie ein bloßes gescheites Spiel." Italics: capitals in the reference.

Ludwig Wittgenstein, 1931. 15

The main points of the last chapters will be to show that axiomatic sys-tems for intricate problems can ad hoc be generated explicitly, while tradi-tionally model based 'systems' are taught and treated more or less 'implic-itly', thus remaining 'instinctive', incomplete and incoherent.

This procedure is in accordance with Euler's advice (1926/21): 20

"8. Die Naturlehre wird also am füglichsten abgehandelt werden, dass man erstlich die allgemeinen Eigenschaften und daher das Wesen der Kör-per erforscht, hernach aber alle besonderen Arten der Körper auf gleiche Weise untersuchet."

The examples are rooted in the professional background of the author. In 25

view of the central topics of this treatise and of the 'general reader' only the essentials will be dealt with, forgetting about the marine folklore necessary for dealing with real problems and in presentations addressed to naval archi-tects.

Rigid bodies in incompressible fluids are considered as an example of 30

systems of forcibly driven bodies. Highlighting another false instinctive be-lief the case of potential flows in viscous fluids will be treated in some de-tail. A special chapter will be devoted to the identification of aggregate properties of bodies in fluids.

If the identification is not based on free motions but on forced motions 35

balances are being used, which have to be calibrated. Deflections due to elasticity have to be accounted for already in the calibration and further in determining the actual motions and positions of the bodies during the tests. This example permits an outlook onto elastic bodies and other solid bodies.



A next chapter will be devoted to bodies propelled in fluids. Rational so-lutions will be described which have been developed for the basic problems of performance evaluation from observations at trial runs and under service conditions. These problems have not and cannot be solved adequately and satisfactorily 'inside' traditional ship theory. 5

In both cases the problems are problems of rational conflict resolution in any particular case requiring an explicit axiomatic approach, a simple ab-stract model permitting the identification of its few parameters from the only few data usually available. As a matter of fact the same procedure ap-plies on any level of research. In classical mechanics the structure of matter 10

and the mechanism of momentum production need not to be specified.

As will be shown the evaluation of steady trials does not require any ship theory at all contrary to the instinctive belief held by naval architects. The principle of objectivity, the Π-theorem of dimensional analysis is com-pletely sufficient for the purpose at hand. For more detailed performance 15

analyses simple models of propeller, current, wind and wave action are nec-essary.

As has been demonstrated in the METEOR project the theory of ideal propulsors in ideal wakes provides the model of an axiomatic theory of hull-propeller interactions. Robust 'local' axioms of thrust deduction and wake 20

fractions provide for interpretations of the concepts of hull resistance and propeller advance speed of full scale ships under service conditions. The same concepts have been shown to provide starting points for the design of non-traditional propulsor configurations.

The solutions proposed for the evaluation of trials, the complete perform-25

ance analysis of propulsive systems and for the design of propulsors will support the departure from the naïve concept of propulsion conceiving pro-pulsors as thrusters overcoming the resistances of the bodies to be propelled. As will be shown it is much more adequate to conceive propulsors as pumps feeding energy into volume flows, preferably drawn from energy wakes. 30

The final chapter will be devoted to motions in ideal wave media exhibit-ing certain features 'similar' to phenomena known from the theories of rela-tivity. The first case considered is that of slender bodies moving in media at rest. In this case the speed of sound plays a similar role as the speed of light in the theory of special relativity. This case is analogous to motions of ship 35

hulls in shallow water and thus of interest to naval architects. The second case considered is that of sound waves in moving media exhibiting certain features of light moving in gravity fields, in 'curved space'.




Classical dynamics will be 'understood' by embedding it into the theories of theories, proto-mechanics and meta-mechanics. Further elementary mechanics is considered as the root of classical mechanics, while continuum mechanics is con-veniently considered as the trunk of all branches of the flourishing tree of classical 5

mechanics. Consequently the plan is to reconstruct classical mechanics in this hi-erarchy of theories.

CONCLUSIONS

Elementary mechanics will be developed as an instance the meta-theory of quan-tities in the original sense of extensities. This will be done on the basis of princi-10

ples, instinctive beliefs, which usually take the form of coherence and invariance postulates constituting the abstract concept of objectivity and providing for its op-erational interpretation in any particular instance.

CLOSURE OF CHAPTER 15


In the introduction great care has been taken to demonstrate the poor state of 'classical' expositions of the foundations of classical mechanics and to highlight many aspects of the problem.

Finally and accordingly the goal of the treatise has been defined: to reconstruct 20

classical mechanics by embedding it into general meta-theories and, in order to reach that goal, a plan has been laid out in detail.

CONCLUSIONS

'The rest is entertainment (at least I hope so) and perhaps some associated edifi-cation' (Bellman). 25

Although the following reconstruction will be rather straightforward, in any case references, quotations and discussions will be provided. Some readers may find this procedure disturbing, but the historical arguments will provide additional aspects, which may help to understand the problems and the solutions. Of course 'as scholars we are only too familiar with footnotes that confuse rather than eluci-30

date the text they profess to explain' (WilliamsJT, 1996/159).

SchillerF in a letter to Goethe 1795 felt '… es ist hohe Zeit, daß ich … die philo-sophische Bude schließe. Das Herz schmachtet nach einem betastbaren Objekt…' (Damm, 2004/223) and many readers will feel that way. But during the present ex-ercise the philosophical stall will have to remain open until the end. Though bodies 35

of ponderable matter are 'betastbare Objekte' par excellence they can be grasped and sold only this way

4.4 'Solutions' proposed

Documents

Transcript of 4.4 'Solutions' proposed