Günther Ludwig Gérald A New Thurler Foundation of Physical (cid:2) Theories (cid:2) (cid:3)(cid:4) ANewFoundationofPhysicalTheories Günther Ludwig Gérald Thurler A New Foundation of Physical Theories ABC Professor Dr.Günther Ludwig Dr. Gérald Thurler (1918–2007) Rue Baulacre 30 1202 Genève Switzerland 1st edition, corrected 2nd printing ISBN978-3-540-30832-4 e-ISBN978-3-540-30833-1 DOI 10.1007/3-540-30833-4 LibraryofCongressControlNumber:2006922616 (cid:1)c 2008, 2006 Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright. 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CoverDesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 9 8 7 6 5 4 3 2 spinger.com Foreword (translation) I was interested by the development of a new edition of the book [1] “Die Grundstrukturen einer physikalischen Theorie.” This has been possible, in spite of my old age, thanks to the contributions of Dr. G. Thurler. Without his indefatigable support and his essential and fundamental propositions, this new edition would not have been possible. The new edition clarifies and formulates more precisely the fundamental ideasofphysicaltheoriesinordertoavoidasmuchaspossibleanyambiguities. One begins theoretical physics with concepts that can be explained with- out theories. Later, one introduces other concepts by theories known as “pre-theories.” Thus it does not make sense to introduce concepts such as “state” without a pre-theory. The field of physics is thus determined by the basic concepts introduced without the use of pre-theories. Also, it does not make sense to speak about the position and speed of an electron at a fixed time. “Reality” is not however only the reality which is described by physical concepts.Thus,forexample,colors,tones,joy,hate,andlovearenotphysical concepts. But the demarcation of the physical concepts, and thus the demarcation of the field of physics makes it possible to know more clearly, and thus to describemoreclearlyinthefuture,thestructureofrealitybeyondthedomain ofphysics.Thefieldoflifeandnotthatofdeathshouldbethegoalofmankind. Thus, I hope that this book can also become another small step for life. Marburg Gu¨nther Ludwig October 2005 Vorwort Ich war daran interessiert, bald eine neue Auflage des Buches ,,Die Grundstrukturen einer physikalischen Theorie” zu entwerfen ([1]). Daß dies trotz meines hohen Alters mo¨glich wurde, habe ich Herrn Dr. G. Thurler zu verdanken. Ohne seine unermu¨dliche Hilfe und seine wesentlichen Vorschla¨geauch in haltlicher Art, wa¨re die Neuauflage nie Zustande gekommen. Diese Neuauflage soll die Grundsa¨tzlichen Ideen kla¨ren und pra¨ziser formulieren, um mo¨glichst jede Fehlentwicklung physikalischer Theorien zu vermeiden. Dazu geh¨ort, daß man die theoretische Physik nur mit Begriffen anfa¨ngt,dieohnejedeTheorieerkla¨rtwerdenko¨nnen.Spa¨terfu¨hrtmandann mit Hilfe von Theorien (sogenannten Vortheorien) weitere Begriffe ein. So machteskeinenSinn,denBegriff,,Zustand”ohneeineVortheorieeinzufu¨hren. Der Umfang der Physik ist damit bestimmt durch die ohne Vortheo- rien eingefu¨hrten Grundbegriffe. Ebenso macht es keinen Sinn, von Ort und Geschwindigkeit eines Elektrons zu einer festen Zeit zu sprechen. Die Wirklichkeit ist aber nicht die allein mit physikalischen Begriffen beschriebene Wirklichkeit.So sindz.B.Farben,To¨ne,Freude,Haß undLiebe keine physikalischen Begriffe. Aber die saubere Abgrenzung der physikalischen Begriffe und damit die saubere Abgrenzung des Bereichs der Physik wird es m¨oglich machen, in der Zukunft auch die Struktur der u¨ber den physikalischen Bereich hinaus- gehenden Wirklichkeit deutlicher zu erfahren und damit auch deutlicher zu beschreiben. Der Bereich des Lebens und nicht der des Todes ist das Ziel des Menschen. So hoffe ich, daß auch dieses Buch eine kleiner Schritt zum Leben werden kann. Marburg Gu¨nther Ludwig Oktober 2005 Preface This book is a revision and expansion of the concept of a physical theory as developed in [1]. In this book, we introduce the following: – A concept of basic language; a descriptive language of simple form in which it is possible to formulate recorded facts. The semantics of this basic language make it possible to clarify the links between linguistic, conceptual,andrealentitiesoftheapplicationdomainofaphysicaltheory. – A new concept of idealization.We know that practicallyall mathematical theories used in the physical theories can only be approximations of the reality,i.e.,thattheycanbeappliedtoanapplicationdomainofaphysical theory only under the assumption of allowing for some degree of approxi- mation or degree of inaccuracy. Weproposeareview(relatedtothenewconceptsintroducedabove)ofthe “notion of relations between various physical theories,” and of the “process allowing to find new concepts” developed in [1]. Theanalysispresentedherewillbelessofadescriptionofthecurrentstate of physics than a suggestion to modify this state. The authors think that a solutioncanbe found amongstthe many difficult problemsof physicssuchas the interpretationof physicaltheories,the relationsbetween varioustheories, and the introduction of physical concepts, when the theories are under the form of an axiomatic basis. The analysis presented here does not claim to be definitive. It should,on the contrary,encouragethe readerto continue the developmentofthefundamentalideasofthiswork.Suchadevelopmentshould contribute to highlight the durable core and growing strength of physical knowledge about the real structures of the world, in addition to the process of the historical development of physics. Ifthisbookwastosuggestsuchadevelopment,itwouldthenhaveachieved its goal. The authors also encourage the reader to correct any possible faults VIII Preface in the text and are convinced that the correction of such errors will not call into question the fundamental ideas of this work. Acknowledgments The authors wish to express their deep thanks to Natacha Carrara for her careful re-reading and linguistic revision of the English manuscript. We are also grateful to Wolf Beiglbo¨ck for his competent advice and for his assistance in the completion of the book. Marburg, Gen`eve Gu¨nther Ludwig October 2005 G´erald Thurler Contents Intention of the Book.......................................... 1 Part I A New Foundation of Physical Theories 1 Reality .................................................... 11 1.1 The Structure of Reality ................................. 11 1.2 The Physical Reality..................................... 12 1.2.1 The Application Domain of a PT ................... 12 1.2.2 The Fundamental Domain of a PT .................. 13 1.2.3 The Reality Domain of a PT ....................... 14 1.2.4 The Reality Domain of all PTs ..................... 14 1.2.5 Remarks ......................................... 15 1.3 Fairy Tales ............................................. 16 2 Building of a Mathematical Theory........................ 17 2.1 Formal Language........................................ 17 2.2 Axioms and Proofs ...................................... 19 2.3 Logics.................................................. 21 2.4 Set Theory ............................................. 27 3 From Reality to Mathematics.............................. 33 3.1 Recording Process ....................................... 34 3.1.1 Basic Language ................................... 34 3.1.2 Application Domain of a PT........................ 44 3.1.3 Recording Rules................................... 44 3.1.4 Facts Recorded in the Basic Language ............... 45 3.2 Mathematization Process................................. 46 3.2.1 The Basic Mathematical Theory .................... 46 3.2.2 The Standard Mathematical Theory ................. 48 3.2.3 Enrichment of MT by A .......................... 50 Θ X Contents 3.2.4 The Finiteness of Physics........................... 52 3.3 Idealization Process...................................... 53 3.3.1 Transition from MT to MT ...................... 53 Θ ∆ 3.3.2 Enrichment of MT by A .......................... 56 ∆ 3.3.3 Fundamental Domain of a PT ...................... 60 4 Species of Structures and Axiomatic Basis of a PT .................................................... 63 4.1 Mathematical Structures ................................. 64 4.2 Deduction of Structures .................................. 67 4.3 Axiomatic Basis and Fairy Tales........................... 73 4.4 Pure Laws of Nature..................................... 76 4.5 Change of the Mathematical Form of an Axiomatic Basis......................................... 78 4.6 Inaccuracy Sets and Uniform Structures.................... 85 4.7 Do the “Laws of Nature” Describe Realities?................ 92 4.8 Classification of Laws of Nature ........................... 96 4.9 Skeleton and Uninterpreted Theories.......................101 5 Relations Between Various PTs ...........................105 5.1 Relations Between Two PTs with the Same Application Domain ................................................106 5.2 Relations Between Two PTs with a Common Part of an Application Domain .....................................111 5.3 Pre-theories.............................................112 5.4 Relations Between PTs with Different Application Domains ..116 5.5 Approximation Theories..................................117 5.6 The Network of PTs.....................................118 6 Real and Possible as Physical Concepts....................121 6.1 Closed Theories .........................................124 6.2 Physical Systems ........................................128 6.3 New Concepts in a PT ...................................131 6.4 Indirect Measurements ...................................133 6.5 Classifications and Interpretations .........................138 6.6 The Reality Domain of a PT .............................143 Part II Examples of Simple Theories A A Description of the Surface of the Earth, or of a Round Table .......................................147 B A Simplified Example of Newton’s Mechanics..............159