DOCSLIB.ORG

Ideas and Explorations: Brouwer's Road to Intuitionism

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ideas and Explorations: Brouwer's Road to Intuitionism IDEAS AND EXPLORATIONS Brouwer’s Road to Intuitionism IDEEEN¨ EN VERKENNINGEN; Brouwer’s weg naar het intu¨ıtionisme (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof.Dr. W.H. Gispen, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op 27 februari 2004 des middags te 16.15 uur door Johannes John Carel Kuiper geboren op 4 december 1935, te Arnhem Promotor Prof.Dr. D. van Dalen Faculty of Philosophy Utrecht University The research for this thesis was made possible by a grant of the Netherlands Organization for Scientific Research (NWO). Voor de vijf vrouwen rondom mij: Janny, Ingeborg, Simone, Jessica en Rita. Acknowledgements Life sometimes takes an unexpected and unforseen turn. After retiring at the age of 57 from 35 years of civil aviation (part of that time having filled with studies in theoretical physics at the Leiden University and in numerous hotel rooms), I started a study in philosophy at the Utrecht University, thereby spe- cializing in logic and the foundations of mathematics. The unexpected turn came with a telephone call from Dirk van Dalen, with the invitation to join the ‘L.E.J. Brouwer-group’ immediately after having completed my philosophy study. He thereby offered me the opportunity to write a doctoral thesis on the early work of Brouwer and on the beginning of his road to intuitionism. The result of four years research, reading, writing, crossing out and rewrit- ing again is this dissertation. Writing a doctoral thesis is often a lonely job, especially when the author is constantly threatened by a hostile computer. It will be obvious that I have to express my gratitude in the first place to my thesis advisor Dirk van Dalen for the numerous hours that he spent on my drafts and on the long discussions and explanations of the many topics that are discussed in this dissertation. This all at the cost of the precious time he needed so badly for writing at the same time two biographies about the life and work of Brouwer (one in Dutch and a scientific one in English). It is of course a relief for me that this dissertation is now completed, but undoubtedly it must be a relief for him too (as well as for his wife Dokie, who is almost just as busy with the work on the biographies). In the second place I want to thank Mark van Atten. He was willing to share his thorough and detailed knowledge of the philosophical foundations of mathematics with me by reading through my drafts and manuscripts, pointing to many smaller and larger imperfections; he even was willing to look at the smallest details in punctuation marks and typing errors. I also owe many thanks to Dennis Dieks for several long discussions about Brouwer’s ideas on the character of physical laws and on the nature and aim of physical practice and research. I thank Theo Verbeek and Maarten van Houte, who were so kind to read through, and comment on, what I had to say about Kant, who was of such great influence on Brouwer’s ideas about the ultimate foundations of all of mathemat- ics. Finally, my gratitude goes to the whole staff of the faculty of philosophy. Despite the rather exceptional combination of being the lower in scientific rank v vi ACKNOWLEDGEMENTS and (almost) the senior in age, I have spent wonderful years here. It should, again, be said that the whole project was also made possible by the NWO, the Netherlands Organisation for Scientific Research, in the form of a grant, which is gratefully acknowledged. Ten slotte het ‘thuisfront’: Lieve Janny, we kunnen, na alle drukte van de afgelopen jaren, natuurlijk niet zeggen dat we ons ‘mijn pensioen anders hadden voorgesteld’, want het was allemaal opgenomen in onze planning, hoewel de laatste vier jaren er natuurlijk als een verrassing extra bij kwamen. Maar jij hebt gelukkig je eigen interesse met je studie kunstgeschiedenis, een onderwerp dat mij ook boeit. Jouw interesse voor wat ik doe blijkt uit het feit dat jij met veel genoegen de Brouwer-biografie hebt gelezen en nu dus over een en ander mee kunt praten. En er was gelukkig nog tijd over voor onze kinderen en kleinkinderen en, afgezien dan misschien van het laatste half jaar, voor geregeld museum- en concertbezoek en voor de vakanties samen. Moge dit alles, naast het verdere werk aan het Brouwer project, nog lang voortgang vinden. Contents Acknowledgements v Preface xi 1 Sets and the continuum before 1907 1 1.1 Cantor (1845–1918) . 2 1.1.1 The discovery of non-equivalent infinite sets . 4 1.1.2 On the equivalence of R1 and Rn .............. 5 1.1.3 On linearly ordered point sets . 6 1.1.4 Irrational numbers, the continuum, the continuum hy- pothesis, reducible and perfect sets . 7 1.1.5 Generation principles (Erzeugungsprinzipien) and the Lim- itation principle (Hemmungsprinzip) for finite and trans- finite numbers . 10 1.1.6 The Cantor-Bernstein theorem and the definition of fur- ther new concepts . 12 1.1.7 Beitrage zur Begr¨undung der transfiniten Mengenlehre . 13 1.1.8 The linear continuum . 14 1.1.9 Order types . 15 1.1.10 Fundamental sequences and perfect sets . 17 1.1.11 Well-ordering . 17 1.2 Dedekind (1831–1916) . 19 1.3 Poincar´e(1854–1912) . 23 1.3.1 The first stage in the construction of the mathematical continuum . 24 1.3.2 The second stage . 24 1.3.3 The measurable quantity . 25 1.3.4 A continuum of the third order? . 25 1.3.5 Mathematical existence and complete induction . 26 1.4 Zermelo (1871–1953) . 27 1.5 Schoenflies (1853–1928) . 28 1.6 Bernstein (1878–1956) . 28 1.7 The paradoxes . 30 1.8 Final remarks . 31 vii viii CONTENTS 2 Brouwer’s ur-intuition of mathematics 33 2.1 Introduction . 33 2.2 The dissertation about the ur-intuition . 35 2.3 Mannoury’s opposition . 39 2.4 Barrau’s objections against the ur-intuition . 40 2.5 The ur-intuition in the Rome and Vienna lectures . 42 2.6 Interpretation of the ur-intuition . 43 2.6.1 The construction of natural numbers . 44 2.6.2 The rational numbers . 46 2.7 The ur-intuition in the inaugural lecture . 48 2.8 The ur-intuition in the notebooks . 50 2.9 On the status of spoken or written signs . 56 2.10 The notebooks on spoken or written signs . 60 2.11 Summary and conclusions of this chapter . 61 3 The continuum, intuitive and measurable 63 3.1 Introduction . 63 3.1.1 Aristotle . 64 3.1.2 Georg Cantor . 65 3.1.3 Hermann Weyl . 65 3.1.4 Otto H¨older . 67 3.1.5 Emile´ Borel . 68 3.1.6 A modern sound . 69 3.2 A few other publications by Brouwer . 69 3.3 Properties of the intuitive continuum . 75 3.4 The notebooks and the intuitive continuum . 84 3.4.1 The continuum, its possible construction from logical prin- ciples alone . 85 3.4.2 The continuum, the intuition . 86 3.4.3 The physical continuum . 88 3.4.4 The mathematical versus the intuitive continuum . 89 3.4.5 The mathematical continuum is the measurable contin- uum with rational scale . 90 3.4.6 The continuum is no point set . 91 3.4.7 The vast majority of the irrational numbers can only be approximated and not defined . 92 3.4.8 The continuum; is there a way to get more grip on it, apart from a constructed scale on it? . 95 3.4.9 The second number class does not exist as a finished totality 99 3.4.10 Attempts to introduce the ‘in practice unmeasurable num- bers’..............................101 3.4.11 Pro and contra the pure and intangible continuum . 105 3.5 Summary and conclusions . 111 CONTENTS ix 4 The possible point sets 113 4.1 Introduction . 113 4.2 Set construction . 114 4.3 The principle of the excluded middle . 121 4.4 The third construction rule . 122 4.5 The review of Schoenflies’ Bericht . 126 4.6 The lecture notes 1915/’16 on Set theory . 128 4.7 Addenda and corrigenda to the dissertation . 130 4.8 The ‘Begr¨undung’ papers, 1918/19 . 133 4.9 Intuitionistische Mengenlehre, 1919 . 135 4.10 The notebooks on set formation . 137 4.10.1 Sets, general . 137 4.10.2 Sets, constructibility as condition for their existence . 139 4.10.3 Sets and the Russell-paradox . 142 4.10.4 Sets, limitations resulting from the method of construction 142 4.10.5 The perfect set cannot be constructed . 144 4.10.6 The second number class . 145 4.10.7 A third and a fourth cardinality . 146 4.10.8 Later developments, suggested in the notebooks . 147 4.11 Concluding remarks about sets . 161 5 The ‘continuum problem’ 165 5.1 Introduction . 165 5.2 The solution in the dissertation . 166 5.3 Two other publications on this problem . 170 5.3.1 The Rome lecture . 170 5.3.2 The inaugural address . 171 5.4 The dissertation again . 173 5.5 The notebooks and the continuum problem . 174 5.6 Concluding remarks . 177 6 Mathematics and experience 179 6.1 Introduction . 179 6.2 Man’s desire for knowledge and control . 181 6.2.1 Comments on Brouwer’s views on physics . 186 6.2.2 The notebooks on causality and man’s desire to rule .
Load more

Recommended publications
  • Some Notes on the Real Numbers and Other Division Algebras
    Some Notes on the Real Numbers and other Division Algebras August 21, 2018 1 The Real Numbers Here is the standard classification of the real numbers (which I will denote by R). Q Z N Irrationals R 1 I personally think of the natural numbers as N = f0; 1; 2; 3; · · · g: Of course, not everyone agrees with this particular definition. From a modern perspective, 0 is the \most natural" number since all other numbers can be built out of it using machinery from set theory. Also, I have never used (or even seen) a symbol for the whole numbers in any mathematical paper. No one disagrees on the definition of the integers Z = f0; ±1; ±2; ±3; · · · g: The fancy Z stands for the German word \Zahlen" which means \numbers." To avoid the controversy over what exactly constitutes the natural numbers, many mathematicians will use Z≥0 to stand for the non-negative integers and Z>0 to stand for the positive integers. The rational numbers are given the fancy symbol Q (for Quotient): n a o = a; b 2 ; b > 0; a and b share no common prime factors : Q b Z The set of irrationals is not given any special symbol that I know of and is rather difficult to define rigorously. The usual notion that a number is irrational if it has a non-terminating, non-repeating decimal expansion is the easiest characterization, but that actually entails a lot of baggage about representations of numbers. For example, if a number has a non-terminating, non-repeating decimal expansion, will its expansion in base 7 also be non- terminating and non-repeating? The answer is yes, but it takes some work to prove that.
    [Show full text]
  • Carnegie Mellon University 21-300/21-600 Fall 2012 Assignment 6 Due October 17, 2012
    Carnegie Mellon University 21-300/21-600 Fall 2012 Assignment 6 Due October 17, 2012 ¬ Let S be a first-order signature. A theory Γ is said to be com- plete if for every S-sentence ', either Γ j= ' or Γ j= :'. Prove that if Γ is complete and M is a model of Γ, then the set of logical consequences of Γ consists of all the sentences that are true in M. Solution. Suppose that Γ is complete and M is a model of Γ. We have to prove that f ' j M j= ' g = f ' j Γ j= ' g: Proof of ⊆: Suppose M j= '. Then Γ j= ', because other- wise, since Γ is complete, we would have Γ j= :', and then, since M is a model of Γ, we would also have M j= :', which is impossible. Proof of ⊇: Suppose Γ j= '. Then M j= ', because otherwise we would have M j= :', and then, since M is a model of Γ and Γ j= ', we would also have M j= ', which is impossible. ­ Let R be the structure (R; +; ·), where + and · are the usual addition and multiplication, respectively. p (a) Exhibit a formula that defines the set f 2g in R. Solution. Let '(x) be x2 = 1 + 1 ^ 9y(y2 = x). (b) Prove that every finite union of intervals in R whose end- points are algebraic numbers is definable in R. (Here, the intervals may be of any kind: open, closed, semiopen, bounded, unbounded, etc.) Solution. We show this in stages. First, let's observe that every rational number is definable.
    [Show full text]
  • The Continuum, Intuitive and Measurable
    Chapter 3 The continuum, intuitive and measurable 3.1 Introduction This chapter covers Brouwer’s analysis of the result of the ur-intuition, viz. the intuitive continuum, its character and its properties.1 We ended the preceding chapter with the remark that the intuitive continuum as the immediate corol- lary of the ur-intuition was emphasized repeatedly in the dissertation, as well as in other publications of later date. Also after the introduction and the develop- ment of the ‘perfect spread’, which gives us the ‘full continuum’ of all unfinished choice sequences on the unit interval, the continuum still remains the immedi- ate result of the ur-intuition. We will see that, in order to prevent confusion, the different concepts about the continuum (like the ‘intuitive’ continuum, the ‘reduced’ continuum, the ‘full’ continuum) have to be distinguished sharply. We find the origin of Brouwer’s continuum concept, as we saw, in the flow of time as the medium of cohesion between two well-separated events. Apart from the construction of the integers and of the rational scale, more properties of the continuum and of these scales can be deduced, still on the basis of the mathematical intuition and of the concept of mathematics as a free creation of the individual human mind. Several properties of the continuum were discussed in the previous chapter, and new ones will come up in the present one, again requiring interpretation about what exactly Brouwer had in mind, like the idea of a ‘freely constructed scale’, or the property of being ‘partlially unknown’ of an infinite sequence approximating a point.
    [Show full text]
  • The Octonionic Projective Plane
    The octonionic projective plane Malte Lackmann 1 Introduction As mathematicians found out in the last century, there are only four normed di- vision algebras1 over R: the real numbers themselves, the complex numbers, the quaternions and the octonions. Whereas the real and complex numbers are very well-known and most of their properties carry over to the quaternions (apart from the fact that these are not commutative), the octonions are very different and harder to handle since they are not even associative. However, they can be used for several interesting topological constructions, often paralleling constructions known for R, C or H. In this article, we will construct OP2, a space having very similar properties to the well-known two-dimensional projective spaces over R, C and H. We will begin, in the second section, by recalling a construction of the octonions and discussing their basic algebraic properties. We will then move on to actually constructing the octonionic projective plane OP2. In the fourth section, we will dis- cuss properties of OP2 and applications in algebraic topology. In particular, we will use it to construct a map S15 −! S8 of Hopf invariant 1. Moreover, it will be ex- plained why there cannot be projective spaces over the octonions in dimensions higher than 2. Malte Lackmann Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, e-mail: [email protected] 1 A division algebra over R is a finite-dimensional unital R-algebra without zero divisors, not necessarily commutative or associative. A normed algebra over R is an R-algebra A together with a map k·k : A ! R coinciding with the usual absolute value on R · 1 =∼ R, satisfying the triangular inequality, positive definiteness and the rule kxyk = kxkkyk.
    [Show full text]
  • Bergsonian Axioms for Physics
    Bergsonian Axioms For Physics Doug McLellan 2017 We show here how a more philosophically satisfying foundation for physics might grow from a century-old insight of Henri Bergson, that the world is the continual creation of new possibilities, rather than the successive realization of pre-existing possibilities. This insight invigorated fields both scientific and humanistic, and helped win Bergson a Nobel prize (Literature, 1927), yet it vanished from the intellectual scene because the mathematics did not exist to make it precise. Today these mathematics do exist; we will use them to translate Bergson's insight into axioms that a physical theory ought to satisfy. The core idea motivating this paper is a mathematical version of Berg- son's insight: new mathematical structures are made possible over time. To demonstrate its appeal we will take up each of four long-standing philosoph- ical puzzles, show why each withstands current lines of attack, and let the core idea suggest a postulate that might resolve it. The puzzles that give rise to the postulates are (1) the arbitrariness of natural laws, (2) the in- famous set-of-all-sets paradox, (3) the arbitrariness of space-time structure, and (4) the independence of the continuum hypothesis from all \normal" mathematical considerations. Our fundamental first postulate will identify each point in space-time with the totality of mathematical possibilities there; the other postulates will de- mand that the growth of new possibilities be \philosophically innocuous" in various senses. Once the postulates are stated it will be a fairly straight- forward matter to translate them into axioms.
    [Show full text]
  • Differential Geometry Revisited by Biquaternion Clifford Algebra Patrick Girard, Patrick Clarysse, Romaric Pujol, Liang Wang, Philippe Delachartre
    Differential Geometry Revisited by Biquaternion Clifford Algebra Patrick Girard, Patrick Clarysse, Romaric Pujol, Liang Wang, Philippe Delachartre To cite this version: Patrick Girard, Patrick Clarysse, Romaric Pujol, Liang Wang, Philippe Delachartre. Differential Ge- ometry Revisited by Biquaternion Clifford Algebra. Lecture Notes in Computer Science - Curves and Surfaces, 9213, pp.216-242, 2015, 978-3-319-22804-4. 10.1007/978-3-319-22804-4_17. hal-01622682 HAL Id: hal-01622682 https://hal.archives-ouvertes.fr/hal-01622682 Submitted on 24 Oct 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. DIFFERENTIAL GEOMETRY REVISITED BY BIQUATERNION CLIFFORD ALGEBRA Patrick R. Girard1, Patrick Clarysse1, Romaric Pujol2, Liang Wang1, and Philippe Delachartre1 1 Universit´ede Lyon, CREATIS; CNRS UMR 5220; Inserm U1044; INSA-Lyon; Universit´eLYON 1, France, B^at.Blaise Pascal, 7 avenue Jean Capelle, F-69621 Villeurbanne, France 2 Universit´ede Lyon, P^olede Math´ematiques,INSA-Lyon, B^at.L´eonardde Vinci, 21 avenue Jean Capelle, F-69621 Villeurbanne, France Abstract. In the last century, differential geometry has been expressed within various calculi: vectors, tensors, spinors, exterior differential forms and recently Clifford algebras. Clifford algebras yield an excellent repre- sentation of the rotation group and of the Lorentz group which are the cornerstones of the theory of moving frames.
    [Show full text]
  • Cauchy-Riemann Operators in Octonionic Analysis
    Cauchy-Riemann Operators in Octonionic Analysis Janne Kauhanen∗ and Heikki Orelma† January 31, 2017 Abstract In this paper we first recall the definition of an octonion algebra and its algebraic properties. We derive the so called e4-calculus and using it we obtain the list of generalized Cauchy-Riemann systems in octonionic monogenic functions. We define some bilinear forms and derive the corresponding symmetry groups. Mathematics Subject Classification (2010). 30G35, 15A63 Keywords. Octonions, Cauchy-Riemann operators, Monogenic functions 1 Introduction The algebra of octonions is a well known non-associative division algebra. The second not so well known feature is, that we may define a function the- ory, in spirit of classical theory of complex holomorphic functions, and study its properties. This theory has its limitations, since the multiplication is neither commutative nor associative. The first part of this paper is a sur- arXiv:1701.08698v1 [math.CV] 30 Jan 2017 vey of known results, where we give a detailed definition for the octonions. We derive the so called ”e4-calculus” on it, to make our practical calcula- tions easier. Then we recall the notion of the Cauchy-Riemann operator. A function in its kernel is called monogenic. To find explicit monogenic func- tions directly from the definitions is too complicated, because the algebraic properties give too many limitations. To give an explicit characterization of ∗Laboratory of Mathematics, Faculty of Natural Sciences, Tampere University of Tech- nology, Finland. Electronic address: [email protected] †Electronic address: [email protected] 1 monogenic functions, we separate variables, or represent the target space as a direct sum of subalgebras.
    [Show full text]
  • Beyond Complex Numbers
    Beyond complex numbers Johanna R¨am¨o July 22, 2010 These notes are written for the Goldsmiths’ Company Mathematics Course for Teachers in July 2010. We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers, then to positive rationals, ir- rational reals, negatives and finally to complex numbers. It has not always been easy to accept new numbers. Negative numbers were rejected for cen- turies, and complex numbers, the square roots of negative numbers, were considered impossible. Complex numbers behave like ordinary numbers. You can add, subtract, multiply and divide them, and on top of that, do things which you cannot do with real numbers. Today, complex numbers have many important ap- plications in mathematics and physics, and scientists could not live without them. What if we take the next step? What comes after the complex numbers? Is there a bigger number system that has the same nice properties as the real numbers and the complex numbers? The answer is yes. In fact, there are two (and only two) bigger number systems that resemble real and complex numbers, and their discovery has been almost as dramatic as that of the complex numbers. 1 Complex numbers Complex numbers where discovered in the 16th century when Italian math- ematicians tried to find a general solution to the cubic equation x3 + ax2 + bx + c = 0. 1 At that time, mathematicians did not publish their results but kept them secret. They made their living by challenging each other to public contests of problem solving in which the winner got money and fame.
    [Show full text]
  • Horadam Octonions
    DOI: 10.1515/auom-2017-0038 An. S¸t. Univ. Ovidius Constant¸a Vol. 25(3),2017, 97{106 Horadam Octonions Adnan KARATAS¸and Serpil HALICI Abstract In this paper, first we define Horadam octonions by Horadam se- quence which is a generalization of second order recurrence relations. Also, we give some fundamental properties involving the elements of that sequence. Then, we obtain their Binet-like formula, ordinary gen- erating function and Cassini identity. 1 Introduction Four dimensional quaternion algebra is defined by W. R. Hamilton in 1843. After the definition of quaternion algebra, J. T. Graves asked Hamilton about higher dimensional algebras. He tried to define eight and sixteen dimensional algebras, but Graves encountered zero divisors on sixteen dimensional algebra. Because of zero divisors, Graves defined only eight dimensional real octonion algebra in 1843. Besides Graves' definition, A. Cayley defined also octonions in 1845. For detailed knowledge reader referred to [1]. Octonion algebra is eight dimensional, non-commutative, non-associative and normed division algebra. This algebra is used in various physical problems such as; supersymmetry, su- per gravity and super strings. In addition octonions are used in many subjects such as, coding, unified fields of dyons and gravito-dyons, electrodynamic, elec- tromagnetism. For background the reader can find their properties in [1, 3, 4, 5, 6, 8]. Key Words: Fibonacci Numbers and Generalization, Octonions 2010 Mathematics Subject Classification: 11B39, 17A20 Received: 19.12.2016 Revised: 3.03.2017 Accepted: 13.03.2017 97 HORADAM OCTONIONS 98 The octonion algebra O form an eight-dimensional real algebra with a ba- sis f1; e1; e2; : : : ; e7g.
    [Show full text]
  • [Math.GM] 29 Jan 2006 OI O EVERYONE for LOGIC Oeta Herrmann A
    LOGIC FOR EVERYONE Robert A. Herrmann arXiv:math/0601709v1 [math.GM] 29 Jan 2006 1 Previous titled “Logic For Midshipmen” Mathematics Department U. S. Naval Academy 572C Holloway Rd. Annapolis, MD 21402-5002 2 CONTENTS Chapter 1 Introduction 1.1 Introduction.................................... 5 Chapter 2 The Propositional Calculus 2.1 ConstructingaLanguagebyComputer . .......9 2.2 ThePropositionalLanguage . .....9 2.3 Slight Simplification, Size, CommonPairs . 13 2.4 ModelTheory—BasicSemantics . ... 15 2.5 ValidFormula .................................. 19 2.6 EquivalentFormula. 22 2.7 The Denial, Normal Form, LogicCircuits . 26 2.8 The Princeton Project, ValidConsequences . 32 2.9 ValidConsequences. 35 2.10 SatisfactionandConsistency . ...... 38 2.11 ProofTheory.................................. 42 2.12 Demonstrations, DeductionfromPremises . 45 2.13 TheDeductionTheorem . 47 2.14 DeducibilityRelations . .... 50 2.15 TheCompletenessTheorem . .. 52 2.16 ConsequenceOperators . .. 54 2.17 TheCompactnessTheorem. .. 57 Chapter 3 Predicate Calculus 3.1 First-OrderLanguage. .. 63 3.2 FreeandBoundVariableOccurrences . ....... 67 3.3 Structures.................................... 70 3.4 Valid Formula in P d. ............................... 76 3.5 ValidConsequencesandModels . .... 82 3.6 FormalProofTheory . 86 3.7 Soundness and Deduction Theorem for P d′ ..................... 87 3.8 Consistency, Negation Completeness, Compactness, Infinitesimals . 91 3.9 UltralogicsandNaturalSystems . ...... 97 Appendix Chapter2 ..................................
    [Show full text]
  • Classical Number Fields
    Physics as Generalized Number Theory: Classical Number Fields M. Pitk¨anen Email: [email protected]. http://tgdtheory.com/public_html/. January 18, 2012 Abstract Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article the connection between standard model symmetries and classical number fields is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose K¨ahlergeometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M 4 × S and the idea is that the symmetries of the K¨ahlermanifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M 8 can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. 8 4 Could one understand S = CP2 number theoretically in the sense that M and H = M ×CP2 be in some deep sense equivalent ("number theoretical compactification” or M 8 − H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M 8 or equivalently of H.
    [Show full text]
  • Matheology.Pdf
    Matheology § 001 A matheologian is a man, or, in rare cases, a woman, who believes in thoughts that nobody can think, except, perhaps, a God, or, in rare cases, a Goddess. § 002 Can the existence of God be proved from mathematics? Gödel proved the existence of God in a relatively complicated way using the positive and negative properties introduced by Leibniz and the axiomatic method ("the axiomatic method is very powerful", he said with a faint smile). http://www.stats.uwaterloo.ca/~cgsmall/ontology.html http://userpages.uni-koblenz.de/~beckert/Lehre/Seminar-LogikaufAbwegen/graf_folien.pdf Couldn't the following simple way be more effective? 1) The set of real numbers is uncountable. 2) Humans can only identify countably many words. 3) Humans cannot distinguish what they cannot identify. 4) Humans cannot well-order what they cannot distinguish. 5) The real numbers can be well-ordered. 6) If this is true, then there must be a being with higher capacities than any human. QED [I K Rus: "Can the existence of god be proved from mathematics?", philosophy.stackexchange, May 1, 2012] http://philosophy.stackexchange.com/questions/2702/can-the-existence-of-god-be-proved-from- mathematics The appending discussion is not electrifying for mathematicians. But a similar question had been asked by I K Rus in MathOverflow. There the following more educational discussion occurred (unfortunately it is no longer accessible there). (3) breaks down, because although I can't identify (i.e. literally "list") every real number between 0 and 1, if I am given any two real numbers in that interval then I can distinguish them.
    [Show full text]