FOUNDATIONS OF EUCLIDEAN CONSTRUCTIVE GEOMETRY MICHAEL BEESON Abstract. Euclidean geometry, as presented by Euclid, consists of straightedge-and- compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathe- matics” leads to the development of a first-order theory ECG of “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. Using Gentzen’s cut-elimination theorem, we show that when ECG proves an existential theorem, then the things proved to exist can be constructed by Eu- clidean ruler-and-compass constructions. In the second part of the paper we take up the formal relationships between three versions of Euclid’s parallel postulate: Euclid’s own formulation in his Postulate 5, Playfair’s 1795 version, which is the one usually used in modern axiomatizations, and the version used in ECG. We completely settle the questions about which versions imply which others using only constructive logic: ECG’s version im- plies Euclid 5, which implies Playfair, and none of the reverse implications are provable. The proofs use Kripke models based on carefully constructed rings of real-valued functions. “Points” in these models are real-valued functions. We also characterize these theories in terms of different constructive versions of the axioms for Euclidean fields.1 Contents 1. Introduction 5 1.1. Euclid 5 1.2. The collapsible vs. the rigid compass 7 1.3. Postulates vs. axioms in Euclid 9 1.4. The parallel postulate 9 1.5. Polygons in Euclid 10 2. Euclid’s reasoning considered constructively 10 2.1. Order on a line from the constructive viewpoint 10 2.2. Logical form of Euclid’s propositions and proofs 14 2.3. Case splits and reductio inessential in Euclid 15 3. Constructions in geometry 16 3.1. The 48 Euclidean constructions 16 3.2. The elementary constructions 16 3.3. Circles and lines eliminable when contructing points 19 3.4. Describing constructions by terms 20 4. Models of the elementary constructions 21 4.1. The standard plane 21 4.2. The recursive model 22 4.3. The algebraic model 23 4.4. The Tarski model 23 1 2 MICHAEL BEESON 5. The geometrization of arithmetic without case distinctions 23 5.1. Uniform perpendicular and projection 24 5.2. Rotation 27 5.3. Addition 28 5.4. Multiplication and division 28 5.5. Square roots 31 5.6. Formalizing the geometrization of arithmetic 32 6. The arithmetization of geometry 32 6.1. Euclidean fields in constructive mathematics 32 6.2. Line-circle and circle-circle continuity over Euclideanfields 34 7. Otheraxiomatizationsofgeometry 36 7.1. Pasch 37 7.2. Pieri and Peano 37 7.3. Hilbert 37 7.4. Tarski and his students 38 7.5. Tarski’stheoryofEuclideangeometry 40 7.6. Borsuk-Szmielew 40 7.7. Avigad, Dean, and Mumma 40 7.8. Heyting and other constructive approaches 41 8. The theory ECG of Euclidean Constructive Geometry 41 8.1. Logic of Partial Terms (LPT) 41 8.2. Replacing LPT and sorts with predicates if desired 43 8.3. Language 44 8.4. Intuitionistic logic and stability 45 8.5. Incidence and intersection axioms 46 8.6. Constructorandaccessoraxioms 47 8.7. Meaning of equality 48 8.8. Betweenness axioms 48 8.9. Sides of a line 51 8.10. Right and Left : handedness and sides of lines 52 8.11. Dimension axioms 54 8.12. Rays and segments 54 8.13. Congruence axioms 55 8.14. Line-circle continuity 56 8.15. Intersections of circles 58 9. Development of neutral geometry in ECG 61 9.1. Why we cannot import negative theorems from Tarski’s theory 62 9.2. Congruence and betweenness lemmas 64 9.3. Ordering of segments 68 9.4. Angles and triangles 70 9.5. Perpendicular lines 74 9.6. Existence of midpoints and perpendiculars 75 9.7. Right angles 76 9.8. Various forms of the Pasch axiom 82 9.9. Inner Pasch implies Outer Pasch and plane separation 84 9.10. The crossbar theorem 85 9.11. Ordering of angles 86 FOUNDATIONS OF EUCLIDEAN CONSTRUCTIVE GEOMETRY 3 9.12. Triangle construction and uniqueness 87 9.13. Euclid Books I, II, and III 92 9.14. Consequencesoftheupperdimensionaxiom 94 9.15. Right and left turns and sides of lines 99 9.16. Definability of IntersectCirclesSame and IntersectCirclesOpp 101 9.17. Rotation and Uniform Reflection 104 9.18. The other intersection point 110 9.19. Defining the order of points on a line 111 10. The parallel postulate 113 10.1. Alternate Interior Angles 113 10.2. Euclid’s parallel postulate 115 10.3. The strong parallel postulate 116 10.4. Playfair’s axiom 118 10.5. Another version of the strong parallel postulate 120 10.6. Coordinates 121 10.7. The strong parallel postulate and stability of IntersectLines 123 10.8. Stability of definedness and ∼= 125 10.9. ECG proves Euclid’s parallel postulate 129 10.10. Variousformsoftheparallelaxiom 131 11. Comparisons of ECG to other formalizations 136 11.1. Comparison of ECG to Hilbert-style theories 136 11.2. Comparison of ECG to Tarski’s theories 136 12. Connections between geometry and Euclidean fields 138 12.1. Signed addition 139 12.2. Signed multiplication 144 12.3. The distributive law 149 12.4. Reciprocals 150 12.5. Interpreting ECG in field theory 151 12.6. Area in Euclid, Hilbert, and ECG 174 12.7. Handedness, cross product, and linear transformations 181 12.8. Interpreting field theory in ECG 192 12.9. Faithfulness of the interpretations 201 13. Classical geometry and constructive geometry compared 219 13.1. The double negation interpretation 219 13.2. The double-negation interpretation applied to ECG 221 14. The relation of ECG toTarski’stheoriesofgeometry 222 14.1. Interpretation of Tarksi’s theory in ECG 225 15. Interpretation of ECG in Hilbert’s and Tarski’s theories 227 15.1. Right and left handedness 227 15.2. Apartness 228 16. Metatheorems 230 16.1. Things proved to exist in ECG canbeconstructed 230 16.2. Disjunction properties 232 17. Independence results for the parallel axioms 233 17.1. Kripke models of ring theory 233 17.2. Euclid 5 does not imply Axiom 58 235 17.3. Playfair does not imply Euclid 5 239 4 MICHAEL BEESON 17.4. Independence of Markov’s principle 240 18. Apartness 240 18.1. Constructions and Apartness 240 18.2. Euclidean fields and apartness 240 18.3. Apartness and the parallel axioms 240 19. Appendix A: Roads not taken 242 19.1. Rigid compass undefinable from the collapsible compass 242 19.2. Rigid compass not definable, even with degenerate circles 244 19.3. Circles of zero radius, the rigid compass, and projection 246 19.4. Constructing the center of a circle 247 20. Appendix B: Possible reductions in the primitive construction mechanisms 248 20.1. TheMohr-Mascheronitheorem 248 20.2. Strommer’s theorem 248 FOUNDATIONS OF EUCLIDEAN CONSTRUCTIVE GEOMETRY 5 §1. Introduction. 1.1. Euclid. Euclid’s geometry, written down about 300 BCE, has been ex- traordinarily influential in the development of mathematics, and prior to the twentieth century was regarded as a paradigmatic example of pure reasoning.1 During those 2300 years, most people thought that Euclid’s theory was about something. What was it about? Some may have answered that it was about points, lines, and planes, and their relationships. Others may have said that it was about methods for constructing points, lines, and planes with certain specified relationships to given points, lines, and planes, for example, construct- ing an equilateral triangle with a given side. In these two answers, we see the viewpoints of pure (classical) mathematics and of algorithmic mathematics rep- resented. Near the end of the nineteenth century, the Italian “Peano school”, and especially Mario Pieri, began to use the “hypothetical-deductive method”, today called the “axiomatic method.” At the same time, and apparently without communication, Hilbert took a similar approach, and published a famous book [15] in the last year of the century. According to the axiomatic method, Euclid’s theories were not about anything at all. Instead of “points, lines, and planes”, one should be able to read “tables, chairs, and beer mugs.” All the reasoning should still be valid. The names of the “entities” were just place holders. That was the viewpoint of twentieth-century axiomatics. In this paper, we re-examine Euclidean geometry from the viewpoint of con- structive mathematics. The phrase “constructive geometry” suggests, on the one hand, that “constructive” refers to geometrical constructions with straightedge and compass. On the other hand, the word “constructive” may suggest the use of intuitionistic logic. We investigate the connections between these two mean- ings of the word. Our method is to keep the focus on the body of mathematics in Euclid’s Elements, and to examine what in Euclid is constructive, in the sense of “constructive mathematics”. Our aim in the first phase of this research was to formulate a suitable formal theory that would be faithful to both the ideas of Euclid and the constructive approach of Errett Bishop. We did achieve this aim, and the resulting theory ECG of “Euclidean constructive geometry” is presented in this paper. In constructive mathematics, if one proves something exists, one has to show how to construct it. In Euclid’s geometry, the means of construction are not arbitrary computer programs, but ruler and compass. Therefore it is natural to look for quantifier-free axioms, with function symbols for the basic ruler-and- compass constructions. The terms of such a theory correspond to ruler-and- compass constructions.