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Axiomatizing changing conceptions of the geometric continuuum II: Archimedes - Descartes-Hilbert -Tarski John T. Baldwin
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Axiomatizing changing conceptions of the geometric continuuum II: Archimedes - Descartes-Hilbert -Tarski John T. Baldwin Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago∗ January 5, 2015
In Part I [Bal14a], we defined the notion of a modest complete descriptive axiomatization and showed that HP5 and EG are such axiomatizations of Euclid’s polygonal geometry and Euclidean circle geometry1 . In this paper we argue: 1) Tarski’s axiom set E 2 is a modest complete descriptive axiomatization of Cartesian geometry 2 (Section 2; 2) the theories EGπ,C,A and Eπ,C,A are modest complete descriptive axiomatizations of Euclidean circle geometry and Cartesian geometry, respectively when extended by formulas computing the area and circumference of a circle (Section 3); and 3) that Hilbert’s system in the Grundlagen is an immodest axiomatization of any of these geometries. As part of the last claim (Section 4), we analyze the role of the Archimdedean postulate in the Grundlagen, trace the intricate relationship between alternative formulations of ‘Dedekind completeness’, and exhibit many other categorical axiomatizations of related geometries.
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Background and Definitions
In [Bal14a], we expounded the following historical description. Remark 1.1 (Background). Euclid founds his theory of area (of circles and polygons) on Eudoxus’ theory of proportion and thus (implicitly) on the axiom of Archimedes. Hilbert shows any ‘Hilbert plane’ interprets a field and recovers Euclid’s theory of polygons in a first order theory. ∗ Research
partially supported by Simons travel grant G5402. axioms are described in Notation 1.3; these data sets are described as specific sets of propositions from the Elements in [Bal14a]. 1 These
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The Greeks and Descartes dealt only with geometric objects. The Greeks regarded multiplication as an operation from line segments to plane figures. Descartes interpreted it as an operation from line segments to line segments. Only in the late 19th century, is multiplication regarded as an operation on points (that is ‘numbers’ in the coordinatizing field). We built in [Bal14a] on Detlefsen’s notion of complete descriptive axiomatization and defined a modest complete descriptive axiomatization of a data set Σ to be collection of sentences that imply all the sentences in Σ and ‘not too many more’. A data set is a collection of propositions about a mathematical topic that are accepted at a given point in time. Of course, there will be further results proved about this topic. But if this set of axioms introduces essentially new concepts to the area or, even worse, contradicts the understanding of the original era, we deem the axiomatization immodest. We illustrate these definitions by specific axiomatizations of various areas of geometry that we now describe. We formulate our system in a two-sorted vocabulary τ chosen to make the Euclidean axioms (either as in Euclid or Hilbert) easily translatable into first order logic. This vocabulary includes unary predicates for points and lines, a binary incidence relation, a ternary betweenness relation, a quaternary relation for line congruence and a 6-ary relation for angle congruence. We need one additional first order postulate2 beyond those in [Hil71]. Postulate 1.2. Circle Intersection Postulate If from points A and B, circles with radius AC and BD are drawn such that one circle contains points both in the interior and in the exterior of the other, then they intersect in two points, on opposite sides of AB. Notation 1.3. We follow Hartshorne[Har00] in the following nomenclature. A Hilbert plane is any model of Hilbert’s incidence, betweenness3 , and congruence axioms4 . We denote this axiom as HP and write HP5 for these axioms plus the parallel postulate. By the axioms for Euclidean geometry we mean HP5 and in addition the circle-circle intersection postulate 1.2. We will abbreviate this axiom set5 as EG. By 2 Moore suggests in [Moo88] that Hilbert may have added the completeness axiom to the second edition specifically because Sommer in his review of the first edition pointed out it did not prove the line-circle intersection principle. 3 These include Pasch’s axiom (B4 of [Har00]) as we axiomatize plane geometry. Hartshorne’s version of Pasch is that any line intersecting one side of triangle must intersect one of the other two. 4 These axioms are equivalent to the common notions of Euclid and Postulates I-V augmented by one triangle congruence postulate, usually taken as SAS since the ‘proof’ of SAS is where Euclid makes illegitimate use of the superposition principle. 5 In the vocabulary here, there is a natural translation of Euclid’s axioms into first order statements. The construction axioms have to be viewed as ‘for all– there exist’ sentences. The axiom of Archimedes as discussed below is of course not first order. We write Euclid’s axioms for those in the original [Euc56] vrs (first order) axioms for Euclidean geometry, EG. Note that EG is equivalent to (i.e. has the same models) as the system laid out in Avigad et al [ADM09], namely, planes over fields where every positive element as a square root). The latter system builds the use of diagrams into the proof rules.
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definition, a Euclidean plane is a model of EG: Euclidean geometry. We write E 2 for Tarski’s geometrical axiomatization of the plane over a real closed field (RCF) (Theorem 2.1).
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From Descartes to Tarski
It is not our intent to give a detailed account of Descartes’ impact on geometry. We want to bring out the changes from the Euclidean to the Cartesian data set. For our purposes, the most important is to explicitly (on page 1 of [Des54]) define the multiplication of line segments to give a line segment which breaks with Greek tradition6 . And later on the same page to announce constructions for the extraction of nth roots for all n. The second of these cannot be done in EG, since it is satisfied in the field which has solutions for all quadratic equations but not those of odd degree7 . Marco Panza [Pan11] formulates in terms of ontology a key observation, The first point concerns what I mean by ‘Euclid’s geometry’. This is the theory expounded in the first six books of the Elements and in the Data. To be more precise, I call it ‘Euclids plane geometry’, or EPG, for short. It is not a formal theory in the modern sense, and, a fortiori, it is not, then, a deductive closure of a set of axioms. Hence, it is not a closed system, in the modern logical sense of this term. Still, it is no8 more a simple collection of results, nor a mere general insight. It is rather a well-framed system, endowed with a codified language, some basic assumptions, and relatively precise deductive rules. And this system is also closed, in another sense ([Jul64] 311-312), since it has sharp-cut limits fixed by its language, its basic assumptions, and its deductive rules. In what follows, especially in section 1, I shall better account for some of these limits, namely for those relative to its ontology. More specifically, I shall describe this ontology as being composed of objects available within this system, rather than objects which are required or purported to exist by force of the assumptions that this system is based on and of the results proved within it. This makes EPG radically different from modern mathematical theories (both formal and informal). One of my claims is that Descartes geometry partially reflects this feature of EPG. In our context we interpret ‘composed of objects available within this system’ model theoretically as the existence of certain starting points and the closure of each model of the system under admitted constructions. 6 His
proof is still based on Eudoxus. section 12 of [Har00]. 8 There appears to be a typo. Probably ‘more a” should be deleted. jb 7 See
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We take Panza’s ‘open’ system to refer to Descartes’ ‘linked constructions’9 considered by Descartes which greatly extend the ruler and compass licensed in EG. Descartes endorses such ‘mechanical’ constructions as the duplication of the cubic as geometric. According to Molland (page 38 of [Mol76]) ”Descartes held the possibility of representing a curve by an equation (specification by property)” to be equivalent to its ”being constructible in terms of the determinate motion criterion (specification by genesis)”. But as Crippa points out (page 153 of [Cri14a]) Descartes did not prove this equivalence; there is some controversy as to whether the 1876 work of Kempe solves the precise problem. Descartes rejects as non-geometric any method for quadrature of the circle. See page 48 of [Des54] for his classification of problems by degree. Unlike Euclid, Descartes does not develop his theory axiomatically. But an advantage of the ‘descriptive axiomatization’ rubric is that we can take as data the theorems of Descartes’ geometry. For our purpose we take the common identification of Cartesian geometry with ”real” algebraic geometry: the study of polynomial equalities and inequalities in the theory of real closed fields. To justify this geometry we adapt Tarski’s ‘elementary geometry’. This move makes a significant conceptual step away from Descartes whose constructions were on segments and who did not regard a line as a set of points while Tarski’s axiom are given entirely formally in a one sorted language of relations on points. In our modern understanding of an axiom set the translation is routine. From Tarski [Tar59] we get: Theorem 2.1. Tarski [Tar59] gives a theory equivalent to the following system of axioms E 2 . It is first order complete for the vocabulary τ . 1. Euclidean geometry 2. Either of the following two sets of axioms which are equivalent over i) (a) An infinite set of axioms declaring that every polynomial of odd-degree has a root. (b) The axiom schema of continuity described just below. The connection with Dedekind’s approach is seen by Tarski’s actual formulation as in [GT99]; the first order completeness of the theory is imposed by an Axiom Schema of Continuity - a definable version of Dedekind cuts: (∃a)(∀x)(∀y)[α(x) ∧ β(y) → B(axy)] → (∃b)(∀x)(∀y)[α(x) ∧ β(y) → B(xby)], where B(x, y, z) is the predicate representing y is between x and z, α, β are first-order formulas, the first of which does not contain any free occurrences of a, b, y nor the second any free occurrences of a, b, x. This schema allows the solution of odd degree polynomials. 9 The types of constructions allowed are analyzed in detail in Section 1.2 of [Pan11] and the distinctions with the Cartesian view in Section 3. See also [Bos01].
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Remark 2.2 (G¨odel completeness). In Detlefsen’s terminology we have found a G¨odel complete axiomatization of (in our terminology Cartesian) plane geometry. This guarantees that if we keep the vocabulary and continue to accept the same data set no axiomatization can account for more of the data. There are certainly open problems in plane geometry [KW91]. But however they are solved the proof will be formalizable in E 2 . Of course, more perspicuous axiomatizations may be found. Or one may discover the entire subject is better viewed as an example in a more general context. In the case at hand, however, there are more specific reasons for accepting the geometry over real closed fields as ‘the best’ descriptive axiomatization. It is the only one which is decidable and ‘constructively justifiable’ (See Theorem 3.3.). Remark 2.3 (Undecidability and Consistency). Ziegler [Zie82] has shown that every nontrivial finitely axiomatized subtheory of RCF10 is not decidable. Thus both to more closely approximate the Dedekind continuum and to obtain decidability we restrict to planes over RCF and thus to Tarski’s E 2 [GT99]. Of course, another crucial contribution of Descartes is coordinate geometry. Tarski provides a converse; his interpretation of the plane into the coordinatizing line [Tar51] underlies our smudging of the study of the ‘geometry continuum’ with axiomatizations of ‘geometry’. The biinterpretability11 between RCF and the theory of all planes over real closed fields yields the decidability of E 2 . The crucial fact that makes decidability possible is that the natural numbers are not first order definable in the real field. The geometry can represent multiplication as repeated addition in the sense of a module over a Z but not with the full ring structure.
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Archimedes: π, circumference and area of circles
The geometry over a Euclidean field (every positive number has a square root) may have no straight line segment of length π, since the model containing only the constructible real numbers does not contain π. We extend by adding π each geometry EG and E 2 and write T when discussing results that apply to either. We want to find a theory which proves the circumference and area formulas for circles. Our approach is to extend the theory EG so as to guarantee that there is a point in every model which behaves as π does. In this section we will show that in this extended theory there is a mapping assigning a straight line segment to the circumference of each circle. This goal definitely diverges from a ‘Greek’ data set and is orthogonal to the axiomatization of Cartesian geometry in Section 2. Given that the entire project is modern, we give the arguments entirely in modern style. 10 RCF abbreviates ‘real closed field’; these are the ordered fields such that every positive element has a square root and every odd degree polynomial has at least one root. The theory is complete and recursively axiomatized so decidable. By nontrivial subtheory, I mean one satisfied by one of C,