# Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski

Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski ABSTRACT In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi’s for Cartesian geometry and new systems for adding $$\pi$$. In contrast we find Hilbert’s full second-order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis. This two-part paper analyzes the axiomatic foundation of the ‘geometric continuum’, the line embdedded in the plane. For this, we built on Detlefsen’s notion of complete descriptive axiomatization and defined in Part I [Baldwin, 2017] a modest complete descriptive axiomatization of a data set $$\Sigma$$ (essentially, of facts in the sense of Hilbert) to be a collection of sentences that imply all the sentences in $$\Sigma$$ and ‘not too many more’. Of course, this set of facts will be open-ended, since over time more results will be proved. But if this set of axioms introduces essentially new concepts to the area and certainly if it contradicts the conceptions of the original era, we deem the axiomatization immodest. Part I dealt primarily with Hilbert’s first-order axioms for polygonal geometry and argued that the first-order systems HP5 and EG (defined below) are ‘modest’ complete descriptive axiomatization of most of Euclidean geometry. Part II concerns areas of geometry, e.g., circles, where stronger assumptions are needed. Hilbert postulated his ‘continuity axioms’ — the Archimedean and completeness axioms in extensions of first-order logic; we pursue weaker axioms. In Section 1, we reprise our organization of various ‘data sets’ for geometry and describe the axiom systems. We contend: (1) that Tarski’s first-order axiom set $$\mathcal{E}^2$$ is a modest complete descriptive axiomatization of Cartesian geometry (Section 2); (2) that the theories EG$$_{\pi,C,A}$$ and $$\mathcal{E}^2_{\pi,C,A}$$ are modest complete descriptive axiomatizations of extensions of these geometries designed to describe area and circumference of the circle (Section 3); and (3) that, in contrast, Hilbert’s full second-order system in the Grundlagen is an immodest axiomatization of any of these geometries but a modest descriptive axiomatization of the late nineteenth-century conception of the real plane (Section 4). We elaborate and place this study in a more general context in our book [Baldwin, 2018]. 1. TERMINOLOGY AND NOTATIONS Part I provided the following quasi-historical description. Euclid founded his theory of area for circles and polygons on Eudoxus’s theory of proportion and thus (implicitly) on the axiom of Archimedes. 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. In the late nineteenth century, multiplication became an operation on points (that is, ‘numbers’ in the coordinatizing field). Hilbert showed any plane satisfying his axioms HP5 (below) interprets a field and recovered Euclid’s results about polygons via a first-order theory. The bi-interpretability between various geometric theories and associated theories of fields is the key to the analysis here. We begin by distinguishing several topics in plane geometry1 that represent distinct data sets in Detlefsen’s sense. In cases where certain axioms are explicit, they are included in the data set. Each set includes its predecessors. Then we provide specific axiomatizations of the various areas. Our division of the data sets is somewhat arbitrary and is made with the subsequent axiomatizations in mind. Euclid I, polygonal geometry: Book I (except I.22), Book II.1–II.13, Book III (except III.1 and III.17), Book VI. Euclid II, circle geometry: I.22, II.14, III.1, III.17 and Book IV. Archimedes, arc length and $$\boldsymbol{\pi}$$: XII.2, (area of circle proportional to square of the diameter), approximation of $$\pi$$, circumference of circle proportional to radius, Archimedes’ axiom. Descartes, higher degree polynomials:$$n$$th roots; coordinate geometry. Hilbert, continuity: The Dedekind plane. In Part I, we formulated our formal system in a two-sorted vocabulary $$\tau$$ 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 collinearity relation, a quaternary relation for line congruence and a 6-ary relation for angle congruence. The circle-circle intersection postulate asserts: if the interiors of two circles (neither contained in the other) have a common point, the circles intersect in two points. The following axiom sets 2 are defined to organize these data sets: (1) first-order axioms HP, HP5: We write HP for Hilbert’s incidence, betweenness, 3 and congruence axioms. We write HP5 for HP plus the parallel postulate. A Pythagorean field is any field associated4 with a model of HP5; such fields are characterized by closure under $$\surd (1+a^2)$$. EG: The axioms for Euclidean geometry, denoted EG,5 consist of HP5 and in addition the circle-circle intersection postulate. A Euclidean plane is a model of EG; the associated Euclidean field is closed under $$\surd a$$ for $$a >0$$. $$\boldsymbol{\mathcal{E}^2}$$: Tarski’s axiom system [Tarski, 1959] for a plane over a real closed field 6 (RCF). EG$$_{\boldsymbol{\pi}}$$ and $$\boldsymbol{{\mathcal E}^2_{\boldsymbol{\pi}}}$$: Two new systems extending EG and $$\mathcal{E}^2$$ to discuss $$\pi$$. (2) Hilbert’s continuity axioms, infinitary and second-order: AA: The sentence in $$L_{\omega_1,\omega}$$ expressing the Archimedean axiom. Dedekind: Dedekind’s second-order axiom that there is a point in each irrational cut in the line. Notation 1.1 Closing a plane under ruler-and-compass constructions corresponds to closing the coordinatizing ordered field under square roots of positive numbers to give a Euclidean field.7 As in Example 4.3.2 of Part I, $$F_s$$ (surd field) denotes the minimal field whose geometry is closed under ruler-and-compass construction. Having named $$0,1$$, each element of $$F_s$$ is definable over the empty set. 8 We referred to [Hartshorne, 2000] to assert in Part I that the sentences of Euclid I are provable in HP5 and the additional sentences of Euclid II are provable in EG. Here we consider the data sets of Archimedes, Descartes, and Dedekind and argue for the following claims. (1) Tarski’s axioms $$\mathcal{E}^2$$ are a modest descriptive axiomatization of the Cartesian data set. (2) EG$$_\pi$$ ($$\mathcal{E}^2_\pi$$) are modest descriptive axiomatizations of the extension by the Archimedean data set of Euclidean Geometry (Cartesian geometry). 2. FROM DECARTES TO TARSKI Descartes and Archimedes represent distinct and indeed orthogonal directions in the project to make geometric continuum a precise notion. These directions can be distinguished as follows. Archimedes goes directly to transcendental numbers while Descartes investigates curves defined by polynomials. Of course, neither thought in these terms, although Descartes’ resistance to squaring the circle shows an awareness of what became this distinction. We deviate from chronological order and discuss Descartes before Archimedes; as, in Section 3, we will extend both Euclidean and Cartesian geometry by adding $$\pi$$. As we emphasized in describing the data sets, the most important aspects of the Cartesian data set are: (1) the explicit definition [Descartes, 1954, p. 1] of the multiplication of line segments to give a line segment, breaking with Greek tradition; 9 and (2) on the same page to announce constructions for the extraction 10 of $$n$$th roots for all $$n$$. Panza [2011, p. 44] describes Euclid’s plane geometry as an open system. But he writes, ‘things are quite different with Descartes’ geometry; this a closed system, equally well-framed as (Euclid’s).’ We follow Rodin’s Rodin’s [2017] exposition of the distinction between closed and open systems. It extends the traditional distinction between ‘theorems’ and ‘problems’ in Euclidean geometry in a precise way. Theorems have truth values; problems (constructions) introduce new objects. A closed system has a fixed domain of objects while the domain expands in an open system. According to Panza, Rodin, and others Euclid’s is an open system. In contrast, we treat both Euclidean and Cartesian geometries as Hilbert-style closed systems. As Rodin [2017] points out, the ‘received notion’ of axiomatics interprets a construction in terms of a $$\forall \exists$$-statement that asserts the construction can be made. Panza illustrates the openness by analyzing many types of ‘mechanical constructions’ in pre-Cartesian geometry. 11 According to Molland [1976, p. 38] ‘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)’. 12 By adding the solutions of polynomial equations Tarski’s geometry $$\mathcal{E}^2$$ (below), guarantees in advance the existence of Descartes’ more general notion of construction. This extension is obscured by Hilbert’s overly generous continuity axiom. Descartes’ proposal to organize geometry via the degree of polynomials [1954, p. 48] is reflected in the modern field of ‘real’ algebraic geometry, i.e., the study of polynomial equalities and inequalities in the theory of real closed ordered fields. To ground this geometry we adapt Tarski’s ‘elementary geometry’. Tarski’s system differs from Descartes’ in several ways. He 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. Tarski’s axioms are given entirely formally in a one-sorted language with a ternary relation on points thus making explicit that a line is conceived as a set of points.13 We will describe the theory in both algebraic and geometric terms using Hilbert’s bi-interpretation of Euclidean geometry and Euclidean fields.14 The algebraic formulation is central to our later developments. With this interpretation we can specify (in the metatheory) a minimal model of Tarski’s theory, the plane over the real algebraic numbers.15 It contains exactly (as we now understand) the objects Descartes viewed as solutions of those problems that it was ‘possible to solve’ [Crippa, 2014b, Ch. 6]. In accordance with Descartes’ rejection as non-geometric any method for quadrature of the circle, this model omits $$\pi$$. Tarski’s elementary geometry. The theory $$\mathcal{E}^2$$ is axiomatized by the following sets of axioms (1) and (2a) using the bi-interpretation, while (2b) can be expressed in Tarski’s vocabulary.16 (1) Euclidean plane geometry17 (HP5); (1) Either of the following two sets of axioms which are equivalent over HP5 (in a vocabulary naming two arbitrary points as $$0,1$$): (a) An infinite set of axioms declaring the field is formally real and that every polynomial of odd degree has a root. (b) Tarski’s axiom schema of continuity. Just as restricting induction to first-order formulas translates Peano’s second-order axioms to first-order, Tarski translates Dedekind cuts to first-order cuts. Require that for any two definable sets $$A$$ and $$B$$, if beyond some point $$a$$ all elements of $$A$$ are below all elements of $$B$$, there there is a point $$b$$ which is above all of $$A$$ and below all of $$B$$. [Givant and Tarski, 1999] formalizes the requirement with the Axiom Schema of Continuity:   $$(\exists a) (\forall x) (\forall y) [\alpha(x) \wedge \beta(y) \rightarrow B(axy)] \rightarrow (\exists b) (\forall x) (\forall y)[\alpha(x) \wedge \beta(y) \rightarrow B(xby)],$$ where $$\alpha , \beta$$ are first-order formulas, the first of which contains no free occurrences of $$a, b, y$$ and the second no free occurrences of $$a, b, x$$. Recalling that $$B(x,z,y)$$ represents ‘$$z$$ is between18$$x$$ and $$y$$’, the hypothesis asserts the solutions of the formulas $$\alpha$$ and $$\beta$$ behave like the $$A,B$$ above. This schema allows the solution of odd-degree polynomials. So (b) implies (a). As the theory of real closed fields (ordered fields satisfying (a)) is complete,19 schema (a) proves schema (b). In Detlefsen’s terminology Tarski has laid out a Gödel-complete axiomatization, that is, the consequences of his axioms are a complete first-order theory of (in our terminology) Cartesian plane geometry. This completeness guarantees that if we keep the vocabulary and continue to accept the same data set no axiomatization20 can account for more of the data. There are certainly open problems in Cartesian plane geometry [Klee and Wagon, 1991]. But however they are solved, the proof will be formalizable in $$\mathcal{E}^2$$. Thus, in our view, the axioms are descriptively complete. The axioms $$\mathcal{E}^2$$, consistently with Descartes’ conceptions and theorems, assert the solutions of certain equations. So they provide a modest complete descriptive axiomatization of the Cartesian data set. 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’. Remark 2.1 (Undecidability and Consistency) Ziegler [1982] has shown that every nontrivial finitely axiomatized subtheory21 of RCF is not decidable. Thus both to approximate more closely the Dedekind continuum and to obtain decidability we restrict to the theory of planes over RCF and so to Tarski’s $$\mathcal{E}^2$$. The bi-interpretability between RCF and the theory of all planes over real closed fields yields the decidability of $$\mathcal{E}^2$$ and a finitary proof of its consistency.22 The crucial fact that makes decidability possible is that the natural numbers are not first-order definable in the real field. As we know, the preeminent contribution of Descartes to geometry is coordinate geometry. Tarski (following Hilbert) provides a converse; his interpretation of the plane into the coordinatizing line [Tarski, 1951] unifies the study of the ‘geometric continuum’ with axiomatizations of ‘geometry’. Three post-Descartes innovations are largely neglected in these papers: (a) higher-dimensional geometry, (b) projective geometry, and (c) definability by analytic functions. Item (a) is a largely nineteenth-century innovation which impacts Descartes’ analytic geometry by introducing equations in more than three variables. We have used Tarski’s axioms for plane geometry from his [1959]. However, they extend by a family of axioms for higher dimensions [Givant and Tarski, 1999] to ground modern real algebraic geometry. This natural extension demonstrates the fecundity of Cartesian geometry. Descartes used polynomials in at most two variables. But once the field is defined, the semantic extension to spaces of arbitrary finite dimension, i.e., polynomials in any finite number of variables, is immediate. Thus, every $$n$$-space is controlled by the field; so the plane geometry determines the geometry of any finite dimension. Although the Cartesian data set concerns polynomials of very few variables and arbitrary degree, all of real algebraic geometry is latent. Projective geometry, (b), is essentially bi-interpretable with affine geometry. So both of these threads are more or less orthogonal to our development here, which concerns the structure of the line (and moves smoothly to higher dimensional or projective geometry). A provocative remark in [Dieudonné, 1970] symbolizes (c). He asserted the only correct usage of ‘analytic geometry’ is as the study of solution sets of analytic functions on real $$n$$-space for any $$n$$. ‘It is absolutely intolerable to use analytical geometry for linear algebra with coordinates, still called analytical geometry in elementary textbooks. Analytical geometry in this sense has never existed. There are only people who do linear algebra badly by taking coordinates ... Everyone knows that analytical geometry is the theory of analytical spaces.’ That there never was such a subject is surely hyperbole and [Dieudonné, 1982] makes pretty clear that his sense of analytic geometry is a twentieth-century creation. But Hilbert did lay the grounds for analytic geometry and mathematical analysis on Dedekind’s reals, denoted $$\Re$$. 3. ARCHIMEDES: $$\boldsymbol{\pi}$$ AND THE CIRCUMFERENCE AND AREA OF CIRCLES We begin with our rationale for placing various facts in the Archimedean data set.23 Three propositions encapsulate the issue: Euclid VI.1 (area of a triangle), Euclid XII.2 (area of a circle), and Archimedes’ proof that the circumference of a circle is proportional to the diameter. Hilbert showed that VI.1 is provable already in the first-order HP5 (Part I). While Euclid implicitly relies on the Archimedean axiom, Archimedes makes it explicit in a recognizably modern form. Euclid does not discuss the circumference of a circle. To deal with that issue, Archimedes develops his notion of arc length. By beginning to calculate approximations of $$\pi$$, Archimedes is moving towards the treatment of $$\pi$$ as a number. Consequently, we distinguish VI.1 (Euclid I) from the Archimedean axiom and the theorems on measurement of a circle, and place the latter in the Archimedean data set. The validation in the theories $$\mathrm{EG}_\pi$$ and $$\mathcal{E}^2_{\pi}$$ set out below of the formulas $$A = \pi r^2$$ and $$C = \pi d$$ answer questions of Hilbert and Dedekind not questions of Euclid though possibly of Archimedes. But we think the theory $$\mathrm{EG}_{\pi}$$ is closer to the Greek origins than Hilbert’s second-order axioms are. Certainly EG$$_{\pi}$$ goes beyond Euclidean geometry by identifying a straight-line segment with the same length as the circumference of a circle (as Dedekind’s or Birkhoff’s postulates, discussed below, demand). This demand contrasts with earlier views such as Eutocius (fourth century), ‘Even if it seemed not yet possible to produce a straight line equal to the circumference of the circle, nevertheless, the fact that there exists some straight line by nature equal to it is deemed by no one to be a matter of investigation.’24 Although Eutocius asserts the existence of a line of the same length as a curve but finds constructing it unimportant, Aristotle has a stronger view. Summarizing his discussion of Aristotle, Crippa [2014a, pp. 34–35] points out that Aristotle takes the impossibility of such equality as the hypothesis of an argument on motion and Crippa cites Averroes as holding ‘that there cannot be a straight line equal to a circular arc’. It is widely understood25 that Dedekind’s analysis is radically different from that of Eudoxus. A principal reason for this, discussed in Section 3.1, is that Eudoxus applies his method to specific situations; Dedekind demands that every cut be filled. Secondly, Dedekind develops addition and multiplication on the cuts. Thus, Dedekind’s postulate should not be regarded as part of either Euclidean data set. But $$\mathrm{EG}_\pi$$ makes a much more restrained demand; as in Eudoxus, a specific problem is solved. 3.1 Formalizing $$\boldsymbol{\pi}$$ in Euclidean Geometry The geometry over a Euclidean field (every positive number has a square root) may have no straight-line segment of length $$\pi$$. E.g., the model over the surd field (Notation 1.1) does not contain $$\pi$$. Neither does the field of real algebraic numbers; so $$\mathcal{E}^2$$ does not resolve the issue. 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 $$\pi$$ does. For Archimedes and Euclid, sequences constructed in the study of magnitudes in the Elements are of geometric objects, not numbers. But, in a modern account, as we saw already while discussing areas of polygons in Part I, we must identify the proportionality constant and verify that it represents a point in any model of the theory.26 Thus this goal diverges from a ‘Greek’ data set and indeed is complementary to the axiomatization of Cartesian geometry by Tarski’s $$\mathcal{E}^2$$. Euclid’s third postulate, ‘describe a circle with given center and radius’, entails that a circle is uniquely determined by its radius and center. In contrast, Hilbert simply defines the notion of circle and proves the uniqueness [Hartshorne, 2000, Lemma 11.1]. In either case we have the basic correspondence between angles and arcs: two segments of a circle are congruent if they subtend the same central angle. We established in Part I that for each model of EG and any line of the model, the surd field $$F_s$$ is embeddable in the field definable on that line. On this basis we can interpret the Greek theory of limits by way of cuts in the ordered surd field $$F_s$$. The following extensions, EG$$_{\pi}$$ and $$\mathcal{E}^2_\pi$$, of the systems EG and $$\mathcal{E}^2$$ guarantee the existence of $$\pi$$ as such a cut. Axioms for $$\boldsymbol\pi$$: Add to the vocabulary a new constant symbol $$\pi$$. Let $$i_n$$ ($$c_n$$) be the perimeter of a regular $$3\times 2^n$$-gon inscribed27 (circumscribed) in a circle of radius $$1$$. Let $$\Sigma(\pi)$$ be the collection of sentences (i.e., a type28)   $$i_n < 2\pi < c_n$$ for $$n< \omega$$. Now, we can define the new theories. (1) EG$$_{\pi}$$ denotes the deductive closure of the following set of axioms in the vocabulary $$\tau$$ augmented by constant symbols $$0, 1,\pi$$. (i) the axioms EG of a Euclidean plane; (ii) $$\Sigma(\pi)$$. (2) $$\mathcal{E}^2_\pi$$ is formed by adding $$\Sigma(\pi)$$ to $$\mathcal{E}^2$$ and taking the deductive closure. Dicta on constants: Here we named a further single constant $$\pi$$. But the effect is very different from naming $$0$$ and $$1$$ (Compare the Dicta on constants just after Theorem 4.1.1 of Part I.) The new axioms specify the place of $$\pi$$ in the ordering of the definable points of the model. So the data set is seriously extended. Theorem 3.1EG$$_{\pi}$$ is a consistent but not finitely axiomatizable29incomplete theory. Proof. A model of EG$$_{\pi}$$ is given by closing $$F_s \cup \{\pi\}\subseteq \Re$$ to a Euclidean field. To see the theory is not finitely axiomatizable, for any finite subset $$\Sigma_0(\pi)$$ of $$\Sigma(\pi)$$ choose a real algebraic number $$p$$ satisfying $$\Sigma_0$$ when $$p$$ is substituted for $$\pi$$; close $$F_s \cup \{p\} \subseteq \Re$$ to a Euclidean field to get a model of EG$$\ \cup\ \Sigma_0$$ which is not a model of EG$$_{\pi}$$. □ Dicta on Definitions or Postulates: We now extend the ordering on segments by adding the lengths of ‘bent lines’ and arcs of circles to the domain. Two approaches30 to this step are: (a) introduce an explicit but inductive definition; or (b) add a new predicate to the vocabulary and new axioms specifying its behavior. This alternative reflects in a way the trope that Hilbert’s axioms are implicit definitions. We take approach (a) in Definitions 3.2, 3.3, etc. using our established geometric vocabulary. Crucially, the following definition of bent lines (and thus the perimeter of certain polygons) is not a single formal definition but a schema of formulas $$\phi_n$$ defining an approximation for each $$n$$. Definition 3.2Let$$n\geq 2$$. By a bent line31$$b = Y_1\ldots Y_n$$we mean a sequence of straight line segments $$Y_iY_{i+1}$$, for $$1\leq i \leq n-1$$, such that each end point of one is the initial point of the next. We specify the length of a bent line $$b = Y_1\ldots Y_n$$, denoted by $$[b]$$, as the length given by the straight-line segment composed of the sum of the segments of $$b$$. Now we say an approximant $$Y_1, \ldots Y_{n+1}$$ to the arc $$X_1\ldots X_n$$ of a circle with center $$P$$, is a bent line satisfying: (1) $$X_1, \ldots X_n, Y_1, \ldots Y_{n}$$ are points such that each $$X_i$$ is on the circle and each $$Y_i$$ is in the exterior of the circle; (2) each of $$Y_iY_{i+1}$$ ($$1< i< n$$) $$Y_n Y_1$$ is a line segment; (3) for $$1< i < n$$, $$Y_iY_{i+1}$$ and $$Y_nY_1$$ are tangent to the circle at $$X_i$$. We obtain the circumference of a circle by requiring $$X_n = X_{1}$$ and $$Y_n=Y_1$$. Definition 3.3Let $${\mathcal S}$$ be the set (of congruence classes of) straight line segments. Let $$\mathcal C_r$$ be the set (of equivalence classes under congruence) of arcs on circles of a given radius $$r$$. Now we extend the linear order on $${\mathcal S}$$ to a linear order $$<_r$$ on $${\mathcal S}\cup \mathcal C_r$$ as follows. For $$s \in {\mathcal S}$$ and $$c\in \mathcal C_r$$ (1) The segment $$s<_r c$$ if and only if there is a chord $$XY$$ of a circular segment $$AB \in c$$ such that $$XY \in s$$. (2) The segment $$s >_r c$$ if and only if there is an approximant $$b =X_1 \ldots X_n$$ to $$c$$ with length $$[b] = s$$ and with $$[X_1 \ldots X_n]>_r c$$. It is easy to see that this order is well-defined, as each chord of an arc is shorter than the arc, and the arc is shorter than any approximant to it. Now, we encode a second approximation of $$\pi$$, using the areas $$I_n, C_n$$ of the approximating polygons rather than their perimeters $$i_n, c_n$$. There are two aspects to transferring the defintion from circumference to area: (1) modifying the development of the area function of polygons described in Section 4.4 of Part I, by extending the notion of figure to include sectors of circles, and (2) formalizing a notion of equal area, including a schema for approximation of circles32 by finite polygons. We omit those details analogous to 3.2–3.3. We carried out the harder case of circumference to emphasize the innovation of Archimedes in defining arc length; unlike area it is not true that the perimeter of a polygon containing a second is necessarily larger than the perimeter of the enclosed polygon. Lemma 3.4Let $$I_n$$ and $$C_n$$ denote the area of the regular $$3 \times 2^n$$-gon inscribed in or circumscribing the unit circle. Then $$EG_\pi$$ proves33each of the sentences $$I_n < \pi < C_n$$ for $$n< \omega$$. Proof. The intervals $$(I_n,C_n)$$ define the cut for $$\pi$$ in the surd field $$F_s$$ and the intervals $$(i_n,c_n)$$ define the cut for $$2\pi$$ and it is a fact about the surd field that one half of any realization of the second cut is a realization of the first. □ To argue that $$\pi$$, as implicitly defined by the theory EG$$_\pi$$, serves its geometric purpose, we add new unary function symbols $$C$$ and $$A$$ mapping our fixed line to itself and satisfying a scheme asserting that the functions these symbols refer to do, in fact, produce the required limits. The definitions are identical except for substituting the area for the perimeter of the approximating polygons. This strategy mimics that in an introductory calculus course of describing the properties of area and proving that the Riemann integral satisfies them. Definition 3.5A unary function $$C(r)$$ ($$A(r)$$) mapping $${\mathcal S}$$, the set of equivalence classes (under congruence) of straight line segments, into itself that satisfies the conditions below is called a circumference function (area function). (1) $$C(r)$$ ($$A(r)$$) is less than the perimeter (area) of a regular $$3\times 2^{n}$$-gon circumscribing a circle of radius $$r$$. (2) $$C(r)$$ ($$A(r)$$) is greater than the perimeter (area) of a regular $$3\times 2^n$$-gon inscribed in a circle of radius $$r$$. We can extend EG$$_{\pi}$$ to include definitions of $$C(r)$$ and $$A(r)$$. The theory EG$$_{\pi, A, C}$$ is the extension of the $$\tau \cup \{0,1,\pi\}$$-theory EG$$_{\pi}$$, obtained by the explicit definitions: $$A(r) =\pi r^2$$ and $$C(r) = 2\pi r$$. In any model of EG$$_{\pi,A,C}$$, for each circle of radius $$r$$ there is an $$s \in {\mathcal S}$$ whose length34$$C(r) = 2\pi r$$ is less than the perimeters of all circumscribed polygons and greater than those of the inscribed polygons. We can verify that by choosing $$n$$ large enough we can make $$i_n$$ and $$c_n$$ as close together as we like (more precisely, for given $$m$$, make them differ by $$< 1/m$$). In phrasing this sentence I follow Heath’s description of Archimedes’ statements: But he follows the cautious method to which the Greeks always adhered; he never says that a given curve or surface is the limiting form of the inscribed or circumscribed figure; all that he asserts is that we can approach the curve or surface as nearly as we please. [Heath, 2011, Ch. 4] Invoking Lemma 3.4, since the $$2I_n (2C_n)$$ converge to the limit of the $$i_n (c_n)$$, they determine the same cut, that of $$2\pi$$: Theorem 3.6In EG$$^2_{\pi,A,C}$$, $$C(r) = 2\pi r$$ is a circumference function and $$A(r) = \pi r^2$$ is an area function. In an Archimedean field there is a unique interpretation of $$\pi$$ and thus a unique choice for a circumference function with respect to the vocabulary without the constant $$\pi$$. By adding the constant $$\pi$$ to the vocabulary we get a formula which satisfies the conditions in every model. But in a non-Archimedean model, any point in the monad35 of $$2\pi r$$ would equally well fit our condition for being the circumference. To sum up, we have extended our descriptively complete axiomatization from the polygonal geometry of Hilbert’s first-order axioms (HP5) to Euclid’s results on circles and beyond. Euclid does not deal with arc length at all and we have assigned straight line segments to both the circumference and area of a circle. It follows that our development would not qualify as a modest axiomatization of Greek geometry but only of the modern understanding of these formulas. However, this distinction is not a problem for the notion of descriptive axiomatization. The facts are given as sentences. The formulas for circumference and area are not the same sentences as the Euclid/Archimedes statements in terms of proportions, but the Greek versions are implied by the modern equational formulations. 3.2. Formalizing $$\boldsymbol{\pi}$$ in Cartesian Geometry We now want to make a similar extension of $$\mathcal{E}^2$$. Dedekind [1963, pp. 37–38] observes that the field of real algebraic numbers is ‘discontinuous everywhere’ but ‘all constructions that occur in Euclid’s Elements can ... be just as accurately effected as in a perfectly continuous space’. Strictly speaking, for constructions this is correct. But the proportionality constant $$\pi$$ between a circle and its circumference is absent; so, it cannot be the case that both a straight line segment of the same length as the circumference and the diameter are in the model.36 We want to find a middle ground between the constructible entities of Euclidean geometry and Dedekind’s postulation that all transcendentals exist. That is, we propose a theory which proves the circumference and area formulas for circles and countable models of the geometry over RCF, one where ‘arc length behaves properly’ by characterizing the only transcendental number known in antiquity. In contrast to Dedekind and Hilbert, Descartes eschews the idea that there can be a ratio between a straight line segment and a curve. Crippa [2014b] writes, ‘Descartes excludes the exact knowability of the ratio between straight and curvilinear segments’; then he quotes Descartes: ... la proportion, qui est entre les droites et les courbes, n’est pas connue, et mesme ie croy ne le pouvant pas estre par les hommes, on ne pourroit rien conclure de là qui fust exact et assuré.37 Hilbert [2004, pp. 429–430] asserts that there are many geometries satisfying his axioms I–IV and V.1 but only one, ‘namely the Cartesian geometry’ that also satisfies V.2. Thus the conception of ‘Cartesian geometry’ changed radically from Descartes to Hilbert; even the symbol $$\pi$$ was not introduced until 1706 (by Jones). One wonders whether it had changed by the time Hilbert wrote. That is, had readers at the turn of the twentieth century already internalized a notion of Cartesian geometry which entailed Dedekind completeness and that was, at best, formulated in the nineteenth century (Bolzano-Cantor-Weierstrass-Dedekind)? We now define a theory $$\mathcal{E}^2_\pi$$ analogous to EG$$_\pi$$ that does not depend on the Dedekind axiom but can be obtained in a first-order way. Given Descartes’ proscription of $$\pi$$, the new system will be immodest with respect to the Cartesian data set. But we will argue at the end of this section that both of our axioms for $$\pi$$ are closer to Greek conceptions than the Dedekind axiom. At this point we need some modern model theory to guarantee the completeness of the theory we are defining. A first-order theory $$T$$ for a vocabulary including a binary relation $$<$$ is o-minimal if every model of $$T$$ is linearly ordered by $$<$$ and every $$1$$-ary formula is equivalent in $$T$$ to a Boolean combination of equalities and inequalities [van den Dries, 1999]. Anachronistically, the o-minimality of the reals is a main conclusion of Tarski [1931]. We can now show: Theorem 3.7Form $$\mathcal{E}^2_{\pi}$$ by adjoining $$\Sigma(\pi)$$ to $$\mathcal{E}^2$$. $$\mathcal{E}^2_{\pi}$$ is first-order complete for the vocabulary $$\tau$$ augmented by constant symbols $$0, 1,\pi$$. Proof. We have established that there is a definable ordered field whose domain is the line through the points $$0, 1$$. By Tarski, the theory of this real closed field is complete. The field is bi-interpretable with the plane [Tarski, 1951]; so the theory of the geometry $$T$$ is complete as well. Further by [Tarski, 1931], the field is o-minimal. Therefore, the type over the empty set of any point on the line is determined by its position in the linear ordering of the subfield $$F_s$$ (Notation 1.1). Each $$i_n, c_n$$ is an element of the field $$F_s$$. The position of $$2\pi$$ in the linear order on the line through $$01$$ is given by $$\Sigma$$. Thus $$T \cup \Sigma(\pi)$$ is a complete theory. □ As we extended $$\mathrm{EG}_n$$, we now extend the theory $$\mathcal{E}^2_\pi$$. Definition 3.8The theory $$\mathcal{E}^2_{\pi,A,C}$$ is the extension of the $$\tau \cup \{0,1,\pi\}$$-theory $$\mathcal{E}^2_{\pi}$$ obtained by adding the explicit definitions: $$A(r) =\pi r^2$$ and $$C(r) = 2\pi r$$. Theorem 3.9The theory $$\mathcal{E}^2_{\pi,A,C}$$ is a complete, decidable extension of EG$$_{\pi,A,C}$$ that is coordinatized by an o-minimal field. Moreover, in $$\mathcal{E}^2_{\pi,A,C}$$, $$C(r) = 2\pi r$$ is a circumference function and $$A(r) =\pi r^2$$ is an area function. Proof. We are adding definable functions to $$\mathcal{E}^2_\pi$$ so o-minimality and completeness are preserved. The theory is recursively axiomatized and complete, so decidable. The formulas continue to compute area and circumference correctly (as in Theorem 3.6) since they extend EG$$_{\pi,A,C}$$. □ The assertion that $$\pi$$ is transcendental is a theorem of the first-order theory $$\mathcal{E}^2_{\pi}$$. We explore the distinction between proving a fact of mathematics and showing it is provable in a first-order theory in Section 4.3. In this case, Lindemann proved that $$\pi$$ does not satisfy a polynomial of degree $$n$$ for any $$n$$. Thus for any polynomial $$p(x)$$ over the rationals $$p(\pi) \neq 0$$ is a consequence of the complete type38 generated by $$\Sigma(\pi)$$ and so is a theorem of $$\mathcal{E}^2_{\pi}$$. We now extend the known fact that the theory of real closed fields is ‘finitistically justified’ (in the list of such results [Simpson, 2009, p. 378]) to $$\mathcal{E}^2_{\pi,A,C}$$. For convenience, we lay out the proof with reference to results39 recorded in [Simpson, 2009]. The theory $$\mathcal{E}^2$$ is bi-interpretable with the theory of real closed fields. And thus it (as well as $$\mathcal{E}^2_{\pi,A, C}$$) is finitistically consistent, in fact, provably consistent in primitive recursive arithmetic (PRA). By Theorem II.4.2 of [Simpson, 2009], RCA$$_0$$ proves the system $$(Q,+,\times,<)$$ is an ordered field and by II.9.7 of [Simpson, 2009], it has a unique real closure. Thus the existence of a real closed ordered field and so Con(RCOF) is provable in RCA$$_0$$. (Note that the construction will imbed the surd field $$F_s$$.) Lemma IV.3.3 of [Friedman et al., 1983] asserts the provability of the completeness theorem (and hence compactness) for countable first-order theories from WKL$$_0$$. Note that since every finite subset of $$\Sigma(\pi)$$ is satisfiable in any RCOF, it follows that the existence of a model of $$\mathcal{E}^2_\pi$$ is provable in WKL$$_0$$. Since WKL$$_0$$ is $$\pi^0_2$$-conservative over PRA, we conclude PRA proves the consistency of $$\mathcal{E}^2_\pi$$. As $$\mathcal{E}^2_{\pi, C,A}$$ is an extension by explicit definitions, its consistency is also provable in PRA, as required. This completes the argument. It might be objected that such minor changes as adding to $$\mathcal{E}^2$$ the name of the constant $$\pi$$, or adding the definable functions $$C$$ and $$A$$ undermines the earlier claim that $$\mathcal{E}^2$$ is descriptively complete for Cartesian geometry. But $$\pi$$ is added because the modern view of ‘number’ requires it and increases the data set to include propositions about $$\pi$$ which are inaccessible to $$\mathcal{E}^2$$. We have so far tried to find the proportionality constant only in specific situations. In the remainder of the section, we introduce a model-theoretic scheme to systematize the solution of families of ‘quadrature’ problems. Crippa describes Leibniz’s distinguishing two types of quadrature, ... ‘universal quadrature’ of the circle, namely the problem of finding a general formula, or a rule in order to determine an arbitrary sector of the circle or an arbitrary arc; and on the other [hand] he defines the problem of the ‘particular quadrature’, ... , namely the problem of finding the length of a given arc or the area of a sector, or the whole circle ... . [Crippa, 2014a, p. 424] Birkhoff [1932] ignores such a distinction with the protractor postulate of his system.40 POSTULATE III. The half-lines $$\ell, m$$, through any point O can be put into $$(1, 1)$$ correspondence with the real numbers $$a(\text{mod}\ 2 \pi)$$, so that, if $$A \neq O$$ and $$B\neq O$$ are points of $$\ell$$ and $$m$$ respectively, the difference $$a_m - a_\ell (\text{mod}\ 2\pi)$$ is $$\angle AOB$$. Furthermore, if the point $$B$$ varies continuously in a line $$r$$ not containing the vertex O, the number $$a_m$$ varies continuously also.41 This axiom is analogous to Birkhoff’s ‘ruler postulate’ which assigns each segment a real number length. Thus, he takes the real numbers as an unexamined background object; at one swoop he has introduced addition and multiplication, and assumed the Archimedean and completeness axioms. So even ‘neutral’ geometries studied on this basis are actually greatly restricted.42 He argues that his axioms define a categorial system isomorphic to $$\Re^2$$. So his system (including an axiomatization of the real field that he has not specified) is bi-interpretable with Hilbert’s. However, the protractor postulate conflates three distinct problems: (i) the rectifiability of arcs, the assertion that each arc of a circle has the same length as a straight line segment; (ii) the claim there is an algorithm for finding such a segment; and (iii) the measurement of angles, that is assigning a measure to an angle as the arc length of the arc it determines. The next task is to find a more modest version of Birkhoff’s postulate, namely, a first-order theory with countable models which assign to each angle a measure between $$0$$ and $$2\pi$$. Recall that we have a field structure on the line through the points $$0,1$$ and the number $$\pi$$ on that line, so we can make a further explicit definition. A measurement of angles function is a map $$\mu$$ from congruence classes of angles into $$[0,2\pi)$$ such that if $$\angle ABC$$ and $$\angle CBD$$ are disjoint angles sharing the side $$BC$$, $$\mu (\angle ABD) = \mu(\angle ABC) + \mu(\angle CBD)$$. If we omitted the additivity property this would be trivial. Given an angle $$\angle ABC$$ less than a straight angle, let $$C'$$ be the intersection of a perpendicular to $$BC$$ through $$A$$ with $$BC$$ and let $$\mu (\angle ABC) = 2\pi\cdot\sin(\angle ABC) = \frac{2 \pi \cdot BC'}{AB}$$. (It is easy to extend to larger angles.) Here we use approach (b) of the Dicta on Definitions rather than the explicit definition approach (a) used for $$C(r)$$ and $$A(r)$$. We define a new theory with a function symbol $$\mu$$ which is ‘implicitly defined’ by the following axioms. Definition 3.10The theory $$\mathcal{E}^2_{\pi,A,C,\mu}$$ is obtained by adding to $$\mathcal{E}^2_{\pi,A,C}$$ the assertion that $$\mu$$ is a continuous additive map from congruence classes of angles to $$[0,2\pi)$$. It is straightforward to express that $$\mu$$ is continuous. If we omitted that requirement in Definition 3.10, $$\mathcal{E}^2_{\pi,A,C,\mu}$$ would be incomplete since $$\mu$$ would be continuous in some models, but not in some non-Archimedean models. Thus, we require continuity. Showing consistency of $$\mathcal{E}^2_{\pi,A,C,\mu}$$ is easy; we can define (in the mathematical sense, not as a formally definable function) in $$\mathcal{E}^2_{\pi,A,C}$$ such a function $$\mu^*$$, say, the restricted arc-cosine.43 Hence, the axioms are consistent and this solves the rectifiability problem. But, merely assuming the existence of a $$\mu$$ does not solve our problem (ii), as we have no idea how to compute $$\mu$$ and a recursive axiomatization is a real mathematical problem. 3.3 Some Countable Models of Geometry Blanchette [2014] distinguishes two approaches to logic — deductivist and model-centric — and argues that Hilbert represents the deductivist school and Dedekind the model-centric. Essentially, the second proposes that theories are designed to try to describe an intuition of a particular structure. We now consider a third direction: are there ‘canonical’ models of the various theories we have been considering? By modern tradition, the continuum is the real numbers and geometry is the plane over it. Is there a smaller model which reflects the geometric intuitions discussed here? For Euclid II, there is a natural candidate, the Euclidean plane over the surd field $$F_s$$. Remarkably, this does not conflict with Euclid XII.2 (the area of a circle is proportional to the square of the diameter). The model is Archimedean and $$\pi$$ is not in the model. But Euclid only requires a proportionality which defines a type $$\Sigma(x)$$, not a realization $$\pi$$ of $$\Sigma(x)$$. Plane geometry over the real algebraic numbers plays the same role for $$\mathcal{E}^2$$. Both are categorical in $$L_{\omega_1,\omega}$$. In the second case, the axiomatization asserts ‘every field element is algebraic’. We have developed a method of assigning measures to angles. Now we argue that the methods of this section better reflect the Greek view than Dedekind’s approach does. Mueller makes an important point distinguishing the Euclid/Eudoxus use from Dedekind’s use of cuts. In broad outline, the following quotation describes the methodology here. One might say that in applications of the method of exhaustion the limit is given and the problem is to determine a certain kind of sequence converging to it, ... Since, in the Elements the limit always has a simple description, the construction of the sequence can be done within the bounds of elementary geometry; and the question of constructing a sequence for any given arbitrary limit never arises. [Mueller, 2006, p. 236] But what if we want to demand the realization of various transcendentals? Mueller’s description suggests the principle that we should only realize cuts in the field order that are recursive over a finite subset. We might call these Eudoxian transcendentals. So a candidate would be a recursively saturated model44 of $$\mathcal{E}^2$$. Remarkably, almost magically,45 this model would also satisfy $$\mathcal{E}^2_{\pi,A,C,\mu}$$. A recursively saturated model is necessarily non-Archimedean. There are however many different countable recursively saturated models depending on which transcendentals are realized. Arguably there is a more canonical candidate for a natural model which admits the ‘Eudoxian transcendentals’; take the smallest elementary submodel of $$\Re$$ closed46 under $$A,C, \mu$$ that contains the real algebraic numbers and all realizations of recursive cuts in $$F_s$$. The Scott sentence47 of this sentence is a categorical sentence in $$L_{\omega_1,\omega}$$. The models in this paragraph are all countable; we cannot do this with the Hilbert model of the plane over the real numbers; it has no countable $$L_{\omega_1,\omega}$$-elementary submodel. We turn to the question of modesty. Mueller’s distinction can be expressed in another way. Eudoxus provides a technique to solve certain problems, which are specified in each application. In contrast, Dedekind’s postulate solves $$2^{\aleph_0}$$ problems at one swoop. Each of the theories $$\mathcal{E}^2_{\pi}$$, $$\mathcal{E}^2_{\pi,A,C}$$, $$\mathcal{E}^2_{\pi,A,C, \mu}$$ and the later search for their canonical models reflect this distinction. Each solves at most a countable number of recursively stated problems. In summary, we regard the replacement of ‘congruence class of segment’, by ‘length represented by an element of the field’ as a modest reinterpretation of Greek geometry. But this treatment of length becomes immodest relative even to Descartes when this length is a transcendental. And most immodest of all is to demand lengths for arbitrary transcendentals. 4. AND BACK TO HILBERT In this section we examine Hilbert’s ‘continuity axioms’. We study the syntactic form of various axioms, their consequences, and their role in clarifying the notion of the continuum. The Archimedean axiom is minimizing; each cut is realized by at most one point; so each model has cardinality at most $$2^{\aleph_0}$$. The Veronese postulate (footnote 50) or Hilbert’s Vollständigkeitsaxiom is maximizing; each cut is realized; in the absence of the Archimedean axiom the set of realizations could have arbitrary cardinality. 4.1 The Role of the Axiom of Archimedes in the Grundlagen A primary aim expressed in Hilbert’s introduction is ‘to bring out as clearly as possible the significance of the groups of axioms’. Much of his book is devoted to this metamathematical investigation. In particular this includes Sections 9–12 (from [Hilbert, 1971]) concerning the consistency and independence of the axioms. Further examples,48 in Sections 31–34, show that without the congruence axioms, the axiom of Archimedes is necessary to prove what Hilbert labels as Pascal’s (Pappus) theorem. In the conclusion to [1962], Hilbert notes Dehn’s work on the necessary role of the Archimedean axiom in establishing over neutral geometry the relation between the number of parallel lines through a point and the sum of the angles of a triangle. These are all metatheoretical results. In contrast, the use of the Archimedean axiom in Sections 19 and 21 to prove equidecomposable is the same as equicomplementable (equal content) (in 2 dimensions) is certainly a proof in the system. But an unnecessary one. As we argued in Section 4.4 of Part I, Hilbert could just have easily defined ‘same area’ as ‘equicomplementable’ (as is a natural reading of Euclid). Thus, we find no geometrical theorems in the Grundlagen that essentially depend on the axiom of Archimedes. Rather, Hilbert’s use of the axiom of Archimedes is (i) to investigate the relations among the various principles, and (ii) in conjunction with the Vollständigkeitsaxiom, to identify the field defined in the geometry with the independently existing real numbers as conceived by Dedekind. With respect to the problem studied here, I contend that these results do not affect the conclusion that Hilbert’s full axiom set is an immodest axiomatization49 of Euclid I or Euclid II or of the Cartesian data set since those data sets contain and are implied by the appropriate first-order axioms. 4.2 Hilbert and Dedekind on Continuity In this section we compare various formulations of the completeness axiom. Hilbert wrote: Axiom of Completeness (Vollständigkeitsaxiom): To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid. [Hilbert, 1971] In this article we have used the following adaptation of Dedekind’s postulate for geometry (DG): DG: Any cut in the linear ordering imposed on any line by the betweenness relation is realized. While this formulation is convenient for our purposes, it misses an essential aspect of Hilbert’s system; in a context with a group, DG implies the Archimedean axiom, while Hilbert was aiming for an independent set of axioms. Hilbert’s axiom does not imply Archimedes’. A variant VER50 on Dedekind’s postulate that does not imply the Archimedean axiom was proposed by Veronese [1889]. If VER replaces DG, those axioms would also satisfy the independence criterion. Hilbert’s completeness axiom in [1971] asserting any model of the rest of the theory is maximal, is inherently model-theoretic. The later line completeness [Hilbert, 1962] is a technical variant.51 Giovannini’s account [2013], which relies on [Hilbert, 2004] includes a number of points already made here and three more. First, Hilbert’s completeness axiom is not about deductive completeness (despite having such consequences), but about maximality of every model (p. 145). Secondly (last line of p. 153) Hilbert expressly rejects Cantor’s axiom on the intersection of a sequence of closed intervals because it relies on a sequence of intervals and ‘sequence is not a geometrical notion’. A third intriguing note is an argument due to Baldus in 1928 that the parallel axiom is an essential ingredient in the categoricity of Hilbert’s axioms.52 Here are two reasons for choosing Dedekind’s (or Veronese’s) version. One is that Dedekind’s formulation, since it is about the geometry, not about its axiomatization, directly gives the kind of information about the existence of transcendental numbers that we observe in this paper. Even more basic is that one cannot formulate Hilbert’s version as a sentence $$\Phi_H$$ in second-order logic53 of geometry with the intended interpretation $$(\Re^2,{\bf G}) \models \Phi_H$$. The axiom requires quantification over subsets of an extension of the model which putatively satisfies it. Here is a second-order statement54$$\Theta$$, where $$\psi$$ denotes the conjunction of Hilbert’s first four axiom groups and the axiom of Archimedes.   $$(\forall X)(\forall Y) (\forall {\bf R}) [[X \subseteq Y \wedge (X,{\bf R} \mathord\restriction X) \models \psi \wedge (Y,{\bf R} ) \models \psi]\rightarrow X=Y],$$ whose validity expresses Hilbert’s V.2 but which is a sentence in pure second-order logic rather than in the vocabulary for geometry. Väänänen [2012, p. 94] investigates this anomaly by distinguishing between $$(\Re^2,{\bf G}) \models \Phi$$, for some $$\Phi$$ and the validity of $$\Theta$$. He expounds in [2014] a new notion, ‘Sort Logic’, which provides a logic with a sentence $$\Phi^\prime_H$$ which, by allowing a sort for an extension, formalizes Hilbert’s V.2 with a more normal notion of truth in a structure. In [Väänänen, 2012; Väänänen and Wang, 2015], Väänänen discusses the categoricity of natural structures such as real geometry when axiomatized in second-order logic. He develops the striking phenomenon of ‘internal categoricity’,55 which as we now argue refutes the view that second-order categoricity ‘depends on the set theory’. Suppose the second-order categoricity of a structure $$A$$ is formalized by the existence of sentence $$\Psi_A$$ such that $$A \models \Psi_A$$ and any two models of $$\Psi_A$$ are isomorphic. If this second clause is provable in a standard deductive system for second-order logic, then it is valid in the Henkin semantics, not just the full semantics. Thus, its truth is independent of the ambient set theory. Philip Ehrlich has made several important discoveries concerning the connections between the two ‘continuity axioms’ in Hilbert and develops the role of maximality. First, he observes [Ehrlich, 1995, p. 172] that Hilbert had already pointed out that his completeness axiom would be inconsistent if the maximality were only with respect to the first-order axioms. Secondly, he [1995; 1997] systematizes and investigates the philosophical significance of Hahn’s notion of Archimedean completeness. Here the structure (ordered group or field) is not required to be Archimedean; the maximality condition requires that there is an extension which fails to extend an Archimedean equivalence class.56 This notion provides a tool (not yet explored) for investigating the non-Archimedean models studied in Section 3. In a sense, our development is the opposite of Ehrlich’s in [2012]. Rather than trying to unify all numbers great and small in a class model we are interested in the minimal collection of numbers that allow the development of a geometry that proved a modest axiomatization of the data sets considered. 4.3. Against the Dedekind Posulate for Geometry Our fundamental claim is that (slight variants on) Hilbert’s first-order axioms provide a modest descriptively complete axiomatization of most of Greek geometry. One goal of Hilbert’s continuity axioms was to obtain categoricity. But categoricity is not part of the data set but rather an external limitative principle. The notion that there was ‘one’ geometry (i.e., categoricity) was implicit in Euclid. But it is not a geometrical statement. Indeed, Hilbert [1962, p. 23] described his metamathematical formulation of the completeness axiom as, ‘not of a purely geometrical nature’. We argued [Baldwin, 2014. §3] against the notion of categoricity as an independent desideratum for an axiom system. We noted there that various authors have proved under $$V=L$$, any countable or Borel structure can be given a categorical axiomatization and that there are no strong structural consequences of the mere fact of second-order categoricity. However, there we emphasized the significance of axiomatizations, such as Hilbert’s, that reveal underlying principles concerning such iconic structures as geometry and the natural numbers. Here we go further, and suggest that even for an iconic structure there may be advantages to a first-order axiomatization that trump the loss of categoricity. We argue now that the Dedekind postulate is inappropriate (in particular immodest) in any attempt to axiomatize the Euclidean or Cartesian or Archimedean data sets for several reasons: (1) The requirement that there be a straight line segment measuring any circular arc contradicts the intent of various pre-Cartesian geometers and of Descartes. It clearly extends the collection of geometric entities appearing in either Euclid or Descartes. (2) The Dedekind postulate is not needed to establish the properly geometrical propositions in the data set. Hilbert [1971, p. 26] closes the discussion of continuity with ‘However, in what is to follow, no use will be made of the “axiom of completeness”.’ Why then did he include the axiom? Earlier in the same paragraph,57 he writes that ‘it allows the introduction of limiting points’ and enables one ‘to establish a one-one correspondence between the points of a segment and the system of real numbers’. Archimedes’ axiom makes the correspondence injective and the Vollständigkeitsaxiom makes it surjective. We have noted here that the grounding of real algebraic geometry (the study of systems of polynomial equations in a real closed field) is fully accomplished by Tarski’s axiomatization. And we have provided a first-order extension to deal with the basic properties of the circle. Since Dedekind, Weierstrass, and others pursued the ‘arithmetization of analysis’ precisely to ground the theory of limits, identifying the geometrical line as the Dedekind line reaches beyond the needs of geometry. (3) Proofs from Dedekind’s postulate obscure the true geometric reason for certain theorems. Hartshorne writes: $$\ldots$$ there are two reasons to avoid using Dedekind’s axiom. First, it belongs to the modern development of the real-number systems and notions of continuity, which is not in the spirit of Euclid’s geometry. Second, it is too strong. By essentially introducing the real numbers into our geometry, it masks many of the more subtle distinctions and obscures questions such as constructibility that we will discuss in Chapter 6. So we include the axiom only to acknowledge that it is there, but with no intention of using it. [Hartshorne, 2000, p. 177] (4) The use of second-order axioms undermines a key proof method — informal (semantic) proof — the ability to use higher-order methods to demonstrate that there is a first-order proof. A crucial advantage of a first-order axiomatization is that it licenses the kind of argument described in Hilbert and Ackerman:58 Derivation of Consequences from Given Premises; Relation to Universally Valid Formulas So far we have used the predicate calculus only for deducing valid formulas. The premises of our deductions, viz Axioms (a) through (f), were themselves of a purely logical nature. Now we shall illustrate by a few examples the general methods of formal derivation in the predicate calculus $$\ldots$$ It is now a question of deriving the consequences from any premises whatsoever, no longer of a purely logical nature. The method explained in this section of formal derivation from premises which are not universally valid logical formulas has its main application in the setting up of the primitive sentences or axioms for any particular field of knowledge and the derivation of the remaining theorems from them as consequences $$\ldots$$ We will examine, at the end of this section, the question of whether every statement which would intuitively be regarded as a consequence of the axioms can be obtained from them by means of the formal method of derivation. We noted that Hilbert proved that every planar proposition deducible in 3-dimensional geometry is provable in 2-dimensional geometry from Desargues’ theorem by this sort of argument [Baldwin, 2013, §2.4], and we exploited this technique in Section 3 to provide axioms for the calculation of the circumference and area of a circle.59 In this article we expounded a procedure [Hartshorne, 2000] to define the field operations in an arbitrary model of HP5. We argued that the first-order axioms of EG suffice for the geometrical data sets Euclid I and II, not only in their original formulation but by finding proportionality constants for the area formulas of polygon geometry. By adding axioms to require that the field is real closed we obtain a complete first-order theory that encompasses many of Descartes’ innovations. The plane over the real algebraic numbers satisfies this theory; thus, there is no guarantee that there is a line segment of length $$\pi$$. Using the o-minimality of real closed fields, we can guarantee there is such a segment by adding a constant for $$\pi$$ and requiring it to realize the proper cut in the rationals. However, guaranteeing the uniqueness of such a realization requires the $$L_{\omega_1,\omega}$$ Archimedean axiom. Hilbert and the other axiomatizers of a hundred years ago wanted more; they wanted to secure the foundations of mathematical analysis. In full generality, this surely depends on second-order properties. But there are a number of directions of work on ‘definable analysis’ [Baldwin, 2018, §6.3] that provide first-order approaches to analysis. The study of o-minimal theories makes major strides. One direction of research in o-minimality has been to prove the expansion of the real numbers by particular functions (e.g., the $$\Gamma$$-function on the positive reals [Speissinger and van den Dries, 2000]). Peterzil and Starchenko [2000] study the foundations of calculus. They approach complex analysis through o-minimality of the real part in [Peterzil and Starchenko, 2010]. The impact of o-minimality on number theory was recognized by a Karp prize (Peterzil, Pila, Wilkie, Starchenko) in 2014. And a non-logician Range [2014] suggests using methods of Descartes to teach calculus. A key feature of the interaction of o-minimal theories with real algebraic geometry has been the absence60 of Dedekind’s postulate for most arguments [Bochnak et al., 1998]. Footnotes 1In the first instance we draw from Euclid: Books I–IV, VI, and XII.1, 2 clearly concern plane geometry; XI, the rest of XII, and XIII deal with solid geometry; V and X deal with a general notion of proportion and with incommensurability. Thus, below we put each proposition of Books I–IV, VI, XII.1, 2 in a group and consider certain geometrical aspects of Books V and X. 2The names HP, HP5, and EG come from [Hartshorne, 2000] and $$\mathcal{E}^2$$ from [Tarski, 1959]. 3These include Pasch’s axiom (B4 of [Hartshorne, 2000]) 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. 4The field $$F$$ is associated with a plane $$\Pi$$ if $$\Pi$$ is the Cartesian plane on $$F^2$$. 5In the vocabulary here, there is a natural translation of Euclid’s axioms into first-order statements. The construction axioms have to be viewed as sentences ‘for all ... there exist’. The axiom of Archimedes is of course not first-order. We write Euclid’s axioms for those in the original [Euclid, 1956] vs (first-order) axioms for Euclidean geometry, EG. 6RCF 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. 7We call this process ‘taking the Euclidean closure’ or adding constructible numbers. 8That is, for each point $$a$$ constructible by ruler and compass there is a formula $$\phi_a(x)$$ such that $$\mathrm{EG}\vdash (\exists ! x) \phi(x)$$. in $$EG$$. That is, there is a unique solution to $$\phi$$. 9His proof is still based on Eudoxus. 10This extraction cannot be done in EG, since EG is satisfied in the field which has solutions for all quadratic equations but not those of odd degree. See [Hartshorne, 2000, §12]. 11See Sections 1.2 and 3 of [Panza, 2011] as well as [Bos, 2001]. Rodin [2014] further develops Panza’s ‘open’ notion as a ‘constructive axiomatic’ system. 12But, as Crippa points out [2014a, p. 153], Descartes did not prove this equivalence and there is some controversy as to whether the 1876 work of Kempe solves the precise problem. 13Writing in 1832, Bolyai [Gray, 2004, appendix] wrote in his ‘explanation of signs’, ‘The straight $$AB$$ means the aggregate of all points situated in the same straight line with $$A$$ and $$B$$’. This is the earliest indication I know of the transition to an extensional version of incidence. William Howard showed me this passage. See the Bolzano quote in [Rusnock, 2000, p. 53]. 14In our modern understanding of an axiom set the translation is routine, but anachronistic. 15That is, a real number that satisfies a polynomial with rational coefficients. A real number that satisfies no such polynomial is called transcendental. 16Translation to our official vocabulary requires quantifiers. We abuse Tarski’s notation by letting $$\mathcal{E}^2$$ denote the theory in the vocabulary with constants $$0,1$$. 17Note that circle-circle intersection is implied by the axioms in (2). 18More precisely in terms of the linear order $$B(xyz)$$ means $$x \leq y \leq z$$. 19Tarski [1959] proves that planes over real closed fields are exactly the models of $$\mathcal{E}^2$$. 20Of 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. 21A nontrivial subtheory is one satisfied in $$\Re$$, Dedekind’s reals. 22The geometric version of this result was conjectured by Tarski in [1959]: The theory RCF is complete and recursively axiomatized, so decidable. For the context of Ziegler’s result and Tarski’s quantifier elimination in computer science see [Makowsky, 2013]. 23This classification is not in any sense chronological, as Archimedes attributes the method of exhaustion to Eudoxus who precedes Euclid. Post-Heath scholarship by Becker, Knorr, and Menn [Menn, forthcoming] has identified four theories of proportion in the generations just before Euclid. [Menn, forthcoming] led us to the three prototypic propositions. 24Taken from his commentary on Archimedes in Archimedes Opera Omnia cum commentariis Eutociis, vol. 3, p. 266. Quoted in [Crippa, 2014b]. 25Stekeler-Weithofer [1992] writes, ‘It is just a big mistake to claim that Eudoxus’s proportions were equivalent to Dedekind cuts’. Feferman [2008] avers, ‘The main thing to be emphasized about the conception of the continuum as it appears in Euclidean geometry is that the general concept of set is not part of the basic picture, and that Dedekind style continuity considerations of the sort discussed below are at odds with that picture.’ Stein [1990], though, gives a long (but to me unconvincing) argument for at least the compatibility of Dedekind’s postulate with Greek thought ‘reasons ... plausible, even if not conclusive – for believing the Greek geometers would have accepted Dedekind’s postulate, just as they did that of Archimedes, once it had been stated’. 26For this reason, Archimedes needs only his postulate while Hilbert would also need Dedekind’s postulate to prove the circumference formula. 27I thank Craig Smorynski for pointing out that is not so obvious that the perimeter of an inscribed $$n$$-gon is monotonic in $$n$$ and reminding me that Archimedes avoided the problem by starting with a hexagon and doubling the number of sides at each step. 28Let $$A \subset M \models T$$. A type over $$A$$ is a set of formulas $$\phi({\bf x},{\boldsymbol{a}})$$, where $${\bf x}$$ ($${\boldsymbol{a}}$$) is a finite sequence of variables (constants from $$A$$) that is consistent with $$T$$. Taking $$T$$ as EG, a type over all $$F_s$$ is a type over $$\{0,1\}$$ since each element of $$F_s$$ is definable over $$\{0,1\}$$ in EG. 29Ziegler [1982, Remark 2.1] shows that EG is undecidable. Since for any $$T$$ and type $$p(x)$$ consistent with $$T$$, the decidability of $$T \cup \{p(c)\}$$ implies the decidability of $$T$$, EG$$_{\pi}$$ is also undecidable. 30We could define $$<$$ on the extended domain or, in style (b), we could add a $$<^*$$ to the vocabulary and postulate that $$<^*$$ extends $$<$$ and satisfies the properties of the definition. 31This is less general than Archimedes [1897, p. 2], who allows segments of arbitrary curves ‘that are concave in the same direction’. 32By dealing with a special case, we suppressed Archimedes’ anticipation of the notion of bounded variation. 33Note that we have not attempted to justify the convergence of the $$i_n,c_n, I_n, C_n$$ in the formal system EG$$_\pi$$. We are relying on mathematical proof, not a formal deduction in first-order logic; we explain this distinction in item (4) of Section 4.3. 34A similar argument works for area and $$A(r)$$. 35The monad of $$a$$ is the collection of points that are an infinitessimal distance from $$a$$. 36Thus, Birkhoff’s protractor postulate (below) is violated. 37Descartes, Oeuvres, Vol. 6, p. 412. Crippa also quotes Averroes as emphatically denying the possibility of such a ratio and notes that Vieta held similar views. 38Recall that $$\Sigma(x)$$ is a consistent collection of formulas in one free variable, which by Tarski’s quantifier elimination are Boolean combinations of polynomial inequalities. 39We use RCOF (real closed ordered field) here for what we have called RCF before. Model theoretically adding the definable ordering of a formally real field is a convenience. Here we want to be consistent with the terminology in [Simpson, 2009]. Note that Friedman [1999] strengthens the results for PRA to exponential-function arithmetic (EFA). Friedman reports Tarski had observed the constructive consistency proof much earlier. The theories discussed here, in increasing proof strength are EFA, PRA, RCA$$_0$$ and WKL$$_0$$. 40This is the axiom system used in virtually all U.S. high schools since the 1960s. 41I slightly modified the last sentence from Birkhoff, in lieu of reproducing the diagram. 42That is, they must be metric geometries. 43 In fact, by coding a point on the unit circle by its $$x$$-coordinate and setting $$\mu((x_1,y_1), (x_2,y_2)) = \cos^{-1}(x_1-x_2)$$ one gets such a function which is definable in the theory of the real field expanded by the cosine function restricted to $$[0,2\pi)$$ and in $$\mathcal{E}^2_{\pi,C,A}$$ by Theorem 4.3.1 of Part 1. This theory is known to be o-minimal [van den Dries, 1999]. But there is no known axiomatization and David Marker tells me it is unlikely to be decidable without assuming the Schanuel conjecture. 44A model is recursively saturated if every recursive type over a finite set is realized [Barwise, 1975]. 45The magic is called resplendency. Every recursively saturated model is resplendent [Barwise, 1975] where $$M$$ is resplendent if any formula $$\exists A \phi(A,{\boldsymbol{c}})$$ that is satisfied in an elementary extension of $$M$$ is satisfied by some $$A'$$ on $$M$$. Examples are the formulas defining $$C,A, \mu$$. 46Interpret $$A,C, \mu$$ on $$\Re$$ in the standard way. 47For any countable structure $$M$$ there is a ‘Scott’ sentence $$\phi_M$$ such that all countable models of $$\phi_M$$ are isomorphic to $$M$$; see [Keisler, 1971, Ch. 1]. 48I thank the referee for pointing to the next two examples and emphasizing Hilbert’s more general goals of understanding the connections among organizing principles. The reference to Dehn was dropped in later editions of the Grundlagen. 49It might seem I could claim immodesty for Archimedes as well, in view of my first-order axioms for $$\pi$$. But that would be a cheat. I restricted that data set to Archimedes on the circle, while Archimedes proposed a general notion of arc length and studied many other transcendental curves. 50The axiom VER (see [Cantù, 1999]) asserts that for a partition of a linearly ordered field into two intervals $$L,U$$ (with no maximum in the lower $$L$$ or minimum in the upper $$U$$) and a third set in between with at most one point, there is a point between $$L$$ and $$U$$ just if for every $$e>0$$, there are $$a\in L, b\in U$$ such that $$b-a<e$$. Veronese derives Dedekind’s postulate from his axiom and Archimedes in [Veronese, 1889] and the independence in [Veronese, 1891]. Levi-Civita [1892–93] shows there is a non-Archimedean ordered field that is Cauchy complete. I thank Philip Ehrlich for the references and recommend Section 12 of the comprehensive [Ehrlich, 2006]. See also the insightful reviews [Pambuccian, 2014b] and [Pambuccian, 2014a], where it is observed that Vahlen [1907] also proved this axiom does not imply Archimedes. 51Since any point is in the definable closure of any line and any one point not on the line, one cannot extend any line without extending the model. Since adding either the Dedekind postulate and or Hilbert completeness gives a categorical theory satisfied by a geometry whose line is order-isomorphic to $$\Re$$, the two axioms are equivalent (over HP5 + Archimedes’ axiom). 52Hartshorne [2000, §§40–43] gives a modern account of Hilbert’s argument that replacing the parallel postulate by the axiom of limiting parallels gives a geometry that is determined by the underlying (definable) field. With Hilbert’s V.2 this gives a categorical axiomatization for hyperbolic geometry. 53Of course, this analysis is anachronistic; the clear distinction between first- and second-order logic did not exist in 1900. By $${\bf G}$$, we mean the natural interpretation in $$\Re^2$$ of the geometric predicates from Section 1. 54I am leaving out many details, $${\bf R}$$ is a sequence of relations giving the vocabulary of geometry and the sentence ‘says’ they are relations on $$Y$$; the coding of the satisfaction predicate is suppressed. 55See [Väänänen and Wang, 2015] and [Baldwin, 2018, Chs 2, 11.2]. 56In an ordered group, $$a$$ and $$b$$ are Archimedes-equivalent if there are natural numbers $$m,n$$ such that $$m |a| > |b|$$ and $$n |b| > |a|$$. 57For a thorough historical description, see [Hilbert, 2004, pp. 426–435]. We focus on the issues most relevant to this paper. 58Chapter 3, §11. 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# Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski

, Volume Advance Article – Nov 23, 2017
28 pages

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Oxford University Press
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0031-8019
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10.1093/philmat/nkx031
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### Abstract

ABSTRACT In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi’s for Cartesian geometry and new systems for adding $$\pi$$. In contrast we find Hilbert’s full second-order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis. This two-part paper analyzes the axiomatic foundation of the ‘geometric continuum’, the line embdedded in the plane. For this, we built on Detlefsen’s notion of complete descriptive axiomatization and defined in Part I [Baldwin, 2017] a modest complete descriptive axiomatization of a data set $$\Sigma$$ (essentially, of facts in the sense of Hilbert) to be a collection of sentences that imply all the sentences in $$\Sigma$$ and ‘not too many more’. Of course, this set of facts will be open-ended, since over time more results will be proved. But if this set of axioms introduces essentially new concepts to the area and certainly if it contradicts the conceptions of the original era, we deem the axiomatization immodest. Part I dealt primarily with Hilbert’s first-order axioms for polygonal geometry and argued that the first-order systems HP5 and EG (defined below) are ‘modest’ complete descriptive axiomatization of most of Euclidean geometry. Part II concerns areas of geometry, e.g., circles, where stronger assumptions are needed. Hilbert postulated his ‘continuity axioms’ — the Archimedean and completeness axioms in extensions of first-order logic; we pursue weaker axioms. In Section 1, we reprise our organization of various ‘data sets’ for geometry and describe the axiom systems. We contend: (1) that Tarski’s first-order axiom set $$\mathcal{E}^2$$ is a modest complete descriptive axiomatization of Cartesian geometry (Section 2); (2) that the theories EG$$_{\pi,C,A}$$ and $$\mathcal{E}^2_{\pi,C,A}$$ are modest complete descriptive axiomatizations of extensions of these geometries designed to describe area and circumference of the circle (Section 3); and (3) that, in contrast, Hilbert’s full second-order system in the Grundlagen is an immodest axiomatization of any of these geometries but a modest descriptive axiomatization of the late nineteenth-century conception of the real plane (Section 4). We elaborate and place this study in a more general context in our book [Baldwin, 2018]. 1. TERMINOLOGY AND NOTATIONS Part I provided the following quasi-historical description. Euclid founded his theory of area for circles and polygons on Eudoxus’s theory of proportion and thus (implicitly) on the axiom of Archimedes. 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. In the late nineteenth century, multiplication became an operation on points (that is, ‘numbers’ in the coordinatizing field). Hilbert showed any plane satisfying his axioms HP5 (below) interprets a field and recovered Euclid’s results about polygons via a first-order theory. The bi-interpretability between various geometric theories and associated theories of fields is the key to the analysis here. We begin by distinguishing several topics in plane geometry1 that represent distinct data sets in Detlefsen’s sense. In cases where certain axioms are explicit, they are included in the data set. Each set includes its predecessors. Then we provide specific axiomatizations of the various areas. Our division of the data sets is somewhat arbitrary and is made with the subsequent axiomatizations in mind. Euclid I, polygonal geometry: Book I (except I.22), Book II.1–II.13, Book III (except III.1 and III.17), Book VI. Euclid II, circle geometry: I.22, II.14, III.1, III.17 and Book IV. Archimedes, arc length and $$\boldsymbol{\pi}$$: XII.2, (area of circle proportional to square of the diameter), approximation of $$\pi$$, circumference of circle proportional to radius, Archimedes’ axiom. Descartes, higher degree polynomials:$$n$$th roots; coordinate geometry. Hilbert, continuity: The Dedekind plane. In Part I, we formulated our formal system in a two-sorted vocabulary $$\tau$$ 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 collinearity relation, a quaternary relation for line congruence and a 6-ary relation for angle congruence. The circle-circle intersection postulate asserts: if the interiors of two circles (neither contained in the other) have a common point, the circles intersect in two points. The following axiom sets 2 are defined to organize these data sets: (1) first-order axioms HP, HP5: We write HP for Hilbert’s incidence, betweenness, 3 and congruence axioms. We write HP5 for HP plus the parallel postulate. A Pythagorean field is any field associated4 with a model of HP5; such fields are characterized by closure under $$\surd (1+a^2)$$. EG: The axioms for Euclidean geometry, denoted EG,5 consist of HP5 and in addition the circle-circle intersection postulate. A Euclidean plane is a model of EG; the associated Euclidean field is closed under $$\surd a$$ for $$a >0$$. $$\boldsymbol{\mathcal{E}^2}$$: Tarski’s axiom system [Tarski, 1959] for a plane over a real closed field 6 (RCF). EG$$_{\boldsymbol{\pi}}$$ and $$\boldsymbol{{\mathcal E}^2_{\boldsymbol{\pi}}}$$: Two new systems extending EG and $$\mathcal{E}^2$$ to discuss $$\pi$$. (2) Hilbert’s continuity axioms, infinitary and second-order: AA: The sentence in $$L_{\omega_1,\omega}$$ expressing the Archimedean axiom. Dedekind: Dedekind’s second-order axiom that there is a point in each irrational cut in the line. Notation 1.1 Closing a plane under ruler-and-compass constructions corresponds to closing the coordinatizing ordered field under square roots of positive numbers to give a Euclidean field.7 As in Example 4.3.2 of Part I, $$F_s$$ (surd field) denotes the minimal field whose geometry is closed under ruler-and-compass construction. Having named $$0,1$$, each element of $$F_s$$ is definable over the empty set. 8 We referred to [Hartshorne, 2000] to assert in Part I that the sentences of Euclid I are provable in HP5 and the additional sentences of Euclid II are provable in EG. Here we consider the data sets of Archimedes, Descartes, and Dedekind and argue for the following claims. (1) Tarski’s axioms $$\mathcal{E}^2$$ are a modest descriptive axiomatization of the Cartesian data set. (2) EG$$_\pi$$ ($$\mathcal{E}^2_\pi$$) are modest descriptive axiomatizations of the extension by the Archimedean data set of Euclidean Geometry (Cartesian geometry). 2. FROM DECARTES TO TARSKI Descartes and Archimedes represent distinct and indeed orthogonal directions in the project to make geometric continuum a precise notion. These directions can be distinguished as follows. Archimedes goes directly to transcendental numbers while Descartes investigates curves defined by polynomials. Of course, neither thought in these terms, although Descartes’ resistance to squaring the circle shows an awareness of what became this distinction. We deviate from chronological order and discuss Descartes before Archimedes; as, in Section 3, we will extend both Euclidean and Cartesian geometry by adding $$\pi$$. As we emphasized in describing the data sets, the most important aspects of the Cartesian data set are: (1) the explicit definition [Descartes, 1954, p. 1] of the multiplication of line segments to give a line segment, breaking with Greek tradition; 9 and (2) on the same page to announce constructions for the extraction 10 of $$n$$th roots for all $$n$$. Panza [2011, p. 44] describes Euclid’s plane geometry as an open system. But he writes, ‘things are quite different with Descartes’ geometry; this a closed system, equally well-framed as (Euclid’s).’ We follow Rodin’s Rodin’s [2017] exposition of the distinction between closed and open systems. It extends the traditional distinction between ‘theorems’ and ‘problems’ in Euclidean geometry in a precise way. Theorems have truth values; problems (constructions) introduce new objects. A closed system has a fixed domain of objects while the domain expands in an open system. According to Panza, Rodin, and others Euclid’s is an open system. In contrast, we treat both Euclidean and Cartesian geometries as Hilbert-style closed systems. As Rodin [2017] points out, the ‘received notion’ of axiomatics interprets a construction in terms of a $$\forall \exists$$-statement that asserts the construction can be made. Panza illustrates the openness by analyzing many types of ‘mechanical constructions’ in pre-Cartesian geometry. 11 According to Molland [1976, p. 38] ‘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)’. 12 By adding the solutions of polynomial equations Tarski’s geometry $$\mathcal{E}^2$$ (below), guarantees in advance the existence of Descartes’ more general notion of construction. This extension is obscured by Hilbert’s overly generous continuity axiom. Descartes’ proposal to organize geometry via the degree of polynomials [1954, p. 48] is reflected in the modern field of ‘real’ algebraic geometry, i.e., the study of polynomial equalities and inequalities in the theory of real closed ordered fields. To ground this geometry we adapt Tarski’s ‘elementary geometry’. Tarski’s system differs from Descartes’ in several ways. He 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. Tarski’s axioms are given entirely formally in a one-sorted language with a ternary relation on points thus making explicit that a line is conceived as a set of points.13 We will describe the theory in both algebraic and geometric terms using Hilbert’s bi-interpretation of Euclidean geometry and Euclidean fields.14 The algebraic formulation is central to our later developments. With this interpretation we can specify (in the metatheory) a minimal model of Tarski’s theory, the plane over the real algebraic numbers.15 It contains exactly (as we now understand) the objects Descartes viewed as solutions of those problems that it was ‘possible to solve’ [Crippa, 2014b, Ch. 6]. In accordance with Descartes’ rejection as non-geometric any method for quadrature of the circle, this model omits $$\pi$$. Tarski’s elementary geometry. The theory $$\mathcal{E}^2$$ is axiomatized by the following sets of axioms (1) and (2a) using the bi-interpretation, while (2b) can be expressed in Tarski’s vocabulary.16 (1) Euclidean plane geometry17 (HP5); (1) Either of the following two sets of axioms which are equivalent over HP5 (in a vocabulary naming two arbitrary points as $$0,1$$): (a) An infinite set of axioms declaring the field is formally real and that every polynomial of odd degree has a root. (b) Tarski’s axiom schema of continuity. Just as restricting induction to first-order formulas translates Peano’s second-order axioms to first-order, Tarski translates Dedekind cuts to first-order cuts. Require that for any two definable sets $$A$$ and $$B$$, if beyond some point $$a$$ all elements of $$A$$ are below all elements of $$B$$, there there is a point $$b$$ which is above all of $$A$$ and below all of $$B$$. [Givant and Tarski, 1999] formalizes the requirement with the Axiom Schema of Continuity:   $$(\exists a) (\forall x) (\forall y) [\alpha(x) \wedge \beta(y) \rightarrow B(axy)] \rightarrow (\exists b) (\forall x) (\forall y)[\alpha(x) \wedge \beta(y) \rightarrow B(xby)],$$ where $$\alpha , \beta$$ are first-order formulas, the first of which contains no free occurrences of $$a, b, y$$ and the second no free occurrences of $$a, b, x$$. Recalling that $$B(x,z,y)$$ represents ‘$$z$$ is between18$$x$$ and $$y$$’, the hypothesis asserts the solutions of the formulas $$\alpha$$ and $$\beta$$ behave like the $$A,B$$ above. This schema allows the solution of odd-degree polynomials. So (b) implies (a). As the theory of real closed fields (ordered fields satisfying (a)) is complete,19 schema (a) proves schema (b). In Detlefsen’s terminology Tarski has laid out a Gödel-complete axiomatization, that is, the consequences of his axioms are a complete first-order theory of (in our terminology) Cartesian plane geometry. This completeness guarantees that if we keep the vocabulary and continue to accept the same data set no axiomatization20 can account for more of the data. There are certainly open problems in Cartesian plane geometry [Klee and Wagon, 1991]. But however they are solved, the proof will be formalizable in $$\mathcal{E}^2$$. Thus, in our view, the axioms are descriptively complete. The axioms $$\mathcal{E}^2$$, consistently with Descartes’ conceptions and theorems, assert the solutions of certain equations. So they provide a modest complete descriptive axiomatization of the Cartesian data set. 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’. Remark 2.1 (Undecidability and Consistency) Ziegler [1982] has shown that every nontrivial finitely axiomatized subtheory21 of RCF is not decidable. Thus both to approximate more closely the Dedekind continuum and to obtain decidability we restrict to the theory of planes over RCF and so to Tarski’s $$\mathcal{E}^2$$. The bi-interpretability between RCF and the theory of all planes over real closed fields yields the decidability of $$\mathcal{E}^2$$ and a finitary proof of its consistency.22 The crucial fact that makes decidability possible is that the natural numbers are not first-order definable in the real field. As we know, the preeminent contribution of Descartes to geometry is coordinate geometry. Tarski (following Hilbert) provides a converse; his interpretation of the plane into the coordinatizing line [Tarski, 1951] unifies the study of the ‘geometric continuum’ with axiomatizations of ‘geometry’. Three post-Descartes innovations are largely neglected in these papers: (a) higher-dimensional geometry, (b) projective geometry, and (c) definability by analytic functions. Item (a) is a largely nineteenth-century innovation which impacts Descartes’ analytic geometry by introducing equations in more than three variables. We have used Tarski’s axioms for plane geometry from his [1959]. However, they extend by a family of axioms for higher dimensions [Givant and Tarski, 1999] to ground modern real algebraic geometry. This natural extension demonstrates the fecundity of Cartesian geometry. Descartes used polynomials in at most two variables. But once the field is defined, the semantic extension to spaces of arbitrary finite dimension, i.e., polynomials in any finite number of variables, is immediate. Thus, every $$n$$-space is controlled by the field; so the plane geometry determines the geometry of any finite dimension. Although the Cartesian data set concerns polynomials of very few variables and arbitrary degree, all of real algebraic geometry is latent. Projective geometry, (b), is essentially bi-interpretable with affine geometry. So both of these threads are more or less orthogonal to our development here, which concerns the structure of the line (and moves smoothly to higher dimensional or projective geometry). A provocative remark in [Dieudonné, 1970] symbolizes (c). He asserted the only correct usage of ‘analytic geometry’ is as the study of solution sets of analytic functions on real $$n$$-space for any $$n$$. ‘It is absolutely intolerable to use analytical geometry for linear algebra with coordinates, still called analytical geometry in elementary textbooks. Analytical geometry in this sense has never existed. There are only people who do linear algebra badly by taking coordinates ... Everyone knows that analytical geometry is the theory of analytical spaces.’ That there never was such a subject is surely hyperbole and [Dieudonné, 1982] makes pretty clear that his sense of analytic geometry is a twentieth-century creation. But Hilbert did lay the grounds for analytic geometry and mathematical analysis on Dedekind’s reals, denoted $$\Re$$. 3. ARCHIMEDES: $$\boldsymbol{\pi}$$ AND THE CIRCUMFERENCE AND AREA OF CIRCLES We begin with our rationale for placing various facts in the Archimedean data set.23 Three propositions encapsulate the issue: Euclid VI.1 (area of a triangle), Euclid XII.2 (area of a circle), and Archimedes’ proof that the circumference of a circle is proportional to the diameter. Hilbert showed that VI.1 is provable already in the first-order HP5 (Part I). While Euclid implicitly relies on the Archimedean axiom, Archimedes makes it explicit in a recognizably modern form. Euclid does not discuss the circumference of a circle. To deal with that issue, Archimedes develops his notion of arc length. By beginning to calculate approximations of $$\pi$$, Archimedes is moving towards the treatment of $$\pi$$ as a number. Consequently, we distinguish VI.1 (Euclid I) from the Archimedean axiom and the theorems on measurement of a circle, and place the latter in the Archimedean data set. The validation in the theories $$\mathrm{EG}_\pi$$ and $$\mathcal{E}^2_{\pi}$$ set out below of the formulas $$A = \pi r^2$$ and $$C = \pi d$$ answer questions of Hilbert and Dedekind not questions of Euclid though possibly of Archimedes. But we think the theory $$\mathrm{EG}_{\pi}$$ is closer to the Greek origins than Hilbert’s second-order axioms are. Certainly EG$$_{\pi}$$ goes beyond Euclidean geometry by identifying a straight-line segment with the same length as the circumference of a circle (as Dedekind’s or Birkhoff’s postulates, discussed below, demand). This demand contrasts with earlier views such as Eutocius (fourth century), ‘Even if it seemed not yet possible to produce a straight line equal to the circumference of the circle, nevertheless, the fact that there exists some straight line by nature equal to it is deemed by no one to be a matter of investigation.’24 Although Eutocius asserts the existence of a line of the same length as a curve but finds constructing it unimportant, Aristotle has a stronger view. Summarizing his discussion of Aristotle, Crippa [2014a, pp. 34–35] points out that Aristotle takes the impossibility of such equality as the hypothesis of an argument on motion and Crippa cites Averroes as holding ‘that there cannot be a straight line equal to a circular arc’. It is widely understood25 that Dedekind’s analysis is radically different from that of Eudoxus. A principal reason for this, discussed in Section 3.1, is that Eudoxus applies his method to specific situations; Dedekind demands that every cut be filled. Secondly, Dedekind develops addition and multiplication on the cuts. Thus, Dedekind’s postulate should not be regarded as part of either Euclidean data set. But $$\mathrm{EG}_\pi$$ makes a much more restrained demand; as in Eudoxus, a specific problem is solved. 3.1 Formalizing $$\boldsymbol{\pi}$$ in Euclidean Geometry The geometry over a Euclidean field (every positive number has a square root) may have no straight-line segment of length $$\pi$$. E.g., the model over the surd field (Notation 1.1) does not contain $$\pi$$. Neither does the field of real algebraic numbers; so $$\mathcal{E}^2$$ does not resolve the issue. 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 $$\pi$$ does. For Archimedes and Euclid, sequences constructed in the study of magnitudes in the Elements are of geometric objects, not numbers. But, in a modern account, as we saw already while discussing areas of polygons in Part I, we must identify the proportionality constant and verify that it represents a point in any model of the theory.26 Thus this goal diverges from a ‘Greek’ data set and indeed is complementary to the axiomatization of Cartesian geometry by Tarski’s $$\mathcal{E}^2$$. Euclid’s third postulate, ‘describe a circle with given center and radius’, entails that a circle is uniquely determined by its radius and center. In contrast, Hilbert simply defines the notion of circle and proves the uniqueness [Hartshorne, 2000, Lemma 11.1]. In either case we have the basic correspondence between angles and arcs: two segments of a circle are congruent if they subtend the same central angle. We established in Part I that for each model of EG and any line of the model, the surd field $$F_s$$ is embeddable in the field definable on that line. On this basis we can interpret the Greek theory of limits by way of cuts in the ordered surd field $$F_s$$. The following extensions, EG$$_{\pi}$$ and $$\mathcal{E}^2_\pi$$, of the systems EG and $$\mathcal{E}^2$$ guarantee the existence of $$\pi$$ as such a cut. Axioms for $$\boldsymbol\pi$$: Add to the vocabulary a new constant symbol $$\pi$$. Let $$i_n$$ ($$c_n$$) be the perimeter of a regular $$3\times 2^n$$-gon inscribed27 (circumscribed) in a circle of radius $$1$$. Let $$\Sigma(\pi)$$ be the collection of sentences (i.e., a type28)   $$i_n < 2\pi < c_n$$ for $$n< \omega$$. Now, we can define the new theories. (1) EG$$_{\pi}$$ denotes the deductive closure of the following set of axioms in the vocabulary $$\tau$$ augmented by constant symbols $$0, 1,\pi$$. (i) the axioms EG of a Euclidean plane; (ii) $$\Sigma(\pi)$$. (2) $$\mathcal{E}^2_\pi$$ is formed by adding $$\Sigma(\pi)$$ to $$\mathcal{E}^2$$ and taking the deductive closure. Dicta on constants: Here we named a further single constant $$\pi$$. But the effect is very different from naming $$0$$ and $$1$$ (Compare the Dicta on constants just after Theorem 4.1.1 of Part I.) The new axioms specify the place of $$\pi$$ in the ordering of the definable points of the model. So the data set is seriously extended. Theorem 3.1EG$$_{\pi}$$ is a consistent but not finitely axiomatizable29incomplete theory. Proof. A model of EG$$_{\pi}$$ is given by closing $$F_s \cup \{\pi\}\subseteq \Re$$ to a Euclidean field. To see the theory is not finitely axiomatizable, for any finite subset $$\Sigma_0(\pi)$$ of $$\Sigma(\pi)$$ choose a real algebraic number $$p$$ satisfying $$\Sigma_0$$ when $$p$$ is substituted for $$\pi$$; close $$F_s \cup \{p\} \subseteq \Re$$ to a Euclidean field to get a model of EG$$\ \cup\ \Sigma_0$$ which is not a model of EG$$_{\pi}$$. □ Dicta on Definitions or Postulates: We now extend the ordering on segments by adding the lengths of ‘bent lines’ and arcs of circles to the domain. Two approaches30 to this step are: (a) introduce an explicit but inductive definition; or (b) add a new predicate to the vocabulary and new axioms specifying its behavior. This alternative reflects in a way the trope that Hilbert’s axioms are implicit definitions. We take approach (a) in Definitions 3.2, 3.3, etc. using our established geometric vocabulary. Crucially, the following definition of bent lines (and thus the perimeter of certain polygons) is not a single formal definition but a schema of formulas $$\phi_n$$ defining an approximation for each $$n$$. Definition 3.2Let$$n\geq 2$$. By a bent line31$$b = Y_1\ldots Y_n$$we mean a sequence of straight line segments $$Y_iY_{i+1}$$, for $$1\leq i \leq n-1$$, such that each end point of one is the initial point of the next. We specify the length of a bent line $$b = Y_1\ldots Y_n$$, denoted by $$[b]$$, as the length given by the straight-line segment composed of the sum of the segments of $$b$$. Now we say an approximant $$Y_1, \ldots Y_{n+1}$$ to the arc $$X_1\ldots X_n$$ of a circle with center $$P$$, is a bent line satisfying: (1) $$X_1, \ldots X_n, Y_1, \ldots Y_{n}$$ are points such that each $$X_i$$ is on the circle and each $$Y_i$$ is in the exterior of the circle; (2) each of $$Y_iY_{i+1}$$ ($$1< i< n$$) $$Y_n Y_1$$ is a line segment; (3) for $$1< i < n$$, $$Y_iY_{i+1}$$ and $$Y_nY_1$$ are tangent to the circle at $$X_i$$. We obtain the circumference of a circle by requiring $$X_n = X_{1}$$ and $$Y_n=Y_1$$. Definition 3.3Let $${\mathcal S}$$ be the set (of congruence classes of) straight line segments. Let $$\mathcal C_r$$ be the set (of equivalence classes under congruence) of arcs on circles of a given radius $$r$$. Now we extend the linear order on $${\mathcal S}$$ to a linear order $$<_r$$ on $${\mathcal S}\cup \mathcal C_r$$ as follows. For $$s \in {\mathcal S}$$ and $$c\in \mathcal C_r$$ (1) The segment $$s<_r c$$ if and only if there is a chord $$XY$$ of a circular segment $$AB \in c$$ such that $$XY \in s$$. (2) The segment $$s >_r c$$ if and only if there is an approximant $$b =X_1 \ldots X_n$$ to $$c$$ with length $$[b] = s$$ and with $$[X_1 \ldots X_n]>_r c$$. It is easy to see that this order is well-defined, as each chord of an arc is shorter than the arc, and the arc is shorter than any approximant to it. Now, we encode a second approximation of $$\pi$$, using the areas $$I_n, C_n$$ of the approximating polygons rather than their perimeters $$i_n, c_n$$. There are two aspects to transferring the defintion from circumference to area: (1) modifying the development of the area function of polygons described in Section 4.4 of Part I, by extending the notion of figure to include sectors of circles, and (2) formalizing a notion of equal area, including a schema for approximation of circles32 by finite polygons. We omit those details analogous to 3.2–3.3. We carried out the harder case of circumference to emphasize the innovation of Archimedes in defining arc length; unlike area it is not true that the perimeter of a polygon containing a second is necessarily larger than the perimeter of the enclosed polygon. Lemma 3.4Let $$I_n$$ and $$C_n$$ denote the area of the regular $$3 \times 2^n$$-gon inscribed in or circumscribing the unit circle. Then $$EG_\pi$$ proves33each of the sentences $$I_n < \pi < C_n$$ for $$n< \omega$$. Proof. The intervals $$(I_n,C_n)$$ define the cut for $$\pi$$ in the surd field $$F_s$$ and the intervals $$(i_n,c_n)$$ define the cut for $$2\pi$$ and it is a fact about the surd field that one half of any realization of the second cut is a realization of the first. □ To argue that $$\pi$$, as implicitly defined by the theory EG$$_\pi$$, serves its geometric purpose, we add new unary function symbols $$C$$ and $$A$$ mapping our fixed line to itself and satisfying a scheme asserting that the functions these symbols refer to do, in fact, produce the required limits. The definitions are identical except for substituting the area for the perimeter of the approximating polygons. This strategy mimics that in an introductory calculus course of describing the properties of area and proving that the Riemann integral satisfies them. Definition 3.5A unary function $$C(r)$$ ($$A(r)$$) mapping $${\mathcal S}$$, the set of equivalence classes (under congruence) of straight line segments, into itself that satisfies the conditions below is called a circumference function (area function). (1) $$C(r)$$ ($$A(r)$$) is less than the perimeter (area) of a regular $$3\times 2^{n}$$-gon circumscribing a circle of radius $$r$$. (2) $$C(r)$$ ($$A(r)$$) is greater than the perimeter (area) of a regular $$3\times 2^n$$-gon inscribed in a circle of radius $$r$$. We can extend EG$$_{\pi}$$ to include definitions of $$C(r)$$ and $$A(r)$$. The theory EG$$_{\pi, A, C}$$ is the extension of the $$\tau \cup \{0,1,\pi\}$$-theory EG$$_{\pi}$$, obtained by the explicit definitions: $$A(r) =\pi r^2$$ and $$C(r) = 2\pi r$$. In any model of EG$$_{\pi,A,C}$$, for each circle of radius $$r$$ there is an $$s \in {\mathcal S}$$ whose length34$$C(r) = 2\pi r$$ is less than the perimeters of all circumscribed polygons and greater than those of the inscribed polygons. We can verify that by choosing $$n$$ large enough we can make $$i_n$$ and $$c_n$$ as close together as we like (more precisely, for given $$m$$, make them differ by $$< 1/m$$). In phrasing this sentence I follow Heath’s description of Archimedes’ statements: But he follows the cautious method to which the Greeks always adhered; he never says that a given curve or surface is the limiting form of the inscribed or circumscribed figure; all that he asserts is that we can approach the curve or surface as nearly as we please. [Heath, 2011, Ch. 4] Invoking Lemma 3.4, since the $$2I_n (2C_n)$$ converge to the limit of the $$i_n (c_n)$$, they determine the same cut, that of $$2\pi$$: Theorem 3.6In EG$$^2_{\pi,A,C}$$, $$C(r) = 2\pi r$$ is a circumference function and $$A(r) = \pi r^2$$ is an area function. In an Archimedean field there is a unique interpretation of $$\pi$$ and thus a unique choice for a circumference function with respect to the vocabulary without the constant $$\pi$$. By adding the constant $$\pi$$ to the vocabulary we get a formula which satisfies the conditions in every model. But in a non-Archimedean model, any point in the monad35 of $$2\pi r$$ would equally well fit our condition for being the circumference. To sum up, we have extended our descriptively complete axiomatization from the polygonal geometry of Hilbert’s first-order axioms (HP5) to Euclid’s results on circles and beyond. Euclid does not deal with arc length at all and we have assigned straight line segments to both the circumference and area of a circle. It follows that our development would not qualify as a modest axiomatization of Greek geometry but only of the modern understanding of these formulas. However, this distinction is not a problem for the notion of descriptive axiomatization. The facts are given as sentences. The formulas for circumference and area are not the same sentences as the Euclid/Archimedes statements in terms of proportions, but the Greek versions are implied by the modern equational formulations. 3.2. Formalizing $$\boldsymbol{\pi}$$ in Cartesian Geometry We now want to make a similar extension of $$\mathcal{E}^2$$. Dedekind [1963, pp. 37–38] observes that the field of real algebraic numbers is ‘discontinuous everywhere’ but ‘all constructions that occur in Euclid’s Elements can ... be just as accurately effected as in a perfectly continuous space’. Strictly speaking, for constructions this is correct. But the proportionality constant $$\pi$$ between a circle and its circumference is absent; so, it cannot be the case that both a straight line segment of the same length as the circumference and the diameter are in the model.36 We want to find a middle ground between the constructible entities of Euclidean geometry and Dedekind’s postulation that all transcendentals exist. That is, we propose a theory which proves the circumference and area formulas for circles and countable models of the geometry over RCF, one where ‘arc length behaves properly’ by characterizing the only transcendental number known in antiquity. In contrast to Dedekind and Hilbert, Descartes eschews the idea that there can be a ratio between a straight line segment and a curve. Crippa [2014b] writes, ‘Descartes excludes the exact knowability of the ratio between straight and curvilinear segments’; then he quotes Descartes: ... la proportion, qui est entre les droites et les courbes, n’est pas connue, et mesme ie croy ne le pouvant pas estre par les hommes, on ne pourroit rien conclure de là qui fust exact et assuré.37 Hilbert [2004, pp. 429–430] asserts that there are many geometries satisfying his axioms I–IV and V.1 but only one, ‘namely the Cartesian geometry’ that also satisfies V.2. Thus the conception of ‘Cartesian geometry’ changed radically from Descartes to Hilbert; even the symbol $$\pi$$ was not introduced until 1706 (by Jones). One wonders whether it had changed by the time Hilbert wrote. That is, had readers at the turn of the twentieth century already internalized a notion of Cartesian geometry which entailed Dedekind completeness and that was, at best, formulated in the nineteenth century (Bolzano-Cantor-Weierstrass-Dedekind)? We now define a theory $$\mathcal{E}^2_\pi$$ analogous to EG$$_\pi$$ that does not depend on the Dedekind axiom but can be obtained in a first-order way. Given Descartes’ proscription of $$\pi$$, the new system will be immodest with respect to the Cartesian data set. But we will argue at the end of this section that both of our axioms for $$\pi$$ are closer to Greek conceptions than the Dedekind axiom. At this point we need some modern model theory to guarantee the completeness of the theory we are defining. A first-order theory $$T$$ for a vocabulary including a binary relation $$<$$ is o-minimal if every model of $$T$$ is linearly ordered by $$<$$ and every $$1$$-ary formula is equivalent in $$T$$ to a Boolean combination of equalities and inequalities [van den Dries, 1999]. Anachronistically, the o-minimality of the reals is a main conclusion of Tarski [1931]. We can now show: Theorem 3.7Form $$\mathcal{E}^2_{\pi}$$ by adjoining $$\Sigma(\pi)$$ to $$\mathcal{E}^2$$. $$\mathcal{E}^2_{\pi}$$ is first-order complete for the vocabulary $$\tau$$ augmented by constant symbols $$0, 1,\pi$$. Proof. We have established that there is a definable ordered field whose domain is the line through the points $$0, 1$$. By Tarski, the theory of this real closed field is complete. The field is bi-interpretable with the plane [Tarski, 1951]; so the theory of the geometry $$T$$ is complete as well. Further by [Tarski, 1931], the field is o-minimal. Therefore, the type over the empty set of any point on the line is determined by its position in the linear ordering of the subfield $$F_s$$ (Notation 1.1). Each $$i_n, c_n$$ is an element of the field $$F_s$$. The position of $$2\pi$$ in the linear order on the line through $$01$$ is given by $$\Sigma$$. Thus $$T \cup \Sigma(\pi)$$ is a complete theory. □ As we extended $$\mathrm{EG}_n$$, we now extend the theory $$\mathcal{E}^2_\pi$$. Definition 3.8The theory $$\mathcal{E}^2_{\pi,A,C}$$ is the extension of the $$\tau \cup \{0,1,\pi\}$$-theory $$\mathcal{E}^2_{\pi}$$ obtained by adding the explicit definitions: $$A(r) =\pi r^2$$ and $$C(r) = 2\pi r$$. Theorem 3.9The theory $$\mathcal{E}^2_{\pi,A,C}$$ is a complete, decidable extension of EG$$_{\pi,A,C}$$ that is coordinatized by an o-minimal field. Moreover, in $$\mathcal{E}^2_{\pi,A,C}$$, $$C(r) = 2\pi r$$ is a circumference function and $$A(r) =\pi r^2$$ is an area function. Proof. We are adding definable functions to $$\mathcal{E}^2_\pi$$ so o-minimality and completeness are preserved. The theory is recursively axiomatized and complete, so decidable. The formulas continue to compute area and circumference correctly (as in Theorem 3.6) since they extend EG$$_{\pi,A,C}$$. □ The assertion that $$\pi$$ is transcendental is a theorem of the first-order theory $$\mathcal{E}^2_{\pi}$$. We explore the distinction between proving a fact of mathematics and showing it is provable in a first-order theory in Section 4.3. In this case, Lindemann proved that $$\pi$$ does not satisfy a polynomial of degree $$n$$ for any $$n$$. Thus for any polynomial $$p(x)$$ over the rationals $$p(\pi) \neq 0$$ is a consequence of the complete type38 generated by $$\Sigma(\pi)$$ and so is a theorem of $$\mathcal{E}^2_{\pi}$$. We now extend the known fact that the theory of real closed fields is ‘finitistically justified’ (in the list of such results [Simpson, 2009, p. 378]) to $$\mathcal{E}^2_{\pi,A,C}$$. For convenience, we lay out the proof with reference to results39 recorded in [Simpson, 2009]. The theory $$\mathcal{E}^2$$ is bi-interpretable with the theory of real closed fields. And thus it (as well as $$\mathcal{E}^2_{\pi,A, C}$$) is finitistically consistent, in fact, provably consistent in primitive recursive arithmetic (PRA). By Theorem II.4.2 of [Simpson, 2009], RCA$$_0$$ proves the system $$(Q,+,\times,<)$$ is an ordered field and by II.9.7 of [Simpson, 2009], it has a unique real closure. Thus the existence of a real closed ordered field and so Con(RCOF) is provable in RCA$$_0$$. (Note that the construction will imbed the surd field $$F_s$$.) Lemma IV.3.3 of [Friedman et al., 1983] asserts the provability of the completeness theorem (and hence compactness) for countable first-order theories from WKL$$_0$$. Note that since every finite subset of $$\Sigma(\pi)$$ is satisfiable in any RCOF, it follows that the existence of a model of $$\mathcal{E}^2_\pi$$ is provable in WKL$$_0$$. Since WKL$$_0$$ is $$\pi^0_2$$-conservative over PRA, we conclude PRA proves the consistency of $$\mathcal{E}^2_\pi$$. As $$\mathcal{E}^2_{\pi, C,A}$$ is an extension by explicit definitions, its consistency is also provable in PRA, as required. This completes the argument. It might be objected that such minor changes as adding to $$\mathcal{E}^2$$ the name of the constant $$\pi$$, or adding the definable functions $$C$$ and $$A$$ undermines the earlier claim that $$\mathcal{E}^2$$ is descriptively complete for Cartesian geometry. But $$\pi$$ is added because the modern view of ‘number’ requires it and increases the data set to include propositions about $$\pi$$ which are inaccessible to $$\mathcal{E}^2$$. We have so far tried to find the proportionality constant only in specific situations. In the remainder of the section, we introduce a model-theoretic scheme to systematize the solution of families of ‘quadrature’ problems. Crippa describes Leibniz’s distinguishing two types of quadrature, ... ‘universal quadrature’ of the circle, namely the problem of finding a general formula, or a rule in order to determine an arbitrary sector of the circle or an arbitrary arc; and on the other [hand] he defines the problem of the ‘particular quadrature’, ... , namely the problem of finding the length of a given arc or the area of a sector, or the whole circle ... . [Crippa, 2014a, p. 424] Birkhoff [1932] ignores such a distinction with the protractor postulate of his system.40 POSTULATE III. The half-lines $$\ell, m$$, through any point O can be put into $$(1, 1)$$ correspondence with the real numbers $$a(\text{mod}\ 2 \pi)$$, so that, if $$A \neq O$$ and $$B\neq O$$ are points of $$\ell$$ and $$m$$ respectively, the difference $$a_m - a_\ell (\text{mod}\ 2\pi)$$ is $$\angle AOB$$. Furthermore, if the point $$B$$ varies continuously in a line $$r$$ not containing the vertex O, the number $$a_m$$ varies continuously also.41 This axiom is analogous to Birkhoff’s ‘ruler postulate’ which assigns each segment a real number length. Thus, he takes the real numbers as an unexamined background object; at one swoop he has introduced addition and multiplication, and assumed the Archimedean and completeness axioms. So even ‘neutral’ geometries studied on this basis are actually greatly restricted.42 He argues that his axioms define a categorial system isomorphic to $$\Re^2$$. So his system (including an axiomatization of the real field that he has not specified) is bi-interpretable with Hilbert’s. However, the protractor postulate conflates three distinct problems: (i) the rectifiability of arcs, the assertion that each arc of a circle has the same length as a straight line segment; (ii) the claim there is an algorithm for finding such a segment; and (iii) the measurement of angles, that is assigning a measure to an angle as the arc length of the arc it determines. The next task is to find a more modest version of Birkhoff’s postulate, namely, a first-order theory with countable models which assign to each angle a measure between $$0$$ and $$2\pi$$. Recall that we have a field structure on the line through the points $$0,1$$ and the number $$\pi$$ on that line, so we can make a further explicit definition. A measurement of angles function is a map $$\mu$$ from congruence classes of angles into $$[0,2\pi)$$ such that if $$\angle ABC$$ and $$\angle CBD$$ are disjoint angles sharing the side $$BC$$, $$\mu (\angle ABD) = \mu(\angle ABC) + \mu(\angle CBD)$$. If we omitted the additivity property this would be trivial. Given an angle $$\angle ABC$$ less than a straight angle, let $$C'$$ be the intersection of a perpendicular to $$BC$$ through $$A$$ with $$BC$$ and let $$\mu (\angle ABC) = 2\pi\cdot\sin(\angle ABC) = \frac{2 \pi \cdot BC'}{AB}$$. (It is easy to extend to larger angles.) Here we use approach (b) of the Dicta on Definitions rather than the explicit definition approach (a) used for $$C(r)$$ and $$A(r)$$. We define a new theory with a function symbol $$\mu$$ which is ‘implicitly defined’ by the following axioms. Definition 3.10The theory $$\mathcal{E}^2_{\pi,A,C,\mu}$$ is obtained by adding to $$\mathcal{E}^2_{\pi,A,C}$$ the assertion that $$\mu$$ is a continuous additive map from congruence classes of angles to $$[0,2\pi)$$. It is straightforward to express that $$\mu$$ is continuous. If we omitted that requirement in Definition 3.10, $$\mathcal{E}^2_{\pi,A,C,\mu}$$ would be incomplete since $$\mu$$ would be continuous in some models, but not in some non-Archimedean models. Thus, we require continuity. Showing consistency of $$\mathcal{E}^2_{\pi,A,C,\mu}$$ is easy; we can define (in the mathematical sense, not as a formally definable function) in $$\mathcal{E}^2_{\pi,A,C}$$ such a function $$\mu^*$$, say, the restricted arc-cosine.43 Hence, the axioms are consistent and this solves the rectifiability problem. But, merely assuming the existence of a $$\mu$$ does not solve our problem (ii), as we have no idea how to compute $$\mu$$ and a recursive axiomatization is a real mathematical problem. 3.3 Some Countable Models of Geometry Blanchette [2014] distinguishes two approaches to logic — deductivist and model-centric — and argues that Hilbert represents the deductivist school and Dedekind the model-centric. Essentially, the second proposes that theories are designed to try to describe an intuition of a particular structure. We now consider a third direction: are there ‘canonical’ models of the various theories we have been considering? By modern tradition, the continuum is the real numbers and geometry is the plane over it. Is there a smaller model which reflects the geometric intuitions discussed here? For Euclid II, there is a natural candidate, the Euclidean plane over the surd field $$F_s$$. Remarkably, this does not conflict with Euclid XII.2 (the area of a circle is proportional to the square of the diameter). The model is Archimedean and $$\pi$$ is not in the model. But Euclid only requires a proportionality which defines a type $$\Sigma(x)$$, not a realization $$\pi$$ of $$\Sigma(x)$$. Plane geometry over the real algebraic numbers plays the same role for $$\mathcal{E}^2$$. Both are categorical in $$L_{\omega_1,\omega}$$. In the second case, the axiomatization asserts ‘every field element is algebraic’. We have developed a method of assigning measures to angles. Now we argue that the methods of this section better reflect the Greek view than Dedekind’s approach does. Mueller makes an important point distinguishing the Euclid/Eudoxus use from Dedekind’s use of cuts. In broad outline, the following quotation describes the methodology here. One might say that in applications of the method of exhaustion the limit is given and the problem is to determine a certain kind of sequence converging to it, ... Since, in the Elements the limit always has a simple description, the construction of the sequence can be done within the bounds of elementary geometry; and the question of constructing a sequence for any given arbitrary limit never arises. [Mueller, 2006, p. 236] But what if we want to demand the realization of various transcendentals? Mueller’s description suggests the principle that we should only realize cuts in the field order that are recursive over a finite subset. We might call these Eudoxian transcendentals. So a candidate would be a recursively saturated model44 of $$\mathcal{E}^2$$. Remarkably, almost magically,45 this model would also satisfy $$\mathcal{E}^2_{\pi,A,C,\mu}$$. A recursively saturated model is necessarily non-Archimedean. There are however many different countable recursively saturated models depending on which transcendentals are realized. Arguably there is a more canonical candidate for a natural model which admits the ‘Eudoxian transcendentals’; take the smallest elementary submodel of $$\Re$$ closed46 under $$A,C, \mu$$ that contains the real algebraic numbers and all realizations of recursive cuts in $$F_s$$. The Scott sentence47 of this sentence is a categorical sentence in $$L_{\omega_1,\omega}$$. The models in this paragraph are all countable; we cannot do this with the Hilbert model of the plane over the real numbers; it has no countable $$L_{\omega_1,\omega}$$-elementary submodel. We turn to the question of modesty. Mueller’s distinction can be expressed in another way. Eudoxus provides a technique to solve certain problems, which are specified in each application. In contrast, Dedekind’s postulate solves $$2^{\aleph_0}$$ problems at one swoop. Each of the theories $$\mathcal{E}^2_{\pi}$$, $$\mathcal{E}^2_{\pi,A,C}$$, $$\mathcal{E}^2_{\pi,A,C, \mu}$$ and the later search for their canonical models reflect this distinction. Each solves at most a countable number of recursively stated problems. In summary, we regard the replacement of ‘congruence class of segment’, by ‘length represented by an element of the field’ as a modest reinterpretation of Greek geometry. But this treatment of length becomes immodest relative even to Descartes when this length is a transcendental. And most immodest of all is to demand lengths for arbitrary transcendentals. 4. AND BACK TO HILBERT In this section we examine Hilbert’s ‘continuity axioms’. We study the syntactic form of various axioms, their consequences, and their role in clarifying the notion of the continuum. The Archimedean axiom is minimizing; each cut is realized by at most one point; so each model has cardinality at most $$2^{\aleph_0}$$. The Veronese postulate (footnote 50) or Hilbert’s Vollständigkeitsaxiom is maximizing; each cut is realized; in the absence of the Archimedean axiom the set of realizations could have arbitrary cardinality. 4.1 The Role of the Axiom of Archimedes in the Grundlagen A primary aim expressed in Hilbert’s introduction is ‘to bring out as clearly as possible the significance of the groups of axioms’. Much of his book is devoted to this metamathematical investigation. In particular this includes Sections 9–12 (from [Hilbert, 1971]) concerning the consistency and independence of the axioms. Further examples,48 in Sections 31–34, show that without the congruence axioms, the axiom of Archimedes is necessary to prove what Hilbert labels as Pascal’s (Pappus) theorem. In the conclusion to [1962], Hilbert notes Dehn’s work on the necessary role of the Archimedean axiom in establishing over neutral geometry the relation between the number of parallel lines through a point and the sum of the angles of a triangle. These are all metatheoretical results. In contrast, the use of the Archimedean axiom in Sections 19 and 21 to prove equidecomposable is the same as equicomplementable (equal content) (in 2 dimensions) is certainly a proof in the system. But an unnecessary one. As we argued in Section 4.4 of Part I, Hilbert could just have easily defined ‘same area’ as ‘equicomplementable’ (as is a natural reading of Euclid). Thus, we find no geometrical theorems in the Grundlagen that essentially depend on the axiom of Archimedes. Rather, Hilbert’s use of the axiom of Archimedes is (i) to investigate the relations among the various principles, and (ii) in conjunction with the Vollständigkeitsaxiom, to identify the field defined in the geometry with the independently existing real numbers as conceived by Dedekind. With respect to the problem studied here, I contend that these results do not affect the conclusion that Hilbert’s full axiom set is an immodest axiomatization49 of Euclid I or Euclid II or of the Cartesian data set since those data sets contain and are implied by the appropriate first-order axioms. 4.2 Hilbert and Dedekind on Continuity In this section we compare various formulations of the completeness axiom. Hilbert wrote: Axiom of Completeness (Vollständigkeitsaxiom): To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid. [Hilbert, 1971] In this article we have used the following adaptation of Dedekind’s postulate for geometry (DG): DG: Any cut in the linear ordering imposed on any line by the betweenness relation is realized. While this formulation is convenient for our purposes, it misses an essential aspect of Hilbert’s system; in a context with a group, DG implies the Archimedean axiom, while Hilbert was aiming for an independent set of axioms. Hilbert’s axiom does not imply Archimedes’. A variant VER50 on Dedekind’s postulate that does not imply the Archimedean axiom was proposed by Veronese [1889]. If VER replaces DG, those axioms would also satisfy the independence criterion. Hilbert’s completeness axiom in [1971] asserting any model of the rest of the theory is maximal, is inherently model-theoretic. The later line completeness [Hilbert, 1962] is a technical variant.51 Giovannini’s account [2013], which relies on [Hilbert, 2004] includes a number of points already made here and three more. First, Hilbert’s completeness axiom is not about deductive completeness (despite having such consequences), but about maximality of every model (p. 145). Secondly (last line of p. 153) Hilbert expressly rejects Cantor’s axiom on the intersection of a sequence of closed intervals because it relies on a sequence of intervals and ‘sequence is not a geometrical notion’. A third intriguing note is an argument due to Baldus in 1928 that the parallel axiom is an essential ingredient in the categoricity of Hilbert’s axioms.52 Here are two reasons for choosing Dedekind’s (or Veronese’s) version. One is that Dedekind’s formulation, since it is about the geometry, not about its axiomatization, directly gives the kind of information about the existence of transcendental numbers that we observe in this paper. Even more basic is that one cannot formulate Hilbert’s version as a sentence $$\Phi_H$$ in second-order logic53 of geometry with the intended interpretation $$(\Re^2,{\bf G}) \models \Phi_H$$. The axiom requires quantification over subsets of an extension of the model which putatively satisfies it. Here is a second-order statement54$$\Theta$$, where $$\psi$$ denotes the conjunction of Hilbert’s first four axiom groups and the axiom of Archimedes.   $$(\forall X)(\forall Y) (\forall {\bf R}) [[X \subseteq Y \wedge (X,{\bf R} \mathord\restriction X) \models \psi \wedge (Y,{\bf R} ) \models \psi]\rightarrow X=Y],$$ whose validity expresses Hilbert’s V.2 but which is a sentence in pure second-order logic rather than in the vocabulary for geometry. Väänänen [2012, p. 94] investigates this anomaly by distinguishing between $$(\Re^2,{\bf G}) \models \Phi$$, for some $$\Phi$$ and the validity of $$\Theta$$. He expounds in [2014] a new notion, ‘Sort Logic’, which provides a logic with a sentence $$\Phi^\prime_H$$ which, by allowing a sort for an extension, formalizes Hilbert’s V.2 with a more normal notion of truth in a structure. In [Väänänen, 2012; Väänänen and Wang, 2015], Väänänen discusses the categoricity of natural structures such as real geometry when axiomatized in second-order logic. He develops the striking phenomenon of ‘internal categoricity’,55 which as we now argue refutes the view that second-order categoricity ‘depends on the set theory’. Suppose the second-order categoricity of a structure $$A$$ is formalized by the existence of sentence $$\Psi_A$$ such that $$A \models \Psi_A$$ and any two models of $$\Psi_A$$ are isomorphic. If this second clause is provable in a standard deductive system for second-order logic, then it is valid in the Henkin semantics, not just the full semantics. Thus, its truth is independent of the ambient set theory. Philip Ehrlich has made several important discoveries concerning the connections between the two ‘continuity axioms’ in Hilbert and develops the role of maximality. First, he observes [Ehrlich, 1995, p. 172] that Hilbert had already pointed out that his completeness axiom would be inconsistent if the maximality were only with respect to the first-order axioms. Secondly, he [1995; 1997] systematizes and investigates the philosophical significance of Hahn’s notion of Archimedean completeness. Here the structure (ordered group or field) is not required to be Archimedean; the maximality condition requires that there is an extension which fails to extend an Archimedean equivalence class.56 This notion provides a tool (not yet explored) for investigating the non-Archimedean models studied in Section 3. In a sense, our development is the opposite of Ehrlich’s in [2012]. Rather than trying to unify all numbers great and small in a class model we are interested in the minimal collection of numbers that allow the development of a geometry that proved a modest axiomatization of the data sets considered. 4.3. Against the Dedekind Posulate for Geometry Our fundamental claim is that (slight variants on) Hilbert’s first-order axioms provide a modest descriptively complete axiomatization of most of Greek geometry. One goal of Hilbert’s continuity axioms was to obtain categoricity. But categoricity is not part of the data set but rather an external limitative principle. The notion that there was ‘one’ geometry (i.e., categoricity) was implicit in Euclid. But it is not a geometrical statement. Indeed, Hilbert [1962, p. 23] described his metamathematical formulation of the completeness axiom as, ‘not of a purely geometrical nature’. We argued [Baldwin, 2014. §3] against the notion of categoricity as an independent desideratum for an axiom system. We noted there that various authors have proved under $$V=L$$, any countable or Borel structure can be given a categorical axiomatization and that there are no strong structural consequences of the mere fact of second-order categoricity. However, there we emphasized the significance of axiomatizations, such as Hilbert’s, that reveal underlying principles concerning such iconic structures as geometry and the natural numbers. Here we go further, and suggest that even for an iconic structure there may be advantages to a first-order axiomatization that trump the loss of categoricity. We argue now that the Dedekind postulate is inappropriate (in particular immodest) in any attempt to axiomatize the Euclidean or Cartesian or Archimedean data sets for several reasons: (1) The requirement that there be a straight line segment measuring any circular arc contradicts the intent of various pre-Cartesian geometers and of Descartes. It clearly extends the collection of geometric entities appearing in either Euclid or Descartes. (2) The Dedekind postulate is not needed to establish the properly geometrical propositions in the data set. Hilbert [1971, p. 26] closes the discussion of continuity with ‘However, in what is to follow, no use will be made of the “axiom of completeness”.’ Why then did he include the axiom? Earlier in the same paragraph,57 he writes that ‘it allows the introduction of limiting points’ and enables one ‘to establish a one-one correspondence between the points of a segment and the system of real numbers’. Archimedes’ axiom makes the correspondence injective and the Vollständigkeitsaxiom makes it surjective. We have noted here that the grounding of real algebraic geometry (the study of systems of polynomial equations in a real closed field) is fully accomplished by Tarski’s axiomatization. And we have provided a first-order extension to deal with the basic properties of the circle. Since Dedekind, Weierstrass, and others pursued the ‘arithmetization of analysis’ precisely to ground the theory of limits, identifying the geometrical line as the Dedekind line reaches beyond the needs of geometry. (3) Proofs from Dedekind’s postulate obscure the true geometric reason for certain theorems. Hartshorne writes: $$\ldots$$ there are two reasons to avoid using Dedekind’s axiom. First, it belongs to the modern development of the real-number systems and notions of continuity, which is not in the spirit of Euclid’s geometry. Second, it is too strong. By essentially introducing the real numbers into our geometry, it masks many of the more subtle distinctions and obscures questions such as constructibility that we will discuss in Chapter 6. So we include the axiom only to acknowledge that it is there, but with no intention of using it. [Hartshorne, 2000, p. 177] (4) The use of second-order axioms undermines a key proof method — informal (semantic) proof — the ability to use higher-order methods to demonstrate that there is a first-order proof. A crucial advantage of a first-order axiomatization is that it licenses the kind of argument described in Hilbert and Ackerman:58 Derivation of Consequences from Given Premises; Relation to Universally Valid Formulas So far we have used the predicate calculus only for deducing valid formulas. The premises of our deductions, viz Axioms (a) through (f), were themselves of a purely logical nature. Now we shall illustrate by a few examples the general methods of formal derivation in the predicate calculus $$\ldots$$ It is now a question of deriving the consequences from any premises whatsoever, no longer of a purely logical nature. The method explained in this section of formal derivation from premises which are not universally valid logical formulas has its main application in the setting up of the primitive sentences or axioms for any particular field of knowledge and the derivation of the remaining theorems from them as consequences $$\ldots$$ We will examine, at the end of this section, the question of whether every statement which would intuitively be regarded as a consequence of the axioms can be obtained from them by means of the formal method of derivation. We noted that Hilbert proved that every planar proposition deducible in 3-dimensional geometry is provable in 2-dimensional geometry from Desargues’ theorem by this sort of argument [Baldwin, 2013, §2.4], and we exploited this technique in Section 3 to provide axioms for the calculation of the circumference and area of a circle.59 In this article we expounded a procedure [Hartshorne, 2000] to define the field operations in an arbitrary model of HP5. We argued that the first-order axioms of EG suffice for the geometrical data sets Euclid I and II, not only in their original formulation but by finding proportionality constants for the area formulas of polygon geometry. By adding axioms to require that the field is real closed we obtain a complete first-order theory that encompasses many of Descartes’ innovations. The plane over the real algebraic numbers satisfies this theory; thus, there is no guarantee that there is a line segment of length $$\pi$$. Using the o-minimality of real closed fields, we can guarantee there is such a segment by adding a constant for $$\pi$$ and requiring it to realize the proper cut in the rationals. However, guaranteeing the uniqueness of such a realization requires the $$L_{\omega_1,\omega}$$ Archimedean axiom. Hilbert and the other axiomatizers of a hundred years ago wanted more; they wanted to secure the foundations of mathematical analysis. In full generality, this surely depends on second-order properties. But there are a number of directions of work on ‘definable analysis’ [Baldwin, 2018, §6.3] that provide first-order approaches to analysis. The study of o-minimal theories makes major strides. One direction of research in o-minimality has been to prove the expansion of the real numbers by particular functions (e.g., the $$\Gamma$$-function on the positive reals [Speissinger and van den Dries, 2000]). Peterzil and Starchenko [2000] study the foundations of calculus. They approach complex analysis through o-minimality of the real part in [Peterzil and Starchenko, 2010]. The impact of o-minimality on number theory was recognized by a Karp prize (Peterzil, Pila, Wilkie, Starchenko) in 2014. And a non-logician Range [2014] suggests using methods of Descartes to teach calculus. A key feature of the interaction of o-minimal theories with real algebraic geometry has been the absence60 of Dedekind’s postulate for most arguments [Bochnak et al., 1998]. Footnotes 1In the first instance we draw from Euclid: Books I–IV, VI, and XII.1, 2 clearly concern plane geometry; XI, the rest of XII, and XIII deal with solid geometry; V and X deal with a general notion of proportion and with incommensurability. Thus, below we put each proposition of Books I–IV, VI, XII.1, 2 in a group and consider certain geometrical aspects of Books V and X. 2The names HP, HP5, and EG come from [Hartshorne, 2000] and $$\mathcal{E}^2$$ from [Tarski, 1959]. 3These include Pasch’s axiom (B4 of [Hartshorne, 2000]) 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. 4The field $$F$$ is associated with a plane $$\Pi$$ if $$\Pi$$ is the Cartesian plane on $$F^2$$. 5In the vocabulary here, there is a natural translation of Euclid’s axioms into first-order statements. The construction axioms have to be viewed as sentences ‘for all ... there exist’. The axiom of Archimedes is of course not first-order. We write Euclid’s axioms for those in the original [Euclid, 1956] vs (first-order) axioms for Euclidean geometry, EG. 6RCF 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. 7We call this process ‘taking the Euclidean closure’ or adding constructible numbers. 8That is, for each point $$a$$ constructible by ruler and compass there is a formula $$\phi_a(x)$$ such that $$\mathrm{EG}\vdash (\exists ! x) \phi(x)$$. in $$EG$$. That is, there is a unique solution to $$\phi$$. 9His proof is still based on Eudoxus. 10This extraction cannot be done in EG, since EG is satisfied in the field which has solutions for all quadratic equations but not those of odd degree. See [Hartshorne, 2000, §12]. 11See Sections 1.2 and 3 of [Panza, 2011] as well as [Bos, 2001]. Rodin [2014] further develops Panza’s ‘open’ notion as a ‘constructive axiomatic’ system. 12But, as Crippa points out [2014a, p. 153], Descartes did not prove this equivalence and there is some controversy as to whether the 1876 work of Kempe solves the precise problem. 13Writing in 1832, Bolyai [Gray, 2004, appendix] wrote in his ‘explanation of signs’, ‘The straight $$AB$$ means the aggregate of all points situated in the same straight line with $$A$$ and $$B$$’. This is the earliest indication I know of the transition to an extensional version of incidence. William Howard showed me this passage. See the Bolzano quote in [Rusnock, 2000, p. 53]. 14In our modern understanding of an axiom set the translation is routine, but anachronistic. 15That is, a real number that satisfies a polynomial with rational coefficients. A real number that satisfies no such polynomial is called transcendental. 16Translation to our official vocabulary requires quantifiers. We abuse Tarski’s notation by letting $$\mathcal{E}^2$$ denote the theory in the vocabulary with constants $$0,1$$. 17Note that circle-circle intersection is implied by the axioms in (2). 18More precisely in terms of the linear order $$B(xyz)$$ means $$x \leq y \leq z$$. 19Tarski [1959] proves that planes over real closed fields are exactly the models of $$\mathcal{E}^2$$. 20Of 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. 21A nontrivial subtheory is one satisfied in $$\Re$$, Dedekind’s reals. 22The geometric version of this result was conjectured by Tarski in [1959]: The theory RCF is complete and recursively axiomatized, so decidable. For the context of Ziegler’s result and Tarski’s quantifier elimination in computer science see [Makowsky, 2013]. 23This classification is not in any sense chronological, as Archimedes attributes the method of exhaustion to Eudoxus who precedes Euclid. Post-Heath scholarship by Becker, Knorr, and Menn [Menn, forthcoming] has identified four theories of proportion in the generations just before Euclid. [Menn, forthcoming] led us to the three prototypic propositions. 24Taken from his commentary on Archimedes in Archimedes Opera Omnia cum commentariis Eutociis, vol. 3, p. 266. Quoted in [Crippa, 2014b]. 25Stekeler-Weithofer [1992] writes, ‘It is just a big mistake to claim that Eudoxus’s proportions were equivalent to Dedekind cuts’. Feferman [2008] avers, ‘The main thing to be emphasized about the conception of the continuum as it appears in Euclidean geometry is that the general concept of set is not part of the basic picture, and that Dedekind style continuity considerations of the sort discussed below are at odds with that picture.’ Stein [1990], though, gives a long (but to me unconvincing) argument for at least the compatibility of Dedekind’s postulate with Greek thought ‘reasons ... plausible, even if not conclusive – for believing the Greek geometers would have accepted Dedekind’s postulate, just as they did that of Archimedes, once it had been stated’. 26For this reason, Archimedes needs only his postulate while Hilbert would also need Dedekind’s postulate to prove the circumference formula. 27I thank Craig Smorynski for pointing out that is not so obvious that the perimeter of an inscribed $$n$$-gon is monotonic in $$n$$ and reminding me that Archimedes avoided the problem by starting with a hexagon and doubling the number of sides at each step. 28Let $$A \subset M \models T$$. A type over $$A$$ is a set of formulas $$\phi({\bf x},{\boldsymbol{a}})$$, where $${\bf x}$$ ($${\boldsymbol{a}}$$) is a finite sequence of variables (constants from $$A$$) that is consistent with $$T$$. Taking $$T$$ as EG, a type over all $$F_s$$ is a type over $$\{0,1\}$$ since each element of $$F_s$$ is definable over $$\{0,1\}$$ in EG. 29Ziegler [1982, Remark 2.1] shows that EG is undecidable. Since for any $$T$$ and type $$p(x)$$ consistent with $$T$$, the decidability of $$T \cup \{p(c)\}$$ implies the decidability of $$T$$, EG$$_{\pi}$$ is also undecidable. 30We could define $$<$$ on the extended domain or, in style (b), we could add a $$<^*$$ to the vocabulary and postulate that $$<^*$$ extends $$<$$ and satisfies the properties of the definition. 31This is less general than Archimedes [1897, p. 2], who allows segments of arbitrary curves ‘that are concave in the same direction’. 32By dealing with a special case, we suppressed Archimedes’ anticipation of the notion of bounded variation. 33Note that we have not attempted to justify the convergence of the $$i_n,c_n, I_n, C_n$$ in the formal system EG$$_\pi$$. We are relying on mathematical proof, not a formal deduction in first-order logic; we explain this distinction in item (4) of Section 4.3. 34A similar argument works for area and $$A(r)$$. 35The monad of $$a$$ is the collection of points that are an infinitessimal distance from $$a$$. 36Thus, Birkhoff’s protractor postulate (below) is violated. 37Descartes, Oeuvres, Vol. 6, p. 412. Crippa also quotes Averroes as emphatically denying the possibility of such a ratio and notes that Vieta held similar views. 38Recall that $$\Sigma(x)$$ is a consistent collection of formulas in one free variable, which by Tarski’s quantifier elimination are Boolean combinations of polynomial inequalities. 39We use RCOF (real closed ordered field) here for what we have called RCF before. Model theoretically adding the definable ordering of a formally real field is a convenience. Here we want to be consistent with the terminology in [Simpson, 2009]. Note that Friedman [1999] strengthens the results for PRA to exponential-function arithmetic (EFA). Friedman reports Tarski had observed the constructive consistency proof much earlier. The theories discussed here, in increasing proof strength are EFA, PRA, RCA$$_0$$ and WKL$$_0$$. 40This is the axiom system used in virtually all U.S. high schools since the 1960s. 41I slightly modified the last sentence from Birkhoff, in lieu of reproducing the diagram. 42That is, they must be metric geometries. 43 In fact, by coding a point on the unit circle by its $$x$$-coordinate and setting $$\mu((x_1,y_1), (x_2,y_2)) = \cos^{-1}(x_1-x_2)$$ one gets such a function which is definable in the theory of the real field expanded by the cosine function restricted to $$[0,2\pi)$$ and in $$\mathcal{E}^2_{\pi,C,A}$$ by Theorem 4.3.1 of Part 1. This theory is known to be o-minimal [van den Dries, 1999]. But there is no known axiomatization and David Marker tells me it is unlikely to be decidable without assuming the Schanuel conjecture. 44A model is recursively saturated if every recursive type over a finite set is realized [Barwise, 1975]. 45The magic is called resplendency. Every recursively saturated model is resplendent [Barwise, 1975] where $$M$$ is resplendent if any formula $$\exists A \phi(A,{\boldsymbol{c}})$$ that is satisfied in an elementary extension of $$M$$ is satisfied by some $$A'$$ on $$M$$. Examples are the formulas defining $$C,A, \mu$$. 46Interpret $$A,C, \mu$$ on $$\Re$$ in the standard way. 47For any countable structure $$M$$ there is a ‘Scott’ sentence $$\phi_M$$ such that all countable models of $$\phi_M$$ are isomorphic to $$M$$; see [Keisler, 1971, Ch. 1]. 48I thank the referee for pointing to the next two examples and emphasizing Hilbert’s more general goals of understanding the connections among organizing principles. The reference to Dehn was dropped in later editions of the Grundlagen. 49It might seem I could claim immodesty for Archimedes as well, in view of my first-order axioms for $$\pi$$. But that would be a cheat. I restricted that data set to Archimedes on the circle, while Archimedes proposed a general notion of arc length and studied many other transcendental curves. 50The axiom VER (see [Cantù, 1999]) asserts that for a partition of a linearly ordered field into two intervals $$L,U$$ (with no maximum in the lower $$L$$ or minimum in the upper $$U$$) and a third set in between with at most one point, there is a point between $$L$$ and $$U$$ just if for every $$e>0$$, there are $$a\in L, b\in U$$ such that $$b-a<e$$. Veronese derives Dedekind’s postulate from his axiom and Archimedes in [Veronese, 1889] and the independence in [Veronese, 1891]. Levi-Civita [1892–93] shows there is a non-Archimedean ordered field that is Cauchy complete. I thank Philip Ehrlich for the references and recommend Section 12 of the comprehensive [Ehrlich, 2006]. See also the insightful reviews [Pambuccian, 2014b] and [Pambuccian, 2014a], where it is observed that Vahlen [1907] also proved this axiom does not imply Archimedes. 51Since any point is in the definable closure of any line and any one point not on the line, one cannot extend any line without extending the model. Since adding either the Dedekind postulate and or Hilbert completeness gives a categorical theory satisfied by a geometry whose line is order-isomorphic to $$\Re$$, the two axioms are equivalent (over HP5 + Archimedes’ axiom). 52Hartshorne [2000, §§40–43] gives a modern account of Hilbert’s argument that replacing the parallel postulate by the axiom of limiting parallels gives a geometry that is determined by the underlying (definable) field. With Hilbert’s V.2 this gives a categorical axiomatization for hyperbolic geometry. 53Of course, this analysis is anachronistic; the clear distinction between first- and second-order logic did not exist in 1900. By $${\bf G}$$, we mean the natural interpretation in $$\Re^2$$ of the geometric predicates from Section 1. 54I am leaving out many details, $${\bf R}$$ is a sequence of relations giving the vocabulary of geometry and the sentence ‘says’ they are relations on $$Y$$; the coding of the satisfaction predicate is suppressed. 55See [Väänänen and Wang, 2015] and [Baldwin, 2018, Chs 2, 11.2]. 56In an ordered group, $$a$$ and $$b$$ are Archimedes-equivalent if there are natural numbers $$m,n$$ such that $$m |a| > |b|$$ and $$n |b| > |a|$$. 57For a thorough historical description, see [Hilbert, 2004, pp. 426–435]. We focus on the issues most relevant to this paper. 58Chapter 3, §11. 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