# Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert

Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert ABSTRACT We give a general account of the goals of axiomatization, introducing a variant (modest) on Detlefsen’s notion of ‘complete descriptive axiomatization’. We describe how distinctions between the Greek and modern view of number, magnitude, and proportion impact the interpretation of Hilbert’s axiomatization of geometry. We argue, as did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable (in their modern interpretation in terms of number) from Hilbert’s first-order axioms. We argue that Hilbert’s axioms including continuity show much more than the geometrical propositions of Euclid’s theorems and thus are an immodest complete descriptive axiomatization of that subject. Consider the following two propositions. Euclid VI.1: Triangles and parallelograms which are under the same height are to one another as their bases. (*) Hilbert1 gives the area of a triangle by the following formula. Hilbert: Consider a triangle $$ABC$$ having a right angle at $$A$$. The measure of the area of this triangle is expressed by the formula $$F(ABC) = \mbox{\frac{1}{2}} AB \cdot AC.{(**)}$$ At first glance these statements seem to carry the same message: the familiar fact about computing the area of triangle. But clearly they are not identical. Euclid tells us that the two-dimensional area of two triangles ‘under the same height’ is proportional to their 1-dimensional bases. Hilbert’s result is not a statement of proportionality; it tells us the 2-dimensional measure of a triangle is computed from a product of the 1-dimensional measures of its base and height. Hilbert’s rule looks like a rule of basic analytic geometry, but it is not. He derived it from an axiomatic geometry similar to Euclid’s, which in no way builds on Cartesian analytic geometry. Although the subject, often called Euclidean geometry, seems the same, clearly much has changed. This paper and its sequel [Baldwin, 2017] aim to contribute to our understanding of some important aspects of this change. One of these aspects is the different ways in which the geometric continuum, the line in the context of the plane, is perceived. Another is a different basis for the notion of proportion. The two papers deal with axiomatizations of the geometric line in logics of differing strengths. This analysis is integrated with a more general discussion of model theory and the philosophy of mathematical practice in [Baldwin, 2018]. Hilbert’s axiomatization is central to our considerations. One can see several challenges that Hilbert faced in formulating a new axiom set in the late nineteenth century: (1) Delineate the relations among the principles underlying Euclidean geometry. In particular, identify and fill ‘gaps’ or remove ‘extraneous hypotheses’ in Euclid’s reasoning. (2) Reformulate propositions such as VI.1 to reflect the nineteenth-century understanding of real numbers as measuring both length and area. (3) Ground the geometry of Descartes, late nineteenth-century analytic geometry, and mathematical analysis. The third aspect of the third challenge is not obviously explicit in Hilbert. We will argue Hilbert’s completeness axiom is unnecessary for the first two challenges and at least for the Cartesian aspect of the third. The gain is that it grounds mathematical analysis (provides a rigorous basis for calculus); that Hilbert desired this is more plausible than that he thoughtlessly assumed too much. For such a judgement we need some idea of the goals of axiomatization and when such goals are met or even exceeded. We frame this discussion in terms of the notion of descriptive axiomatization from [Detlefsen, 2014], which is discussed in the first section of this paper. The main modification to Detlefsen’s framework addresses the concern that the axioms might be too strong and obscure the ‘cause’ for a proposition to hold. We introduce the term ‘modest’ descriptive axiomatization to denote one which avoids this defect. That is, one which meets a certain clear aim, but does not overshoot by too much. We describe several ‘data sets’, explicit lists of propositions from Euclid, and draw from [Hartshorne, 2000] to link specified subsets of Hilbert’s axioms that justify them. Recall that Hilbert groups his axioms for geometry into five classes. The first four are first-order. Group V, Continuity, contains Archimedes’ axiom, which can be stated in the logic2$$L_{\omega_1,\omega}$$, and a second-order completeness axiom equivalent (over the other axioms) to Dedekind completeness3 of each line in the plane. The side-splitter theorem asserts that a line parallel to the base cuts the sides of a triangle proportionately. To prove it Euclid passes through a theory of area that is based on a theory of proportion that implicitly uses a form of Archimedes’ axiom. Bolzano called this path an atrocious detour. Hilbert met what we call Bolzano’s challenge by developing proportionality, similar triangles, and then area on the basis of his first-order axioms. We conclude that Hilbert’s first-order axioms provide a modest complete descriptive axiomatization for most of Euclid’s geometry. In the sequel we argue that the second-order axioms aim at results that are beyond (and even in some cases antithetical to) the Greek and even the Cartesian view of geometry. So Hilbert’s axioms are immodest as an axiomatization of traditional geometry. This conclusion is no surprise to Hilbert4 although it may be to many readers.5 In the preface to [Hilbert, 1962] the translator Townsend writes, ‘it is shown that the whole of the Euclidean geometry may be developed without the use of the axiom of continuity’. Hilbert concludes his introduction of the continuity axioms with: From a theoretical point of view, the value of this axiom is that it leads indirectly to the introduction of limiting points, and, hence, renders it possible to establish a one-to-one correspondence between the points of a segment and the system of real numbers. However, in what is to follow, no use will be made of the ‘axiom of completeness’. [Hilbert, 1962, p. 26] How should one compare such statements as (*) and (**)? We lay out the relations among three perspectives on a mathematical topic. After clarifying the notion of data set in the next section, the two papers focus on aligning the latter two perspectives for various data sets. (1) A data set [Detlefsen, 2014], a collection of propositions about the topic. (2) A system of axioms and theorems for the topic. (3) The different conceptions of various terms used in the area at various times. In Section 1, we consider several accounts of the purpose of axiomatization. We adjust Detlefsen’s definition of complete descriptive axiomatization to guarantee some ‘minimality’ of the axioms by fixing on a framework for discussing the various axiom systems: a modest descriptively complete axiomatization. One of our principal tools is Detlefsen’s notion of ‘data set’, a collection of sentences to be accounted for by an axiomatization. ‘The data set for area $$X$$’ is time-dependent; new sentences are added; old ones are reinterpreted. Section 2 lists data sets (collections of mathematical ‘facts’), then specific axiom systems, and asserts the correlation. In Section 3, we consider the changes in conception of the continuum, magnitude, and number. In particular, we analyze the impact of the distinction between ratios in the language of Euclid and segment multiplication in [Hilbert, 1962] or multiplication6 of ‘numbers’. With this background, we sketch in Section 4 Hilbert’s theory of proportions and area, focusing on Euclidean propositions that might appear to depend on continuity axioms. In particular, we outline Hilbert’s definition of the field in a plane and how this leads to a good theory of area and proportion, while avoiding the axiom of Archimedes. This paper expounds the consequences of Hilbert’s first-order axioms and argues they form a modest descriptive axiomatization of Euclidean geometry. The sequel extends the historical analysis from Euclid and Hilbert to Descartes and Tarski, and explores several variants on Dedekind’s axiom and the role of first-order, infinitary, and second-order logic. Further, it expounds a first-order basis for the formulas for area and circumference of a circle. 1. THE GOALS OF AXIOMATIZATION In this section, we place our analysis in the context of recent philosophical work on the purposes of axiomatization. We explicate Detlefsen’s notion of ‘data set’ and investigate the connection between axiom sets and data sets of sentences for an area of mathematics. Hilbert begins the Grundlagen [1971] with: The following investigation is a new attempt to choose for geometry a simple and complete set of independent axioms and to deduce from them the most important geometrical theorems in such a manner as to bring out as clearly as possible the significance of the groups of axioms and the scope of the conclusions to be derived from the individual axioms. Hallett [2008, p. 204], delineates the meaning of facts in this context, ‘simply what over time has come to be accepted, for example, from an accumulation of proofs or observations. Geometry, of course, is the central example $$\ldots$$’. Hallett [Hilbert, 2004, p. 434] presaged the emphasis here on ‘data sets’. Thus completeness appears to mean [for Hilbert] ‘deductive completeness with respect to the geometrical facts’. $$\ldots$$ In the case of Euclidean geometry there are various ways in which ‘the facts before us’ can be presented. If interpreted as ‘the facts presented in school geometry’ (or the initial stages of Euclid’s geometry), then arguably the system of the original Festschrift [i.e., the 1899 French version] is adequate. If, however, the facts are those given by geometrical intuition, then matters are less clear. Hilbert described the general axiomatization project in 1918. When we assemble the facts of a definite, more or less comprehensive field of knowledge, we soon notice these facts are capable of being ordered. This ordering always comes about with the help of a certain framework of concepts [Fachwerk von Begriffen] in the following way: a concept of this framework corresponds to each individual object of the field of knowledge, a logical relation between concepts corresponds to every fact within the field of knowledge. The framework of concepts is nothing other than the theory of the field of knowledge. [Hilbert, 2005, p. 1107] Detlefson [2014] describes such a project as a descriptive axiomatization. He motivates the notion with this remark by Huntington: [A] miscellaneous collection of facts $$\ldots$$ does not constitute a science. In order to reduce it to a science the first step is to do what Euclid did in geometry, namely, to select a small number of the given facts as axioms and then to show that all other facts can be deduced from these axioms by the methods of formal logic. [Huntington, 1911, emphasis original] Detlefsen uses the term data set (i.e., facts7) to describe a local descriptive axiomatization as an attempt to organize a data set of commonly accepted sentences deductively. The axioms are descriptively complete if all elements of the data set are deducible from them. This raises two questions. What is a sentence? Who commonly accepts? From the standpoint of modern logic, a natural answer to the first question would be to specify a logic and a vocabulary and consider all sentences in that language. Detlefsen argues [2014, pp. 5–7] that this is the wrong answer. He thinks Gödel errs in seeing the problem as completeness in the now standard sense of a first-order theory.8 Rather, Detlefsen presents us with an empirical question. We (at some point in time) look at the received mathematical knowledge in some area and want to construct a set of axioms from which it can all be deduced. In our case we want to compare the commonly accepted sentences from 300 BC with a twentieth-century axiomatization. As we see below, new interpretations for terms arise. Nevertheless, a specific data set delineates a certain area of inquiry. The data set is inherently flexible; conjectures are proven (or refuted) from time to time. The analysis in this paper tries to distinguish propositions that are simply later deductions about the same intuitions and those which invoke radically different assumptions. By analyzing the interpretations in particular cases, we can specify a data set. Comparing geometry at various times opens a deep question worthy of more serious exploration than there is space for here. In what sense do (*) and (**) opening this paper express the same thought, concept etc.? Rather than address the issue of what is expressed, we will simply show how to interpret (*) (and other propositions of Euclid) as propositions in Hilbert’s system. Working in a framework of formal logic, we return to our question, ‘What is a sentence?’ The first four groups of Hilbert’s axioms are sentences of first-order logic: quantification is over individuals and only finite conjunctions are allowed. As noted in footnote 2, Archimedes’ axiom can be formulated in $$L_{\omega_1,\omega}$$. But the Dedekind postulate in any of its variants is a sentence of a kind of second-order logic.9 All three logics have deductive systems, and the first- and second-order systems allow only finite proofs; so the set of provable sentences is recursively enumerable. Second-order logic (in the standard semantics) fails the completeness theorem but, by the Gödel and Karp [1959] completeness theorems, every valid sentence of $$L_{\omega,\omega}$$ or $$L_{\omega_1,\omega}$$ is provable. Adopting this syntactic view, there is a striking contrast between the data set of geometry in earlier generations and the axiom systems advanced near the turn of the twentieth century. Except for the Archimedean axiom, the earlier data sets are expressed in first-order logic. Geometry is an example of what Detlefsen calls a local as opposed to a foundational (global) descriptive axiomatization. Beyond the obvious difference in scope, Detlefsen points out several other distinctions. In particular, Detlefsen [2014, p. 5] asserts that axioms of a local axiomatization are generally among the given facts while those of a foundational axiomatization are found by (paraphrasing Detlefsen) tracing each truth in a data set back to the deepest level to which it can be properly traced. Hilbert’s geometric axioms have a hybrid flavor. Through the analysis of the concepts involved, Dedekind arrived at a second-order axiom.10 that is not in the data set but formed the capstone. An aspect of choosing axioms seems to be missing from the account so far. Hilbert [2005] provides the following insight into how axioms are chosen: If we consider a particular theory more closely, we always see that a few11 distinguished propositions of the field of knowledge underlie the construction of the framework of concepts, and these propositions then suffice by themselves for the construction, in accordance with logical principles, of the entire framework. $$\ldots$$ These underlying propositions may from an initial point of view be regarded as the axioms of the respective field of knowledge $$\ldots$$ By a modest axiomatization of a given data set,12 we mean one that implies all the data and not too much more.13 Of course, ‘not too much more’ is a rather imprecise term. One cannot expect a list of known mathematical propositions to be deductively complete. By ‘too much more’, we mean the axioms introduce essentially new concepts and concerns or add additional hypotheses proving a result that contradicts the explicit understandings of the authors of the data set (Section 3 in the sequel). In this paper, we are investigating modern axiomatizations for an ancient data set. As we shall see below, using Notation 2.2, Hilbert’s first-order axioms (HP5) are a modest axiomatization of the data (Euclid I): the theorems in Euclid about polygons (not circles) in the plane. We give an example later showing that HP5 + CCP (circle-circle intersection postulate, defined in Section 2), while modest for Euclid II, is an immodest first-order axiomatization of polygonal geometry. In the twentieth century the process of formalization is usually attentive to modesty; so it is a bit hard to find non-artificial examples. However, to study complex exponentiation, Zilber [2005] defined a quasi-minimal excellent class. The models admitted a closure relation satisfying the axioms of a combinatorial geometry as well as certain homogeneity conditions. Although his axioms were informal they can be formalized14 in $$L_{\omega_1,\omega}(Q)$$. Quite unexpectedly, [Bays et al., 2014] showed ‘excellence’ was not needed for the main result. Thus, the original axiomatization was immodest. No single axiom is modest or immodest; the relation has two arguments: a set of sentences is a modest axiomatization of a given data set. If the axioms are contained in the data set the axiomatization is manifestly modest, and this is just a mathematical fact that can be clarified by formalization. But some other subset might later be taken as axioms that imply the whole set. This might just happen by a clever proof. But the cases studied here are more subtle. New interpretations of the basic concepts (of multiplication and number) developed over time so that the sentences attained essentially new meanings. As (**) illustrates, such is the case with Euclid’s VI.1. Distinct philosophical issues arise in checking modesty and immodesty. For modesty, an historical investigation, as in this paper, must interpret the new conception (**) into the original data set (*). For immodesty, the conceptual content of the new axioms must be compared with that of the data set. In our view, modesty and purity are distinct, though related, notions. We just saw that formalization is useful to check modesty. Hallett [2008], Arana and Mancosu [2012], Detlefsen and Arana [2011], and Baldwin [2013a] argue that purity asks about specific arguments for a proposition. [Baldwin, 2013a] emphasized that the same theorem can have both pure and impure proofs and [Baldwin, 2018] extends the analysis of modesty versus purity. In contrast, modesty concerns the existence of proofs and appropriateness of hypotheses. In the quotation above, Hilbert takes the axioms to come from the data set. But this raises a subtle issue about what composes the data set. For examples such as geometry and number theory, it was taken for granted that there was a unique model. Even Hilbert adds his completeness axiom to guarantee categoricity and to connect with the real numbers. So one could argue that the early twentieth-century axiomatizers took categoricity as part of the data.15 But such an intuition is inherently metatheoretic and so dissimilar to the other data. It is certainly not in the data set of the Greeks for whom ‘categoricity’ is meaningless. 2. SOME GEOMETRIC DATA SETS AND AXIOM SYSTEMS This section is intended to lay out several topics in plane geometry that represent distinct data sets in Detlefsen’s sense.16 In cases where certain axioms are explicit, they are included in the data set. Although we describe five sets here, only polygonal geometry and circle geometry are considered in this paper; the others are treated in the sequel. Each set includes its predecessors; the description is of the added propositions. Notation 2.1 (5 data sets of geometry) Euclid I, polygonal geometry: Book I (except I.1, I.22), Book II.1-II.13, Book III (except III.1 and III.17), Book VI. Euclid II, circle geometry: CCP (circle-circle intersection postulate, defined below), I.1, 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. Our division of the data sets is somewhat arbitrary and is made with the subsequent axiomatizations in mind. The importance of Euclid II appears already in Proposition I of Euclid, where Euclid makes the standard construction of an equilateral triangle on a given base.17 Why do the two circles intersect? While some18 regard the absence of an axiom guaranteeing such intersections as a gap in Euclid, Manders [2008, p. 66] asserts: ‘Already the simplest observation on what the texts do infer from diagrams and do not suffices to show the intersection of two circles is completely safe’.19 The circle-circle intersection axiom resolves those continuity issues involving intersections of circles and lines.20 As noted, this proposition is in the data set; so adding this one axiom in our axioms EG (below), which does not appear explicitly in either the Grundlagen or Euclid, does not detract from the modesty of our axioms. It is a first-order consequence of the Dedekind postulate which plays an essential role in Euclidean geometry. Hilbert is aware of that fact; he chooses to resolve the issue (implicitly) by his completeness axiom. Circle-Circle Intersection Postulate (CCP): If from distinct points $$A$$ and $$B$$, circles with radius $$AC$$ and $$BD$$ are drawn such that one circle contains points in the interior and in the exterior of the other, then they intersect in two points, on opposite sides of $$AB$$. We have placed Euclid XII.2 (area of a circle is proportional to the square of the diameter) with Archimedes rather than Euclid’s other theorems on circles. The crux is the different resources needed to prove VI.1 (area of a parallelogram) and XII.2. The first is provable in EG; the sequel contains a first-order extension of EG in which XII.2 is provable. Note also that we consider only a fraction of Archimedes, his work on the circle. We analyze the connections among Archimedes, Descartes, and Tarski in the sequel. Showing a particular set of axioms is descriptively complete is inherently empirical. One must check whether each of a certain set of results is derivable from a given set of axioms. Hartshorne [2000] carried out this project independently from Detelfsen’s analysis; we organize his results at the end of this section. We identify two levels of formalization in mathematics. By the Euclid-Hilbert style we mean the axiomatic approach of Euclid along with the Hilbert insight that postulates are implicit definitions of classes of models.21 By the Hilbert-Gödel-Tarski style, we mean that that syntax and semantics have been identified as mathematical objects. Gödel’s completeness theorem is a standard tool; so methods of modern model theory can be applied.22 We will give our arguments in English, but we will be careful to specify the vocabulary and the postulates in such a way that the translation to a first-order theory is transparent. We will frequently switch from syntactic to semantic discussions; so we stipulate precisely the vocabulary in which we take the axioms above to be formalized. We freely use defined terms such as collinear, segment, and angle in giving the reading of the relevant relation symbols. The fundamental relations of plane geometry make up the following vocabulary $$\tau$$. (1) Two-sorted universe: points ($$P$$) and lines ($$L$$). (2) Binary relation $$I(A,\ell)$$: Read: a point is incident on a line; (3) Ternary relation $$B(A,C,D)$$: Read: $$C$$ is between $$A$$ and $$D$$ (and $$A,C,D$$ are collinear). (4) Quaternary relation, $$C(A,B,A',B')$$: Read: two segments are congruent, in symbols $$\overline{AB}\cong \overline {A'B'}$$. (5) 6-ary relation $$C'(A,B,D,A',B',D')$$: Read: the two angles $$\angle ABD$$ and $$\angle A'B'D'$$ are congruent, in symbols $$\angle{ ABD}\cong \angle {A'B'D'}$$. Notation 2.2 Consider the following axiom sets.23 (1) First-order axioms: HP, HP5: We write HP for Hilbert’s incidence, betweenness,24 and congruence axioms. We write HP5 for HP plus the parallel postulate. EG: The axioms for Euclidean geometry, denoted EG,25 consist of HP5 and in addition the circle-circle intersection postulate CCP. $$\boldsymbol{{\mathcal E}^2}:$$ Tarski’s axiom system for a plane over a real closed field (RCF26). $$\mathbf{EG}_{\boldsymbol{\pi}}$$ and $$\boldsymbol{{\mathcal E}^2_\pi}$$: Two new systems, which extend EG and $${\mathcal E}^2$$, will be described and analyzed in the sequel. (2) Hilbert’s continuity axioms, infinitary and second-order, will also be examined in detail in the sequel. Archimedes: The sentence in the logic $$L_{\omega_1,\omega}$$ expressing the Archimedean axiom. Dedekind: Dedekind’s second-order axiom27 that there is a point in each irrational cut in the line. With these definitions we align various subsystems of Hilbert’s geometry with certain collections of propositions in Euclidean geometry as spelled out in Hartshorne.28 With our grouping, Hartshorne shows the following results. They imply that the statements in HP5 or EG are either theorems or implicit (CCP) in Euclid; they are a modest axiomatization of Euclid I and II. First-order Axiomatizations (1) The sentences of Euclid I are provable in HP5. (2) The additional sentences of Euclid II are provable in EG. In this framework we discuss the changing conceptions of the continuum, ratio, and number from the Greeks to modern times and sketch some highlights of the proof of this result to demonstrate the philosophical modesty of the axiomatization. The sequel contains modest descriptive axiom systems29 for the data sets of Archimedes and Descartes and argues that the full Hilbert axiom set is immodest for any of these data sets. 3. CHANGING CONCEPTIONS OF THE CONTINUUM, MAGNITUDE, AND NUMBER In Section 3.1, we distinguish the geometric continuum from the set-theoretic continuum. In Section 3.2 we sketch the background shift from the study of various types of magnitudes by the Greeks to the modern notion of a collection of real numbers, which can measure any sort of magnitude Section 3.3 concerns Bolzano’s challenge. In Section 3.4 we contrast the current goal of an independent basis for geometry with the nineteenth-century arithmetization project. 3.1. Conceptions of the Continuum In this section, we motivate our restriction to the geometric continuum; we defined it as a linearly ordered structure that is situated in a plane. Sylvester30 describes the three divisions of mathematics: There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all of them combined, every mathematical truth admits of being referred; these are the three cardinal notions, of Number, Space and Order. This is a slightly unfamiliar trio. We are all accustomed to the opposition between arithmetic and geometry. While Newton famously founded the calculus on geometry [Detlefsen and Arana, 2011] the ‘arithmetization of analysis’ in the late nineteenth century reversed the priority. From the natural numbers the rational numbers are built by taking quotients and the reals by some notion of completion. And this remains the normal approach today. We want here to consider reversing the direction again: building a firm grounding for geometry and then finding first the field and then some completion and considering incidentially the role of the natural numbers. In this process, Sylvester’s third cardinal notion, order, will play a crucial role. The notion that one point lies between two others will be fundamental and an order relation will naturally follow; the properties of space will generate an ordered field and the elements of that field will be numbers albeit not numbers in the Greek conception. There are different conceptions of the continuum (the line); hence different axiomatizations may be necessary to reflect these different conceptions. These conceptions are witnessed by such collections as [Ehrlich, 1994; Salanskis and Sinaceur, 1992] and further publications concerned with the constructive continuum and various non-Archimedean notions of the continuum. In his [2008], Feferman lists six31 different conceptions of the continuum: (i) the Euclidean continuum, (ii) Cantor’s continuum, (iii) Dedekind’s continuum, (iv) the Hilbertian continuum, (v) the set of all paths in the full binary tree, and (vi) the set of all subsets of the natural numbers. For our purposes, we will identify (ii), (v), and (vi) as essentially cardinality-based as they have lost the order type imposed by the geometry; so they are not in our purview. We want to contrast two essentially geometrically based notions of the continuum: those of Euclid and Hilbert/Dedekind. Hilbert’s continuum differs from Dedekind’s as it has the field structure derived from the geometric structure of the plane, while Dedekind’s field is determined by continuity from known field operations on the rationals. Nevertheless they are isomorphic as ordered fields. We stipulated that ‘geometric continuum’ means ‘the line situated in the plane’. One of the fundamental results of twentieth-century geometry is that any (projective32 for convenience) plane can be coordinatized by a ‘ternary field’. A ternary field is a structure with one ternary function $$f(x,y,z)$$ such that $$f$$ has the properties that $$f(x,y,z) = xy +z$$ would have if the right-hand side were interpreted in a field. In dealing with Euclidean geometry here, we assume the axioms of congruence and the parallel postulate; this implies that the ternary field is actually a field. But these geometric hypotheses are necessary. In [Baldwin, 1994], I constructed an $$\aleph_1$$-categorical projective plane where the ternary field is as wild as possible (in the precise sense of the Lenz-Barlotti classification in [Dembowski, 1977]: the ternary function cannot be decomposed into an addition and a multiplication). 3.2. Ratio, Magnitude, and Number In this section we give a short review of Greek attitudes toward magnitude and ratio. Fuller accounts of the transition to modern attitudes appear in such sources as [Mueller, 2006; Euclid, 1956; Stein, 1990; Menn, 2017]. We by no means follow the ‘geometric algebra’ interpretation decried in [Grattan-Guinness, 2009]. Rather, we attempt to contrast the Greek meanings of propositions with Hilbert’s understanding. When we rephrase a sentence in algebraic notation we try to make clear that this is a modern formulation and often does not express the intent of Euclid. Euclid develops arithmetic in Books VII–IX. What we think of as the ‘number one’, was ‘the unit’; a number (Definition VII.2) was a multitude of units. These are counting numbers. So from our standpoint (considering the unit as the number 1) Euclid’s numbers (in the arithmetic) can be thought of as the ‘natural numbers’. The numbers33 are a discretely ordered collection of objects. Following Mueller [2006, p. 21] we work from the interpretation of magnitudes in the Elements as ‘abstractions from geometric objects which leave out of account all properties of those objects except quantity: length for lines, area of plane figures, volume of solid figures, etc.’ Mueller emphasizes the distinction between the properties of proportions of magnitudes developed in Book V and those of number in Book VII. The most easily stated is implicit in Euclid’s proof of Theorem V.5; for every $$m$$, every magnitude can be divided in $$m$$ equal parts. This is of course, false for the (natural) numbers. There is a second use of ‘number’ in Euclid. It is possible to count unit magnitudes, to speak of, e.g., four copies of a unit magnitude. So (in modern language) Euclid speaks of multiples of magnitudes by positive integers. Magnitudes of the same type are also linearly ordered and between any two there is a third.34 Multiplication of line segments yields rectangles. Ratios are not objects; equality of ratios is a 4-ary relation between two pairs of homogenous magnitudes.35 Some key points from Euclid’s discussion of proportion in Book V are: (1) Definition V.4 of Euclid [1956]: Magnitudes are said to have a ratio to one another, which are capable, when multiplied, of exceeding one another. (2) Definition V.5, ‘sameness of two ratios’ (in modern terminology): The ratio of two magnitudes $$x$$ and $$y$$ are proportional to the ratio of two others $$z, w$$ if for each $$m,n$$, $$mx> ny$$ implies $$mz > nw$$ (and also replacing $$>$$ by $$=$$ or $$<$$). (3) Definition V.6: Let magnitudes which have the same ratio be called proportional. (4) Proposition V.9: ‘Same ratio’ is, in modern terminology, a transitive relation. Apparently Euclid took symmetry and reflexivity for granted and treats proportionality as an equivalence relation. 3.3. Bolzano’s Challenge In a discussion of the foundations of geometry Bolzano discusses the ‘dissimilar objects’ found in Euclid and finds Euclid’s approach fundamentally flawed. His general position is that one must analyze conceptually prior notions (line) before more complex notions (plane). See [Russ, 2004, p. 33] and [Rusnock, 2000, p. 53] for discussions of Bolzano’s claim that the study of the line, as ‘more fundamental’ must precede that of the plane. More specifically, he objects to Euclid’s proof in VI.2 that similar triangles have proportional sides: Firstly triangles, that are already accompanied by circles which intersect in certain points, then angles, adjacent and vertically opposite angles, then the equality of triangles, and only much later their similarity, which however, is derived by an atrocious detour [ungeheuern Umweg], from the consideration of parallel lines, and even of the area of triangles, etc.36 [Bolzano, 1810] What we call Bolzano’s challenge has two aspects: (a) the evil of using two-dimensional concepts to understand the line, and (b) the ‘atrocious detour’ to similarity (VI.2). We consider the plane essential to understanding the geometric continuum; so (a) is irrelevant to our project; Sections 4.1–4.3 sketch how Hilbert meets the challenge by avoiding the detour. The side-splitter theorem, VI.2. is taken to represent similarity. Euclid deduces VI.2 from VI.1 (see Figure 1). In his proof of VI.1 (*) Euclid applies Definition V.5 (above) to deduce that the areas of two triangles with the same height are proportional to their bases. But this assumes that any two lengths (or any two areas) have a ratio in the sense of Definition V.4. This is an implicit assertion of Archimedes’ axiom for both area and length.37 Fig. 1. View largeDownload slide Euclid’s proof of VI.1. Fig. 1. View largeDownload slide Euclid’s proof of VI.1. If, for example, $$BC$$, $$GB$$, and $$HG$$ are congruent segments then the area of $$ACH$$ is triple that of $$ABC$$. But without assuming $$BC$$ and $$BD$$ are commensurable, Euclid calls on Definition V.5 to assert that $$ABD: ABC :: BD : BC$$. In VI.2, he uses these results to show that similar triangles have proportional sides. From VI.2, Euclid constructs in VI.12 the fourth proportional to three lines but does not regard it as a definition of multiplication of segments. As we will see in more detail, Hilbert’s treatment of area and similarity has no such dependence on Archimedes’ axiom. By interpreting the field with segment arithmetic, he defines proportionality directly. In contrast, Descartes defines the multiplication of line segments to give another segment,38 but he still relies on Euclid’s theory of proportion to justify the multiplication. Hilbert’s innovation is use to segment multiplication to gain the notion of proportionality, which is defined in Section 4.2. 3.4. From Arithmetic to Geometry or From Geometry to Algebra? On the first page of Continuity and the Irrational Numbers, Dedekind wrote: Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful from the didactic standpoint $$\ldots$$ But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. [Dedekind, 1963] I do not contest Dedekind’s claim. I quote this passage to indicate that Dedekind’s motivation was to provide a basis for analysis, not geometry. But I will argue that the second-order Dedekind completeness axiom is not needed for the geometry of Euclid and indeed for the grounding of the algebraic numbers, although it is in Dedekind’s approach. Dedekind provides a theory of the continuum (the continuous) line by building up in stages from the structure that is fundamental to him: the natural numbers under successor. This development draws on second-order logic in several places. The well-ordering of the natural numbers is required to define addition and multiplication by recursion. Dedekind completeness is a second appeal to a second-order principle. Perhaps in response to Bolzano’s insistence, Dedekind constructs the line without recourse to two-dimensional objects and from arithmetic. Thus, he succeeds in the ‘arithmetization of analysis’. We proceed in the opposite direction for several reasons. Most important is that we are seeking to ground geometry, not analysis. Further, we adopt as a principle that the concept of line arises only in the perception of at least two-dimensional space. Dedekind’s continuum knows nothing of being straight or breadthless. Hilbert’s proof of the existence of the field is the essence of the geometric continuum. By virtue of its lying in a plane, the line acquires algebraic properties. Moreover, the distinction between the arithmetic and geometric intuitions of multiplication is fundamental. The basis of the first is iterated addition; the basis of the second is scaling or proportionality. The late nineteenth-century developments provide a formal reduction of the second to the first but the reduction is only formal; the ‘scaling’ intuition is lost. In this paper we view both intuitions as fundamental and develop the second (Section 4.1) with the understanding that development of the first through the Dedekind-Peano treatment of arithmetic is in the background. 4. AXIOMATIZING THE GEOMETRY OF POLYGONS AND CIRCLES Section 4.1 sketches the key step in making an independent foundation of geometry: Hilbert’s definition of a field in a geometry. Section 4.2 describes the transition from segments to points as the domain of that field and gives Hilbert’s definition of proportional. Section 4.3 distinguishes the role of the CCP and analyzes several problems that can be approached by limits but have uniform solutions in any ordered field; order completeness of the field is irrelevant. We return to Bolzano’s challenge and derive Theorem 4.3.4, the properties of similar triangles and, in Section 4.4, the area of polygons. 4.1. From Geometry to Segment Arithmetic to Numbers One of Hilbert’s key innovations is his segment arithmetic and his definition of the semi-field39 of segments with partial subtraction and multiplication. We assume the axiom system we called HP5 in Notation 2.2. The details can be found in, e.g., [Hilbert, 1971; Hartshorne, 2000; Baldwin, 2018; Giovannini, 2016]. Note that congruence ($$\cong$$) forms an equivalence relation on line segments. Fix a ray $$\ell$$ with one end point $$0$$ on $$\ell$$. For each equivalence class of segments, we consider the unique segment $$0A$$ on $$\ell$$ in that class as the representative of that class. We will often denote the segment $$0A$$ (ambiguously its congruence class) by $$a$$. We say a segment $$CD$$ (on any line) has length $$a$$ if $$CD \cong 0A$$. Following Hartshorne [2000], here is our official definition of segment multiplication.40 Fix a unit segment class, $$1$$. Consider two segment classes $$a$$ and $$b$$. To determine their product, define a right triangle with legs of length $$1$$ and $$a$$. Denote the angle between the hypoteneuse and the side of length $$1$$ by $$\alpha$$. Now construct another right triangle with base of length $$b$$ with the angle between the hypoteneuse and the side of length $$1$$ congruent to $$\alpha$$. The product $$ab$$ is defined to be the length of the vertical leg of the triangle. Fig. 2. View largeDownload slide Segment multiplication. Fig. 2. View largeDownload slide Segment multiplication. Note that we must appeal to the parallel postulate to guarantee the existence of the point $$F$$. It is clear from the definition that there are multiplicative inverses; use the triangle with base $$a$$ and height $$1$$. Hartshorne has a roughly three-page proof41 that shows multiplication is commutative, associative, distributes over addition, and respects the order. It uses only the cyclic quadrilateral theorem and connections between central and inscribed angles in a circle. To summarize the effect of the axiom sets, we introduce two definitions. (1) An ordered field $$F$$ is Pythagorean if it is closed under addition, subtraction, multiplication, division and for every $$a\,{\in}\, F$$, $$\surd (1+a^2) \,{\in}\, F$$. (2) An ordered field $$F$$ is Euclidean if it is closed under addition, subtraction, multiplication, division and for every positive $$a \,{\in}\, F$$, $$\surd a \,{\in}\, F$$. When the model is taken as geometry over the reals, it is easy42 to check that the multiplication defined on the positive reals by this procedure is exactly the usual multiplication on the positive reals because they agree on the positive rational numbers. As in Section 21 of [Hartshorne, 2000], we have: Theorem 4.1.1 (1) HP5 is bi-interpretable with the theory of ordered Pythagorean fields. (2) Similarly EG is bi-interpretable with the theory of ordered Euclidean fields. Formally bi-interpretability means there are formulas in the field language defining the geometric notions (point, line, congruence, etc) and formulas in the geometric language (plus constants) defining the field operations ($$0,1, +,\times$$) such that interpreting the geometric formulas in a Pythagorean field gives a model of HP5 and conversely. See [Hodges, 1993, Ch. 5] for general background on interpretability. With this information we can explain why Proposition I.1 (equilateral triangle) is in Euclid I rather than II. Both it and I.22 (construct a triangle given three lines with the sum of the length of two greater than the length of the third) use circle-circle intersection in Euclid. However, Hilbert proves the first in HP5, but the second requires the field to be Euclidean and so uses CCP. Dicta on Constants: Note that to fix the field we had to add constants $$0$$, $$1$$. These constants can name any pair of points in the plane.43 But this naming induces an extension of the data set. We have in fact specified the unit. This specification has little effect on the data set but a major change in view from either the Greeks or Descartes. 4.2. Points and Numbers, Multiplication and Proportionality Hilbert shows the multiplication on segments of a line through points $$0,1$$ satisfies the semi-field axioms. He defines segment multiplication on the ray from $$0$$ through $$1$$ as in Section 4.1. But to get negative numbers one must reflect through $$0$$. Then addition and multiplication can be defined on directed segments of the line44 through $$0,1$$ and thus all axioms for a field are obtained. The next step is to identity the points on the line and the domain of an ordered field by mapping $$A$$ to $$0A$$. This naturally leads to thinking of a segment as a set of points, which is foreign to both Euclid and Descartes. Although in the context of the Grundlagen, Hilbert’s goal is to coordinatize the plane by the real numbers, his methods open the path to thinking of the members of any field as ‘numbers’ that coordinatize the associated geometries. Boyer traces the origins of numerical coordinates to 1827–1829 and writes It is sometimes said that Descartes arithmetized geometry but this is not strictly correct. For almost two hundred years after his time coordinates were essentially geometric. Cartesian coordinates were line segments $$\ldots$$ The arithmetization of coordinates took place not in 1637 but in the crucial years 1827–1829. [Boyer, 1956, p. 242] Boyer points to Bobillier, Möbius, Feurbach, and most critically Plücker as introducing several variants of what constitute numerical (signed distance) barycentric coordinates of a point. Multiplication is not repeated addition: We now have two ways in which we can think of the product $$3a$$. On the one hand, we can think of laying 3 segments of length $$a$$ end to end. On the other, we can perform the segment multiplication of a segment of length $$3$$ (i.e., $$3$$ segments of length $$1$$ laid end to end) by the segment of length $$a$$. It is an easy exercise to show these give the same answer. But these distinct constructions make an important point. The (inductive) definition of multiplication by a natural number is indeed ‘multiplication as repeated addition’. But the multiplication by another field element is based on similarity and has multiplicative inverses; so from a modern standpoint they yield very different structures. No extension of natural number arithmetic is decidable but important theories of fields are. The first notion of multiplication in the last paragraph, where the multiplier is a natural number, is a kind of scalar multiplication by positive integers that can be viewed mathematically as a rarely studied object: a semiring (the natural numbers) acting on a semigroup (positive reals under addition). There is no uniform definition45 of this binary operation of scalar multiplication within the semiring action. A mathematical structure more familiar to modern eyes is obtained by adding the negative numbers to get the ring $${\mathbb Z}$$, which has a well-defined notion of subtraction. The scalars are now in the ring $$({\mathbb Z},+,\cdot)$$ and act on the module $$(\Re, +)$$. Now we can multiply by $$-17$$ but the operation is still not uniform but given by a family of unary functions. Looking more closely, we have three different referents for segment multiplication: Euclid’s computation of the area of rectangle, his construction of the fourth proportional, and Hilbert’s definition. These three computations give the ‘same answer’. We now consider the fourth proportional in $$1 : a :: b : x$$ to measure the area of the rectangle with sides of length $$a$$ and $$b$$. In view of this we do not view Hilbert as introducing a new concept of multiplication — but as reinterpreting the notion and indeed the same geometric construction as applying to line segments, which we now read as numbers, rather than mapping from linear magnitudes to planar magnitudes. With this in hand, Hilbert redefines proportionality: Proportionality: We write the ratio of CD to CA is proportional to that of CE to CB, $$CD:CA :: CE: CB$$ which is defined as $$CD \times CB = CE \times CA,$$ where $$\times$$ is taken in the sense of segment multiplication as defined in Section 4.1. While in Book V Euclid provides a general account of proportionality, Hilbert’s ability to avoid the Archimedean axiom depends both on the geometrical construction of the field and the reinterpretation of ‘number’. 4.3. Field Arithmetic and Basic Geometry In this section we investigate some statements from: (1) Euclid’s geometry that depended in his development on the Archimedean axiom and some from (2) Dedekind’s development of the properties of real numbers that he deduces from his postulate. In each case, they are true in any field associated with a geometry modeling HP5. We established in Section 4.1 that one could define an ordered field $$F$$ in any plane satisfying HP5. The converse is routine: the ordinary notions of line and incidence in $$F^2$$ creates a geometry over any Pythagorean ordered field, which is easily seen to satisfy HP5. We now exploit this equivalence to show some important algebraic facts using our defined operations, thus basing them on geometry. First, note that taking square root commutes with multiplication for algebraic numbers. Dedekind [1963, p. 22] wrote ‘$$\dots$$ in this way we arrive at real proofs of theorems (as, e.g., $$\surd 2 \cdot \surd 3 = \surd 6$$), which to the best of my knowledge have never been established before’. Note that this commutativity is a problem for Dedekind but not for Descartes. Euclid had already, in constructing the fourth proportional, constructed from segments of length $$1$$, $$a$$ and $$b$$, one of length $$ab$$; but he does not regard this operation as multiplication. When Descartes interprets this procedure as multiplication of segments, the reasoning above shows multiplying square roots is not an issue. But Dedekind has presented the problem as multiplication in his continuum and so he must prove a theorem which allows us to find the product as a real number; that is, he must show the limit operation commutes with product. But, in an ordered field, for any positive $$a$$, if there is an element $$b>0$$ with $$b^2 =a$$, then $$b$$ is unique (and denoted $$\surd a$$). Moreover, for any positive $$a,c$$ with square roots, $$\surd a \cdot \surd c = \surd ac$$, since each side of the equality squares to $$ac$$. This fact holds for any field coordinatizing a plane satisfying HP5. Thus, the algebra of square roots in the real field is established without any appeal to limits. The usual (e.g., [Spivak, 1980; Apostol, 1967]) developments of the theory of complete ordered fields follow Dedekind and invoke the least-upper-bound principle to obtain the existence of the roots although the multiplication rule is obtained by the same algebraic argument as here. Hilbert’s approach contrasts with Dedekind’s.46 The justification here for either the existence of, or operations on, roots does not invoke limits. Hilbert’s treatment is based on the geometric concepts and in particular regards ‘congruence’ and ‘number’ as equally fundamental notions. In short, the shift here is from ‘proportional segments’ to ‘product of numbers’. Euclid had a rigorous proof of the existence of a line segment which is the fourth proportional of $$1:a = b:x$$. Dedekind demands a product of numbers; Hilbert provides this by a combination of his interpretation of the field in the geometry and the geometrical definition of multiplication. Euclid’s proof of Pythagoras’s theorem I.47 uses the properties of area that we will justify in Section 4.4. His second proof (Lemma for X.33) uses the property of similar triangles that we prove in Theorem 4.3.4. In both cases Euclid depends on the theory of proportionality (and thus implicitly on Archimedes’ axiom) to prove the Pythagorean theorem; Hilbert avoids this. appeal.47 Similarly, since the right-angle trigonometry in Euclid concerns the ratios of sides of triangles, the field multiplication justifies basic right-angle trigonometry. We have: Theorem 4.3.1The Pythagorean theorem as well as the law of cosines (Euclid II.12, II.13) and the law of sines [Maor, 1993, pp. 217–217] hold in HP5. Hartshorne [2000] describes two instructive examples, connecting the notions of Pythagorean and Euclidean planes. Example 4.3.2 (1) The Cartesian plane over a Pythagorean field may fail to be closed under square root.48 (2) Hartshorne49 observes that the smallest ordered field closed under addition, subtraction, multiplication, division, and square roots of positive numbers and satisfying the CCP is a Euclidean field, denoted by $$F_s$$ for surd field. Note that if HP5 + CCP were proposed as an axiom set for polygonal geometry it would be a complete descriptive but not modest axiomatization since it would prove CCP which is not in the polygonal geometry data set. In a Euclidean plane every positive element of the coordinatizing field has a square root; so Heron’s formula ($$A =\sqrt{s(s-a)(s-b)(s-c)}$$, where $$s$$ is half the perimeter and $$a,b,c$$ are the side lengths) computes the area of a triangle from the lengths of its sides. This fact demonstrates the hazards of the kind of organization of data sets attempted here. The geometric proof of Heron does not involve the square roots of the modern formula50 [Heath, 1921, p. 321]. But since in EG we have the field and we have square roots, the modern form of Heron’s formula can be proved from EG. Thus, as in the shift from (*) to (**) at the beginning of the paper, the different means of expressing the geometrical property requires different proofs. In each case considered in this section, Greeks give geometric constructions for what in modern days becomes a calculation involving the field operations and square roots. However, we still need to complete the argument that HP5 is descriptively complete for polygonal Euclidean geometry. In particular, is our notion of proportional correct? The test question is the similar triangle theorem. We turn to this issue now. Definition 4.3.3Two triangles $$\triangle ABC$$ and $$\triangle A'B'C'$$ are similar if under some correspondence of angles, corresponding angles are congruent; i.e., $$\angle A' \cong\angle A$$, $$\angle B' \cong\angle B$$, $$\angle C' \cong\angle C$$. Various texts define ‘similar’ as we did, or focus on corresponding sides being proportional, or require both (Euclid). We now meet Bolzano’s challenge by showing that in Euclidean Geometry (without the continuity axioms) the choice does not matter. We defined ‘proportional’ in terms of segment multiplication in Section 4.2. Hartshorne [2000, pp. 176–177] proves the fundamental result (Euclid VI.2). Theorem 4.3.4 (EG). Two triangles are similar if and only if corresponding sides are proportional. Euclid VI.2 follows. There is no assumption that the field is Archimedean or satisfies any sort of completeness axiom. There is thus no appeal to approximation or limits. We have avoided Bolzano’s ‘atrocious detour’ through area. But area is itself a vital geometric notion, and that is the topic of the next section. 4.4. Area of Polygonal Figures Hilbert [1971, p. 57] wrote ‘We $$\ldots$$ establish Euclid’s theory of area for the plane geometry and that independently of the axiom of Archimedes’ (emphasis original). We now sketch Hartshorne’s [2000] exposition of this topic, stressing the connections with Euclid’s Common Notions. We show that the notions defined here are expressible in first-order logic, which supports our fourth objection in Section 4.3 of the sequel: although these arguments are not carried out as direct deductions from the first-order axioms, the results are derivable by a direct deduction. That is, we develop area in first-order logic and even though some of the arguments are semantical the conclusions are theorems of first-order logic. This is further evidence of the immodesty of the second-order axiomatization. Informally, those configurations whose areas are considered in this section are figures, where a figure is a subset of the plane that can be represented as a finite union of disjoint triangles. There are serious issues concerning the formalization in first-order logic of such notions as figure or polygon that involve quantification over integers; such quantification is strictly forbidden within a first-order system. We can approach these notions with axiom schemes51 and sketch a uniform metatheoretic definition of the relevant concepts to prove that the theorems hold in all models of the axioms. Hilbert raised a pseudogap in Euclid52 by distinguishing area and content. In Hilbert, two figures have (1) equal area (are equidecomposable) if they can each be decomposed into a finite number of triangles that are pairwise congruent; (2) equal content (are equicomplementable) if we can transform one into the other by adding and subtracting congruent triangles. Hilbert showed that under the Archimedean axiom the two notions are equivalent; and that without it they are not. Euclid treats the equality of areas as a special case of his Common Notions. The properties of equal content, described next, are consequences for Euclid of the Common Notions and need no justification. We introduce the notion of area function to show they hold in all models of HP5. Fact 4.4.1 (Properties of Equal Content) The following properties of area are used in Euclid I.35 through I.38 and beyond. (1) Congruent figures have the same content. (2) The content of two ‘disjoint’ figures (i.e., meet only in a point or along an edge) is the sum of the contents of the two figures. The analogous statements hold for difference and half. (3) If one figure is properly contained in another then the area of the difference (which is also a figure) is positive. More precisely, Definition 4.4.2 (Equal content) Two figures $$P,Q$$ have equal content (are equi-complementable) in $$n$$ steps if there are figures $$P'_1 \ldots P'_n$$, $$Q'_1 \ldots Q'_n$$ such that none of the figures overlap, each pair $$P'_i$$ and $$Q'_i$$ are equi-decomposable and $$P=P'_1\cup P'_2 \ldots \cup P'_n$$ is equi-decomposable with $$Q=Q'_1\cup Q'_2 \ldots \cup Q'_n$$. Reading equal content for Euclid’s ‘equal’, Euclid’s I.35 (for parallelograms) and the derived I.38 (triangles) become Theorem 4.4.3 and in this formulation Hilbert accepts Euclid’s proof. Theorem 4.4.3 (Euclid/Hilbert) If two parallelograms (triangles) are on the same base and between parallels they have equal content in 1 step. Fig. 3. View largeDownload slide Parallelograms between the same parallels. Fig. 3. View largeDownload slide Parallelograms between the same parallels. Proof. $$P =ADCB$$ has the same content as $$P'=EFCB$$ in one step as letting $$Q=Q'=DGE$$, $$P+Q$$ and $$P'+Q'$$ which are equidecomposable by $$CBG +ABE$$ and $$CBG +CDF$$ since $$ABE \cong DCF$$. □ To show area is well-defined for figures, varying Hilbert, Hartshorne [2000, §§19–23] shows Fact 4.4.1 in the first-order axiom system EG (Notation 2.2). The key tool is: Definition 4.4.4An area function is a map $$\alpha$$ from the set of figures, $$\mathcal {P}$$, into an ordered additive abelian group with $$0$$ such that (1) For any nontrivial triangle $$T$$, $$\alpha(T) > 0$$. (2) Congruent triangles have the same content. (3) If $$P$$ and $$Q$$ are disjoint figures $$\alpha(P \cup Q) = \alpha(P) +\alpha(Q)$$. Semiformally, the idea is straightforward. Argue (a) that every $$n$$-gon is triangulated into a collection of disjoint triangles, and (b) establish an area function on such collections. The complication for formalizing is describing the area function by first-order formulas. For this,53 add to the vocabulary for geometry a $$3$$-ary function $$\alpha_1$$ from points into line segments that satisfies (1) and (2). For $$n>1$$, add a $$3n$$-ary function $$\alpha_n$$ and assert that $$\alpha_n$$ maps $$n$$-disjoint triangles into the field of line segments by summing $$\alpha_1$$ on the individual triangles; this will satisfy (3). This handles disjoint unions of triangles. Hartshorne (§§23.2–23.5) gives an inductive proof showing every polygon can be decomposed into a finite number of disjoint triangles and the area does not depend on the decomposition. It is evident that if a plane admits an area function then the conclusions of Fact 4.4.1 hold. This obviates the need for positing separately De Zolt’s axiom, (Fact 4.4.1 (3)). In particular this implies Common Notion 4 for ‘area’. As we just saw, Hilbert established the existence of an area function and thus a theory of area for any plane satisfying HP5. Now, letting $$F(ABC)$$ be an area function as in Definition 4.4.4 (** from the second page), VI.1 is interpreted as a variant of (**): $$F(ABC) =\mbox{\frac{1}{2}}\alpha \cdot AB \cdot AC.$$ But the cost is that while Euclid does not specify what we now call the proportionality constant, Hilbert must. In (**) Hilbert assigns $$\alpha$$ to be one. A short argument [Hartshorne, 2000, §23.3] shows that the formula $$A = \frac{bh}{2}$$ does not depend on the choice of the base and height. Thus, Hilbert proves (**) without recourse to the axiom of Archimedes. 5. CONCLUSION In this paper we defined the notion of a modest descriptive axiomatization to emphasize that the primary goal of an axiomatization is to distill what is ‘really going on’. One can axiomatize the first-order theory of any structure by taking as axioms all the first-order sentences true in it; such a choice makes a farce of axiomatizing. Historically, we stress one of Hilbert’s key points. Hilbert eliminated the use of the axiom of Archimedes in Euclid’s polygonal and circle geometry (except for area of circle), thus exhibiting a first-order modest descriptive axiomatization of that geometry. His proof is impure [Baldwin, 2013a] as he introduces the concept of field; but it remains modest; the axioms are in the data set. As we elaborate in Chapter 12 of [Baldwin, 2018], extensions by explicit first-order definition are (often unconsciously) standard mathematical tools that may impair purity but not provability. He [Hilbert, 1971] was finding the ‘distinguished propositions of the field of knowledge that underlie the construction of the framework of concepts’ and showing what we now call first-order axioms sufficed. The sequel (1) expounds Tarski’s system $${\mathcal E}^2$$ as a modest descriptive axiomatization of Cartesian geometry, (2) analyzes the distinctions between the completeness axioms of Dedekind and Hilbert and argues that Hilbert’s continuity axioms are overkill for strictly geometric propositions, and (3) supports conclusion (2) by providing a first-order theory to justify the formulas for circumference and area of a circle. Footnotes † I very much appreciate helpful comments by Neil Barton, Rose Cherubin, Davide Crippa, Philip Ehrlich, Eduardo Giovannini, Michael Hallett, Robin Hartshorne, William Howard, Juliette Kennedy, Barry Mazur, John Mumma, Julian Narboux, Victor Pambuccian, Marco Panza, Craig Smorynski, and Jouko Väänänen. I especially thank the referees. Research was partially supported by Simons travel grant G5402. 1Hilbert does not state this result as a theorem; I have excerpted the statement from an application on page 66 of [Hilbert, 1962]. 2In the logic, $$L_{\omega_1,\omega}$$, quantification is still over individuals but now countable conjunctions are permitted; so it is easy to formulate Archimedes’ axiom: $$\forall x,y (\bigvee_{m\in \omega} mx>y)$$. By switching the roles of $$x$$ and $$y$$ we see each is reached by a finite multiple of the other. 3Dedekind defines the notion of a cut in a linearly ordered set $$I$$ (a partition of $${\mathbb{Q}}$$ into two intervals $$(L,U)$$ with all elements of $$L$$ less than all elements of $$U$$). He postulates that each cut has unique realization, a point above all members of $$L$$ and below all members $$U$$ — it may be in either $$L$$ or $$U$$ [Dedekind, 1963, p. 20]. If either the $$L$$ contains a least upper bound or the upper interval $$U$$ contains a greatest lower bound, the cut is called ‘rational’ and no new element is introduced. Each of the other (irrational) cuts introduces a new number. It is easy to see that the unique realization requirement implies the Archimedean axiom. By Dedekind completeness of a line, I mean the Dedekind postulate holds for the linear ordering of that line. See the sequel [Baldwin, 2017]. 4Hilbert lectured on geometry several summers in the 1890s and his notes (in German) with extremely helpful introductions (in English) appear in [Hilbert, 2004]. The first Festschrift version of the Grundlagen does not contain the continuity axioms. I draw primarily on the second (Townsend) edition of [Hilbert, 1962] and on the tenth [Hilbert, 1971]. 5The first ten urls from a google search for ‘Hilbert’s axioms for Euclidean geometry’ contained eight with no clear distinction between the geometries of Hilbert and Euclid and two links to Hartshorne, who distinguishes them. 6That is, a multiplication on points rather than segments. See [Heyting, 1963]; the most thorough treatment is in [Artin, 1957]. 7There is an interesting subtlety here (perhaps analogous to the Shapiro’s algebraic and non-algebraic theories). The data set for group theory could be interpreted as (a) formal consequences of the axioms for groups in the vocabulary $$(+,0)$$, or (b) theorems proved by group theorists. For Euclid-Hilbert geometry these meanings coalesce. 8We dispute some of his points just before Remark 2.1 in the sequel. 9See the caveats on ‘second-order’ (e.g., sortal) in the sequel. 10We discuss the ‘equivalence’ of Dedekind’s and Hilbert’s formulation of completeness in the sequel. 11Often, few is interpreted as finite. Whatever Hilbert meant, we should now be satisfied with a small finite number of axioms and axiom schemes. At the beginning of the Grundlagen, Hilbert adds ‘simple, independent, and complete’. Such a list including schemes is simple. 12We considered replacing ‘modest’ by ‘precise’ or ‘safe’ or ‘adequate’. We chose ‘modest’ rather than one of the other words to stress that we want a sufficient set and one that is as necessary as possible. As the examples show, ‘necessary’ is too strong. Later work finds consequences of the original data set undreamed by the earlier mathematicians. Thus just like ‘descriptively complete’, ‘modest’ is a description, not a formal definition. 13This concept describes normal work for a mathematician. ‘I have a proof; what are the actual hypotheses so I can convert it to a theorem.’ 14The $$Q$$ is the quantifier ‘there exist uncountably many’. 15Huntington [1911] invokes Dedekind’s postulate when axiomatizing the complex field. 16In 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. 17Proposition I.1 is omitted from Euclid I, because it is not possible to construct an equilateral triangle with side length 1 in HP5. Proposition I.2 (transferring a length) is an axiom for Hilbert and of course follows from CCP. 18E.g., Veblen [1914, p. 4]. 19Manders emphasized the use of diagrams as a coherent mathematical practice. Properties that are not changed by minor variations in the diagram such as subsegment, inclusion of one figure in another, the intersection of two lines, betweenness, are termed coexact. Properties that can be changed by minor variations in the diagram, such as whether a curve is a straight line, congruence, a point is on a line, are termed exact. We can rely on reading coexact properties from the diagram. The difficulty in turning this insight into a formal deductive system is that, depending on the particular diagram drawn, after a construction the diagram may have different coexact properties. Independently, Miller [2007] and Avigad et al. [2009] formulated deductive systems whose rules are intended to capture properties that formerly were read off from diagrams. Miller treats diagrams as topological objects; this leads to difficulties bounding the number of cases. The system in [Avigad et al., 2009] incorporates metric information and provides a complete and sound deductive system for a collection of $$\forall \exists$$ sentences on models of EG; these include Euclid I–IV, but do not appear to include the theorems relating area and proportionality that are central here. 20Circle-circle intersection implies line-circle intersection. Hilbert shows that circle-circle intersection holds in a Euclidean plane, as pointed out in [Hilbert, 2004, pp. 204–206]. See Section 4.3. 21The priority for this insight is assigned to such slightly earlier authors as Pasch, Peano, Fano, in works such as [Freudenthal, 1957] as commented on in [Bos, 1993] and [Gray, 2011, ch. 24]. 22See [Baldwin, 2014] and the sequel for further explication of this method. 23The names HP, HP5, and EG come from [Hartshorne, 2000] and $${\mathcal E}^2$$ from [Tarski, 1959]. In fact, Tarski also studies EG under the name $${\mathcal E}''_2$$. 24These 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. 25In the vocabulary here, there is a natural translation of ‘Euclid’s axioms’ into first-order statements. The construction axioms have to be viewed as ‘for all – there exist’ sentences. The axiom of Archimedes is of course not first-order. We write Euclid’s axioms for those in the original as opposed to modernized (first-order) axioms for Euclidean geometry, EG. Note that EG is equivalent to (i.e., has the same models) as the system laid out in [Avigad et al., 2009]. 26A field is real closed if it is formally real ($$-1$$ is not a sum of squares) and every odd degree polynomial has a solution. 27Hilbert added his Vollstandigkeitsaxiom to the French translation of the first edition and it appears from then on. In Section 4.2 of the sequel we explore the connections between various formulations of completeness. We take Dedekind’s formulation (Footnote 3) as emblematic. 28See Theorems 10.4, 12.3–12.5 in Section 12 and Sections 20–23 of [Hartshorne, 2000]. 29In [Baldwin and Mueller, 2012] and [Baldwin, 2013b] we give an equivalent set of postulates to EG, which returns to Euclid’s construction postulates and stresses the role of Euclid’s axioms (Common Notions) in interpreting the geometric postulates. While not spelled out rigorously, our aim is to consider the diagram as part of the argument. For pedagogical reasons the system used SSS rather than SAS as the basic congruence postulate, as it more easily justifies the common-core approach to similarity through dilations and makes clear that the equality axioms in logic, as in Euclid’s Common Notions, apply to both algebra and arithmetic. This eliminates the silly 6-step arguments in high-school texts reducing subtraction of segments to the axioms of the real numbers. 30As quoted in [Mathias, 1992]. 31Smorynski [2008] notes that Bradwardine already reported five in the fourteenth century. Reeder [2017] analyzes five different philosophical approaches to the notion of ‘infinite divisibility’. 32That is, any system of points and lines such that two points determine a line, any two lines intersect in a point, and there are 4 non-collinear points. 33More precisely, for the Greeks, natural numbers greater than 1. 34The Greeks accepted only potential infinity. From a modern perspective, the natural numbers are ordered in order type $$\omega$$, and any collection of homogeneous magnitudes (e.g., areas) are in a dense linear order (which is necessarily infinite); however, this imposition of completed infinity is not the understanding of the Greeks. 35Homogeneous pairs means magnitudes of the same type. Ratios of numbers are described in Book VII while ratios of magnitudes are discussed in Book V. 36This quotation is taken from [Franks, 2014]. 37Euclid’s development of the theory of proportion and area requires the Archimedean axioms. Our assertion is one of many descriptions of the exact form and location of the dependence among such authors as [Euclid, 1956; Mueller, 2006; Stein, 1990; Fowler, 1979; Smorynski, 2008]. Since our use of Euclid is as a source of sentences, not proofs, the details are not essential to our argument. 38He refers to the construction of the fourth proportional (‘ce qui est même que la multiplication’ [Descartes, 1954]. See also [Bos, 2001, §21, p. 296]. 39In a semi-field there is no requirement of an additive inverse. 40Hilbert’s definition goes directly via similar triangles. The clear association of a particular angle with right multiplication by $$a$$ recommends Hartshorne’s version. 41See [Baldwin and Mueller, 2012] or [Hartshorne, 2000, p. 170], which explain the background cyclic quadrilateral theorem. That theorem asserts: Let $$ACED$$ be a quadrilateral. The vertices of $$ACED$$ lie on a circle (the ordering of the name of the quadrilateral implies $$A$$ and $$E$$ are on opposite sides of $$CD$$) if and only if $$\angle EAC \cong \angle CDE$$. Hilbert uses Pascal’s theorem, a relative of Desargues’ theorem. 42One has to verify that segment multiplication is continuous but this follows from the density of the order since the addition respects order. 43The automorphism group of the plane acts 2-transitively on the plane (any pair of distinct points can be mapped by an automorphism to any other such pair); this can be proven in HP5. This transitivity implies that a sentence $$\phi(0,1)$$ holds just if either or both of $$\forall x \forall y \phi(x,y)$$ and $$\exists x \exists y \phi(x,y)$$ hold. 44Hilbert had done this in lecture notes in 1894 Hilbert [2004]. Hartshorne constructs the field algebraically from the semi-field rather than in the geometry. 45Instead, there are infinitely many formulas $$\phi_n(x,y)$$ defining unary operations $$nx = y$$ for each $$n> 0$$. 46Dedekind [1963, p. 9] objects to the introduction of irrational numbers by measuring an extensive magnitude in terms of another of the same kind. 47However, Hilbert does not avoid the parallel postulate since he uses it to establish multiplication and thus similarity. Note also that Euclid’s theory of area depends heavily on the parallel postulate. It is a theorem in ‘neutral geometry’ in the metric tradition that the Pythagorean theorem is equivalent to the parallel postulate (See Theorem 9.2.8 of [Millman and Parker, 1981].) But this approach basically assumes the issues dealt with in these papers; as, the ‘ruler postulate’ (Birkhoff’s Postulate III in the sequel) also provides a multiplication on the ‘lengths’ (since they are real numbers). Julien Narboux pointed out the issues in stating the Pythagorean theorem in the absence of the parallel postulate. 48See [Hartshorne, 2000, Exercises 39.30, 39.31]. This was known to Hilbert [Hilbert, 2004, pp. 201–202]. 49Hartshorne and Greenberg [2010, p. 146] call this the constructible field, but given the many meanings of constructible, we use Moise’s term: surd field. 50Heron’s proof invokes a peculiar squaring of areas. But Taisbak [2014] argues that this operation can be replaced by inferences from VI.1 or X.53. 51In order to justify the application of the completeness theorem we have to produce a scheme giving the definition of an $$n$$-decomposable figure as the disjoint union of an $$(n-1)$$-decomposable figure $$A$$ with an appropriately placed triangle. The axioms for $$\pi$$ in the sequel illustrate such a scheme. 52Any model with infinitessimals shows the notions are distinct and Euclid I.35 and I.36 (triangles on the same (congruent) base(s) and same height have the same area) fail for what Hilbert calls area. Since Euclid includes preservation under both addition and subtraction in his Common Notions, his term ‘area’ clearly refers to what Hilbert calls ‘equal content’, I call this a pseudogap. 53See Definition 3.2 of the sequel for a more detailed discussion in a slightly different context. REFERENCES Apostol T. [ 1967 ]: Calculus. New York : Blaisdell . Arana A. , and Mancosu P. [ 2012 ]: ‘On the relationship between plane and solid geometry’, Review of Symbolic Logic 5 , 294 – 353 . Google Scholar Crossref Search ADS Artin E. [ 1957 ]: Geometric Algebra. New York : Interscience . Avigad J. , Dean E. and Mumma J. [ 2009 ]: ‘A formal system for Euclid’s Elements’, Review of Symbolic Logic 2 , 700 – 768 . 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Amsterdam: North-Holland . Veblen O. [ 1914 ]: ‘The foundations of geometry’, in Young J. ed., Monographs of Modern Mathematics Relevant to the Elementary Field , pp. 1 – 51 . New York : Longmans, Green . Zilber B. [ 2005 ]: ‘A categoricity theorem for quasiminimal excellent classes’, in Blass Andreas and Zhang Yi eds, Logic and its Applications , pp. 297 – 306 . Contemporary Mathematics . Providence, R.I. : American Mathematical Society . © The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Philosophia Mathematica Oxford University Press

# Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert

, Volume 26 (3) – Oct 1, 2018
29 pages

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Oxford University Press
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0031-8019
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1744-6406
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10.1093/philmat/nkx030
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### Abstract

ABSTRACT We give a general account of the goals of axiomatization, introducing a variant (modest) on Detlefsen’s notion of ‘complete descriptive axiomatization’. We describe how distinctions between the Greek and modern view of number, magnitude, and proportion impact the interpretation of Hilbert’s axiomatization of geometry. We argue, as did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable (in their modern interpretation in terms of number) from Hilbert’s first-order axioms. We argue that Hilbert’s axioms including continuity show much more than the geometrical propositions of Euclid’s theorems and thus are an immodest complete descriptive axiomatization of that subject. Consider the following two propositions. Euclid VI.1: Triangles and parallelograms which are under the same height are to one another as their bases. (*) Hilbert1 gives the area of a triangle by the following formula. Hilbert: Consider a triangle $$ABC$$ having a right angle at $$A$$. The measure of the area of this triangle is expressed by the formula $$F(ABC) = \mbox{\frac{1}{2}} AB \cdot AC.{(**)}$$ At first glance these statements seem to carry the same message: the familiar fact about computing the area of triangle. But clearly they are not identical. Euclid tells us that the two-dimensional area of two triangles ‘under the same height’ is proportional to their 1-dimensional bases. Hilbert’s result is not a statement of proportionality; it tells us the 2-dimensional measure of a triangle is computed from a product of the 1-dimensional measures of its base and height. Hilbert’s rule looks like a rule of basic analytic geometry, but it is not. He derived it from an axiomatic geometry similar to Euclid’s, which in no way builds on Cartesian analytic geometry. Although the subject, often called Euclidean geometry, seems the same, clearly much has changed. This paper and its sequel [Baldwin, 2017] aim to contribute to our understanding of some important aspects of this change. One of these aspects is the different ways in which the geometric continuum, the line in the context of the plane, is perceived. Another is a different basis for the notion of proportion. The two papers deal with axiomatizations of the geometric line in logics of differing strengths. This analysis is integrated with a more general discussion of model theory and the philosophy of mathematical practice in [Baldwin, 2018]. Hilbert’s axiomatization is central to our considerations. One can see several challenges that Hilbert faced in formulating a new axiom set in the late nineteenth century: (1) Delineate the relations among the principles underlying Euclidean geometry. In particular, identify and fill ‘gaps’ or remove ‘extraneous hypotheses’ in Euclid’s reasoning. (2) Reformulate propositions such as VI.1 to reflect the nineteenth-century understanding of real numbers as measuring both length and area. (3) Ground the geometry of Descartes, late nineteenth-century analytic geometry, and mathematical analysis. The third aspect of the third challenge is not obviously explicit in Hilbert. We will argue Hilbert’s completeness axiom is unnecessary for the first two challenges and at least for the Cartesian aspect of the third. The gain is that it grounds mathematical analysis (provides a rigorous basis for calculus); that Hilbert desired this is more plausible than that he thoughtlessly assumed too much. For such a judgement we need some idea of the goals of axiomatization and when such goals are met or even exceeded. We frame this discussion in terms of the notion of descriptive axiomatization from [Detlefsen, 2014], which is discussed in the first section of this paper. The main modification to Detlefsen’s framework addresses the concern that the axioms might be too strong and obscure the ‘cause’ for a proposition to hold. We introduce the term ‘modest’ descriptive axiomatization to denote one which avoids this defect. That is, one which meets a certain clear aim, but does not overshoot by too much. We describe several ‘data sets’, explicit lists of propositions from Euclid, and draw from [Hartshorne, 2000] to link specified subsets of Hilbert’s axioms that justify them. Recall that Hilbert groups his axioms for geometry into five classes. The first four are first-order. Group V, Continuity, contains Archimedes’ axiom, which can be stated in the logic2$$L_{\omega_1,\omega}$$, and a second-order completeness axiom equivalent (over the other axioms) to Dedekind completeness3 of each line in the plane. The side-splitter theorem asserts that a line parallel to the base cuts the sides of a triangle proportionately. To prove it Euclid passes through a theory of area that is based on a theory of proportion that implicitly uses a form of Archimedes’ axiom. Bolzano called this path an atrocious detour. Hilbert met what we call Bolzano’s challenge by developing proportionality, similar triangles, and then area on the basis of his first-order axioms. We conclude that Hilbert’s first-order axioms provide a modest complete descriptive axiomatization for most of Euclid’s geometry. In the sequel we argue that the second-order axioms aim at results that are beyond (and even in some cases antithetical to) the Greek and even the Cartesian view of geometry. So Hilbert’s axioms are immodest as an axiomatization of traditional geometry. This conclusion is no surprise to Hilbert4 although it may be to many readers.5 In the preface to [Hilbert, 1962] the translator Townsend writes, ‘it is shown that the whole of the Euclidean geometry may be developed without the use of the axiom of continuity’. Hilbert concludes his introduction of the continuity axioms with: From a theoretical point of view, the value of this axiom is that it leads indirectly to the introduction of limiting points, and, hence, renders it possible to establish a one-to-one correspondence between the points of a segment and the system of real numbers. However, in what is to follow, no use will be made of the ‘axiom of completeness’. [Hilbert, 1962, p. 26] How should one compare such statements as (*) and (**)? We lay out the relations among three perspectives on a mathematical topic. After clarifying the notion of data set in the next section, the two papers focus on aligning the latter two perspectives for various data sets. (1) A data set [Detlefsen, 2014], a collection of propositions about the topic. (2) A system of axioms and theorems for the topic. (3) The different conceptions of various terms used in the area at various times. In Section 1, we consider several accounts of the purpose of axiomatization. We adjust Detlefsen’s definition of complete descriptive axiomatization to guarantee some ‘minimality’ of the axioms by fixing on a framework for discussing the various axiom systems: a modest descriptively complete axiomatization. One of our principal tools is Detlefsen’s notion of ‘data set’, a collection of sentences to be accounted for by an axiomatization. ‘The data set for area $$X$$’ is time-dependent; new sentences are added; old ones are reinterpreted. Section 2 lists data sets (collections of mathematical ‘facts’), then specific axiom systems, and asserts the correlation. In Section 3, we consider the changes in conception of the continuum, magnitude, and number. In particular, we analyze the impact of the distinction between ratios in the language of Euclid and segment multiplication in [Hilbert, 1962] or multiplication6 of ‘numbers’. With this background, we sketch in Section 4 Hilbert’s theory of proportions and area, focusing on Euclidean propositions that might appear to depend on continuity axioms. In particular, we outline Hilbert’s definition of the field in a plane and how this leads to a good theory of area and proportion, while avoiding the axiom of Archimedes. This paper expounds the consequences of Hilbert’s first-order axioms and argues they form a modest descriptive axiomatization of Euclidean geometry. The sequel extends the historical analysis from Euclid and Hilbert to Descartes and Tarski, and explores several variants on Dedekind’s axiom and the role of first-order, infinitary, and second-order logic. Further, it expounds a first-order basis for the formulas for area and circumference of a circle. 1. THE GOALS OF AXIOMATIZATION In this section, we place our analysis in the context of recent philosophical work on the purposes of axiomatization. We explicate Detlefsen’s notion of ‘data set’ and investigate the connection between axiom sets and data sets of sentences for an area of mathematics. Hilbert begins the Grundlagen [1971] with: The following investigation is a new attempt to choose for geometry a simple and complete set of independent axioms and to deduce from them the most important geometrical theorems in such a manner as to bring out as clearly as possible the significance of the groups of axioms and the scope of the conclusions to be derived from the individual axioms. Hallett [2008, p. 204], delineates the meaning of facts in this context, ‘simply what over time has come to be accepted, for example, from an accumulation of proofs or observations. Geometry, of course, is the central example $$\ldots$$’. Hallett [Hilbert, 2004, p. 434] presaged the emphasis here on ‘data sets’. Thus completeness appears to mean [for Hilbert] ‘deductive completeness with respect to the geometrical facts’. $$\ldots$$ In the case of Euclidean geometry there are various ways in which ‘the facts before us’ can be presented. If interpreted as ‘the facts presented in school geometry’ (or the initial stages of Euclid’s geometry), then arguably the system of the original Festschrift [i.e., the 1899 French version] is adequate. If, however, the facts are those given by geometrical intuition, then matters are less clear. Hilbert described the general axiomatization project in 1918. When we assemble the facts of a definite, more or less comprehensive field of knowledge, we soon notice these facts are capable of being ordered. This ordering always comes about with the help of a certain framework of concepts [Fachwerk von Begriffen] in the following way: a concept of this framework corresponds to each individual object of the field of knowledge, a logical relation between concepts corresponds to every fact within the field of knowledge. The framework of concepts is nothing other than the theory of the field of knowledge. [Hilbert, 2005, p. 1107] Detlefson [2014] describes such a project as a descriptive axiomatization. He motivates the notion with this remark by Huntington: [A] miscellaneous collection of facts $$\ldots$$ does not constitute a science. In order to reduce it to a science the first step is to do what Euclid did in geometry, namely, to select a small number of the given facts as axioms and then to show that all other facts can be deduced from these axioms by the methods of formal logic. [Huntington, 1911, emphasis original] Detlefsen uses the term data set (i.e., facts7) to describe a local descriptive axiomatization as an attempt to organize a data set of commonly accepted sentences deductively. The axioms are descriptively complete if all elements of the data set are deducible from them. This raises two questions. What is a sentence? Who commonly accepts? From the standpoint of modern logic, a natural answer to the first question would be to specify a logic and a vocabulary and consider all sentences in that language. Detlefsen argues [2014, pp. 5–7] that this is the wrong answer. He thinks Gödel errs in seeing the problem as completeness in the now standard sense of a first-order theory.8 Rather, Detlefsen presents us with an empirical question. We (at some point in time) look at the received mathematical knowledge in some area and want to construct a set of axioms from which it can all be deduced. In our case we want to compare the commonly accepted sentences from 300 BC with a twentieth-century axiomatization. As we see below, new interpretations for terms arise. Nevertheless, a specific data set delineates a certain area of inquiry. The data set is inherently flexible; conjectures are proven (or refuted) from time to time. The analysis in this paper tries to distinguish propositions that are simply later deductions about the same intuitions and those which invoke radically different assumptions. By analyzing the interpretations in particular cases, we can specify a data set. Comparing geometry at various times opens a deep question worthy of more serious exploration than there is space for here. In what sense do (*) and (**) opening this paper express the same thought, concept etc.? Rather than address the issue of what is expressed, we will simply show how to interpret (*) (and other propositions of Euclid) as propositions in Hilbert’s system. Working in a framework of formal logic, we return to our question, ‘What is a sentence?’ The first four groups of Hilbert’s axioms are sentences of first-order logic: quantification is over individuals and only finite conjunctions are allowed. As noted in footnote 2, Archimedes’ axiom can be formulated in $$L_{\omega_1,\omega}$$. But the Dedekind postulate in any of its variants is a sentence of a kind of second-order logic.9 All three logics have deductive systems, and the first- and second-order systems allow only finite proofs; so the set of provable sentences is recursively enumerable. Second-order logic (in the standard semantics) fails the completeness theorem but, by the Gödel and Karp [1959] completeness theorems, every valid sentence of $$L_{\omega,\omega}$$ or $$L_{\omega_1,\omega}$$ is provable. Adopting this syntactic view, there is a striking contrast between the data set of geometry in earlier generations and the axiom systems advanced near the turn of the twentieth century. Except for the Archimedean axiom, the earlier data sets are expressed in first-order logic. Geometry is an example of what Detlefsen calls a local as opposed to a foundational (global) descriptive axiomatization. Beyond the obvious difference in scope, Detlefsen points out several other distinctions. In particular, Detlefsen [2014, p. 5] asserts that axioms of a local axiomatization are generally among the given facts while those of a foundational axiomatization are found by (paraphrasing Detlefsen) tracing each truth in a data set back to the deepest level to which it can be properly traced. Hilbert’s geometric axioms have a hybrid flavor. Through the analysis of the concepts involved, Dedekind arrived at a second-order axiom.10 that is not in the data set but formed the capstone. An aspect of choosing axioms seems to be missing from the account so far. Hilbert [2005] provides the following insight into how axioms are chosen: If we consider a particular theory more closely, we always see that a few11 distinguished propositions of the field of knowledge underlie the construction of the framework of concepts, and these propositions then suffice by themselves for the construction, in accordance with logical principles, of the entire framework. $$\ldots$$ These underlying propositions may from an initial point of view be regarded as the axioms of the respective field of knowledge $$\ldots$$ By a modest axiomatization of a given data set,12 we mean one that implies all the data and not too much more.13 Of course, ‘not too much more’ is a rather imprecise term. One cannot expect a list of known mathematical propositions to be deductively complete. By ‘too much more’, we mean the axioms introduce essentially new concepts and concerns or add additional hypotheses proving a result that contradicts the explicit understandings of the authors of the data set (Section 3 in the sequel). In this paper, we are investigating modern axiomatizations for an ancient data set. As we shall see below, using Notation 2.2, Hilbert’s first-order axioms (HP5) are a modest axiomatization of the data (Euclid I): the theorems in Euclid about polygons (not circles) in the plane. We give an example later showing that HP5 + CCP (circle-circle intersection postulate, defined in Section 2), while modest for Euclid II, is an immodest first-order axiomatization of polygonal geometry. In the twentieth century the process of formalization is usually attentive to modesty; so it is a bit hard to find non-artificial examples. However, to study complex exponentiation, Zilber [2005] defined a quasi-minimal excellent class. The models admitted a closure relation satisfying the axioms of a combinatorial geometry as well as certain homogeneity conditions. Although his axioms were informal they can be formalized14 in $$L_{\omega_1,\omega}(Q)$$. Quite unexpectedly, [Bays et al., 2014] showed ‘excellence’ was not needed for the main result. Thus, the original axiomatization was immodest. No single axiom is modest or immodest; the relation has two arguments: a set of sentences is a modest axiomatization of a given data set. If the axioms are contained in the data set the axiomatization is manifestly modest, and this is just a mathematical fact that can be clarified by formalization. But some other subset might later be taken as axioms that imply the whole set. This might just happen by a clever proof. But the cases studied here are more subtle. New interpretations of the basic concepts (of multiplication and number) developed over time so that the sentences attained essentially new meanings. As (**) illustrates, such is the case with Euclid’s VI.1. Distinct philosophical issues arise in checking modesty and immodesty. For modesty, an historical investigation, as in this paper, must interpret the new conception (**) into the original data set (*). For immodesty, the conceptual content of the new axioms must be compared with that of the data set. In our view, modesty and purity are distinct, though related, notions. We just saw that formalization is useful to check modesty. Hallett [2008], Arana and Mancosu [2012], Detlefsen and Arana [2011], and Baldwin [2013a] argue that purity asks about specific arguments for a proposition. [Baldwin, 2013a] emphasized that the same theorem can have both pure and impure proofs and [Baldwin, 2018] extends the analysis of modesty versus purity. In contrast, modesty concerns the existence of proofs and appropriateness of hypotheses. In the quotation above, Hilbert takes the axioms to come from the data set. But this raises a subtle issue about what composes the data set. For examples such as geometry and number theory, it was taken for granted that there was a unique model. Even Hilbert adds his completeness axiom to guarantee categoricity and to connect with the real numbers. So one could argue that the early twentieth-century axiomatizers took categoricity as part of the data.15 But such an intuition is inherently metatheoretic and so dissimilar to the other data. It is certainly not in the data set of the Greeks for whom ‘categoricity’ is meaningless. 2. SOME GEOMETRIC DATA SETS AND AXIOM SYSTEMS This section is intended to lay out several topics in plane geometry that represent distinct data sets in Detlefsen’s sense.16 In cases where certain axioms are explicit, they are included in the data set. Although we describe five sets here, only polygonal geometry and circle geometry are considered in this paper; the others are treated in the sequel. Each set includes its predecessors; the description is of the added propositions. Notation 2.1 (5 data sets of geometry) Euclid I, polygonal geometry: Book I (except I.1, I.22), Book II.1-II.13, Book III (except III.1 and III.17), Book VI. Euclid II, circle geometry: CCP (circle-circle intersection postulate, defined below), I.1, 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. Our division of the data sets is somewhat arbitrary and is made with the subsequent axiomatizations in mind. The importance of Euclid II appears already in Proposition I of Euclid, where Euclid makes the standard construction of an equilateral triangle on a given base.17 Why do the two circles intersect? While some18 regard the absence of an axiom guaranteeing such intersections as a gap in Euclid, Manders [2008, p. 66] asserts: ‘Already the simplest observation on what the texts do infer from diagrams and do not suffices to show the intersection of two circles is completely safe’.19 The circle-circle intersection axiom resolves those continuity issues involving intersections of circles and lines.20 As noted, this proposition is in the data set; so adding this one axiom in our axioms EG (below), which does not appear explicitly in either the Grundlagen or Euclid, does not detract from the modesty of our axioms. It is a first-order consequence of the Dedekind postulate which plays an essential role in Euclidean geometry. Hilbert is aware of that fact; he chooses to resolve the issue (implicitly) by his completeness axiom. Circle-Circle Intersection Postulate (CCP): If from distinct points $$A$$ and $$B$$, circles with radius $$AC$$ and $$BD$$ are drawn such that one circle contains points in the interior and in the exterior of the other, then they intersect in two points, on opposite sides of $$AB$$. We have placed Euclid XII.2 (area of a circle is proportional to the square of the diameter) with Archimedes rather than Euclid’s other theorems on circles. The crux is the different resources needed to prove VI.1 (area of a parallelogram) and XII.2. The first is provable in EG; the sequel contains a first-order extension of EG in which XII.2 is provable. Note also that we consider only a fraction of Archimedes, his work on the circle. We analyze the connections among Archimedes, Descartes, and Tarski in the sequel. Showing a particular set of axioms is descriptively complete is inherently empirical. One must check whether each of a certain set of results is derivable from a given set of axioms. Hartshorne [2000] carried out this project independently from Detelfsen’s analysis; we organize his results at the end of this section. We identify two levels of formalization in mathematics. By the Euclid-Hilbert style we mean the axiomatic approach of Euclid along with the Hilbert insight that postulates are implicit definitions of classes of models.21 By the Hilbert-Gödel-Tarski style, we mean that that syntax and semantics have been identified as mathematical objects. Gödel’s completeness theorem is a standard tool; so methods of modern model theory can be applied.22 We will give our arguments in English, but we will be careful to specify the vocabulary and the postulates in such a way that the translation to a first-order theory is transparent. We will frequently switch from syntactic to semantic discussions; so we stipulate precisely the vocabulary in which we take the axioms above to be formalized. We freely use defined terms such as collinear, segment, and angle in giving the reading of the relevant relation symbols. The fundamental relations of plane geometry make up the following vocabulary $$\tau$$. (1) Two-sorted universe: points ($$P$$) and lines ($$L$$). (2) Binary relation $$I(A,\ell)$$: Read: a point is incident on a line; (3) Ternary relation $$B(A,C,D)$$: Read: $$C$$ is between $$A$$ and $$D$$ (and $$A,C,D$$ are collinear). (4) Quaternary relation, $$C(A,B,A',B')$$: Read: two segments are congruent, in symbols $$\overline{AB}\cong \overline {A'B'}$$. (5) 6-ary relation $$C'(A,B,D,A',B',D')$$: Read: the two angles $$\angle ABD$$ and $$\angle A'B'D'$$ are congruent, in symbols $$\angle{ ABD}\cong \angle {A'B'D'}$$. Notation 2.2 Consider the following axiom sets.23 (1) First-order axioms: HP, HP5: We write HP for Hilbert’s incidence, betweenness,24 and congruence axioms. We write HP5 for HP plus the parallel postulate. EG: The axioms for Euclidean geometry, denoted EG,25 consist of HP5 and in addition the circle-circle intersection postulate CCP. $$\boldsymbol{{\mathcal E}^2}:$$ Tarski’s axiom system for a plane over a real closed field (RCF26). $$\mathbf{EG}_{\boldsymbol{\pi}}$$ and $$\boldsymbol{{\mathcal E}^2_\pi}$$: Two new systems, which extend EG and $${\mathcal E}^2$$, will be described and analyzed in the sequel. (2) Hilbert’s continuity axioms, infinitary and second-order, will also be examined in detail in the sequel. Archimedes: The sentence in the logic $$L_{\omega_1,\omega}$$ expressing the Archimedean axiom. Dedekind: Dedekind’s second-order axiom27 that there is a point in each irrational cut in the line. With these definitions we align various subsystems of Hilbert’s geometry with certain collections of propositions in Euclidean geometry as spelled out in Hartshorne.28 With our grouping, Hartshorne shows the following results. They imply that the statements in HP5 or EG are either theorems or implicit (CCP) in Euclid; they are a modest axiomatization of Euclid I and II. First-order Axiomatizations (1) The sentences of Euclid I are provable in HP5. (2) The additional sentences of Euclid II are provable in EG. In this framework we discuss the changing conceptions of the continuum, ratio, and number from the Greeks to modern times and sketch some highlights of the proof of this result to demonstrate the philosophical modesty of the axiomatization. The sequel contains modest descriptive axiom systems29 for the data sets of Archimedes and Descartes and argues that the full Hilbert axiom set is immodest for any of these data sets. 3. CHANGING CONCEPTIONS OF THE CONTINUUM, MAGNITUDE, AND NUMBER In Section 3.1, we distinguish the geometric continuum from the set-theoretic continuum. In Section 3.2 we sketch the background shift from the study of various types of magnitudes by the Greeks to the modern notion of a collection of real numbers, which can measure any sort of magnitude Section 3.3 concerns Bolzano’s challenge. In Section 3.4 we contrast the current goal of an independent basis for geometry with the nineteenth-century arithmetization project. 3.1. Conceptions of the Continuum In this section, we motivate our restriction to the geometric continuum; we defined it as a linearly ordered structure that is situated in a plane. Sylvester30 describes the three divisions of mathematics: There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all of them combined, every mathematical truth admits of being referred; these are the three cardinal notions, of Number, Space and Order. This is a slightly unfamiliar trio. We are all accustomed to the opposition between arithmetic and geometry. While Newton famously founded the calculus on geometry [Detlefsen and Arana, 2011] the ‘arithmetization of analysis’ in the late nineteenth century reversed the priority. From the natural numbers the rational numbers are built by taking quotients and the reals by some notion of completion. And this remains the normal approach today. We want here to consider reversing the direction again: building a firm grounding for geometry and then finding first the field and then some completion and considering incidentially the role of the natural numbers. In this process, Sylvester’s third cardinal notion, order, will play a crucial role. The notion that one point lies between two others will be fundamental and an order relation will naturally follow; the properties of space will generate an ordered field and the elements of that field will be numbers albeit not numbers in the Greek conception. There are different conceptions of the continuum (the line); hence different axiomatizations may be necessary to reflect these different conceptions. These conceptions are witnessed by such collections as [Ehrlich, 1994; Salanskis and Sinaceur, 1992] and further publications concerned with the constructive continuum and various non-Archimedean notions of the continuum. In his [2008], Feferman lists six31 different conceptions of the continuum: (i) the Euclidean continuum, (ii) Cantor’s continuum, (iii) Dedekind’s continuum, (iv) the Hilbertian continuum, (v) the set of all paths in the full binary tree, and (vi) the set of all subsets of the natural numbers. For our purposes, we will identify (ii), (v), and (vi) as essentially cardinality-based as they have lost the order type imposed by the geometry; so they are not in our purview. We want to contrast two essentially geometrically based notions of the continuum: those of Euclid and Hilbert/Dedekind. Hilbert’s continuum differs from Dedekind’s as it has the field structure derived from the geometric structure of the plane, while Dedekind’s field is determined by continuity from known field operations on the rationals. Nevertheless they are isomorphic as ordered fields. We stipulated that ‘geometric continuum’ means ‘the line situated in the plane’. One of the fundamental results of twentieth-century geometry is that any (projective32 for convenience) plane can be coordinatized by a ‘ternary field’. A ternary field is a structure with one ternary function $$f(x,y,z)$$ such that $$f$$ has the properties that $$f(x,y,z) = xy +z$$ would have if the right-hand side were interpreted in a field. In dealing with Euclidean geometry here, we assume the axioms of congruence and the parallel postulate; this implies that the ternary field is actually a field. But these geometric hypotheses are necessary. In [Baldwin, 1994], I constructed an $$\aleph_1$$-categorical projective plane where the ternary field is as wild as possible (in the precise sense of the Lenz-Barlotti classification in [Dembowski, 1977]: the ternary function cannot be decomposed into an addition and a multiplication). 3.2. Ratio, Magnitude, and Number In this section we give a short review of Greek attitudes toward magnitude and ratio. Fuller accounts of the transition to modern attitudes appear in such sources as [Mueller, 2006; Euclid, 1956; Stein, 1990; Menn, 2017]. We by no means follow the ‘geometric algebra’ interpretation decried in [Grattan-Guinness, 2009]. Rather, we attempt to contrast the Greek meanings of propositions with Hilbert’s understanding. When we rephrase a sentence in algebraic notation we try to make clear that this is a modern formulation and often does not express the intent of Euclid. Euclid develops arithmetic in Books VII–IX. What we think of as the ‘number one’, was ‘the unit’; a number (Definition VII.2) was a multitude of units. These are counting numbers. So from our standpoint (considering the unit as the number 1) Euclid’s numbers (in the arithmetic) can be thought of as the ‘natural numbers’. The numbers33 are a discretely ordered collection of objects. Following Mueller [2006, p. 21] we work from the interpretation of magnitudes in the Elements as ‘abstractions from geometric objects which leave out of account all properties of those objects except quantity: length for lines, area of plane figures, volume of solid figures, etc.’ Mueller emphasizes the distinction between the properties of proportions of magnitudes developed in Book V and those of number in Book VII. The most easily stated is implicit in Euclid’s proof of Theorem V.5; for every $$m$$, every magnitude can be divided in $$m$$ equal parts. This is of course, false for the (natural) numbers. There is a second use of ‘number’ in Euclid. It is possible to count unit magnitudes, to speak of, e.g., four copies of a unit magnitude. So (in modern language) Euclid speaks of multiples of magnitudes by positive integers. Magnitudes of the same type are also linearly ordered and between any two there is a third.34 Multiplication of line segments yields rectangles. Ratios are not objects; equality of ratios is a 4-ary relation between two pairs of homogenous magnitudes.35 Some key points from Euclid’s discussion of proportion in Book V are: (1) Definition V.4 of Euclid [1956]: Magnitudes are said to have a ratio to one another, which are capable, when multiplied, of exceeding one another. (2) Definition V.5, ‘sameness of two ratios’ (in modern terminology): The ratio of two magnitudes $$x$$ and $$y$$ are proportional to the ratio of two others $$z, w$$ if for each $$m,n$$, $$mx> ny$$ implies $$mz > nw$$ (and also replacing $$>$$ by $$=$$ or $$<$$). (3) Definition V.6: Let magnitudes which have the same ratio be called proportional. (4) Proposition V.9: ‘Same ratio’ is, in modern terminology, a transitive relation. Apparently Euclid took symmetry and reflexivity for granted and treats proportionality as an equivalence relation. 3.3. Bolzano’s Challenge In a discussion of the foundations of geometry Bolzano discusses the ‘dissimilar objects’ found in Euclid and finds Euclid’s approach fundamentally flawed. His general position is that one must analyze conceptually prior notions (line) before more complex notions (plane). See [Russ, 2004, p. 33] and [Rusnock, 2000, p. 53] for discussions of Bolzano’s claim that the study of the line, as ‘more fundamental’ must precede that of the plane. More specifically, he objects to Euclid’s proof in VI.2 that similar triangles have proportional sides: Firstly triangles, that are already accompanied by circles which intersect in certain points, then angles, adjacent and vertically opposite angles, then the equality of triangles, and only much later their similarity, which however, is derived by an atrocious detour [ungeheuern Umweg], from the consideration of parallel lines, and even of the area of triangles, etc.36 [Bolzano, 1810] What we call Bolzano’s challenge has two aspects: (a) the evil of using two-dimensional concepts to understand the line, and (b) the ‘atrocious detour’ to similarity (VI.2). We consider the plane essential to understanding the geometric continuum; so (a) is irrelevant to our project; Sections 4.1–4.3 sketch how Hilbert meets the challenge by avoiding the detour. The side-splitter theorem, VI.2. is taken to represent similarity. Euclid deduces VI.2 from VI.1 (see Figure 1). In his proof of VI.1 (*) Euclid applies Definition V.5 (above) to deduce that the areas of two triangles with the same height are proportional to their bases. But this assumes that any two lengths (or any two areas) have a ratio in the sense of Definition V.4. This is an implicit assertion of Archimedes’ axiom for both area and length.37 Fig. 1. View largeDownload slide Euclid’s proof of VI.1. Fig. 1. View largeDownload slide Euclid’s proof of VI.1. If, for example, $$BC$$, $$GB$$, and $$HG$$ are congruent segments then the area of $$ACH$$ is triple that of $$ABC$$. But without assuming $$BC$$ and $$BD$$ are commensurable, Euclid calls on Definition V.5 to assert that $$ABD: ABC :: BD : BC$$. In VI.2, he uses these results to show that similar triangles have proportional sides. From VI.2, Euclid constructs in VI.12 the fourth proportional to three lines but does not regard it as a definition of multiplication of segments. As we will see in more detail, Hilbert’s treatment of area and similarity has no such dependence on Archimedes’ axiom. By interpreting the field with segment arithmetic, he defines proportionality directly. In contrast, Descartes defines the multiplication of line segments to give another segment,38 but he still relies on Euclid’s theory of proportion to justify the multiplication. Hilbert’s innovation is use to segment multiplication to gain the notion of proportionality, which is defined in Section 4.2. 3.4. From Arithmetic to Geometry or From Geometry to Algebra? On the first page of Continuity and the Irrational Numbers, Dedekind wrote: Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful from the didactic standpoint $$\ldots$$ But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. [Dedekind, 1963] I do not contest Dedekind’s claim. I quote this passage to indicate that Dedekind’s motivation was to provide a basis for analysis, not geometry. But I will argue that the second-order Dedekind completeness axiom is not needed for the geometry of Euclid and indeed for the grounding of the algebraic numbers, although it is in Dedekind’s approach. Dedekind provides a theory of the continuum (the continuous) line by building up in stages from the structure that is fundamental to him: the natural numbers under successor. This development draws on second-order logic in several places. The well-ordering of the natural numbers is required to define addition and multiplication by recursion. Dedekind completeness is a second appeal to a second-order principle. Perhaps in response to Bolzano’s insistence, Dedekind constructs the line without recourse to two-dimensional objects and from arithmetic. Thus, he succeeds in the ‘arithmetization of analysis’. We proceed in the opposite direction for several reasons. Most important is that we are seeking to ground geometry, not analysis. Further, we adopt as a principle that the concept of line arises only in the perception of at least two-dimensional space. Dedekind’s continuum knows nothing of being straight or breadthless. Hilbert’s proof of the existence of the field is the essence of the geometric continuum. By virtue of its lying in a plane, the line acquires algebraic properties. Moreover, the distinction between the arithmetic and geometric intuitions of multiplication is fundamental. The basis of the first is iterated addition; the basis of the second is scaling or proportionality. The late nineteenth-century developments provide a formal reduction of the second to the first but the reduction is only formal; the ‘scaling’ intuition is lost. In this paper we view both intuitions as fundamental and develop the second (Section 4.1) with the understanding that development of the first through the Dedekind-Peano treatment of arithmetic is in the background. 4. AXIOMATIZING THE GEOMETRY OF POLYGONS AND CIRCLES Section 4.1 sketches the key step in making an independent foundation of geometry: Hilbert’s definition of a field in a geometry. Section 4.2 describes the transition from segments to points as the domain of that field and gives Hilbert’s definition of proportional. Section 4.3 distinguishes the role of the CCP and analyzes several problems that can be approached by limits but have uniform solutions in any ordered field; order completeness of the field is irrelevant. We return to Bolzano’s challenge and derive Theorem 4.3.4, the properties of similar triangles and, in Section 4.4, the area of polygons. 4.1. From Geometry to Segment Arithmetic to Numbers One of Hilbert’s key innovations is his segment arithmetic and his definition of the semi-field39 of segments with partial subtraction and multiplication. We assume the axiom system we called HP5 in Notation 2.2. The details can be found in, e.g., [Hilbert, 1971; Hartshorne, 2000; Baldwin, 2018; Giovannini, 2016]. Note that congruence ($$\cong$$) forms an equivalence relation on line segments. Fix a ray $$\ell$$ with one end point $$0$$ on $$\ell$$. For each equivalence class of segments, we consider the unique segment $$0A$$ on $$\ell$$ in that class as the representative of that class. We will often denote the segment $$0A$$ (ambiguously its congruence class) by $$a$$. We say a segment $$CD$$ (on any line) has length $$a$$ if $$CD \cong 0A$$. Following Hartshorne [2000], here is our official definition of segment multiplication.40 Fix a unit segment class, $$1$$. Consider two segment classes $$a$$ and $$b$$. To determine their product, define a right triangle with legs of length $$1$$ and $$a$$. Denote the angle between the hypoteneuse and the side of length $$1$$ by $$\alpha$$. Now construct another right triangle with base of length $$b$$ with the angle between the hypoteneuse and the side of length $$1$$ congruent to $$\alpha$$. The product $$ab$$ is defined to be the length of the vertical leg of the triangle. Fig. 2. View largeDownload slide Segment multiplication. Fig. 2. View largeDownload slide Segment multiplication. Note that we must appeal to the parallel postulate to guarantee the existence of the point $$F$$. It is clear from the definition that there are multiplicative inverses; use the triangle with base $$a$$ and height $$1$$. Hartshorne has a roughly three-page proof41 that shows multiplication is commutative, associative, distributes over addition, and respects the order. It uses only the cyclic quadrilateral theorem and connections between central and inscribed angles in a circle. To summarize the effect of the axiom sets, we introduce two definitions. (1) An ordered field $$F$$ is Pythagorean if it is closed under addition, subtraction, multiplication, division and for every $$a\,{\in}\, F$$, $$\surd (1+a^2) \,{\in}\, F$$. (2) An ordered field $$F$$ is Euclidean if it is closed under addition, subtraction, multiplication, division and for every positive $$a \,{\in}\, F$$, $$\surd a \,{\in}\, F$$. When the model is taken as geometry over the reals, it is easy42 to check that the multiplication defined on the positive reals by this procedure is exactly the usual multiplication on the positive reals because they agree on the positive rational numbers. As in Section 21 of [Hartshorne, 2000], we have: Theorem 4.1.1 (1) HP5 is bi-interpretable with the theory of ordered Pythagorean fields. (2) Similarly EG is bi-interpretable with the theory of ordered Euclidean fields. Formally bi-interpretability means there are formulas in the field language defining the geometric notions (point, line, congruence, etc) and formulas in the geometric language (plus constants) defining the field operations ($$0,1, +,\times$$) such that interpreting the geometric formulas in a Pythagorean field gives a model of HP5 and conversely. See [Hodges, 1993, Ch. 5] for general background on interpretability. With this information we can explain why Proposition I.1 (equilateral triangle) is in Euclid I rather than II. Both it and I.22 (construct a triangle given three lines with the sum of the length of two greater than the length of the third) use circle-circle intersection in Euclid. However, Hilbert proves the first in HP5, but the second requires the field to be Euclidean and so uses CCP. Dicta on Constants: Note that to fix the field we had to add constants $$0$$, $$1$$. These constants can name any pair of points in the plane.43 But this naming induces an extension of the data set. We have in fact specified the unit. This specification has little effect on the data set but a major change in view from either the Greeks or Descartes. 4.2. Points and Numbers, Multiplication and Proportionality Hilbert shows the multiplication on segments of a line through points $$0,1$$ satisfies the semi-field axioms. He defines segment multiplication on the ray from $$0$$ through $$1$$ as in Section 4.1. But to get negative numbers one must reflect through $$0$$. Then addition and multiplication can be defined on directed segments of the line44 through $$0,1$$ and thus all axioms for a field are obtained. The next step is to identity the points on the line and the domain of an ordered field by mapping $$A$$ to $$0A$$. This naturally leads to thinking of a segment as a set of points, which is foreign to both Euclid and Descartes. Although in the context of the Grundlagen, Hilbert’s goal is to coordinatize the plane by the real numbers, his methods open the path to thinking of the members of any field as ‘numbers’ that coordinatize the associated geometries. Boyer traces the origins of numerical coordinates to 1827–1829 and writes It is sometimes said that Descartes arithmetized geometry but this is not strictly correct. For almost two hundred years after his time coordinates were essentially geometric. Cartesian coordinates were line segments $$\ldots$$ The arithmetization of coordinates took place not in 1637 but in the crucial years 1827–1829. [Boyer, 1956, p. 242] Boyer points to Bobillier, Möbius, Feurbach, and most critically Plücker as introducing several variants of what constitute numerical (signed distance) barycentric coordinates of a point. Multiplication is not repeated addition: We now have two ways in which we can think of the product $$3a$$. On the one hand, we can think of laying 3 segments of length $$a$$ end to end. On the other, we can perform the segment multiplication of a segment of length $$3$$ (i.e., $$3$$ segments of length $$1$$ laid end to end) by the segment of length $$a$$. It is an easy exercise to show these give the same answer. But these distinct constructions make an important point. The (inductive) definition of multiplication by a natural number is indeed ‘multiplication as repeated addition’. But the multiplication by another field element is based on similarity and has multiplicative inverses; so from a modern standpoint they yield very different structures. No extension of natural number arithmetic is decidable but important theories of fields are. The first notion of multiplication in the last paragraph, where the multiplier is a natural number, is a kind of scalar multiplication by positive integers that can be viewed mathematically as a rarely studied object: a semiring (the natural numbers) acting on a semigroup (positive reals under addition). There is no uniform definition45 of this binary operation of scalar multiplication within the semiring action. A mathematical structure more familiar to modern eyes is obtained by adding the negative numbers to get the ring $${\mathbb Z}$$, which has a well-defined notion of subtraction. The scalars are now in the ring $$({\mathbb Z},+,\cdot)$$ and act on the module $$(\Re, +)$$. Now we can multiply by $$-17$$ but the operation is still not uniform but given by a family of unary functions. Looking more closely, we have three different referents for segment multiplication: Euclid’s computation of the area of rectangle, his construction of the fourth proportional, and Hilbert’s definition. These three computations give the ‘same answer’. We now consider the fourth proportional in $$1 : a :: b : x$$ to measure the area of the rectangle with sides of length $$a$$ and $$b$$. In view of this we do not view Hilbert as introducing a new concept of multiplication — but as reinterpreting the notion and indeed the same geometric construction as applying to line segments, which we now read as numbers, rather than mapping from linear magnitudes to planar magnitudes. With this in hand, Hilbert redefines proportionality: Proportionality: We write the ratio of CD to CA is proportional to that of CE to CB, $$CD:CA :: CE: CB$$ which is defined as $$CD \times CB = CE \times CA,$$ where $$\times$$ is taken in the sense of segment multiplication as defined in Section 4.1. While in Book V Euclid provides a general account of proportionality, Hilbert’s ability to avoid the Archimedean axiom depends both on the geometrical construction of the field and the reinterpretation of ‘number’. 4.3. Field Arithmetic and Basic Geometry In this section we investigate some statements from: (1) Euclid’s geometry that depended in his development on the Archimedean axiom and some from (2) Dedekind’s development of the properties of real numbers that he deduces from his postulate. In each case, they are true in any field associated with a geometry modeling HP5. We established in Section 4.1 that one could define an ordered field $$F$$ in any plane satisfying HP5. The converse is routine: the ordinary notions of line and incidence in $$F^2$$ creates a geometry over any Pythagorean ordered field, which is easily seen to satisfy HP5. We now exploit this equivalence to show some important algebraic facts using our defined operations, thus basing them on geometry. First, note that taking square root commutes with multiplication for algebraic numbers. Dedekind [1963, p. 22] wrote ‘$$\dots$$ in this way we arrive at real proofs of theorems (as, e.g., $$\surd 2 \cdot \surd 3 = \surd 6$$), which to the best of my knowledge have never been established before’. Note that this commutativity is a problem for Dedekind but not for Descartes. Euclid had already, in constructing the fourth proportional, constructed from segments of length $$1$$, $$a$$ and $$b$$, one of length $$ab$$; but he does not regard this operation as multiplication. When Descartes interprets this procedure as multiplication of segments, the reasoning above shows multiplying square roots is not an issue. But Dedekind has presented the problem as multiplication in his continuum and so he must prove a theorem which allows us to find the product as a real number; that is, he must show the limit operation commutes with product. But, in an ordered field, for any positive $$a$$, if there is an element $$b>0$$ with $$b^2 =a$$, then $$b$$ is unique (and denoted $$\surd a$$). Moreover, for any positive $$a,c$$ with square roots, $$\surd a \cdot \surd c = \surd ac$$, since each side of the equality squares to $$ac$$. This fact holds for any field coordinatizing a plane satisfying HP5. Thus, the algebra of square roots in the real field is established without any appeal to limits. The usual (e.g., [Spivak, 1980; Apostol, 1967]) developments of the theory of complete ordered fields follow Dedekind and invoke the least-upper-bound principle to obtain the existence of the roots although the multiplication rule is obtained by the same algebraic argument as here. Hilbert’s approach contrasts with Dedekind’s.46 The justification here for either the existence of, or operations on, roots does not invoke limits. Hilbert’s treatment is based on the geometric concepts and in particular regards ‘congruence’ and ‘number’ as equally fundamental notions. In short, the shift here is from ‘proportional segments’ to ‘product of numbers’. Euclid had a rigorous proof of the existence of a line segment which is the fourth proportional of $$1:a = b:x$$. Dedekind demands a product of numbers; Hilbert provides this by a combination of his interpretation of the field in the geometry and the geometrical definition of multiplication. Euclid’s proof of Pythagoras’s theorem I.47 uses the properties of area that we will justify in Section 4.4. His second proof (Lemma for X.33) uses the property of similar triangles that we prove in Theorem 4.3.4. In both cases Euclid depends on the theory of proportionality (and thus implicitly on Archimedes’ axiom) to prove the Pythagorean theorem; Hilbert avoids this. appeal.47 Similarly, since the right-angle trigonometry in Euclid concerns the ratios of sides of triangles, the field multiplication justifies basic right-angle trigonometry. We have: Theorem 4.3.1The Pythagorean theorem as well as the law of cosines (Euclid II.12, II.13) and the law of sines [Maor, 1993, pp. 217–217] hold in HP5. Hartshorne [2000] describes two instructive examples, connecting the notions of Pythagorean and Euclidean planes. Example 4.3.2 (1) The Cartesian plane over a Pythagorean field may fail to be closed under square root.48 (2) Hartshorne49 observes that the smallest ordered field closed under addition, subtraction, multiplication, division, and square roots of positive numbers and satisfying the CCP is a Euclidean field, denoted by $$F_s$$ for surd field. Note that if HP5 + CCP were proposed as an axiom set for polygonal geometry it would be a complete descriptive but not modest axiomatization since it would prove CCP which is not in the polygonal geometry data set. In a Euclidean plane every positive element of the coordinatizing field has a square root; so Heron’s formula ($$A =\sqrt{s(s-a)(s-b)(s-c)}$$, where $$s$$ is half the perimeter and $$a,b,c$$ are the side lengths) computes the area of a triangle from the lengths of its sides. This fact demonstrates the hazards of the kind of organization of data sets attempted here. The geometric proof of Heron does not involve the square roots of the modern formula50 [Heath, 1921, p. 321]. But since in EG we have the field and we have square roots, the modern form of Heron’s formula can be proved from EG. Thus, as in the shift from (*) to (**) at the beginning of the paper, the different means of expressing the geometrical property requires different proofs. In each case considered in this section, Greeks give geometric constructions for what in modern days becomes a calculation involving the field operations and square roots. However, we still need to complete the argument that HP5 is descriptively complete for polygonal Euclidean geometry. In particular, is our notion of proportional correct? The test question is the similar triangle theorem. We turn to this issue now. Definition 4.3.3Two triangles $$\triangle ABC$$ and $$\triangle A'B'C'$$ are similar if under some correspondence of angles, corresponding angles are congruent; i.e., $$\angle A' \cong\angle A$$, $$\angle B' \cong\angle B$$, $$\angle C' \cong\angle C$$. Various texts define ‘similar’ as we did, or focus on corresponding sides being proportional, or require both (Euclid). We now meet Bolzano’s challenge by showing that in Euclidean Geometry (without the continuity axioms) the choice does not matter. We defined ‘proportional’ in terms of segment multiplication in Section 4.2. Hartshorne [2000, pp. 176–177] proves the fundamental result (Euclid VI.2). Theorem 4.3.4 (EG). Two triangles are similar if and only if corresponding sides are proportional. Euclid VI.2 follows. There is no assumption that the field is Archimedean or satisfies any sort of completeness axiom. There is thus no appeal to approximation or limits. We have avoided Bolzano’s ‘atrocious detour’ through area. But area is itself a vital geometric notion, and that is the topic of the next section. 4.4. Area of Polygonal Figures Hilbert [1971, p. 57] wrote ‘We $$\ldots$$ establish Euclid’s theory of area for the plane geometry and that independently of the axiom of Archimedes’ (emphasis original). We now sketch Hartshorne’s [2000] exposition of this topic, stressing the connections with Euclid’s Common Notions. We show that the notions defined here are expressible in first-order logic, which supports our fourth objection in Section 4.3 of the sequel: although these arguments are not carried out as direct deductions from the first-order axioms, the results are derivable by a direct deduction. That is, we develop area in first-order logic and even though some of the arguments are semantical the conclusions are theorems of first-order logic. This is further evidence of the immodesty of the second-order axiomatization. Informally, those configurations whose areas are considered in this section are figures, where a figure is a subset of the plane that can be represented as a finite union of disjoint triangles. There are serious issues concerning the formalization in first-order logic of such notions as figure or polygon that involve quantification over integers; such quantification is strictly forbidden within a first-order system. We can approach these notions with axiom schemes51 and sketch a uniform metatheoretic definition of the relevant concepts to prove that the theorems hold in all models of the axioms. Hilbert raised a pseudogap in Euclid52 by distinguishing area and content. In Hilbert, two figures have (1) equal area (are equidecomposable) if they can each be decomposed into a finite number of triangles that are pairwise congruent; (2) equal content (are equicomplementable) if we can transform one into the other by adding and subtracting congruent triangles. Hilbert showed that under the Archimedean axiom the two notions are equivalent; and that without it they are not. Euclid treats the equality of areas as a special case of his Common Notions. The properties of equal content, described next, are consequences for Euclid of the Common Notions and need no justification. We introduce the notion of area function to show they hold in all models of HP5. Fact 4.4.1 (Properties of Equal Content) The following properties of area are used in Euclid I.35 through I.38 and beyond. (1) Congruent figures have the same content. (2) The content of two ‘disjoint’ figures (i.e., meet only in a point or along an edge) is the sum of the contents of the two figures. The analogous statements hold for difference and half. (3) If one figure is properly contained in another then the area of the difference (which is also a figure) is positive. More precisely, Definition 4.4.2 (Equal content) Two figures $$P,Q$$ have equal content (are equi-complementable) in $$n$$ steps if there are figures $$P'_1 \ldots P'_n$$, $$Q'_1 \ldots Q'_n$$ such that none of the figures overlap, each pair $$P'_i$$ and $$Q'_i$$ are equi-decomposable and $$P=P'_1\cup P'_2 \ldots \cup P'_n$$ is equi-decomposable with $$Q=Q'_1\cup Q'_2 \ldots \cup Q'_n$$. Reading equal content for Euclid’s ‘equal’, Euclid’s I.35 (for parallelograms) and the derived I.38 (triangles) become Theorem 4.4.3 and in this formulation Hilbert accepts Euclid’s proof. Theorem 4.4.3 (Euclid/Hilbert) If two parallelograms (triangles) are on the same base and between parallels they have equal content in 1 step. Fig. 3. View largeDownload slide Parallelograms between the same parallels. Fig. 3. View largeDownload slide Parallelograms between the same parallels. Proof. $$P =ADCB$$ has the same content as $$P'=EFCB$$ in one step as letting $$Q=Q'=DGE$$, $$P+Q$$ and $$P'+Q'$$ which are equidecomposable by $$CBG +ABE$$ and $$CBG +CDF$$ since $$ABE \cong DCF$$. □ To show area is well-defined for figures, varying Hilbert, Hartshorne [2000, §§19–23] shows Fact 4.4.1 in the first-order axiom system EG (Notation 2.2). The key tool is: Definition 4.4.4An area function is a map $$\alpha$$ from the set of figures, $$\mathcal {P}$$, into an ordered additive abelian group with $$0$$ such that (1) For any nontrivial triangle $$T$$, $$\alpha(T) > 0$$. (2) Congruent triangles have the same content. (3) If $$P$$ and $$Q$$ are disjoint figures $$\alpha(P \cup Q) = \alpha(P) +\alpha(Q)$$. Semiformally, the idea is straightforward. Argue (a) that every $$n$$-gon is triangulated into a collection of disjoint triangles, and (b) establish an area function on such collections. The complication for formalizing is describing the area function by first-order formulas. For this,53 add to the vocabulary for geometry a $$3$$-ary function $$\alpha_1$$ from points into line segments that satisfies (1) and (2). For $$n>1$$, add a $$3n$$-ary function $$\alpha_n$$ and assert that $$\alpha_n$$ maps $$n$$-disjoint triangles into the field of line segments by summing $$\alpha_1$$ on the individual triangles; this will satisfy (3). This handles disjoint unions of triangles. Hartshorne (§§23.2–23.5) gives an inductive proof showing every polygon can be decomposed into a finite number of disjoint triangles and the area does not depend on the decomposition. It is evident that if a plane admits an area function then the conclusions of Fact 4.4.1 hold. This obviates the need for positing separately De Zolt’s axiom, (Fact 4.4.1 (3)). In particular this implies Common Notion 4 for ‘area’. As we just saw, Hilbert established the existence of an area function and thus a theory of area for any plane satisfying HP5. Now, letting $$F(ABC)$$ be an area function as in Definition 4.4.4 (** from the second page), VI.1 is interpreted as a variant of (**): $$F(ABC) =\mbox{\frac{1}{2}}\alpha \cdot AB \cdot AC.$$ But the cost is that while Euclid does not specify what we now call the proportionality constant, Hilbert must. In (**) Hilbert assigns $$\alpha$$ to be one. A short argument [Hartshorne, 2000, §23.3] shows that the formula $$A = \frac{bh}{2}$$ does not depend on the choice of the base and height. Thus, Hilbert proves (**) without recourse to the axiom of Archimedes. 5. CONCLUSION In this paper we defined the notion of a modest descriptive axiomatization to emphasize that the primary goal of an axiomatization is to distill what is ‘really going on’. One can axiomatize the first-order theory of any structure by taking as axioms all the first-order sentences true in it; such a choice makes a farce of axiomatizing. Historically, we stress one of Hilbert’s key points. Hilbert eliminated the use of the axiom of Archimedes in Euclid’s polygonal and circle geometry (except for area of circle), thus exhibiting a first-order modest descriptive axiomatization of that geometry. His proof is impure [Baldwin, 2013a] as he introduces the concept of field; but it remains modest; the axioms are in the data set. As we elaborate in Chapter 12 of [Baldwin, 2018], extensions by explicit first-order definition are (often unconsciously) standard mathematical tools that may impair purity but not provability. He [Hilbert, 1971] was finding the ‘distinguished propositions of the field of knowledge that underlie the construction of the framework of concepts’ and showing what we now call first-order axioms sufficed. The sequel (1) expounds Tarski’s system $${\mathcal E}^2$$ as a modest descriptive axiomatization of Cartesian geometry, (2) analyzes the distinctions between the completeness axioms of Dedekind and Hilbert and argues that Hilbert’s continuity axioms are overkill for strictly geometric propositions, and (3) supports conclusion (2) by providing a first-order theory to justify the formulas for circumference and area of a circle. Footnotes † I very much appreciate helpful comments by Neil Barton, Rose Cherubin, Davide Crippa, Philip Ehrlich, Eduardo Giovannini, Michael Hallett, Robin Hartshorne, William Howard, Juliette Kennedy, Barry Mazur, John Mumma, Julian Narboux, Victor Pambuccian, Marco Panza, Craig Smorynski, and Jouko Väänänen. I especially thank the referees. Research was partially supported by Simons travel grant G5402. 1Hilbert does not state this result as a theorem; I have excerpted the statement from an application on page 66 of [Hilbert, 1962]. 2In the logic, $$L_{\omega_1,\omega}$$, quantification is still over individuals but now countable conjunctions are permitted; so it is easy to formulate Archimedes’ axiom: $$\forall x,y (\bigvee_{m\in \omega} mx>y)$$. By switching the roles of $$x$$ and $$y$$ we see each is reached by a finite multiple of the other. 3Dedekind defines the notion of a cut in a linearly ordered set $$I$$ (a partition of $${\mathbb{Q}}$$ into two intervals $$(L,U)$$ with all elements of $$L$$ less than all elements of $$U$$). He postulates that each cut has unique realization, a point above all members of $$L$$ and below all members $$U$$ — it may be in either $$L$$ or $$U$$ [Dedekind, 1963, p. 20]. If either the $$L$$ contains a least upper bound or the upper interval $$U$$ contains a greatest lower bound, the cut is called ‘rational’ and no new element is introduced. Each of the other (irrational) cuts introduces a new number. It is easy to see that the unique realization requirement implies the Archimedean axiom. By Dedekind completeness of a line, I mean the Dedekind postulate holds for the linear ordering of that line. See the sequel [Baldwin, 2017]. 4Hilbert lectured on geometry several summers in the 1890s and his notes (in German) with extremely helpful introductions (in English) appear in [Hilbert, 2004]. The first Festschrift version of the Grundlagen does not contain the continuity axioms. I draw primarily on the second (Townsend) edition of [Hilbert, 1962] and on the tenth [Hilbert, 1971]. 5The first ten urls from a google search for ‘Hilbert’s axioms for Euclidean geometry’ contained eight with no clear distinction between the geometries of Hilbert and Euclid and two links to Hartshorne, who distinguishes them. 6That is, a multiplication on points rather than segments. See [Heyting, 1963]; the most thorough treatment is in [Artin, 1957]. 7There is an interesting subtlety here (perhaps analogous to the Shapiro’s algebraic and non-algebraic theories). The data set for group theory could be interpreted as (a) formal consequences of the axioms for groups in the vocabulary $$(+,0)$$, or (b) theorems proved by group theorists. For Euclid-Hilbert geometry these meanings coalesce. 8We dispute some of his points just before Remark 2.1 in the sequel. 9See the caveats on ‘second-order’ (e.g., sortal) in the sequel. 10We discuss the ‘equivalence’ of Dedekind’s and Hilbert’s formulation of completeness in the sequel. 11Often, few is interpreted as finite. Whatever Hilbert meant, we should now be satisfied with a small finite number of axioms and axiom schemes. At the beginning of the Grundlagen, Hilbert adds ‘simple, independent, and complete’. Such a list including schemes is simple. 12We considered replacing ‘modest’ by ‘precise’ or ‘safe’ or ‘adequate’. We chose ‘modest’ rather than one of the other words to stress that we want a sufficient set and one that is as necessary as possible. As the examples show, ‘necessary’ is too strong. Later work finds consequences of the original data set undreamed by the earlier mathematicians. Thus just like ‘descriptively complete’, ‘modest’ is a description, not a formal definition. 13This concept describes normal work for a mathematician. ‘I have a proof; what are the actual hypotheses so I can convert it to a theorem.’ 14The $$Q$$ is the quantifier ‘there exist uncountably many’. 15Huntington [1911] invokes Dedekind’s postulate when axiomatizing the complex field. 16In 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. 17Proposition I.1 is omitted from Euclid I, because it is not possible to construct an equilateral triangle with side length 1 in HP5. Proposition I.2 (transferring a length) is an axiom for Hilbert and of course follows from CCP. 18E.g., Veblen [1914, p. 4]. 19Manders emphasized the use of diagrams as a coherent mathematical practice. Properties that are not changed by minor variations in the diagram such as subsegment, inclusion of one figure in another, the intersection of two lines, betweenness, are termed coexact. Properties that can be changed by minor variations in the diagram, such as whether a curve is a straight line, congruence, a point is on a line, are termed exact. We can rely on reading coexact properties from the diagram. The difficulty in turning this insight into a formal deductive system is that, depending on the particular diagram drawn, after a construction the diagram may have different coexact properties. Independently, Miller [2007] and Avigad et al. [2009] formulated deductive systems whose rules are intended to capture properties that formerly were read off from diagrams. Miller treats diagrams as topological objects; this leads to difficulties bounding the number of cases. The system in [Avigad et al., 2009] incorporates metric information and provides a complete and sound deductive system for a collection of $$\forall \exists$$ sentences on models of EG; these include Euclid I–IV, but do not appear to include the theorems relating area and proportionality that are central here. 20Circle-circle intersection implies line-circle intersection. Hilbert shows that circle-circle intersection holds in a Euclidean plane, as pointed out in [Hilbert, 2004, pp. 204–206]. See Section 4.3. 21The priority for this insight is assigned to such slightly earlier authors as Pasch, Peano, Fano, in works such as [Freudenthal, 1957] as commented on in [Bos, 1993] and [Gray, 2011, ch. 24]. 22See [Baldwin, 2014] and the sequel for further explication of this method. 23The names HP, HP5, and EG come from [Hartshorne, 2000] and $${\mathcal E}^2$$ from [Tarski, 1959]. In fact, Tarski also studies EG under the name $${\mathcal E}''_2$$. 24These 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. 25In the vocabulary here, there is a natural translation of ‘Euclid’s axioms’ into first-order statements. The construction axioms have to be viewed as ‘for all – there exist’ sentences. The axiom of Archimedes is of course not first-order. We write Euclid’s axioms for those in the original as opposed to modernized (first-order) axioms for Euclidean geometry, EG. Note that EG is equivalent to (i.e., has the same models) as the system laid out in [Avigad et al., 2009]. 26A field is real closed if it is formally real ($$-1$$ is not a sum of squares) and every odd degree polynomial has a solution. 27Hilbert added his Vollstandigkeitsaxiom to the French translation of the first edition and it appears from then on. In Section 4.2 of the sequel we explore the connections between various formulations of completeness. We take Dedekind’s formulation (Footnote 3) as emblematic. 28See Theorems 10.4, 12.3–12.5 in Section 12 and Sections 20–23 of [Hartshorne, 2000]. 29In [Baldwin and Mueller, 2012] and [Baldwin, 2013b] we give an equivalent set of postulates to EG, which returns to Euclid’s construction postulates and stresses the role of Euclid’s axioms (Common Notions) in interpreting the geometric postulates. While not spelled out rigorously, our aim is to consider the diagram as part of the argument. For pedagogical reasons the system used SSS rather than SAS as the basic congruence postulate, as it more easily justifies the common-core approach to similarity through dilations and makes clear that the equality axioms in logic, as in Euclid’s Common Notions, apply to both algebra and arithmetic. This eliminates the silly 6-step arguments in high-school texts reducing subtraction of segments to the axioms of the real numbers. 30As quoted in [Mathias, 1992]. 31Smorynski [2008] notes that Bradwardine already reported five in the fourteenth century. Reeder [2017] analyzes five different philosophical approaches to the notion of ‘infinite divisibility’. 32That is, any system of points and lines such that two points determine a line, any two lines intersect in a point, and there are 4 non-collinear points. 33More precisely, for the Greeks, natural numbers greater than 1. 34The Greeks accepted only potential infinity. From a modern perspective, the natural numbers are ordered in order type $$\omega$$, and any collection of homogeneous magnitudes (e.g., areas) are in a dense linear order (which is necessarily infinite); however, this imposition of completed infinity is not the understanding of the Greeks. 35Homogeneous pairs means magnitudes of the same type. Ratios of numbers are described in Book VII while ratios of magnitudes are discussed in Book V. 36This quotation is taken from [Franks, 2014]. 37Euclid’s development of the theory of proportion and area requires the Archimedean axioms. Our assertion is one of many descriptions of the exact form and location of the dependence among such authors as [Euclid, 1956; Mueller, 2006; Stein, 1990; Fowler, 1979; Smorynski, 2008]. Since our use of Euclid is as a source of sentences, not proofs, the details are not essential to our argument. 38He refers to the construction of the fourth proportional (‘ce qui est même que la multiplication’ [Descartes, 1954]. See also [Bos, 2001, §21, p. 296]. 39In a semi-field there is no requirement of an additive inverse. 40Hilbert’s definition goes directly via similar triangles. The clear association of a particular angle with right multiplication by $$a$$ recommends Hartshorne’s version. 41See [Baldwin and Mueller, 2012] or [Hartshorne, 2000, p. 170], which explain the background cyclic quadrilateral theorem. That theorem asserts: Let $$ACED$$ be a quadrilateral. The vertices of $$ACED$$ lie on a circle (the ordering of the name of the quadrilateral implies $$A$$ and $$E$$ are on opposite sides of $$CD$$) if and only if $$\angle EAC \cong \angle CDE$$. Hilbert uses Pascal’s theorem, a relative of Desargues’ theorem. 42One has to verify that segment multiplication is continuous but this follows from the density of the order since the addition respects order. 43The automorphism group of the plane acts 2-transitively on the plane (any pair of distinct points can be mapped by an automorphism to any other such pair); this can be proven in HP5. This transitivity implies that a sentence $$\phi(0,1)$$ holds just if either or both of $$\forall x \forall y \phi(x,y)$$ and $$\exists x \exists y \phi(x,y)$$ hold. 44Hilbert had done this in lecture notes in 1894 Hilbert [2004]. Hartshorne constructs the field algebraically from the semi-field rather than in the geometry. 45Instead, there are infinitely many formulas $$\phi_n(x,y)$$ defining unary operations $$nx = y$$ for each $$n> 0$$. 46Dedekind [1963, p. 9] objects to the introduction of irrational numbers by measuring an extensive magnitude in terms of another of the same kind. 47However, Hilbert does not avoid the parallel postulate since he uses it to establish multiplication and thus similarity. Note also that Euclid’s theory of area depends heavily on the parallel postulate. It is a theorem in ‘neutral geometry’ in the metric tradition that the Pythagorean theorem is equivalent to the parallel postulate (See Theorem 9.2.8 of [Millman and Parker, 1981].) But this approach basically assumes the issues dealt with in these papers; as, the ‘ruler postulate’ (Birkhoff’s Postulate III in the sequel) also provides a multiplication on the ‘lengths’ (since they are real numbers). Julien Narboux pointed out the issues in stating the Pythagorean theorem in the absence of the parallel postulate. 48See [Hartshorne, 2000, Exercises 39.30, 39.31]. This was known to Hilbert [Hilbert, 2004, pp. 201–202]. 49Hartshorne and Greenberg [2010, p. 146] call this the constructible field, but given the many meanings of constructible, we use Moise’s term: surd field. 50Heron’s proof invokes a peculiar squaring of areas. But Taisbak [2014] argues that this operation can be replaced by inferences from VI.1 or X.53. 51In order to justify the application of the completeness theorem we have to produce a scheme giving the definition of an $$n$$-decomposable figure as the disjoint union of an $$(n-1)$$-decomposable figure $$A$$ with an appropriately placed triangle. 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