The second half of the nineteenth and the beginning of the twentieth centuries brought about a dramatic change in our understanding of the nature of geometry. Until then, Kant’s theories of geometry had dominated a significant portion, if not all, of the philosophical scene. Kant believed that the statements of geometry were synthetic a priori truths, grounded in pure spatial intuition. Two discoveries in geometry shattered this view, however. The non-Euclidean geometries of Bolyai and Lobatchevsky raised doubts as to the a priori validity of Euclidean geometry, while the construction of geometry as a purely formal deductive system that culminated in Hilbert’s work Grundlagen der Geometrie showed that appeal to spatial intuition was not necessary in order to derive geometrical propositions.1  Drawing the consequences from these developments, Einstein formulated a decidedly anti-Kantian point of view on the nature of geometry, one that was defended by the logical positivists and became a standard view.2  He believed that we had to distinguish between pure (formal) and applied (physical) geometry. Pure geometry was a branch of pure mathematics and as such a purely formal deductive system; the terms occurring in its sentences did not refer to either real or ideal objects.3  As a consequence, pure geometry did not appeal to intuition; its sentences were true a priori simply because they followed from a consistent system of axioms. Intuition entered into the picture only when such geometrical constructs were applied to reality, when their terms were interpreted with concepts referring to real elements like light-rays or pencil-lines. The truths of the resulting geometry, that is, of applied geometry, became empirical statements. Here, too, there was no need to appeal to anything like pure intuition.

Rudolf Carnap also rejected Kant’s idea of a priori synthetic judgments.4  Yet, in his first work, his doctoral dissertation Der Raum from 1922, he pursued a project that was strongly influenced by the neo-Kantianism of the Marburg school and by Husserl’s mature philosophy of science.5  In this work, Carnap attempted to reconcile the new results of geometric research with Kant’s idea of spatial intuition as an a priori source of geometric knowledge. Like Husserl, he believed that this could be done by assuming in addition to pure and applied geometry a third type of geometry. In order to realize this project, Carnap adopted Husserl’s general typology of theoretical sciences that the latter had developed most fully in Ideen I (1913).6  Husserl had categorized these sciences according to their degree of abstraction and the corresponding epistemic status of their knowledge claims. The most general type was characterized by the category "formal ontology" (formale Ontologie). The sciences falling under it provided a general formal structure and yielded analytic a priori knowledge. The second type of science was concerned with ideal objects and their relations to each other. These sciences fell under the category "material ontology" (materiale Ontologie) and yielded synthetic a priori knowledge. Finally, the "sciences of fact" (Tatsachenwissenschaften) were concerned with actual objects of experience and yielded empirical knowledge.7  Carnap changed the terminology slightly, but also distinguished between three types of geometry, namely formal, intuitive, and physical, and asserted that they correlated with Husserl’s three types of science.8  In particular, intuitive geometry was a synthetic a priori science or, as Husserl would say, a "material geometry."

I believe that it is difficult to escape the appeal of Kant’s idea of a geometry whose propositions are neither purely formal consequences following from an arbitrarily accepted system of axioms, nor empirical truths in the sense of applied geometry and to simply dismiss Carnap’s and Husserl’s suggestions of a material geometry. My goal in this paper is to formulate a coherent notion of such a geometry. I will first criticize Carnap’s account of a material geometry in his doctoral dissertation and show that it fails for two reasons: intuition is restricted to a limited region of space and material geometry is constructed as an axiomatic system in the contemporary sense.9  In the second part of this paper, I will then show that Husserl had earlier provided an account of spatial intuition that allows us to avoid Carnap’s first problem. In the last part of this paper, I will suggest a non-standard interpretation of Euclid’s method in the Elements that circumvents Carnap’s second problem.10 

Within the context of the standard view of geometry and its clear distinction between pure (formal) and applied (physical) geometry, the main challenge for a philosopher who wants to do justice to the intuitions underlying Kant’s idea is to formulate a notion of a material geometry that is truly distinct and does not collapse into either of these two geometries. In his doctoral dissertation Der Raum, Carnap establishes a distinction between the three types of geometry by ascribing to them separate subject matters that function as different sources of spatial knowledge. Formal space is the subject matter of formal geometry, physical space the subject matter of applied geometry, and intuitive space the subject matter of material geometry.11  Accordingly, in order to assess his suggestion, we have to understand these three concepts of space and the way in which they function as sources of spatial knowledge.12 

If we understand a general structure of relations not as one consisting of relations between [individually] determined objects of a sensible or non-sensible domain, but rather between completely undetermined relational terms of which we know only that from a connection of a certain kind we may infer a connection of another kind, then formal space is a general system of relations of a special kind.13 

 

Whereas geometric relations are commonly said to hold between specifically geometric objects like points, lines, and planes, the relations of formal geometry contain only empty spots, or place-holders, as their relata. They are considered independently of any actual or ideal subject matter. Such a formal system of relations defines a structure that can be filled or interpreted with any isomorphic system of objects. In order to illustrate his exposition of the notion of formal space, Carnap points out that this concept of a formal geometry was first worked out by Hilbert in his Grundlagen der Geometrie in 1899.14  In this work, Hilbert established a system of axioms that implicitly defined various geometric relations such as ‘to lie on’ and ‘between,’ independently of any particular system of geometric objects. The set of axioms was complete in such a way that Hilbert could deduce from it all the propositions of Euclidean geometry by purely logical means. But in contrast to Hilbert, Carnap did not restrict his concept of formal space only to Euclidean geometry. Rather, he included a multiplicity of spaces in this category by distinguishing between formal spaces with different numbers of dimension on the one hand and between topological, projective, and metric formal spaces on the other. In effect, he recognized an infinite number of formal spaces. In order to do justice to this multiplicity of constructible spaces, Carnap diverted from Hilbert’s method in the Grundlagen and took the more general approach of a theory of continuous series of more that one dimension as suggested by Russell in his The Principles of Mathematics.15  Carnap believed that only this method was general enough to provide an adequate tool for constructing all the different types of formal space as purely logical systems of relations.

Those matters of fact, as, for example, the observation that the edge of this body stands in a certain spatial relation to a particular edge of another body, constitute the system of relations of physical space.16 

 

We understand intuitive space as the structure of the relations between ‘spatial’ forms understood in the usual sense, that is, the line-, surface-, and space-elements whose determinate peculiarities we apprehend on the occasion of perception or also in mere imagination. These peculiarities do not yet concern spatial facts present in empirical reality. Rather, they concern only the ‘essence’ of these objects, which can be recognized in any of their representatives.17 

 

We can understand Carnap’s characterization of intuitive space better if we unpack the reference to Husserl’s notion of Wesensschau, or eidetic seeing, implicit in this quote.18  Carnap believes that Wesensschau is a capacity of the human mind that allows a geometer to apprehend the essences of empirical spatial objects, like points, lines, and surfaces, by observing or imagining individual instances of them. For Husserl, the essences given in acts of Wesensschau do not merely represent general features of empirical reality. Rather, they are idealities in the sense of ideal objects.19  That Carnap agreed with this Husserlian view of the nature of essences becomes apparent in a later part of his doctoral dissertation where he states that the objects of intuitive space are apprehended as specifically spatial objects but not as objects of actual empirical intuition.20  The only remaining option is that they are ideal objects. Intuitive space is then a system of relations between ideal points, lines, planes, etc. that represents the essential form of the space of actual experience. In other words, it is an idealization constituted in acts of Wesensschau.

But Carnap believes that Wesensschau alone is insufficient to constitute intuitive space as a whole, i.e., as a total system of relations (Gesamtgefüge). The reason for this is that empirical intuition allows us to "derive knowledge only about spatial objects of a limited size."21  Since the idealizations of Wesensschau always depart from perceived or imagined empirical objects, they can never reach beyond a limited region of space. This leads Carnap to constitute intuitive space in two steps. He departs from the axiomatic system of geometry presented in Hilbert’s Grundlagen der Geometrie and shows which of its axioms are valid within an intuitable region of space. Afterwards, he establishes a number of principles or demands (Forderungen) that allow him to extend these axioms to a system that comprises a totality of spatial relations. I will not explicate the details of Carnap’s analysis, but rather simply state the results. Carnap finds that Hilbert’s axioms of Euclidean geometry can be verified by intuition only in small areas, or more precisely, only in infinitesimally small areas.22  Intuitive space as a whole is thus characterized by what Riemann had called "flatness in the smallest areas."23  Riemann has shown that this type of space leaves open the particular curvature values of each point. Accordingly, Carnap’s concept of intuitive space so far is more general than Kant’s and contains Euclidean space as a special case.

Carnap broadens his concept of intuitive space even further. He believes that the concept of three-dimensional metric space can be generalized in various ways and that the resulting spaces also deserve to be called intuitive. Further dimensions can be added to three-dimensional space, thus constituting n-dimensional metrical spaces. One can alternatively abstract from the congruence of lines and angles and consider only the system consisting of the basic concepts "point," "line," and "plane," thus constituting a three-dimensional projective space. One can further abstract from the geometric concepts "straight line" and "Euclidean plane" and consider only the relations between lines and surfaces in general, thus constituting a three-dimensional topological space. Finally, one can lift the restriction of three dimensions in projective and topological spaces, thus constituting n-dimensional projective and topological spaces.24  An n-dimensional topological space is the most general structure common to this infinite multiplicity of spaces. Carnap believes that this structure represents the general condition for the perception of the matters of fact and, therefore, calls it the a priori form of spatial intuition.25 

The relation of R [formal space] to R’ [intuitive space] is that of a species of systems with specific properties of order but undetermined objects to a system with the same properties but with determined objects, namely intuitable spatial objects. The relation of R’ [intuitive space] to R’’ [physical space] is that of a form of intuition to a system of this form containing objects of actual experience.26 

 

Accordingly, the three types of space are distinct because they contain objects belonging to different ontological realms. Formal space is a purely formal species; its objects are entirely undetermined with respect to their non-formal properties. Intuitive space is a system of specifically spatial, but ideal, objects – ideal points, lines, etc. Physical space is a system of relations between objects of actual experience. Given this, formal, intuitive, and physical space function as sources of spatial knowledge in the following way: The epistemic function of formal space is exhausted by the demand that it can be defined as a continuous series of more than one dimension. The construction of such a purely formal-logical system is not restricted by any external constraints. In this sense, it is independent from experience and thus a priori. Since its theorems are derived from purely logical laws, Carnap calls them "analytic." The properties of the ideal objects of intuitive space, on the other hand, are intuitively accessible through Wesensschau and, therefore, provide a source of synthetic knowledge. This knowledge is a priori because the Wesensschau departs from the observation or imagination of one instance and does not need further empirical input.27  Finally, the structure of physical space must be discovered through observation of actual sensible objects and thus gives empirical knowledge.

In order to evaluate the coherence of Carnap’s suggestion for a material geometry, we have to ask whether intuitive space as he defines it can in principle serve as a source of a priori synthetic knowledge. In his dissertation, Carnap claims that this is the case because the different types of intuitive space retain an intimate connection to the intuition of a limited spatial region. He specifies four different relations between a limited region of space and the various types of intuitive space: ‘whole-part’, ‘being-contained-in’, ‘being-built-up-of’, and ‘general-particular.’ Three-dimensional Riemannian space is intuitive because it contains intuitively accessible regions as its parts. N-dimensional metric space is intuitive because "all intuitable objects that are known to be part of three-dimensional metrical space [are also contained] in an n-dimensional metric space; and . . . all these higher level objects are built up of parts given in intuition."28  Finally, projective and topological spaces deserve to be called intuitive, because they relate to other intuitive spaces as a general to a particular.29  Of these four relations, the whole-part relation is the crucial one, because the extension of directly intuitable space to the full space depends on it and the latter serves as the starting point for the further generalizations. If the whole-part relation cannot account for the intuitive nature of the different types of three-dimensional Riemannian space, then the other relations will also be unable to do this for their respective spaces. Since my goal is to criticize Carnap’s notion of intuitive space, it is thus sufficient to consider the whole-part relation alone and to ask whether intuitive space as constituted by Carnap really is a composite of idealizations of intuitively accessible parts.

Intuitive space is an order structure of which we can certainly circumscribe conceptually its formal type, but, like everything intuitable, not its particular properties. Here we can only point to contents of experience, namely to intuitive-spatial objects and relations: points, line-segments, plane-elements, volume-elements, the lying of a point on a line or in a volume, the intersection of two lines, etc.30 

 

We can illustrate the fact that the objects of the intuitively accessible region belong to a different category than those in the space constructed through Carnap’s extension by means of the following example. Carnap believed that Hilbert’s axiom I,1 was verifiable through Wesensschau. The axiom states that "for any two points there exists (at least) one line that contains each of them."31  Wesensschau allows the geometer to verify axiom I,1 because for any two intuited points he/she can also intuit a line on which they both lie. One of the principles by which Carnap extends his axioms to a total system demands that axiom I,1 should also hold for space as a whole. Since intuition is always restricted to a limited region of space, there exist points in the total space that are further apart than the limits of any such region. For any two points of this kind, the geometer can no longer intuit a line which contains both of them. He/she therefore cannot verify axiom I,1. Carnap’s demand simply stipulates that such a line exist and that the axiom be true. As a result, the points and lines are characterized only by the relational properties defined through the axioms. Whereas points and lines were intuitable objects before the extension, after it they are purely formal objects. On the basis of this criticism, we can state a first condition for a coherent notion of a material geometry: it must be based on a concept of spatial intuition as a capacity that grants human beings access to the structure of space as a whole.

The question now is whether this view allows us to recognize an essential difference between Carnap’s material geometry and the formal system constructed from it. The way in which the former is said to differ from the latter is that the geometer understands its terms as speaking about certain objects given through Wesensschau. This, however, is irrelevant to the resulting structure. As we have seen in the previous paragraph, material and formal systems are constructed from axioms according to the same principles, i.e., according to the principles of logical deduction, and thus have the same structure. Accordingly, the difference between the conceptual structure represented by Carnap’s material geometry and an isomorphic formal axiomatic system lies only in the geometer’s intention of "seeing" the former as being about one particular intuitable reality. But this does not amount to an essential difference between the two, because the formal system also represents perfectly well the logical structure of this particular reality. Consequently, we have to say that the material geometry suggested by Carnap is a formal axiomatic system which is interpreted by certain ideal objects accessible through Wesensschau.32  Thus, the assumption that material geometry is an axiomatic system in the contemporary sense also leads to its collapse into formal geometry.33  We can now formulate a second condition for a coherent notion of a material geometry: it must not be understood as an axiomatic system in which the propositions are derived from the axioms by purely formal deductions.

By representation of space, we may first mean the space of intuition, that is, the space of extra-scientific experience, the space which everyone, children or adults, scholars or laymen, are able to experience in lived perception and fantasy.34 

 

Husserl’s term ‘intuitive space’ here refers to what Carnap had called an intuitively accessible limited region of space, i.e., to perceptual space.35  In his lecture course Ding und Raum from 1907, Husserl describes how perceptual space is experienced:

One actual bodily objectivity is seen, but it leaves open infinitely many possibilities for further objectivities in the "between." The "between" however is constituted because discrete extensions, no matter what they are, can be mediated by continuous extensions in different ways and finally continuously. Although we cannot say that empty space is seen, we have the between as an empty space, that can be filled continuously, as a mere possibility of actual mediations that are characterized by laws. We can see only bodies and with them we see the between. Space is rather implied in actual perception.36 

 

Since perceptual space is founded in the experience of concrete spatial objects, Husserl first tries to understand how the latter are given in experience or, as he says, "constituted." The central notion is what one could call a "perspectival system." Let us assume a situation in which an observer perceives a given spatially extended visual object from a fixed point in space. In this case, the object is seen only from one side. As Husserl would say, it is given only by means of a certain fixed adumbrated representation or appearing side (Abschattung). Since the object is experienced as a spatial object, however, this perception of one side entails a system of other possible sides which the observer would see, if he/she moved in certain ways. So, for example, if an observer perceives a car from the front, he/she will expect that successively different points of observation will give successively different pictures or sides, as Husserl calls them, like the car’s doors, then its back, etc. Being only in one single position at a time, the observer will not have a fully specified idea of how these sides will actually look. Nevertheless, they will be the doors and the back of a car, of this particular car. This means that there is a law-like connection, not between concrete changes of the observer’s position and concrete changes of the adumbrated representation, but rather between certain types of observer movements and certain types of adumbrational changes. According to Husserl, these types are specified a priori by the essence of the given spatial object. A spatial object is thus constituted as the correlate of a system of rule-governed typical perspectival changes. One perspective is actually adumbrated and the others – an infinite manifold – are given as expectations. In this sense, a perceptual spatial object is experienced as an ideality.37  Perceptual space as the system of possible places for these objects is then constituted as a generalized perspectival system comprising all possible concrete perspectival systems.38 

Husserl argues that we can not specify a priori the particular law that governs the connection between the two types of change explicated in the previous paragraph; but he believes that we can show that this law has to fulfill a general condition. The most important presupposition of his argument is the fact that human beings experience rigid, or, as Husserl says, "thing-like" objects. The constitution of such objects, presupposes that an observer make a clear distinction between merely perspectival changes on the one hand and physical changes on the other.39  An observer can do so only on the basis of experiences of exclusively perspectival changes, which can be of two types. First, an observer experiences purely perspectival changes if the modifications in the appearance (the changes in the adumbrated representation) of an object which is moved are completely reversed when it returns to its point of departure. Second, an observer experiences purely perspectival changes if the modifications in the appearance of an object that are caused by the observer’s movements are completely reversed when he/she returns to his/her point of departure. The former presupposes that an object can freely move around in space and the latter presupposes that this is possible for the observer him/herself. By appealing to the results of Helmholz’s and Lie’s mathematical analysis of free mobility, Husserl is able to derive a following general condition for the law governing the two types of change:40 

The forms of this law-like connection [that is, the laws that underlie the adumbrational changes] are bounded by the further requirement that this objectivity has to be thing-like, that is, has to be such that the manifolds contained in it exhibit fixed relations, which leave open the possibility of movement and change. But identity in motion requires a continuum of places, one that is congruent in itself.41 

 

The fact that perceptual space is founded in the experience of individual spatial objects seems to imply that the perspectival structure applies only to one part of this space, namely, to the part that falls within the reach of the visual system. Indeed, this seems to have been Husserl’s view.42  Yet, his analysis of the constitution of space commits him to the opposite point of view. Husserl’s description of the perception of a spatial object showed that the latter includes expectations concerning the perspectival changes that would follow upon certain changes in the observer’s position. Thus, spatial perception includes an implicit knowledge that I want to call "spatial intuition." Since, as Husserl argued, the expectations determine space as a structure with constant curvature, spatial intuition comprises space as a whole. As a result, an individual spatial object and a limited region of space can only be experienced as parts of a total space.43  This fulfills the first condition derived in the previous section. The geometer does not need to introduce a stipulation extending the visually accessible facts to a system comprising space as a totality.

The second condition that I derived from Carnap’s account of a material geometry requires that it not be understood as an axiomatic system in which the propositions are derived by purely formal deductions. In other words, material geometry must be based on inferences which can explicate and carry non-formal content. In order to fulfill this condition, I want to suggest that we understand material geometry as a practice that allows a geometer to explore the perspectival structure of perceptual space by means of certain constructed spatial objects, namely geometric diagrams. I believe that this is accomplished by Euclid’s method as exhibited in the Elements.44  Most contemporary textbooks on Euclidean geometry understand it as a precursor of axiomatic geometry. As such, it is said to contain some confusions typical for such a pioneering project, as, for example, hidden assumptions or the reliance on the method of superposition.45  Against this view, Ian Mueller has argued that there is a significant difference between Euclid’s method and modern axiomatic geometry as presented by Hilbert – a difference that is not adequately characterized by saying that the former is simply a precursor of the latter. More specifically, Mueller writes:

For Hilbert geometric axioms are characterized by an existent system of points, straight lines, etc. At no time in the Grundlagen is an object brought into existence, constructed. Rather its existence is inferred from the axioms. In general Euclid produces, or imagines produced, the objects he needs for a proof.... It seems fair to say then that in the geometry of the Elements there is no underlying system of points, straight lines, etc. which Euclid attempts to characterize. Rather, geometric objects are treated as isolated entities about which one reasons by bringing other entities into existence and into relation with the original objects and one another.46 

 

According to Mueller, a Euclidean proof is based on a visual object that is constructed according to certain admissible procedures which are stated in Euclid’s first three postulates.47  If this reading of Euclid is correct, we can say that the constructive system defined by his postulates allows him to explore the structure of perceptual space, because the diagrams are nothing other than objects in it. Moreover, Euclidean geometry thus interpreted would retain an intimate connection to spatial perception and would in this way differ from formal geometry.

Knowledge of the propositions derived by Euclid’s method is absolutely certain because the latter restricts the reader’s understanding of a given diagram in such a way that it is uniquely fixed and allows him/her to derive a proposition from it by purely logical means. I believe that this first point has been sufficiently confirmed by the long history of Euclid’s Elements. Questions about the particular understanding of the diagrams rarely arise, and when they do, they are easily answered. Thus, the text itself provides its readers with adequate clues for interpreting the diagrams in the right way. Mueller suggested that this is mainly done by Euclid’s definitions.48  I believe that they accomplish this by focusing the reader’s interest exclusively on readily recognizable qualitative features of a given diagram. This happens in two ways. On the one hand, the definitions pick out certain visual features of a diagram like lines and points, thus characterizing them as specifically geometric objects. On the other hand, the definitions show how those visual aspects of a given diagram that serve as a source of spatial knowledge are to be idealized. In the context of my suggestion, the main problem here is to explain how the definitions characterize an ideal geometric object in purely qualitative terms. If this was not the case, the properties of the geometric object would have to be approximated as an object of physics is, and geometry would end up becoming an empirical science. In order to explain how Euclid’s definitions function, I will consider one example.49  Euclid’s first definition states that a "point is that which has no parts." This is often interpreted as defining a point as a limit-object, i.e., an ideal object that results from a process of continually diminishing some visual object or property.50  Mueller’s argument allows us to view this definition in a different way. A point is part of an already drawn and lettered diagram – it is given through a letter which either determines, more or less precisely, a certain place on a visual line or signifies a given intersection of different lines. Given as a feature of a diagram, a point is ambiguous with respect to its size, however. Without further information, the geometer would not know how many lines could be drawn through the same point in the same direction, for example. Thus, in order to be able to generate unique results, the geometer requires additional information that disambiguates the relation between points and lines.

In order to show the specific way in which Euclid’s first definition achieves this, I want to consider two alternative definitions of a point. Heron writes that "a point is that which has no parts or an extremity without extension, or the extremity of a line."51  His first characterization of a point coincides with Euclid’s. Yet, Heron seems to consider it as equivalent to the second and third definitions. Accordingly, he believes that Euclid disambiguated the notion of a point by ascribing to it zero extension. Aristotle, on the other hand, defines a point as an indivisible magnitude which has position. Thus, whereas Heron’s definition appeals to the concept of extension, Aristotle’s appeals to the process of division. Euclid’s definition, in contrast, avoids any references to measurable notions and processes by appealing to a readily recognizable figurative feature of a visual point, namely the fact that it contains parts. It specifies criteria for the correct interpretation of a point (given by a lettered diagram) by requiring the geometer to abstract away from a certain visual feature. This does not involve approximation to a limit or measurement and appeals only to observable qualitative features.52 

We can give further evidence for this reading of Euclid’s definition of a point. First, Euclid formulated this definition in a purely negative way. This was already pointed out in antiquity by Proclus, for example, who believed that this showed that Euclid did not want to define ideal objects and was not a Platonist. If Euclid or whoever introduced the definitions into his text wanted to specify a point as an ideal object one would expect that he would have specified either the property that belonged essentially to this ideal object or the process according to which one could form the idea of such an object. If the goal of the definitions was to present criteria that disambiguated a given visual object, however, they would have had to have been formulated negatively. Second, Euclid seems to have thought of the part-whole relation as one that was readily recognizable in a given diagram. This can be seen from the way in which he applied Common Notion 5, which states that "the whole is greater than the part."53  This notion establishes a relationship between the two relations ‘greater than’ and ‘being-part-of.’ From the way in which Euclid applies this common notion, we can see that he considered the ‘part-whole’ relation as primary and that he also thought that it could be recognized immediately in a given diagram. The same must then hold of a particular point: it is immediately recognized as an object containing parts. It is this feature that has to be disregarded in the actual proof. The remaining definitions function similarly to the first and also appeal to only qualitative features of the diagram.54 

valid deductive inference is often described as the extraction or making explicit of information that is only implicit in information already obtained. Modern logic builds on this intuition by modeling inference as a relation between sentences of a formal language like the first-order predicate calculus. In particular, it views deductive proofs as structures built out of such sentences by means of certain predetermined formal rules. But of course language is just one of many forms in which information can be couched. Visual images, whether in the form of geometrical diagrams, maps, graphs, or visual scenes of real-world situations, are other forms.55 

 

Suppose you are a tourist in San Francisco’s Chinatown, and a motorist stops and asks how to get to China Basin. You take out your map, find both Chinatown and China Basin, and tell him what route he could take.56 

Here is a clear sense in which you have engaged in a valid piece of deductive reasoning, one whose assumptions consist in part on the information provided by the map and whose conclusion consists in the claim that a certain route will take the motorist to China Basin.57 

 

Let me now turn to the second point, the requirement that the proofs yield general results, despite the fact that they start from one individual object. Mueller himself believes that the Greeks never solved the problem of mathematical generality and that Euclid’s proofs simply do not show that the propositions are true of all the objects they speak about.58  More recently, however, Reviel Netz has argued that there is a way of interpreting Euclid’s method that saves the generality of its proofs. Netz points out that the first interesting lesson about generality can be drawn from proofs like that of Proposition III.1, which construes the center of a circle.59  Euclid gives the appropriate construction and then proves that the center can not be any other point than the point F, i.e., the one given by this procedure. In order to show this, he assumes that another point G be the center and demonstrates that this leads to a contradiction. After showing this for a given point G, Euclid concludes: "Similarly we can prove that neither is any other point [the center of the circle] except F." Accordingly, Euclid secures the validity of this proposition by excluding any other point within the circle from being its center. He does this simply by saying that an analogous argument can be given for any other point (except for point F). This is somewhat curious, because Euclid appeals here to a potentially infinite number of proofs. Netz concludes from this that the generality relevant to the Elements is likely to be generality with respect to provability, rather than with respect to the geometric results: a proof is universally valid, if it can be given for any adequate diagram. Generality thus is not truth about a domain of objects, but repeatability of proofs.

[apodeixis (proof)] Then, since DA is equal to AC,

the angle ADC is also equal to the angle ACD;

therefore the angle BCD is greater that the angle ADC.

And since DCB is a triangle having the angle BCD greater than the angle BDC,

and the greater angle is subtended by the greater side, therefore

DB is greater than BC.

But DA is equal to AC;

therefore BA, AC, are greater than BC.

Similarly we can prove that AB, BC are also greater than CA, and BC, CA that AB.

[sumperasma (conclusion)] Therefore, in any triangle two sides taken together in any manner are greater than the remaining one. Q.E.D.60 

 

The proof consists of an enunciation or general statement that makes a conditional claim whose antecedent describes a constructed geometric situation and whose consequent states something that is supposed to follow from this situation. Netz represents this general conditional as C(x) P(x). This is followed by the setting-out, which states a particular situation, symbolized as C(a). Given this particular situation, Euclid states the definition-of-goal (diorismos): P(a). The construction then extends the given situation C(a) in such a way that the apodeixis can derive the particular conclusion P(a). The achievements of the construction and the apodeixis together can be symbolized as C(b), . . . , C(n), P(b), . . . , P(n). Finally, the conclusion repeats the enunciation C(x) P(x). Given this representation of the structural elements of a Euclidean proof, Netz states his theory of generality as follows: Construction and apodeixis prove a particular case, namely P(a). As such they prove the definition-of-goal (diorismos), rather than the conclusion (sumperasma).61  Now, setting-out (ekthesis) and definition of goal (diorismos) together are able to support the general claim C(x) P(x). This follows from the fact that construction and apodeixis do not only show that P(a) follows from C(a), but also the provability of P(a) from C(a). In other words, by giving a proof for a particular geometric object, it is shown that "the same proof must be repeatable for any other object as long as the same ekthesis applies to that objects."62  This repeatability then shows the generality of the claim C(x) P(x).

Even if my interpretation of Euclid’s method is correct and its results are uniquely determined, material geometry thus understood could still collapse into applied geometry. If human beings could constitute intuitive space in a different way, then material geometry as the science that explicates its structure would be empirical. But intuition specifies the structure of intuitive space as a whole. If human beings have a representation of space, i.e., if they clearly distinguish between spatial change and other types of change, space must necessarily include a constant curvature.63  Accordingly, Euclid’s geometry is a priori at least in one sense, namely in that it is a special case of the more general Riemannian space with constant curvature.

In this paper I argued that Carnap’s notion of a material geometry was internally inconsistent and collapsed into pure (formal) geometry. The reason for this was that he constructed it as an axiomatic system in the contemporary sense and based it on a notion of spatial intuition that was restricted to a limited region of space. Shortly after the completion of his doctoral dissertation, Carnap gave up the idea of a material geometry and adopted the standard view.64  It is likely that he changed his mind, because he realized the problems with his own early account. In sections three and four, I showed that we can define both the notion of spatial intuition and the concept of a material geometry in a way that prevents its collapse into pure or applied geometry. I therefore conclude that Carnap gave up the idea of a material geometry too soon and that it can be saved if we clarify both the concept of spatial intuition and Euclid’s method in the Elements along the lines suggested.

  1. David Hilbert, "Grundlagen der Geometrie," 1st. ed., published in Festschrift zur Feier der Enthüllung des Gauß-Weber-Denkmals in Göttingen, Leipzig: Teubner Verlag, 1899, pp. 3-92.

  4. A summary of his view on geometry can be found in Rudolf Carnap, An Introduction to the Philosophy of Science, ed. Martin Gardner, New York: Dover Publications, Inc., 1995, Part III, pp. 125-183. This text is based on a lecture course given by Carnap in 1958 at the University of California at Los Angeles.

  18. Carnap makes this implicit reference to Husserl’s notion of Wesensschau fully explicit later on in his dissertation. Ibid., p. 22.

  34. "Unter Raumvorstellung kann fürs erste gemeint sein der Raum der Anschauung, ich meine den Raum des außerwissenschaftlichen Bewußtseins, den Raum, wie ihn alle, ob Kinder oder Erwachsene, ob Gelehrte oder Laien, in lebendiger Wahrnehmung oder Phantasie vorfinden." Edmund Husserl, "Mehrfache Bedeutung des Terminus Raum," p. 271.

  44. Heath, Euclid: The Thirteen Books of The Elements, op. cit.

  56. Jon Barwise and John Etchemendy, "Visual Information and Valid Reasoning," p. 162.