A Quasianalytical Constitution of Physical Space

Thomas Mormann, Donostia-San Sebastián, Spain

1.	Attacks on Carnap's Constitutional Theory 
2.	The Geometric Background of the Aufbau
3.	Quasianalysis and Synthetic Geometry
4.	The Affine Plane as a Similarity Structure
5.	A Quasianalytical Constitution of Physical Space
6.	Concluding Remarks

1.	Attacks on Carnap's Constitutional Theory.

Carnap's quasianalytical constitution theory of Aufbau has been subjected to many criticisms on quite different levels. Let us mention just three: (i) Goodman attacked the roots of Carnap's account claiming that the constitutional method of the Aufbau is doomed to fail from its very beginning, since the constitution of qualities from elementary experiences is fatally flawed (cf. Goodman 1951); (ii) Quine attacked the constitutional theory of the Aufbau on on an intermediate level contending that, when it came to the constitution of space-time, Carnap was not able to constitute spacetime. Instead, Quine objected, Carnap changed the method of constitution without clearly announcing it, introducing a new undefined connective "is at" (cf. Quine 1951); (iii) recently Friedman contended  that even if the quasianalytical method of  constitutions properly worked, it would not deliver what Carnap expected from it, to wit, a complete structuralization of empirical knowledge (cf. Friedman 1999). 
I think, there are good reasons to  assume that Goodman's criticisms can be defused  (cf. Proust 1989, Mormann 2002). Thus, I will say nothing about Goodman’s criticism nor I will treat Friedman's objections dealing with difficulties concerning Carnap's notion of foundedness (cf. Aufbau  §154 - §155, Friedman 1999). In this paper I only want to deal with Quine's criticism concerning the constitution of spacetime. Quine maintained that the Aufbau account of the constitution of the physical world is principally flawed: 

"Statements of the form "Quality q is at point-instant x;y;z;t" were, according to [Carnap's] canons, to be apportioned truth values in such a way as to maximize and minimize certain over-all features, and with growth of experience the truth values were to be progressively revised in the same spirit. I think this is a good schematization ... of what science really does; but it provides no indication, not even the sketchiest, of how a statement of the form "Quality q is at x;y,z;t" could ever be translated into Carnap's initial language of sense data and logic. The connective "is at" remains an added undefined connective; the canons counsel us in its use but not in its elimination" (Quine 1951, p. 40). 

In this paper I'd like to provide such a sketch. Admittedly, something like this cannot be found in Carnap's original account, at  least not explicitly. Nevertheless, I claim that my constitution of the "is at"-connective is fully in line with the spirit of Carnap's approach. That is to say, Carnap could have constituted physical space by quasianalytical methods alone. He wasn't forced to fall back on other, conventionalist constitutional methods.
Although Quine's empiricist interpretation of the Aufbau has recently been critized from many quarters, his thesis of the alleged break in the methodology of the Aufbau and the resulting unreducibility of the coordinating connective "is at" has remained unchallenged up to now (cf. Richardson 1998). Against this common wisdom, I'd like to show how the notorious connective "is at"  may indeed be eliminated. The outline of this paper is as follows: 
In section 2 we sketch the geometric background of the constitutional theory of the Aufbau. Relying on the rather neglected relationship between Aufbau and Der Raum (Carnap 1922) it is shown that the basic intuition for the constitutional theory is to be found in the realm of synthetic geometry. In section 3 it is argued that the constitutional method of quasianalysis may be interpreted as a genuine geometric method designed to treat appropriate relational structures (similarity structures) by the methods of  synthetic geometry. The aim of section 4 is to show that the affine Euclidean plane may be conceived as a similarity structure for which a quasianalysis may be set up that yields the original affine incidence relation. Applying a fundamental theorem of synthetic geometry, the so called coordinatization theorem, this implies that an appropriate quasianalysis of an affine similarity structure yields a sort of auto-coordinatization of this structure. In section 5 it is argued that this autocoordinatization may be interpreted in Carnapian terms as the desired constitution of physical space. We close with some remarks on the general relevance of this result in section 6. 

2. The Geometric Background of the Aufbau 

As has been shown  in (Mormann 2002) geometry was an important source of inspiration of Carnap's philosophical thought. To a large extent, the constitutional theory of Der Logische Aufbau der Welt  was inspired by the relational systems of synthetic geometry. The first traces of a constitutional theory of conceptual systems can already be found in Der Raum. Since the Aufbau program may be said to have been decisive for Carnap's philosophy überhaupt, one  may say that geometry had a substantial influence  on his philosophy in its entirety. In this paper I do not want to dwell upon the historical details of this "geometric" interpretation of the Aufbau, rather I'd like to recall with as less fuss as possible the basic conceptual ideas of synthetic geometry necessary to understand the quasianalytical constitution of physical space Carnap could have carried out if only he had paid more attention to the expressive power of geometry. 
The leitmotif of synthetic geometry is order. As Carnap put it, geometry is a general theory of Ordnungsgefüge (complexes of order stipulations). Carnap understood Ordnungsgefüge  in a semi-technical sense intended to mean something like "relational structure" or "structured set". By conceiving a domain as a possible application for the theory of Ordnungsgefüge one imposes some order on it. This is achieved by certain  Ordnungssetzungen (stipulations of order). Hence, as a theory of Ordnungsgefüge synthetic geometry has a strong applicative dimension. In Der Raum Carnap explains this fact for projective geometries at great length. According to him, synthetic geometry is designed to offer an arsenal of possible conceptual schemes applicable to many domains. This leads to the following two characteristic features of geometry: 
 
(1)  Space (and other geometric notions such as points and lines) are conceived as general notions having many different instantiations. Geometry studies them all without blinders. It does not aim to single out one  geometric system as the "true" one.  

(2) Synthetic geometry is relational: the objects of geometric systems are determined by a net of implicit relational definitions. The ontological status of a geometric object is determined by its relational position within a certain relational system. 
 
Understanding geometry as a general theory of Ordnungsgefüge gives Ordnungssetzungen  a crucial role. The most important Ordnungssetzungen are lines. To put it bluntly, lines are the entities that establish geometric order. It is surprising that this simple idea, the imposition of order by lines, is sufficient to constitute all concepts of geometry. That is to say, points and lines are the basic building blocks for all other geometrical concepts. Of course, one has to subscribe to a general concept of line in order that lines can play this almost universal role of Ordnungssetzungen. Lines in the sense of synthetic geometry need not look like the lines we are  accostumed to. For instance, in projective geometry "lines" have the structure of "circles". The point is that the "lines" of geometric systems function as lines.  
The upshot of all this is the following: a geometric system in the sense of synthetic geometry may be defined as a triple (V, L, I): V is  the set of points,  L the set of lines and I is the so-called  incidence relation I  V x L. Obviously, the incidence relation I fully characterizes the geometric system (V, L, I). Synthetic Geometry , then, is the theory of incidence relations (cf. Buekenhout 1995). More specifically, the incidence relation I determines which points are related to which lines. Intuitively stated, it determines which points are on which lines: (x, m)  I is to be interpreted as the fact that in the geometric system defined by the incidence relation I "the point x is on the line m". In the same vein, two lines m and k are said to be said to intersect if and only if there is a point x, which belongs to both of them, i.e., there are ordered pairs (x, m), (x, k) I; two lines m and k are parallel (m || k) if and only if there do not have a common point; two points x and y are collinear if and only if there is a line m such that (x, m), (y, m) I. Depending on the axioms imposed on I different types of geometric systems are obtained. Traditionally, the most important ones are affine and projective systems, but in contemporary geometry, many other systems are studied as well (cf. Buekenhout 1995). 
For later use let us note that the systems (V, L, I) of synthetic geometry are extensional systems in the following sense: 

(2.1) Definition (Extensional Geometric Systems) . Let S = (V, L, I) be a system of synthetic geometry.  For m  L denote  by V(m):= {x; (x, m) I} the set of points of the line m. Then S is an extensional system iff two lines m and n are equal iff they their point sets V(m) and V(n) coincide.

Extensional systems (V, L, I) may be cast in a canonical form that eliminates lines as primitive: denote the power set of V by PV. Then an extensional geometric system is isomorphic to a system of the form (V, PV, I) by identifying lines with their sets of points. In the following it is assumed throughout that all geometric systems are extensional, although we do not denote this always explicitly. 
Now let us begin with the endevour of connecting synthetic  geometry as the theory of incidence structures with the constitutional theory of the Aufbau.  First let us note a rather curious evidence for such a connection. In Der Raum Carnap considers some geometric systems (P, C, I) that may be considered  as a primitive forerunner of the constitutional systems of the Aufbau  (for details see Mormann (2002): P is a set of objects (Gegenstände),  C a set of concepts (Begriffe) such that (p, c)  I iff the object p can be subsumed under c. Or, in other words, (p, c)  I if and only if p is a case of c or p can be subsumed under c. For systems of this kind, which he calls "conceptual geometries", Carnap requires the following axioms to hold:  

(2.2) Conceptual Geometries (Der Raum, p. 14). "Let us assume that the objects P1, P2, ... fall under the concept P such that the following conditions are satisfied: there is a concept G, under which not objects are subsumed but concepts g1, g2, g3, ... such that the following requirements are satisfied:
 
(1)		At least  three P-objects fall under any g-concept, but not all P-objects can be 	subsumed under one g-concept.
(2)		For two different P-objects there is always one and only one g-concept under 	which they fall, their "common" concept.
(3)		If P1, P3, P'2 fall under g1, P2, P3, P'1 under g2, and g1 and g2 are 	diffe	rent, then there exists an object P4 that falls under the common concept of P1 	and P'1 and under the common concept of P2 and P'2; moreover there is a 	concept g3, which subsumes P1, but no object of g2."

The first two axioms of this system of conceptual geometry are easily understood, even if they may not appear very plausible for concepts. The third, as it stands appears hopelessly abstruse.  In geometric terms it essentially tells us that the space of the conceptual geometry has (at least) three dimensions. 
In sum, what Carnap is doing here is just taking the familiar axiom system of 3-dimensional  projective space   and replacing the standard geometric interpretations "point", "line", and "is incident with" by the expressions "object", "concept", and "falls under" ("is subsumed to"), respectively. At first look, the "projective conceptual geometry" obtained by this procedure may appear to be nothing but an amusing idea devoid of any deeper meaning. This, however, would be a misunderstanding. As is shown in (Mormann 2002), the conceptual geometries are the primitive precursors of the constitutional systems of the Aufbau. They may be considered as powerful intuition pumps for the constitutional theory of the Aufbau  (cf. § 70, 72). One may say that Carnap took the constitutional systems of the Aufbau and the conceptual geometries of Der Raum as being of the same ilk.
In this paper I want to show that synthetic geometry not only belongs to the Aufbau’s prehistory, rather it should be regarded as its conceptual core, which helps elucidate the true nature of the Aufbau program. In order to render plausible this contention, we have to recall an  important result of 19th century mathematics, to wit, the coordinatization theorem of affine geometries. We will use this theorem to  show that the real number space R2 can be constituted from a purely qualitative base, or, to use Quine's wording, from Carnap's initial language of sense data and logic. In the context of the present paper, this base will be a relational structure defined by a set ofElementarerlebnisse endowed with a purely qualitative similarity relation (cf. Aufbau). In other words, we may consider qualitative, non-metrical geometry, as  a foundational theory for number systems. Today, philosophers of mathematics do not attribute any philosophical relevance to this fact. Rather unanimously they consider geometry as reducible to linear algebra in such a way that geometry as a mathematical discipline loses any genuine philosophical interest. It is remarkable, and mathematically non-trivial, that the direction of reduction may be reversed.  The proof of this crucial result is long and involved, and need not concern us here. It may be suffice to present an axiom system which shows, at least in principle, how number systems such as the real numbers R can be reconstructed in the framework of synthetic geometry:
 
(2.3) Affine Incidence Structures: Let A = (V, L, I) a system of synthetic geometry.   A is called an affine Pappus incidence structure (AP-structure) iff the following conditions are satisfied:

(PA1)	There exist at least three non-collinear points. (Nontriviality)
(PA2)	Any two distinct points x and y lie on exactly one line. Hence, we may denote the line determined by x and y with xy.(Linearity)
(PA3)	Given a point x and a line m, there is exactly one line k that passes x and  	is parallel to m. (Parallelity axiom)
(PA4)	If x, y, z is a triple of points on m, and x', y', z' points of m' such that 		xy' ll  x'y   and xz' ll x'z then yz' ll y'z (Pappus’s axiom):
 
The axioms (PA1) – (PA4) suffice to ensure that the lines of on AP-incidence structure have a quite rich algebraic structure, to wit, they are fields in the sense of mathematics (cf. Goldblatt 1987). That is to say, for the points of a line n  L,  operations of addition and multiplication can be defined which obey the laws of associativity, commutativity, distributivity etc thereby rendering n a field K. We need not study these operations in detail, their definition can be found in any textbook of synthetic geometry (cf. Buekenhout 1995, Coppel 1998). Rather, we are content to recall the addition of collinear points in the special case of the Euclidean plane for which the field K is the familiar field R of real numbers:

FIGURE 1

Choose two distinct lines m and n which intersect in a point 0. Fix some point w on m different from 0. The line through w parallel to n is denoted by n'. Let x, y n. Let the line through y parallel to m meet n' at y', and then let the line through x parallel to 0y' meet n' at z'. Then the line parallel to m through z' meets n at z. Declare x + y = z. It can be shown that this operation + renders n a commutative group, i.e. addition on n is associative, commutative, has a neutral element 0 etc. In a similar one can define a commutative multiplication • on n and show that it obeys the laws a multiplication of a field has to satisfy. In sum, these geometrically defined operations + and  • render (n, +, ) a field.  This construction may be expanded to a coordinatisation of the AP-structure  A as follows: 

FIGURE 2

Choose  points e(m) and e(n) on m and n respectively, both different from 0. The assignment of b(n) to b(m) (defined by the line b(m)b(n) parallel to e(m)e(n)) establishes a bijective correspondence between the point of n and of m. Then, given a point a in the plane, let the lines through a parallel to n and m meet m at a(m) and n and a’(n). Let a(n) correspond to a’(n) on n. Then the ordered pair (a(n), a(m)) are the coordinates of a in the coordinatisation of A defined by the lines n, m, and the points e(m), e(n). Of course, choosing different basic lines and points, one obtains a different coordinatisation. But, as is well known, all coordinatisations obtained in this way are isomorphic (cf. Goldblatt 1987, ch.2).
In the rest of this paper I intend to show that this coordinatisation provides the base for a quasianalytical constitution of spacetime in the sense of the Aufbau.  A first evidence that the constitutional theory of the Aufbau  are related the systems of synthetic geometry were the "conceptual geometries" of Der Raum. In order to substantiate this relation we have to delve deeper into the technicalities of the  Aufbau.
Stating it beforehand, the result will be that in the constitutional theory incidence structures (E, Q, I) will play an important role. Here, E is an already constituted domain, and the next higher level of constitution can be described in terms of the system (E, Q, I). This is a strong evidence that something like a coordinatization of AP-planes may be carried out in the conceptual framework of the constitutional theory of the Aufbau. The aim of the following sections is to do exactly this. More precisely it will be shown that Carnap’s quasianalysis may be interpreted as a method whose task is to construct appropriate incidence structures (E, Q, I) for certain "inhomogeneous" sets E, as Carnap called them. 

3. Quasianalysis and Synthetic Geometry

The Aufbau  is a complex work that has been interpreted in many different, sometimes incompatible ways (Friedman 1999, Proust 1989, Richardson 1998). In this article I assume that the main aim of the Aufbau is not to present a full-fledged constitutional system based on Elementarerlebnisse, rather I assume that the Aufbau was intended to exemplify a new general philosophical discipline, called Konstitutionstheorie (constitutional theory) which had the task of "investigating all possible forms of stepwise definitional systems of concepts" (Friedman 1999, p. 115, Aufbau § 46). Carnap intended to create a scientific successor discipline of traditional epistemology and philosophy of science that remained neutral with respect to the futile metaphysical quarrels that had plagued the traditional accounts.  As the core of this new kind of philosophy of science Carnap  considered not the constitutional system he had sketched in the Aufbau  but the method by which it was constituted.  This constitutional method is the method of quasianalysis. Hence, if the thesis of the geometric origins of the Aufbau is to be taken seriously, quasianalysis should fit into the conceptual framework of synthetic geometry. In this section I want to  show that this is indeed the case. More precisely I contend the following: 
  
(3.1) Quasianalysis in the Framework of Synthetic Geometry. Let E be an inhomogeneous set. Quasianalyzing E  is taking the elements of E as points of a geometric system (E, Q, I). The system (E, Q, I) may be considered as a complex of Ordnungssetzungen, i.e., as a geometrical Ordnungsgefüge. The Ordnungsgefüge  set up by (E, Q, I) may be interpreted as an externalisation of the inhomogeneities of E.

In order to unfold this succinct characterization of quasianalysis the following remarks  may be in order. The key term of (3.1) is "inhomogeneous set". Hence, first we have to explain what is to be understood by that term. Then we will explain what is meant by embedding an inhomogeneous set E in a geometric frame (E, Q, I). 
For Carnap,  the foremost examples of non-homogeneous sets   E are similarity structures. A similar structure, denoted by (E, ~), is a set E endowed with a binary similarity relation ~. Two elements e and e' of E related by the relation ~ are said to be similar to each other (e ~ e’). The relation ~ is assumed to be reflexive and symmetric, i.e. each element is assumed to be similar to itself (e ~ e) and e ~ e' implies e' ~ e. The relation need  not be transitive, however. This is plausible, since if e is similar to e', and e' is similar to e'' then e need not be similar to e''. A non-homogeneous set (E, ~) may be represented more perspicuously as a graph: 

   

 FIGURE 3  

Here, two distinct similar elements of E that are connected by a straight line, two elements that are not similar are not directly connected by a straight line. Actually, the concepts of simple graph and similarity structure are strictly equivalent. Hence, according to the Aufbau, the world (or some part of it) may be conceived of as a huge graph: the vertices of this graph are the "elementary experiences" and the edges are formed by the  pairs of similar elements.  
In an unpublished manuscript with the programmatic title Quasizerlegung - Ein Verfahren zur Ordnung nichthomogener Mengen mit den Mitteln der Beziehungslehre (Quasianalysis - A Method to Order Non-homogeneous Sets by Means of the Theory of Relations) (Carnap 1922/23) the task of Quasizerlegung, i.e., quasianalysis for similarity structures, is described as follows:

“Suppose there is given a set of elements, and for each element the specification to which it is similar. We aim at a description of the set which only uses this information but ascribes to these elements quasicomponents or quasi-properties in such a way that it is possible to deal with each element separately using only the quasiproperties, without reference to other elements.” (Quasizerlegung, p.4) 

As Quasizerlegung  makes clear, the method of quasianalysis is a purely formal method.  It may be applied to any nonhomogeneous set (E, ~), not just to Elementarerlebnisse as in the Aufbau.  Submitting (E, ~) to a quasianalysis means to impose certain Ordnungssetzungen   on it in order to unfold its structure. Thereby appropriate invariants may be found which characterize its structure in a succinct manner. As modern mathematics teaches, the finding of characteristic invariants is an unending task. Even simple structures give rise to a profusion of invariants. Although Carnap concentrates on the quasianalysis of similarity structures, he mentions the possibility to submitting all kinds of relational structures to a quasianalytical constitution process (Aufbau  § 104). Hence, I think it would not be totally out of place to interpret quasianalysis in a generalized sense as mathematical constitution überhaupt. Be this as it may, in the following we will concentrate on the quasianalysis of similarity structures. In Quasizerlegung (p. 4) Carnap axiomatically describes this kind of quasianalysis as the assignation of quasiproperties to the objects to be quasianalysed, whereby these quasiproperties function as Ordnungssetzungen such that the similarity structure of E is taken into account:

(3.2) Definition . Let (E, ~) be a non-homogeneous set. The quasiproperties of the elements of E are entities which have to satisfy the following requirements:

(C1) 	If two elements are similar they share at least one quasiproperty.
(C2) 	If two elements are not similar they do not share any quasiproperty.
(C3)	If two elements are similar to exactly the same elements they have the same 	quasiproperties.
(C4)	No quasiproperty can be removed unless (C1) - (C3) are violated. 

The general intention  of these requirements is to regulate the relation between "being similar" on the one hand and "sharing a property" on the other. (C1) - (C3) express characteristics of this relation that may be considered as more or less common sensical asserting that similarity and properties co-vary. The requirement (C4) is a principle of parsimony which intends to bar superfluous properties not needed for an economic description of the similarity relation. 
Let us postpone for the moment the question of what quasiproperties "really are" simply assuming that there is a set Q := {q; q quasiproperty of (E, ~)}. With the help of Q we may define an incidence relation I  E x Q by

(3.3)		 	(e, q) I := the element e has the quasiproperty q. 

In this way the relation between elements of E and their quasiproperties q may be succinctly described by the incidence relation I. If the resulting system (E, Q, I) of incidence geometry satisfies (C1) - (C4) it is called a quasianalysis of the non-homogeneous set (E, ~).  This construction of a geometric system is analogous to that of conceptual geometries Carnap discussed in Der Raum.  In contrast to the axioms for "conceptual geometries" in Der Raum (which are simply copied from the axioms for projective spaces) the requirements (C1) - (C4) for (E, Q, I) are much better adapted to the  intuitive requirements one entertains for property distributions. Hence, we are still in the realm of incidence structure although Carnap no longer maintains the (rather absurd) thesis that conceptual geometries are to be modelled after the patterns of projective geometry. 
It will be expedient to define a quasianalysis in still another but equivalent way. As is easily checked, every relation I  E x Q gives rise to a mapping 

(3.4) 		rI: E -----> PQ    defined by rI(e) := {q; (e, q) I}

Obviously, I and rI determine each other in a 1-1 fashion. Hence we may characterize a quasianalysis either by (3.3) as an incidence relation, or by (3.4) as a (structure preserving) map (cf. Mormann 1994). The version (3.4) is particularly convenient if one wants to check if structure preserving conditions such as (C1) - (C4) are satisfied. Summarizing we end up with the following characterization of a quasianalysis of similarity structures (or, in terms of Quasizerlegung, "inhomogeneous sets"):

(3.5) Definition of Quasanalysis . Let (E, ~) be a similarity structure. A quasianalysis for E is an incidence structure (E, Q, I) as defined in (3.3), or equivalently, as a representation rI: E ----> PQ satisfying the requirements (C1) - (C4) as defined in (3.5).
 
Now let us come back to the question of what quasiproperties "really are". Since the constitutional theory of the Aufbau is extensional we may assume that the quasianalytical systems (E, Q, I) are extensional systems in the sense of (2.1) so that they have the form (E, PE, I) and a quasiproperty q is identified with the set {e; (e, q) I}. The set Q of quasiproperties may be considered as a subset Q  PPE of subsets of subsets of E. Hence the general format of a quasianalysis of a similarity structure (E, ~) is

(3.6)  			I  E x PE           or            rI: E ----> PPE

such that (C1) - (C4) are satisfied.  This description of quasiproperties is still rather vague. In the following  we will give a more specific characterization of quasiproperties for the so called quasianalysis of the first kind, i.e., we will characterize quasiproperties (E, ~) as special subsets of E by taking into account the  similiarity structure of (E, ~). For this purpose we need  the following preparatory definition:

(3.7) Definition. Let (E, ~) be a similarity structure. A similarity circle T (Ähnlichkeitskreis) is a subset T  E which satisfies the requirements

(i)		(x) (y) (x,y T -----> x ~ y) 
(ii)		(x) y ( x  T ------> y T and  NOT(x ~ y))

Denote the set  of similarity circles of (E, ~) by SCE. Conceiving a similarity structure (E, ~) as a graph, similiarity circles T may be characterized as maximal subgraphs of (E, ~) all of whose elements are similar to each other. Hence, for all elements x not belonging to T there is a y of T such that Not (x ~ y) obtains. The concept of similarity circles gives rise to the following definition:

(3.8) Definition (Aufbau § 69, Mormann 1994) . A quasianalysis (E, Q, I) of the similarity structure (E, ~) is of the first kind if and only if all its quasiproperties are elements of SCE, i.e. Q PSCE. In mapping form this is expressed by the requirement that a quasianalysis of the first kind has the form

			rI: E ------> PSCE   rather than rI: E ------> PPE

In the following we will only consider quasianalysis of the first kind. Similarity circles may be used to define  a "standard pseudo-quasianalysis" in the following way:

(3.9) Proposition . Let (E, ~) be a similarity structure. Define a map rI: E ---> PSCE by rI(x) := {T: T  SCS(E) and  x T}. Then rI  satisfies the conditions (C1) - (C3).

In order to make  the preceding chain of abstract definitions a little bit more vivid let us consider the following elementary example (Goodman 1951): 	 

FIGURE 4

The similarity structure defined by this graph has four similarity circles: a = {1, 2, 3}, b = {2, 3, 5},  c = {3, 4, 5},  and d = {5, 6}.  Thus for this graph we get the following property list: 

			1.	a				4.	c
(3.10)		2.	ab				5.	bcd
			3.	abc				6.	d

A list of this kind is to be read as "1 has the property a", "2 has the properties a and b", etc. In this way we see that 1 and 2 share the property a, 2 and 3 share the properties a and b etc. As is easily seen the property distribution provided by the list (3.11) satisfies Carnap’s requirements (C1) - (C4). 
In the tradition of Carnap and Goodman the virtues and vices of the quasianalytical approach have been discussed almost exclusively in terms of small examples such as (3.10) (cf. Goodman 1951, Mormann 1994). The geometric  reinterpretation of this method shows that the domain of applications of the quasianalysis approach is not exhausted by these rather elementary cases. This will be shown in the following section.

4. The Affine Plane as a Similarity Structure

 A new field of applications for quasianalytical constitutional theory is opened when we take seriously the fact that quasianalysis of similarity structures are incidence  structures (E, Q, I) of synthetic geometry. A particularly important example is the incidence structure of the familiar plane of Euclidean geometry. This example will bring us close to our ultimate destination, the quasianalytical constitution of physical space time. The details are as follows. Let A be the Euclidean plane. Denote its points  by x,y, z, .. and its lines by k, m, n, etc. As is well known, the geometric structure of A may be codified in terms of an incidence relation I  A x PA.   Since in A any two different points x and y determine exactly one line, a line m may be denoted by xy, x and y being two different points of m.
Now, our task is to characterize the Euclidean plane A (endowed with its standard incidence structure) as a similarity structure (A, ~). First, let us define an appropriate similarity relation. For this purpose, choose a class B of parallel lines of A. Depending on B we will define a similarity relation ~B on A. Hence, the resulting similarity structure should be denoted by (A, ~B). In order not to overload denotation, however, we will denote it simply by (A, ~). This is justified since for different B the resulting similarity structures turn out to be canonically isomorphic. For reasons of intuitive vividness we may refer to the lines of B as horizontal lines. Having chosen B two points x and y are defined to be similar iff they are equal or are on a line m not belonging to B:

(4.1)			x ~ y :=  (x ≠ y and xy B) or x = y

The following diagram exhibits the geometrical meaning of this definition:

FIGURE 5

Obviously, the relation ~ is reflexive and symmetric, but not transitive. Hence, (A, ~) is a similarity structure. Define the complementary similarity structure (A, ~*) by x ~* y := (xy  B or x = y)). (A, ~*) is a very special similarity structure, to wit, it is an equivalence structure, whose equivalence classes are just the lines of B. Obviously, (A, ~) and (A, ~*) determine each other, and all considerations dealing with ~* could be formulated in terms of ~, and vice versa. Hence, dealing with (A, ~) and (A, ~*) (instead of (A, ~) or (A, ~*) alone) does not add  anything new. After these preparations we are able to construct the following quasianalysis of A: 

(4.2) Lemma .  Let (A, ~) be the similarity structure defined by (4.1). Then define Q: A ------> PPA by Q(x) := {xy; xy  B}. Then the map Q is a quasianalysis of A of the first kind.
 
Proof: Geometrically, the quasiproperties attributed to x by Q  are just the lines of A through x not belonging to B. First we show that Q satisfies (C1) and (C2). For x ≠ y we have x ~ y iff xy B. Hence xy Q(x)  Q(y) ≠ Ø. On the other hand, if m Q(x)   Q(y) we have m B. Since x and y are on m we may write m = xy, hence x ~ y. Moreover,  since any non-horizontal line m may be characterized as m = xy for some points x and y satisfying x ~ y, removing m would amount to a violation of C1.  Hence, Q satisfies (C2). The conditions (C3) and (C4) are obvious.
In order to prove that Q is of the first kind we have to show that any line q not belonging to B is a similarity circle of the similarity structure (A, ~). Let m  Q(x). For any y  m and x ≠ y we have xy = m. Hence, all points of m are similar to x. Suppose z  m and z ~ x. Then there is a unique horizontal line k through z which meets m at, say, z'. Hence, by definition, z ~* z'. Hence m  SC(A, ~) and Q is of the first kind. The representation Q has some properties which deserve to be singled out, since they will be crucial for the following (cf. (2.3)):
 
(4.3) Lemma . Let (A, ~) be the Euclidean plane endowed with the similarity relation defined by (4.1) endowed with the quasianalysis defined by (4.2). Then the following holds:

(PA1)’	There exist at least three non-collinear points.
(PA2)’	Any two distinct similar points x and y lie on exactly one line. Hence, we 	may denote the line determined by x and y with xy. 
(PA3)’	Given a point x and a line m, there is exactly one line k that passes x and  		is parallel to m.
(PA4)’	Suppose that x, y, z is a triple of points on m, and x', y', z' a triple of 		points of m' such that x ~ y',  x' ~y,  x ~ z',  x' ~ z,  y ~ z', y' ~z. If xy' 		ll  x'y   and xz' ll x'z then yz' ll y'z.
 
Lemma (4.3) asserts that the quasianalysis Q of (A, ~) renders (A, ~) essentially an AP-plane in the sense of (2.3) changing (P1) - (P4) to (P1)’ - (P4)’ since the lines of B have to be left out.
The incidence relation IQ defined by Q is not quite the incidence relation of the affine Euclidean plane we are looking for, since the lines of B are missing. In order to include them we proceed as follows. First note that the complementary similarity relation ~* is an equivalence relation whose equivalence classes are just the lines of B. Hence, for the complement similarity structure (A, ~*) we have a canonical mapping Q*: A -----> PPA which maps x to the singleton {q}, q being the unique line of B with x  q. Q* satisfies the condition (C1), (C2), and (C4). Denote the incidence relation defined by Q* by IQ*. Then we define the set theoretical union IQQ*  A x PA of IQ* and IQ* by 
		
(4.4)			(x, m)  IQQ* := (x, m)  IQ or (x, m)  IQ*

This is the relation we need for the construction of a coordination mapping rQQ*: A  -----> R2. As is easily seen, IQQ* defines indeed an AP-plane in the sense of (2.3). Imposing some further axioms on IQQ* as we will do in the next section one can ensure  that the affine plane defined by IQQ* is an Euclidean plane. Then the coordinatisation procedure sketched in section 2 ensures that A can be mapped onto R2 in such a way that the incidence structure IQQ* on A is isomorphically mapped onto the standard real affine structure I  R2 x PR2.  
Before we come to this task let us observe that the quasianalytical construction of an affine plane achieved so far is unique up to isomorphism. This is seen as follows: if we had chosen another family B' of parallels we had obtained a different similarity structure (A, ~'). But then the similarity structures (A, ~) and (A, ~') are isomorphic, since for any pair B and B' of parallels one can find an affine map which maps B onto B' preserving the affine structure, to wit, incidence relation and parallelism. This map defines an isomorphism between the similarity structures (A, ~) and (A, ~').
The last step to get the real numbers is to impose some further axioms on the incidence relation I in order to ensure that the field is indeed R. The crucial point in the proof that the field of the plane A is indeed R is the observation that R is distinguished from other fields in that it is a Dedekind complete ordered field. That is to say, the elements of a line of the real affine plane can be ordered in such a way that we may talk about positive and negative elements in a sense to be specified. In particular, this order allows us to define a triadic relation of betweenness for collinear points x, y, z. Thus, in order to construct the real affine plane from a similarity structure (S, ~) one has to assume the existence of an order on the similarity circles T   Q(SC(A,~))  Q*(SC(A, ~*)). This  leads to the following definition:

(4.5) Definition . Let (E, ~) be a similarity structure. A quasianalysis Q: E ----> SCE is an ordered quasianalysis if and only if all similarity circles T  Q(E) SCE are endowed with an linear order. 

As is shown the real plane R2 ordered quasianalysis of (A, ~) and (A, ~*) exist that are compatible with the field structure defined on the lines m, i.e., on every m we may distinguish between positive elements (0 < x) and negative elements (x < 0) in such a way that addition and multiplication are compatible with the relation <, i.e., the sum and the product of positive elements are again positive etc. Hence we may assume that the similarity circles of a similarity structure (E, ~) having an ordered quasianalysis in the sense of (4.1) are ordered fields. Now we are almost done. The last requirement we need to obtain the real affine plane is to stipulate that the ordered fields of our lines are Dedekind complete (Goldblatt 1987, p.69/70): 

(4.6) Definition .  Let K be an ordered field. Assume that K is the union of two non-empty sets C and D such that x < y for all x  C and y  D. K is Dedekind complete if and only if there is some z  K such that x ≤ z ≤ y.

As is well known, the structure of an ordered Dedekind complete field is categorical, i.e., up to isomorphism, there is only one type of Dedekind complete ordered field, to wit, the field of real numbers R. 
Summarizing we have obtained the result that we may describe the affine Euclidean plane as a similarity structure (E, ~), which has an ordered, Dedekind complete AP-quasianalysis as defined by (4.2) - (4.6). In the next section we will show that this result may be "read backwards" leading to a quasianalytical coordinatisation of a physical domain P that may be interpreted as physical space in the  sense of Carnap.

5. The Constitution of Physical Space

Now we have gathered all pieces we need to tackle our main task, the constitution of physical space along the lines of the quasianalytical constitution theory of the Aufbau.  To make clear what is going on let us succinctly recall what, according to the canons of the Aufbau,  is to be constituted and what are the assumptions under which this constitution is carried out.
To begin with, let us note that Carnap had a rather peculiar conception of physical space which essentially differs from that of common sense. Before one can embark on the task of constituting it by a quasianalysis it has to be explained what we are after when we are in pursuing the constitutional endevour. For this purpose, we have to begin with Carnap's own constitution. 
Carnap started the constitution of physical space with the presupposition that we already have the four-dimensional Minkowski vector space R4 as a purely mathematical (or logical) object. He was entitled to do so, since the constitution theory is based on the assumption that the Aufbauer has available for his purposes the full resources of logic and mathematics (cf. also Quine 1951). The mathematical object R4 as mathematical object cannot be regarded as physical space, of course, and Carnap does not make this assertion. Rather, according to him, the mathematical object R4 acquires its status as physical space through the coordinatization of physical qualities such as colours to the points of this heretofore purely logical object. So we may say, physical space is the physically interpreted mathematical space R4. This interpretation has to satisfy certain requirements, for example, stability conditions but this need not concern us for the moment. The problem is, if this physical interpretation of the purely mathematical object R4 can be carried out in the constitutional system of the Aufbau by using the method of quasianalytical constitution. Quine claimed that Carnap’s constitutional sketch failed to do this and went on to contend that the quasianalytical constitution of spacetime principally impossible. He took this as a conclusive argument against the feasibility of empiricist reductionism. Up to this very day, Quine’s verdict has been accepted rather unanimously. That is to say, even those who maintain that Quine’s empiricist interpretation of the Aufbau is untenable agree with him, not only that Carnap did not provide a sketch of the constitution of spacetime but also that this is actually impossible to do in the framework of the quasianalytical program. It is this second thesis that I want to refute in the following, i.e., I want to show that, although Carnap has failed to provide a quasianalytical constitution he could have done it, i.e., a quasianalytical constitution is feasible. 
For this purpose it is expedient first to recall Quine’s full argument in which he spotted a change of method in Carnap’s constitutional entreprise. He describes Carnap’s procedure as follows:

"He [Carnap]explained spatio-temporal point-instants as quadruples or real numbers and envisaged assignment of sense qualities to point-instants according to certain canons. Roughly summarized, the plan was that qualities should be assigned to point-instants in such a way as to achieve the laziest world compatible with our experience. The principle of least action was to be our guide in constructing a world from experience.
Carnap did not seem to recognize, however, that his treatment of physical objects fell short of reduction not merely through sketchiness, but in principle. Statements of the form "Quality q is at point-instant x;y,x,t" were, according to his (Carnap's) canons, to be apportioned truth values in such a way as to maximize certain over-all features, and with growth of experience the truth values were to be progressively revised in the same spirit. I think this is a good schematization (deliberately oversimplified, to be sure) of what science really does; but it provides no indication, not even the sketchiest, of how a statement of the form "Quality q is at x; y; z; t" could ever be translated into Carnap's initial language of sense data and logic. The connective "is at" remains an added undefined connective; the canons counsel us in its use but not in its elimination." (Quine 1951, p.40)

As I said, even those, who reject Quine's empiricist interpretation of the Aufbau, concede him this point. I think, this is not necessary, Carnap can be saved from Quine's attack, or so I want to argue. Strictly speaking, Quine does not offer any argument why it may be impossible to provide a translation of the desired kind. Rather, he is content to (correctly) point out that Carnap does not provide such a translation. This, however, does not imply that such a translation is impossible.
Bringing to work the apparatus of synthetic geometry developed in the preceding sections this can be done as follows: The fatal flaw of Carnap's attempt to constitute physical space quasianalytically resides in the fact that he separates what should not be separated. That is to say, he starts with the ready-made mathematical object R4 on the one hand, and the physical object, i.e. the quasianalytically constituted domain of physical qualities P on the other hand. Then, the problem is to bring together these two separated domains. This can, obviously, be done only by fiat, i.e., one has to stipulate that there is an 1-1-assignment between quadruples of real numbers r R4 and physical qualities q P. Even we grant that this is possible following certain (empiristically acceptable) "canons", this assignment of number quadruples to qualities cannot be considered as a quasianalytical constitution. That is to say, we got a fatal gap between the physical and the mathematical that cannot be bridged by quasianalytical constitution. Rather, as Quine correctly observes, one has to rely on the un-quasianalytical extra primitive relation "is at" to assert statements of the type "Quality q  is at  (x,y,z;t)". So far, so good. The question is, if Carnap’s deviation from the path of true quasianalytical constitution was unavoidable. It is possible to saving Carnap against himself and Quine and his followers. It is not necessary to assume  that the ready-made mathematical structure R4 is already there, waiting to be related or interpreted empirically. Rather, we may constitute physical space, i.e., a physically interpreted R4 in one fell swoop, so to speak. This amounts to a sort of auto-coordinatisation of the already constituted domain P of qualities along the lines sketched in the previous section. 
As already announced in the introduction for avoiding unnecessary technicalities, let us replace R4 by R2. Hence, the problem is to provide a quasianalytically acceptable coordinatization of the already constituted domain P. 
This can be done as follows: The system P is assumed to be a similarity structure (E, ~). It does not matter whether the elements of E are intuitively interpreted as Elementarerlebnisse or qualities or whatever, since the "nature" of the elements does not play a role in constitution theory. According to (3.5) a quasianalysis of (E, ~) is a geometric system (E, PE, I) which satisfies the requirements (C1) - (C4). Using the results of section 4 we will construct a quasianalysis of the first kind that gives rise to a coordinatisation of E in the sense that each element e of E can be uniquely named by an ordered pair (a, b) of elements of a similarity circle of E. Then the coordinatisation theorem tells us that E is the 2-dimensional real number space R2. That is to say, in contrast to Carnap’s flawed coordinatisation, which first separated the physical and the mathematial, and later attempted to bring them together again by the notorious "is at"-relation, our approach constitutes the coordinating numerical structure directly from the physical structure. Let us start from the following definition:

(5.1) Affine Pappian Similarity Structure . Let (E, ~) be a similarity structure. E is called an affine Pappian similiarity structure (AP-similarity structure) iff the following holds:

(1) 	E has a quasianalysis (E, SCE, I) of the first kind satisfying (C1) - (C4).
(2)	The complementary similarity structure (E, ~*) is an equivalence relation.
(3)	I satisfies the requirements (P1)’ - (P4)’.
(4)	For J : = I I* the system (E, SCE SCE*, J) is an affine Pappian plane.

As has been shown in the previous section, AP-similarity structures exist. In particular, it should be noted that the definition of an AP-similarity structure is defined fully in terms of what Quine called "Carnap’s initial language of sense data and logic" (Quine 1951, p. 40). 
Now a quasianalytical coordinatisation of P = (E, ~) is at  hands: choose three non-collinear points r,s, and t. With the help of the intersecting lines rs and rt one may construct an internal coordinatisation of E which renders it isomorphic to the 2-dimensional plane K2 of some commuative field K. Thus, the statement “Quality point x is at (p,q)“ has the meaning “with respect to the coordinatisation based on r, s, and t the quality point a is represented by the ordered pair of  (p,q) of elements p and q of K. This may not be a coordinatisation by real numbers, since we cannot be be sure that the field K is the field R of real numbers. In order to ensure this some further axiomatic requirements has to be imposed on E. This can be done in several ways. maybe the simplest one is the following (cf. Goldblatt 1987):

(5.2) Ordered Field.  An ordered field K is a field with a distinguished subset K+ closed under addition and multiplication such that for each x Kexactly one of the conditions x K, -x K, x = 0 is true. K+ is to be thought as the set of positive elements of K. With the help of K+ one can defined an order on K by x < y := (y - x) K+.

The last step to ensure that K is the real number field R is done by imposing the further axiom on K that it also satisfies the Dedekind continuity axiom (cf. Goldblatt 1987, p. 69/70). Noting that the Dedekind axiom can be expressed in "Carnap’s initial language of sense data and logic" we are done: it can be proved that K is (up to isomorphism) just R. Hence, the coordinatisation  as described in section 2 yields that P can be identified with R2. 
Summarizing we may say, that for AP-similarity structures (E, ~) whose lines are linearly ordered Dedekind-complete point sets, there exists a quasi-analytically constituted coordinatisation, which  assigns real  number coordinates to the elements of E. Of course, this coordinatisation is not unique: Given an AP-quasianalysis (E, SCE, I) of E  one may choose another tripel of non-collinear elements of r’, s’ t’ yielding an other coordinatisation (E, SCE, I’) related to the former by a unique linear isomorphism. Moreover, given (E, ~) the choice of an AP-quasianalysis (E, SCE, I) may not be unique. It may happen that for a given (E, ~) there exist several, different quasianalysis (E, SCE, I) and (E, SCE, I’). This gives ample leeway for conventional choices. That is to say, for some reason or other the Aufbauer may prefer one coordinatisation against the other, since one allows to maximize certain desirable over-all features, and the other does not. Thereby the quasianalytical constitution of physical space, i.e. the constitution of a spatio-temporal coordinatisation of the physical domain P = (E, ~) is seen to follow at least schematically, what science really does (cf. Aufbau  § 135, 136, Richardson 1998, pp.70ff).  

6. Concluding Remarks

Summarizing we conclude that the program of quasianalytical constitution is not bound to break down when it comes to the quasianalytical constitution of physical space. It may founder at other occasions, or may be considered not to be attractive for other reasons, but there is no deep reason why it has to founder at the foundations of physical space  - as Carnap understood this notion. 
One may wonder, why Quine and so many philosophers following him were thus confident that Carnap’s failure of providing a sketch of a quasianalytical constitution of spacetime was tantamount to the impossibility of achieving this task in general. Quine never gave any indication, even the sketchiest one, why this constitution should be impossible in principle. A not unplausible answer seems to be that he and many philosophers underestimated the expressive power of synthetic geometry of the 19th century, to say nothing of its modern achievements (cf. Buekenhout 1995, Coppel 1998). It is nothing but a common-sense prejudice that a qualitative language like Carnap’s initial language is principally unable to cope with the quantitative as it crops up in the real number coordinatisation of physical space. 
In a general vein one may say that modern synthetic geometry, which in this paper we used only in a very elementary way may well have the potential to support rational reconstruction programs such as the Aufbau ’s. Hence, some basic knowledge of synthetic geometry may still be useful also for philosophers of the 21st century.

References:

Buekenhout, F. 1995, Handbook of Incidence Geometry (Ed.), Amsterdam, Elsevier.
Carnap, R. 1922, Der Raum. Ein Beitrag zur Wissenschaftslehre, KANT STUDIEN, ERGÄNZUNGSHEFTE NR. 56.
Carnap, R. 1922/23, Quasizerlegung, Ein Verfahren zur Ordnung nichthomogener Mengen mit den Mitteln der Beziehungslehre, Unpublished Manuscript, Carnap Archive, University of Pittsburgh, RC-081-04-01.
Carnap, R., 1961(1928), Der Logische Aufbau der Welt, Hamburg, Meiner.
Cassirer, E.1985(1910), Substanzbegriff und Funktionsbegriff, Darmstadt, Wissenschaftliche Buchgesellschaft.
Coppel, W.A., 1998, Foundations of Convex Geometry, Cambridge, Cambridge University Press.
Friedman,M.,1999, Reconsidering Logical Positivism, Cambridge, Cambridge University Press.
Goldblatt, R., 1987, Orthogonality and Spacetime Geometry, New York and Wien, Springer Verlag.
Goodman, N. 1951, The Structure of Appearance, Bobbs-Merrill, Indianapolis
Goodman, N. 1972, Seven Strictures on Similarity, Projects and Problems, Bobbs-Merrill, Indianapolis.
Hilbert, D. 1971(1899), Foundations of Geometry, La Salle, The Open Court.
Mormann, A Representational Reconstruction of Carnap’s Quasianalysis, PSA 1994, vol. 1, 96 - 104, 1994.
Mormann, Th., 1995, Incompatible Empirically Equivalent Theories: a Structural Explication, Synthese 103, 203 - 249.
Mormann, Th., A Quasianalytical Reconstruction of Spacetime, Talk presented at the Biennial conference of the PSA, Vancouver 2000.
Mormann, Th., 2002, Synthetic Geometry and Aufbau, to appear in Th. Bonk (ed.), Language, Logic and Truth, The Philosophy of Rudolf Carnap, Vienna Circle Collection, Dordrecht, Kluwer.
Proust, J., 1989, Questions of Form. Logic and the Analytic Proposition from Kant to Carnap, Minneapolis, University of Minnesota Press.
Quine, W.V., 1953 (1961), Two Dogmas of Empiricism, in From a Logical Point of View, Cambridge/Mass., Harvard University Press, 20 - 46. 
Richardson, A.W., 1998, Carnap's Construction of the World, The Aufbau and the Emergence of Logical Empiricism, Cambridge, Cambridge University Press. 
Robb, A.A., 1936, A Theory of Time and Space, Revised Edition, Cambridge, Cambridge University Press.