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Texto de nota al final;}{\s31\widctlpar\adjustright \f12\fs20\lang3082\cgrid \sbasedon0 \snext31 Encabezado de tda;}{\s32\qj\widctlpar\adjustright \f7\fs18\up2\lang3082\cgrid \snext32 WP_Fu\'a7note;}{\s33\qj\widctlpar\adjustright
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{\revtim\yr2002\mo10\dy7\hr14\min26}{\printim\yr2002\mo9\dy27\hr12}{\version2}{\edmins0}{\nofpages2}{\nofwords8028}{\nofchars45763}{\*\company Carnegie Mellon University}{\nofcharsws56200}{\vern115}}
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\qj\widctlpar\adjustright \f12\fs22\lang3082\cgrid {\f7\fs18 Actually the assumption that the similarity relation ~ is the symmetrization of an asym\-\-me\-tric relation < is unnecessary. That is to say, starti
ng with another, quite differently defined symmetric similarity relation it is possible to construct the an ordered affine plane and thereby the real numbers }{\b\f7\fs18 R}{\f7\fs18 .
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\par A Quasianalytical Constitution of Physical Space
\par
\par
\par }{\f7\fs20 Thomas Mormann, Donostia-San Sebasti\u225\'87n, Spain
\par
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 1.\tab Attacks on Carnap's Constitutional Theory }{\i\f7\fs20
\par }\pard \qj\fi-700\li700\ri18\sl360\slmult0\widctlpar\tx700\adjustright {\f7\fs20 2.\tab The Geometric Background of the }{\i\f7\fs20 Aufbau}{\f7\fs20
\par 3.\tab Quasianalysis and Synthetic Geometry
\par 4.\tab The Affine Plane as a Similarity Structure
\par 5.\tab A Quasianalytical Constitution of Physical Space
\par 6.\tab Concluding Remarks
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx700\adjustright {\f7\fs20
\par }\pard \qc\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qc\fi-360\li360\ri18\sl360\slmult0\widctlpar\tx360\adjustright {\b\f7\fs20 1.\tab Attacks on Carnap's Constitutional Theory.
\par }\pard \qc\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\fi360\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 Carnap's quasianalytical con\-sti\-tu\-tion theory of the }{\i\f7\fs20 Aufbau }{\f7\fs20 has been subjected to many criticisms on quite different le\-
vels. Let us mention just three: (i) Goodman attacked the roots of Carnap's account claiming that the consti\-tu\-ti\-o\-nal method of the }{\i\f7\fs20 Aufbau}{\f7\fs20
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 }{\i\f7\fs20 Aufbau}{\f7\fs20
on on an intermediate level contending that, when it came to spacetime, Carnap was not able to constitute it. Instead, Quine objected, Carnap changed the method of constitution without clearly announcing it, int
roducing a new undefined connective "is at" (cf. Quine 1951); (iii) recently Friedman contended that even if the quasianalytical method of constitutions worked properly throughout, it would not deliver what Carnap expected from it, to wit, a complete st
ructuralization of empirical knowledge (cf. Friedman 1999).
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab I think there are good reasons to assume that Goodman's cri\-ticisms can be defused (cf. Proust 1989, Mormann 2002). Thus, I will say nothing about them, nor will I treat Fried\-
man's objections dealing with difficulties concerning Carnap's notion of foundedness (cf. }{\i\f7\fs20 Aufbau }{\f7\fs20 \u167\'a4154 - \u167\'a4
155, Friedman 1999). In this paper I only want to deal with Quine's criticism concerning the constitution of spacetime. Quine maintained that the }{\i\f7\fs20 Aufbau }{\f7\fs20 ac\-count of the con\-\-
stitution of the physical world is principally flawed:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\li720\ri598\widctlpar\tx-709\adjustright {\f7\fs20 "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 ma\-\-ximize and minimiz
e certain over-all features, and with growth of ex\-pe\-\-ri\-\-
ence 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 sketchi
est, of how a statement of the form "Quality q is at x;y,z;t" could ever be translated into Carnap's initial lan\-gu\-age of sense data and logic. The connective "is at" remains an added un\-
defined connective; the canons counsel us in its use but not in its elimi\-na\-tion" (Quine 1951, p. 40).
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par \tab In this pa\-per I'd like to provide such a sketch. Admittedly, something like this cannot be found in Carnap's ori\-ginal account, at least not explicitly. Nevertheless, I claim that my constitution of the "is at"-conne
ctive is fully in line with the spirit of Carnap's approach. That is to say, Carnap }{\i\f7\fs20 could}{\f7\fs20 have constituted physical space by qua\-si\-ana\-lytical methods alone. He wasn't forced to fall back on other, conventionalist con\-sti\-tu
\-ti\-onal methods. Although Quine's empiricist interpretation of the }{\i\f7\fs20 Aufbau}{\f7\fs20 has recently been critized from many quarters, his thesis of the alleged break in the methodology of the }{\i\f7\fs20 Aufbau }{\f7\fs20
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.}{\cs16\super
\chftn {\footnote \pard\plain \qj\ri18\widctlpar\adjustright \f12\fs22\lang3082\cgrid {\cs16\super \chftn }{ }{\f7\fs18 In order not to overburden the paper with ma\-\-the\-\-\-matical techicalities I will prove only a sim\-plified version
of this contention, to wit, how a statement of the form "Quality q is at (x,y)" (x,y) }{\f3\fs18 \uc1\u-3890\'ce\u-4064\'20}{\b\f7\fs18 R}{\f7\fs18\up6 2}{\f7\fs18 , can be translated into a statement using only the basic terms of a con\-sti\-tu\-
tional system. Here, of course, }{\b\f7\fs18 R}{\f7\fs18\up6 2 }{\f7\fs18 is the 2-dimensional vector space over the real numbers }{\b\f7\fs18 R}{\f7\fs18 . The generalization to 4-dimensional spacetime is not too difficult.}}}{\f7\fs20
The outline of this paper is as follows:
\par }\pard \qj\fi709\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 In section 2 we sketch the geometric background of the constitutional theory of the }{\i\f7\fs20 Aufbau}{\f7\fs20 . Relying on the rather neglected relationship between }{\i\f7\fs20
Aufbau }{\f7\fs20 and }{\i\f7\fs20 Der Raum}{\f7\fs20
(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 me\-
thod designed to treat appropriate relational structures (similarity struc\-tures) 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 wh
ich a quasianalysis may be set up that yields the original affine incidence relation. Applying a fundamental theorem of syn\-thetic geometry, the so called coor\-di\-na\-ti\-zation theorem, this implies that an appropriate qua\-\-
sianalysis of an affine similarity structure yields a sort of auto-co\-\-ordinatization of this structure. In section 5 it is argued that this auto-\-co\-ordinatization may be in\-ter\-preted in Carnapian terms as the desired constitution of phy\-
sical space. We close with some remarks on the general relevance of this re\-sult in section 6.
\par }\pard\plain \s26\qj\ri18\sl360\slmult0\widctlpar\adjustright \f12\lang3082\cgrid {\f7\fs20\ul
\par
\par }\pard\plain \qc\ri18\sl360\slmult0\widctlpar\adjustright \f12\fs22\lang3082\cgrid {\b\f7\fs20 2. The Geometric Background of the }{\b\i\f7\fs20 Aufbau }{\b\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab As has been shown in (Mor\-mann 2002) geometry was an important source of inspiration of Car\-nap's philo\-\-so\-\-phical thought. To a large extent, the con\-sti\-tut\-i\-o\-
nal theory of the }{\i\f7\fs20 Aufbau}{\f7\fs20 was inspired by the re\-la\-tional systems of synthetic geometry. The first traces of a constitutional theory of conceptual systems can already be found in }{\i\f7\fs20 Der Raum}{\f7\fs20 . Since the }{
\i\f7\fs20 Aufbau}{\f7\fs20 program may be said to have been decisive for Carnap's philosophy }{\i\f7\fs20 \u252\'9fberhaupt}{\f7\fs20
, 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 }{\i\f7\fs20 Aufbau}{\f7\fs20 , ra\-ther I'd like to
recall with as little fuss as possible the basic conceptual ideas of syn\-the\-tic geometry ne\-cessary to understand the quasianalytical constitution of physical space that Car\-
nap could have carried out, if only he had paid more attention to the expressive power of geo\-me\-try}{\i\f7\fs20 .}{\cs16\super \chftn {\footnote \pard\plain \s19\qj\widctlpar\adjustright \f12\fs20\lang3082\cgrid {\cs16\super \chftn }{\f7\fs18
For a fuller account, see (Mormann 2002).}}}{\f7\fs20
\par \tab The }{\i\f7\fs20 leitmotif}{\f7\fs20 of synthetic geometry is order. As Carnap put it, geometry is a general theory of }{\i\f7\fs20 Ordnungsgef\u252\'9fge }{\f7\fs20 (complexes of order stipulations)}{\i\f7\fs20 . }{\f7\fs20 Carnap under\-stood }{
\i\f7\fs20 Ordnungsgef\u252\'9fge }{\f7\fs20 in a semi-technical sense intended to mean some\-thing like "re\-lational structure" or "structured set". By conceiving a domain as a possible application for the theory of }{\i\f7\fs20 Ordnungsgef\u252\'9fge}
{\f7\fs20 one imposes some order on it. This is achieved by certain }{\i\f7\fs20 Ord\-nungs\-setzun\-gen}{\f7\fs20 (stipulations of order). Hence, as a theory of }{\i\f7\fs20 Ordnungsgef\u252\'9fge}{\f7\fs20 synthetic ge\-o\-me\-try has a strong ap\-
pli\-cative dimension. In }{\i\f7\fs20 Der Raum}{\f7\fs20 Carnap explains this fact for projective ge\-o\-metries at great length. According to him, synthetic geometry is designed to offer an arsenal of possible con\-
ceptual schemes applicable to many domains. This leads to the following two characteristic features of geometry:
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx0\adjustright {\f7\fs20 (1) Space (and other geometric notions such as points and lines) are conceived as general notions havi
ng many different instantiations. Geometry studies them all without blinders. It does not aim to single out one geometric system as the "true" one.
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx0\adjustright {\f7\fs20 (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 ob\-\-
ject is determined by its relational position within a certain relational system.
\par }\pard \qj\ri18\widctlpar\tx0\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab Understanding geometry as a general theory of }{\i\f7\fs20 Ordnungsgef\u252\'9fge}{\f7\fs20 gives }{\i\f7\fs20 Ordnungs\-setz\-un\-gen }{\f7\fs20
a crucial role. The most important }{\i\f7\fs20 Ordnungssetzungen }{\f7\fs20 are lines. To put it bluntly, lines are the entities that establish geometric order. It is sur\-\-
prising that this simple idea, the imposition of order by lines, is sufficient to con\-sti\-tute }{\i\f7\fs20 all}{\f7\fs20 concept
s 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 ge\-\-neral concept of line in order that lines can play this almost universal role of }{\i\f7\fs20
Ordnungs\-setzungen}{\f7\fs20 . 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 geo\-me\-
tric systems }{\i\f7\fs20 function as}{\f7\fs20 lines.
\par }\pard \qj\fi567\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 The upshot of this is the following: a geometric system in the sense of synthetic geo\-me\-
try 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 }{\f3\fs20 \u-3891\'cd}{\f7\fs20 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 det
ermines which points are on which lines: (x, m) }{\f3\fs20 \u-3890\'ce}{\f7\fs20
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 in\-tersect if an
d only if there is a point x that belongs to both of them, i.e., there are or\-dered pairs (x, m), (x, k) }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20
I; two lines m and k are parallel (m || k) if and only if they 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) }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20
I. Depending on the axioms imposed on I dif\-ferent types of geometric systems are obtained. Traditionally, the most important ones are }{\i\f7\fs20 affine}{\f7\fs20 and }{\i\f7\fs20 projective}{\f7\fs20
systems, but in contemporary geometry, many other systems are studied as well (cf. ibidem). For later use let us note that the systems (V, L, I) of synthetic geometry are}{\i\f7\fs20 exten\-sio\-nal}{\f7\fs20 in the following sense:
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\i\f7\fs20 (2.1) Definition (Extensional Geometric Systems) .}{\f7\fs20 Let S = (V, L, I) be a system of syn\-thetic geometry. For m }{\f3\fs20 \u-3890\'ce}{\f7\fs20 L denote by V(m):= \{x; (x, m) }
{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20 I\} the set of points of the line m. Then S is an }{\i\f7\fs20 extensional}{\f7\fs20 system iff two lines m and n are equal iff their point sets V(m) and V(n) coincide.
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab Extensional systems (V, L, I) may be cast in a canonical form that eliminates lines as primitive: de\-
note the power set of V by PV. Then an extensional geometric system is isomorphic to a system of the form (V, L, I) where L }{\f7\fs20 {\field{\*\fldinst SYMBOL 205 \\f "Symbol" \\s 10}{\fldrslt\f3\fs20}}}{\f7\fs20
PV 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.
\par \tab Now let us begin with the endevour of connecting synthetic geometry as the theory of in\-ci\-dence structures with the constitutional theory of the }{\i\f7\fs20 Aufbau}{\f7\fs20
. First let us note a rather curious piece of evidence for such a connection. In }{\i\f7\fs20 Der Raum }{\f7\fs20 Carnap considers some geometric systems (P, C, I) that may be considered as primitive forerunners of the constitutional systems of the }{
\i\f7\fs20 Aufbau }{\f7\fs20 (for details see Mormann (2002): P is a set of objects (}{\i\f7\fs20 Gegen\-st\u228\'8ande}{\f7\fs20 ), C a set of concepts (}{\i\f7\fs20 Begriffe}{\f7\fs20 ) such that (p, c) }{\f3\fs20 \u-3890\'ce}{\f7\fs20
I iff the object p can be subsumed under c. Or, in other words, (p, c) }{\f3\fs20 \u-3890\'ce}{\f7\fs20 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:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }{\i\f7\fs20 (2.2) Conceptual Geometries (Der Raum, p. 14).}{\f7\fs20 "Let us assume that the objects P1, P2, ... fall under the concept P such that the followin
g 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:
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\fi-560\li560\ri18\sl360\slmult0\widctlpar\tx560\adjustright {\f7\fs20 (1)\tab \tab At least three P-objects fall under any g-concept, but not all P-objects can be \tab subsumed under one g-concept.
\par (2)\tab \tab For two different P-objects there is always one and only one g-concept under \tab which they fall, their "common" concept.
\par (3)\tab \tab If P1, P3, P'2 fall under g1, P2, P3, P'1 under g2, and g1 and g2 are dif\-fe-\-\tab rent, then there exists an object P4 that falls under the common concept of P1 \tab and P'1 and under the common concept of P2 and P'2; moreover there is a
\tab concept g3, which subsumes P1, but no object of g2."
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab The first two axioms of this system of conceptual geometry are eas
ily 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, wh
a
t 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 s
ubsumed"), respecti\-
vely. At first glance, 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 con\-cep\-tu\-
al geometries are the primitive precursors of the con\-sti\-tu\-ti\-onal systems of the }{\i\f7\fs20 Auf\-bau}{\f7\fs20 . They may be considered as powerful intuition pumps for the constitutional theory of the }{\i\f7\fs20 Aufbau}{\f7\fs20 (cf. \u167
\'a4 70, 72). One may say that Carnap took the con\-st\-\-i\-tu\-\-tional systems of the }{\i\f7\fs20 Aufbau }{\f7\fs20 and the conceptual geometries of }{\i\f7\fs20 Der Raum}{\f7\fs20 as being of the same ilk.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab In this paper I want to show that synthetic geometry not only belongs to the }{\i\f7\fs20 Aufbau\rquote s }{\f7\fs20 pre\-
history, rather it should be regarded as its conceptual core, which helps elucidate the true nature of the }{\i\f7\fs20 Aufbau}{\f7\fs20 program. In order to render plausible this contention, we have to recall an im\-por\-
tant result of 19th century mathe\-\-ma\-tics, to wit, the coordinatization theorem of affine geometries. We will use this theorem to show that the real number space }{\b\f7\fs20 R}{\f7\fs18\up6 2}{\f7\fs20 can be constituted from a purely qua\-li\-ta\-
tive base, or, to use Quine's wording, from Carnap's initial language of sense data and logic. In the context of the present pa\-per, this base will be a relational structure defined by a set of }{\i\f7\fs20 Elementarerlebnisse }{\f7\fs20 en\-\-
dowed with a purely qualitative similarity relation (cf. }{\i\f7\fs20 Auf\-bau}{\f7\fs20 \u167\'a4 108ff). In other words, we may consider qualitative, non-metrical geometry, as a foun\-da\-tional theory for number systems. Today, ph
ilosophers of mathematics do not at\-tri\-bute any philosophical relevance to this fact. Rather unanimously they consider ge\-ometry as reducible to linear algebra in such a way that ge\-o\-metry as a mathematical dis\-
cipline loses any genuine philosophical interest. It is re\-markable, and mathe\-ma\-ti\-cal\-ly non-trivial, that the direction of reduction may be re\-\-
versed. The proof of this crucial result is long and involved, and need not concern us here. It may suffice to pre\-sent an axiom system which shows, at least in principle, how number systems such as the real numbers }{\b\f7\fs20 R}{\f7\fs20
can be reconstructed in the framework of synthetic geometry:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }{\i\f7\fs20 (2.3) Affine Incidence Structures}{\f7\fs20 . Let A = (V, L, I) be a system of synthetic geometry. A is called an }{\i\f7\fs20 affine Pappus in\-ci\-dence structure}{\f7\fs20 (AP-structure) iff the following con\-di\-tions are satisfied:}{
\f7\fs20\ul
\par }{\f7\fs20
\par (PA1)\tab There exist at least three non-collinear points. (Nontriviality)
\par }\pard \qj\fi-700\li700\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20 (PA2)\tab Any two distinct points x and y lie on exactly one line. Hence, we may de\-note the line determined by x and y with xy. (Linearity)
\par (PA3)\tab Given a point x and a line m, there is exactly one line k that passes through x and is parallel to m. (Parallel axiom)
\par }\pard \qj\fi-560\li560\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20 (PA4)\tab If x, y, z is a triple of points on m, and x', y', z' points of m' such that \tab \tab xy' ll x'y and xz' ll x'z then yz' ll y'z (Pappus\rquote s axiom).
\par }\pard \qj\ri18\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab The axioms (PA1) \endash (PA4) suffice to ensure that the lines of an AP-incidence struc\-ture have a quite rich algebraic structure; they are fields in the sense of ma\-\-
the\-ma\-tics (cf. Goldblatt 1987). That is to say, for the points of a line n }{\f3\fs20 \u-3890\'ce}{\f7\fs20 L, operations of addition and mul\-\-ti\-pli\-ca\-
tion can be defined which obey the laws of associativity, commutativity, distributivity etc., thereby rendering n a field }{\b\f7\fs20 K}{\f7\fs20 . We need not study these operations in detail, their definition can be found in any textbook of syn\-the\-
tic 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 }{\b\f7\fs20 K}{\f7\fs20 is the familiar field }{\b\f7\fs20 R}{\f7\fs20
of real numbers:
\par }\pard \qc\li1020\ri1078\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20 \page
\par }\pard \qj\li1020\ri1078\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \tx-709\adjustright {\f7\fs20
\par
\par
\par }\pard \qc\li1020\ri1078\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \tx-709\adjustright {\f7\fs20
\par FIGURE 1
\par }\pard \qj\li1020\ri1078\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab 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 }{\f3\fs20
\u-3890\'ce\u-4064\'20}{\f7\fs20
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 com\-mu\-\-ta\-
\-tive group, i.e., addition on n is associative, commutative, has a neutral element 0, etc. In a similar way one can define a commutative mul\-ti\-plication \bullet on n and show that it obeys the laws a multi\-pli\-\-ca\-\-\-tion of a field has to sa\-
tisfy. In sum, these geo\-\-metrically defined operations + and \bullet render (n, +, }{\f3\fs20 \u-3913\'b7}{\f7\fs20 ) a field. This construction may be expanded to a coordinatization of the AP-structure A as follows:
\par }\pard \qc\li1020\ri1078\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\li1020\ri1078\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \tx-709\adjustright {\f7\fs20
\par
\par }\pard \qc\li1020\ri1078\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \tx-709\adjustright {\f7\fs20
\par
\par FIGURE 2
\par }\pard \qj\li1020\ri1078\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab Choose distinct points e(m) and e(n) on m and n respectively, both different from 0. The as\-signment of b(n) to b(m) (defined by the line b(m)b(n) parallel to e(m)e(n)) esta
\-blishes 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\rquote (n). Let a(n) correspond to a\rquote
(n) on n. Then the ordered pair (a(n), a(m)) are the coordinates of a in the coordinatization of A defined by the lines n, m, and the points e(m), e(n). Of course, choosing different basic lines and points, one ob
tains a different coordinatization. But, as is well known, all coordinatizations obtained in this way are isomorphic (cf. Goldblatt 1987, ch.2).
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab In the rest of this paper I intend to show that this coordinatization provides the base for a quasianalytical constitution of spacetime in the sense of the }{\i\f7\fs20 Aufbau. }{
\f7\fs20 A first bit of evi\-dence that the constitutional theory of the }{\i\f7\fs20 Aufbau}{\f7\fs20 is related the sys\-tems of synthetic geometry are the "conceptual geometries" of }{\i\f7\fs20 Der Raum}{\f7\fs20
. In order to substantiate this relation we have to delve deeper into the technicalities of the }{\i\f7\fs20 Aufbau}{\f7\fs20 .
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 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 constitu
tion can be described in terms of the system (E, Q, I). This is strong evidence that something like a \-co\-or\-di\-na\-tization of AP-planes may be carried out in the conceptual framework of the constitutional theory of the }{\i\f7\fs20 Aufbau}{\f7\fs20
. The aim of the following sections is to do exactly this. More precisely it will be shown that Carnap\rquote
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.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx360\tx6663\adjustright {\f7\fs20
\par }\pard \qc\ri18\sl360\slmult0\widctlpar\adjustright {\b\f7\fs20 3. Quasianalysis and Synthetic Geometry}{\b\f7\fs20\ul
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx360\tx6663\adjustright {\f7\fs20 \tab The }{\i\f7\fs20 Aufbau }{\f7\fs20
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 }{\i\f7\fs20 Aufbau}{\f7\fs20 is }{\i\f7\fs20 not}{\f7\fs20
to present a full-fledged constitutional system based on }{\i\f7\fs20 Elementar\-er\-lebnisse}{\f7\fs20 . Rather I assume that the }{\i\f7\fs20 Aufbau}{\f7\fs20 was intended to exemplify a new ge\-neral philosophical dis\-\-cipline, called }{\i\f7\fs20
Kon\-stitu\-ti\-ons\-theorie}{\f7\fs20 (constitutional theory), which had the task}{\i\f7\fs20 }{\f7\fs20 of}{\i\f7\fs20 }{\f7\fs20 "investigating all possible forms of step\-wise definitional sys\-\-tems of concepts" (Friedman 1999, p. 115, }{
\i\f7\fs20 Aufbau \u167\'a4 }{\f7\fs20 46). Carnap inten\-ded to create a sci\-en\-ti\-fic successor discipline of tra\-di\-tional epistemology and philosophy of science that re\-mai
ned neutral with respect to the futile metaphysical quarrels that had plagued the traditional accounts.}{\cs16\super \chftn {\footnote \pard\plain \qj\widctlpar\adjustright \f12\fs22\lang3082\cgrid {\cs16\super \chftn }{\f7 }{\f7\fs18
In this paper, I do not want to argue for this sweeping claim, rather I\rquote d like to rely on the interpretative efforts of authors such as Friedman, Proust, or Ri\-chard\-son who have supplied ample evidence for this contention (cf. Fried\-\-
man 1999, Proust 1986, Richardson 1998). According to their interpretations, the topics of phenomenalism, ge\-stalt theory, or reductionism do }{\i\f7\fs18 not}{\f7\fs18 lie at the heart of the }{\i\f7\fs18 Aufbau}{\f7\fs18 . }}}{\f7\fs20
As the core of this new kind of philosophy of science Carnap considered not the constitutional system he had sketched in the }{\i\f7\fs20 Aufbau}{\f7\fs20 but the }{\i\f7\fs20 method}{\f7\fs20
by which it was constituted. This constitutional method is the method of quasianalysis. Hence, if the thesis of the geometric origins of the }{\i\f7\fs20 Aufbau}{\f7\fs20
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:
\par }\pard \qj\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\li20\ri18\sl360\slmult0\widctlpar\adjustright {\i\f7\fs20 (3.1) Quasianalysis in the Framework of Synthetic Geometry.}{\f7\fs20 Let E be an in\-ho\-mo\-ge\-neous set. Quasi\-ana\-ly\-
zing E is taking the elements of E as points of a geometric system (E, Q, I). The system (E, Q, I) may be con\-si\-dered as a complex of }{\i\f7\fs20 Ordnungssetzungen}{\f7\fs20 , i.e., as a ge\-ometrical }{\i\f7\fs20 Ordnungsgef\u252\'9fge}{\f7\fs20
. The }{\i\f7\fs20 Ord\-nungsgef\u252\'9fge}{\f7\fs20 set up by (E, Q, I) may be inter\-preted as an externalisation of the inhomogeneities of E.
\par }{\f7\fs20\ul
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab In order to unfold this succinct characterization of quasianalysis the following re\-mar
ks may be in order. The key term of (3.1) is "non-homogeneous 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-homo\-geneous sets E are similarity struc\-tures. A similarity structure, denoted by (E, ~), is a set E endowed with a binary si\-mi\-la\-ri\-ty relation ~. Two ele\-
ments e and e' of E related by the relation ~ are said to be si\-milar to each other (e ~ e\rquote ). The re\-lation ~ is assumed to be reflexive and sym\-me\-
tric, i.e. each element is assumed to be similar to itself (e ~ e) and e ~ e' implies e' ~ e. The re\-\-lation need not be transitive, however. This is plausible, since if e is similar to e', and e' is si\-mi\-\-
lar to e'' then e need not be similar to e''. A non-homo\-geneous set (E, ~) may be re\-pre\-\-sen\-\-\-ted more per\-spi\-cuously as a graph:
\par }\pard \qc\ri1278\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qc\li1780\ri1278\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \adjustright {\f7\fs20
\par
\par
\par
\par
\par FIGURE 3
\par }\pard \qc\li1780\ri1278\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab Here, two distinct similar elements of E are connected by a straight line, and two ele\-ments that are not similar are not directly connected by a straight line. Actually, the con\-
\-cepts of simple graph and simi\-\-la\-ri\-ty structure are strictly equivalent. Hence, ac\-cor\-ding to the }{\i\f7\fs20 Aufbau}{\f7\fs20 , 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.
\par \tab In an unpublished manuscript with the programmatic title }{\i\f7\fs20 Quasizerlegung - Ein Verfahren zur Ordnung nichthomogener Mengen mit den Mit\-\-teln der Be\-\-zieh\-\-ungs\-lehre}{\f7\fs20 (Quasianalysis - A Method to Order Non-Homo\-\-ge\-
neous Sets by Means of the Theory of Relations) (Carnap 1922/23) the task of }{\i\f7\fs20 Quasizerlegung}{\f7\fs20 , i.e., quasianalysis for similarity structures, is described as follows:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\li851\ri823\widctlpar\adjustright {\f7\fs20 \ldblquote Suppose there is given a set of elements, and for each ele\-ment the spe\-\-cification to which it is similar. We aim at a des\-c\-ription of the set which only uses this information but as
\-cribes to these elements qua\-si\-com\-po\-nents or quasi\--proper\-ties in such a way that it is possible to deal with each element se\-pa\-rately using only the quasiproperties, without refe\-rence to other ele\-ments.\rdblquote (}{\i\f7\fs20
Quasizerlegung}{\f7\fs20 , p.4)
\par }\pard \qj\li1276\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab As }{\i\f7\fs20 Quasizerlegung }{\f7\fs20 makes clear, the method of quasianalysis is a purely formal method. It may be applied to any nonhomogeneous set (E, ~), not just to }{
\i\f7\fs20 Elementarerlebnisse}{\f7\fs20 as in the }{\i\f7\fs20 Aufbau}{\f7\fs20 . Submitting (E, ~) to quasianalysis means imposing certain }{\i\f7\fs20 Ord\-nungss\-etzungen }{\f7\fs20 on it in
order to unfold its structure. Thereby appropriate invariants may be found which characterize its structure in a suc\-cinct manner. As modern ma\-the\-\-
matics teaches us, the finding of characteristic invariants is an unending task. Even apparently simple st
ructures give rise to a profusion of invariants. Although Carnap concentrates on the quasianalysis of similarity structures, he mentions the possibility of sub\-mit\-ting }{\i\f7\fs20 all}{\f7\fs20 kinds of re\-la\-\-\-tional struc\-
tures to a quasianalytical constitution process (}{\i\f7\fs20 Aufbau }{\f7\fs20 \u167\'a4 104). Hence, I think it would not be to\-tal\-ly off to interpret quasianalysis in a ge\-ne\-ralized sense as mathe\-ma\-tical constitution }{\i\f7\fs20 \u252\'9f
berhaupt}{\f7\fs20 . Be that as it may, in the following we will concentrate on the quasianalysis of similarity structures. In }{\i\f7\fs20 Quasi\-zer\-legung }{\f7\fs20 (p. 4) Carnap describes this kind of quasianalysis axiomatically as the as\-sig\-
nation of quasi\-pro\-perties to the objects to be quasianalysed, whereby these qua\-si\-pro\-perties function as }{\i\f7\fs20 Ordnungssetzungen }{\f7\fs20 such that the similarity structure of E is taken into account:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }{\i\f7\fs20 (3.2) Definition .}{\f7\fs20 Let (E, ~) be a non-homogeneous set. A set of quasi\-pro\-per\-ties of the ele\-ments of E is a set of entities which satisfy the following requirements}{\cs16\f7\fs20\super \chftn {\footnote \pard\plain
\s19\widctlpar\adjustright \f12\fs20\lang3082\cgrid {\cs16\super \chftn }{ }{\f7\fs18 As is shown in Mormann (1994) there are indeed similarity structures that have essentially different sets of quasiproperties.}}}{\f7\fs20 :
\par
\par (C1) \tab If two elements are similar they share at least one quasi\-pro\-perty.
\par (C2) \tab If two elements are not similar they do not share any quasiproperty.
\par (C3)\tab If two elements are similar to exactly the same elements they have the same \tab quasiproperties.
\par (C4)\tab No quasiproperty can be removed without violating unless (C1) - (C3).
\par
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab The general intention of these requirements is to regulate the relation between "being similar" on the one ha
nd 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.
\par \tab Let us postpone for the moment the question of what quasiproperties "really are", sim\-ply assuming that there is a set Q := \{q; q quasiproperty of (E, ~)\}. With the help of Q we may define an incidence relation I }{\f3\fs20 \u-3891\'cd}{\f7\fs20
E x Q by
\par }\pard \qj\fi13\li20\ri18\widctlpar\adjustright {\f7\fs20
\par (3.3)\tab \tab \tab (e, q) }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20 I := the element e has the quasiproperty q.
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 In this way the relation between elements of E and their quasiproperties q may be succinctly described by the inciden
ce 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 discu
ssed in }{\i\f7\fs20 Der Raum}{\f7\fs20 . In contrast to the axioms for "conceptual geometries" in }{\i\f7\fs20 Der Raum}{\f7\fs20
(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 enter\-
tains 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.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab It will be expedient to define a quasianalysis in still another but equivalent way. As is evident, every relation I }{\f3\fs20 \u-3891\'cd}{\f7\fs20 E x Q gives rise to a mapping
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par (3.4) \tab \tab r}{\f7\fs20\dn4 I}{\f7\fs20 : E -----> PQ defined by r}{\f7\fs18\dn4 I}{\f7\fs20 (e) := \{q; (e, q) }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20 I\}
\par
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 Obviously, I and r}{\f7\fs18\dn4 I}{\f7\fs20 determine each other uniquely. Hence we may characterize a quasi\-ana\-
lysis either by (3.3) as an incidence relation, or by (3.4) as a suitable map (cf. Mormann 1994). The version (3.4) is particularly convenient if one wants to check if conditions such as
(C1) - (C4) are satisfied. Summarizing we end up with the following characterization of a quasi\-ana\-ly\-sis of similarity structures (or, in terms of }{\i\f7\fs20 Quasizerlegung}{\f7\fs20 , "non-homogeneous sets"):
\par
\par }{\i\f7\fs20 (3.5) Definition of Quasianalysis .}{\f7\fs20 Let (E, ~) be a similarity structure. A }{\i\f7\fs20 quasianalysis}{\f7\fs20 for E is an incidence structure (E, Q, I) as defined in (3.3), or equivalently, as a representation r}{\f7\fs18\dn4 I
}{\f7\fs20 : E ----> PQ satisfying the requirements (C1) - (C4) as defined in (3.3).
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab Now let us come back to the question of what quasiproperties "really are". Since the constitutional theory of the }{\i\f7\fs20 Aufbau }{\f7\fs20
is extensional we may assume that the quasi\-ana\-ly\-ti\-cal systems (E, Q, I) are extensional systems in the sense of (2.1) where the quasiproperty q is identified with the set \{e; (e, q) }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20 I\}
. A set Q of quasiproperties may thus be considered as a subset Q }{\f3\fs20 \u-3891\'cd\u-4064\'20}{\f7\fs20 PE of subsets of E. Hence the general format of a quasianalysis of a similarity structure (E, ~) is
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par (3.6) \tab \tab \tab I }{\f3\fs20 \u-3891\'cd}{\f7\fs20 E x PE or r}{\f7\fs18\dn4 I}{\f7\fs20 : E ----> PPE
\par
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 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 o
f (E, ~) as special subsets of E by taking into account the similiarity structure of (E, ~). For this purpose we need the following preparatory definition:
\par
\par }{\i\f7\fs20 (3.7) Definition.}{\f7\fs20 Let (E, ~) be a similarity structure. A similarity circle T }{\i\f7\fs20 (\u196\'80hnlichkeitskreis}{\f7\fs20 ) is a sub\-\-set T }{\f3\fs20 \u-3891\'cd}{\f7\fs20 E which satisfies the requirements
\par
\par (i)\tab \tab (x) (y) (x,y }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20 T -----> x ~ y)
\par (ii)\tab \tab (x) }{\f3\fs20 \u-4060\'24}{\f7\fs20 y ( x }{\f3\fs20 \u-3889\'cf}{\f7\fs20 T ------> y }{\f3\fs20 \u-3890\'ce}{\f7\fs20 T and NOT(x ~ y))
\par
\par Denote the set of similarity circles of (E, ~) by SCE. Conceiving a similarity struc\-ture (E, ~) as a graph, similiarity circles T may be characterized as maximal sub\-\-\-graphs of (E, ~) all of whose elements are si\-
milar 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:
\par
\par }{\i\f7\fs20 (3.8) Definition (Aufbau \u167\'a4 69, Mormann 1994) .}{\f7\fs20 A quasianalysis (E, Q, I) of the similarity structure (E, ~) is }{\i\f7\fs20 of the first kind}{\f7\fs20 if and only if all its quasiproperties are elements of SCE, i.e. Q }{
\f3\fs20 \u-3891\'cd\u-4064\'20}{\f7\fs20 PSCE. In mapping form this is expressed by the requirement that a quasianalysis of the first kind has the form
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par \tab \tab \tab r}{\f7\fs18\dn4 I}{\f7\fs18 : }{\f7\fs20 E ------> PSCE rather than r}{\f7\fs18\dn4 I}{\f7\fs18 : }{\f7\fs20 E ------> PPE
\par
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 In the following we will only consider quasianalysis of the first kind.}{\cs16\super \chftn {\footnote \pard\plain \s19\qj\widctlpar\adjustright \f12\fs20\lang3082\cgrid {\cs16\super \chftn }{
}{\f7\fs18 In the }{\i\f7\fs18 Aufbau}{\f7\fs18 , Carnap also considers a quasianalysis of the second kind which has formally less satisfying properties.}}}{\f7\fs20
Similarity circles may be used to define a "standard pseudo-quasianalysis" in the following way:
\par
\par }{\i\f7\fs20 (3.9) Proposition .}{\f7\fs20 Let (E, ~) be a similarity structure. Define a map r}{\f7\fs18\dn4 I}{\f7\fs18 : }{\f7\fs20 E ---> PSCE by r}{\f7\fs18\dn4 I}{\f7\fs20 (x) := \{T: T }{\f3\fs20 \u-3890\'ce}{\f7\fs20 SCS(E) and x }{\f3\fs20
\u-3890\'ce}{\f7\fs20 T\}. Then r}{\f7\fs18\dn4 I}{\f7\fs18 }{\f7\fs20 satisfies the conditions (C1) - (C3).}{\cs16\super \chftn {\footnote \pard\plain \s19\qj\widctlpar\adjustright \f12\fs20\lang3082\cgrid {\cs16\super \chftn }{\f7\fs18
It may be called "pseudo-quasianalysis", since, as is shown in (Mormann 1994) it does not necessarily satisfy (C4).}}}{\f7\fs20
\par
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab In order to make the preceding chain of abstract definitions a little bit more vivid let us consider the following elementary example (Goodman 1951): \tab
\par }\pard \qc\li1020\ri1078\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par \page
\par }\pard \qj\li1020\ri1078\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \tx-709\adjustright {\f7\fs20
\par
\par
\par
\par }\pard \qc\li1020\ri1078\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \tx-709\adjustright {\f7\fs20 FIGURE 4
\par }\pard \qj\li1020\ri1078\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 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:
\par
\par \tab \tab \tab 1.\tab a\tab \tab \tab \tab 4.\tab c
\par (3.10)\tab \tab 2.\tab ab\tab \tab \tab \tab 5.\tab bcd
\par \tab \tab \tab 3.\tab abc\tab \tab \tab \tab 6.\tab d
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 A list of this kind is to be read as "1 ha
s 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\rquote
s requirements (C1) - (C4).
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab
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 o
f 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.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par
\par }\pard \qc\ri18\sl360\slmult0\widctlpar\adjustright {\b\f7\fs20 4. The Affine Plane as a Similarity Structure
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab 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 par\-\-
ticularly 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 it
s 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 }{\f3\fs20 \u-3891\'cd}{\f7\fs20 A x PA. Since in A any two different points x and y de\-ter\-
mine exactly one line, a line m may be denoted by xy, x and y being two different points of m.
\par \tab Now, our task is to characterize the Euclidean plane A (endowed with its standard incidence structure) as a simi\-la\-rity structure (A, ~). First, let us define an appropriate similarity relation. For this purpose, choose a class B o
f parallel lines of A. Depending on B we will define a similarity relation ~}{\f7\fs16\dn4 B}{\f7\fs20\dn4 }{\f7\fs20 on A. Hence, the resulting similarity structure should be denoted by (A, ~}{\f7\fs16\dn4 B}{\f7\fs20
). 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 in\-tui\-\-
tive vividness we may refer to the lines of B as horizontal lines. Having chosen B two points x and y are de\-fined to be similar iff they are equal or are on a line m }{\i\f7\fs20 not}{\f7\fs20 belonging to B:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par (4.1)\tab \tab \tab x ~ y := (x \u8800\'ad y and xy }{\f3\fs20 \u-3889\'cf\u-4064\'20}{\f7\fs20 B) or x = y
\par
\par The following diagram exhibits the geometrical meaning of this definition:
\par }\pard \qc\li1020\ri1078\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\li1020\ri1078\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \tx-709\adjustright {\f7\fs20
\par
\par }\pard \qc\li1020\ri1078\sl360\slmult0\widctlpar\box\brdrs\brdrw15 \tx-709\adjustright {\f7\fs20
\par
\par FIGURE 5
\par }\pard \qj\li1020\ri1078\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab Obviously, the relation ~ is reflexive and symmetric, but not tran\-sitive. Hence, (A, ~) is a similarity structure. Define the com\-ple\-men\-tary similarity struc\-\-
ture (A, ~*) by x ~* y := (xy }{\f3\fs20 \u-3890\'ce}{\f7\fs20 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. Ob\-vi\-ous\-
ly, (A, ~) and (A, ~*) determine each other, and all con\-si\-de\-rations dealing with ~* could be formulated in terms of ~, and vice versa. Hence, dealing wi
th (A, ~) and (A, ~*) (instead of (A, ~) or (A, ~*) alone) does not add anything new. After these preparations we are able to construct the following quasi\-ana\-ly\-sis of A:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }{\i\f7\fs20 (4.2) Lemma .}{\f7\fs20 Let (A, ~) be the similarity structure defined by (4.1). Then define Q: A ------> PPA by Q(x) := \{xy; xy }{\f3\fs20 \u-3889\'cf}{\f7\fs20 B\}. Then the map Q is a quasianalysis of A of the first kind.
\par
\par }{\i\f7\fs20 Proof:}{\f7\fs20 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 \u8800\'ad y we have x ~ y iff xy }{\f3\fs20 \u-3889\'cf
\u-4064\'20}{\f7\fs20 B. Hence xy }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20 Q(x) }{\f3\fs20 \u-3897\'c7}{\f7\fs20 Q(y) \u8800\'ad \u216\'af. On the other hand, if m }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\f7\fs20 Q(x) }{\f3\fs20 \u-3897\'c7}{\f7\fs20
Q(y) we have m }{\f3\fs20 \u-3889\'cf\u-4064\'20}{\f7\fs20 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.
\par 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 }{\f3\fs20 \u-3890\'ce}{\f7\fs20 Q(x). For any y }{\f3\fs20 \u-3890\'ce}{\f7\fs20 m and x
\u8800\'ad y we have xy = m. Hence, all points of m are similar to x. Suppose z }{\f3\fs20 \u-3890\'ce}{\f7\fs20 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 }{\f3\fs20
\u-3890\'ce}{\f7\fs20 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)):
\par
\par }{\i\f7\fs20 (4.3) Lemma .}{\f7\fs20 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:
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20 (PA1)\rquote \tab There exist at least three non-collinear points.
\par }\pard \qj\fi-700\li700\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20 (PA2)\rquote \tab Any two distinct similar points x and y lie on exactly one line. Hence, we \tab may de\-note the line determined by x and y with xy.
\par }\pard \qj\fi-560\li560\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20 (PA3)\rquote \tab Given a point x and a line m, there is exactly one line k that passes x and \tab \tab is parallel to m.
\par (PA4)\rquote \tab Suppose that x, y, z is a triple of points on m, and x', y', z' a triple of \tab \tab points of m' such that x ~ y', x' ~y, x ~ z', x' ~ z, y ~ z', y' ~z. If xy' \tab \tab ll x'y and xz' ll x'z then yz' ll y'z.
\par }\pard \qj\ri18\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 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)\rquote - (P4)\rquote
since the lines of B have to be left out.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab The incidence relation I}{\f7\fs20\dn4 Q}{\f7\fs20 defined by Q is not quite the incidence relation of the affine Eu\-
clidean 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 class
es 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 }{\f3\fs20 \u-3890\'ce}{\f7\fs20 q. Q* sa\-
tisfies the condition (C1), (C2), and (C4). Denote the incidence relation defined by Q* by I}{\f7\fs16\dn4 Q*}{\f7\fs20 . Then we define the set theoretical union I}{\f7\fs20\dn4 QQ* }{\f3\fs20 \u-3891\'cd}{\f7\fs20 A x PA of I}{\f7\fs16\dn4 Q}{\f7\fs20
and I}{\f7\fs16\dn4 Q*}{\f7\fs20 by
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20 \tab \tab
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 (4.4)\tab \tab \tab (x, m) }{\f3\fs20 \u-3890\'ce}{\f7\fs20 I}{\f7\fs16\dn4 QQ* }{\f7\fs20 :=}{\f7\fs20\dn4 }{\f7\fs20 (x, m) }{\f3\fs20 \u-3890\'ce}{\f7\fs20 I}{\f7\fs16\dn4 Q}{\f7\fs20
or (x, m) }{\f3\fs20 \u-3890\'ce}{\f7\fs20 I}{\f7\fs16\dn4 Q*}{\f7\fs20
\par }\pard \qj\ri18\widctlpar\adjustright {\f7\fs20\ul
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 This is the relation we need for the construction of a coordination mapping r}{\f7\fs20\dn4 QQ*}{\f7\fs20 : A -----> }{\b\f7\fs20 R}{\b\f7\fs13\up10 2}{\f7\fs20 . As is easily seen, I}{
\f7\fs16\dn4 QQ*}{\f7\fs20\dn4 }{\f7\fs20 indeed defines indeed an AP-plane in the sense of (2.3). Imposing some further axioms on I}{\f7\fs16\dn4 QQ*}{\f7\fs20\dn4 }{\f7\fs20
as we will do in the next section one can ensure that the affine plane defined by I}{\f7\fs16\dn4 QQ*}{\f7\fs20\dn4 }{\f7\fs20 is an Euclidean plane. Then the coordinatization procedure sketched in section 2 en\-sures that A can be mapped onto }{
\b\f7\fs20 R}{\b\f7\fs13\up10 2}{\b\f7\fs20\up6 }{\f7\fs20 in such a way that the incidence structure I}{\f7\fs20\dn4 QQ* }{\f7\fs20 on A is isomorphically mapped onto the standard real affine structure I }{\f3\fs20 \u-3891\'cd}{\f7\fs20 }{\b\f7\fs20 R}
{\b\f7\fs13\up10 2}{\f7\fs20 x P}{\b\f7\fs20 R}{\b\f7\fs13\up10 2}{\f7\fs20 .
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab Before we come to this task let us observe that the quasianalytical con\-\-\-struc\-\-tion of an affine plane achieved so far is unique up to iso\-
morphism. This is seen as follows: if we had chosen another family B' of parallels, we would have obtained a different si\-milarity structure (A, ~'). But then the similarity struc\-\-
tures (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 pa\-ra\-l\-le\-lism. This map defines an iso\-mor\-\-
phism between the similarity struc\-\-tures (A, ~) and (A, ~').
\par \tab 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 }{\b\f7\fs20 R. }{\f7\fs20 The crucial point in the proof that the field of the plane A is indeed }{
\b\f7\fs20 R }{\f7\fs20 is the observation that }{\b\f7\fs20 R }{\f7\fs20 is distinguished from other fields in that it is a }{\i\f7\fs20 Dedekind complete ordered}{\f7\fs20 field. That is to say, the ele\-\-
ments 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 col\-
linear points x, y, z. Thus, in order to con\-st\-ruct the real af\-fine plane from a similarity structure (S, ~) one has to assume the existence of an order on the si\-mi\-\-\-\-larity circles T }{\f3\fs20 \u-3890\'ce}{\f7\fs20 Q(SC(A,~)) }{\f3\fs20
\u-3896\'c8}{\f7\fs20 Q*(SC(A, ~*)). This leads to the following definition:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }{\i\f7\fs20 (4.5) Definition .}{\f7\fs20 Let (E, ~) be a similarity structure. A quasianalysis Q: E ----> PSCE is an ordered quasianalysis if and only if the similarity circles T }{\f3\fs20 \u-3890\'ce}{\f7\fs20 Q(E) }{\f3\fs20 \u-3891\'cd\u-4064\'20}{
\f7\fs20 SCE are endowed with an linear order.
\par
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab It can be shown that the real plane }{\b\f7\fs20 R}{\f7\fs20\up6 2}{\b\f7\fs20 }{\f7\fs20
(conceived as a similarity structure (A, ~) via its standard affine structure) has an ordered quasia
nalysis that is compatible with the field structure defined on the lines m. 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 <. He
n
ce 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 th
at the ordered fields of our lines are }{\i\f7\fs20 Dedekind complete}{\f7\fs20 in the standard sense (e.g. Goldblatt 1987, p.69/70):
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }{\i\f7\fs20 (4.6) Definition .}{\f7\fs20 Let }{\b\f7\fs20 K}{\f7\fs20 be an ordered field. Assume that }{\b\f7\fs20 K}{\f7\fs20 is the union of two non-empty sets C and D such that x < y for all x }{\f3\fs20 \u-3890\'ce}{\f7\fs20 C and y }{\f3\fs20
\u-3890\'ce}{\f7\fs20 D. }{\b\f7\fs20 K}{\f7\fs20 is Dedekind complete if and only if there is some z }{\f3\fs20 \u-3890\'ce}{\f7\fs20 }{\b\f7\fs20 K}{\f7\fs20 such that x \u8804\'b2 z \u8804\'b2 y.
\par
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab As is well known, the structure of a Dedekind ordered complete field is cate\-\-go\-\-ri\-\-cal, i.e., up to isomorphism, there is only one Dedeki
nd complete ordered field, to wit, the field of real numbers }{\b\f7\fs20 R}{\f7\fs20
. Summarizing we have obtained the result that we may describe the affine Euclidean plane as a similarity structure (E, ~), which has a Dedekind complete ordered 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 coordinatization of a physical domain }{\b\f7\fs20 P }{\f7\fs20 that may be interpreted as physical space in the sense of Carnap.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par
\par }\pard \qc\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\b\f7\fs20 5. The Constitution of Physical Space
\par }{\i\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab Now we have gathered all the pieces we need to tackle our main task, the constitution of physical space along the lines of the quasianalytical constitution theory of the }{
\i\f7\fs20 Aufbau}{\f7\fs20 . To make clear what is going on let us succinctly recall what, according to the canons of the }{\i\f7\fs20 Aufbau}{\f7\fs20 , is to be constituted and what are the assumptions under which this constitution is carried out.
\par \tab
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 system.
\par \tab Carnap started the constitution of physical space with the presupposition that we al\-ready have the four-dimensional Minkowski vector space }{\b\f7\fs20 R}{\f7\fs18\up6 4}{\b\f7\fs20 }{\f7\fs20 as a purely ma\-the\-ma\-
tical (or logical) object. He was entitled to do so, since the constitution theory is based on the assumption that the }{\i\f7\fs20 Aufbauer}{\f7\fs20 has available for his purposes the full resources of logic and mathematics (cf. also Quine
1951). The mathematical object }{\b\f7\fs20 R}{\f7\fs18\up6 4}{\b\f7\fs20 }{\f7\fs20 cannot be regarded as physical space, of course, and Carnap does not make this assertion. Rather, according to him, the mathematical object }{\b\f7\fs20 R}{
\f7\fs18\up6 4}{\b\f7\fs20 }{\f7\fs20 acquires its status as physical space through the coordinatization of physical qualities such as colours by the points of this heretofore purely logical object.}{\b\f7\fs20 }{\f7\fs20
So we may say that physical space is the physically interpreted mathematical space }{\b\f7\fs20 R}{\f7\fs18\up6 4}{\b\f7\fs20 . }{\f7\fs20 This interpretation has to satisfy certain requirements, for example, stability condit
ions, but this need not concern us for the moment. The problem is, if this physical interpretation of the purely mathe\-ma\-ti\-cal object }{\b\f7\fs20 R}{\f7\fs18\up6 4}{\b\f7\fs20 }{\f7\fs20 can be carried out }{\i\f7\fs20 in}{\f7\fs20
the constitutional system of the }{\i\f7\fs20 Aufbau}{\f7\fs20 by using the method of quasianalytical constitution only, without the introduction of new undefined primitives as the notorious }{\i\f7\fs20 is at}{\f7\fs20
relation. Quine claimed that Carnap\rquote s constitutional sketch failed to do this and went on to contend that the quasianalytical constitution of spacetime is principally impossible
. He took this as a conclusive argument against the feasibility of empiricist reductionism. Up to this day, Quine\rquote s verdict has been accepted almost unanimously. That is to say, even those who maintain that Quine\rquote
s empiricist interpretation of the }{\i\f7\fs20 Aufbau }{\f7\fs20 i
s 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 want to show that, although Carnap has failed to provide a qua\-si\-\-analytical constitution, he }{\i\f7\fs20 could}{\f7\fs20 have done it. A quasianalytical constitution }{\i\f7\fs20 is}{\f7\fs20
feasible. For this purpose it is expedient first to recall Quine\rquote s full argument, in which he spot\-ted a change of method in Carnap\rquote s constitutional entreprise. He describes Carnap\rquote s procedure as follows:
\par }\pard \qj\ri18\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\li680\ri738\widctlpar\tx-709\adjustright {\f7\fs20 "He [Carnap] explained spatio-temporal point-instants as quadruples of real numbers and en\-vi\-saged assignment of sense qualities to point-in
stants 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 con
structing a world from experience.
\par 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" wer
e, 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 schema
t
ization (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 s
ense 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)
\par }\pard \qj\ri18\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab As I have said, even those who reject Quine's empiricist interpretation of the }{\i\f7\fs20 Aufbau}{\f7\fs20 con\-\-
cede 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 trans\-lation. This, however, does not imply that such a translation is impossible.
\par \tab Bringing to bear the apparatus of synthetic geometry de\-ve\-loped in the preceding sections, this can be done as follows: The fatal flaw of Carnap's at\-
tempt 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 ma\-thematical object }{\b\f7\fs20 R}{\f7\fs18\up6 4}{\f7\fs20 on the one hand, an
d the physical object, i.e. the quasianalytically constituted domain of physical qualities }{\b\f7\fs20 P}{\f7\fs20
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 qua\-druples of real numbers r }{\f3\fs20 \u-3890\'ce
\u-4064\'20}{\b\f7\fs20 R}{\f7\fs18\up6 4}{\f7\fs18 }{\f7\fs20 and physical qualities q }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\b\f7\fs20 P}{\f7\fs20 . Even if we grant that this is pos\-\-
sible following certain (empiristically acceptable) "canons", this assign\-ment of num\-\-ber quadruples to quali\-ties cannot be considered as a quasiana\-ly\-ti\-cal con\-sti\-tution. That is to say, we have a fatal gap between the physical and the ma\-
thematical that cannot be bridged by quasi\-analytical constitution. Rather, as Quine correctly observes, one has to rely on the un-quasi\-ana\-ly\-\-tical extra primitive relation "is at" to assert state\-ments of the type "Quality q }{\i\f7\fs20 is at}
{\f7\fs20 (x,y,z;t)". So far, so good. The question is, whether Carnap\rquote s deviation from the path of true quasianalytical constitution was unavoidable. Indeed, it is possible to save Carnap from himself and Quine and his fol\-lo\-
wers. It is not necessary to assume that the ready-made mathematical structure }{\b\f7\fs20 R}{\f7\fs18\up6 4}{\b\f7\fs20 }{\f7\fs20 is already there, waiting to be related or in\-terpreted empirically. Rather, we may constitute physical spac
e, i.e., a physically in\-terpreted }{\b\f7\fs20 R}{\f7\fs18\up6 4}{\b\f7\fs20 }{\f7\fs20 in one fell swoop, so to speak. This amounts to a sort of auto-coordinatization of the already constituted domain }{\b\f7\fs20 P }{\f7\fs20
of qualities along the lines sketched in the pre\-vious section.
\par \tab As already announced in the introduction to avoid unnecessary technicalities, let us re\-place }{\b\f7\fs20 R}{\f7\fs18\up6 4 }{\f7\fs20 by }{\b\f7\fs20 R}{\f7\fs18\up6 2}{\f7\fs18 . }{\f7\fs20
Hence, the problem is to provide a quasianalytically acceptable co\-or\-dinatization of the already constituted do\-main }{\b\f7\fs20 P}{\f7\fs20 . This can be done as follows: The system }{\b\f7\fs20 P}{\f7\fs20 is assumed to be a simila
rity structure (E, ~). It does not matter whether the elements of E are intuitively interpreted as }{\i\f7\fs20 Elementarerlebnisse}{\f7\fs20
or qualities or whatever, since the "nature" of the elements does not play a role in constitution theory. According to (3.5) a quasianaly
sis 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 coordi\-na\-ti\-
zation 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 coordinatization theorem tells us that E is the 2-dimensional real number space }{\b\f7\fs20 R}{
\f7\fs18\up6 2}{\f7\fs20 . That is to say, in contrast to Carnap\rquote s flawed coordinatization, which
first separated the physical and the mathematical, 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 t
he following definition:
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }{\i\f7\fs20 (5.1) Affine Pappian Similarity Structure .}{\f7\fs20 Let (E, ~) be a similarity structure. E is called an }{\i\f7\fs20 affine Pappian similiarity}{\f7\fs20 structure (AP-similarity structure) iff the fol\-lo\-wing holds:
\par }\pard \qj\ri18\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20 (1) \tab E has a quasianalysis (E, SCE, I) of the first kind satisfying (C1) - (C4).
\par (2)\tab The complementary similarity structure (E, ~*) is an equivalence relation.
\par (3)\tab I satisfies the requirements (P1)\rquote - (P4)\rquote .
\par (4)\tab For J : = I }{\f3\fs20 \u-3896\'c8\u-4064\'20}{\f7\fs20 I* the system (E, SCE }{\f3\fs20 \u-3896\'c8\u-4064\'20}{\f7\fs20 SCE*, J) is an affine Pappian plane.}{\cs16\super \chftn {\footnote \pard\plain \s19\qj\widctlpar\adjustright
\f12\fs20\lang3082\cgrid {\cs16\super \chftn }{\f7\fs18 Here, of course, SCE* is the class of similarity circles of the similarity structure (E, ~), intuitively to be interpreted as the class of deleted parallels.}}}{\f7\fs20
\par
\par As has been shown in the previous section, AP-similarity structures exist. In par\-ti\-cular, it should be noted that the definition of an AP-similarity structure is defined fully in terms of what Quine called "Carnap\rquote
s initial language of sense data and logic" (Quine 1951, p. 40).
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab Now a quasianalytical coordinatization of }{\b\f7\fs20 P}{\f7\fs20 = (E, ~) is at hand: choose three
non-collinear points r,s, and t. With the help of the intersecting lines rs and rt one may construct an internal coordinatization of E which renders it isomorphic to the 2-dimensional plane }{\b\f7\fs20 K}{\f7\fs20\up6 2}{\f7\fs20
of some commuative field }{\b\f7\fs20 K}{\f7\fs20 ." Thus, the statement \ldblquote Quality point x is at (p,q)\ldblquote have the meaning \ldblquote
with respect to the coordinatization 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 }{\b\f7\fs20 K}{\f7\fs20 . This may not be a coordinatization by real numbers, since we cannot be be
sure that the field }{\b\f7\fs20 K}{\f7\fs20 is the field }{\b\f7\fs20 R}{\f7\fs20
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):
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }{\i\f7\fs20 (5.2) Definition .}{\f7\fs20 An }{\i\f7\fs20 ordered field}{\f7\fs20 }{\b\f7\fs20 K}{\f7\fs20 is a field with a distinguished subset }{\b\f7\fs20 K}{\f7\fs20\up6 +}{\f7\fs20 closed under addition and multiplication such that for each x }{
\f3\fs20 \u-3890\'ce\u-4064\'20}{\b\f7\fs20 K}{\f3\fs20 \u-4052\'2c\u-4064\'20}{\f7\fs20 exactly one of the con\-ditions x }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\b\f7\fs20 K}{\f7\fs20 , -x }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\b\f7\fs20 K}{\f7\fs20
, x = 0 is true. }{\b\f7\fs20 K}{\f7\fs20\up6 + }{\f7\fs20 is to be thought as the set of positive ele\-ments of }{\b\f7\fs20 K}{\f7\fs20 . With the help of }{\b\f7\fs20 K}{\f7\fs20\up6 + }{\f7\fs20 one can defined an order on }{\b\f7\fs20 K }{\f7\fs20
by x < y := (y - x) }{\f3\fs20 \u-3890\'ce\u-4064\'20}{\b\f7\fs20 K}{\f7\fs20\up6 +}{\f7\fs20 .
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab The last step to ensure that }{\b\f7\fs20 K }{\f7\fs20 is the real number field }{\b\f7\fs20 R }{\f7\fs20 is done by imposing the further axiom on }{\b\f7\fs20 K }{\f7\fs20
that it also satisfies the Dedekind completeness axiom (4.6). Noting that the Dedekind axiom can be expressed in "Carnap\rquote s initial language of sense data and logic" we are done: it can be proved that }{\b\f7\fs20 K }{\f7\fs20
is (up to isomorphism) just }{\b\f7\fs20 R}{\f7\fs20 . Hence, the coordinatization as described in section 2 yields that }{\b\f7\fs20 P }{\f7\fs20 can be identified with }{\b\f7\fs20 R}{\b\f7\fs13\up10 2}{\b\f7\fs20 .}{\cs16\super \chftn {\footnote
\pard\plain \s19\qj\widctlpar\adjustright \f12\fs20\lang3082\cgrid {\cs16\super \chftn }{\f7\fs18 Another, possibly more elegant way to cope with the problem of endowing the lines with an order structure would be to pursue the approach by Robb (1934) (cf
. also Goldblatt 1987). Robb\rquote s reconstruction of spacetime is based on a single primitive "y is after z" which formally (but not intuitively) corresponds to Carnap\rquote s }{\i\f7\fs18 \uc1\u196\'80hnlichkeitserinnerung}{\f7\fs18 (cf. }{
\i\f7\fs18 Aufbau }{\f7\fs18 \uc1\u167\'a4 110) whose symmetrization is the similarity relation ~. Robb\rquote
s original presentation is difficult, a more accessible modern account is to be found in (Goldblatt 1987, Appendix B). The constructions are still complicated and I cannot go into the details. Nevertheless it should be said that the constructions
are quite compatible with the spirit of quasianalytical constitution. As it seems Carnap did not know Robb\rquote s work. Although Robb\rquote
s account has been treated by some authors in recent years, nobody seems to have studied more closely its possible relations with the }{\i\f7\fs18 Aufbau}{\f7\fs18 -approach.}}}{\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab Summarizing we may say that for AP-similarity structures (E, ~) whose lines are Dedekind-complete li\-nearly ordered point sets, there exists a quasi-analytically con\-sti\-\-
tuted coordinatization, which assigns real number coordinates to the elements of E. Of course, this coordinatization is not unique: Given an AP-quasianalysis (E, SCE, I) of E one may choose another triple of non-collinear elements of r\rquote , s\rquote
, t\rquote yielding an\-other coordinatization (E, SCE, I\rquote ) related to the former by a unique linear iso\-mor\-phism. More\-
over, 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\rquote ).}{\cs16\super \chftn {\footnote \pard\plain
\s19\qj\widctlpar\adjustright \f12\fs20\lang3082\cgrid {\cs16\super \chftn }{\f7\fs18 For an argument (contra Goodman) that the non-uniqueness of quasianalysis should not be con\-\-sidered as a fatal flaw, see (Mormann 2002). }}}{\f7\fs20
This gives ample leeway for conventional choices. That is to say, for some reason or other the }{\i\f7\fs20 Aufbauer}{\f7\fs20 may prefer one coordinati
zation over the other, since one allows maximizing certain desirable over-all features, and the other does not. Thereby the quasianalytical constitution of physical space, i.e. the consti\-tu\-ti\-
on of a spatio-temporal coordinatization of the physical domain }{\b\f7\fs20 P}{\f7\fs20 = (E, ~), is seen to follow, at least schematically, what science actually does (cf. }{\i\f7\fs20 Aufbau }{\f7\fs20 \u167\'a4 135, 136, Richardson 1998, pp.70ff).
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par
\par }\pard \qc\ri18\sl360\slmult0\widctlpar\adjustright {\b\f7\fs20 6. Concluding Remarks
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\adjustright {\f7\fs20
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx-709\tx567\adjustright {\f7\fs20 \tab Summarizing we conclude that the program of quasianalytical constitution is not bound to bre
ak down when it comes to the quasianalytical constitution of physical space. It may founder at other points, or may be considered to be unattractive for other reasons, but there is no deep reason why it has to fail at the foundations of physical space, a
s Carnap understood this notion.
\par \tab One may well wonder why Quine and so many philosophers following him were confident that Carnap\rquote s failure providing a sketch of a quasianalytical consti\-tu\-tion of spacetime was tantamount to the impossibility of achieving
this task in general. Quine ne\-
ver 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 g
eometry 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\rquote
s initial language is principally unable to cope with the quantitative as it crops up in the real number coordinatization of physical space.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\tx567\adjustright {\f7\fs20 \tab 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 reconstruc
tion pro\-\-grams such as the }{\i\f7\fs20 Aufbau}{\f7\fs20 's. Hence, some basic knowledge of synthetic geometry may still be useful also for philosophers of the 21st century.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20
\par }{\b\f7\fs20 References:}{\f7\fs20\ul
\par }{\f7\fs20
\par }\pard \widctlpar\adjustright {\f7\fs20 Buekenhout, F. }{\i\f7\fs20 Handbook of Incidence Geometry}{\f7\fs20 (Ed.), Amsterdam: Elsevier, 1995.
\par }\pard \qj\ri18\sl360\slmult0\widctlpar\adjustright {\f7\fs20 Carnap, R. "Der Raum: Ein Beitrag zur Wissenschaftslehre" }{\i\f7\fs20 Kant Studien, Erg\u228\'8anzungshefte }{\f7\fs20 56. 1922.
\par Carnap, R. }{\i\f7\fs20 Quasizerlegung, Ein Verfahren zur Ordnung nicht\-ho\-\-mo\-\-ge\-ner Mengen mit den Mitteln der Beziehungslehre}{\f7\fs20 . Unpublished Manuscript, Car\-nap Archive, University of Pittsburgh, RC-081-04-01, 1922/23.
\par Carnap, R. }{\i\f7\fs20 Der Logische Aufbau der Welt}{\f7\fs20 . Hamburg: Felix Meiner Verlag, 1961(1928).
\par Cassirer, E. }{\i\f7\fs20 Substanzbegriff und Funktionsbegriff}{\f7\fs20 . Darmstadt: Wissen\-schaft\-liche Buchgesellschaft, 1985(1910).
\par Coppel, W.A. 1998, }{\i\f7\fs20 Foundations of Convex Geometry}{\f7\fs20 . Cambridge: Cambridge Univer\-si\-\-ty Press, 1998.
\par Friedman,M. }{\i\f7\fs20 Reconsidering Logical Positivism}{\f7\fs20 . Cambridge: Cam\-bridge Uni\-ver\-sity Press, 1999.
\par Goldblatt, R. }{\i\f7\fs20 Orthogonality and Spacetime Geometry}{\f7\fs20 . New York and Wien: Springer Verlag, 1987.
\par Goodman, N. }{\i\f7\fs20 The Structure of Appearance}{\f7\fs20 . Bobbs-Merrill: Indianapolis, 1951.
\par Goodman, N. }{\i\f7\fs20 Seven Strictures on Similarity, Projects and Problems}{\f7\fs20 . Bobbs-Merrill: Indianapolis, 1972.
\par Hilbert, D. }{\i\f7\fs20 Foundations of Geometry}{\f7\fs20 : La Salle, The Open Court: 1971(1899).
\par Mormann, Th. "A Representational Reconstruction of Carnap\rquote s Quasianalysis", }{\i\f7\fs20 PSA}{\f7\fs20 1994, vol. 1, 96 - 104.
\par Mormann, Th. "Incompatible Empirically Equivalent Theories: a Structural Expli\-ca\-tion." }{\i\f7\fs20 Synthese}{\f7\fs20 103(1995), 203 - 249.
\par Mormann, Th. "A Quasianalytical Reconstruction of Spacetime." Talk presented at the Bi\-\-ennial conference of the PSA, Vancouver 2000.
\par Mormann, Th. 2002, "Synthetic Geometry and }{\i\f7\fs20 Aufbau"}{\f7\fs20 , to appear in Th. Bonk (ed.), }{\i\f7\fs20 Language, Logic and Truth, The Philosophy of Rudolf Carnap}{\f7\fs20 , Vienna Circle Collection, Dordrecht: Kluwer, 2002.
\par Proust, J., 1989}{\i\f7\fs20 , Questions of Form. Logic and the Analytic Proposition from Kant to Carnap}{\f7\fs20 , Minneapolis: University of Minnesota Press, 1989.
\par Quine, W.V.O. "Two Dogmas of Empiricism", in }{\i\f7\fs20 From a Logical Point of View}{\f7\fs20 , Cambridge/Massachusetts: Harvard University Press, 20 - 46, 1953(1961).
\par Richardson, A.W. }{\i\f7\fs20 Carnap's Construction of the World, The Aufbau and the Emergence of Logical Empiricism}{\f7\fs20 . Cambridge: Cambridge University Press, 1998.
\par Robb, A.A. }{\i\f7\fs20 A Theory of Time and Space}{\f7\fs20 . Cambridge: Cambridge University Press, 1936.
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