This paper is devoted to some mathematical considerations on the geometrical ideas contained in *PNK*, *CN* and, successively, in *PR*. Our main point will be to emphasize that these ideas give very promising suggestions for a modern point-free foundation of geometry.

### 1. Introduction

Recently research in point-free geometry has received increasing interest in areas as diverse as computability theory, lattice theory, and computer science. The basic ideas of point-free geometry were firstly formulated by A. N. Whitehead in *PNK* and *CN *where the *extension relation* between events is proposed as a primitive. The points, the lines and all the “abstract” geometrical entities are defined by suitable abstraction processes. Indeed, as observed in Casati and Varzi 1997,^{[1]} the approach proposed in these books is a basis for a *“mereology” *(i.e. an investigation about the *part-whole* relation) rather than for a point-free geometry. Indeed, the inclusion relation is set-theoretical and not topological in nature and this generates several difficulties. As an example, the definition of point is unsatisfactory (see Section 6). So, it is not surprising that some years after the publication of *PNK* and *CN*, Whitehead in *PR *proposed a different approach in which the primitive notion is the one of *connection* *relation*. This idea was suggested in de Laguna 1922.^{[2]}

The aim of this paper is not to give a precise account of geometrical ideas contained in these books but only to emphasize their mathematical potentialities. So, we translate the analysis of Whitehead into suitable first-order theories and examine these theories from a logical point of view. Also, we argue that multi-valued logic is a promising tool to reformulate the approach in *PNK* and *CN*.

In the following we refer to first-order logic. If *L *is a first order language, *a *a formula whose free variables are among *x*_{1},…,*x _{n}* and

*I*an interpretation of

*L*with domain

*S*, then we write

*I*£

*a*[

*d*

_{1},…,

*d*] to say

_{n}*a*is satisfied in

*I*by the elements

*d*

_{1},…,

*d*. Given a relation

_{n}*R*Í

*S*, we say that

^{n}*R*is

*defined by*

*a*or that

*R*is

*the extension of*

*a*in

*I*,provided that

*R*= {(

*d*

_{1},…,

*d*):

_{n}*I*£

*a*[

*d*

_{1},…,

*d*]}. For example, the extension of the formula $

_{n}*r*(

*r*£

*x*Ù

*r*£

*y*) is the overlapping relation.

### 2. A Mathematical Formulation of the Inclusion Based Approach

In *PNK *and *CN *one considers as primitives *events *and a binary relation named *extension. *Indeed, Whitehead says:

The fact that event a extends over event b will be expressed by the abbreviation aKb. Thus ‘K’ is to be read ‘extends over’ and is the symbol for the fundamental relation of extension.

Moreover, Whitehead in *PNK *lists the following properties of the extension relation.

Some properties of K essential for the method of extensive abstraction are,

i) aKb implies that a is distinct from b, namely, ‘part’ here means ‘proper part’:

ii) Every event extends over other events and is itself part of other events: the set of events which an event e extends over is called the set of parts of e:

iii) If the parts of b are also parts of a and a and b are distinct, then aKb:

iv) The relation K is transitive, i.e. if aKb and bKc, then aKc:

v) If aKc, there are events such as b where aKb and bKc:

vi) If a and b are any two events, there are events such as e where eKa and eKb.

We adopt a slightly different notation which is related in a more strict way to recent research in point-free geometry. So, in accordance with *PR*, we use the word “region” instead of “event.” Also, we call inclusion relation the converse of the extension relation and we refer to the partial order £ rather than to the strict partial order. In accordance, we can reformulate the list of properties proposed by Whitehead into a simple first order theory whose language L_{£} contains only a binary relation symbol £.

Definition 2.1. We call inclusion based point-free geometry the first order theory defined by the following axioms:

(i)“x(x£x) (reflexive)

(ii)“x“y“z((x£zÙz£y) Þx£y) (transitive)

(iii)“x“xy(x£yÙy£xÞx=y) (anti-symmetric)

(iv)“z$x$y(x<z<y) (there is no minimal or maximal region)(

v) “x“y(x<yÞ$z(x<z<y)(dense)(

vi) “x“y$z(x£zÙy£z)(upward-directed)(

vii) “x“y(“x’(x’<xÞx’<y)Þx£y).

We call *inclusion space *any model of this theory.

Then, if we denote by £ the interpretation of £, an inclusion space is a structure (*S*,£) where *S *is a nonempty set and £ an order relation with no minimal or maximal element which is dense, upward-directed and such that, for every region *x*, *x* = *sup*{*x’*Î*S:* *x’*<*x*}.

The existence of suitable mathematical models for the inclusion-based point-free geometry is a basic question, obviously (in spite of the fact that Whitehead seems to understimate it). To argue how this should be done, recall that in the case of non-Euclidean geometry, Poincaré, Klein and others authors proposed models which were defined from the usual Euclidean spaces. In the same way, we are justified in defining models of point-free geometry by starting from a *n-*dimensional(point-based) Euclidean space *R ^{n}* (where

*R*denotes the real numbers set). In accordance, we have to propose a suitable class of subsets of

*R*to represent the notion of region. To do this, usually in literature one refers to the regular subsets of

^{n}*R*.

^{n}Definition 2.2. We call *closed regular* any subset *x *of *R ^{n}* such that

*x*=

*cl*(

*int*(

*x*)) where

*cl*and

*int*denotes the closure and interior operators, respectively. We denote by

*RC*(

*R*) the class of all the closed regular subsets of

^{n}*R*.

^{n}There are several reasons in favour of the notion of regular set. As an example, in accordance with our intuition, the closed balls and the cubes of three-dimensional geometry are regular sets. Instead, points, lines and surfaces are not regular (in accordance with Whitehead’s aim to define these geometrical notions by abstraction processes). Moreover the class *RC*(*R ^{n}*) defines a very elegant algebraic structure. Indeed it is a complete atom-free Boolean algebra with respect to the inclusion relation. More precisely, to obtain a model of Whitehead’s axioms we have to refer to a suitable subclass of

*RC*(

*R*). In fact, Axiom

^{n}*iv*) says that the whole space

*R*and the empty set Æ are not seen as regions by Whitehead. Also, it is evident that Whitehead refers only to bounded regions. Otherwise, for example, the proposed notion of “point” (see Section 4) should be not able to exclude points “at infinity”. This leads to assume as natural models of the notion of region the bounded nonempty closed regular sets.

^{n}Theorem 2.3. Let *Re *be the class of all closed nonempty bounded regular subsets of *R ^{n}*. Then (

*Re*,Í) is an inclusion space we call

*the canonical inclusion space*.

*Proof.* Properties (*i*), (*ii*), (*iii*)and (*vi*) are trivial. To prove (*iv*)*, *let *x*Î*Re*, then, since *int*(*x*) ≠Æ, an open ball with radius *r *and centre *P *contained in *x *exists. We denote by* x*_{1} the closed ball with centre *P *and radius *r*/2. Then *x*_{1}Î*Re* and* x*_{1 }Ì *x*. Also, since *x* is bounded, a closed ball with centre *P *and radius *r’ *containing *x *exists. Let *x*_{2} the closed ball with centre *P *and radius 2×*r’*. Then *x*_{2}Î*Re* and *x *Ì* x*_{2}*.*

To prove (*v*), assume that *x *Ì* y *and let *P*Î* int*(*y*)such that *P*Ï*x*. Then there is a closed ball *y’*, *y’*Ì*y* and *y’*Ç*x *= Æ. By setting *z* = *y’*È*x* we obtain a bounded regular subset such that *x *Ì* z *Ì* y*. To prove *(vii), *assume that all the regular proper subsets of *x *are contained in *y *and that *x* is not contained in *y*. Then *int*(*x*)is not contained in *y*, too. So there is a point *P*Î* int*(*x*) and a real positive number *r *such that the closure of the ball with centre *P* and radius *r *is contained in *int*(*x*)and disjoint from *y*: a contradiction.

Notice that in literature inclusion spaces are obtained also from the class *RO*(*R ^{n}*)of

*open regular subsets,*i.e. the sets

*x*such that

*x*=

*int*(

*cl*(

*x*)) (see for example Pratt 2006). The arising model is isomorphic with (

*Re*,Í).

Proposition 2.4. Denote by *Re’ *the class of bounded nonempty open regular subsets. Then the structure (*Re’*,Í) is isomorphic with the structure (*Re*,Í) and therefore is an inclusion space.

*Proof. *We observe only that the map *int*: *RC*(*R ^{n}*)→

*RO*(

*R*)is an order-isomorphism between (

^{n}*RC*(

*R*),Í) and (

^{n}*RO*(

*R*),Í). Moreover,

^{n}*x*is bounded if and only if

*int*(

*x*) is bounded.

Finally, observe that perhaps the class *Re *is still too large. Indeed Whitehead’s intuition refers to the *connected *(in a topological sense) elements in *Re*. Also, for example, in Pratt 2006 several possible subsets of *Re *are considered.

### 3. A Mathematical Formulation of the Connection Based Approach

Some years later the publication of *PNK*, and *CN*, Whitehead in *PR *proposed a different idea in which the primitive notion is the one of *connection* *relation*:

the terms ‘connection’ and ‘connected’ will be used […]. The term ‘region’ will be used for the relata which are involved in the scheme of ‘extensive connection’. Thus, in the shortened phraseology, regions are the things which are connected (

PR, Chapter II, Section I).

Now, Whitehead is not interested to formulate the properties of this relation as a system of axioms and to reduce them at a logical minimum.

No attempt will be made to reduce these enumerated characteristic to a logical minimum from which the remainder can be deduced by strict deduction. There is not a unique set of logical minima from which the rest can be deduced. There are many such sets. The investigation of such sets has great logical interest, and has an importance which extends beyond logic. But it is irrelevant for the purpose of this discussion (*PR*, Chapter II, Section I).

So a very long list of “assumptions” is proposed. As an example in Chapter II Whitehead included as many as thirty one assumptions! Nevertheless, it is possible to try to reduce these assumptions to a sufficiently small set of axioms. For instance, Gerla and Tortora have proven that the first twelve assumptions are equivalent to the following first-order theory (1992). We refer to a language *L _{C}* with a binary relation symbol

*C*to represent the connection relation and we write

*x*£

*y*to denote the formula “

*z*(

*zCx*Þ

*zCy*) and

*x*<

*y*to denote the formula (

*x*£

*y*)Ù(

*x*≠

*y*).

Definition 3.1. We call a *connection theory *the first-order theory whose axioms are:

*C1 *“*x*“*y*(*xCy *Þ*yCx*)(symmetry)

*C2 *“*x*$*y*(*x*<*y*) (there is no maximum for £)

*C3* “*x*“*y*$*z*(*zCx*Ù*zCy*)

*C4 *“*x*(*xCx*)

*C5 *“*z*(*zCx* Û *zCy*) Þ *x=y*

*C6 *“*z*$*x*$*y*((*x*£*z*)Ù(*y*£*z*)Ù(Ø*xCy*))

where *x*£*y *denotes the formula “*z*(*zCx *Þ *zCy*) and *x*<*y *the formula (*x*£*y*)Ù(*x*≠*y*). We call a* connection space *any model of such a theory.

We denote by *C *the interpretation of the relation symbol *C *and by £ the interpretation of £. So we write (*S*,*C*)to denote a connection space. It is easy to prove that in any connection space £ is an order relation. As in the case of inclusion spaces, we can define “canonical” connection spaces.

Theorem 3.2. (Gerla and Tortora 1996) Define in *Re *the relation *C *by setting

*XCY *Û *X*Ç*Y **≠ *Æ.

Then (*Re,C*) is a connection space we call *canonical connection space*. Moreover, the associated order relation coincides with the usual set theoretical inclusion.

An analogous definition can be given by referring to the regular open sets and by putting *XCY *if and only if *cl*(*X*)Ç*cl*(*Y*) ≠ Æ.

### 4. Abstractive Classes

In order to define the points and the lines and other “abstract” entities, Whitehead in *PNK *considers the following basic notion.

Definition 4.1. Given an inclusion space, we call *abstractive class *any totally ordered class *G* of regions such that

*i*) *G *is totally ordered

*ii*) no region exists which is contained in all the regions in *G*.

We denote by *AC* the set of abstractive classes.

The idea is that an abstractive class represents an “abstract object” which is the limit (the intersection, in a sense) of the regions in the abstractive class. Condition *ii*) means that these objects have a dimension lower than the one of the regions. As an example, in the canonical Euclidean plane an abstractive class is intended to represent either a point or a line. Now, it is possible that two different abstractive classes represent the same object. For example, let *G*_{1} be the class of closed balls with centre in *P *and let *G*_{2} be the class of closed squares with the same centre. Then while *G*_{1} ≠*G*_{2}, our intuition says that both *G*_{1} and *G*_{2} represent *P*. To tackle such a question, we define a pre-order relation and a corresponding equivalence relation.

Definition 4.2. The *covering *relation £* _{c}* is defined by setting, for any

*G*

_{1}and

*G*

_{2}in

*AC*,

*G*_{1}£_{c}G_{2} Û “*x*Î*G*_{2} $*y*Î*G*_{1} *x*>*y*.

The covering relation £* _{c}* is a pre-order in

*AC*, i.e. it is reflexive and transitive, and therefore it defines an equivalence.

Proposition 4.3. Define in *AC *the relation

*G*_{1 }º *G*_{2} Û *G*_{1}£_{c }G_{2} and *G*_{2 }£_{c}*G*_{1}.

Then º is an equivalence relation.

We can consider the quotient *AC*/º and the partial order relation £* _{c}* in

*AC*/º defined by setting, for every pair [

*G*

_{1}], [

*G*

_{2}] of elements in

*AC*/º,

[*G*_{1}] £* _{c}* [

*G*

_{2}] Û

*G*

_{1 }£

_{c}*G*

_{2}.

At this point it is possible to give the notion of geometrical element. Since the definition in *PNK* and *CN* is uselessly complicated, we refer to the equivalent definition adopted in *PR*.

Definition 4.4. We call *geometrical element* any element of the quotient *AC/**º*, i.e. any complete class of equivalence modulo º. Also, we call *point* any geometrical element which is minimal in the ordered set (*AC/*º, £* _{c}*).

Analogous definitions can be given by referring to the connection spaces provided that we modify the notion of abstractive class by involving the topological notion of non-tangential inclusion.

Definition 4.5. Two connected regions are called *externally connected* if they do not overlap.^{[3]} A region *y* is *non-tangentially included* in a region *x, *if

(*j*) *y* is included in *x*,

(*jj*) no region exists which is externally connected with both *x *and *y*.

If we denote by *xOy* the formula $*r*(*r*£*x*Ù*r*£*y*) expressing the overlapping relation, we can represent the non-tangential inclusion in a very simple way.

Proposition 4.6. The non-tangential inclusion is the relation “ defined by the formula “*z*(*zCy *Þ *zOx*)*.*

*Proof. *We have to prove that, under the hypothesis *y*£*x*, the conditions

*a*) every region *z* which is externally connected with *y* is not externally connected with *x*,

*b*) if a region *z* is connected with *y*, then *z* overlaps *x*,

are equivalent. Assume *a*) and observe that, in account of the inclusion *y *£*x*, any region *z *which is connected with *y *is also connected with *x*. Assume that *z *is connected with *y*. Then if *z *overlaps *y* it is trivial that *z* overlaps *x*. Otherwise, *z *is externally connected with *y *and therefore, by *a*), it is not externally connected with *x*. So since *z *is connected with *x*, then *z *overlaps *x*.

Conversely, assume *b*). Then since *z *overlaps *x *entails that *z* is connected with *x*, *y*£*x *by definition. Assume that *z *is externally connected with *y*. Then, since *z *is connected with *y*, it overlaps *x*. Thus, *z *is not externally connected with *x*.

The relation “ is on the basis of the notion of abstractive class.

Definition 4.7. An *abstractive class* is a set *G* of regions such that

*j*) *G* is totally ordered by the non-tangential inclusion,

*jj*) no region exists which is contained in all the regions in *G*.

The *geometrical elements *and, in particular, the *points* are defined as in Definition 4.4. The reference to the non-tangential inclusion will be motivated in Section 6.

### 5. Ovals to Define Geometrical Notions

The question of defining the basic notion of straight segment arises.^{[4]} Now Whitehead in Chapter III of *PR *criticizes Euclid’s definition of a line (“A straight line is any line which lies evenly with the points on itself”) since “evenly” requires definition and since “nothing has been deduced from it.” In alternative, a good definition “should be such that the uniqueness of the straight segment between two points can be deduced from it.” In accordance, an attempt of giving an adequate definition in terms of the “extensive notions” is proposed. More precisely Whitehead assumes that in the space of the regions we can isolate a class of regions whose elements are called *ovals.* The underlying idea is perhaps that the ovals are suitable convex regions of an Euclidean space (a set *x *of points is *convex *if for every *P *and *Q *in *x *the segment *PQ *is contained in *x*)*. *The interest of the convex sets lies in the fact that the straight segment *PQ *is the intersection of all the convex sets containing *P *and *Q. *Obviously, Whitehead lists suitable properties for the class of ovals.^{[5]} It is an open question to translate these properties into a suitable system of axioms. The following is a reformulation, in mathematical terms, of Whitehead’s definition if straight segment.

Definition 5.1. We call *convex *a geometrical element represented by an abstractive class whose elements are ovals. The *straight segment *between two points *P* and *Q* is the convex geometrical element containing *P* and *Q* and which is minimal with respect to this condition.

Whitehead proves that there is one straight segment between two points. We conclude this section by emphasizing that Whitehead’s addition of the notion of oval as a primitive to the one of connection is a necessary step from a mathematical point of view. In fact, taking account of the topological nature of the notion of connection, we know that all the notions we can define in a canonical model are invariant with respect to the topological transformations. As a consequence, there is no possible definition of straight segment based on the connection relation.

### 6. Mathematical Motivations for the Passage from *PNK *and *CN *to *PR*

Surely there are philosophical motivations on the basis of Whitehead’s passage from the inclusion-based approach proposed in the books *PNK* and *CN* to the connection-based approach proposed in *PR*. In such a section we will argue that, in any case, there are also mathematical reasons (we do not know whether Whitehead was completely aware of them or not). The first one is related with the definition of point.^{[6]} Indeed, consider in *R*^{2}the abstractive classes *G _{–}*

_{0,}

*G*

_{0}and

*G*

_{+0}defined by sequences of balls with radius 1/

*n*and centre in (-1

*/n*,0), (0,0) and (1

*/n*,0), respectively. Then, since

*G*

_{-0,}

*G*

_{0}and

*G*

_{+0}are not equivalent, they represent different geometrical elements. As a matter of fact, the class

*G*

_{0}covers both the classes

*G*

_{-0}and

*G*

_{+0}. So, since

*G*

_{0}is not minimal, it cannot represent a point. Obviously should be intriguing to imagine an universe in which an Euclidean point as

*P*= (0,0) splits in three different “points”

*P*

_{–}_{0}

*=*[

*G*

_{-0}]

_{,}*P*

_{0}= [

*G*

_{0}],

*P*

_{+0}= [

*G*

_{+0}] (as a matter of fact into a cloud of infinite points). A similar phenomenon occurs in non-standard analysis. However, this is surely far from the aim of Whitehead.

Instead these difficulties do not occur in the case of the canonical connection spaces. In fact the sequences *G*_{-0} and *G*_{+0} (differently from *G*_{0})are not abstractive classes since they are not ordered with respect to the non-tangential inclusion. As a matter of fact, we can prove the following proposition giving a strong reason in favour of the connection-based approach.

Proposition 6.1. Consider a canonical connection space (*Re,C*) in an Euclidean space *R ^{n}*

_{. }Then the points in (

*Re,C*) defined by the abstractive classes “coincide” with the usual points in

*R*(i.e. with the elements of

^{n}*R*).

^{n}Another reason is related with the strength of the two approaches. Indeed, the following theorem holds true.

Theorem 6.2. It is not possible to define the connection relation in a canonical inclusion space (*Re*,Í). So, the connection-based approach is strictly more potent than the inclusion-based one.

*Proof.* Theorem 3.2 shows that in a canonical connection space the inclusion relation is definable by the formula “*z*(*zCx *Þ *zCy*) involving only the connection relation. Then the connection-based approach is either equivalent or more potent than the inclusion-based one. Consider an automorphism *f:* *Re*→*Re*, i.e. a map such that

*d*_{1}Í*d*_{2} Û *f*(*d*_{1})Í*f*(*d*_{2}).

Then from a general result in model theory we have that

(*Re*,Í) £ *a* [*d*_{1},*d*_{2}] Û (*Re*,Í) £ *a* [*f*(*d*_{1}),*f*(*d*_{2})] (6.1)

for any formula *a *whose free variables *x*_{1} and *x*_{2}and for any *d*_{1}, *d*_{2}in *Re*. In particular, if *a *is able to define the connection relation *C*, then

*d*_{1}* C d*_{2}* *Û *f*(*d*_{1})* C f*(*d*_{2}) (6.2)

for any automorphism *f. *Consider the case *n *= 2, set

*r _{y} *= {(

*x*,

*y*)Î

*R*:

^{2}*x*= 0};

*P*= {(

^{<}*x*,

*y*)Î

*R*:

^{2}*x<*0};

*P*= {(

^{>}*x*,

*y*)Î

*R*:

^{2}*x>*0}

and define the map *g:* *R ^{2}*→

*R*by setting

^{2}*g*((*x*,*y*)) = (*x*,*y*+1) if *x*Î*r _{y}*È

*P*

^{>}*g*((*x*,*y*)) = (*x*,*y*) otherwise.

We can visualize this map as a *cut *of the Euclidean plane along the *y-*axis *r _{y} *and a vertical translation of the half-plane

*r*È

_{y}*P*. Now,if

^{>}*X*Î

*Re*,then

*g*(

*X*) is not regular, in general. Nevertheless, we have that

*int*(

*g*(

*X*))≠Æ and therefore that

*reg*(

*g*(

*X*)) is a regular bounded non-empty subset of

*R*. In fact, since

^{2}*int*(

*X*) ≠Æ, either

*int*(

*X*)Ç

*P*≠Æ

^{>}*or*

*int*(

*X*)Ç

*P*≠Æ and therefore either

^{<}*g*(

*int*(

*X*)Ç

*P*) or

^{>}*g*(

*int*(

*X*)Ç

*P*) is a non-empty open set contained in

^{>}*g*(

*X*).We claim that the map

*f*:

*Re*→

*Re*defined by setting

*f*(*X*) = *reg*(*g*(*X*))

is an automorphism. In fact, it is evident that *X*Í*Y *entails *f*(*X*) Í *f*(*Y*). To prove the converse implication assume that *f*(*X*) Í *f*(*Y*) and, by absurdity, that *X *is not contained in *Y. *Then *int*(*X*) is not contained in *Y *and a closed ball *B* exists such that *B *Í *int*(*X*) and *B*Ç*Y *= Æ. Also, it is not restrictive to assume that *B *is either completely contained in *P ^{>}* or completely contained in

*P*and therefore that

^{<}*f*(

*B*)

*= g*(

*B*).Now, since

*g*is injective and

*B*Ç

*Y*= Æ, we have

*g*(

*B*)Ç

*g*(

*Y*) = Æ and therefore

*int*(

*g*(

*B*))Ç

*g*(

*Y*) = Æ. On the other hand

*int*(*g*(*B*)) Í *g*(*B*)* = f*(*B*) Í* f*(*X*) Í *f*(*Y*) Í* r _{y}*È

*g*(

*Y*).

Therefore, *int*(*g*(*B*))Í*r _{y}*, an absurdity. This proves that

*f*is an automorphism. On the other hand, for example, two closed balls which are tangent in the same point in

*r*are connected while their images are not connected. This contradicts (6.2).

_{y}Note that analogous results were proved in a series of basic papers of I. Pratt. Anyway, in these papers Pratt one refers to a different notion of canonical space in which also unbounded regions are admitted (and this is far from Whitehead’s ideas).

### 7. Multi-Valued Logic to Reformulate Whitehead’s Inclusion-Based Approach

We have just argued about the inadequateness of the inclusion-based approach to point-free geometry. Nevertheless, in our opinion, we can get around this inadequateness by reconsidering this approach in the framework of multi-valued logic (see Gerla and Miranda 2004). Indeed, consider the first two axioms in Definition 2.1 in a language *L _{Incl} *with a predicate symbol

*Incl*:

*A1* “*x*(*Incl*(*x*,*x*))*;* *A2* “*x*“*y*“*z*((*Incl*(*x*,*z*)Ù*Incl*(*z*,*y*))Þ *Incl*(*x*,*y*)).

But, unlike Section 2, let us interpret these axioms in a multi-valued logic. For example, we can consider the *product *logic (see for example Hájek 1998) whose set of truth values is [0,1] and in which

—the conjunction is interpreted by the usual product in [0,1],

—the implication by the operation → defined by setting *x*®*y *= 1 if *x*£*y* and *x*®*y *= *y/x *otherwise,

—the equivalence by the operation « defined by setting *x*«*y* = 1 if *x *= *y *and *x*«*y* = (*x*Ù*y*)/(*x*Ú*y*) otherwise,

—the universal quantifier by the greatest lower bound.

In such a case an *interpretation* of *L _{Incl}* is a pair

*I =*(

*S*,

*incl*) such that

*S*is a nonempty set and

*incl*:

*S*

^{2}→[0,1]

*is a fuzzy relation to interpret*

*Incl.*As in the classical case, given a formula

*a*whose free variables are among

*x*

_{1},…,

*x*and

_{n}*d*

_{1},…,

*d*in

_{n }*S*,the truth value

*Val*(

*I*,

*a*,

*d*

_{1},…,

*d*) Î [0,1] of

_{n}*a*in

*d*

_{1},…,

*d*is defined. This enables us to associate

_{n}*a*with its

*extension in*I, i.e. the

*n-*ary fuzzy relation

*I(*

*a*)

*:*

*S*→[0,1] defined by setting

^{n }*I(**a*)(*d*_{1},…,*d _{n}*) =

*Val*(

*I*,

*a*,

*d*

_{1},…,

*d*)

_{n}for every *d*_{1},…,*d _{n} *in

*S*.Also, (

*S*,

*incl*) is a model of

*A1*and

*A2*if and only if

*a1* *incl*(*x*,*x*) = 1; *a2 incl*(*x*,*y*)×*incl*(*y*,*z*) £* incl*(*x*,*z*),

for every *x*, *y*, *z* Î *S*. In order to express the anti-symmetric property, we assume that in our logic there is a modal operator *Cr *such that *Cr*(*a*) means *“**a** is completely true” *and that this operator is interpreted by the function *cr:* [0,1]→[0,1] such that *cr*(*x*) = 1 if *x *= 1 and *cr*(*x*) = 0 otherwise. Then we can consider the axiom

*A3 Cr*(*Incl*(*x*,*y*))Ù*Incl*(*y*,*x*)) → *x* = *y.*

A fuzzy interpretation (*S*, *incl*)satisfies *A3 *if and only if

*a3* (*incl*(*x*,*y*) = *incl*(*y*,*x*) = 1) Þ *x *= *y*.

Definition 7.1. Denote by *x*£*y *the formula *Cr*(*Incl*(*x*,*y*)) and by £ its extension in a given interpretation. Then £ is called *the crisp inclusion associated with incl*. Denote by *Pl*(*x*) the formula “*x’*(*x’*£*x* Þ *Incl*(*x,x’*)) and by *pl *its extension. Then the fuzzy set *pl *expresses the *pointlikeness* *property*.

Trivially, the crisp inclusionis defined by,

*x*£*y* Û *incl*(*x*,*y*) = 1 (7.1)

and the pointlikeness property is defined by,

*pl*(*x*) = *inf*{*incl*(*x*,*x’*): *x’*£*x*}. (7.2)

Such a property is a graded counterpart of the definition

*“x is a point provided that every part of x coincides with x”*.

The next axiom says that if the regions *x* and *y* are (approximately) points, then the graded inclusion is (approximately) symmetric.

*A4*) *Pl*(*x*)Ù*Pl*(*y*)→(*Incl*(*x*,*y*) ↔* Incl*(*y*,*x*)).

Then such an axiom is satisfied if and only if

*a4*)*pl*(*x*)×*pl*(*y*)£(*incl*(*x*,*y*)↔* incl*(*y*,*x*)) (7.3)

or, equivalently,

*pl*(*x*)×*pl*(*y*)×*incl*(*x*,*y*)£*incl*(*y*,*x*).

Definition 7.2. We call *graded inclusion space *any model of *A1*, *A2*, *A3*, *A4*.

In any graded inclusion space we can define a notion of point as follows.

Definition 7.3. Given a graded inclusion space (*S*, *incl*), we call *nested abstraction process *any order-reversing sequence <*p _{n}*>

_{n}_{ÎN}

*of regions such that*

*lim _{n}*

_{®∞ }

*pl*(

*p*) = 1.

_{n}We denote by *Nr* the class of the nested abstraction processes.

We can give to the set *Nr *a structure of pseudo-metric space.

Proposition 7.4. Let (*S*, *incl*) be a graded inclusion space such that *Nr *≠ Æ, then the map *d*: *Nr*´*Nr*®*R ^{+}*

^{ }obtained by setting

*d*(<*p _{n}*>

_{n}_{ÎN},<

*q*>

_{n}

_{n}_{ÎN}) = –

*lim*

_{n}_{®∞}

*Log*(

*incl*(

*p*,

_{n}*q*)), (7.4)

_{n}defines a pseudo-metric space (*Pr*, *d*).

As it is usual in the theory of pseudo-metric spaces, we can associate (*Pr*, *d*) with a metric space.

Proposition 7.5.The relation º in *Nr *defined by setting <*p _{n}*>

_{n}_{ÎN}º <

*q*>

_{n}

_{n}_{ÎN}if

*d*(<

*p*>

_{n}

_{n}_{ÎN},<

*q*>

_{n}

_{n}_{ÎN}) = 0 is an equivalence relation. In the quotient

*Nr*/º we can define a metric

*d*by setting

*d*([<*p _{n}*>

_{n}_{ÎN}], [<

*q*>

_{n}

_{n}_{ÎN}]) =

*d*(<

*p*>

_{n}

_{n}_{ÎN},<

*q*>

_{n}

_{n}_{ÎN}).

We call *points *the elements in *Nr*/º, i.e. the complete equivalence classes

[<*p _{n}*>

_{n}_{ÎN}] = {<

*q*>

_{n}

_{n}_{ÎN}

*Î*

*Nr:*

*d*(<

*p*>

_{n}

_{n}_{ÎN},<

*q*>

_{n}

_{n}_{ÎN}) = 0}.

Observe that the “pathological” abstractive classes *G*_{-0}, *G*_{0} and *G*_{+0} defined in Section 6 are equivalent nested abstraction processes and therefore they represent the same point.

There is no difficulty to define canonical spaces in the Euclidean space *R ^{n}*. In fact if d denotes the usual distance in

*R*and

^{n}*x*,

*y*are nonempty bounded subsets of

*R*, then we define the

^{n}*excess function*

*e*by setting,

*e*(*x*,*y*) = *sup _{P}*

_{∈}

_{x}*inf*

_{Q}_{∈}

*d(*

_{y}*P*,

*Q*). (7.5)

Theorem 7.6. Let *Re* be the class of all nonempty bounded closed regular subsets of *R ^{n}* and define

*incl*:

*Re*´

*Re*→[0,1] by setting

*incl*(*x*,*y*) = 10^{– e(x,y)}. (7.6)

Then (*R ^{n}*,

*incl*) is a graded inclusion space we call

*canonical graded inclusion space*. In such a space

*pl*(

*x*) = 10

^{-|x|}.

It is possible to see that in these spaces the inclusion and the connection relations are definable by the two formulas *Cr*(*Incl*(*x*,*y*)) and *Cr*($*z*(*Pl*(*z*)Ù(*Incl*(*z*,*x*)Ù*Incl*(*z*,*y*))), respectively. Moreover the points coincide with the usual points in the Euclidean metric space *R ^{n}*. This suggests that the notion of graded inclusion space looks to be a good candidate to reformulate Whitehead’s point-free geometry as proposed in

*PNK*and

*CN.*

### Notes

[1] “Mereology can hardly serve the purpose of spatial representation even if we confine ourselves to very basic patterns. Not only is it impossible to capture the notion of one-piece wholeness; mereologically one cannot even account for such basic notions as, say, the relationship between an object and its surface, or the relation of something being inside, abutting, or surrounding something else. These and similar notions are arguably fundamental for spatial reasoning (for type (i) theories as well as for type (ii) theories). Yet they run afoul of plain part-whole relations, and their systematic account seems to require an explicit topological machinery of some sort” (p. 77).

[2] As a matter of fact Whitehead proposed a theory of the events and not a point-free geometry. Instead in De Laguna one refers in an explicit way to the geometrical notion of “solid”: The following pages contain a series of definitions of geometrical concepts, based upon the assumed entity “solid” and the assumed relation “can connect”. More precisely he considers regions since he claims The “solid”, then, may be said to be the space occupied by a physical solid.

[3] Such a definition plays a basic role in the definition of abstraction process and therefore of point. Whitehead is aware of the importance of such a notion which is possible only in the De Laguna approach. Two regions are externally connected when (i) they are connected, and (ii) they do not overlap. The possibility of this definition is another of the advantages gained from the adoption of Professor de Laguna’s starting point, ‘extensive whole and extensive part’ (Definition 7, *PR*, p. 349)

[4] In the inclusion based geometry we are at a mereological level. By introducing the connection structures we pass at a topological level. In order to reach a geometric level, we have to introduce the geometrical notion of straight line. Whitehead refers to the Euclid’s Elements (*PR*, p. 354): The first definition of Euclid’s Elements runs, “A point is that of which there is no part.” The second definition runs, “A line is breadthless length.” The fourth definition runs, “A straight line is a line which lies evenly with the points on itself.”

[5] This is evident from the list of the properties Whitehead assign to the class of ovals. For example, consider the first two properties (*PR*, p. 356):Any two overlapping regions of the ovate class have an unique intersect which also belongs to that ovate class. Any region, not a member of the ovate class, overlaps some members of that class with “multiple intersection”.

[6] “Since that date Professor T. de Laguna has shown that the somewhat more general notion of ‘extensive connection’ can be adopted as the starting point for the investigation of extension; and that the more limited notion of ‘whole and part’ can be defined in terms of it. In this way, as Professor de Laguna has shown, my difficulty in the definition of a point, without recourse to other considerations, can be overcome” (*PR*, p. 338).

### Works Cited and Further Readings

Casati, Roberto & Varzi, Achille. 1997. “Spatial Entities,” in *Spatial and Temporal Reasoning*, edited by Oliviero Stock (Dordrecht, Kluwer), 73-96.

Gerla, Giangiacomo & Miranda, Annamaria. 2004. “Graded inclusion and point-free geometry,” *International Journal of Pure and Applied Mathematics*, 11, 63-81.

Gerla, Giangiacomo & Tortora, Roberto. 1992. “La relazione di connessione in A. N. Whitehead: aspetti matematici,” *Epistemologia*, 15, 341-54.

Pratt, Ian. 2006. “First-Order Mereotopology,” Draft chapter in forthcoming book on *Spatial Logics*.

### Author Information

Giangiacomo Gerla

Dipartimento di Matematica e Informatica

Università di Salerno, Via Ponte Don Melillo, Fisciano, 84084, Italy

gerla@unisa.it

Annamaria Miranda

Dipartimento di Matematica e Informatica

Università di Salerno, Via Ponte Don Melillo, Fisciano, 84084, Italy

amirada@unisa.it

### How to Cite this Article

Gerla, Giangiacomo, and Annamaria Miranda, “Mathematical Features of Whitehead’s Point-free Geometry”, last modified 2008, *The Whitehead Encyclopedia*, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <http://encyclopedia.whiteheadresearch.org/entries/thematic/mathematics-and-logic/mathematical-features-of-whiteheads-point-free-geometry/>.