# Mathematical Features of Whitehead’s Point-free Geometry

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, 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.

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 x1,…,xn and I an interpretation of L with domain S, then we write I £ a [d1,…,dn] to say a is satisfied in I by the elements d1,…,dn. Given a relation RÍSn, we say that R is defined by a or that R is the extension of a in I,provided that R = {(d1,…,dn): I £ a [d1,…,dn]}. For example, the extension of the formula \$r(r£xÙr£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)xyz ((x£zÙz£y) Þ x£y) (transitive)

(iii)xxy(x£yÙy£x Þ x = y) (anti-symmetric)

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

(v) “xy(x<y Þ \$z (x<z<y)(dense)

(vi) “xy\$z(x£zÙy£z)(upward-directed)

(vii) “xy(“x’(x’<x Þx’<yx£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’ÎS: 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 Rn (where R denotes the real numbers set). In accordance, we have to propose a suitable class of subsets of Rn to represent the notion of region. To do this, usually in literature one refers to the regular subsets of Rn.

Definition 2.2. We call closed regular any subset x of Rn such that x = cl(int(x)) where cl and int denotes the closure and interior operators, respectively. We denote by RC(Rn) the class of all the closed regular subsets of Rn.

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(Rn) 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(Rn). In fact, Axiom iv) says that the whole space Rn 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.

Theorem 2.3. Let Re be the class of all closed nonempty bounded regular subsets of Rn. 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 xÎRe, then, since int(x) ≠Æ, an open ball with radius r and centre P contained in x exists. We denote by x1 the closed ball with centre P and radius r/2. Then x1ÎRe and x1 Ì x. Also, since x is bounded, a closed ball with centre P and radius r’ containing x exists. Let x2 the closed ball with centre P and radius 2×r’. Then x2ÎRe and x Ì x2.

To prove (v), assume that x Ì y and let PÎ int(y)such that PÏx. Then there is a closed ball y’, y’Ìy and y’Çx = Æ. By setting z = y’Èx 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 PÎ 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(Rn)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(Rn)→RO(Rn)is an order-isomorphism between (RC(Rn),Í) and (RO(Rn),Í). Moreover, 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 LC 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(xy).

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

C1 xy(xCy ÞyCx)(symmetry)

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

C3xy\$z(zCxÙzCy)

C4 x(xCx)

C5 z(zCx Û zCy) Þ x=y

C6 z\$x\$y((x£z)Ù(y£z)Ù(ØxCy))

where x£y denotes the formula “z(zCx Þ zCy) and x<y the formula (x£y)Ù(xy). 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 XÇY 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(Xcl(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 G1 be the class of closed balls with centre in P and let G2 be the class of closed squares with the same centre. Then while G1G2, our intuition says that both G1 and G2 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 G1 and G2 in AC,

G1£cG2 Û “xÎG2 \$yÎG1 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

G1 º G2 Û G1£c G2 and G2 £c G1.

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 [G1], [G2] of elements in AC/º,

[G1] £c [G2] Û G1 £c G2.

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. 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£xÙr£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. 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. 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. Indeed, consider in R2the abstractive classes G0, G0 and G+0defined 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, G0and G+0 are not equivalent, they represent different geometrical elements. As a matter of fact, the class G0 covers both the classes G-0 and G+0. So, since G0 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” P0=[G-0], P0 = [G0], 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 G0)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 Rn. Then the points in (Re,C) defined by the abstractive classes “coincide” with the usual points in Rn (i.e. with the elements of Rn).

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: ReRe, i.e. a map such that

d1Íd2 Û f(d1f(d2).

Then from a general result in model theory we have that

(Re,Í) £ a [d1,d2] Û (Re,Í) £ a [f(d1),f(d2)]                                                                   (6.1)

for any formula a whose free variables x1 and x2and for any d1, d2in Re. In particular, if a is able to define the connection relation C, then

d1 C d2 Û f(d1) C f(d2)                                                                                                   (6.2)

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

ry = {(x,yR2: x = 0}; P< = {(x,yR2: x<0}; P> = {(x,yR2: x>0}

and define the map g: R2R2 by setting

g((x,y)) = (x,y+1) if xÎryÈP>

g((x,y)) = (x,y) otherwise.

We can visualize this map as a cut of the Euclidean plane along the y-axis ry and a vertical translation of the half-plane ryÈP>. Now,if X ÎRe,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 R2. In fact, since int(X) ≠Æ, either int(XP>≠Æ or int(XP<≠Æ and therefore either g(int(XP>) or g(int(XP>) 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 XÍY 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 BÇY = 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 BÇY = Æ, we have g(Bg(Y) = Æ and therefore int(g(B))Çg(Y) = Æ. On the other hand

int(g(B)) Í g(B) = f(B) Í f(X) Í f(Y) Í ryÈg(Y).

Therefore, int(g(B))Íry, 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 ry are connected while their images are not connected. This contradicts (6.2).

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 LIncl with a predicate symbol Incl:

A1x(Incl(x,x)); A2xyz((Incl(x,zIncl(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 = (xÙy)/(xÚy) otherwise,

—the universal quantifier by the greatest lower bound.

In such a case an interpretation of LIncl is a pair I = (S,incl) such that S is a nonempty set and incl: S2→[0,1] is a fuzzy relation to interpret Incl. As in the classical case, given a formula a whose free variables are among x1,…,xn and d1,…,dn in S,the truth value Val(I,a,d1,…,dn) Î [0,1] of a in d1,…, dn is defined. This enables us to associate a with its extension in I, i.e. the n-ary fuzzy relation I(a): Sn →[0,1] defined by setting

I(a)(d1,…,dn) = Val(I,a,d1,…,dn)

for every d1,…,dn in S.Also, (S,incl) is a model of A1 and A2 if and only if

a1 incl(x,x) = 1; a2 incl(x,yincl(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))ÙIncl(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(xPl(y)→(Incl(x,y) ↔ Incl(y,x)).

Then such an axiom is satisfied if and only if

a4)pl(xpl(y)£(incl(x,y)↔ incl(y,x))                                                                               (7.3)

or, equivalently,

pl(xpl(yincl(x,yincl(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 <pn>nÎN of regions such that

limn® pl(pn) = 1.

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(<pn>nÎN,<qn>nÎN) = –limn®∞ Log(incl(pn,qn)),                                                               (7.4)

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 <pn>nÎN º <qn>nÎN if d(<pn>nÎN,<qn>nÎN) = 0 is an equivalence relation. In the quotient Nr/º we can define a metric d by setting

d([<pn>nÎN ], [<qn>nÎN ]) = d(<pn>nÎN,<qn>nÎN).

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

[<pn>nÎN ] = {<qn>nÎN ÎNr: d(<pn>nÎN,<qn>nÎN) = 0}.

Observe that the “pathological” abstractive classes G-0, G0 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 Rn. In fact if d denotes the usual distance in Rn and x, y are nonempty bounded subsets of Rn, then we define the excess function e by setting,

e(x,y) = supPx inf Qy d(P,Q).                                                                                       (7.5)

Theorem 7.6. Let Re be the class of all nonempty bounded closed regular subsets of Rn and define incl: Re´Re →[0,1] by setting

incl(x,y) = 10e(x,y).                                                                                                        (7.6)

Then (Rn,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,xIncl(z,y))), respectively. Moreover the points coincide with the usual points in the Euclidean metric space Rn. 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

 “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).

 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.

 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)

 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.”

 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”.

 “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