=Paper=
{{Paper
|id=None
|storemode=property
|title=Prolegomena to an Ontology of Shape
|pdfUrl=https://ceur-ws.org/Vol-1007/invited4.pdf
|volume=Vol-1007
|dblpUrl=https://dblp.org/rec/conf/shapes/Galton13
}}
==Prolegomena to an Ontology of Shape==
https://ceur-ws.org/Vol-1007/invited4.pdf
Prolegomena to an Ontology of Shape
Antony GALTON
College of Engineering, Mathematics and Physical Sciences
University of Exeter, UK.
Email: apgalton@ex.ac.uk
Abstract. Influenced by the four-category ontology of Aristotle, many
modern ontologies treat shapes as accidental particulars which (a) are
specifically dependent on the substantial particulars which act as their
bearers, and (b) instantiate accidental universals which are exemplified
by those bearers. It is also common to distinguish between, on the one
hand, these physical shapes which form part of the empirical world and,
on the other, ideal geometrical shapes which belong to the abstract
realm of mathematics. Shapes of the former kind are often said to ap-
proximate, but never to exactly instantiate, shapes of the latter kind.
Following a suggestion of Frege, ideal mathematical shapes can be given
precise definitions as equivalence classes under the relation of geometri-
cal similarity. One might, analogously, attempt to define physical shape
universals as equivalence classes under a relation of physical similarity,
but this fails because physical similarity is not an equivalence relation. In
this talk I will examine the implications of this for the ontology of shape
and in particular for the relationship between mathematical shapes and
the shapes we attribute to physical objects.
Keywords. shape ontology; mathematical vs physical shape; intrinsic vs
embedded shape
1. Introduction
What are shapes, and how are shapes related to things which are not shapes? Are
there indeed such things as shapes at all, entities of some sort that have a place
in an inventory of the world’s contents? Or can we explain talk about shapes in
terms of an ontology in which shapes do not feature as entities of any kind?
There seem to be two distinct kinds of shapes: physical shapes, which we en-
counter in the physical world as the shapes of entities that exist in space, and
mathematical shapes, which we encounter in geometry, the shapes of abstract
mathematically-defined constructions. In both cases it seems evident that shapes
are ontologically dependent on the objects whose shapes they are (their bearers),
but the relationship between the two kinds of shape is not necessarily straight-
forward. I shall defer till later a discussion of mathematical shapes, and for the
moment concentrate on physical shapes, the shapes of physical objects.
Granted, then, that physical shapes are always shapes of things, what kinds
of things have shapes in this sense? A brief catalogue might be as follows:
29
1. Material objects, including chunks of matter (e.g., a pebble), organisms
(e.g., a penguin), and assemblies (e.g., a bicycle).
2. Non-material physical objects such as holes, faces, and edges.
3. Aggregates such as a flock of birds or a cluster of buildings.
The boundary between material objects and aggregates is not sharp, since even
a chunk of matter is, at submicroscopic resolution, an aggregate of atoms. With
aggregates it is often not easy to determine an exact boundary, or therefore an
exact shape [5], and to the extent that objects may be similarly indeterminate,
it may likewise be impossible to assign exact shapes to them. I shall take this
issue up later in the discussion of shape approximation. We should not assume
uncritically that all material objects have shapes; Stroll [13] suggests that not all
material objects have surfaces, and possession of a surface seems to be strongly
associated with possession of a shape, even if not a necessary condition for it.
Phrases of the form “the shape of X” and “X has such-and-such a shape”
attest to the intimate relation between an object and its shape, characterised as
an ontological dependence of the latter on the former. Other key elements of the
object–shape relationship, to be accounted for in an ontology of shape, include
two objects having the same shape, and an object changing shape, expressed using
the sentence forms:
(1) x and y have the same shape at time t
(2) x changed shape between times t1 and t2
In what follows, we will pay careful attention to these notions.
2. The dependency of shape upon objects
Shape-words in language typically come both as nouns and adjectives: in English
we have, alongside nouns such as “circle”, “triangle”, “sphere”, and “cylinder”,
the respective adjectives “circular”, “triangular”, “spherical”, and “cylindrical”.
Cases in which the nominal and the adjectival functions are borne by the same
word—e.g., “square” and “oblong”—are the exception rather than the rule. We
also freely form compound adjectives such as “pear-shaped” and “heart-shaped”,
and in some cases the noun forming the first part of the compound refers, not
to the physical object which it normally designates, but to some mathematical
shape conventionally abstracted from it—e.g., the “heart” symbol ~ only very
approximately resembles the complex three-dimensional shape of an anatomical
heart.
Shape adjectives point to the notion of shape as a property of objects, whereas
shape nouns point to shapes as entities in their own right. Which of these two
pictures enjoys logical or ontological primacy over the other, and how are the two
pictures related?
Ontological parsimony suggests that shape-as-property should take priority
over shape-as-entity. Looking around us, we see physical objects, each with its
own shape, but to suggest that we see the shapes as well as the objects smacks
of ontological over-abundance. It is more natural, when in a parsimonious mood,
to say that each object is shaped in such-and-such a way, where this notion is
30
expressed using a shape adjective. Thus we can say that the table is square, rather
than that it stands in some relation to a shape entity which is a square.
A logical analysis of this view will invoke shape predicates, leading to pred-
ications of the form Square(x) or Circular(y)—or rather, allowing for the fact
that objects can change shape, Square(x, t), etc. A major disadvantage lurks be-
hind the attractive simplicity of this scheme: if we want to generalise over shapes,
we have to quantify over predicates, and this requires the use of second-order
logic, with all the difficulties that that brings in its wake. Thus to express the
sentence-forms (1) and (2) we would have to write something like
(1a) 8 (ShapeP roperty( ) ! ( (x, t) $ (y, t))),
(2a) 9 1 9 2 (ShapeP roperty( 1 ) ^ ShapeP roperty( 2) ^
1 (x, t1 ) ^ 2 (x, t2 ) ^ ¬ 1 (x, t2 ) ^ ¬ 2 (x, t1 )).
A standard way of reducing such second-order predications to first-order form
is by reifying the properties expressed by the predicates which are being quan-
tified over [4]. By this means we introduce terms designating shape entities, and
introduce a first-order HasShape predicate to relate objects to the shapes that
they have: thus instead of Square(x), say, we would write HasShape(x, square).
In e↵ect, this is to accord priority to shape-nouns over shape-adjectives. Our
sentences (1) and (2) can now be expressed in first-order form as
(1b) 8s(HasShape(x, s, t) $ HasShape(y, s, t)),
(2b) 9s1 9s2 (HasShape(x, s1 , t1 ) ^ HasShape(x, s2 , t2 ) ^
¬HasShape(x, s2 , t1 ) ^ ¬HasShape(x, s1 , t2 ))
On this view, it is natural to regard shapes—that is, the entities designated by
the s variables in (1b) and (2b)—as generically dependent entities. They are
dependent, since a shape only exists insofar as it has bearers, and this dependence
is generic because a shape is not dependent on the existence of a unique bearer
but can be multiply realised in di↵erent bearers having the same shape.
Modern information systems ontologies such as BFO [7] and DOLCE [9] do
not take this line; instead, they treat an object’s shape as specifically dependent
on that object, meaning that the shape belongs uniquely to that object and
cannot be shared with any other. In DOLCE, shapes, along with such things as
colours, volumes, weights, and densities, are classified as qualities. The identity
of an object’s shape is tied to the identity of the object itself: the shape comes
into existence when the object comes into existence, and endures for as long as
the object does. This does not mean that an object cannot change shape, though;
what happens, according to DOLCE, is not that the object assumes a di↵erent
shape, but that the object’s shape assumes a di↵erent value. The values that
may be assumed by a quality are entities of another kind, called qualia, which
collectively constitute a domain known as a quality space—in the case of shape,
we could speak of shape qualia in shape space. These quality spaces are similar to
the conceptual spaces of Gärdenfors [6].
On this picture, variability of shape shows up as a time-dependency, not of the
shape on its bearer, but of the value of the shape on the shape. Writing shape(o)
to refer to the shape which uniquely inheres in the object o, we have the rule
31
shape(x) = shape(y) ! x = y,
and our formulae now come out as:
(1c) value(shape(x), t) = value(shape(y), t)
(2c) value(shape(x), t1 ) 6= value(shape(x), t2 )
The shape-as-quality view embraced by DOLCE has a solid pedigree in the
Aristotelian four-category ontology that is encapsulated in the ontological square
[8,12], which presents a cross-classification of the entities of an ontology along the
dimensions of universal vs particular (distinguishing types from their instances)
and substance vs accident (distinguishing independent from dependent entities).
Thus the roundness of this ball is an accidental particular inhering in (and thus
dependent on) the substantial particular this ball ; and these two particular entities
are instances of the accidental universal roundness and the substantial universal
ball respectively. The ball itself is said to exemplify roundness.
The roundness of this ball is not quite the same as the shape of the ball
conceived as a quality in DOLCE. The former is a trope, i.e., a specific instance
of a property inhering in an object. In DOLCE terms, the “property” in question
is not just a quality but a quality’s having a particular value. Thus a trope could
be regarded as a quality/value pair. If the quality changes value (e.g., an object
changes shape), then the previous trope is superseded by a new one. When the
value of some quality changes continuously, there is a continuous succession of
di↵erent tropes.
3. The primacy of “same shape” over “shape”
The reified analyses discussed above are predicated on the assumption that there
are such things as shapes, whether universals or particulars, with a bona fide exis-
tence that must be accounted for by according them a place within our ontology.
This can, however, be questioned. Consider again the two main ways in which we
describe the shape of an object:
1. Using a descriptive adjective such as “square”, “round”, or a combination
of adjectives such as “long and thin”;
2. By means of a comparison with some other object whose shape is assumed
known, e.g., “heart-shaped”, “hourglass-shaped”.
In neither of these cases is there an explicit reference to shapes per se: we can
understand “square” as a descriptive adjective without having to postulate any
entity that is a square shape distinct from the square object we are talking about;
and in saying that something is heart-shaped we are not saying that its shape is
a heart, but rather that it is similarly-shaped to a heart. We do, of course, use
the expression “it has the same shape as a heart”, which seems to suppose the
existence of the shape as something distinct from the heart, but it may be argued
that this too is a misunderstanding, the locution “has the same shape as” being
more correctly paraphrased as “is shaped the same as”.
This way of arguing has venerable roots. It presents shape as one of a group
of concepts X for which the notion of X itself is logically dependent on a prior
32
notion which, once we have the concept X at our disposal, it is natural to express
using the words “has the same X as”. This latter notion is an equivalence relation
which can be defined without any reference to the concept X itself. This idea is
due to Frege [2], who noted that the concepts of number, direction, and shape can
all be derived in this way.
In the case of (cardinal) number, the relevant relation is defined as follows:
• Set X has the same number as set Y if and only if there is a bijection (i.e.,
an exhaustive one-to-one correspondence) between the elements of X and
the elements of Y .
Notice that bijections can be defined without reference to number; but on the
other hand, according to Frege’s argument, number cannot be defined without
having the prior notion of “same number” to establish an identity criterion for the
new concept. Thus “same number” is shown to be logically prior to “number”.
Instead of “has the same number as”, we can use the term “equipollent”.1 We
now define number in terms of equipollence as follows: The number of set X (i.e.,
the number of its elements, its cardinality) is the set of all sets equipollent to
X.2 This set is what we would now recognise as an equivalence class under the
equipollence relation.3
Similarly, “direction” is logically dependent on the relation “has the same
direction as”—which we routinely express as “is parallel to”—, and “shape” is
logically dependent on “has the same shape as”, i.e., is geometrically similar
to. In particular, the shape of an object can be defined as the equivalence class
comprising all objects which have the same shape as it. This definition works well
so long as (a) a domain of “objects” is established for the relation to be defined
on, and (b) within this domain “same shape” can be defined as an equivalence
relation. In the next section I consider as candidate domains, first, geometrical
constructions, and second, physical objects; I then go on to consider what it means
to say that a physical object has the same shape as a geometrical construction.
4. Definition of the “same shape” relation
At the end of the previous section I glossed the relation “has the same shape as”
as “is geometrically similar to”. The latter relation, however, is first and foremost
defined as a relation on geometrical objects—which, for the moment, we may
understand, in standard mathematical fashion, as subsets of Rn , for some n 2 Z+ .
We therefore need to ask in what way this relation can be applied to the very
di↵erent domain of physical objects: very di↵erent because physical space is not
a set of real-number triples, the usefulness of R3 as a model for physical space
1 Frege’s term was gleichzahlig i.e., “equal-numbered”.
2 Frege did not himself formulate this in terms of sets: he spoke of the number which belongs to
the concept F (die Anzahl, welche dem Begri↵e F zukommt) and equated this to the extension
of the concept “equipollent to the concept F ” (der Umfang des Begri↵es “gleichzahlig dem
Begri↵e F”).
3 This is not unproblematic: What set is this relation defined on? Frege supposed this could be
the set of all sets, but as Russell pointed out to him, this notion leads inexorably to devastating
paradoxes. An adequate discussion of this point would take us well out of scope of this paper.
33
being rather that it a↵ords constructions which capture at least some parts of the
abstract essence of phenomena in physical space that we wish to model.
4.1. Similarity of geometrical objects
Considering first the notion of geometrical similarity as it applies to objects in
geometrical space, the key notion is that of distance, which serves as a measure
of the separation between two points. Writing (p, q) for the distance between
points p, q 2 Rn , defined by the usual Pythagorean rule, we have:
Definition of geometric similarity between figures in Euclidean space.4 Two subsets
X and Y of Rn are geometrically similar if and only if there is a bijection from the
points of X to the points of Y such that, for some constant 2 R+ , the following
relation holds:
8x, x0 2 X. ( (x), (x0 )) = (x, x0 ).
In other words, distances between points in X are multiplied by a constant factor
when the points are mapped by into their images in Y . This is straightforward
and familiar. It is of particular importance to note that the relation thereby
defined is an equivalence relation, and it is this that enables the Fregean move
by which the shape of a figure can be identified with the equivalence class of all
figures having the same shape as it.
4.2. Similarity of physical objects
When we turn from Rn to the physical world, things are less straightforward.
Whereas distance in Rn can be defined mathematically, in physical space the
notion of distance is inextricably tied up with that of measurement, and the key
fact about measurement here is that all measurement has finite precision. This
means that whereas in Rn , since distances can be arbitrary non-negative real
numbers, the space of possible distances is simply R+ [{0}, the space of measured
distances in physical space cannot take this form. To see this, note that we cannot
meaningfully ask whether the length of a rod in metres is rational or irrational.
Given that in physical space we can only characterise distances in terms of
measurement, and that measurement always has a finite precision, corresponding
to the resolving power of the measuring instrument, it follows that geometrical
similarity for physical objects can only be defined relative to a specified level of
resolution. Consider two objects whose shapes we wish to compare, say P and
Q, where Q is at least as big as P . Suppose the volume of P is v and that the
resolving power of our measuring instrument is such that the smallest distance
we can distinguish is h (I shall describe this as “resolution h”). Then in principle,
within the physical space occupied by P we can distinguish, say, n ⇡ v/h3 points,
4 Here I am only dealing with Euclidean space—complications arise when we turn to non-
Euclidean spaces. For example, on the surface of the sphere, figures cannot be similar without
also being congruent. This is because, in this space, the sum of the interior angles of a triangle
exceed 2⇡ by an amount that is proportional to the area of the triangle, and hence no figure
can be expanded or contracted without changing shape. Thus only in Euclidean space is shape
completely independent of size.
34
and to each pair x, y of these points we can assign a distance h (x, y) that is
some multiple kh of the minimum discernible distance.5 Let Sh (P ) be this set of n
discernible points in P ; we may think of them, if we wish, as “blobs” of diameter
h, though this is not really how they seem to us as observers.
To say that objects P and Q have the same shape is to say that the points
we can discern in P at resolution h can be mapped onto some set of points we
can discern in Q at that resolution, such that, first, the distances between pairs
of the latter set of points are not discernibly di↵erent, at resolution h, from some
constant multiple of the distances between the pairs of points from the former
set to which they correspond under the mapping; and second, every point in Q
discernible at resolution h is “sufficiently near” one of the points corresponding
to a discernible point in P . In other words:
Definition of “same shape” for physical objects. Physical objects P and Q (where Q
is at least as big as P ) have the same shape, at resolution h, if, for some constant
1, the set Sh (P ) of points discernible in P at resolution h can be mapped into
the set Sh (Q) of such points of Q by means of an injective mapping , such that
the following relations hold:
1. 8x, y 2 Sh (P ). | h ( (x), (y)) h (x, y)| 12 h
2. 8x 2 Sh (Q). 9y 2 Sh (P ). h (x, (y)) 12 h
This is perhaps as near as we can come to the notion of geometric similarity in a
physical setting, in which the idea of “exact distance” gains no purchase.
An immediate consequence of this is that a pair of objects which come out
as having the same shape at one level of resolution may have di↵erent shapes
at a finer level of resolution. For physical objects, the concept of “same shape”
is inescapably tied to the level of resolution at which the objects are examined;
and since, according to the Fregean argument, the concept of “shape” is logically
dependent on the concept of “same shape”, it follows that the notion of shape is
also tied to levels of resolution. This is, of course, a familiar idea in Computer
Science, where the notion of resolution, which we handled in a very crude manner
here, has been considerably refined, e.g., in the technique of multiscale represen-
tation in which by convolving an original image with Gaussian kernels of di↵erent
variances we obtain a series of images at di↵erent resolution levels (see [1, Ch.7]).
Unfortunately, the Fregean construction cannot be achieved in this instance,
because the “same shape” relation on physical objects, as defined above, is not
an equivalence relation. It is perfectly possible to have three objects A, B, and
C, such that, at some resolution h, A has the same shape as B and B has the
same shape as C, but A does not have the same shape as C. This is essentially
because the “same shape” relation, as here defined, is not capturing a notion of
identical shape so much as a notion of indiscernible (at resolution h) shape; and
it is a familiar fact that unlike identity relations, indiscernibility relations are
not transitive. The crucial implication of this, for us, is that there is no coherent
5 This is somewhat oversimplified since in practice the resolution of our observations will not
harmonise with the levels of resolution available in the system of units used for recording them—
e.g., given resolving power (for lengths) of 0.03mm, recording to 1 decimal place is too coarse
and recording to two decimal places is too fine. For an attempt to deal with this issue in an
ontological framework, see [10].
35
notion of “exact shape” for a physical object, only that of objects being more or
less approximately the same shape as other objects.
4.3. Similarity between physical and geometrical objects
It is often said that the shapes of physical objects can also approximate to the
shapes of ideal geometrical objects (compare [11]). Thus there are many approxi-
mate spheres in the physical world, but no geometrically exact spheres, the exact
sphere being an inhabitant of mathematical, not physical space. This is true so
far as it goes, but we can explain what is being said here more carefully using
resolution-based shape comparison. An explanation is needed since on the face
of it there is something paradoxical about comparing something physical with
an abstract mathematical construction: the two seem to belong to such entirely
di↵erent realms that any notion of comparison ought to be out of the question.
If we are to compare the shapes of, say, a sheet P of A4 paper and a certain
rectangle R defined in R2 , then we need some way of matching up points in the
former with points in the latter. There are, at least on the orthodox view, un-
countably many points in R, but there does not seem to be any meaningful sense
in which we can attribute uncountably many points to P . There is already some-
thing dubious about the notion of attributing infinitely many points to P , since
as noted earlier, P can only be observed at all at some finite level of resolution,
and at any such level only finitely many points can be distinguished within it.
One might, of course, entertain the notion that, if there is no limit to how fine the
resolution level can be, then there is no limit to how many points we can discern
in P , so that the number of points in P , if not actually infinite, is at least poten-
tially infinite. But it is far from clear that, in the physical world, resolution could
be made indefinitely fine. For example, in the state of our current understanding
of physics, the Planck length (approximately 1.6 ⇥ 10 35 m) is believed to provide
a lower bound for the resolution of any possible technique of measurement.
But there is another problem we need to face, which is that while the physical
piece of paper P does have an actual (albeit indeterminate) size, the mathematical
rectangle R does p not. How
p wide is the rectanglepwhose cornersp are at the points
(0, 0), (0, 1), ( 2, 0), (p 2, 1)? That’s easy: it’s 2! But 2 is a number, not a
length. Well, then, it’s 2 units. But what is a unit? How does a unit compare with
a millimetre or an inch? It is a meaningless question: Objects in the mathematical
world do not have actual sizes that can be compared directly to those of physical
objects.
In fact this is not as serious a problem as it may seem at first sight, since
in assessing geometric similarity we are only interested in relative length, not
absolute length. The scale factor can take care of di↵erences in absolute length,
so long as the relative lengths remain unchanged. Thus in the ideal geometric
rectangle, considered aspa set of points in R2 , we can follow the usual practice of
recording the width as 2 units and the height as 1 unit, even though “unit” does
not designate any actual physical length, since what matters for our purposes
is only the ratio between the widthpand the height. And clearly all geometric
rectangles with sides in the ratio 1 : 2 are geometrically similar to one another.
What, then, does it mean to say that a piece of paper, considered at resolution
h, has the shape of a rectangle with sides in a given ratio? We cannot use the
36
definition of strict geometric similarity for geometric figures here, nor can we use
the definition of “same shape at resolution h” for physical objects. Instead, we
modify the criterion by combining elements from the two definitions as follows:
Definition of a physical object having the “same shape” as a geometrical object. At
resolution h, a physical object P has the same shape as a geometrical object Q if
there is an injective mapping from the set of points Sh (P ) discernible in P at
resolution h into the set of points in Q such that, for some constant > 0:
1. 8x, y 2 Sh (P ). ( (x), (y)) = h (x, y)
2. 8x 2 Q. 9y 2 Sh (P ). (x, (y)) 12 h.
The point here is that the objects in the pure geometric world, being given to us
by thought rather than by observation, can be specified with infinite precision: in
particular, the distances between points can be arbitrary real numbers and are
not constrained to being multiples of some minimal discernible distance.
5. Intrinsic vs Embedded Shape
Up to this point in our discussion, there has been an implicit assumption that
the shape of an object can be identified with the shape of the portion of space, or
region, which is occupied by that object. This fits in with a general presumption
that the spatial properties of objects are nothing other than the properties of the
portions of space they occupy. For many purposes this is not an unreasonable
presumption, and has the advantage that all such properties can then be handled
purely within a theory of space itself, without our having to worry about other
physical properties of the objects.
In the case of shape, this does not always accord with our everyday ways of
thinking. A rectangular sheet of paper, for example, is still, essentially, a rectan-
gular piece of paper if it is folded along a diagonal, or screwed into a ball. A single
long strand of wool retains this character whether it is coiled into in a skein or
knitted into a scarf. A tall, thin person is still a tall thin person whatever posture
he adopts; and more generally, the shape of a human body might be understood
to be “something which is invariant across all the various postures that the body
is capable of assuming” [3, p.201]. In such cases, we cannot identify the shape of
the object with the shape of the space it occupies, since the shape, understood
in this more general sense, may remain unchanged even as the object occupies a
succession of di↵erently-shaped spatial regions.
Let us distinguish between, on the one hand, the intrinsic shape of an object—
which is the more general, abstract notion of shape described in the previous
paragraph—and, on the other, its embedded shape, which is the shape of the region
of space that it currently occupies. The idea is that the embedded shape of an
object may change while its intrinsic shape remains constant. If we consider a
square sheet of paper, and all the myriad origami figures which it can be folded
into without tearing, we can say that across this range of figures the paper retains
its intrinsic shape (i.e., square) but assumes di↵erent embedded shapes.
Like embedded shape, the notion of intrinsic shape is logically dependent
on a prior notion of same shape: same embedded shape or same intrinsic shape
37
respectively. Both of these notions can be defined in terms of the distances be-
tween points in the respective objects. The di↵erence is that whereas with “same
embedded shape” the distances are measured with reference to the space within
which the objects are embedded, with “same intrinsic shape” they must be mea-
sured within the object itself. Given a physical object P , I define the P -intrinsic
distance between two points x and y in P , written P (x, y), as the length of the
shortest path between x and y which lies wholly within P . I contrast this with the
embedded distance (x, y) used previously. Note that, for points within a convex
object, the intrinsic and embedded distances are the same.
We can now give a rough definition of “same intrinsic shape” as the existence
of a bijective mapping between the points of P and the points of Q such that,
for any pair of points x and x0 in P we have Q ( (x), (x0 )) = P (x, x0 ), for
some constant 2 R+ . This is only a rough definition because, since we are here
dealing with physical objects, we have to take into account the resolution of the
distance measurements, just as we did in the embedded case. When a person
adopts di↵erent bodily postures, although their intrinsic shape remains approxi-
mately the same, we will always find, if measuring sufficiently precisely, that there
are di↵erences resulting from muscular contractions and extensions which distort
the shapes of individual parts of the body. Similarly, when a piece of paper is
folded, at the fold there will be minute tears or stretches which disrupt the exact
relationships between the intrinsic distances. But by measuring at a sufficiently
coarse resolution these small-scale disruptions will disappear from view, so that
intrinsic shape, at that resolution level, remains constant.
For a more exact definition, then, we need to introduce the notation P,h (x, y)
to mean the P -intrinsic distance between x and y at resolution h. We then have:
Definition of “same intrinsic shape” for physical objects. Physical objects P and Q
(where Q is at least as big as P ) have the same intrinsic shape, at resolution h, if,
for some constant 1, the set Sh (P ) of points discernible in P at resolution h
can be mapped into the set Sh (Q) of such points of Q by means of an injection ,
such that the following relations hold:
1. 8x, y 2 Sh (P ). | Q,h ( (x), (y)) P,h (x, y)| 12 h
2. 8x 2 Sh (Q). 9y 2 Sh (P ). Q,h (x, (y)) 12 h
What this definition does not tell us is how widely applicable the notion
of intrinsic shape is. The only examples I have given so far concern sheets of
paper, strands of wool, and human bodies, but for many objects the notion of an
underlying shape which is retained even as the object occupies di↵erently-shaped
regions of space does not seem to apply. We can arrive at a rough characterisation
of the kinds of object for which the notion of intrinsic shape can do useful work
by considering what kinds of transformations can change the embedded shape of
an object while retaining its intrinsic shape.
Rigid motions—rotations, translations, and reflections—preserve both intrin-
sic and embedded shape. For rigid bodies, therefore, intrinsic shape does not con-
vey any information beyond embedded shape. The same applies to magnification;
we can see this, at least approximately, in the case of a spherical rubber ball that
is being inflated; as it gets bigger, all the inter-point distances, whether measured
across the embedding space or within the material of the ball itself, expand at
the same rate, thus preserving both embedded and intrinsic shape.
38
All these transformations preserve the topology of the object, and it is cer-
tainly true that transformations that alter the topology, such as tearing or frac-
turing, will also alter the intrinsic shape—thus two sheets of A4 paper lying flat
on the table, one of which is intact and the other has a razor-thin slit in the middle
(making it topologically a torus), exhibit the same embedded shape but di↵erent
intrinsic shapes. However, topological equivalence is a much coarser relation than
having the same intrinsic shape: in general stretching and compression disrupt in-
trinsic shape while preserving topology. As noted above, transformations such as
folding or “reposturing” always involve some such disruptive transformations, but
because these are small compared with the objects in which they occur, intrinsic
shape can be preserved even under reasonably fine resolution.
Can we characterise exactly those types of object which typically undergo
transformations of a kind that result in changes of embedded shape while pre-
serving intrinsic shape at an appropriate level of resolution? Because notions such
as “typically” and “appropriate” are inherently inexact, we will never find such
an exact characterisation; but it would be interesting at any rate to find a more
exact characterisation than we have at present.
6. Conclusion
As with number and direction, the ontological status of shape is problematic
because of its dependent character: shapes do not exist “in their own right”,
but only as qualities of objects. As Frege observed, for the shapes of geometrical
figures characterised mathematically as subsets of Rn , the relation of geometrical
similarity provides a robust criterion of identity which, by establishing the notion
of “same shape” as an equivalence relation, can support the notion of shapes as
entities that could be included within an ontology.
By contrast, the “same shape” relation for physical objects, since it can only
be defined relative to some finite resolution level, fails to be an equivalence re-
lation, and therefore cannot provide a criterion of identity for a notion of phys-
ical shape. This casts doubt on the ontological integrity of the notion of shape,
and we are left with the intransitive “same shape at resolution h” relation as the
primary vehicle for shape-attribution to physical objects. This is reflected in the
fact that, in practice, when we ascribe a shape to an object or object part it is al-
ways by comparison with something else —either another object or a geometrical
figure—and relative to some (often implicit) level of resolution.
As a final observation, it should be noted that since “same shape” relations
are founded on the comparison of distances amongst the points within the objects
to be compared, it follows that a notion of shape should be, in principle, available
in any domain where some notion of “distance” is applicable.
The notion of intrinsic, as opposed to embedded shape arises as a result of
reinterpreting what is meant by distance: instead of distances measured across
the space in which the objects under consideration are embedded, we measure
distance along paths which are constrained to lie within the objects themselves.
In this case “distance” is still essentially spatial, but by extending this term
to measures of separation along non-spatial dimensions we obtain metaphorical
extensions to the notion of shape.
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One example is in the temporal domain, where we often speak of the profile of
a process, meaning by this its temporal “shape”—typically rendered visible in the
form of the spatial shape of some graphical representation of the process in which
time is the independent variable and the dependent variables measure one or more
qualities whose values are modified by the process. If we consider the process itself
rather than its graphical representation, we are faced with the problem that the
time dimension is not commensurable with the dimensions along which the other
variables are measured: thus, for example, given a scale model of a railway train,
where the non-temporal dimensions are spatial, there is no determinate answer
to the question how fast the model should be run to preserve the spatio-temporal
“shape” of the process in order to secure maximum verisimilitude.
An interesting extension of this is the idea of the “shape” of a musical phrase.
In music, there are two dimensions that can provide analogues of distance, namely
time and pitch.6 Since these have di↵erent measurement scales, distances along
these dimensions cannot be compared with each other, and this means that we
can reasonably regard, e.g., augmentations or compressions along either the time
axis or the pitch axis as in some sense shape-preserving. An examination of music
from many di↵erent styles will furnish numerous examples of composers exploiting
the expressive potential of such phenomena.
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6 Leaving aside loudness here as a third candidate.
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