Is Space-Time Discrete or Continuous? An Empirical Question

Is Space-Time Discrete or Continuous? An Empirical QuestionAuthor(s): Peter ForrestSource: Synthese, Vol. 103, No. 3 (Jun., 1995), pp. 327-354Published by: SpringerStable URL: http://www.jstor.org/stable/20117405 .

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PETER FORREST

IS SPACE-TIME DISCRETE OR CONTINUOUS? - AN EMPIRICAL

QUESTION*

ABSTRACT. In this paper I present the Discrete Space-Time Thesis, in a way which enables me to defend it against various well-known objections, and which extends to the

discrete versions of Special and General Relativity with only minor difficulties. The point of this presentation is not to convince readers that space-time really is discrete but rather

to convince them that we do not yet know whether or not it is. Having argued that it is an

open question whether or not space-time is discrete, I then turn to some possible empirical

evidence, which we do not yet have. This evidence is based on some slight differences

between commonly occurring differential equations and their discrete analogs.

In this paper I describe possible evidence for and against the Discrete

Space-Time Thesis, namely that the number of points in any sensibly

shaped (say convex) region of space-time of finite volume is finite.1 My chief conclusion is that we do not yet have much evidence either way. But

that is not because it is a matter of purely metaphysical speculation, even

less because we already know the answer without evidence. No, I hope to

show that the issue could be discussed, although not perhaps permanently

settled, in an empirical fashion.

There are many who think that the issue has already been settled in

favour of the non-discrete character of space-time. So most of this paper is taken up by pointing to some of the advantages of the Discrete Space

Time Thesis, and offering a defence against some objections to it. My aim,

however, is not to convince readers that space-time is discrete, but rather

to convince them that this is an open question. We should be suspending

judgement, I say. Having reached that conclusion it is then fairly straight forward to discuss what evidence there might be which would enable us to

stop suspending judgement. I have divided the paper into four sections: a case for discrete space,

the presentation of some of the details of a theory of discrete space, replies to various objections to discrete space as I have presented it, and, finally, a discussion of how we might decide empirically whether or not space is

discrete.

Synthese 103: 327-354, 1995.

? 1995 Kluwer Academic Publishers. Printed in the Netherlands.



328 PETER FORREST

1. A CASE FOR DISCRETE SPACE

1.1. The Charm of Being Discrete

The theories of Special and General Relativity provide extra problems for

the Discrete Space-Time Thesis. Initially, therefore, I shall concentrate

on the Discrete Space Thesis for three dimensional space, ignoring time.

Likewise, I shall initially assume that the alternative to discrete space is

standard Euclidean space, ignoring, among other things the possibility of

infinitesimal distances.2

Gr?nbaum (1973), concentrating on the one dimensional case, noted, that in discrete space the distance along a line between P and Q could

be defined in terms of the number of points between P and Q? The

generalisation for three dimensions is to define the volume of a region in

discrete space as the number of points it contains. But knowing volumes is

not enough to give us interesting geometry. That is because any permutation of the points, however drastic, would preserve volume. For interesting finite

geometry we need to know what is to count as a straight line. One way to

introduce lines is to take collinearity as a three place primitive term. But a

more elegant approach is to characterise the whole of geometry in terms of

the single dyadic relation of spatial nextness. To avoid awkward English, I shall call points which are next to each other adjacent. That will cause

no trouble provided we think of adjacency as meaning having no points between rather than as sharing a common boundary.

We can define the distance between two points P and Q as the smallest

number of 'links' in a chain of points connecting PtoQ each one of which

is adjacent to the previous one. We can then characterise the straight lines

(geodesies) in terms of distance. We shall require, in a way I make precise in Subsection 2.3, that this discrete space converges to Euclidean space in

the macrogeometric limit.

Discrete space may lack something when it comes to mathematical

tractability. But, I say, God did not just consider the convenience of mathe

maticians in deciding to create a universe. And to get the whole of geometry out of just one dyadic relation is surely elegant. Indeed if we could only combine geometrodynamics with discrete space the whole of the physical

world could be described just by saying which points are adjacent and

which times are adjacent. What a wonderful world it would be!

By contrast, the geometry of continuous space may well require a

distance function, which, assuming the unit of length turns out to be con

ventional, is given by a five place relation, which holds between P, Q,

R, S, and x if the distance from P to Q stands to the distance from R

to S in the ratio x. Even worse, one of the relata, x, is a number, while



IS SPACE-TIME DISCRETE OR CONTINUOUS? 329

the other four, P, Q, R and S are points. To avoid having a number as a

relatum, we might follow Field (1980), and rely on Hubert's axioms for

Euclidean Geometry. But they require two primitive relations, one the tri

adic betweenness relation, and the other a tetradic congruence relation. It

is more elegant if we only need the single dyadic relation of adjacency. A further theoretical advantage of discrete space is that it enables us

to explain why distance satisfies the triangle inequality, namely that the

distance from P\ to P3 cannot exceed the sum of the distances from P\ to Pj and P2 to P3. To be sure we would not call a quantitative relation distance

unless it satisfied the triangle inequality. So if there is such a relation

as distance it is no wonder that it satisfies the inequality.4 What is worth

explaining, though, is the occurrence of a quantitative relation of theoretical

significance satisfying the axioms for distance, and in particular the triangle

inequality. The triangle inequality follows from the way distance is defined

in terms of adjacency in a discrete space, but just has to be accepted without

explanation in continuous space. This is an advantage for the Discrete

Space Thesis.

There is a lot to be said, then, for the Discrete Space Thesis even before

we look at the details. And some of the details look good. For instance, much of classical mechanics concerned itself with differential equations,

which, if unpacked using the orthodox, Weierstrassian e/6 definition are

extremely messy.5 But expressed in terms of discrete space they are much

simpler. (See Appendix One for examples.) I take it, then, that there is an initial case for discrete space. Before I

consider the case against it, I shall provide further details of what a discrete

space might be like.

2. TOWARDS A THEORY OF DISCRETE SPACE

2.1. The Regular Tile Paradigm

The most common idea of what discrete space would be like is that there

would be small regions of space which could not be further divided. These

regions are then thought of as non-overlapping tiles, covering space. I

suppose the tiles could be shaped in a fairly random fashion, but we tend

to think of them as having triangular, hexagonal, or, most naturally of all, a

square shape. We then say that squares with a side in common are adjacent and so distance one (very small) unit apart. The distance between two tiles

P and Q is then the least number of 'links' in a chain of pairwise adjacent tiles the first of which is P and the last of which is Q.

This regular tile paradigm suffers from Pythagoras trouble, namely the

objection due to Weyl (Gr?nbaum 1973, p. 336) that the resulting geometry



330 PETER FORREST

would not even be approximately Euclidean, because Pythagoras' Theorem

would not even be approximately satisfied.

For instance, consider the square tiling, according to which the points of space are identified with, or at least represented by, very small squares.

Squares touching at a corner will then be distance 2 units apart, and there

could be a right angle triangle with sides TV, TV, and 2TV units, however

large TV is. According to Pythagoras' Theorem the hypotenuse should be

of length (a/2) TV not TV. The proportionate error, a/2, clearly does not

decrease as TV increases and so cannot be treated as a microgeometrical

curiosity, which vanishes in the macrogeometric limit. All this should be

familiar enough to anyone who lives in a city with a regular grid of streets.

The tiles are the city blocks, and the distance between blocks defined in

terms of adjacency is the distance you have to walk. It can be up to a/2 times the distance the crow flies.

Pythagoras trouble is not removed by considering, say, hexagonal tiles.

For as Rogers has pointed out (1968, p. 122) the trouble arises in that case

because there are 'privileged' paths, namely the paths along which the

hexagons are arranged in straight lines. These can be contrasted with the

'ordinary' paths along which the hexagons are arranged in a zigzag. If one

of the sides of a right angled triangle is on a privileged path, the other two

cannot be, and it is easy to check that Pythagoras Theorem does not even

approximately hold.

There are two ways of retaining the tile paradigm without running into

Pythagoras trouble. One is not to define distance in terms of adjacency

(Van Bendegem 1987, pp. 295-302). 1 reject that because, as I pointed out

in the previous subsection, the great appeal of discrete space is that we can

characterise everything just in terms of a single primitive dyadic relation.

The other way of retaining the tile paradigm is to allow the number of tiles

adjacent to a given tile to vary. Think of a flat surface tiled with a mixture

of (irregular) quadrilaterals pentagons and hexagons. We should be able to

prevent there being any privileged paths and hence, we might well hope, avoid Pythagoras trouble.

Rather than adopt either of these ways of avoiding Pythagoras trouble

I think we should re-examine the tile paradigm itself. If we take it literally it would require the points to have extension. But that seems to imply that part of a point is some distance from some other part of a point. But

we are defining distance in terms of a relation between points. We should

not therefore take the tile paradigm literally, but rather as just a way of

representing one possible discrete geometry using Euclidean space. While

I do not know the precise diagnosis of Pythagoras trouble, it shows at least

that the true discrete geometry cannot be represented both so that points




get represented by regularly shaped regions, and so that adjacent points are represented by regions sharing a line boundary.

I see no reason, however, to expect discrete geometry to be representable in that way. It suffices that its intrinsic structure be suitably simple, and

that in the macrogeometrical limit it approximates Euclidean space. Ease

of representation is a nice bonus if we can get it. And, in Subsection

2.3, 1 shall provide examples of discrete spaces which can be represented

quite nicely in Euclidean Spaces, or, with slightly greater convenience, in the vector space of n-tuples of real numbers. These will, however, be

representations by points rather than by regions.

2.2. The Characterisation of Discrete Space

A discrete space is a finite or countably infinite set of elements (points) with a single symmetric irreflexive relation (adjacency). Because I am

concerned with discrete analogs of finite dimensional Euclidean space, I

shall assume that the finiteness condition holds, namely that only finitely

many points are adjacent to a given point. Given this austere mathematical structure we can define the distance

between two points P and Q as the smallest integer n such that there is

a chain of points with n links each member of which is adjacent to the

previous one. That is, the distance is the smallest integer n such that there

are points Po, P\, ..., Pn, where P0 = P, where Pn -

Q, and where for

i ? 1,..., n, Pt is adjacent to P?_ i.

Thus far we have merely an abstract mathematical structure. The

'points' could be people and the relation of 'adjacency' could be that

of being at one time married. So what makes it a genuine space? I shall

write down some conditions which are, I submit, jointly sufficient con

ditions for a quantitative relation between points to count as distance. If

the distance relation between points defined in terms of adjacency satisfies

these conditions, then we have indeed a theory of discrete space, not just a system with the same mathematical structure as discrete space. Here are

the conditions:

(i) The relation satisfies the appropriate axioms.

(ii) The relation plays a non-redundant role in an adequate theoret

ical account of the physical world.

(iii) The relation is the relation which we are observing when we

are commonly said to observe distance.

Condition (ii) is non-redundant because it serves to exclude infinitely

many quantitative relations which happen to agree with distance in the

macrogeometric limit.6 Condition (iii) is not intended as the requirement



332 PETER FORREST

that the relation has to be the way we think distance is. It is that, whether we

know it or not, it is what we are observing when we observe distance.

The Discrete Space Thesis is the hypothesis that there is a symmetric irreflexive relation of adjacency which results in a quantitative relation

satisfying the requirements for distance.

Having characterised distance in terms of adjacency, the next step is to

say what the lines or, more generally, the geodesies are. Let us characterise

betweenness as the triadic relation which holds of P\, P2 and P3 in that

order, just in case the distance from Pi to P3 equals the sum of the distances

from P\ to Pi and Pi to P3. (In that case Pi is between Pi and P3.) A line

is then a set of points such that of any three at least one lies between the

other two.

We have distance and we have lines. What else should a decent space have? Volume, for a start. Here we can define the volume of a region intrin

sically as the number of points in that region. Alternatively we could con

sider a region which approximated a cube in three dimensional Euclidean

space and insist that its volume is the third power of its side. The examples of discrete space which I give are such that in the macrogeometric limit

these two definitions agree up to a scale factor. I do not think it a drawback

that we have this scale factor rather than exact agreement.

Something else which we would like are vector-analogs. We may per

haps be forced to consider discrete spaces which only have approximations to vector-analogs. But it is worth considering what these vector-analogs are. Now a vector can be thought of as a mapping or transformation which

sends points to points. Consider, for the moment, Euclidean space. A trans

formation T of the space corresponds to a vector just in case:

(i) T preserves distance, that is the distance between T(P) and

T(Q) is the same as the distance between P and Q.1

(ii) The distance between a point and its transform is the same for

all points. That is, the distance between T(P) and P does not

depend on the point P. (This distance is the length or magnitude of the vector.)8

On a given discrete space there will be a group of vector-analogs, namely the transformations satisfying (i) and (ii).9 If the discrete space is espe

cially well-behaved, then this group will play a role like that of vectors in

continuous space. This is easy to check in the case of the discrete spaces I

shall describe in the next subsection.

I have assumed in this subsection that there is a determinate matter of

fact whether or not two points are adjacent. It would be interesting, but




beyond the scope of this paper, to discuss indeterminate or fuzzy discrete

spaces.10

2.3. A family of Discrete Spaces

We need some examples. I shall provide a family of them, the discrete

spaces En?m for any positive integers n and m. The points of En?m are in

one to one correspondence with the n-tuples of integers, and two points are

adjacent if they are distinct and their Euclidean distance is no greater than

m. For simplicity, let us concentrate on the n ? 2 case. The points are then

in one to one correspondence with the pairs of integers, and adjacent points

correspond to pairs (w, v) and (#, y) such that (u -

x)2 + (v -

y)2 < m2.

The square tile space is then an instance of E2,i .n.

We shall now consider in what sense a discrete space D can approximate n-dimensional Euclidean space. Here we are working with two distance

functions, the one on the discrete space defined in terms of adjacency, call that d and the one on the Euclidean space we are approximating, call that e. Now the discrete nature of space will only be significant for

very small distances. So 1 cm will approximately correspond to some very

large number K of units in the discrete space. Bearing this in mind, I

submit that a sufficient condition for the discrete space to approximate the

Euclidean space is that there is a (many to one) mapping F which assigns to each point in the Euclidean space a point in the discrete space, and

which approximately preserves distance, given the K to 1 cm conversion.

That is, we require that the ratio between the distance between points P

and Q in Euclidean space and their images under F approximate K. For

the precise statement, and proof, of this Approximation Theorem I refer

readers to Appendix Two.

The Approximation Theorem shows that we can ensure as accurate an

approximation of n-dimensional Euclidean space by En?m as we please,

by taking the integer m to be large enough. The square tile model ran into

Pythagoras trouble because it is E2,i when we should have been looking at E2,m, where m is, say, 10 to the power 30.

Let us now consider the volume, or rather, since we are looking at

the two-dimensional case, the area. The number of points in a region

represented by a square of side mK is m2K2. But if we use the distance

defined by the adjacency relation the side of the region is only K units.

So there is indeed a scale factor of m2 between the intrinsic area of square

region and its area defined in terms of the length of a side. Likewise in

three dimensions the scale factor would be m3.



334 PETER FORREST

3. OBJECTIONS AND REPLIES

3.1. Is the Description of Discrete Space Parasitic?

I have already considered one objection to discrete space, namely trouble

with Pythagoras ' Theorem. But this is not an objection to the sort of discrete

space which I am proposing. Nor shall I discuss Zeno's Stadium Paradox, which has, I think, been shown to be fallacious (Rogers 1968, pp. 119-20). But this still leaves a fair number of interesting objections. I begin with

the objection that the elegance of discrete space is outweighed by its par asitic character.12 In the previous subsection I considered discrete spaces

which approximated Euclidean space. Later I shall consider approxima tions to Minkowski space time. And by patching together approximations to Minkowski space time we could obtain approximations to the curved

space time posited by General Relativity. But in all cases, it seems, I am

describing the discrete space as an approximation to a non-discrete space. What is required, the objection goes, is some simple intrinsic characteri

sation of the sort of space which will then be shown to approximate the

Euclidean, or the Minkowski, as the case may be.

My reply to this objection is to take up the challenge and provide a

characterisation which is not parasitic on Euclidean Space. Any discrete

space will have a group of automorphisms, that is one to one mappings of

the space onto itself which preserve adjacency. Because of the way distance

is defined in terms of adjacency, the automorphisms are precisely the one

to-one onto mappings which preserve distance. So we can refer to them as sometries. In the spirit of Klein's Erlanger program, and in accordance

with van Fraassen's emphasis on symmetry considerations in Science (Van

Fraassen, 1989, Ch. 11), it is appropriate, if we can, to characterise a

discrete space, up to isomorphism, in terms of its isometries.

First, we restrict our attention to the vector-analogs, where an isometry T is a vector-analog if the distance between P and T(P) is the same for all

points P. I now say that a discrete space is vectorial if:

(i) The combination of any two vector-analogs is itself a vector

analog.13

(ii) For any two points P and Q there is some vector-analog, T, such that g =

T(P).

It is now possible to write down various intrinsic characteristics of the

group of vector-analogs of a vectorial discrete space which ensure that it

is isomorphic to En,m and hence approximates n dimensional Euclidean

Space if m is large enough. (See Appendix Three).




The result is that various symmetry properties, namely various charac

teristics of the group of vector-analogs, ensure that a discrete space will

approximate Euclidean space, so we do not need to characterise a discrete

space by saying it approximates Euclidean Space. The same will hold for

Minkowski space. In the case of a discrete space which in fact approximates a curved space, we would rely on the procedure used by cartographers to

map the Earth. We would divide the whole of space into fairly small over

lapping regions which approximate regions in discrete spaces satisfying the conditions discussed in Appendix Three. In that way, no doubt, the

characterisation of the discrete analog of a curved space would be parasitic on the characterisation of the discrete spaces which they locally approxi

mate. But that parasitism is no worse than that accepted when we describe

curved spaces by means of the way they have local approximations to

portions of Euclidean Space. I conclude that the charge of parasitism has

been rebutted.

Although the charge of parasitism has been rebutted for the case of

discrete spaces which approximate Euclidean manifolds, we might still be

concerned about whether the dimension of a discrete space can be defined

intrinsically, that is without considering the topology of a continuous space which it approximates.14 To meet this concern I shall now provide such an

intrinsic characterisation of dimension. An initial obstacle is that dimen

sion is usually characterised in terms of the topology of a space, but the

topologies defined on a discrete space are either trivial or highly non

standard.15 To overcome this obstacle I shall offer a definition of what I

call the metric dimension of a metric space. This will require only a metric

and not a suitable topology for its characterisation. Hence it provides an

intrinsic characterisation of the dimension of a discrete space. The idea of the metric dimension is to consider the maximum number of

points which are equidistant from each other. In n-dimensional Euclidean

space this maximum is (n + 1) and the points are arranged as the vertices

of a regular n-simplex. Thus in three dimensional Euclidean space we can

have four points equidistant from each other arranged as the vertices of

a regular 3-simplex (i.e. a tetrahedron). But we cannot find five points

equidistant from each other.

We cannot simply define the metric dimension of a space as one less

than the maximum number of points equidistant from each other, for that

maximum might vary, depending on what the equal distance is. Consider, for example, a circle of radius 1 with the distance between points being the length of the shorter arc between them. Its topological dimension is

1, but the maximum number of points distance x apart varies. If x is less

than 27r/3, the maximum is 2. But if x = 2it/3 there are three points on



336 PETER FORREST

the circle distance x apart. If x is greater than 27r/3 but no greater than 7r,

again there the maximum number of equidistant points is 2. Finally if x is

greater than 7r, the maximum drops to 1.

If we were only interested in continuous metric spaces we might char

acterise the metric dimension by considering sufficiently small distances, but in a discrete space it is not just fairly large distances which cause

problems, peculiarities occur with very small distances. For instance, it is

easy to find nine points in E23 all distance 1 from each other. (Consider all the points represented by pairs of integers {u,v) where u and v take the

values ?1,0 and +1. All nine of these points are adjacent to each other.) But we do not want to say that E23 has at least eight dimensions, at least

not without further qualification. I shall, therefore, characterise the metric dimension in a scale-relative

fashion. In this way, the dimension could, as in the example of the circle,

change with scale. For continuous metric spaces we can define a non

relative dimension as the dimension for small enough scales. For discrete

metric spaces there is no precise non-relative dimension, but we might well

attach special significance to the dimension for scales which are neither

too large nor too small.

I say, then, that a metric space has metric dimension at least n relative

to scale a just in case (n +1 ) points can be found each of which is distance

a from each other. It has dimension exactly n relative to scale a if it has

dimension at least n relative to scale a but does not have dimension at least

n+l relative to scale a. It is infinite-dimensional relative to scale a if it

has dimension at least n relative to scale a for all positive integers n.

Euclidean n-dimensional space can be shown to have metric dimension

n relative to any (positive) scale. But what of En?m? The least that we

would expect to be able to show is that, relative to scales which are neither

too large nor too small, En>m has metric dimension n, provided n is not

itself too large, and m not too small. A proof of this will be provided in

Appendix Four. I leave it as an open question whether the requirement that

the scale not be too large is redundant.

We could also define the number of dimensions relative to an approx imate scale <7 ? 7ncr. The definitions are obtained from the above by

replacing the phrase 'distance a' by 'distance within jna of a\ Here the

7n are suitably small positive numbers chosen, of course, to ensure that En has dimension n relative to the approximate scale a?/yna, for any positive a.16 I leave readers to adjust the results of Appendix Four to obtain results

concerning the metric dimension of En?m relative to approximate scales. It

can be shown that, for suitable 7n, the metric dimension of En,m relative

to a ? ")na is n, provided a is not too small, n is not too large, and m is




not too small. (So in this case we definitely do not require that a not be too

large.)

3.2. The Jerky Motion Objection

The next objection is based on an observation of Russell's (1927, p. 375) and goes like this.17 Assuming that both space and time are discrete, then

(the analog18 of) continuous motion is motion for which at the next moment

of time a particle is either at the same point or at an adjacent point. If it

moves to a point distance two units away, then it would have jumped over

a point, as it were, and the motion would not be continuous. Hence there is

a maximum speed for a particle with continuous motion, namely uniform

motion in which at every moment the particle is displaced one unit. That

is good news because it provides a way of understanding why no particle can go faster than some maximum speed which we can identify with that

of light. But it does lead to the problem of how particles can go slower

than the speed of light. One solution to this problem is to say that a particle can move backwards

and forwards. If it goes two paces forward, one unit back, two units forward

etc, then it averages a speed of one third that of light. Alternatively, it

could mark time for say two units of time, then go one unit forward, then

mark time again and so on. However it is done, there is something jerky about the motion and this might seem implausible, even though we would

not notice any lack of smoothness in the motion.19 The implausibility, I

suggest, is because in that case there would be no such thing as a strictly uniform velocity, or even smoothly varying velocity, except for the special case of the speed of light. This might seem to prevent there being any

well-defined momentum, and so might seem to violate the conservation of

momentum and, because kinetic energy is defined in terms of momentum, the conservation of energy.

A minor point here is that, as I have stated the objection, I have tacitly assumed that space and time are like E3J and E14 respectively. However,

much the same problem will arise provided we take space and time as like

E3?m and Ei?m, for the same large integer ra.20

Some solutions to the jerkiness problem involve Quantum Theory, but

I shall provide one which does not. The problem was that momentum is

usually defined as mass times velocity, but if the motion is jerky, then there

is no velocity. My solution is that momentum need not be defined as mass

times velocity, but rather should be understood as a measure of the tendency of a particle to move. In discrete space we may take a momentum to be

characterised by a direction, (corresponding to a translation by one unit) and a magnitude which is just the mass times the propensity to move in that



338 PETER FORREST

direction. The particle either stays still or moves along but its propensity to move could be constant. That would be the analog of uniform motion.

The analog of the rate of change of momentum would then be a measure

of the propensity for the momentum to change.21 If there was no independent reason for treating momentum as a disposi

tion to move, then this would be ad hoc. But there is an independent reason,

namely that physicists in formulating new theories have considered them

selves free to alter the formula connecting momentum and velocity. It takes

a different form in Special Relativity from that in Newtonian mechanics.

That shows that they are thinking of momentum as a quantity which is not

merely mass times velocity. However, it surely has some connection with

velocity. So it is quite plausible that momentum is thought of as a measure

of the tendency to move.

3.3. The Anisotropy Problem

The discrete spaces En?m lack isotropy; that is, there are a number of priv

ileged directions. If we represent En?m by points with integer coordinates, then these privileged directions are just the coordinate axes. But a priv

ileged direction can be characterised intrinsically as corresponding to a

vector-analog T which maximises A(T), where ?(T) is the largest integer K such that TK P is adjacent to P for all points P.

Euclidean space is, however, isotropic. And this might be said to be a

distinct advantage it has over discrete space. To be sure General Relativity has accustomed us to the idea of curved space, and curved space might lack isotropy. But this scarcely affects the issue, for a curved space will be

locally isotropic, in the sense that isotropy is approached as we consider

smaller and smaller regions. Discrete analogs of curved space for which

the small regions approximated En?m would not be locally isotropic. The anisotropy of En?m is a microgeometric phenomenon, which, pro

vided m is large enough, becomes less and less noticeable in the macro

geometric limit, that is, as we consider distances which are large num

bers of units. (These could still be very small regions from our point of

view. Thus if 1030 units is about 1 cm, there are 1015 units, a large number

indeed, in 10-15 cm, a very small distance.) What I mean by the anisotropy

vanishing in the macrogeometric limit is that if we look for translations

which preserve distance to within a small proportion e of K units, where

K is large, we shall be able to find them in many different directions.

Because the anisotropy of En?m is a microgeometric phenomenon, it

should not surprise us that we have never noticed the anisotropy of space.

Therefore, anisotropy is not a refutation. However, we might well consider

that the isotropy of Euclidean space is a point in its favour, balancing the




advantages I have already noted for discrete space -

bringing the score to

one all, as it were.

At least, that is what we should say if discrete space must be anisotropic. In fact I consider that the best way of handling the anisotropy problem is to

resort to 'fuzzy' or indeterminate discrete spaces. These would arise quite

naturally in the context of Quantum Discrete General Relativity. But that

is beyond the scope of this paper.

3.4. The Problem with Relativity

If we ignore Special and General Relativity, the Discrete Space Thesis looks

in fairly good shape, with the only slight worry being anisotropy. There

are, however, some difficulties in developing a discrete relativistic theory.

However, I think they can be overcome. First consider points of space time separated in a space-like fashion. How far apart they are depends on

a frame of reference, but we can consider the relation of being adjacent

if simultaneous, which is short for: adjacent in a frame of reference for

which they are simultaneous. Likewise for time-like separated points in

space-time we can consider the relation of being next if co-located which

is short for: temporally adjacent and after in a frame of reference in which

they are at the same location in space. Those seem the natural relativistic

analogs of spatial and temporal adjacency. However, we face an immediate

problem when we try to define the metric on space-time. We can no longer

say that if Pi is adjacent to P2 and P2 is adjacent to P3 then Pi and P3 are

at most two units apart. To be sure Pi and P2 are adjacent if simultaneous, and P2 and P3 are adjacent if simultaneous, but we cannot in general obtain

a frame of reference in which P\, Pi and P3 are all three simultaneous.

How then do we characterise the metric for space-time? First let

us define a small space-like shift to be a transformation of space-time which:

(i) Sends a pair of adjacent if simultaneous points either to the

same point or to adjacent if simultaneous points.

(ii) Sends a pair of next if co-located points either to the same point or to next if co-located points.

(iii) Always sends a point to the same point or to an adjacent if

simultaneous point.

Two points which are space-like separated are now no further than r units

apart if r iterations of some small space-like shift transforms one to the

other. The exact distance is the minimum number r for which r iterations

of some small space-like shift transforms one to the other. (There is an



340 PETER FORREST

analogous definition of small time-like shifts and of the distance apart of

time-like separated points.) There is a family of discrete spaces Mm obtained by providing suit

able coordinates in Minkowski space and considering only the points with

integer coordinates. Points which are space-like separated and distance at

most m apart if simultaneous will be considered adjacent if simultane

ous. (Likewise for time-like separated points.) I conjecture that the metric

defined in the above manner will approximate the metric on Minkowski

space.22 There is, however, no simple proof of this result, for it is not easy to survey the small shifts. But if the conjecture is refuted with a weird

looking small shift, a bit of monster-barring would be in order.

4. EMPIRICAL TESTS

4.1. Deciding Whether Space is Discrete or Continuous

I have presented an account of some advantages of the Discrete Space Time Thesis, and some replies to objections. This was not intended as

a case for that hypothesis, but rather to keep the issue open. How then

should we decide whether space-time is discrete or continuous? As I shall

now relate, there is the possibility of a quite striking confirmation of the

thesis that space is discrete, as well as the possibility of a rather weaker

disconfirmation.

First consider the possible empirical confirmation. Physicists might pro

pose a theory which could be formulated in a vector-free fashion. However, on performing very accurate measurements they might find a series of small

discrepancies between predictions and observations. These discrepancies

might then be explained as due to small quantities previously ignored when arguing that the discrete space formulation of the laws coincides

with the continuous space formulation. For example, suppose the theory were formulated using some familiar second-order differential operators

(such as d2/dx2 + d2/dy2 + d2/dz2) then the discrete space analogs would

agree with them only if we ignored fourth powers of some small number.

(See Appendix One.) The discrepancies between prediction and observa

tion might then be accounted for by incorporating correction terms, which

would correspond to fourth-order differential operators (d4/dx4 + ). It

would then be ad hoc for the defenders of continuous space to complicate their laws by putting in these correction terms, whose occurrence followed

automatically from the discrete space formulation. But worse could follow.

Further discrepancies might be discovered which could be accounted for

by the way we had ignored the sixth powers of the small number, and so




on. That would be a near conclusive refutation of the continuous space

hypothesis. The case for discrete space would become even stronger if two inde

pendent physical theories, dealing say with different fields, supported the

Discrete Space-Time Thesis in this fashion, with, of course, the same value

in both cases being obtained for the number of units per cm, K. That would

then provide a Whewellian consilience, making discrete space-time one of

the best established scientific theories.

Notice that this way of arguing for Discrete Space-Time Thesis would

not be metaphysical except in the way in which all theory choice involves

metaphysical considerations of elegance, the avoidance of the ad hoc and

so on. The nature of space-time would be a genuine physical theory. The possibility of confirming the discrete character of space also results

in a rather weaker kind of disconfirmation. We could discover that the

laws which best explain the phenomena do not contain the correction

terms mentioned above. This would not refute the conjecture that space is discrete, but it would put a lower bound on the number K of units of

discrete distance per cm. If increasingly accurate measurements repeatedly increased this lower bound, this would, both on intuitive and on Bayesian

grounds, lower the probability that space is discrete.23

5. APPENDIX ONE: DISCRETE ANALOGS OF SOME DIFFERENTIAL EQUATIONS

Consider a discrete space D. We shall be interested in the discrete analogs of derivatives of functions from D to a vector space V.24 Given any trans

formation T of D, we can define the difference operator dj thus:

dT4>{P) = 4>(TP) -

<?(P), for any function cb from D to V.

If T is a small translation then it is not surprising that dj is approximated

by a differential operator. For simplicity, consider the one dimensional

discrete space Ei?m. Then the points correspond to the integers and we

have vector-analogs T^j which send the integer r to r + j. If we assume

the function qb is represented by an analytic function of real numbers, f, such that 4>(r)

= f(re) for some small c, then

dxy] is represented by the

operator d3 where:

djf(re) =

i(re + je)-f(re).

Ignoring second and higher powers of e, we have:

?jf(re) = jef'(re).



342 PETER FORREST

This is a good approximation provided j is not too large. Hence small

vector-analogs are approximated by the operation of taking derivatives.

Therefore, we can expect any differential equation to have a discrete ana

log, provided there is a suitable group of vector-analogs. What may be more surprising is that even without a group of vector

analogs we can find nice discrete analogs for the most common differential

operators. Consider the spatial average difference operator SAD defined

as follows:

For any function <\) from D to V, SAD0(P) is equal to the

average of </>(X) -

</>(P) for all points X spatially adjacent to P.

If we take D to be space-time, we can define the temporal average differ

ence operator TAD:

For any function (f) from D to V, TAD</>(P) is equal to the

average of (j){X) -

(?>(P) for all points X temporally adjacent to P.

Or if we restrict our attention to the next after relation we have the future

average difference operator FAD:

For any function (j) from D to V, FAD0(P) is equal to the

average of </>(X) -

<?>{P) for all points X temporally adjacent to and after P.

I think you will agree that these are natural operators. It is significant, therefore that in the macrogeometric limit these averaging operators corre

spond to commonly occurring differential operators. This can be illustrated

most easily in the one dimensional case, in which discrete space has the

structure Eijm so the points correspond to integers. Then we may think of

the function <f> as corresponding to an analytic function fin such a way that

for each integer j, </>(j) =

f(je) for some small e. Therefore:

SADf (j) =

[f (?e -

me) + f (je - me + e) + + f (je

- e)

+f(je + ) + f (je + me-e) + f (je + me)}

/2m-/(je).

Ignoring fourth and higher powers of e, this becomes (2m+1 ) e2f " (je)/'2.

The factor (2m + l)/2 is a result of using the Eijfn discrete space.

Generalising to the three dimensional case, we find that SAD corresponds to a differential operator a(d2/dx2 + d2/dy2 + d2/dz2) where a is of order e2. The exact formula for a could be computed for En>m, but there

is no point in doing so, since we are interested in the En,m more as




suggestive models than as accurate accounts of the structure of discrete

space. Likewise TAD corresponds to ?d2/dt2 and FAD corresponds to

7<9/<9?, where ? is of order e2 and 7 is of order e.

We are now in a position to write down neat discrete analogs of a variety of differential equations. For example, consider Schroedinger's equation for a particle of mass ??:25

y/(-l)(dF/dt) =

-(d2F/dx2 + d2F/dy2 + d2F/dz2)/2? +VF.

This is approximated by:

V(-l)FAD$/7 =

SAD$/2a/z + V$.

Provided, then, we stick to suitable equations there are discrete analogs even without a group of translations.

6. APPENDIX TWO: THE APPROXIMATION THEOREM

We are considering an n-dimensional Euclidean Space En, whose Euclidean

distance function is e. The Euclidean space En is represented by the vector

space En of n-tuples of real numbers, whose distance apart is given by the usual formula. We are going to approximate it by a discrete space Sm,

which is represented by the members of SL, a countable subset of En, such that every vector in En is distance at most L from some member of

SL. Here L is any positive number. If Sm is to be one of the spaces En?m introduced in Subsection 2.3, we should make a special choice of the set

of n-tuples SL, namely the set of all n-tuples with integer coordinates. In

that case L ? ^Jn. It is this special case which I rely on in the paper. But

the slightly more general result which I prove in this appendix may be of

interest to those who think there is some advantage (e.g. alleviating the

anisotropy problem) in a different choice for SL.

The distance function, d, on Sm is defined in terms of adjacency, where

two points are adjacent if they are distinct and they are represented by

n-tuples no further than distance m apart. The numbers L and m will

eventually be selected to provide the required degree of approximation. The idea behind the proof is that the n-tuples of reals, representing the

points in Euclidean space, are approximated by the n-tuples in S?, each

n-tuple being approximated by the (or a) nearest n-tuple in S^. Before I state and prove the theorem, I shall introduce some notation:



344 PETER FORREST

(a) The n-tuples will be thought of as vectors, and the length of an

n-tuple ? will be denoted by ||?||, where if ? is (x\,..., xn), then ||?||

= V(xl H-?~xn)' Therefore the distance between

n-tuples ? and rj is ||f -

77I|.

(b) If x and y are real numbers, then I write x ? y ? A just in case

x and y differ by at most A.

(c) I shall assume we have coordinates for the n-dimensional

Euclidean space En. I designate the n-tuple of reals forming the coordinates of P by P*.

THE APPROXIMATION THEOREM: Let n be any positive integer, and L any positive real number. Our aim is to approximate the n-dimensional

Euclidean space, En by a discrete space Sm. Given any positive real number

e less than 2, and any positive real number 6 then, there is some positive M

such that for any integer m > M, we can find a positive real number K, and a mapping F from En to Sm, where Sm is represented by SL, described

above, such that:

If P and Q are any points in En, for which e(P, Q) > ?>, then the ratio d(F(P), F(Q))/Ke(P, Q) differs from 1 by less than

e.26

Proof Let 0 < e < 2; let 6 > 0; let M = 8L/e; let m be an integer

greater than M (so 2L/m < 1/2); let J = 2m/e8\ and let K =

2/e6 (so K =

J/m). Then F is the mapping defined by: For any point P in En,

F(P) is one of the nearest n-tuples in SL to JP*.

Because every n-tuple is within L of a member of SL, we have:

(1) If P is in En, ||F(P) -

JP*\\ < L.

Therefore:

(2) If P and Q are in En, ||F(P)-F(Q)|| ? J||P* -

Q*|| ?2L.

Now F(P) and F(Q) are adjacent just in case ||F(P) -

F(Q)|| < m.

Therefore, from (2) we have:

(3) If P and Q are in En, and if Je(P, Q) < m - 2L, then F(P)

and F(Q) are adjacent.

(4) If P and Q are in En, and F(P) and F(Q) are adjacent, then

Je(P, Q) <m + 2L.

I now provide some inequalities relating e(P, Q) and d(F(P), F(Q)). First we take e(P, Q) as given and estimate d(F(P), F(Q)). Let r be




the largest integer such that: r - 1 < Je(P,Q)/(m -

2L) < r. Then

we can find a chain of r points linking P to Q whose images under

F are pairwise adjacent. It follows that F(P) and F(Q) are no more

than distance r apart. More formally, select points Po,..., Pr, such that

P0 =

P,Pr =

Qand,fori =

1,2,... ,r -

1, e(P?,P?+i) =

e(P,Q)/r. It follows that Je(Pi, Pi+i) < m - 2L. So, by (3), F(P?) is adjacent to

F(P?+i). Therefore, by the definition of distance in Sm,d(F(P),F(Q)) < r. But r - 1 < Je(P, Q)/(m

- 2L), so:

(5) d(F(P), F(Q)) < Je(P, Q)/(m -

2L) + 1.

Next we take d(F(P), F(Q)) as given and estimate e(P, Q). Now d(F(P), F(Q)) is some integer t, and by the definition of distance on Sm, there will

be some (not necessarily unique) chain of (t + 1 ) pairwise adjacent points in Sm connecting F(P) to F(Q). These points are themselves the images under F of points Po to Rt in En. And we can (over)estimate e(P, Q), the

Euclidean distance between P and Q, by summing the Euclidean distances

between the P?. More precisely, because d(F(P), F(Q)) = t, there are n

tuples ?o, - . ,& in SL such that ?0 = F(P), ?t = F(Q), and such that

||& ~

C?+i|| < m- F?r * ? 1,..., ? ? 1, let i2i be the point in En with

coordinates &/J. Let Po = P, and let P?

= Q. Then, &

= F(P?), and

F(Ri) is adjacent to F(Pi+i ). Therefore, by (4), Je(P?, Ri+i ) < m + 2L.

Using the triangle inequality for Euclidean distance, we have:

(6) If P and Q are in En, Je(P, Q) <(m + 2L)d(F(P), F(Q)).

From (5) and (6), we have the following estimate for the ratio d(F(P), F(Q))/e(P, Q), for any P and Q in En:

(7) J/(m + 2L) < d(F(P), F(Q))/e(P,Q) < J/(m -

2L) + l/e(P,Q).

Using elementary algebra (remembering that 2L/m < 1/2) we obtain:

(8) md(F(F), F(Q))/Je(P, Q) ? 1 ? A, where A = 4L/m +

m/Je(P, Q).

Because of the choice of M, J, K and L and because m > M, we find

that 4L/m < 4L/M =

e/2, and m/Je(P, Q) < m/J6 =

e/2. So:

(9) A < e.

Now J/m = K, so from (8) and (9) we have the required result, namely

that the ratio d(F(P), F(Q))/Ke(P, Q) differs from 1 by less than e. Hence we have shown that the Euclidean distance e(P, Q) is, when multiplied by



346 PETER FORREST

a suitable constant K, approximated by the distance on the discrete space of the points F(P) and ?(Q) which approximate P and Q.

It is also useful to have a result which shows that the approximation holds provided d(F(P), F(Q)) is sufficiently large. So I state and prove the following.

COROLLARY TO THE APPROXIMATION THEOREM: Suppose the numbers e, ?>, M and K are as in the Approximation Theorem, and suppose

m > M.Ifd(F(P),F(<2)) > 2(1 + l/e) thene(P,Q) > ?andsod(F(P), F(Q))/Ke(P, Q) differs from 1 by less than e.

Proof of Corollary: The proof of the Approximation Theorem proceeds until (8) without requiring the assumption that e(P, Q) > 8. From (8) and the inequality ALjm < AL/M

= e/2, we obtain:

(90 (d(F(P), F(Q)) -

l)/Ke(P, Q) < 1 + e/2.

From (90 we have:

(10) If d(F(P), F(Q)) > 2(1 + 1/e), then

Ke(P, Q) > (1 + 2/e)/(l + e/2) = 2/e. But K =

2/e8. Therefore (10) implies the required result.

7. APPENDIX THREE: THE INTRINSIC CHARACTERISATION OF En,m

The space En,m can be characterised up to isomorphism by various intrinsic

characteristics, especially those involving the group of vector analogs. Let

us consider, then, a vectorial discrete space whose group of isometries is

isomorphic to the group of n-tuples of integers. (That is, it is a torsion-free

Abelian group with n, but no fewer, generators.) Because this is an Abelian

group I shall denote the combination of S and T by: S + T. In this group there is a subset Bp consisting of all the vector-analogs T such that T(P) is adjacent to P. I call this the adjacency ball for P. Because the space is vectorial it is easy to check that adjacency ball Bp is the same for all

points P. So let us call it the adjacency ball B.

A necessary condition for D to be isomorphic to En,m is that there are

generators T\,..., Tn for the group of vector-analogs such that k\T\ +

-h knTn is in B just in case k2-\-h k2 < m2. If this condition holds

I say that the adjacency ball has Euclidean radius m.

This condition is sufficient as well as necessary. For take any point P

in the space D. Then there is a mapping H from n-tuples of integers to

D, defined by H((ku..., A:n)) =

T(P), where T = kxTx + + knTn.

Because D is vectorial it is easy to check that H is 1 to 1 and onto. We have




to show that if T = kxTx + + knTn and S = jxTx + + jnTn, then

T(P) is adjacent to S(P) just in case (fci ?

ji)2 H-\-(kn- jn)2 < m2. But T(P) is the result of the action of the operation (T

- S) on S(P). So

T(P) is adjacent to S(P) just in case T - 5 is in BS(p) = P. And that

holds just in case (fci ?

j\)2 -\-+ (kn -

jn)2 < m2. We have, then, a not too complicated intrinsic characterisation of the

space En,m- First we require the space to be vectorial. Next we require that the group of vector-analogs be the right sort of group (isomorphic to

the n-tuples of integers). Finally we require that the adjacency ball has

Euclidean radius m.

8. APPENDIX FOUR: ON THE METRIC DIMENSION OF En, m

My aim is to show that En,m has metric dimension at most n for any scale

a greater than 20 x 3n~3, provided m > l?^n; and metric dimension

at least n for any (integer) scale greater than and less than 3m/\6y/n.

Therefore, if n is fairly small (say no greater than 4) and m is fairly large

(say 1015) En,m has metric dimenson equal to n for scales which are neither

too small nor too large (between 60 and about 1014.) I begin with some results for En (standard n-dimensional Euclidean

space) which will be needed to establish the required results for En,m. First, for any positive integer n, I define a regular n-simplex of edge a to

be a set of n + 1 points distance a from each other. These n + 1 points are said to be vertices of the simplex. I assume that for any positive real

number a, En contains a regular n-simplex of edge a. I also assume that

this regular n-simplex has a unique centre, that is a point inside the n

simplex equidistant from all the vertices. This is obvious for n = 2 or

3, the cases of greatest interest, and may be established for all n, using mathematical induction.

In order to discuss the dimension of the discrete space En?m, I need some

results about n-simplices in the Euclidean Space, En. I begin with:

THE HEIGHT THEOREM:27 Let the height of a regular n-simplex of

edge a, i.e. the length of a perpendicular from a vertex to the opposite

(n -

l)-simplex, be una. Then u2 = 1 - l/4u2_{ and u\ ?

3/4. (So

u\ =

2/3, u\ =

5/8, etc.) Proof. By elementary trigonometry u2 =

3/4. To show that u2 ?

1 - 1 /4i?2 _ x, we consider a vertex A of a regular n-simplex of edge a, and

let point P be the centre of the opposite (n - 1 )-simplex (i.e. the (n

- 1 )

simplex contained in the n-simplex but not containing A). Then AP can

be shown to be perpendicular to the (n - 1 )-simplex opposite A. We then



348 PETER FORREST

consider another vertex B and extend the line BP until it intersects the

(n ?

2) simplex which contains neither A nor B. The point of intersection, call it C, will be the centre of that (n

- 2) simplex. It can be shown that

both AC and PC are perpendicular to that (n -

2) simplex. The point P

lies in the plain containing A, B, and C, as in the accompanying diagram. And the following hold for any triangle ABC with sides a, 6, c as in the

diagram.

(1) h2 = AP2 = AB2 - PP2 = c2 - x2

(2) x = a/2 + (c2

- 62)/2a

For the triangle being considered, a = un-\a,b

= un-\a,c = a, and

h = una. Substituting these values in (2), we obtain:

(3) x = a/2un.

From(l) and (3):

(4) u2na2 = a2 -a2/4ul_{.

The required result now follows by dividing by a2.

COROLLARY: For all n greater than 1, un > 1/^2. (The proof is by induction on n.)

Next I show that there cannot be an approximately regular (n + 1)

simplex in En.

THE APPROXIMATELY REGULAR SIMPLEX THEOREM: For any integer n greater than 1, for any positive number a, and for any positive e less than a/(20 x 3n~3), there cannot be n + 2 points in En, the n

dimensional Euclidean space, distant from each other between a and a ?

e.




LEMMA: Let e be some positive number no greater than a/10. Let ABC

be any triangle with sides of length a, b and c between a and a/y/2. And

let A'B'C be a triangle whose sides a\ b', and c! are also between a and

a/x/2 and which differ from a, 6, and c by less than e. Finally, suppose the

the height h of ABC is greater than a/>/2. Then the height t? of A'B'C1 differs from the height /i of APC by less than 3e.

Proof of Lemma: Because all three sides of the triangle ABC lie

between a and a/2, the triangle is acute angled. So the perpendicular from A to the side BC is AP where P is between B and C. So the triangle

ABC is as in the diagram. Therefore:

(1) h2 = c2 -

x2.

Likewise, using the obvious notation:

(2) t?2 = c'2 -

x'2.

Now x' + y' = a'>a-e = x + y-e.

Therefore, either x' > x - e/2 or y' > y

- e/2. Without loss of

generality we may suppose the former:

(3) x'>x- e/2.

Now c' < c + e. This inequality, together with (2) and (3), shows that

(4) t?2 <(c + e)2 -(x- e/2)2 = c2 - x2 + (2c + x)e + 3e2/4.

Now c and x are both less than a, and e < a/10. So from (4):

(5) ha < c2 -

x2 + 3.1 x ae = t?2 + 3.1 x ae.

Therefore:

(6) t?2 -h2 < 3.1 x ae.

If t? > h, then t? > h > a/y/2. Soh + t? > ol^/2 > 1.4 x a. Therefore, from (6):

(7) /i'-/i<3.1x ae/1.4 x a < 3a.

A similar calculation shows that:

(8) h2 - t?2 < 3ae - 3e2/4.

If t? < h, we can obtain a crude est?mate for h-t? by using the inequality h + t? > h> a I y/2 and then inferring from (8) that h- t? < 3e^/2. To obtain a more accurate estimate we then use the crude estimate to show

that h + t? > av"2 -

3c>/2. So, from (8):



350 PETER FORREST

(9) h - t? < (3ae -

3e2/4)/(ay/2 - 3 >/2) =

(3/V2)(l -

e/4a)/(l -

3e/a).

Since e < a/10, (1 -

e/4a)/(l -

3e/a) < 39/28 < y/2. Therefore:

(10) h-t? < 3e.

(7) and (10) show that t? differs from h by less than 3e.

Proof of the Approximately Regular Simplex Theorem: Consider a reg ular n-simplex ABC etc. with edge a. And compare it with another n

simplex A'B'C etc. with edges between a and a - e. As in the proof of

Result One, the height una of ABC etc. is the height of a triangle whose

sides are a, un-\a and un-\a. Let the height of A'B'C etc. be u'na. We

can prove by induction on n, using the Lemma, that u'n differs from un

by less than 3n~2e/a. But 3n~3e = a/20, so 3n~2e/a

= 3/20. Hence

u'n > un -

3/20. Now consider a set of n + 2 points in En. This set

comprises n points in an n - 1 dimensional subspace and two points U and

V on either side of that space. So the distance between U and V is no less

than the sum of the heights of the two n-simplices formed by omitting first

U and then V from the set of n + 2 points. So if all the distances between

the (n + 2) points are between a and a - e, the distance between U and

V is at least 2u'na, where u'n > un -

3/20. Now 2^n > y/2 > 1.4. So

2u'n > 1. Therefore U and V are more than a apart. This establishes the

required result.

COROLLARY: The metric dimension of En, the n dimensional Euclidean

space, is equal to n, relative to any scale.

Proof of Corollary: I have assumed that there is a regular n-simplex with edge any length. That shows that the metric dimension is at least n.

To show it is exactly n we need to show that, for no positive a is there

a point X distance a from all n + 1 vertices of a regular n-simplex of

edge a. That is just a special case of the Approximately Regular Simplex Theorem.

We are now in a position to show that En,m has metric dimension n for

suitable scales. As in the proof of the Approximation Theorem of Appendix Two it is less confusing to prove the result for the slighly more general case of the space Sm. (If Sm

= En,m, then L =

y/n.)

THE DIMENSION THEOREM: Let SL be a subset of the set of n-tuples of reals chosen so that every n-tuple of reals is Euclidean distance at most

L from some member of SL. And let the discrete space Sm be characterised

as in Appendix Two. Then:




(i) Sm has metric dimension at most n for any scale greater than

20 x 3n_3, provided m > 16L.

(ii) Sm has metric dimension at least n for any (integer) scale a

suchthat 1 < er < 3m/\6L.

Proof of (i): If n = 1, and a > 1, it is obvious that Sm had dimension

n relative to a. So we may concentrate on the case in which n > 1. We

show that Sm cannot have dimension greater than n relative to scale a

greater than 20 x 3n~3. For this purpose we rely on the Corollary to the

Approximation Theorem with 8 = 4 and e = 1/2. (So K =

1.) Suppose, that there were n + 2 points in Sm distance a apart. Then we may suppose

F to be chosen so that they were the points F(P0), F(Pi),..., F(Pn+i). Since a > 2(1 + 1/e), by the Corollary to the Approximation Theorem

d(F(P?), F(Pj))/Ke(Pi, Pj) would differ from 1 by less than e. Therefore the e(Pi,Pj) would differ from a/K by less than e/K(\

- e) =

1/2 and so would lie between a and a - 1 where a =

a/K + 1/2. By the Approximately Regular Simplex Theorem, this is impossible, because

a/K + 1/2 = a + 1 /2 > 20 x 3n"3. This shows that there cannot be n + 2

points in Sm an equal distance a apart, completing the proof.

Proof of (ii): Given any integer a greater than 1, let 8 = a, and let

e = 3/2a. So e < 3/4 < l.Becausecr < 3ra/16L, it follows that m > M,

where M = 8L/e

= \6La/3. Let F be the mapping defined in the proof of

the Approximation Theorem. Then, provided all the Euclidean distances

considered are greater than 8, the proof of the Approximation Theorem still

holds. However the inequalities concerning the comparison of e(P, Q) with

d(F(P), F(Q)), for points P and Q in En, established in that proof are not quite strong enough for my present purpose. We need to consider the

special case in which Je(P, Q)/(m -

2L) =

Ke(P, Q)/(\ -

2L/m) is itself an integer r'. In that case we can find a chain of r' points linking P

to Q whose images under F are pairwise adjacent. It follows that:

(5') d(F(P), F(Q)) < MP, Q)/(m -

2L) =

Ke(P,Q)/(l-2L/m).

As in the proof of the Approximation Theorem, we also have:

(6) Ke(P, Q)/(l + IL/m) = Je(P, Q)/(m + 2L) <d(F(P),F(Q)).

From (5') and (6) we have, for this special case:

(7) d(F(P), F(Q))/Ke(P, Q) ? 1 ? [1/(1 -

2L/m) -

1].



352 PETER FORREST

Now there are n + 1 points, Po, Pi,..., Pn in En, Euclidean distance

a/K(\ -

2L/m) apart. Because a = 8, this distance is greater than 8. By

(7) the points F(P0), F(Pi ),..., F(Pn) are distance apart within a[\/(\ -

2L/m) -

1] of a/(I -2L/m) and hence within 2cr[l/(l -2L/m) -

1] of a. Now L/m < e/8, and e < 1, so 2a[l/(l

- 2L/m)

- 1] < 2cre/3

= 1. But the points F(Po), F(Pi ),..., F(Pn) are integer distances apart and we

have just shown that these integers differ from the integer a by less than

1. So they are exactly a apart. This shows that Sm has dimension at least

n relative to scale a provided m > M = SL/e

= 16La/3.

NOTES

*I would like to express my thanks to John McKie with whom I have had several inspiring conversations about discrete space. I would also like to thank the audience of a paper on

this topic which I read in October 1991 at the College Park campus of the University of

Maryland. Finally I would like to thank the referees of Synthese for their comments. One of them, in particular, should be thanked especially for help in improving Appendix Two. lrThis is quite different from having the discrete topology, in which any set of points is a

neighbourhood of every point in that set.

2If space-time does turn out to be continuous there is much to be said for introducing

infinitesimal distances. It is, however, convenient to concentrate on one controversial topic at a time.

3The distance being one less than the number of points including P and Q themselves. 4Nor is it a wonder that we can define relations which do satisfy the triangle inequality.

For given any non-negative quantitative dyadic relation R we may define a distance rela

tion dR in terms of R thus: dR(P,P) = 0 for any point P; and, for any two points P

and Q, dR(P,Q) is the greatest lower bound of all R-sums along chains connecting P and Q. Here the R-sum is the sum of the values of R(P?, Pi-\) for i =

1,..., n where

P = Po, Pi, , Pn =

Q is the chain connecting P to Q. Provided this greatest lower

bound is never zero, dR is easily seen to satisfy the axioms for a distance function. How

ever, we have no reason to expect that if R is of theoretical significance so will be dR. 5 And if that does not seem too complicated, remember that very often we require second

order derivatives.

6In particular we could select some small distance S. Then if d is distance we could define

d* by d*(P, Q) =

d(P, Q), unless 0 < d(P, Q) < 6, in which case d*(P, Q) = 6.

7Condition (i) is not redundant. Give the points coordinates and let T send points with

integer coordinates one unit to the left, and the other points one unit to the right. T does

not correspond to a vector but does satisfy condition (ii). 8Condition (ii) is not redundant. Rotations satisfy (i) but not (ii). 9But in unfavourable cases the group of translations is the trivial group, containing only

the null transformation.

10Quantum Theory might suggest that there is some indeterminacy in which point is next

to which. The most straightforward indeterminate discrete space would be specified by a set of points indexed by positive integers and an assignment of a positive number 7rn,m no

greater than one to the points indexed by n and m. The number 7rn,m could be thought of

as the degree of truth of the proposition that the points indexed by n and m are adjacent.

It is convenient to consider the matrix whose (n,m) entry is 7rn,m. Call this matrix n.




Making a plausible independence assumption, the proposition that the points indexed by n

and m are distance k apart has a degree of truth which is the (n, m) entry in the matrix

n\ the kth power of II. 11 As a referee has pointed out, there are duals to the tile models, in which the vertices of

the tiles, rather than the tiles themselves, represent points. The dual of the square tile space

is another instance of E2,i.

12I would like to express my thanks to the anonymous referee of an earlier article of mine

for raising the objection.

13It is trivial that the inverse of a vector-like isometry is vector-like. So this condition entails

that the vector-like isometries form a group.

14The problem of providing an intrinsic characterisation of the dimension of a discrete

space was raised by one of the referees.

15The standard discrete topology is trivial in that all sets are open. Using it, the topological dimension of any discrete space would be zero. A more interesting topology is that in

which, for any point P, any set of points containing P and all the points adjacent to P

is a neighbourhood of P. This is useful for some definitions in topology. But we do not

have the usual relation between open sets and neighbourhoods. For if an open set is defined

as a set which contains a neighbourhood of any point in it, then, on En,m for instance,

there are only two open sets, the whole space and the empty set. Hence a neighbourhood

of P cannot be characterised as any set containing an open set containing P. The moral

is that for discrete spaces, neighbourhoods are more fundamental than open sets. But that

interferes with those definitions, such as of dimension, which seem to make essential use

of the open sets (Nagata 1965, pp. 8-10). 16For the sake of definiteness we could take jn to be l/(10)n. 17But Russell did not treat this as a serious objection. He merely remarked on the disconti

nuity of motion in discrete space. 18To ensure this is topological continuity, and not just an analog, we need a suitable

topology. For discrete space we obtain a not quite trivial topology, as above, by taking a

neighbourhood of a point P to be any set which includes P and all the points adjacent to

it. With this definition continuity turns out to be as I have characterised it. 19If there are 1030 points per cm, and the speed of light is taken as unity, then, since the

speed of light is about 3 x 1012 cm/sec, there would be about 3 x 1042 temporal instants per

second. Something travelling as slowly as 1 cm per million years (about 3 cm per 1014 sec)

would move forward to the next point, then pause for about 1026 instants before moving

again. But these 1026 instants would only occupy 10"16 of a second, and the amount moved,

when there is motion, would only be 10~30 cm. So it is not surprising that we do not notice

the jerky motion.

20A rather ad hoc solution to the jerkiness problem is to take space as E3,m and time as

Ei)Tn/ where m is very much greater than ml'. 21 What happens if there are no vector-analogs, i.e. translations? We may still consider the

propensity for a particle to move to a given adjacent point. But we no longer have an analog

of uniform motion.

22As a referee has pointed out to me, the discrete space will not be even approximately invariant under all Lorentz transformations. If we squash distances along one light-like

line by a large enough factor, say 1030 and stretch it by the same factor along another

light-like line, then the discrete structure will interfere with invariance. But quite generally we cannot expect approximations to symmetric structures to be approximately invariant

under approximations to all the symmetries.

23On the assumption that space is discrete, we can make an initial estimate of a lower bound



354 PETER FORREST

for K. Call that Ko. Then, prior to further empirical investigation, the probability that K is

at least M, on the assumption that space is discrete, will be less the greater M is, provided M is at least Ko. And that probability will tend to zero as M tends to infinity. I take it that

the initial probability that space is discrete is less than 1 (and less than 1 by more than an

infinitesimal). The required result now follows from Bayes Theorem. 240ver the rationals or some field such as the reals in which the rationals can be imbedded. Indeed it suffices that the values be in a torsion-free Abelian group, because this can be

uniquely embedded in a rational vector space.

25Putting Planck's constant equal to 2tt.

26The numbers e and 6 are thought of as extremely small. The requirement that e be

less than 2 is thus of no significance. It is required in the proof because I assume that

1/(1 -

e/4) < 1 + e/2. This holds provided e < 2.

27Presumably this result is well known.

REFERENCES

Field, Hartry H.: 1980, Science without Numbers: A Defense of Nominalism, Princeton

University Press, Princeton.

Gr?nbaum, Adolph: 1973, in Robert S. Cohen and Marx W. Wartofsky (eds.), Philosophical Problems of Space and Time. Boston Studies in the Philosophy of Science. Vol XII, D.

Reidel, Dordrecht.

Nagata, Jun-Iti: 1965, Modern Dimension Theory, Groningen, Noordhoff.

Rogers, Ben: 1968, 'On Discrete Spaces', American Philosophical Quarterly 5, 117-124.

Russell, Bertrand: 1927, The Analysis of Matter, Kegan Paul, London.

Van Bendegem, Jean Paul: 1987, 'Zeno's Paradoxes and the Tile Argument', Philosophy

of Science 54, 295-302. Van Fraassen, Bas C: 1989, Laws and Symmetry, Oxford University Press, Oxford.

Philosophy Department University of New England Armidale,NSW2351 Australia



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Is Space-Time Discrete or Continuous? An Empirical Question