onphilosophicalproblemsinthefoundationsofchaos eory · a nowledgements...

On Philosophical Problems in the Foundations of Chaos eory

by

Christopher Armand Belanger

A thesis submied in conformity with the requirements

for the degree of Doctor of Philosophy

Institute for the History and Philosophy of Science and Technology

University of Toronto

© Copyright by Christopher Armand Belanger (2015)

On Philosophical Problems in the Foundations of Chaos eoryChristopher Armand Belanger

Doctor of Philosophy

Institute for the History and Philosophy of Science and TechnologyUniversity of Toronto

2015

Abstract

is dissertation examines several philosophical issues in the foundations of chaos

theory and fractal geometry. In Chapter 1, I argue that our epistemological and on-

tological investigations would be beer served by looking at the particular successes

and failures of individual chaotic models, rather than focussing on broad questions

of approximate truth. e rest of the dissertation can then be seen as a set of at-

tempts to put this program into practice. In Chapter 2 I consider the prospects for

instrumental fractal models of non-fractal physical objects. Although philosophers

have contended that such models must always be inferior to non-fractal models, I

argue that in some cases fractal models can be vastly epistemologically superior to

their non-fractal rivals. In Chapter 3 I take up questions of ontology, and consider

the prospects for the existence of fractals in physical space. Although philosophers

have argued that physical fractals are an impossibility, I argue that classical me-

chanics and chaotic models could entail the existence of interesting fractal regions

of space. In Chapter 4 I consider two definitions of observational equivalence for

chaotic models, and ague that they fail to meet acceptability criteria.

ii

Anowledgements

is dissertationwas onlymade possible through the help, support, and encouragement of

many people. I am deeply indebted to my supervisor Joseph (Jossi) Berkovitz for suggesting

this project, for innumerable hours of insightful discussion, and for reading and commenting

on countless dras. Jossi has been unfailingly generous with his time and energy, and he has

been instrumental in helpingme to achievemy fullest potential. Along with Jossi, I would like

to recognize my two other supervisory commiee members, Robert Baerman and Charloe

Werndl. I thank Robert for his feedback and friendly advice, and for encouraging me to think

about how my work connects to broader issues. I owe many thanks to Charloe, who helped

me greatly with both the writing process and with many mathematical technicalities. Her

enthusiasm, keen suggestions, and positive feedback helped me and meant more to me than

she may have realized.

I was fortunate to have an excellent examination commiee, who both challenged me

and helped me to see my own work from a new point of view: Craig Fraser, William Seager,

Joseph Berkovitz, Chen-Pang Yeang, and Roman Frigg. I must single out Roman Frigg for his

generosity and graciousness in taking time out of his research leave and travelling to serve

as my external examiner. Roman’s carefully reasoned comments brought a new perspective

to my dissertation, and pointed out new avenues with will certainly inform my future work

on this material.

During my years at the IHPST I have benefited in some way from helpful conversations

with almost everyone, but of those not already named I must mention Denis Walsh, Anjan

Chakravary, Denise Horsley, and Muna Salloum. Many thanks to you all.

e graduate students of the IHPST past and present are a great bunch, and it’s been a

privilege to work with and among you. In particular I tip my hat to theesis Support Group:

Greg Lusk, Alex Djedovic, Aaron Wright, Paul Greenham, Mike Stuart, and Cory Lewis. You

trudged through some awful first dras and saved me from myself, so the next round’s on

iii

me. Near the end of the process I benefited greatly from discussions with Ari Gross and Isaac

Record, who helped me wrap my head around what exactly it is to finish a dissertation.

My family has been unflaggingly supportive, and my sincerest love and thanks go out to

all of you. My parents Karin and Richard, and my sister and brother Caroline and David have

steadfastly assured me that it’s cool to get a PhD. Particular thanks to David for helping me

with my math homework. anks also to my uncles, Roger and Chris, and my grandparents,

Dorothy and Jules, and Graham and Barbara. It is thanks to your love and support that I have

been able to achieve my goals.

anks also to the open-source community, whose volunteer work made this dissertation

possible. Every task involved in the creation of this dissertation, including writing (gedit),

typeseing (LTEX), drawing (InkScape), and simulating (Processing) was done entirely using

free and open-source soware available on Ubuntu Linux.

Lastly and most importantly, my surest and most constant source of strength has been the

love and support of my wife Anne. ere is no possible world where I could have done this

without you.

20 October 2014

iv

http://sourceforge.net/projects/gedit/

http://www.latex-project.org/

http://www.inkscape.org/

http://processing.org/

http://www.ubuntu.com

Outline

0 Introduction 10.1 Motivations: Why Chaos and Fractals? . . . . . . . . . . . . . . . . . . . . . . 10.2 What is Chaos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 What is Fractal Geometry? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.4 Review of Selected Philosophical Works on Chaos and Fractals . . . . . . . . . 11

0.4.1 Chaos and Approximate Truth . . . . . . . . . . . . . . . . . . . . . . 120.4.2 Prediction, Determinism, and Randomness . . . . . . . . . . . . . . . . 13

Prediction and Classical Determinism . . . . . . . . . . . . . . . . . . 13Chaos and antum Unpredictability . . . . . . . . . . . . . . . . . . 15Chaos and Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . 16

0.4.3 Determinism and Observational Equivalence . . . . . . . . . . . . . . 180.4.4 Fractals, from Initial Optimism to a Negative Consensus . . . . . . . . 20

0.5 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1 Chaos and Approximate Truth: A Reappraisal 251.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.2 A Brief Mathematical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.2 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.3 Continuous and discrete dynamical systems . . . . . . . . . . . . . . . 291.2.4 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.3 Example 1: e “Standard Map” of the Kicked Rotator . . . . . . . . . . . . . . 321.3.1 e Classical Kicked Rotator . . . . . . . . . . . . . . . . . . . . . . . . 321.3.2 An Extension: e antum Kicked Rotator . . . . . . . . . . . . . . . 351.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.4 Example 2: e Belousov-Zhabotinsky Reaction . . . . . . . . . . . . . . . . . 381.4.1 A Continuous Model of the Belousov-Zhabotinsky Reaction . . . . . . 381.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.5 ree Approaches to the Approximate Truth of Chaotic Models . . . . . . . . 421.5.1 Peter Smith’s Geometrical-Modelling Account . . . . . . . . . . . . . . 431.5.2 Jeffrey Koperski’s Truesdell-Inspired Account . . . . . . . . . . . . . . 481.5.3 A Popperian Verisimilitude Approach . . . . . . . . . . . . . . . . . . 52

1.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

v

2 Fractals as Instrumental Models 582.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.2 A Few Words on Instrumentalism and Models . . . . . . . . . . . . . . . . . . 592.3 Objections to Fractal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.4 e Trouble with Tribology: Instrumentally Useful Fractals Models . . . . . . 652.5 Fractal Models as Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.6 Fractals or Pre-Fractals? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Fractal Geometry is a Geometry of Nature 753.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2 What is a Fractal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3 Two Standard Objections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3.1 e Constructivist Objection . . . . . . . . . . . . . . . . . . . . . . . 783.3.2 e Atomic Objection . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4 What is a ‘Geometry of Nature?’ . . . . . . . . . . . . . . . . . . . . . . . . . . 813.5 Fractals in Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.6 Fractals in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4 Chaos and Observational Equivalence 994.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.1.1 Background and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 994.1.2 Deterministic Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 1004.1.3 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.1.4 Finite-valued observation function . . . . . . . . . . . . . . . . . . . . 1034.1.5 Deterministic Representation of a Stochastic Process . . . . . . . . . . 103

4.2 Two Mathematical Definitions of Observational Equivalence . . . . . . . . . . 1044.2.1 Manifest Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.2.2 ε-Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5 Epilogue 122

A Appendix: Some Details of Fractal Geometry 127A.1 e Koch Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127A.2 Similarity Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.3 e Hausdorff-Besicovitch Dimension . . . . . . . . . . . . . . . . . . . . . . 132

References 135

Index 145

vi

List of Figures

0.1 Plot of a non-chaotic system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2 Plot of a chaotic system with sensitive dependence on initial conditions. . . . 50.3 A series of curves converging to the Koch curve. . . . . . . . . . . . . . . . . . 70.4 Self-similarity of the Koch curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 80.5 A random Koch curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1 Simple harmonic motion in phase space. . . . . . . . . . . . . . . . . . . . . . 301.2 Discrete motion in phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . 301.3 Schematic drawing of a kicked rotator. . . . . . . . . . . . . . . . . . . . . . . 321.4 Phase portraits of the standard kicked rotator for different kicking strengths. . 341.5 Potentially geometrically similar phase-space trajectories. . . . . . . . . . . . 44

2.1 Example Weierstrass-Mandelbrot functions. . . . . . . . . . . . . . . . . . . . 67

3.1 A series of curves converging to the Koch curve. . . . . . . . . . . . . . . . . . 783.2 Representation of a Julia set, a quasi-circular basin boundary. . . . . . . . . . . 843.3 Fractal phase space basins of araction of a particle in a double potential well. 863.4 Basin of araction and spatial basin of araction (v = 0) for undriven particle

in double-well potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.5 Basins of araction and spatial basins of araction (v = 0) for driven particle

in double-well potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.6 Schematic drawing of the magnetic pendulum system. . . . . . . . . . . . . . 913.7 A trajectory of a magnetic pendulum, as seen from above. . . . . . . . . . . . 943.8 Spatial basins of araction for magnetic pendula with varying coefficients of

friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.9 Progressive zooms of a magnetic pendulum’s basins of araction. . . . . . . . 96

4.1 An example of a finite partition of a metric space. . . . . . . . . . . . . . . . . 1044.2 A schematic diagram of Werndl’s observational equivalence criterion. . . . . . 107

4.3 e bump function B(x) = e−1

1−x2 . . . . . . . . . . . . . . . . . . . . . . . . . 1164.4 e bump function P(x) which problematizes ε-congruence. . . . . . . . . . . 1174.5 e bump function P∗(x) further problematizes ε-congruence. . . . . . . . . . 119

A.1 A series of curves converging to the Koch curve. . . . . . . . . . . . . . . . . . 128

vii

.. 0Introduction

0.1 Motivations: Why Chaos and Fractals?Chaos theory and fractals have a unique place in both popular and scientific culture. In the

late 1980s and early 1990s, science popularizers and scientists penned countless popular and

academic articles connecting fractals and chaos to subjects as disparate as biology (Dewdney

1989), earthquake geology (Sahimi, Robertson, and Sammis 1993), managerial organization

(iétart and Forgues 1995), and computation (Gibbs 1998). e academic presses published

books purportedly linking chaos to subjects ranging from clinical psychology (Chamberlain

and Butz 1999) to warfare (James 1996). And James Gleick, in his Pulitzer-nominated book on

chaos, famously lauded chaos theory as “the century’s third great revolution in the physical

sciences” (Gleick 1987, 6). Chaos theory and fractals even penetrated the popular conscious-

ness enough for the best-selling novel Jurassic Park to have a fractal heading for every chapter,

and for a chaos theorist, played by Hollywood superstar Jeff Goldblum, to have a prominent

role in its blockbuster movie adaptation—and, really, a branch of science could domuchworse

than to have a shirtless Jeff Goldblum as its public ambassador.

Philosophical work on chaos and fractals also reached a crescendo in the 1990s. Some

speculated that it would cause us to boldly revise our views on subjects such as scientific

prediction and explanation (e.g. Suppes 1985; Smith 1998b), realism (e.g. Shenker 1994; Smith

1

0.2. WHAT IS CHAOS? 2

1998b), and perhaps even determinism (e.g. Suppes 1993; Winnie 1998). Recently, however,

the philosophical work on chaos has slowed down. is is, in my opinion, a shame, because

many of these earlier analyses contain insights that are not fully developed, or make propos-

als that might not quite work out. In this dissertation, my goal is to reconsider some of these

older analyses, and to advance several novel proposals, thereby demonstrating that that philo-

sophical analysis of fractals and chaos theory may yield important insights about the way the

world could be and for the philosophy of science.

Chaos and fractals may at first appear to be unrelated topics, since the former is dynamical

and the laer is geometrical. ey are, however, intricately intertwined, and fractals are

ubiquitous in the study of chaos. Chaotic systems are frequently associated with fractals,

whether in their trajectories, phase-space aractors, or basins of araction. And, as we shall

see, these fractals are at the root of many of the philosophical puzzles surrounding chaos.

e remainder of this introductory chapter will proceed as follows. First, I will present

an introduction to the mathematics of chaotic dynamical systems, then the basics of fractal

geometry. Next, I will review the philosophical literature relating to chaos and fractals. In

the final section, I will outline the structure and arguments of the dissertation.

0.2 What is Chaos?ere is currently no single accepted definition of chaos. is is certainly not for lack

of trying, as numerous scientists, mathematicians, and philosophers have both proposed and

criticized candidate definitions (e.g. Crutchfield et al. 1986; Hunt 1987; Devaney 1992; Banks et

al. 1992; Winnie 1992; Baerman 1993; Baerman et al. 1996; ornton and Marion 2004; Tél

and Gruiz 2006; Werndl 2009c). Explicating and evaluating these definitions in detail would

be well outside the scope of this introduction. Fortunately, there are some generally accepted

features of chaos which are both simple to present and intuitively compelling, and we will

focus on these. For our purposes, we will say that chaos is a property of dynamical systems,

and we will tentatively call systems chaotic if their trajectories are bounded, non-periodic,


....0.

10.

20.

30.

40.

50.0 .

0.2

.

0.4

.

0.6

.

0.8

.

1

..

Iteration, n

.

xn

.

. ..x0 = 0.7

. ..x0 = 0.8

Figure 0.1: Plot of the first 50 iterations of the relation xn+1 = 0.9xn for two sets ofinitial conditions. Even though the two initial states differ by 10−1, their trajectoriesremain close to each other. is system is not chaotic.

and if they exhibit SDIC. Let us examine each of these aspects in turn.

Chaos, as we will use the term, is a feature of mathematical dynamical systems. Although

dynamical systems can be defined completely abstractly, in general they are oen taken as

mathematical representations of how values change over time (see, e.g. Devaney 1992, 1). If

the dynamical system is taken to correspond to some part of the real world, then we will

say that it is being used as a representation of its target system. e nature of scientific mod-

els, their target systems, and the representation relation between them are highly contested

issues in the philosophy of science (e.g., Bailer-Jones 2003; Giere 2004; Fraassen 2006; Frigg

2006; Bolinska 2013). However, without wishing to downplay their import, I will bracket these

issues for the present. We need to understand what chaos is before we can think about how

it relates to the world.

In this dissertation I will examine both continuous dynamical systems, which evolve con-

tinuously through time, and discrete dynamical systems, which jump from one value to an-

other in discrete time steps. For simplicity’s sake, however, in this introduction let us focus

on discrete dynamical systems which evolve through iteration. e same general consider-

ations will also apply to continuous dynamical systems, and the similarities and differences

between the two classes of systems will be discussed in more detail in Chapter 1.


To iterate means simply to apply the same function over and over, using the output of one

application as the input to the next. In such a system we have a function f (xn) and an initial

condition x0. We plug x0 into the function to generate a new number x1, which we plug back

in to generate x2, and then x2 gives x3, and so on. e subscript is called the index, and we

use n to represent an arbitrary index. e process of plugging in numbers to generate new

numbers is called iteration, and we will call the set of numbers generated a trajectory. is is

an example of one kind of dynamical system (Devaney 1992, 9).

Many dynamical systems behave in a regular way. Consider the following dynamical

system:

xn+1 = 0.9xn (0.1)

To get the next iteration of this system, we take the current value of xn and multiply it by

0.9. Unsurprisingly, all trajectories tend to zero. Furthermore, trajectories of this system that

start out nearby therefore remain close to each other, and get even closer with each iteration.

If we look at a graph of this behaviour (see Figure 0.1), it is easy to imagine physical systems

it could potentially represent: perhaps the height of an airplane as it comes in to land, the

population of a species approaching extinction, or the speed of a bicycle as it rolls to a stop.

is system is regular, well behaved, and definitely not chaotic.

Consider now the following dynamical system:

xn+1 = αxn(1 − x2n

)(0.2)

Let us consider what happens if we start this system from two initial conditions that differ

slightly. But while to generate Figure 0.1 we used initial conditions that differed by 10−1 let’s

start this time with initial conditions that differ by only 10−6. Metaphorically, if the first two

initial conditions differed by the width of a finger, these new conditions differ by less than

the width of a human hair. As we can see in Figure 0.2, this system is quite different from the

previous one (see Figure 0.2).


....0.

10.

20.

30.

40.

50.0 .

0.2

.

0.4

.

0.6

.

0.8

.

1

..

Iteration, n

.

xn

.

. ..x0 = 7.000000

. ..x0 = 7.000001

Figure 0.2: Plot of the first 50 iterations of the relation xn+1 = αxn(1 − x2n

)with α = 2.5

for two sets of initial conditions. Even though the two initial states differ only by 10−6,their trajectories are noticeably different aer 30 iterations, and they diverge completelyaer 40 iterations. is is evidence of sensitive dependence on initial conditions, a hall-mark of chaos.

ere are three things to notice about this new system. First, we might expect these two

initial conditions to give rise to near-identical trajectories, and for the first 25 iterations or

so this is what we see (see Figure 0.2). However, by around 30 iterations the two trajectories

begin to diverge noticeably, and aer 40 iterations they are entirely uncorrelated. is is evi-

dence of sensitive dependence on initial conditions (SDIC), a hallmark of chaos. In general, any

two trajectories evolved from arbitrarily similar but different initial conditions will diverge,

usually on time-scales that are quite short (e.g. Tél and Gruiz 2006, 159). e type of trajec-

tory divergence necessary for chaos is a maer of dispute. Typically, chaos theorists have

assumed that trajectories must diverge exponentially (e.g Tél and Gruiz 2006, 156). Recently,

however, Werndl (2009a) has argued that chaos is possible given sufficiently rapid polynomial

divergence. In either case, SDIC guarantees that trajectories will eventually diverge greatly,

although it is silent about the rate of divergence.

Technically, the rate of exponential divergence between nearby trajectories is oen given

in terms of a system’s Lyapunov exponents. Note that a system’s Lyapunov exponents can

vary at different points, since trajectories may diverge (or even converge) at different rates at


different times, and it is therefore common to consider a system’s average Lyapunov exponent.

A positive average Lyapunov exponent implies SDIC, and so is oen taken as a mark of chaos

(e.g. Tél and Gruiz 2006, 117–20). Of coursemerely having SDIC cannot guarantee chaos, since

unbounded systems like xn+1 = x2n can have trajectories which diverge rapidly as well (for

initial conditions greater than 1, in this case). is is why it is important for chaotic systems

to be bounded: their trajectories diverge, but remain confined to the same region of phase

space.

e second thing to notice about this system is that if we have such a system in front of us

but are not aware of its precise initial conditions, its long-term behaviour will be very difficult

to predict. If we estimate its initial conditions at all incorrectly, even if we are extremely close,

SDIC will cause our predictions to diverge substantially from the actual system’s behaviour.

is has implications for scientific modelling. At least prima facie, if a physical system is

governed by chaotic dynamical equations, then, provided there is some unavoidable error in

our measurement of initial conditions, eventually our predictions will be completely wrong.

ird, although Equation 0.2may appear simple, it generates quite complicated and random-

seeming behaviour which is certainly not periodic. is “randomness” arises even though the

system in question is deterministic, in the sense that a given input will always produce the

same output. If we turn this around, this means that apparently “random” or “noisy” signals,

whichwemight thinkwould require very complicatedmathematics to generate, can be gener-

ated by simple mathematical expressions. is also has implications for scientific modelling,

because although Figure 0.2 is much more irregular than Figure 0.1, we can also imagine real-

world systems it might represent: say, the daily rainfall in a region, or the hourly price of a

stock-market index. e intuition here is that certain unpredictable-seeming systems might

in fact be governed by chaotic systems with simple mathematical expressions.

To sum up, although there is no generally accepted definition of chaos, we have identi-

fied some salient features of chaotic systems that will suffice for the present investigations.

Chaos, for our purposes, is a property of dynamical systems. We call systems chaotic if their

0.3. WHAT IS FRACTAL GEOMETRY? 7

(a) K0 (b) K1

(c) K2 (d) K5

Figure 0.3: A series of curves converging to the Koch curve. Figure 0.3a is a straight lineK0, corresponding to the 0ᵗʰ iteration. Figure 0.3b shows the line K1 aer one iteration,figure 0.3c shows the result K2 aer two iterations, and figure 0.3d shows the Koch curveK5 aer five iterations.

trajectories are bounded, non-periodic, and if they exhibit SDIC. Chaotic systems oen have

simple equations but complicated trajectories, and predicting their long-term behaviour is

difficult. In addition, there are suggestive reasons to think that some physical systems might

be governed by chaotic equations.

0.3 What is Fractal Geometry?In this section I will present some basic elements of fractal geometry, with a much more

thorough discussion relegated to Appendix A. Our case study will be the Koch curve, an ex-

emplary fractal. To construct the Koch cure, we begin with a straight line of unit length

(figure 0.3a). e construction procedure is to take each straight line segment—of which, to

begin, there is only one—and to divide it into three equal segments, remove the middle third

segment, and replace it with a tent shape whose sides are the same length as the piece we

just removed (figure 0.3b). Alternatively, we could think of placing an equilateral triangle

over the middle third segment, and then removing the boom of the triangle. e resulting

figure, K1, represents the first iteration of a construction process that leads to a fractal. Now

we repeat the operation, this time performing it on each of the straight lines in K1, which

gives us the more complicated shape K2, the second iteration of the process (figure 0.3c). is


Figure 0.4: e Koch curve is self-similar and contains smaller replicas of itself at all levelsof magnification.

process can be repeated as oen as desired. Each iteration will deliver a more complicated

shape with more kinks in the line. In the infinite limit of iterations we get the Koch curve.

Since the construction procedure takes an infinite number of steps, we can never carry

it out to completion in practice. On this basis, some Contructivists have argued that it is

incoherent to speak of the final product as an object, and that instead we should think of

fractals as processes (e.g. Shenker 1994). I will consider this argument more in Section 3.3.1,

and in the meantime for the sake of simplicity I will adopt the Platonist’s language and speak

of fractals as objects, albeit objects that oen are the results of infinite construction processes.

e Koch curve is self-similar, in that it contains smaller perfect replicas of itself at all

levels of magnification. If, for example, we take the “peak” of the curve of length 13 , upon

magnifying it by a power of three we recover the whole of the original curve (see figure 0.4).

is process can be repeated indefinitely, and no maer how many times we magnify the

picture, more and more details will be revealed. e notion of “detail” at work here is rough

and anthropocentric, but note for the present that familiar shapes like circles and sine curves

do boom out in terms of interesting details under magnification. If we continue to zoom in

on one region of a straight line, for example, no unseen interesting features will be revealed.

I will refer to this property of having interesting details at all level of magnification as infinite

intricacy, but postpone further discussion of it for the moment.

A shape need not be strictly self similar to be a fractal, and it is also possible for shapes to

be statistically self similar (SSS). Such shapes contain reduced-scale images of the whole “in

a statistical sense.” At each construction step of the Koch curve, for example, we could flip a


Figure 0.5: A random Koch curve aer five steps, where at each construction step a coinis flipped to determine whether the new peak will go above or below the starting line.

coin to determine whether the new kink in the line should point up or down. In the infinite

limit we will obtain a random Koch curve (see figure 0.5), which is a statistically self-similar

fractal (Falconer 2003, 245).

e notion of dimension is crucial to fractal geometry. Our everyday intuitive notion of

dimension is close to the topological dimension of Rn: a point is zero dimensional, a line is one

dimensional, and so on.¹ Fractals, however, oen seem ill-classified by topological dimension.

e Koch curve, for example, is topologically one dimensional since it is a continuous curve,

but because of its infinite intricacy it seems ‘bigger’ than a regular curve. Textbook authors

will oen use this point to argue that we need a more refined notion of fractal dimension to

capture these intuitions (e.g. Devaney 1992, 185).

e similarity dimension is one kind of fractal dimension, and it is easy to calculate for

self-similar shapes. Recall that we call a shape self-similar if it can be divided into smaller

pieces that, when magnified, yield a reproduction of the whole.² e similarity dimension D

is defined as the ratio of the logarithms of the number of pieces the shape is broken into to

the magnification factor required to regain the initial shape (Devaney 1992, 188):

D =log(number of pieces)

log(magnification factor)(0.3)

¹ere are actually several types of topological dimension, the definitions of which are quite involved, butthis gloss suffices for present purposes (cf. Edgar 1992, Ch. 3).

²For a more technical discussion of self-similarity, see Edgar (1992, 185).


More concretely, if we chop the Koch curve along the bends introduced in its first iteration,

we get four pieces which, when magnified by a factor of 3, yield a replica of the whole curve

(see figure 0.4). us, for the Koch curve:

DKoch curve =log 4log 3

' 1.262 (0.4)

We can easily see that the similarity dimension of more familiar shapes will oen be equal

to their topological dimension. A square, for instance, is a highly self-similar shape, and so

has a similarity dimension. If we divide a square into four equal parts, we get four smaller

squares, each of which, when magnified by a factor of two, yields the original square. Apply-

ing equation 0.3, we find that a square has similarity dimension 2. Similarly a straight line

will have similarity dimension 1, a cube 3, and so on.

What is important for our purposes is that from some points of view fractals seem ill-

classified by topological dimension, and that we can define new notions of dimension that

beer reflect some fractal properties. Using these new notions some fractals will, as we saw,

have non-integral dimensions. For further details on fractal geometry, including technical

discussions of fractal dimension, the reader is referred to Appendix A.

Much like chaos, there is no uncontroversial and clear way of demarcating between frac-

tals and non-fractals. One of the earliest suggestions was that fractal is a set whose fractal

dimension is strictly greater than its topological dimension (Mandelbrot 1982, 15), (Devaney

1992, 186). However, Mandelbrot found this definition inadequate because it excluded “bor-

derline fractals” (Edgar 1992, 211-2). Some authors have focussed on the infinite intricacy of

fractals. Orly Shenker, for example, suggested that a necessary and sufficient condition for

being a fractal is to have “infinitely many details within a finite volume of the embedding

space” (Shenker 1994, 970), and Edgar, writes that “roughly speaking, a fractal is a set that is

more ‘irregular’ than the sets considered in classical geometry” (Edgar 1992, VII). While these

last definitions seem intuitively correct, they lack mathematical rigour. Mandelbrot may have

0.4. REVIEW OF SELECTED PHILOSOPHICAL WORKS ON CHAOS AND FRACTALS 11

recognized the difficulty of giving a clear definition of the term, because he eventually—with

just a hint of sour grapes—decided to “leave the term ‘fractal’ without a pedantic definition”

(Mandelbrot 1982, 459).

Some mathematicians have, following Mandelbrot and Shenker, accepted the notion that

we may have to rely on a vague definition of the term “fractal.” Falconer, for example, notes

that many perfectly good scientific definitions are not clear cut (Falconer 2003, xxv). Similarly,

Falconer argues, any precise definition of the term “fractal” will almost certainly rule out some

interesting cases, and so a beer approach is to use a vague definition. When we refer to a

set F as a fractal, Falconer says, we will typically have the following properties in mind:

(i) F has a fine structure, i.e. detail on arbitrarily small scales.

(ii) F is too irregular to be described in traditional geometrical language, both locally andglobally.

(iii) Oen F has some form of self-similarity, perhaps approximate or statistical.

(iv) Usually, the ‘fractal dimension’ of F (defined in some way) is greater than its topologicaldimension.

(v) In most cases of interest F is defined in a very simple way, perhaps recursively” (Falconer2003, xxv).

Giving a precise definition of the term “fractal” is clearly a challenging project at the inter-

section of mathematics and philosophy. Fortunately, in this dissertation we will be looking at

sets that are uncontroversially fractal, and sowe able to bracket this problem for themost part.

Falconer’s “know-it-when-you-see-it” approach will work well enough for our purposes.

0.4 Review of Selected Philosophical Works on Chaos and

FractalsIn a perfect world this literature review would cover all philosophical work on chaos;

however, the literature on chaos theory is so enormous as to be unwieldy. In the late 1980s

and 1990s, chaos theory was applied to a dizzying array of fields ranging from scientific ex-

planation (e.g. Suppes 1985; Smith 1998b), to model confirmation (e.g. Rueger and Sharp 1996;


Koperski 1998; Harrell and Glymour 2002), to the compatibility of science and miracles (e.g.

Polkinghorne 1991, 1996; Colwell 2000; Saunders 2002). Limitations of scope prevent us from

covering all of the philosophical topics to which chaos has been applied.

is review will therefore focus on four main areas relevant to the dissertation: approx-

imate truth; prediction, determinism, and randomness; observational equivalence; and the

nature and scientific status of fractals.

0.4.1 Chaos and Approximate TruthOver the last several decades, the consensus in the philosophy of science has been that

scientific scientific theories and models cannot be strictly true, since they involve approx-

imations and idealizations that do not accurately reflect their targets (e.g. Redhead 1980;

Cartwright 1983; McMullin 1985; Chakravary 2001; Teller 2001). For many philosophers,

however, there is a powerful intuition that science does manage to somehow reflect the na-

ture of reality, if imperfectly. e question is oen framed in terms of approximate truth, and

if and how scientific theories and models could be strictly false yet approximately true (e.g.

Chakravary 2014, and references therein). e debate over whether chaotic models specifi-

cally could be approximately true has been dominated by Peter Smith and Jeffrey Koperski.

Smith proposed a geometrical notion of approximate truth tailored to dynamical models

(Smith 1991, 1998a, 1998b). Smith’s proposal is that approximate truth for dynamical theories

can be reduced to a notion of geometrical closeness, which he views as less metaphysically

problematic. Dynamical theories, on this view, can be represented in terms of particular

phase-space trajectories, and physical systems can also be represented in terms of the phase-

space trajectories that correspond to their actual and possible behaviour. Dynamical theories

are approximately true on Smith’s account to the extent that their trajectories are geometri-

cally similar to a physical system’s actual phase-space trajectories. Spelling out the precise

nature of approximate truth in this context therefore reduces to describing the correct no-

tion of geometrical closeness, and “there is no deep conceptual problem about a claim that

one geometric structure approximates to another” (Smith 1998a, 259). is account has been


criticized by Anjan Chakravary, who argues that is at best an instrumental account of ap-

proximate truth (Chakravary 2001, 340). Smith’s account, and Chakravary’s criticisms of

it, will be considered in more depth in Section 1.5.1.

Koperski (2001) takes a different view of the approximate truth of chaotic models. Kop-

erski begins with an apparent conflict between atomic theory and chaotic models in the mi-

croscopic realm, where chaotic models make false assumptions about the ontology of their

target systems on small scales. Given this apparent conflict, he asks how we “can… treat

such a model realistically” (Koperski 2001, 684). Drawing inspiration from Clifford Trues-

dell’s work in continuum mechanics, Koperski argues that this conflict is only an illusion,

because chaotic models do not actually imply anything about the microscopic realm. Instead,

Koperski argues, “the small-scale structure is ignored” (Koperski 2001, 697) and “the question

is whether [the target system] behaves that way at the scale the modeller is interested in”

(Koperski 2001, 698). Since these models do not imply anything about the microscopic phys-

ical realm, there cannot be a contradiction with other physical theories. Koperski’s position

gives a good description of standard scientific and engineering practice. However, there is a

tension between being a realist about chaos and disregarding its micro-states which Koperski

does not resolve.

0.4.2 Prediction, Determinism, and RandomnessPrediction and Classical Determinism

Several authors have argued that the predictive difficulties associated with chaos may have

implications for determinism.³ Hunt (1987) argued that chaotic systems are deterministic but

fundamentally unpredictable, because a continuous change to a chaotic system’s initial condi-

tions causes a discontinuous change in its output (Hunt 1987, 131-2). However, as Baerman

(1993) pointed out, it is a theorem that any system of chaotic ordinary differential equations

(ODEs) will be “well-posed,” and so any continuous change in initial conditions will result in a

³Determinism itself is, of course, a multi-faceted and extremely difficult concept (cf. Earman 1986)


continuous change in its trajectory (Baerman 1993, 54).⁴ Smith (1998a), in response, argued

that a chaotic system can be well-posed and yet have the kind of properties Hunt identi-

fied near phase-space basin boundaries. Near the boundary, Smith wrote, “between any two

points whose path ends later at [one point] there will be a point whose path ends at [another

point] and vice versa” (Smith 1998b, 57). But while it is true that the magnetic pendulum has

a complicated fractal basin boundary structure, it is not true that it has the property Smith

aributes to it (Peitgen, Jürgens, and Saupe 2004, 714).

Several other authors have argued that chaotic systems are intuitively deterministic but

fail to be predictable in one of two ways. Stone (1989), Holt and Holt (1993), and Suppes (1993)

argue that, because of SDIC, the measurement error in our initial conditions will quickly

make our model predictions inaccurate. Stone concludes, “for any input there will always

be some distance over which error will be sufficiently amplified such that all accuracy is

effectively lost. us in a strong sense chaotic systems are not predictable even though they

are deterministic” (Stone 1989, 127). Stone (1989) and Suppes (1993) give a second argument

for the determinism but non-predictability of chaotic systems, based on the fact that they do

not have “closed-form solutions” (Stone 1989, 126). Roughly, an open-form solution is one

which becomes more complicated as we simulate our system for a longer period of time,

and a closed-form solution is one which does not. To sum the first N integers, 1 + 2 +

. . .+ N is an open-form solution, and N(N +1)/2 is a closed-form solution. Stone identifies

prediction with the use of a closed-form model, whereas all we can do is inspect an open-form

model (Stone 1989, 126). Since many chaotic models cannot be solved analytically and must

be approximated numerically, chaotic models therefore do not provide predictions. Stone and

Suppes have identified a salient feature of chaotic models, but their conclusion relies on the

intuition that an open-form model does not provide predictions.

More recently, Werndl (2009c) has argued that the implications of chaos for predictability

⁴An initial value problem is well-posed, a notion first introduced by Hadamard, if it has a unique solutionwhich depends continuously on the initial conditions (Baerman 1993, 53). Hunt seems to be claiming thatchaotic systems are not well-posed in this sense, whereas in fact they must be if they are given by ODEs.


are best thought of in terms of the probabilistic relevance of past events to predictions of

future events. Drawing on work by Berkovitz, Frigg, and Kronz (2006) (discussed below in

Section 0.4.2), Werndl argues that a class of dynamical systems called mixing systems both

satisfies our intuitive criteria for chaos, and is extensionally correct as a definition of chaos.

As a consequence, Werndl argues, when predicting future events in a chaotic system, “all

sufficiently past events are approximately probabilistically irrelevant” (Werndl 2009c, 215).

is irrelevance is only approximate, because oen correlations between events will remain

nonzero, and it applies only to sufficiently past events, because the decay of correlations must

occur in the long run, with no guarantees as to its speed.

Chaos and antum Unpredictability

Several authors have proposed that quantum mechanics and chaos theory jointly entail the

failure of determinism. ese authors assume that the world is genuinely indeterminis-

tic on the microscopic scale—itself a controversial assumption (e.g. Albert 1992; Goldstein

2013)—and then claim that this microscopic indeterminism will propagate up to the macro-

scopic world through SDIC. Small indeterministic events, on this view, will greatly affect

macroscopic events, and thus induce large-scale indeterminism in a meaningful sense. Au-

thors who have presented arguments along these lines include Hobbs (1991), Barone et al.

(1993), and Colwell (2000). Indeed, Barone et al. (1993)’s title, “Newtonian chaos + Heisen-

berg uncertainty = macroscopic indeterminacy,” represents perhaps the purest expression of

this argument in the literature.

However, while combining quantum indeterminism with chaotic SDIC has intuitive ap-

peal, this project has been very difficult in practice. On one influential view, chaos theory

and quantum mechanics impose incompatible mathematical structures on the world, and so

for one to be true the other must be false, and any argument that relies on both theories being

true therefore entails a contradiction (Ford 1989). is position has been vigorously debated,

and the arguments are oen extremely philosophically and mathematically technical (e.g.


Belot and Earman 1997, and references therein). A rigorous blending of chaos and quantum

mechanics may very well be possible, but to my knowledge no such combination yet exists.

At present, it therefore seems inadvisable to draw strong ontological conclusions from the

combination of these two theories.

Chaos and Randomness

Chaotic dynamics is oen referred to as “random” or “complex,” and several authors have tried

to make these intuitions precise. One helpful distinction is that between process and product

randomness (Eagle 2005), where process randomness is usually taken to involve some kind

of stochasticity or unpredictability of dynamics, and product randomness is a property of

outputs (Smith 1998b, 149).

One of the earliest and most influential aempts to link chaos and randomness was made

by Joseph Ford, who argued that “in its strictest technical sense, chaos is merely a synonym for

randomness as the algorithmic complexity theory of Andrei Kolmogorov, Gregory Chaitin,

and Ray Solomonov so clearly reveals” (Ford 1989, 350). e algorithmic complexity of a se-

quence is a notion of product randomness, and is roughly defined as the length of the shortest

computer program that will print it out. On this view non-complex sequences can be gener-

ated by short algorithms, and complex sequences can only be generated by long algorithms,

possibly algorithms as long as the sequences themselves. Algorithmic information theory is

itself complex, and the reader is referred to resources such as Chaitin (2003) or Chaitin (2006)

for details. Ford’s claim, then, is that chaotic dynamical systems are complex in this algorith-

mic information sense, since their orbits cannot be computed by an algorithm appreciably

shorter than manually instructing the computer to print them out one by one (Ford 1989,

350).

Baerman (1993) offered a strong criticism this view. Baerman conceded that there is

a theorem, by the mathematician A. A. Brudno, which says roughly that for almost all of a

system’s trajectories, the algorithmic complexity equals the system’s Kolmogorov-Siani (KS)


entropy (Baerman 1993, 61). Positive KS-entropy is oen taken as a hallmark of chaos, so

this would seem to strengthen Ford’s position. However, while chaotic systems may pro-

duce algorithmically complex outputs, Baerman argued that non-chaotic systems can also

give algorithmically complex outputs under certain circumstances. For example, the outputs

of non-chaotic deterministic dynamical systems can be unpredictable if there is uncertainty

about their initial conditions. But then, Baerman argues, “the randomness is due to some-

thing other than the dynamics—to our ignorance of the exact initial conditions” (Baerman

1993, 64). By ignoring a system’s dynamics and focussing on its outputs, Ford’s definition

risks misclassifying classical integrable non-chaotic systems as chaotic. We might say that

Ford has focussed on a notion of product randomness, and that Baerman is arguing that a

process notion is more appropriate to characterize chaos. Any adequate definition of chaos

must, on this view, focus on dynamics. Baerman and White (1996) strengthened this posi-

tion with a sophisticated mathematical treatment.

Frigg (2004) provided a formal demonstration that positive KS-entropy is connected to

positive information or Shannon entropy, thereby giving a justification for linking random

behaviour with positive Kolmogorov-Sinai entropy. Frigg noted that although positive KS-

entropy was commonly taken as a hallmark of chaos and random behaviour, and although

several authors had drawn similar connections before (e.g. Petersen 1983), there was no ac-

ceptable formal justification for this link in the literature (Frigg 2004, 413). As a result of his

analysis, Frigg concluded, “if an automorphism has positive KSE, then whatever the past his-

tory of the system, we are on average not able to predict with certainty in what cell of the

partition the system’s state will lie next” (Frigg 2004, 429). One corollary of this analysis is

that “product and process randomness are extensionally equivalent” in that whenever a sys-

tem’s outputs are unpredictable (i.e. has positive communications-theory entropy) almost all

of its trajectories will have product randomness (i.e. positive algorithmic complexity) (Frigg

2004, 431).

e relationship between randomness and chaotic dynamics was further investigated in


Berkovitz, Frigg, and Kronz (2006)’s study of the ergodic hierarchy. e ergodic hierarchy is a

ranking of classes of deterministic dynamical systems which is oen presented as a hierarchy

of random behaviour, but the sense of this randomness had not been clarified. e authors

argued that the ergodic hierarchy is best understood as a hierarchy of random behaviour if

randomness is explicated in terms of unpredictability. Unpredictability is explicated in terms

of the type and strength of decay of correlations, in terms of probabilistic relevance, between

past states and futures states of a system. Importantly, this does not imply that “more random”

systems exhibit faster rates of decay, since systems within classes can be constructed to as to

have as fast or slow a rate of decay as desired (Berkovitz, Frigg, and Kronz 2006, 687). It does,

however, suggest that chaos may come in degrees, in the precise sense of being ranked lower

or higher in the ergodic hierarchy (Berkovitz, Frigg, and Kronz 2006, 688-9).

0.4.3 Determinism and Observational EquivalenceAnother debate which concerns us here is about the purported observational indistin-

guishability of deterministic chaotic models and indeterministic stochastic models, and the

potential ramifications for determinism. e debate began with Ornstein and Weiss (1991)’s

definition of ε-congruence, which is a mathematical relation betweenmathematical models. A

full technical treatment of ε-congruence will be given in Chapter 4. For our present purposes,

if we consider two mathematical models and their associated phase spaces, these models are

ε-congruent if each trajectory in one phase space has a corresponding trajectory in the other,

and if these trajectories are close to each other most of the time. e value ε is a parameter

that determines how close the trajectories must be, so as ε becomes small, the predictions

given by the two models become more similar. And if ε is below the threshold of observ-

ability, whatever that might be in a given context, then we cannot determine on the basis

of observation alone which of the two systems we are watching. It is in this way that the

systems are claimed to be observationally equivalent.

One of Ornstein and Weiss (1991)’s surprising findings was that deterministic chaotic

models and indeterministic stochastic models can be ε-congruent. If to be ε-congruent is to


be observationally equivalent, then this means that deterministic and indeterministic systems

can be indistinguishable. Furthermore, this result appears to imply that deterministic and

indeterministic systems can have arbitrarily similar behaviour in some cases. Ornstein and

Weiss suggested the following provocative interpretation:

is may mean that there is no philosophical distinction between processes

governed by roulee wheels and processes governed by Newton’s laws … we are

comparing, in a strong sense, Newton’s laws and coin flipping. (Ornstein and

Weiss 1991, 39)

Responding to this suggestion, Patrick Suppes argued Ornstein andWeiss’s results imply that

no observational evidence could determine whether a particular systemwere deterministic or

indeterministic, and so a strong form of underdetermination obtains. He concluded that the

true deterministic or indeterministic nature of the world must necessarily “transcend experi-

ence” (Suppes 1993; Suppes and de Barros 1996). John Winnie, in a reply, conceded that the

ε-congruence results show that some deterministic and stochastic models are observationally

equivalent. However, Winnie argued, there are inductive reasons to prefer deterministic mod-

els, since a given deterministic model “outstrips any single Markov model in its conceptual

and predictive power” (Winnie 1998, 317).

Most recently, Charloe Werndl has wrien a series of papers examining and extending

this earlier work on chaos and observational equivalence, which will be addressed more fully

in the body of the dissertation (Werndl 2009a, 2011).

ere is, however, a fundamental weakness in the aforementioned literature: there has

been, as of yet, no rigorous justification for interpreting ε-congruence as observational equiv-

alence. Instead, the authors simply assume that ε-congruence implies observational equiva-

lence as a starting premise. ere is, therefore, a need to analyze this assumption, and I will

provide such an analysis in Chapter 4 of this dissertation.


0.4.4 Fractals, from Initial Optimism to a Negative Consensuse philosophical literature on fractals is very thin on the ground indeed, but it is, I think,

safe to say that the received wisdom is that there are no fractals in nature. e standard-

bearer for the received view is Orly Shenker’s 1994 paper entitled, appropriately, “Fractal

geometry is not the geometry of nature” (Shenker 1994). Shenker argued emphatically that

fractals could not exist, and that characterizing natural objects as fractals is, in general, a

mistake. Also in 1994, B. Jack Copeland considered fractal-boundaried objects tomake a larger

point about vague objects. While Copeland seemed to accept the in-principle possibility of

fractal objects, he concluded that there “is no reason to think that the real world contains

any [fractal] boundaries” (Copeland 1994, 94). Peter Smith, in his 1998 monograph on chaos

theory, endorsed Shenker’s arguments (Smith 1998b). Both Smith and Shenker were cited

favourably by Nicholas Saunders in a survey of the literature on chaos, fractals, and ontology

(Saunders 2002, 201—2). But here the conversation seems, as far as I can tell, to have ended.

Smith and Shenker’s arguments sparked no debates that I have found; their position remains

unchallenged.

is negative consensus may be surprising if we consider how much enthusiasm there

was surrounding fractals in the 1970s and 1980s. Benoît Mandelbrot, who coined the name

“fractal,” was the earliest proponent of fractals in science. In a seminal paper of 1967, Mandel-

brot, one of Shenker’s chief targets, asserted that coastlines (among countless other objects)

are statistically self-similar fractals, and so are either of indeterminate or else infinite length

(Mandelbrot 1967). In his later book e Science of Fractal Images Mandelbrot reaffirmed this

position, stating that “the typical coastline’s length is very large and so ill determined that it

is best considered infinite,” and that it is “impossible to avoid” introducing fractal concepts to

discuss them (Mandelbrot 1982, 25). A veritable coage industry sprang up in Mandelbrot’s

wake, calculating fractal dimensions of physical objects ranging from fractured steel (Mu et

al. 1993) to earthquake epicentres (Sahimi, Robertson, and Sammis 1993).⁵ In early textbooks

⁵For an exhaustive survey of scientific studies purporting to measure fractal dimensions of physical objects


too we find dubiously strong claims about the applicability of fractal geometry. Michael F.

Barnsley’s Fractals Everywhere, for example, breathlessly informs us that fractal geometry

“can be used to make precise models of physical structures from ferns to galaxies,” and that

by using it one “can describe the shape of a cloud as precisely as an architect can describe a

house” (Barnsley 1993, 1).

Despite this early euphoria, or perhaps in reaction to it, a counter-movement of fractal

skeptics developed in the late 1980s and early 1990s. Scientists like Leo Kadanoff had been

critical of fractals in science as early as 1986, but by the late 1990s the counter-movement was

in full force. In particular, Malcai et al. (1997)’s comprehensive survey of scientific work on

fractals revealed that the vast majority of so-called “empirical fractals” were only fractal-like

over a very limited scale. Based on this empirical work, and with explicit reference to the

arguments of Shenker (1994), Avnir, Bihan, and Malcai (1998) argued that claims about the

fractal nature of reality were highly overblown. In the face of this onslaught, Mandelbrot felt

compelled to respond that he had “stressed… fractals are not a panacea; they are not every-

where” (Mandelbrot et al. 1998, 783). He even agreed that some claims in the literature would

be “best understood as unfortunate side effects of enthusiasm, imperfectly controlled by refer-

eeing” (Mandelbrot et al. 1998, 783). What exactly Mandelbrot meant by his earlier statements

about fractals is not our concern here, but this last admission certainly represented an about-

face. Mandelbrot’s public recantation may have signalled the demise of the respectability of

the fractal geometry of nature.

is consensus was solidified by Smith (1998a)’s definitive monograph on the philosoph-

ical significance of chaos theory and fractals. Smith examined the benefits of using fractal

models to represent physical objects, but ultimately decided that the nonphysical infinite de-

tail in fractal models must always weigh against them. For, according to Smith, “we always

need to ask: won’t a beer description of nature in fact be provided by prefractals that lack

the infinite excess detail? To which the answer seems invariably ‘yes’” (Smith 1998a, 33).

published up to 1997, see Malcai et al. (1997).

0.5. OUTLINE OF THE DISSERTATION 22

e philosophical consensus has since been that fractal models are somehow taboo, and that

prefractal models, that is, models which are fractal-like on only a limited number of scales, are

always to be preferred. As we will see in Chapters 2 and 3, this consensus can be challenged

on a number of fronts.

0.5 Outline of the Dissertationis dissertation has been designed to be broadly modular, in that each chapter consists

largely of a self-contained analysis and includes a discussion of the mathematical and philo-

sophical prerequisites. One downside to this is that there is a small amount of overlap in

the technical sections of some chapters; on the other hand, readers with a particular inter-

est can skip to any of the chapters without worrying about missing crucial definitions or

sub-arguments. at said, the chapters are related and they build on each other thematically.

I will begin in the first chapter by considering whether chaotic models can be considered

approximately true. is question is a subtle and complicated one, and I will argue that, at

the end of the day, focussing on broad honourifics like “approximate truth” may actually lead

us away from more interesting and important ontological and epistemological questions. I

will begin by examining two specific chaotic models, a continuous model of the Belousov-

Zhabotinsky chemical reaction, and a discontinuous model of a kicked rotator. I will argue

that each of these models has representational strengths and weaknesses, since they provide

both clear and distorted representations of their targets in different domains. I will then con-

sider three separate proposals for what it is for a chaotic model to be approximately true:

Peter Smith’s geometrical-modelling account, Jeffrey Koperski’s continuum-mechanics ac-

count, and a Popperian verisimilitude account. I will argue that none of these three options

can do the job they are required to, and that in general the kind of classification these kinds of

schemes imply are unlikely to lead to new knowledge about scientific models. However, I will

argue, the kind of in-depth examination I provided of the two example models can provide

new and useful knowledge about our models and their applicability. e result, I propose, is


that oentimes honourifics like approximate truth actually distract us from more meaningful

philosophical examinations, and that a case-by-case examination of a model’s strengths and

weaknesses, while messy and unglamourous, can teach us more in the end.

In Chapter 2 I consider imperfect fractal models in detail, and argue that despite their un-

physicality they can yet be good instrumental models. Specifically, I consider fractal models

of non-fractal physical objects. ese models have been broadly and explicitly rejected in the

philosophical literature, and the consensus is that non-fractal models should always be prag-

matically and philosophically preferable. I will challenge this consensus, first by addressing

the published objections to fractal models, and then by providing a positive argument based

on a case study from the engineering literature. I argue that fractal representations of non-

fractal physical objects are instrumentally useful, and sometimes even desirable.

In Chapter 3, I will argue that the philosophical arguments for the impossibility of spatial

fractals are not conclusive. Furthermore, as I will demonstrate using several examples of

chaoticmodels and computer simulations, there are positive reasons to believe that some parts

of the physical world could have fractal structure. at is, I will argue that although fractal

geometry is clearly unsuitable for describing a great many things, if various chaotic models

or (something like them) are true there are (or could be) actual things in the world which have

truly fractal shapes. In the literature, opposition to this view is sometimes summarized in the

slogan that “fractal geometry is not the geometry of nature” (Shenker 1994). In the form of a

counter-slogan, then, in this chapter I argue that fractal geometry is a geometry of nature.

In Chapter 4 I will consider chaotic systems in light of several accounts of observational

indistinguishability. ese accounts purportedly establish that some deterministic chaotic

models will always be observationally indistinguishable from indeterministic stochastic pro-

cesses, and this result has been used to undergird arguments about metaphysical determinism

(Suppes 1993; Suppes and de Barros 1996). In this chapter I examine two mathematical def-

initions of observational equivalence. e first was proposed by Charloe Werndl and is

based on manifest isomorphism, which is a special case of general measure-theoretic isomor-


phism. e second notion of observational equivalence is based on Ornstein and Weiss’s

ε-congruence, which was outlined above in Section 0.4.3. I argue, for two related reasons, that

neither one of these two mathematical definitions of observational equivalence is adequate.

First, each definition permits of counterexamples; second, overcoming these counterexamples

will introduce non-mathematical premises about the systems in question. Accordingly, the

prospects for a broadly applicable and purely mathematical definition of observational equiv-

alence are unpromising. Despite this critique, I suggest that Werndl’s proposals are valuable

because they clarify the distinction between provable and unprovable elements in arguments

for observational equivalence.

.. 1Chaos and Approximate Truth:

A Reappraisal

Smith argues that chaotic models are, in an important sense,more complex and intricate than the physical systems beingmodeled. How then can one treat such a model realistically?

Jeffrey Koperski, Has chaos been explained? p. 684

1.1 IntroductionChaos theory seems to pose particular problems for philosophers interested in the ques-

tion of truth. Any tractable scientific model must leave out or distort some details about its

target, and so in general we should not expect our scientific models to be perfectly true rep-

resentations (Teller 2001). But many scientific models, including chaotic models, seem to get

things right enough that many philosophers intuitively believe that they are somehow ap-

proximately true. e question is how to make this intuition precise, or indeed whether this

is at all possible.

In this chapter wewill investigate the motivations and prospects for an account of approx-

imate truth for chaotic models. I will argue for both a negative and a positive proposal. e

25

1.1. INTRODUCTION 26

negative proposal is that the accounts of approximate truth on offer are incomplete, that com-

pleting themwill be very difficult, and that working to complete themmay actually distract us

from more fruitful investigations. e positive proposal is that we can advance our epistemic

and ontological investigations without a general notion of approximate truth by engaging

in fine-grained case-by-case examinations of the strengths and weaknesses of chaotic mod-

els. A more general account may certainly be forthcoming, but it should arise naturally from

investigating the particulars, rather than forcing the particulars to conform to the general.

Much of the past literature on chaotic models has been concerned with adjudicating their

approximate truth: that is, with coming up with criteria for success, and excuses for failure,

that will allow us to bestow the honourific of “approximately true” on a model. Based on

a close examination of two particular chaotic models, and three previous approaches to ap-

proximate truth, I will argue that perhaps the label “approximate truth” is not that important

in this case. Instead, what is more important is determining which parts of our models we

can trust, and how much we can trust them. By leing go of “approximate truth” as central

to the investigation of the epistemic status of models, and focussing on the study of which

parts of our chaotic models are more or less adequate representations of their target systems

or phenomena, some form of progress can be made. And based on the examples and the pre-

vious discussions we will consider, it will appear that this progress is best made when models

are examined on a case-by-case basis.

e structure of the chapter is as follows. First, I will give a brief and non-technical in-

troduction to the elements of chaotic dynamics needed to motivate the discussion. Second,

we will consider two examples of chaotic models which seem to get some things right and

other things wrong. Specifically, we will look at the Belousov-Zhabotinsky chemical reac-

tion, which uses a continuous model to represent discrete phenomena, and the kicked rotator,

which uses a discrete mapping dynamics to represent continuous motion. Each of these mod-

els has domains where it is successful and domains where it is not, and I will argue that it is

difficult to know how to interpret these successes and failures in terms of approximate truth.

1.2. A BRIEF MATHEMATICAL OVERVIEW 27

Next, we will consider three past approaches to the approximate truth of chaotic models and I

will argue that each faces challenges. First, Peter Smith’s account reduces approximate truth

to the intuitive notion of geometrical similarity, but I argue that precisely explicating this

intuitive notion will be very difficult. Second, Jeffrey Koperski’s approach is broadly in line

with standard scientific and engineering practice, but I will argue that it involves a combina-

tion of instrumental and realist commitments that are difficult to maintain simultaneously.

ird and finally, it is sometimes stated in the literature on chaos that a Popperian notion of

verisimilitude could serve as an analysis of approximate truth. However, upon examination

it will become clear that verisimilitude and approximate truth are distinct concepts, and so

the former cannot explicate the laer.

In the discussion section, I will propose that the debates surrounding chaotic models could

be greatly simplified by seing aside the honourific title “approximate truth.” e accounts of

approximate truth on offer are incomplete at present, and the prospects for their completion

are dim. It will be clear frommy analysis that different chaotic models have different strengths

andweaknesses, and should be takenmore or less seriously as a representations of their target

systems or phenomena in different domains. In other words, titles like “approximate truth”

are not important and moreover they tend to be counterproductive in that they frequently

oversimplify and drive one away from the important questions. Rather than engaging in an

abstract general discussion of whether models are approximately true, the best approach is

to engage in case-by-case examinations of their strengths and weaknesses.

1.2 A Brief Mathematical Overview

1.2.1 Dynamical Systemse purpose of this section is to give a brief, non-technical introduction to the aspects of

chaos theory necessary to frame the discussion in later sections. Our focus here will be on

two different classes of chaotic systems: those that exhibit dissipative non-Hamiltonian chaos,

and those that exhibit non-dissipative Hamiltonian chaos. A dissipative system is one that


loses energy over time, like a mechanical clock winding down. A non-dissipative system is

one that does not lose energy over time. Such systems cannot exist in nature, but they are of

theoretical interest as a limit case, and of empirical interest because some physical systems,

such as the orbits of the planets, are approximately non-dissipative.

Chaos is associated with nonlinear dynamical systems, which are sets of equations that

can be used to represent how values change over time. is technique is very general, and

the values of our equations could be interpreted as speed and position, animal populations,

or anything else. e precise definition of chaos is controversial (Werndl 2009c), but in broad

strokes chaotic equations can be simple in appearance and surprisingly complicated in be-

haviour. Indeed, for all practical purposes, some chaotic systems can be nearly impossible to

predict exactly. One of the central features of chaos is that systems exhibit sensitive depen-

dence on initial conditions (SDIC), which roughly means that making a very small change to

a system’s starting state can dramatically change its future evolution. Two systems that be-

gin in nearly identical states will oen quickly diverge. Since we can never know a system’s

starting state with perfect accuracy, this means that systems with SDIC are very difficult to

predict. However, there is empirical evidence that chaotic models can apply fruitfully to as-

pects of natural systems, ranging from how pendulums swing (Shinbrot et al. 1992), to the

way fluids flow (Rahal, Cerisier, and Abid 2007; Zhu, Duan, and Kang 2013), to the fluctu-

ations of laser beams (ornburg, Möller, and Roy 1997; Syvridis et al. 2009; Uchida 2012).

Some physical systems seem to behave chaotically.

1.2.2 Phase SpaceWhen discussing chaos, it is helpful to talk about a system’s phase space, which gives us

a geometrical representation how the system changes over time. Each point in the phase

space represents a set of values for the system. As the system changes it will move from

point to point and trace out a path through phase space called an orbit or trajectory. If our

systemmodels the position and momentum of an ideal pendulum constrained to swing in one

dimension, for example, its phase space will be a two-dimensional plane with position and


momentum as its axes. It is crucial to note that time is not an axis of the phase space; phase

space is like a timeless, “frozen” representation of all possible values a system can take on, and

how it can move from one state to the next. Returning to our pendulum, it will have some

position and momentum to begin with, and so its initial state will be represented by some

point in phase space. As the pendulum swings its position and momentum will change, and

so its state will trace out an orbit in phase space. If the pendulum experiences no frictional

forces, it will swing forever, endlessly tracing an ellipse in phase space.

Of course, chaotic systems have behaviourmuchmore complicated than that of our simple

pendulum, and so they have correspondingly more complicated trajectories through phase

space. Since they exhibit sensitive dependence on initial conditions, trajectories of chaotic

systems that start out close together will eventually—and oen quite soon—end up tracing

completely different paerns through phase space. ere is, however, some semblance of or-

der to the proceedings. As dissipative chaotic systems evolve, they tend to be drawn to a re-

gion of phase space called an aractor. Conservative systems, on the other hand, do not have

aractors. In both cases, although the systems may seem “random” in the sense of being dif-

ficult to predict, a system’s initial conditions fully determine its future evolution (Berkovitz,

Frigg, and Kronz 2006). ere are no chancy, stochastic, or indeterministic elements in the

classical chaotic models we will consider here, and if you plug the same numbers into a sys-

tem’s governing equations twice, you will always get the same answer.

1.2.3 Continuous and discrete dynamical systemse dynamical systems we will consider cleave neatly into two types, which we will call

continuous and discrete, and which differ in their mathematical, dynamical, and geometric

properties. Continuous systems are defined mathematically by sets of differential equations,

and they evolve continuously as some variable, usually time, changes. Geometrically, the

trajectories of a continuous system trace out continuous lines in phase space. For example,

simple harmonic motion, such as that of the pendulum mentioned above, is a continuous

dynamical system defined by the following differential equation:


x

p

Figure 1.1: Trajectories representing simple harmonic motion in phase space.

d2xdt2

= −k x (1.1)

Where k is a constant. And if we were to plot this phase-space system’s trajectories, we

would see that they correspond to an infinite number of perfect circles centred on the origin

(see Figure 1.1).

A discrete system, on the other hand, is typically defined mathematically as an iterated

function or set of functions. When a function is iterated, this means that it is applied over and

over again, with the output of the last time used as the input the next time. Oen a variable

will have a subscript indexing its iterations, such that for example xn is the nᵗʰ iteration, and

xn+1 is the next. Geometrically, the phase-space trajectories of discrete systems are generally

sets of disconnected points: with each iteration, they jump discontinuously from one phase

space to another (see Figure 1.2).

x

p

Figure 1.2: A set of points representing a discrete orbit in phase space.


ere is, of course, muchmore that could be said about discrete and continuous dynamical

systems. For more information on different kinds of dynamical systems, the reader is directed

to such sources as Devaney (1992, 9-16) and Tél and Gruiz (2006, 90-109).

1.2.4 Fractalse aractors of many classical dissipative chaotic systems are fractals. e precise defi-

nition of “fractal” is a maer of some dispute, but in the present context we can get by with

the non-technical observation that fractals display “fine detail” on all levels of magnification

(see Appendix A for a more thorough discussion). Most familiar geometrical shapes are well

behaved upon magnification: eventually they yield smooth curves, or else they are limited

in their jaggedness. A triangle, for example, consists of three straight lines joined at three

points, and if we view any part of it more closely we will see only the same three straight

lines and the same three joints. Fractals, in contrast, reveal more andmore details as we exam-

ine them more closely. ey are, in a rough sense, infinitely detailed. Many famous chaotic

aractors are fractals, but there is no necessary connection between chaotic dynamics and

fractal aractors. ere are non-fractal chaotic aractors, and chaotic non-fractal aractors

(Grebogi et al. 1984). Still, the connection between chaos and fractal aractors holds in most

paradigmatic cases.

1.2.5 SummaryIn summary, then, chaos theory provides a way of mathematically modelling physical

systems. ese systems are oen visualized in an abstract phase space, wherein each point

represents a possible state of the system. Chaotic systems are deterministic, in the sense that

their governing equations will always generate the same output for a given set of initial con-

ditions. However, because these equations exhibit SDIC, arbitrarily similar initial conditions

will give rise to very different trajectories. In practice this means that chaotic systems are

extraordinarily difficult to predict. Dissipative chaotic systems evolve toward fractal phase-

space aractors. Furthermore, these fractal aractors are oen responsible for the character-

1.3. EXAMPLE 1: THE “STANDARD MAP” OF THE KICKED ROTATOR 32

v

x

Figure 1.3: Schematic drawing of a kicked rotator, with angular displacement x and an-gular velocity v. Image by the author, aer Tél and Gruiz (2006, Fig. 7.6).

istic behaviour of the chaotic systems.

1.3 Example 1: e “Standard Map” of the Kied Rotator

1.3.1 e Classical Kied RotatorIn this subsection we will consider a system called the kicked rotator.¹ In essence, this

system consists of a rotating body which is occasionally subjected to a discontinuous change

in velocity (a “kick”). In some circumstances such a body will behave quite simply; in other

circumstances, as we will see, the kicked rotator can behave chaotically.

Consider a body rotating freely about a fixed axis, such as a ball fixed to a weightless rod

which is aached to a rotating sha (see Figure 1.3). Let x be the ball’s angle of deflection as

measured from some fixed arbitrary point, and let v be the ball’s angular velocity in radians

per second. x is therefore periodic in 2π, and if le to evolve continuously the system would

rotate endlessly, completing one rotation every 2π/v seconds.

Consider now what happens if the rotator receives a “kick,” characterised by an instanta-

neous change in velocity, every T seconds. e rotator will swing inertially for T seconds,

at which point it will experience a sudden discontinuous change in velocity, and then swing

¹Only general results are presented here. For further details, including derivations and analytic investigationsof the rotator’s dynamics, consult Tél and Gruiz (2006, 235-42); for an extensive discussion of the kicked rotatorconsult Chirikov (1979).


inertially for T seconds until it is kicked again. e system resembles, in some sense, a game

of tetherball, where the participants hit the ball at regular intervals. Although the system’s

velocity changes discontinuously and its acceleration is undefined during the instant of the

kick, its position changes continuously at all times.

Let us now consider how this system could be modelled as a discrete map. Let the position

and velocity of the system aer the nᵗʰ kick be denoted by xn and vn respectively. Let the

strength of the kick depend on the position of the rotator, and represent it by the function

f (x). If we consider the state of the system every T seconds immediately aer the kick, we

can model this behaviour using the following map:

xn+1 = xn + vnT, vn+1 = vn + u f (xn+1) (1.2)

where u is a dimensional constant. en if we measure velocity in units of u and distance

in units of uT , the dimensionless form of the kicked rotator map is given by the following:

xn+1 = xn + vn , vn+1 = vn + f (xn+1) (1.3)

A special case of the kicked rotator is the so-called standard map, first presented by

Chirikov (1979), which uses the sinusoidal kicking function f (x) = a sin x:

xn+1 = xn + vn , vn+1 = vn + a sin xn+1 (1.4)

e behaviour of the rotator depends strongly on the kicking amplitude a. If the rotator

is not kicked at all, in which case a = 0, then the system will rotate with constant velocity

vn (see Figure 1.4a). Since the model is discrete, if the velocity vn and time-step T are both

constant then the position co-ordinate changes by the same amount in each step. e map’s

dynamics therefore depend strongly on the precise value of vn. If vn = π, for example, the

system’s positionwill cycle back and forth between two values xn and xn+π. As an extension,

if vn = 2π/q, where q is some natural number, then the system’s position will cycle through


0

0 π

π

2π

2π

vn

xn

(a) a = 0

0

0 π

π

2π

2π

vn

xn

(b) a = 0.25

0

0 π

π

2π

2π

vn

xn

(c) a = 0.9

0

0 π

π

2π

2π

vn

xn

(d) a = 1.5

0

0 π

π

2π

2π

vn

xn

(e) a = 3.0

0

0 π

π

2π

2π

vn

xn

() a = 10

Figure 1.4: Phase portraits of the standard kicked rotator for different kicking strengths.Initial conditions v0 ranged from 0 to 2π in increments of 0.3, x0 ranged from 0 to 2π insteps of 2π/5, and 5000 iterations were performed from each starting point. Images bythe author.


q different values before repeating itself. If, however, vn = 2πr , where r is irrational, then

the system will never return to its initial position, and the orbit will trace out the entire line

segment vn = 2πr (Tél and Gruiz 2006, 237). Note that the dynamics remain discrete, but the

system never returns exactly to its initial position, and in the long term the discrete points

reached by the system trace out a solid line. is kind of motion is called quasiperiodic (Tél

and Gruiz 2006, 252).

In the presence of kicks (a > 0), the system’s behaviour changes dramatically. e math-

ematical explanation for this change is complicated and beyond the scope of the present dis-

cussion (see Tél and Gruiz (2006, 236-7)), but its effect can be clearly seen in Figure 1.4. If a is

small then some fixed points and two-cycles remain, but they are now surrounded by closed

trajectories (see Figure 1.4b). ese closed trajectories correspond to the system swinging

back and forth over a given angle, much like a pendulum or a compass needle (Tél and Gruiz

2006, 239). Trajectories farther from the fixed points still cover the entire x-range, albeit with

non-uniform velocity, which corresponds to a full rotation around the vertical sha. How-

ever, despite the new complexity in the trajectories, they remain quasiperiodic, and they still

form solid continuous paths through phase space.

As the kicks grow stronger, some of these solid trajectories in Figure 1.4b break down

and speckled bands begin to appear in phase space (see Figures 1.4c-1.4e). ese regions are

chaotic, and the trajectories within them jump randomly over the entire two-dimensional

area, and come arbitrarily close to each point in the entire region (Tél and Gruiz 2006, 239).

Past a certain critical value ac, the last smooth curves corresponding to continuous trajecto-

ries disappear. e chaotic region covers the entire phase space (see Figure 1.4f), and for all

initial conditions the system will wander over the entire two-dimensional phase space with

a uniform distribution (Tél and Gruiz 2006, 239).

1.3.2 An Extension: e antum Kied RotatorAlthough our main focus here is on classical systems, it is worth noting that the kicked

rotator has also been immensely fruitful in the field of quantum chaos, which applies quantum


and classical analytic techniques to the study of quantum analogues of classically chaotic

systems. Such systems are oen semiclassical, since they are derived from both classical and

quantum theory but technically inconsistent with both. ere is a vast literature on the kicked

quantum rotator, both theoretically and empirically, and it remains a topic of interest (e.g.

Moore et al. 1995; Altland and Zirnbauer 1996; Ammann et al. 1998; Lopez et al. 2013). In

the present discussion, however, we will keep our scope narrow and focus on Raizen et al.

(1996)’s classic experimental study, which provided the first experimental realization of a

kicked quantum rotator.

e “rotator” in Raizen et al. (1996)’s experimental set-up was a cluster of approximately

105 laser-cooled sodium atoms. ese atoms were kept in a very low-pressure vacuum cham-

ber, and stabilized using a precise arrangement of three laser beams and an artificially in-

duced magnetic field. e “kicks” came from pulses emied by another laser beam, which

were reflected off a mirror to create a standing wave. Aer each kick, the atoms were le to

dri briefly and then frozen in so-called “optical molasses” created by the trapping beams.

e atoms were then induced to fluoresce, and a two-dimensional image, reminiscent in

appearance of an elliptical galaxy, was recorded. By integrating over the vertical axis the

authors were able to generate a one-dimensional atomic momentum distribution (Raizen et

al. 1996, 691). is amounts essentially to generating a momentum distribution by calculat-

ing backwards from position using time-of-flight. Note that this last calculation is explicitly

semiclassical, since the quantum position and momentum operators do not commute, and so

any assumption of simultaneous position and momentum measurement is illegitimate from

a quantum perspective (e.g. Shankar 1994, 131).

e formalism of the kicked quantum rotator differs somewhat from that of the classical

kicked rotator, because position and momentum in the classical model must be replaced by

their corresponding quantum operators. is can give rise to some effects without classical

analogues when the kicking period is set to certain values, but overall the model’s predic-

tions agree extremely well with empirical results. In particular, the statistical distribution


of the particles’ momenta was found to be well within 10% of the distribution predicted for

dynamical localization in the kicked rotor. As the authors discuss, uncertainty in laser in-

tensity calibration introduces an error bar of approximately 10%, and so the empirical results

should be considered in agreement with the theory (Raizen et al. 1996, 691). In the years since

this result has held up well and been very influential. Raizen et al. (1996)’s work is still cited

as a landmark in work on quantum chaos, and one commentator recently described the ex-

perimental procedure they pioneered as “one of the workhorses for studies of experimental

quantum chaos” (Shrestha 2013, 18).

1.3.3 Discussione standard map of the kicked rotator might seem innocent enough as a model of a

continuous classical system, but the dynamical mismatch between the discrete model and the

continuous target system requires some untangling. First of all, if we accept that the model

is correct in its predictions of position and momentum, we have to admit that it is still less

than perfectly informative. e continuous system has a position and momentum at all times,

whereas the discontinuous model says nothing at all about what happens between kicks. e

model will apply equally well to systems with different dynamics between kicks, including

systems with arbitrary evolutions. Even if the model is correct in what it does say, in other

words, it is not telling us the whole truth about the target system.

Furthermore, the discrete map has some dynamical properties that the continuous system

does not. A kicked continuous rotator has energy added to or subtracted from it with each

kick, and so is not a conservative system. e standard map, on the other hand, is a con-

servative Hamiltonian system. To see this mathematically, note that the determinant of the

map’s Jacobian is equal to 1.² In dynamical terms, this means that the system portrayed by the

discrete model conserves energy, whereas that portrayed by the continuous system does not.

²To get this result, first we substitute the value of xn+1 into the equation for vn+1, and get xn+1 = xn + vn ,vn+1 = vn + a sin (xn + vn). e Jacobian matrix of this map is

[1 1

a cos(xn+vn) 1+a cos(xn+vn)

], whose determi-

nant can easily be shown to equal 1.

1.4. EXAMPLE 2: THE BELOUSOV-ZHABOTINSKY REACTION 38

In terms of phase space, this means that the discrete map transforms regions of phase space

to other regions of equal volume, whereas regions of phase space of the continuous system

will shrink or expand. No maer how we frame it, the discrete map seems to get something

fundamentally wrong in terms of the system’s dynamics.

Although our main focus is on the classical case, note that the situation is arguably even

less clear in the case of the quantum kicked rotator. As we saw, experimental results are in

good agreement with model predictions for large ensembles of particles. But these results are

statistical and apply only to aggregates of thousands of particles, which does not guarantee

agreement at the individual level of any one particle. ere are, furthermore, the standard

unresolved problems of quantum theory, such as the ontology of quantum states and the

interpretation of the wave function. More pressingly, because the kicked quantum rotator

is semi-classical, it makes assumptions that simultaneously conflict with both classical and

quantummechanics. e kicked quantum rotator therefore seems to provide good predictions

for large ensembles of particles under certain conditions, but it is unclear which aspects of

the model we should take seriously, and how seriously we should take them.

e standard map of the kicked rotator therefore does have some real strengths, but it is

also clear that, in many ways, it misrepresents its target. Given this spoy record, it is not

immediately clear how we should interpret this model.

1.4 Example 2: e Belousov-Zhabotinsky Reaction

1.4.1 A Continuous Model of the Belousov-Zhabotinsky ReactionIn the philosophical literature on chaos, the Belousov-Zhabotinsky (BZ) chemical reac-

tion is oen cited as an example of a physical process that can be successfully modelled using

continuous chaotic dynamics (Smith 1998b; Rueger and Sharp 1996; Saunders 2002). e BZ

reaction occurs in a liquid or gel soup of up to two dozen different reagents, and for demon-

stration purposes it can be realized in a petri dish on a tabletop. It is an unusual reaction for

two reasons. First, it is quite colourful: two of the major chemical constituents are markedly


different in colour, and their transformation is manifested as a change in the colour of the

soup. Second, the reaction is oscillatory: key reactions happen both forward and backward,

transforming one chemical into another and then back, until the essential reagents are con-

sumed. As the reaction progresses, the shiing chemical concentrations have the effect of

creating spiral-shaped waves that propagate across the container. Combined with the above-

mentioned colour changes, in appearance the BZ reaction is reminiscent of a 60s rock-and-roll

light show.³

is reaction is monstrously complex, with up to 25 different chemicals (or more, on some

accounts) reacting with each other and varying in concentration over time. Research into the

BZ reaction continues, in both the empirical and theoretical domains. Empirically, for exam-

ple, researchers are investigating its potential for converting mechanical energy to chemical

energy (Chen et al. 2012), alternative chemical forms of the reaction (Ueki, Watanabe, and

Yoshida 2012), and possible applications for so-called “smart materials” (Yoshida 2010).

e large number of chemicals involved, and the correspondingly large phase space, might

make an analytic model of the BZ reaction seem unlikely. Fortunately, a technique known as

phase space reconstruction allows researchers to study such a system’s dynamics using mea-

surements of the concentration of only one chemical at a single point. Given an empirical

time series B(t) of a chemical’s concentration, an m-dimensional vector can be constructed

by using the delay parameter τ: B(t), B(t + τ), B(t + 2τ), ..., B(t + (m − 1)τ). For suitable

choices of B(t) and τ, the m-dimensional phase space corresponding to this new vector will

have the same dynamical properties as the original phase space (Roux, Simoyi, and Swinney

1983, 258). Using this technique, several researchers were able to measure the concentration

of one particular chemical over time and reconstruct a full phase space for the reaction (Roux,

Simoyi, and Swinney 1983; Simoyi, Wolf, and Swinney 1982). is full phase space had many

hallmarks of chaos, including positive Lyapunov exponents and strange aractors. e dy-

³A video of the BZ Reaction, produced by the University of Toronto’s Dr. Stephen Morris, is available online:hp://youtu.be/3JAqrRnKFHo.

http://youtu.be/3JAqrRnKFHo


namics of the chemical concentration at a single point in the BZ reaction can, therefore, be

fruitfully modelled as a continuous dynamical system.

e BZ reaction continues to be a subject of research on the straightforwardly theoretical

front as well. Budronia, Rusticia, and Tiezzi (2011), for example, recently provided a new

model of the BZ reaction by coupling the so-called “Oregonator”—essentially a simplified

chemical model of the BZ reaction—with the Navier-Stokes equations which govern fluid

dynamics. e result is a model of the BZ reaction in terms of nonlinear partial differential

equations. e authors note that this may be controversial, since the BZ reaction is typically

conceived of as a purely chemical process without fluid dynamical influences, but early results

are promising.

In sum, continuous models of the BZ reaction have been created both in what we might

call a more data-driven fashion through phase-space reconstruction, and alternately in a more

theory-driven way using chemical and fluid-dynamical methods. ese models are both ex-

perimentally confirmed and chaotic, and so there is strong evidence that the BZ reaction can

be fruitfully modelled as a continuous chaotic system.

1.4.2 DiscussionLet us examine the BZ reaction and its continuousmodelsmore closely, beginningwith the

model’s implicit ontology of chemicals on a small scale. e model’s variables are supposed

to represent chemical concentration at a point, and so each point in phase space represents a

different possible set of concentrations-at-a-point. But what could we mean by a chemical’s

concentration at a point? A point, aer all, contains no volume, and so there can be no

chemicals at all in a given point. ere is something deeply problematic here.

ere are, of course, other ways of defining chemical concentration at a point, but all

face serious problems. One option would be to define concentration-at-a-point as the ratio

of particles in some fixed but small volume, such as a small sphere, and then let this volume

tend to zero. is definition is analogous to how we define concentration for larger volumes

of chemicals, and if we set the sphere’s volume intelligently then we can be assured that it


will contain enough particles to give sensible results. However, Peter Smith argues that this

definition leads to absurdity on the small scales imposed by the model (Smith 1991, 256).⁴ If we

shrink our volume to or below themolecular scale, it is quite likely not to contain any particles

at all, and once we shrink our volume to a point it will certainly be empty. If we accept this

definition of chemical concentration at a point then no chemicals have any concentration

anywhere, which would certainly be news to most chemists. Similar results can be derived

for any number of macroscopic physical quantities which feature in chaotic models (Smith

1991, 256). e problem, Smith says, is that many physical quantities cannot take on infinitely

precise values at infinitely precise locations: many parts of nature are coarse-grained.

Beyond the limit definition, the only option Smith considers is to define concentration-at-

a-point as the proportions of the particles within some small but finite volume.⁵ He dismisses

this approach for two reasons. First, any such volume must be selected “arbitrarily,” and

therefore will not provide a “principled way of completely precisifying the quantity” of in-

terest (Smith 1998b, 40). Second, it will not yield a precise value of concentration at a point,

because quantum effects will make it impossible to tell whether some particles are inside or

outside of the volume. In response, note first that a definition can be arbitrary without be-

ing whimsical or useless: the distance denoted by “one metre” is arbitrary, aer all, but also

scientifically useful and meaningful. To Smith’s quantum objection, it is not surprising that

quantum particles will not fit classical concepts like “being inside a definite area.” A quantum

analogue could perhaps be constructed, but since the present discussion is framed in classical

terms we will not pursue it further. However, note that the constant-volume interpretation

of concentration-at-a-point is not perfect, and also has odd side-effects: for example, if the

chemical mixture is contained in a vessel, the concentration near the edge of the vessel will

taper off to zero. So while some interpretations may be beer than others, none considered

so far is perfect.

⁴Smith’s original argument involves fluid velocity, but it is parallel to the one presented here.⁵Again, Smith’s original discussion was in the context of circulation velocity at a point rather than concen-

tration.

1.5. THREE APPROACHES TO THE APPROXIMATE TRUTH OF CHAOTIC MODELS 42

e conclusion is that it is difficult to give a sensible interpretation of chemical concen-

tration at a point. According to the limit definition of chemical concentration at a point, no

chemicals have any concentration anywhere; philosophers like Peter Smith have argued that

a fixed-volume definition will be arbitrary or useless; and even if we reject Smith’s arguments

and adopt a fixed-volume definition, we get non-physical model artifacts like concentration

tapering off near the walls of a container. It is unlikely, in other words, that the system could

really have the chaotic structure the model aributes to it.

On the other hand, the continuous BZ model does seem to capture other elements of the

chemical reaction’s dynamics. If we focus on large-scale phase space properties, then the

model performs quite well: it delivers aractors, Lyapunov exponents, and other general dy-

namical properties that match the system quite well empirically. On a large scale the system

seems to behave chaotically, and the model captures some elements of the BZ reaction’s dy-

namical structure reasonably well. From a broader point of view, then, it does seem that we

can interpret some elements of the model more realistically.

On the one hand, the BZ model is almost certainly completely wrong in the ontology

of chemical concentration it posits. On the other hand, it seems to correctly describe many

large-scale elements of the reaction’s dynamics, and it is certainly instrumentally useful in this

domain. e result of this analysis is that the BZ model seems to represent different parts of

the target system with different degrees of success, so it is very difficult to get a clear picture

of how the continuous BZ model connects to its target. Accounting for this instrumental

success coupled with fundamental misrepresentations remains an open problem.

1.5 ree Approaes to the Approximate Truth of Chaotic

ModelsChaotic models, as we have seen, tend to misrepresent their targets in certain ways. As a

result we can say that chaotic models will almost certainly not be true, in the sense of provid-


ing perfect representations. However, a good deal of effort has been expended in aempting

to articulate a notion of approximate truth that could apply to chaotic models. e basic in-

tuition is that although these models are imperfect, perhaps they are somehow close enough

to the truth to deserve special recognition. In this section we will examine three such pro-

posals, one from Peter Smith, one from Jeffrey Koperski, and one based on Popper’s notion

of verisimilitude. I will argue that each of these accounts faces serious challenges.

1.5.1 Peter Smith’s Geometrical-Modelling AccountPeter Smith has proposed a geometrical notion of approximate truth tailored to dynamical

models. Although chaotic models will strictly misrepresent natural systems, Smith argues

that chaotic models get it ‘right enough,’ in a precise geometrical way, that we can call them

approximately true.

In this section I will introduce Smith’s account, consider some objections to this account

from Anjan Chakravary and aempt to defend Smith’s account against them, and then pro-

pose novel difficulties for Smith’s account. ese difficulties are not intended to refute Smith’s

account, but to motivate a different approach outlined in the next section. Aer examining

Smith’s account here, I conclude that this account has strengths but is incomplete, and that it

looks like it would be very difficult to complete. First, it is not clear that Smith’s geometrical

account will work if the physical systems involved are coarse-grained; and second, even if the

target systems are continuous, it is not clear that Smith’s account will work if they are chaotic,

and it is possible that it will be too ad-hoc to support the kind of metaphysical demands made

by realists like Chakravary.

Smith’s account of approximate truth is not meant to apply to all theories, but is tailored

specifically for what he calls “geometrical modelling theories” (or “GM-theories” for short)

(Smith 1998a, 258). A GM-theory consists of two parts, labelled M and A. M is a mathemat-

ical structure, for our purposes defined by a set of equations, and A provides the necessary

interpretation to claim that M replicates the structure found in some real-world system. For

a GM-theory to be true is for M to be precisely the same as the structure in the real world.


v

x

(a) Circular orbits.

v

x

(b) Oval orbits.

v

x

(c) Irregular orbits.

Figure 1.5: e GM-account of approximate truth relies on the geometrical similarityof trajectories in phase space. Here, for example, are three sets of trajectories that areplausibly similar.

To say that a GM-theory is approximately true is to say that “there is an abstract structure

of a certain kind, and in a straightforward sense this approximately replicates a structure in

the real [system]’s spectrum of possible behaviours” (Smith 1998a, 260). us, Smith claims,

approximate truth is “just a maer of one geometric structure approximating another” (Smith

1998a, 261). is provides an aractive reduction of the somewhat mysterious notion of ap-

proximate truth to the more familiar notion of geometrical similarity.

To illustrate this notion, consider the following simplified example. Imagine we have a

model of a pendulum which says it moves with simple harmonic motion. Mathematically,

if we let x represent its position and c be some constant of motion, our model involves the

following simple equation:

x = −cx (1.5)

And this equation defines a set of circular trajectories through phase space corresponding to

regular oscillations about the origin (Figure 1.5a).

Smith then assumes that there is another structure in phase space corresponding to the

actual system’s spectrum of possible behaviours, and our model is approximately true to

the extent that the circles in Figure 1.5a are geometrically similar to the trajectories in this

actual phase space. ere are many ways that the actual trajectories could differ from the

model trajectories: they could be slightly elliptical (e.g. Figure 1.5b), or even be perturbed


in some non-uniform way (e.g. Figure 1.5c). e point is that judging whether one shape

approximates another is supposed to be a transparent process, possibly even amenable to

mathematical treatment. And the process is supposed to be effectively the same for more

complicatedmodels, such as the kicked rotator and Belousouv-Zhabotinsky chemical reaction

we examined in the previous section.

Anjan Chakravary has criticized Smith’s account on two points. I will consider them

both, and aempt to defend Smith’s account. First, Chakravary argues that if Smith’s ac-

count focuses only on observable elements then it will be at best instrumental. If an approx-

imately true GM-theory restricts itself to observable phenomena then it may make accurate

predictions, but “this by itself makes no commitment with respect to ontology, and in partic-

ular, with respect to the “unobservable”” (Chakravary 2001, 340). However, Smith explicitly

rejects the empiricist view that model-world correspondences are limited to observable fea-

tures (Smith 1998a, 262). For Smith, approximate truth must also capture modal features of a

system’s “spectrum of possible behaviours” (Smith 1998a, 260). As such, while Chakravary’s

point is well taken, Smith’s account is flexible enough to side-step the objection.

Chakravary’s second objection is that if a GM-theory does incorporate unobservable

elements, “geometrical closeness of model parameters and worldly measurements does not

entail that such counterparts exist” (Chakravary 2001, 341). “If they do not,” Chakravary

continues, “such theories will not count as approximately true for the realist” (Chakravary

2001, 341). Chakravary is correct, but this problem is epistemological, not ontological. We

evaluate approximate truth by evaluating the closeness of two geometrical structures, but

if we do not know what one of the structures is, we cannot evaluate closeness. erefore

Smith’s account is difficult to apply because of our limited knowledge. However, this does

not mean that no real-world geometrically similar counterparts to a model’s structures could

exist, or that they could not meet the rest of Smith’s criteria for approximate truth. We can

agree that this is a problem for Smith’s notion of approximate truth in practice, but not in

principle.


I propose a new critical analysis of Smith’s account. In summary, Smith’s account relies

on an unanalyzed notion of “geometrical closeness,” and we run into conceptual problems

when we try to make it precise. Any single notion of closeness is unlikely to cover all cases,

and a more complicated account involving many notions of closeness loses the simplicity that

made Smith’s approach aractive to begin with. However, without a clear definition of this

term, Smith’s account is incomplete. I will spell this argument out in more detail.

Any one notion of geometrical closeness is unlikely to work well in all cases, for both

general and specific reasons. In general terms, as I argue elsewhere, precise mathematical

definitions of intuitive concepts can be prone to counterexamples.⁶ If we choose any one

notion of closeness, it will plausibly not apply in all cases where it intuitively should. e

more specific reason I propose is that mismatches in terms of structure type or dimension

between models and the world will cause problems for one single definition of geometrical

closeness.

Let us consider the impact of mismatches between the geometrical structures of our mod-

els and the world. Plausibly, nature either is or is not continuous in the way our models are, so

we will consider each of these scenarios in turn. e first case is that nature is not continuous

in the way our models are. Smith himself has argued that many of the properties we aribute

to nature are “coarse grained” (Smith 1998b, 39-41). ere are many ways this could happen.

For example, on a small scale, instead of being continuous nature could be somehow “pixe-

lated,” with properties defined only on a grid of finite size. Or properties could be defined at

continuous points, but take on only discrete values. In either case the structure of the “natural

geometrical space” will be very different from that of our models. If the relevant quantities

in nature have discrete rather than continuous properties, the structures in the models and

nature will have qualitatively very different natures. Accordingly, a simple comparison of the

distance between the values of the quantities in the model and the corresponding quantities

⁶In particular, see my argument in Chapter 4 regarding mathematical definitions of observational equiva-lence.


in nature is misguided. ere is no obvious standard of geometrical similarity that will cover

all of these situations.

e second possibility is that nature is continuous in the way our models are. is would

make it in-principle possible for the structures in our models to approximate the real struc-

tures in nature. However, there would still be two sets of conceptual problems. First, scientists

Judd and Smith identify theoretical reasons to think that model structures will differ greatly

from natural structures (Judd and Smith 2004). eir analysis is highly technical and covers

several different types of world-model mismatch, but for our purposes we will focus on what

they call “ignored-subspace model inadequacy.” In essence, a model has an ignored-subspace

inadequacy when its target system has a component of its dynamics that is unknown or not

included in the model (Judd and Smith 2004, 226). In more technical terms, this means that

the physical system’s phase space consists of two subspaces, K and K′, which are d- and

d′-dimensional Euclidean spaces respectively, but the model only captures K . It is, first of

all, not clear how trajectories in different-dimensional spaces can approximate each other: is

a straight line in one-dimension, for example, more similar to a square or a triangle in two

dimensions? And furthermore, when we view the projection of the full chaotic K × K′ sys-

tem into K-space, its trajectories will necessarily cross each other, whereas by assumption

the trajectories in our models cannot. ere is no obvious standard of geometrical similarity

that applies in such cases.

One final possibility is that nature has continuous and coarse-grained elements. Indeed,

this may be the most likely case, if space and time are continuous but other properties like

chemical concentration are coarse-grained. en the difficulties from the previous two situa-

tions combine, and there is almost certainly no single definition of geometrical closeness that

will work.

One response would be to adopt more than one notion of geometrical closeness, and use

the appropriate notion in a given situation. But then the account becomes both more compli-

cated and less compelling. Smith himself thinks there is probably no one-size-fits all account


of approximate truth, but now we are no longer articulating one notion of approximate truth

for GM theories: we are articulating different notions for different types of GM-theories

(Smith 1998b, 71). is makes his account less compelling because, in the absence of pre-

existing justifications, adding more definitions of closeness to the account seems more like

an ad-hoc way of saving it than explicating what we mean by approximate truth.

In defense of Smith’s account, one might point out that it may not be a problem that there

is no single definition for all cases provided that the various criteria for similarity are based

on a similar idea that is spelled out in differently in different cases. In order to pose a serious

challenge, one would need to show that this is not the case. is defense has merit, but my

goal here is not to demonstratively refute Smith’s entire approach. My two more modest

points are, first of all, that it is not clear how to spell this account out differently in different

cases, and lacking this information we have more a description of an account of approximate

truth rather than a complete account; and, second, the more general but less informative such

an approach is, the less aractive it is as a well-articulated philosophical account.

In conclusion, Smith’s account has an aractive simplicity to it but is incomplete, and

there are conceptual reasons why it will be very difficult to complete. Smith’s account has

an aractive simplicity because it reduces the mysterious notion of approximate truth to the

intuitive notion of geometrical similarity. However, making this intuitive notion precise will

be very difficult. is is not a refutation, but it does show that Smith’s account has serious

weaknesses. Smith focussed his account on GM-theories, rather than scientific theories in

general, because he thought it was “doubtful that there is any story to be told [about approxi-

mate truth] which is both substantive and general” (Smith 1998b, 71). Unfortunately, it seems

that similar considerations apply to an account even limited to GM-theories.

1.5.2 Jeffrey Koperski’s Truesdell-Inspired AccountIn this section I will examine a second account of the approximate truth of chaotic models

given by Jeffrey Koperski (2001). Koperski begins with an apparent conflict between atomic

theory and chaotic models in the microscopic realm, like we saw in our examination of the BZ


reaction, where models make false assumptions about their target’s ontology on small scales.

Given this apparent conflict, he asks how we “can… treat such a model realistically” (Koper-

ski 2001, 684). Koperski’s thesis is that there is no conflict, because chaotic models actually do

not imply anything about the microscopic realm. Since these models do not imply anything

about the microscopic physical realm, there cannot be a contradiction with other physical

theories. Here I will examine Koperski’s motivations and position in more detail. I will ar-

gue that Koperski’s position gives a good description of standard scientific and engineering

practice. However, there is a tension between treating a model’s chaoticity realistically and

disregarding its micro-states.

Koperski begins his investigation with the observation that chaotic models are unphysi-

cal in a way that non-chaotic models are not. Specifically, “[u]nlike textbook physical models

such as the ideal pendulum, chaotic models appear to add an infinite amount of surplus struc-

ture that does not exist in reality” because of the infinite intricacy in their fractal phase-space

aractors (Koperski 2001, 696). Chaotic models are frequently in conflict with the “underlying

physics,” by which Koperski means the atomic theory of maer (Koperski 2001, 687). Koperski

is concerned to “justify this nonphysical structure” (Koperski 2001, 686).

Koperski’s approach to this problem draws on work in continuum mechanics, which, like

chaotic dynamics, uses real-valued variables to represent discrete physical quantities. is

means, for example, that a continuum-mechanical treatment of fluid flow does not recognize

the molecular nature of fluids. Prima facie, continuum-mechanical models appear to entail

that maer is continuous, rather than particulate, and so impose similar “surplus structure”

that prevents us from interpreting them realistically in a straightforward way.

e solution to his dilemma, Koperski argues, can be found in the work of continuum-

mechanicist Clifford Truesdell. Truesdell denies that continuum-mechanical models have

anything to say about the micro-structure of their targets at all:

Rather than imposing a false microstructure on the world, continuum models

like those in fluid mechanics ignore the small-scale facts… Strictly speaking, there


can be no mismatch between facts in the macroscopic model and facts in the

microscopic world. For there to be a mismatch, continuum models would have

to imply something about the structure of maer. However, nothing is presumed

and nothing is implied. (Koperski 2001, 697)

e proposed solution, in other words, is that the model actually implies nothing about a

system’s micro-structure or ontology. Since themodel implies nothing about micro-structure,

we can therefore believe everything it does imply without commiing ourselves to micro-

ontology. According to Koperski, the reason that chaotic models do not imply anything about

the microscopic realm is that model users are not interested in this realm: “the small-scale

structure is ignored” (Koperski 2001, 697) and “the question is whether [the target system]

behaves that way at the scale the modeler is interested in” (Koperski 2001, 698).

Koperski takes an instrumental viewpoint about the microscopic realm. From a descrip-

tive point of view this position matches up well with actual scientific and engineering prac-

tice, but the question is then what part or aspects of the model are supposed to have a realist

interpretation. Here there are two concerns.

First, this response may be in tension with Koperski’s original motivation to treat the

system “realistically.” Certainly we can believe imperfect models are approximately true, but

Koperski is very permissive about what might exist in the microscopic realm: “At boom

there might just as well be Newtonian corpuscles, Boscovichian point masses, or Leibnizian

monads, so long as they behave in large numbers like a continuous fluid” (Koperski 2001,

697). And he is clearly focussed entirely on macroscopic predictions: “atoms, point masses,

or monads, it does not maer, so long as macroscopically detectable changes in [the property

of interest] are continuous rather than discrete” (Koperski 2001, 698). If the only things we

treat realistically are macroscopically detectable changes, then this is not approximate truth

in the traditional sense. More would need to be said to make this a compelling notion of

approximate truth.

Second and more seriously, it is not clear in what sense we can say that a system is (ap-


proximately) chaotic if it is assumed that its postulated ontology is false, yet the claim that it

is chaotic is based on this ontology. We assume various microscopic properties of a system to

derive chaoticity; so if we then deny those microscopic features, on what basis can we con-

tinue to assert that it is even approximately chaotic? Even more plainly, if we assume A to

derive B, and then deny A, the status of B is highly suspect. We derive that a model is chaotic

based on features like infinitely intricate fractal aractors and the long-term behaviour of

initially infinitely close trajectories, and it is not clear how we can accept the result of this

derivation while simultaneously denying its core assumptions.

In the case of the fluid-dynamical models Koperski considers, for example, we derive the

chaoticity of the models from continuity assumptions. If we were to interpret these assump-

tions, they would imply that the target physical system is non-atomic. But if we reject these

model assumptions, or at least remain agnostic about them, it is no longer clear howwe justify

the claim that the model is approximately true. e model cannot be truly chaotic without

these features, and it is not clear what notion of “approximately chaotic” Koperski appeals to

and whether it really captures the feature that are usually associated with approximate truth

could apply.

We might consider alternate sources of justification for the claim that a system is chaotic.

Claims of chaoticity cannot be justified merely by appealing to predictive usefulness, because

non-chaotic models with very different mechanisms (e.g. Landau–Hopf turbulence, or even

stochastic evolution) might be just as predictively useful and so just as justifiable. Chaotic

models may be simpler than rival models, but simplicity is controversial as a marker of truth

or approximate truth (Forster and Sober 1994; Sober 1996). An alternate source of justification

for claims of chaoticity could perhaps be found, but there is not currently one on offer, and

the future prospects are unclear.

At boom, Koperski has argued that the microscopic disagreement between chaotic mod-

els and atomic models does not prevent us from constructing empirically successful scientific

models or pieces of physical technology, and so it is not a pragmatic problem. And on this


point, he is absolutely correct. Koperski’s account also sheds much light on actual scientific

and engineering practice. But to the extent that there is a peculiarly philosophical problem

of realist interpretations and approximate truth, Koperski’s argument is incomplete as a jus-

tification of “realistic aitudes,” or approximate truth, for chaotic models.

1.5.3 A Popperian Verisimilitude Approae final approach to the approximate truth of chaotic models that we will consider is

based on verisimilitude. Several authors have argued that verisimilitude is intended as a notion

of approximate truth, and Peter Smith has explicitly considered its prospects as an account of

approximate truth for chaotic dynamical models (Smith 1998b, 82-3). Although verisimilitude

has several nice features, we will see that it is not intended, and nor will it work, as an analysis

of approximate truth.

e verisimilituditarian approach (henceforth “VS”) began with Karl Popper’s aempts to

reconcile his deductivist conjectures-and-refutations view of science with his belief in scien-

tific progress (Popper 1963, 1972). e intuitive idea is that even a false theory can be verisim-

ilar, or truthlike, if it also says enough true things about a domain of inquiry, and that a series

of strictly false theories can nonetheless say more true things and so increase in verisimili-

tude. However, Miller (1974) and Tichy (1974) independently discovered serious logical flaws

in Popper’s original approach. In subsequent years Niiniluoto (1984, 1987) and Kuipers (1987)

developed neo-Popperian accounts of VS that overcame these problems. Despite new and

recent challenges to VS (Bird 2007, 2008), its supporters maintain that the approach is sound

in general (Cevolani and Tambolo 2013).⁷

ere are many accounts of verisimilitude on offer, but for our purposes we can consider

the general features of VS. Informally, a theory is said to be highly verisimilar if it “says

many things about the target domain, and if many of those things are (almost exactly) true”

(Cevolani and Tambolo 2013, 924). is notion depends on both the content and accuracy of a

theory, or in other words on both how much is said and how much of that is true. Following

⁷For more thorough surveys of the literature on verisimilitude, see Niiniluoto (1998) and Oddie (2008).


Cevolani and Tambolo (2013)’s example, consider n logically independent atomic propositions

in a given language L, and let the conjunction of these propositions (p1 ∧ p2 ∧ . . . ∧ pn) be

the maximally informative true description of the part of the world to which this language

is intended to apply. Let us assume that a theory is a logical connection of propositions.

en the singular propositions (p1) and (¬p2) are each theories, albeit very short ones. ey

are equally informative, because each makes the same number of atomic claims, but the for-

mer is more truthlike because it makes more true claims. Similarly, (p1) and (p1 ∧ p2) are

equally true—indeed, each is completely true—but (p1∧ p2) is more informative, and so more

verisimilar. One important consequence of this approach is that a false theory may be more

verisimilar than a true theory if the false theory is informative enough (Cevolani and Tambolo

2013, 925). (p1) is truer than (p1 ∧ p2 ∧ . . .∧¬p2), but the laer is so much more informative

that it is more verisimilar. Of course, there remains the all-important epistemic problem of

how we estimate verisimilitude, but it makes sense for some false theories to be beer than

some true theories (Cevolani and Tambolo 2013, 928-9). A tautology is universally true, af-

ter all, but also devoid of content. e VS approach is intended to capture the intuition that

false but informative theories can be more truthlike—that is, beer scientific theories—than

tautologies (Cevolani and Tambolo 2013, 925).

It is fair to ask whether the VS approach will be applicable to chaotic dynamical systems,

since if dynamical models are geometrical or mathematical then they may not have the re-

quired propositional content. Fortunately, there are reasons to think that dynamical models

involve some linguistic elements. Recall, for example, that Peter Smith divided GM-theories

into mathematical components M and linguistic components A (see Section 1.5.1 above). On

this view, even if M remains stubbornly non-linguistic, the VS approach could be applied to

the content of A. Furthermore, Chakravary (2001) has argued that realism requires our the-

ories to have some linguistic element, even if it is only a straightforward claim that theoretical

elements correspond to certain parts of the world. It is therefore plausible that chaotic models

include some linguistic content, in which case they will be amenable to a VS treatment.

1.6. DISCUSSION 54

Unfortunately, despite verisimilitude’s apparent advantages, it cannot function as an anal-

ysis of approximate truth. In fact, proponents of VS are clear on this point, since conflating

approximate truth and truthlikeness has led to confusion in the literature (Niiniluoto 1998, 18-

9)(Cevolani and Tambolo 2013, 931). Much could be said on this point, but two observations

should suffice to make the distinction clear. First, truthlikeness and approximate truth can-

not be the same because they differ extensionally. Consider the two true theories T1 = (p1)

and T2 = (p1 ∧ . . . ∧ p10) in the language introduced above. T1 and T2 are both true, and

so are equally approximately true. But as we saw above, they differ in their verisimilitude

because T2 is much more informative than T1. Verisimilitude cannot, therefore, be the same

as approximate truth. Second, accounts of verisimilitude oen incorporate an independent

notion of approximate truth. is is because many of the propositions in any actual scientific

theory, as opposed to an ideal scientific theory, will only be approximately true. Indeed, ac-

cording to Cevolani’s recent defense of the VS approach, “verisimilitude is a combination of

(approximate) truth and content” (Cevolani and Tambolo 2013, 930). An account of verisimil-

itude therefore assumes a working account of approximate truth, and we cannot interpret VS

as approximate truth on pain of circularity. us, despite its apparent advantages, we cannot

use verisimilitude as an account of approximate truth for chaotic models.

1.6 DiscussionSeveral points emerge from this analysis. e first is that chaotic models seem to provide

both very good and very bad representations of their targets in different domains. rough

the examples of the kicked rotator and the BZ reaction we saw various types of representa-

tional success and failure in action. ese misrepresentations make it highly unlikely that

any chaotic model will ever be straightforwardly true, but these models do seem to get some

things right. Even if misrepresentation is the norm, chaotic models do seem informative and

useful.

e second point is that developing a coherent account of approximate truth for chaotic

1.6. DISCUSSION 55

models will be an extremely challenging project. Of the three accounts we considered, Smith’s

and Koperski’s both faced serious metaphysical and epistemological problems. e third ac-

count, based on a Popperian notion of verisimilitude, initially seemed more promising. How-

ever, it cannot function as an account of approximate truth because it presupposes an account

of approximate truth. e result is that if we want to insist that chaotic models can be ap-

proximately true, there is no aractive option on the table to define what it is we even mean

by “approximate truth.”

But what is approximate truth even supposed to accomplish in these cases? Presumably

it is meant to be a success term that we bestow upon models because they perform well

in some areas, and despite the fact that they perform poorly in other areas. On all of the

accounts we considered, however, the designation “approximately true” can only be applied

to a model aer we have already undertaken an in-depth investigation into its strengths and

weaknesses. e disagreements primarily revolve around what kind of strengths and how

many are necessary, and what kind of weaknesses and how many we are willing to overlook.

When viewed in this way, the arguments over approximate truth seem almost like an aer-

thought, since they all take place aer a model’s epistemological and ontological implications

have been thoroughly investigated.

On this view, investigations of approximate truth are actually at best ancillary to more

fundamental questions of a model’s epistemological and ontological consequences. In this

best-case scenario, we first perform a in-depth analysis of such factors as a given model’s

predictive strengths and weaknesses, its areas of agreement and conflict with other scientific

models, and so on. en, once we have this data, we come up with schemes to adjudicate

approximate truth. In actual fact, however, it seems that a good deal of time and energy could

be devoted to coming up with adjudication schemes before we have a good idea of what

kinds of strengths and weaknesses our models have. Focussing on approximate truth can, in

other words, actually lead us away from important investigations into the applicability and

foundations of our scientific models.

1.6. DISCUSSION 56

With this in mind, the third point is that it might be both beer and simpler to move

away from arguments about approximate truth, and to focus on the investigation of a model’s

epistemological and ontological merits. ose involved in the approximate truth debate are,

aer all, already performing some of this kind of analysis already. Furthermore, in practical

terms, these considerations are what actually maer. We learn something substantial when

we learn which parts of a model we believe more or less correspond to a real part of the

world, and which parts do not. On the other hand, it is not clear what we gain in adjudicating

whether a model is approximately true beyond the evaluations of the model’s virtues and

vices. All that we learn is that a given model meets a given set of criteria.

Focussing on the strengths and weaknesses of individual models also allows us to take a

more nuanced view of model success. Approximate truth itself may have been intended as

a way to take a more nuanced view of truth and falsity, by providing a kind of “grey area”

between strict truth and falsity. But in practice, we can see that too oen approximate truth

theorists treat dynamical systems in a binary fashion, wherein they are either approximately

true or not. e nuance is gone. Focussing on the complexities in a model can bring back this

kind of nuance: rather than worrying about whether a model is simply approximately true

or not, we are free to investigate the myriad ways in which it represents and misrepresents

its target.

A full, complete, consistent, and nuanced account of the approximate truth of chaotic

dynamical models may well be possible. If we do decide to take a criterion of approximate

truth seriously, it is important that this kind of judgement call be made explicit, transparent,

and open to debate. In the meantime, however, if we insist on a one-size-fits-all account of

approximate truth then we risk glossing over these differences, and leading us away from

investigating the epistemological and ontological implications of our models.

1.7. CONCLUSION 57

1.7 ConclusionIn this chapter I examined several philosophical accounts of approximate truth for chaotic

models and two case studies of chaotic models. rough the case studies I argued that chaotic

models have a complicated relationship with their targets, oen representing and misrepre-

senting them in different domains and in subtle ways. Any account of approximate truth will

have to take these complexities into account. I then examined three accounts of approximate

truth, and argued that each faces difficulties, and that none is acceptable in its current form. I

argued that based on the complexities of chaotic representations, and the poor prospects for a

coherent account of their approximate truth, we would be beer to avoid approximate-truth

talk at all concerning chaotic models. A beer use of our philosophical resources, I argued,

would be to focus on investigating the particular successes and failures of different models in

different contexts.

.. 2Fractals as Instrumental Models

For we always need to ask: won’t a beer description of nature infact be provided by prefractals that lack the infinite excess detail?To which the answer seems invariably ‘yes’.

Peter Smith, Explaining Chaos, p.33

2.1 IntroductionIn this chapter I will argue that fractal representations of material objects can be good in-

strumental models. e question of whether fractals might be instrumentally useful is simple

enough, but it has largely been ignored in the published philosophical literature. As we will

see, the main focus in the literature has been whether or not fractals can be interpreted real-

istically, and instrumental fractal models have generally been deemed inadequate by authors

with strongly realist intuitions. We will return to questions of realism in the next chapter, but

for the moment let us put any realist intuitions on hold and consider whether fractal models

might, aer all, be instrumentally useful.

I will begin by considering the three main objections to fractal models: that they cannot

be interpreted realistically; that they will never be beer than non-fractal models; and that, in

a sense I will explain, they are not properly scientific. I will pursue two strategies to overcome

58

2.2. A FEW WORDS ON INSTRUMENTALISM AND MODELS 59

these objections. First, I will argue that in some presentations each objection relies on realist

assumptions, and so begs the question against instrumentalism. Second, I will argue that

to the extent that these objections do not make realist assumptions, they raise the empirical

question of whether fractal models are scientifically useful. Since these worries are raised in

the abstract, without looking at any particular cases, we might naturally ask whether fractal

models of material objects are in fact useful.

To show that fractal models can be instrumentally useful, I will give a case example from

tribology, the study of rough surfaces. First I will describeMajumdar and Tien (1990)’s general

approach to modelling rough surfaces as fractals. en, I will give an in-depth examination

of Chen et al. (2009)’s recent model of fluids flowing through very small channels. is, I will

argue, is a good instrumental fractal model. is may not constitute the “physics of fractals”

some critics have demanded, but it certainly shows that there are good instrumental fractal

models in physics.

2.2 A Few Words on Instrumentalism and ModelsSince this section is about instrumental interpretations of models, before continuing I

should clarify what I mean by “instrumentalism” and “models.” Although the past thirty years

or so has seen an explosion in the literature on scientific models, there is still considerable

disagreement about what they are, what they do, and how they do it (for an excellent review,

see Frigg and Hartmann 2009). It is not my aim to give a comprehensives account of scientific

models, so I hope I can squeak by with a minimally objectionable account. My focus here is

on models in the physical sciences. Specifically, I will focus on mathematical models, like the

simple harmonic pendulum, rather than physical models, like a balsa-wood model airplane in

a real wind tunnel. Whether or not models are constituted by equations, I take it to be the case

that in this domain models somehow involve equations. Furthermore, at least some variables

of these equations are taken by scientists to correspond to physical magnitudes. is means

that if an ocean wave were, somehow, to leave an imprint of the Navier-Stokes equations on

2.2. A FEW WORDS ON INSTRUMENTALISM AND MODELS 60

the sea shore it would not itself constitute a model; but if a vacationing fluid dynamicist were

to notice and use these markings to calculate the improbability of their very existence, then

they might very well be a model, or at least a part of one.

In sum, the models I am concerned with here use math, and some parts of that math are

interpreted by scientists as physical magnitudes. is is certainly not the whole story about

models, but it is all I need here.

e instrumentalist, as I shall use the term, is a metaphysical mercenary who cares only

about scientific models inasmuch as they are useful for generating predictions. A scientist’s

model, from this perspective, is like a mechanic’s wrench: it is a tool for a practical job.

Mechanisms and explanations are not concerns for either mechanics or instrumentalists. If

you ask a mechanic why or how a wrench works, you will probably get either a puzzled stare

or an answer along the lines of “it just does.” Asking an instrumentalist how or why a model

works will merit a similar response, because this is simply not what instrumentalists care

about.

If we combine these two definitions, we get an instrumental view of models: models use

math; some parts of that math are interpreted by scientists as physical magnitudes; and we

only care about how well the physically interpreted parts of the models predict our measure-

ments. I take this to be a minimal position, because inasmuch as scientific practice is con-

cerned with prediction and control it is at least instrumentalist in this sense. Furthermore,

we could bolt elements of a more realist metaphysics onto this basic chassis. A semantic re-

alist, for example, might give physical interpretations for more of our model elements, while

an epistemic realist might give conditions for belief in a model’s truth above and beyond its

utility. In this section, however, we will focus on the bare form of instrumentalism outlined

above.

2.3. OBJECTIONS TO FRACTAL MODELS 61

2.3 Objections to Fractal ModelsIn this section I will consider three main objections to the use of fractals in science to

see whether they exclude an instrumentalist view of fractal models. I will argue that most of

these objections do not refute this minimalist view of fractal models, since theymake strongly

realist assumptions that beg the question against instrumentalism. e question of whether

they necessarily rule out a more realistic view of fractals will be taken up in the next chapter.

e first objection is that fractal models have essentially unphysical features, and so have

no realistic interpretation. e lightning rod for this argument is usually Mandelbrot’s as-

sertion that the coastline of Britain is a fractal. In brief, Mandelbrot (1967) argued that the

empirical evidence shows that as we measure the coastline of Britain more precisely, its total

length increases without bound. is is certainly not how measurements of ordinary objects

behave. If we were to measure the length of my table ever more precisely, for example, our

results would converge on some finite value. However, we would expect to find a divergent

result if we tried to measure a fractal like the Koch curve (See Appendix A for details). If

we first estimated its length using a metre stick of length 1, we would get a result of 1; if we

increased our precision and estimated its length using a metre stick of length 1/3, the total

measured length would increase to 4/3; and so on. Mandelbrot’s major claim is that there is

empirical evidence that the coastline of Britain is more like the Koch curve than a table. Man-

delbrot then estimates the fractal dimension D of several coastlines, to show that his findings

are not restricted to Britain. He ends by endorsing the “positive interpretation stated at the

beginning of this report” (Mandelbrot 1967, 638)—that coastlines are really fractals.

In response, one might object that fractal models of material objects cannot have a real-

istic interpretation, since material objects cannot have fractal structure on all levels of detail.

Peter Smith, for example, stated that “[t]he idea that the coastline has detail at every scale is

absurd” (Smith 1998a, 32). Along similar lines, Orly Shenker objected that any claims about

objects being fractals must be “inconsistent with modern science,” and that Mandelbrot’s sug-

gestions must be “interpreted within a radically new, non-atomistic science” (Shenker 1994,


979). In general, this line of argument runs that fractal models of material objects cannot be

interpreted realistically because they will, on a small enough scale, conflict with the atomic

theory of maer.

is objection only poses a problem if we place standards on our scientific models that

go well beyond instrumentalist commitments. We can agree that material objects are made

of atoms, but this is only a problem for a fractal model per se if it is taken to be a completely

accurate representation of nature. ere is widespread agreement between scientists and

philosophers that this is an unrealistic and perhaps impossible standard (e.g., Teller 2001; Judd

and Smith 2004). Even a realist about models could meet this challenge by responding that the

fractal model is only approximately true. A metaphysically deflationary instrumentalist, who

is not interested in a realistic interpretation of the model at all, would therefore be uerly

unmoved by the point. We can use the atomic theory when we’re dealing with atoms, and

the fractal theory when we’re dealing with coastlines, and as long as our predictions work

out we need not trouble ourselves with ontology.

A second and related objection is that pre-fractal models will always be preferable to frac-

tal models. A pre-fractal, in this sense, is a finitely intricate non-fractal shape that shows

fractal-like paerns on finitely many scales. Smith, for example, argued that “prefractal mod-

ellings must be at least as good as fractal ones,” since “prefractals lack all the extra infinitely

intricate tail with no empirical content; and if anything, modellings that lack redundant con-

tent are to be preferred when available” (Smith 1998a, 32). e rationale is that fractal models

will always have “excess structure” not found in the target system, whereas pre-fractal models

will not.

Clearly redundant content should be excluded, but the question is whether, and in what

sense, the infinite intricacy in fractal models is redundant. If the argument is that model

components without physical correlates are redundant, then it is essentially the same as the

first objection considered above. Pre-fractal models may indeed meet strong realist criteria

of “goodness” beer than fractal models, but this will clearly not trouble an instrumentalist.


However, on a more moderate reading it is not clear that non-empirical content must be

avoided at all cost. If we also consider a model’s predictive utility, non-empirical content

may be permissible as long as it makes the model easy to use for its intended purpose. And

it stands to reason that if a model’s sole purpose is predictive utility, then this “extra” content

need not even be extra in any substantial sense.

To state it plainly, this “nonempirical” content may not be redundant from an instrumental

point of view if it makes prediction easier. By way of analogy, imagine we believed the true

equation governing some process near the value x = 0 to be:

f (x) = 1 + x − x2

2+

x3

6+ · · · + x10

10!(2.1)

We could certainly work with this equation. However, we could also recognize that it consists

of the first eleven terms of the Taylor expansion of ex near x = 0. So instead we could work

with f (x) = ex , which would make our calculations much simpler without sacrificing much

accuracy. Since a Taylor expansion is an infinite sum, in one sense we have added an infinite

amount of nonphysical surplus structure to our original equation. Furthermore, the new

formula gives very bad predictions for values of x that are too far from zero. On the other

hand, our new function is much easier to deal with, and the predictions within our area of

interest are accurate enough. From an instrumental point of view, this infinite nonphysical

structure is justified.

ere is no in-principle reason why the situation could not be the same with a fractal

model. If we think that a physical structure is self similar over at least a few scales, then it

could be useful to model it as a fractal. Adding an infinite amount of structure might give us

a model that’s simpler, easier to solve analytically, and that makes good predictions. I will

discuss an example along these lines below in Section 2.4.

e third objection to fractal models is that the purportedly scientific work on fractals is

actually nothing more than naïve paern finding, with no relation to prediction, explanation,


or any other features of real science. For example, Peter Smith admied the coastline of

Britain might have fractal-like properties on some length scales. But, he wondered, “why

suppose that this is more than a mildly diverting accident” (Smith 1998a, 33)? Geology and

environment vary widely across Britain, aer all. Smith argued that “it will be luck if any

scaling law applies (even approximately) right down the coast,” and advised us not to draw

any further conclusions (Smith 1998a, 33). Leo Kadanoff put the point more strongly. He

argued that early proponents of fractals offered only a “zoology of interesting specimens and

facile classifications” which was “superficial and even slightly pointless” (Kadanoff 1986, 7).

Presumably, merely finding a large number of fractal-like paerns in nature is not physics. At

best, it is mere classification; at worst, it is crankish paern finding. To rehabilitate fractal

science, Kadanoff suggested that we need a “substantial theoretical base in which geometrical

form is deduced from the mechanisms that produce it.” Only then, he concluded, will we have

a “physics of fractals” (Kadanoff 1986, 7).

Let us consider this objection in an instrumentalist light. ere are two common threads:

first, fractal models do not provide scientific explanations of the mechanisms whereby fractal

paerns arise; and second, they are oen “pointless.” It is certainly true that a fractal descrip-

tion of a surface-level regularity with no appeal to generating mechanisms will not tell uswhy

this regularity obtains. Smith seems to be asking for an explanation of why a fractal-like reg-

ularity holds, and particularly some kind of causal-mechanical explanation. Kadanoff, with

his use of the terms “facile” and “superficial,” seems concerned that fractal classifications do

not capture the deeper elements that give rise to fractal appearances. But this worry is based

on a particular view of what science does, namely that it provides explanations and discovers

causal mechanisms. is view of science makes the most sense from a realist perspective. In-

strumentalists are concerned only with predictions, and so the demand for explanations and

underlying mechanisms is misplaced.

It is also true that naïve paern finding may lead us to focus on “mildly diverting” or

“pointless” regularities at the expense of other, more fruitful models. But this is a worry the

2.4. THE TROUBLE WITH TRIBOLOGY: INSTRUMENTALLY USEFUL FRACTALS MODELS 65

instrumentalist can agree with: aer all, if a pointless model has no predictive utility, then it

will not be instrumentally acceptable. So this objection hinges on an empirical question: are

fractal models predictively useful? Above, I argued that they could be. In the next section, I

will argue that they are.

2.4 e Trouble with Tribology: Instrumentally Useful

Fractals ModelsTribology, broadly speaking, is the study of “all aspects of moving surfaces and the trans-

mission and dissipation of energy and materials in mechanical systems” (Czichos 1978, 2).

More plainly, it is the study of how moving bodies interact with each other and the environ-

ment through friction and wear.

Most surfaces, even those that appear very smooth to the naked eye, are in fact quite

jagged on a microscopic scale. e typical way of measuring a surface’s roughness is by

dragging a diamond-tipped stylus across it, like a phonograph needle on an LP, and recording

the stylus’s vertical displacement. is gives a one-dimensional record of a surface’s topog-

raphy, called its profile. When magnified and graphed, these profiles are oen surprising:

even surfaces that seem very smooth to the touch will have profiles that look like tiny moun-

tain ranges, with sharp and dramatic peaks and valleys. is microscopic surface roughness

affects the frictional forces the surface experiences when brought into contact with other

objects, and so, along with being of general scientific interest, is important in a variety of

manufacturing and engineering contexts.

One of the fundamental issues in tribology is how best to characterize this surface rough-

ness. e so-called “classical” approach treats the deviation of the surface profile from its

mean as a stochastic signal. e surface itself, of course, is unchanging, but the idea is that by

treating the profile as if it were a signal from an unpredictable random source we can measure

the statistical features of the surface topography as a whole. is allows scientists to give a


complete statistical characterization of the surface topography using only a few variables.

ere are several ways of modelling a rough surface classically. Whitehouse and Archard

(1970), for example, propose that the peak height follows a Gaussian distribution and that the

autocorrelation of the surface profile decays exponentially. is exponential decay ensures

that there are no repeated paerns in the profile, and so imposes a sort of randomness require-

ment. On this account, the Gaussian variance and autocorrelation distance together suffice

to characterize the surface. Greenwood and Williamson (1966) propose a different stochastic

model which characterizes the peak distribution in terms of yet other variables. e precise

details need not concern us here, and it is enough to know that the classical approach models

surface roughness as random variations from a mean value.

In the early 1990s, Arunava Majumdar and Chan-Lin Tien proposed a new approach to

surface roughness whereby the profile is characterized as a fractal (Majumdar and Tien 1990).

Taking their cue directly from Mandelbrot, Majumdar and Tien argued that there were good

theoretical and empirical reasons to think that rough surfaces were statistically self similar,

and thus amenable to a fractal treatment. Specifically, Majumdar and Tien proposed using

the Weierstrass-Mandelbrot (W-M) function to model the profiles of rough surfaces:

z(x) = A(D−1)∞∑

n=n1

cos 2πγnxγ(2−D)n

(2.2)

Where the parameters D and γ are constrained such that 1 < D < 2, and γ > 1.

eW-M function is an infinite sum of trigonometric functions, similar to a Fourier series.

e parameter γ determines the relative periods of each of trigonometric function in the sum.

Majumdar and Tien set it to 1.5 to ensure that none of these individual periods will overlap,

thus guaranteeing that the full function z(x)will be non-periodic. e other three parameters,

D, A, and n1, can be used to characterize the roughness of surfaces. D is the function’s fractal

dimension, A is a characteristic length which varies with the surface, and n1 is the low cut-off

frequency of the profile, which is related to the length of the sample being characterized.


-1.5

-1

-0.5

0

0.5

1

1.5

1 1.2 1.4 1.6 1.8 2

x

(a) D = 1.1

-1.5

-1

-0.5

0

0.5

1

1.5

1 1.2 1.4 1.6 1.8 2

x

(b) D = 1.3

-1.5

-1

-0.5

0

0.5

1

1.5

1 1.2 1.4 1.6 1.8 2

x

(c) D = 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

1 1.2 1.4 1.6 1.8 2

x

(d) D = 1.7

Figure 2.1: Several Weierstrass-Mandelbrot functions with varying values for D, all withA = 1, γ = 1.5, and n1 = 0.

e most important parameter for our purposes is the fractal dimension D, because it

concisely captures the “jaggedness” of the microscopic profile. Figure 2.1 shows graphs of

several W-M functions generated with increasing values of D. As D increases, the profile

becomes more jagged, representing a rougher surface. D therefore determines the larger-

scale frictional forces experienced by the surface.

A concrete example of the fractal approach to rough surfaces is provided by Chen et al.

(2009)’s study of fluid flow through microchannels. In most everyday fluid mechanical con-

texts, the roughness of a channel wall has no effect on fluid flow. For example, when water

flows through a large copper pipe, the smoothness of the pipe’s interior makes no difference

to the water’s flow speed. However, if we were to shrink the pipe, its surface roughness would

start to havemore of an effect on thewater’s movement. When the channel width decreases to

the “microchannel” range of 10 µm to 3mm, surface roughness can have a very large effect on

fluid pressure. When experimental measurements are performed on fluid flowing throughmi-

crochannels, much larger pressure drops are recorded than the standard, roughness-ignoring

models predict.


Scientists responded to this mismatch between theory and experiment by incorporating

the roughness of the microchannel surfaces into their models. Aempts prior to Chen et al.

(2009) modelled microchannel surface roughness in the “classical” way, as per Whitehouse

and Greenwood, as a random variation from a level mean. e main parameter used to clas-

sify surfaces under this scheme is the root-mean-square peak height σ, which determines the

height and depth of the peaks and valleys. However, experiments showed that microchan-

nel surfaces with the same values of σ could generate very different pressure drops. ese

“classical” models were, therefore, empirically inadequate.

Motivated by these problems, Chen et al. proposed to model the rough surfaces of mi-

crotubes as fractals instead. ey used the W-M equation and focused in particular on what

happens when the control parameter D, the fractal dimension, is varied. A higher D corre-

sponds to a more jagged surface, with more and sharper peaks on peaks (on peaks), and a

lower D corresponds to a gentler and smoother-looking curve (see Figure 2.1).

Chen et al. argue that, while σ is undoubtedly relevant, the fractal dimension D also

gives a useful characterization of the channel surface. ere is oen a correlation between

the measured pressure drop and σ, but Chen et al. argue on both numerical and experimental

grounds that the pressure drop also varies with D. Furthermore, since σ encodes the average

height of the profile peaks and D determines how jagged those peaks are, two surface profiles

with the same σ can have different values of D. D therefore contains “information” not

included in σ, and numerical simulations indicate that if we hold σ constant we can generate

different pressure drops by varying D.

In order to apply this new model to a physical surface, we need to determine suitable val-

ues of our parameters. First we measure the surface profile, which will allow us to calculate

the mean height of the surface roughness σ as before. en we perform a Fourier transform

on the measured profile which tells us, roughly, which sine curves of what height we would

need to sum up to produce the original profile. e heights of these sine curves give us the

profile’s power spectrum. D can either be determined through a direct comparison of the

2.5. FRACTAL MODELS AS INSTRUMENTS 69

profile’s power spectrum and that of the W-M function, or through more involved mathe-

matical methods. e low cut-off frequency n1 is derived from the sample length L through

the relation n1 = 1/L, and the scaling constant G is fixed by the parameters σ, D, and n1.

e details are unimportant here; the point is that the process of constructing a fractal model

for a specific surface requires no measurements beyond those needed by the classical model,

and takes only a few extra calculations.

Chen et al. (2009) also demonstrate that their fractal model gives good predictions. To

validate their models, Chen et al. refer to experimental results from an earlier paper by Pfund

et al. (2000). Pfund et al. performed experiments on microchannel fluid flow, and tried to

account for their measurements using the classical approach. Chen et al. demonstrate that

the new fractal model not only matches the experimental data to within the limits of error,

but it actually provides a closer fit to the data than Pfund et al.’s classical models. is is a

clear case of a fractal model not only working, but actually working beer than a non-fractal

rival.

Perhaps unsurprisingly, this new fractal approach to characterizing surface roughness has

generated controversy. Some engineers have expressed skepticism about the utility of fractal

models. Indeed, Greenwood and Whitehouse themselves have argued that in some cases

there are reasons to think that fractal models should provide poor predictions (Whitehouse

2001; Greenwood 2002). Since Whitehouse and Greenwood’s complaints predate Chen et al.’s

model, it is fair to say that this dispute is far from seled. Unfortunately, a full discussion of

the scientific and sociological factors at play would take us far afield, and so must be set aside.

Our focus here is on the instrumental virtues of fractal tribology, to which I now turn.

2.5 Fractal Models as InstrumentsI wish to argue that Chen et al. (2009)’s specific model of microchannel walls, based on

Majumdar and Tien (1990)’s approach to modelling rough surfaces as fractals, is a good instru-

mental model. Criteria for theory and model evaluation are notoriously thorny, but here I will

2.5. FRACTAL MODELS AS INSTRUMENTS 70

consider three minimal conditions I take to be jointly sufficient for being a good instrumental

model. First, the model must not be intended to be interpreted fully realistically, in the sense

outlined above in Section 2.2, since otherwise the model would not be instrumental for our

purposes and the discussion would be moot. Second, the model must be predictively accurate

enough to be useful in its intended domain; and third, it must satisfy pragmatic requirements

like being simple to use. I take these last two conditions to be the marks of a good instrumen-

tal model because instrumentalists are concerned with easily generated accurate predictions.

Any model that is unusable, or gives unreasonable predictions, could scarcely be considered

good or useful for its intended purpose.

First, Majumdar and Tien’s introduction to the fractal approach to modelling surfaces

made it clear that it was intended for instrumental purposes, and these fractal models were

not intended to be given a fully realistic interpretation. Real fractals exhibit either true or

statistical self-similarity over all length scales. Physical surfaces, of course, cannot be self-

similar on all length scales, since any statistical paerns will break down on the atomic scale.

Majumdar and Tien explicitly recognized this limitation of their model, but proposed that for

engineering studies, the perfect continuity of maer was a valid assumption (Majumdar and

Tien 1990, 314). e emphasis on engineering, with its focus on prediction and control, and

the explicit recognition of the nonphysical assumptions underlying the models suggest that

these models were not intended to be given a completely realistic interpretation.

Second, these fractal models can be predictively accurate. As Chen et al. (2009) demon-

strated, fractal models can predict the pressure drop a fluid will experience when travelling

through a rough microtubule. ese predictions were accurate to within experimental error.

Furthermore, the fractal model gave predictions markedly beer than the classical model.

ird, fractal models of surface roughness have several pragmatic virtues. Although the

W-M equation is an infinite sum and somewhat mathematically unwieldy, it is also simple in

that it has very few control parameters. is, in fact, was one of the major points Majumdar

and Tien presented in favour of fractal models, since the difficulty of accurately represent-

2.6. FRACTALS OR PRE-FRACTALS? 71

ing rough surfaces had led to an overabundance of variables that they termed a “parameter

rash” (Majumdar and Tien 1990, 314). Reducing the required parameters to D, σ, A, and n1

helps keep the models easy to use, especially since each of these variables can be calculated

without too much trouble. A and n1 are trivially related to the length of the sample being

profiled, which can be measured simply enough, and D and σ are both easily extracted from

the profile itself. No specialized techniques are required except for determining D from the

power spectrum of the profile, which is easy to do with a computer. Fractal models of surface

roughness are therefore intended to be used instrumentally, are predictively accurate, and are

pragmatically simple to use. ey are good instrumental models.

As an aside, note that it would also be natural to read the classical approach to surface

roughness, wherein the surface profile is treated as a stochastic signal, as instrumental. e

mathematics of stochastic signals may be useful for encoding statistical information about

a surface’s profile, but we do not believe that the profile actually is a stochastic process in

some ontological, indeterministic sense. Nor do we believe that a physical surface can be

completely and accurately characterized as such. e fractal model, from this point of view,

is not much different from the classical model.

2.6 Fractals or Pre-Fractals?An opponent of fractal models might still wonder whether it is necessary to use full frac-

tals, or whether we could get away without their metaphysically objectionable infinite intri-

cacy. It might be argued that the reference to fractals is eliminable, and we should instead

model rough surfaces as pre-fractals, finitely complex shapes which are self-similar on only

a finite number of scales. Empirically, aer all, the dimension D is determined by looking

at a profile’s power spectrum over a finite scale, and there are strong reasons to believe D

will not apply outside of that scale. Philosophically, one might worry that the fractal model

is somehow less simple than a pre-fractal model, either because the true fractal introduces

infinitely many details, or because the W-M function involves an infinite sum. Surely, one

2.6. FRACTALS OR PRE-FRACTALS? 72

might argue, we would be beer off with a more empirically and philosophically justified

pre-fractal model.

However, the empirical part of this argument is misdirected. e model is not, aer all,

intended to provide predictions about the micro-structure of surfaces, but about how fluids

flow past these surfaces. It is therefore entirely beside the point that the physical surface

may not have a fractal dimension D defined for all length scales. Once we shi our focus to

predictions about fluid flow, rather than surface structure, the instrumentalist may reply by

reemphasizing how successful the model is in this domain.

Furthermore, it is not clear that moving to a pre-fractal model will really remove non-

empirical fine structure. If our model used only the first ten terms of the W-M equation

instead of the full infinite expression, it would aer all still be continuous. It would still impose

a smooth and continuous structure on what is ultimately a discrete and atomic surface. is

surplus structuremay be less interesting than fractal structure, but it is there. Such a prefractal

W-M model should therefore be just as objectionable as the original fractal W-M model. A

discrete and atom-friendly model might in principle overcome these problems, but there is

currently no such model on offer.

Finally, the objection that pre-fractal models are preferable because they are simpler will

fall on deaf ears, because every indication is that the model’s pragmatic simplicity comes from

its full fractal structure. It is common for a mathematical problem to be simpler in the infinite

case. We could, aer all, replace derivatives and limits with finite operations, but we do not

because the infinite case is simpler. We could replace differential equations with finite-step

difference equations, but again we oen do not because the infinite case is easier to workwith.

e situation is not much different with our fractal model. On an experimental level, assum-

ing that the fractal structure continues at all levels allows us to estimate D from a finite data

set without having to observe a surface at all levels of magnification. On a theoretical level,

features like the fractal dimension D, which play a key roles in the generating predictions,

are defined only for full fractals. e W-M equation’s infinite sum is also exploited when de-

2.7. CONCLUSION 73

riving its autocorrelation, which is then used to find its power spectrum (Majumdar and Tien

1990, 316). With some ingenuity, I am sure that we could come up with a discrete pre-fractal

model that would be reasonably empirically successful—but if the fractal model is empirically

successful, simpler to use, and bere of physical interpretation anyway, why bother? Using

the full fractal, with its infinite intricacy and infinite W-M sum, is instrumentally pragmatic.

ese models also go beyond a mere “zoology of interesting specimens and facile clas-

sifications” (Kadanoff 1986, 7). Certainly we are classifying interesting surface specimens as

particular types of W-M-fractals, but this classification allows us to do real predictive work.

Does this, however, constitute the “physics of fractals” Kadanoff demanded? If by a “physics

of fractals” Kadanoff has some realist commitments in mind, then obviously an instrumental-

ist can never have such a physics. If, however, he were to lower his metaphysical standards

slightly, then perhaps the answer could be yes. ere is no denying that rather than just as-

sembling fractal-looking objects into groups, these models actually use fractals as scientific

tools. Whether or not we have a physics of fractals, Chen et al. (2009) have certainly put

fractals into engineering—and perhaps even into physics as well.

2.7 ConclusionIn this section I have argued that fractals can be good instrumental models of material

objects, and that fractal representations of material objects are both philosophically and sci-

entifically respectable. I argued that the main objections to the use of fractals in scientific

models are based on realist assumptions unacceptable from a strictly instrumentalist point

of view. I then presented a case study of a class of fractal representations, from the study of

rough surfaces, and a particular model of fluid flow through small rough channels. I argued

that this model is an unambiguous case of a fractal characterization of nature serving as a

legitimate scientific tool. First of all, it is empirically adequate, since the predictions given

fall within the error limits of experimental results. Second, the fractal characterization allows

the construction of a model, where in this case the surface profile is modelled according to

2.7. CONCLUSION 74

theW-M equation. ird, this model allows us to make predictions: given two microchannels

with the same relative roughness, we can predict that the one with the larger fractal dimen-

sion will experience a greater pressure drop. Fractal representations work, and whether or

not we have a physics of fractals, there are fractals in—or at least very near to—the practice

of physics.

.. 3Fractal Geometry is a Geometry of

Nature

A physics of fractals cannot then be born, because fractalgeometry is not the geometry of nature.

Orly Shenker, Fractal Geometry is not the Geometry of Nature,p.980

3.1 IntroductionCould there be fractals in nature? e consensus in the philosophical literature is “no”

(Copeland 1994; Shenker 1994; Smith 1998b; Saunders 2002), a view Orly Shenker summa-

rized in the slogan “fractal geometry is not the geometry of nature.” Here I will challenge this

orthodoxy: I will argue that the published objections to natural fractals are not definitive, and

then provide positive arguments for the possibility of fractals in nature. ere are, it turns

out, reasons to think that we need fractal geometry to describe aspects of many dynamical

systems. First, it is common for dynamical systems to have fractals in their phase spaces,

which encode objective dynamical properties. Furthermore, I will argue that there are rea-

sons to think that purely spatial features of dynamical systems might also be fractals. us,

my thesis is that fractal geometry could be a geometry of nature.

75


e structure of the paper is as follows. In Section 3.2 I give a working definition of the

word “fractal” and an intuitive example to motivate the discussion. In Section 3.3 I introduce

and contest the two main arguments against the fractal geometry of nature. e first argu-

ment, which I will call the constructivist objection, denies the possibility of natural fractals

on the grounds that fractals cannot be constructed in any finite amount of time. e sec-

ond argument, which I will refer to as the atomic objection, concerns the incompatibility of

infinitely intricate fractal shapes with the atomic theory of maer. Proponents of this objec-

tion are correct that atomic objects cannot have essentially non-atomic properties. However,

the atomic objection does not rule out the existence of fractal regions of a dynamical system’s

phase space. Nor does it eliminate the possibility that fractal geometry might be necessary

to characterize scientifically interesting fractal regions of physical space. In the remainder of

the paper I explore these possibilities.

To develop the idea that fractal geometry could apply to nature, in Section 3.4 I consider

what it means to be a “geometry of nature.” I argue that limiting this concept to atomic objects,

as the atomic objection seems to, does not do justice to scientific practice or our intuitions.

In Section 3.5 I then argue that fractal geometry is necessary to characterize the phase-space

basins of araction of many abstract dynamical systems. In Section 3.6 I introduce the notion

of a dynamical system’s spatial basins of araction, defined roughly as initial spatial conditions

which lead to particular long-term behaviours. I then use analytic and numerical results to

argue that the boundaries between these spatial basins of araction can be fractals. It is

difficult to prove the fractal nature of basin boundaries in physically possible scenarios, and

the models I present are necessarily idealized, but these results make it plausible that purely

spatial features of physical systems can be fractals. I conclude that, contrary to the received

wisdom, it is possible that fractal geometry could apply to nature.

3.2. WHAT IS A FRACTAL? 77

3.2 What is a Fractal?What is a fractal? Mathematicians have proposed many definitions of the term “fractal,”

but each of these definitions face serious challenges (e.g., Devaney 1992; Banks et al. 1992;

Baerman 1993). It is a maer of some controversy whether a single precise definition is

even possible (Falconer 2003, xxv). e reader is referred to Appendix A for further details.

Fortunately, we can get by here with only an intuitive notion of what it is to be a fractal.

For our purposes, we will call something a fractal if it is infinitely intricate, which is to say

that when examined more closely it reveals more and more fine details without end. Fractals

are therefore enormously complex in one sense, but can oen be generated by simple recur-

sive algorithms. Recursively generated fractals are oen self-similar, which means that the

structure of the whole can be found replicated, sometimes with distortion, on smaller scales.

Sometimes the resemblance is only statistical, in which case the fractal exhibits statistical self

similarity (Falconer 2003, 246-7).

Two examples may help the reader to get an intuitive handle on this. A chessboard, for

example, is not infinitely intricate in the present sense. From a distance it might appear

grey, but as we approach the board we will see alternating black and white squares. Upon

magnification, we might say, we perceive some additional intricacy in the chessboard. But

when we look even closer, these squares yield no further details: the black and white regions

are coloured solidly no maer how closely we look, and the borders between the squares are

straight and simple. ere is, in other words, a finite amount of intricacy in a chessboard’s

paern.

To get an idea of what infinite intricacy might look like, consider the following construc-

tion process. Say we begin with a straight line 1 unit long (Fig. 3.1a). If we remove the middle

third section of this line and replace it with a tent shape of the same side length as the piece

we removed, we get a line with a kink in the middle (Fig. 3.1b). is process can be repeated

on each smaller line segment, every time removing its middle third and replacing it with a

tent. If we started at step 0, at each step n we have 4n line segments of length 3−n. As we

3.3. TWO STANDARD OBJECTIONS 78

(a) K0 (b) K1

(c) K2 (d) K5

Figure 3.1: A series of curves converging to the Koch curve. Figure 3.1a is a straight lineK0, corresponding to the 0ᵗʰ iteration. Figure 3.1b shows the line K1 aer one iteration,figure 3.1c shows the result K2 aer two iterations, and figure 3.1d shows the Koch curveK5 aer five iterations.

repeat the process the whole figure gets crinklier and crinklier (Figs. 3.1c and 3.1d). Glossing

over some technicalities, in the infinite limit we have an infinitely crinkly line that is infinitely

long, continuous, and nowhere differentiable. is shape is called the Koch curve, and it is a

paradigmatic fractal. No maer how much we zoom in on it, we will always reveal more and

smaller kinks in the line, forever without end: it is infinitely intricate.

e question of whether there could be fractals in nature then becomes, for our purposes,

the question of whether anything in nature could be infinitely intricate in the way the Koch

curve is. In a moment I will argue that this is indeed the case, but first we will consider two

standard and influential objections to natural fractals.

3.3 Two Standard Objections

3.3.1 e Constructivist ObjectionOrly Shenker (1994) gives a clear and forceful presentation of the constructivist objection

to fractals in nature. Shenker adopts a constructivist stance, and posits that only objects con-

structible in finite time exist. Fractals, she continues, are different from other geometrical

entities like points and lines, since fractals’ infinite and non-terminating construction pro-

cesses cannot be completed in finite time. Since fractals cannot be finitely constructed, and


by assumption only finitely constructible objects exist, Shenker concludes that fractal objects

do not exist. True fractal objects are, according to this view, a mathematical impossibility,

and so their physical impossibility follows as a maer of course. Talk about fractals as objects

should therefore be taken as elliptical for talk about their generating processes.

Shenker’s objection relies crucially on a constructivist metaphysics according to which

only objects constructible in finite time exist. is position is known as strict finitism. Since

strict finitism represents an extreme form of constructivism, anyone with a more permissive

aitude toward mathematical objects will likely not share Shenker’s fractal scruples. Shenker

herself provides no reasons to think that nature contains only finitely constructible objects,

and indeed prima facie there is no connection between finite constructibility and physical ex-

istence. Many would, I suspect, find it difficult to accept that the universe could not contain

an infinite number of particles simply because no human could ever count them all. In ad-

dition to these intuitive objections, strict finitism may not even be internally consistent (e.g.,

Dumme 1975). ese factors may lead some readers simply to reject strict finitism, in which

case the constructivist objection loses its force.

On the other hand, if we do accept strict finitism, then we will probably have to reject

much more than fractals. For instance, the kinds of numbers that exist according to strict

finitism are very restricted. Shenker says only rational numbers exist, since the rest of the

real numbers are not finitely constructible (Shenker 1994, 971). is poses a problem since, at

least prima facie, almost all of modern sciencemakes use of the rest of the real numbers, either

through calculus, differential equations, or irrational values. Rejecting the real numbers and

infinity may also lead us to reject simple continuous shapes like lines and curves, since on a

naïve view a line contains uncountably many points. It’s true that we might be able to fix this

naïve view, for example by saying that lines are finite objects and denying that they are made

of points. We might also be able to recast physics so that it does not use calculus. is might

be feasible, but it amounts to reframing all of science on a strict finitist footing. As such,

the constructivist objection goes well beyond the question of the acceptability of fractals and


implies that we have to completely reframe much of modern science as it stands.

We might hope to rescue the spirit of the constructivist objection by dropping finitism

and accepting in-principle constructibility. is would bring the project into line with more

traditional forms of constructivism, but brings infinity and the real numbers back into the

picture. Bishop and Bridges (1985)’s influential textbook on constructivist mathematics, for

example, admits in-principle constructibility, and contains a constructivist definition of the

real numbers (Bishop and Bridges 1985, 17,18); furthermore, these constructivist real num-

bers are uncountable (Bridges and Richman 1987, 20-1). What might be considered one form

of ‘mainstream constructivism’ is therefore consistent with both countable and uncountable

infinities.

is is a problem for Shenker, because we can use infinity and in-principle constructibility

to perform infinite construction processes that lead to fractals. e Koch curve, to take one

example, can be defined as the convergent limit of a Cauchy sequence of continuous functions

on a complete metric space (Edgar 1992, 60-4). Since each of these concepts is well defined in

the constructivist programme, the Koch curve is constructively acceptable (see the first proof

in Ch. 2 Bridges and Richman 1987).e upshot is that if we accept a moderate constructivism

instead of strict finitism, some fractals are constructible aer all.

e constructivist objection can therefore be challenged on three fronts. First, many will

find its base metaphysical assumptions unacceptable, since they imply that only finitely con-

structible objects are physically possible. is entails, for example, that the universe is nec-

essarily finite, spatially discrete, and contains only finitely many objects. Even if, however,

we approach the objection in a constructivist spirit, we encounter a dilemma. If we reject

fractals as on the basis of their finite unconstructibility, then we must also reject—or at least

completely reconstruct—a good deal of modern science. A strict finitist reframing of science

may be possible, but this raises challenges that go well beyond the acceptability of fractals.

If, on the other hand, we drop strict finitism and embrace a more traditional constructivism,

then fractals are constructible. In sum, there are metaphysical and scientific reasons to re-

3.4. WHAT IS A ‘GEOMETRY OF NATURE?’ 81

ject strict finitism, and more moderate varieties of constructivism permit fractal construction

processes.

3.3.2 e Atomic Objectione atomic objection to natural fractals is based on the fact that material objects are made

of atoms. Fractals are, we have assumed, infinitely intricate, and display fine detail at all

levels of detail. Anything made of atoms, however, cannot have this kind of intricacy. ings

like maple trees may have fractal-like self-similar paerns at the macroscopic scale, as their

branches split into smaller branches which themselves split into twigs. But with material

objects, these paerns will not continue past a certain level; twigs, for example, grow leaves,

which don’t grow anything. In general, any fractal-like paern in a material object will stop

at the atomic level. Indeed, if to be a fractal is to be infinitely intricate, and to be atomic is to

be finitely intricate, then an atomic fractal is a logical contradiction. erefore, this objection

concludes, the notion of fractal objects in nature is contradictory. Variations on this line of

reasoning can be found in Shenker (1994, 978-9), Smith (1998b, 32-3), and Saunders (2002,

201-2).

While this argument is straightforward, it establishes only the logical impossibility of

atomic fractals. e objection therefore leaves open the possibility that fractals in nature

might exist in other forms. In the rest of this paper I will consider this possibility.

3.4 What is a ‘Geometry of Nature?’Arguments about natural fractals are frequently framed in terms of the geometry of na-

ture. But what is it to be a geometry of nature?

In one sense of the term, the geometry of nature might be the overall structure of space-

time. On this reading, different theories propose different geometries of nature. For example,

classical physics, general relativity, and quantum mechanics each impose different geometri-

cal constraints on what the world could be like. But fractal geometry could not be a geometry

3.4. WHAT IS A ‘GEOMETRY OF NATURE?’ 82

of nature in this sense, because a fractal is an object in a geometrical space, not a kind of geo-

metrical space. We can certainly define fractals in Euclidean or Riemannian spaces, but they

can also be defined in highly abstract metric spaces with lile or no intuitive connection to

nature (Edgar 1992). Rather than a type of geometrical space, fractal geometry is a set of tools

and techniques for working with objects in various geometrical spaces. It would therefore be

a category error to suppose that fractal geometry was a serious rival to Euclidean or Rieman-

nian geometry, or that it could give the overall structure of space-time. To avoid ambiguity,

I will use the term “fractal geometry” to refer to a set of tools and techniques which can be

applied in any number of geometrical frameworks.

Perhaps, on a less grandiose scale, to be a geometry of nature is to characterize objects

in and regions of geometrical spaces that correspond to nature. is makes it plausible that

more than one set of tools might be necessary to capture the totality of nature. An ellipse, for

example, is not a straight line, and neither can an ellipse be constructed from a finite number

of lines; but, to paraphrase Shenker’s anti-fractal thesis, the elliptical orbits of the planets do

not establish that ‘straight-line geometry is not the geometry of nature.’ Nor can an ellipse

be constructed out of discrete atoms, but the atomic nature of maer does not show that

‘elliptical geometry is not the geometry of nature.’ e tools of fractal geometry, similarly,

are inappropriate to deal with discrete atomic objects, but surely there is more to nature than

the brute existence of material objects and the arrangements of their constituent parts. At

the very least, these objects have positions and velocities; indeed, according to nonlinear

dynamics, these objects also have modal properties encoded in their phase spaces.

For fractal geometry to be a geometry of nature, on this reading, is for the tools of fractal

geometry to be capable of accurately capturing some of these other aspects of nature. It is for

there to really be elements of nature that can only be captured using fractal geometry. If, in

other words, there were fractals in phase-space or spatial properties of systems, then perhaps

fractal geometry could be a geometry of nature aer all.

3.5. FRACTALS IN PHASE SPACE 83

3.5 Fractals in Phase SpaceIn this section I will consider whether some aspects of nature, specifically structures in

phase space, might be fractals. I will begin by introducing the notion of phase space, and then

introducing phase space basins of araction. I will then present a class of dynamical systems

called complex iterated maps whose phase space basins of araction are oen fractals. Since

complex iteratedmaps are somewhat abstract, I will then present a model of a more physically

realistic system. Specifically, I will examine a driven particle in a two-well potential, whose

phase space basins of araction also have fractal boundaries.

A phase space is an abstract space, associated with a dynamical system, which may en-

code information about abstract mathematical and physical systems as well as real physical

systems. Each axis of the phase space corresponds to one of the system’s dynamical variables.

As a dynamical system evolves, it traces out a path through phase space called a trajectory.

Dissipative dynamical systems will tend to approach certain steady states: a clock, for exam-

ple, will eventually wind down and stop, and a pendulum driven at a certain frequency may

sele into a steady rate of swing. Since systems tend to approach these states, their trajecto-

ries in phase space will tend toward the corresponding points. ese aracting sets of points

are appropriately called aractors, and the set of points whose trajectories tend toward an

aractor are called the aractor’s basin of araction.

Basins of aractions can be very simple, but they can also have complex and even frac-

tal structures. In a classic paper, McDonald et al. (1985) identified and investigated several

types of fractal basin boundaries. In the years since, scientists have expanded their classifi-

cation schemes to include fractal basins with names as exotic as their properties, including

riddled basins, intermingled basins, and wada basins (e.g., Aguirre, Viana, and Sanjuán 2009;

Alexander et al. 1992; Camargo, Viana, and Anteneodo 2012).e technical details are beyond

the scope of the present discussion, but fractal basin boundaries are common in dynamical

systems (Lai and Tél 2010, 147).

Some of the simplest systems with fractal basin boundaries are complex iterated maps.


Figure 3.2: e black and white regions are the basins of araction of the iterated complexmap zn+1 = z2n − 0.5 − 0.5i. e border between them is a Julia set, a fractal basinboundary. Image by the author.

Consider what happens to points on the complex plane as we iterate the map f (z) = z2 + c,

where c is a fixed complex number with absolute value less than one. If we let f n(z) denote

the nᵗʰ iteration of f on z, for certain values of z, f n(z)will grow without bound. We say that

these points approach an aractor ‘at infinity.’ For other values of z, f n(z)will always remain

bounded and approach some stable point or periodic orbit. Such systems will therefore have

at least two aractors, and the boundary between their finite and infinite basins of araction

will typically be a continuous but nowhere-differentiable fractal curve (McDonald et al. 1985,

150). Figure 3.2 shows an example. In special cases the basin boundary can be determined

analytically and proved to be a fractal, but this is difficult at the best of times, and impossible

at other times (McDonald et al. 1985, 144-6). Despite their abstract nature, complex iterated

maps provide good examples of simple dynamical systems with fractal basin boundaries.

We can also find fractal basin boundaries in more realistic physical models. Consider,

for example, the case of a particle in a two-well potential subjected to periodic forcing. is


system is described by the following equations of motion:¹

dxdt = y

dydt = −γy + 1

2 x(1 − x2) + f0 cosωt(3.1)

Here γ is a damping factor, ω is the driving frequency, and f0 is the amplitude of the driving

force, which we take to be small. is system has two equilibrium points at x = ±1, and it

has two aractors corresponding to periodic motions about these points (Moon 1992, 379). If

we fix γ, ω, and f0, all points on the plane will tend toward one of these two aractors, and

so the system has two basins of araction.

e border between the two basins of araction will change depending on the system’s

parameters. In the undriven case, where f = 0, the boundaries are smooth and regular (see

Figure 3.3a). If the system is driven a small amount, the boundaries will deform slightly but

remain smooth. However, as we drive the system harder, the boundary between these two

basins will become a fractal. e reason, in brief, is this. e boundary itself can be shown to

be identical to the stable manifold of the system’s Poincaré map. At a certain critical value f0

the stable manifold and unstable manifold of the Poincaré map touch, forming a homoclinic

point—roughly, a point which both aracts some trajectories and repels others. e existence

of one homoclinic point entails the existence of an infinite number of them, and so above this

critical value the two manifolds must touch an infinite number of times. is “results in an

infinite folding of the stable manifold and hence an infinite folding of the basin boundary

and the resulting fractal properties” (Moon 1992, 382). e two basins are folded around each

other an infinite number of times, creating an infinitely intricate boundary in phase space.

While this purely theoretical result establishes the fractal nature of the boundary, it does

not give us an analytic expression for it, or tell us exactly what it will look like. To get

an idea of the shape of these boundaries, we can use a computer to perform a numerical

¹For further details, see Moon (1992, 377—85) and Moon and Li (1985).


(a) f = 0, −2.5 < x < 2.5, −1.5 < y <1.5

(b) f = 0.159, −2.5 < x < 2.5,−1.5 < y < 1.5

(c) f = 0.159, 0.280 < x < 0.380,−0.384 < y < −0.348.

Figure 3.3: Phase space basins of araction for a particle in a double potential well subjectto harmonic forcing of period ω = 0.833, resistance γ = 0.1, and varying f . e x-axisis position, and the y-axis is velocity. Trajectories beginning from black points approachthe aractor at x = 1, and those from white points approach the aractor at x = −1.Boundaries rounded to three decimal places. Images by the author.

simulation.² To do so, we divide the plane up into a grid of initial conditions. Following Moon

and Li (1985), we then use a fourth-order Runge-Kua integration algorithm to compute the

system’s evolution from each of these initial conditions. We evolve each initial condition for

approximately five driving periods, and then determine whether the it is orbiting the positive

²Simulation created by the author.

3.6. FRACTALS IN SPACE 87

or negative fixed point by considering a time average of x(t)’s recent history. If the system

is orbiting the negative point we colour the initial grid location white; if it is orbiting the

positive point we colour it black.

e results of this simulation can be seen in Figure 3.3. Figure 3.3a shows the non-driven

case, and the border between the two basins of araction is smooth. Figure 3.3b shows a more

complicated case where the system is driven with amplitude of f = 0.159, which is well over

the system’s critical value. e basin boundaries are convoluted and intricate. Figure 3.3c is

an enlargement of a small phase-space region, and we can see that the striations between the

black and white regions continue on a smaller scale. ese images are approximations limited

by computing power, and so will not show true fractal detail on all levels of magnification.

Still, they provide a vivid illustration of the theoretical justification for this system’s fractal

basin boundaries.

e example of the driven double-well particle shows that physically plausible models can

have phase-space basins of aractionwith fractal boundaries. Fractal geometry is therefore an

appropriate tool for characterizing this system’s behaviour, and if such a systemwere actually

instantiated, fractal geometry would characterize some aspects of nature. Smith and Shenker

might, given that their primary concern was with fractals in physical space, concede the

existence of phase-space fractals but deny that these fractals are, in a substantial sense, a part

of nature. However, phase space does encode information about the physical properties of

objects, including their potential behaviour in given circumstances. If there could be fractals

in phase space, and if the properties encoded in phase space are part of nature, then fractal

geometry could be a geometry of nature.

3.6 Fractals in SpaceI would now like to go further and argue that we may find fractals in certain purely spatial

aspects of nature. To circumvent the atomic objection, I will argue that there might be scien-

tifically interesting regions of space that can best be characterized using the tools of fractal


geometry. e regions I have in mind are a system’s spatial basins of araction, which I define

below. I will then argue that a particle in a double-well potential has one-dimensional fractal

spatial basins of araction. I will present the more colourful example of a magnetic pendulum

swinging over several fixed magnets, which exhibits fractal spatial basin boundaries in two

dimensions. e models I present are idealizations of physical phenomena, but they make it

plausible that there are physically possible systems with fractal spatial basins of araction. I

will conclude that there could be actual fractal-bounded scientifically interesting regions of

physical space.

In the previous section we considered a dynamical system’s phase space basins of arac-

tion, defined roughly as sets of phase space points which tend toward particular aractors.

Here I want to consider spatial basins of araction, defined roughly as the set of spatial points

from which, for specific values of the remaining dynamical variables, a system will tend to-

ward particular aractors. If a phase space basin of araction tells us for which values of

position and velocity a system will approach an aractor, a spatial basin of araction tells us

for which values of position, given some velocity, a system will approach an aractor. Since

a phase space will in general have both spatial and dynamical axes, this amounts to consid-

ering its lower-dimensional subspaces with purely spatial axes. We can think of this process

as taking ‘slices’ of the phase space corresponding to particular dynamical initial conditions.

Here I will consider cases where the system begins from rest.

To see this in action, consider the undriven particle in a double-well potential from the

previous section. Its full phase-space basins of araction are shown again in Figure 3.4a, and

Figure 3.4b shows its purely spatial basins of araction for initial velocity zero. One can see

it as the ‘slice’ of Figure 3.4a along the line y = 0. e spatial basin of araction tells us

for which values of x the system will approach each aractor when released from rest. e

basins in this case are regular line intervals.

e driven case can be very different. Recall that when the particle is subjected to a

periodic driving force above a certain amplitude, it develops a homoclinic point which leads


(a) f = 0, s −2.5 < x < 2.5, −1.5 < y <1.5

(b) f = 0, −2.5 < x < 2.5, y = 0

Figure 3.4: Phase space and spatial basins of araction for a particle in a double potentialwell subject to harmonic forcing of periodω = 0.833, resistance γ = 0.1, and f = 0.159.e x-axis is position, and the y-axis is velocity. Trajectories beginning from black pointsapproach the aractor at x = 1, and those from white points approach the aractor atx = −1.

to an infinitely intricate intertwining of its stable and unstable manifolds, and likewise of its

basin boundary (see Figure 3.5a). If we determine the spatial basin of araction of a driven

systemwith zero initial velocity, we again get what appears to be a collection of line segments

(see Figure 3.5b). However, these line segments are a good deal more irregular than those of

Figure 3.4b, and they appear to have more fine structure. If we magnify the region in the red

box, we see in Figure 3.5c that this is indeed the case, and further magnification (Figure 3.5d)

shows yetmore fine structure. e spatial basins of araction thus appear to form an irregular

Cantor-style fractal, where the interval is divided into finer and finer regions without end.

e mathematical explanation for this is based on the infinite basin folding brought about

by the homoclinic point. If the basin boundary is folded back on itself an infinite number of

times, then if the boundary crosses a line once we should expect it to cross an infinite number

of times. is gives rise to the fractal structure of the spatial basins of araction.

What this means is that there is a fractal structure to the spatial properties of a forced

particle in a double-well potential. e spatial basins of araction tell us where a particle will


(a) −2.5 < x < 2.5, −1.5 < y < 1.5 (b) −2.5 < x < 2.5 , y = 0

(c) −2.05 < x < −1.3 , y = 0 (d) −1.501 < x < −1.4275 ,1 y = 0

Figure 3.5: Phase space and spatial basins of araction for a particle in a double potentialwell subject to harmonic forcing of periodω = 0.833, resistance γ = 0.1, and f = 0.159.e x-axis is position, and the y-axis is velocity. Trajectories beginning from black pointsapproach the aractor at x = 1, and those from white points approach the aractor atx = −1. Images by the author.

end up if we release it from rest. But these basins of araction have a fractal structure, and so

in order to fully describe this aspect of the system we need to use fractal geometry. In other

words, we need fractal geometry to describe the relevant regions of space around the particle.

As a final example, fractal basins of araction can also be found in prosaic dynamical


Figure 3.6: A schematic drawing of themagnetic pendulum system. e pendulum (black)swings freely over the three magnets (red, blue, and green) arranged on a plane. AerPeitgen, Jürgens, and Saupe (2004), Fig. 12.72.

systems like the magnetic pendulum.³ Consider an ideal damped pendulum of fixed length

suspended over a flat table, affected only by friction and gravity. In the absence of perturbing

factors, when swung this pendulum would simply move back and forth until frictional forces

brought it to a stop. But if we let the pendulum bob be affected bymagnetic forces, and arrange

threemagnets on the surface of the table, the pendulum’s behaviour changes dramatically (see

Fig. 3.6). If lied and dropped, the pendulum will swing and loop erratically before coming

to rest nearly, but not quite, over one of the magnets. As long as the magnets are strong

enough and near enough to the centre of the plane, this system will have one aractor, and

so one spatial basin of araction, associated with each magnet. is kind of pendulum is

easy enough to construct, and its unpredictable motions have made it equally popular as an

undergraduate physics lab and an executive desktop toy.

Following Tél and Gruiz (2006, 224), we can model the pendulum’s motion as follows.

First, let the pendulum be suspended directly over the origin, and let the variables x and

y represent the displacement of the centre of the pendulum bob in the x- and y-directions

respectively. Without magnets the pendulum will oscillate about the origin at its natural

frequency of ω0 =√

g/l, where g is the force of gravity and l is the pendulum’s length.

We assume the pendulum experiences damping forces linearly proportional to the product

³is example is from Smith (1998b, 58)’s discussion of a separate point.


of its velocity and the coefficient of friction α. We make the simplifying assumption that

the pendulum is very long compared to its angle of swing, so it moves in a horizontal plane

at a constant height d above the plane of the magnets. We also assume that the pendulum

interacts with the magnets as point sources, rather than through more complicated dipole

interactions. For each magnet i of strength γi located at (xi , yi), the distance between the

pendulum and the magnet will be:

Di =√(xi − x)2 + (yi − y)2 + d2 (3.2)

And the equations of motion for the magnetic pendulum in the presence of N magnets are

given by:

x = −ω20x − α x +

i=N∑i=1

γi(xi − x)D3

i

(3.3)

y = −ω20y − α y +

i=N∑i=1

γi(yi − y)D3

i

To understand the magnetic pendulum, we need to introduce the Wada property. e

Wada property applies to three or more open subsets of a region if every point on the bound-

ary of any subset is on the boundary of every subset (Kennedy and Yorke 1991, 213). is

is difficult to visualize, since in everyday cases only a few points on a boundary will border

three or more regions. If we slice a pizza in the usual way, for example, the only boundary

point that touches all the pieces will be right in the middle. Similarly, on a map of Canada

there is only one point that simultaneously borders British Columbia, Alberta, and the North-

west Territories. For three or more subsets of a region to have the Wada property, every point

of their boundaries must border all three. e basins must be intricately intermingled with

a fractal structure. Wada basins are counter-intuitive, but are well studied and have been

proven to exist in many dynamical systems (Kennedy and Yorke 1991; Nusse and Yorke 1996;


Aguirre, Viana, and Sanjuán 2009; Lai and Tél 2010).

emagnetic pendulum system is too complicated to solve exactly, but there are reasons to

to believe that its basins are fractals with the Wada property. First of all, experts believe these

basins are Wada (e.g., Camp 2002, 50). Second, systems whose basins appear qualitatively

similar to the magnetic pendulum can be proven to be Wada (e.g., Lai and Tél 2010, 167-70).

In practice we should expect to see the basins have a Cantor-like structure similar to that of

the double-well particle: what appears at first glance to be a simple boundary becomes, upon

closer examination, two boundaries. As we look more closely, we should find boundaries

between boundaries at all levels of magnification. Nusse and Yorke (1996) give a numerically

reliable way to prove that basins areWada. Actually working through the process is, however,

a formidable task, and beyond the scope of this paper. I have not been able to find a proof

that the magnetic pendulum’s basins are Wada, and such a proof escapes me at the moment.

In what follows I will assume the boundaries are Wada, with the understanding that this

assumption is at best tentative.⁴

Although themagnetic pendulummay not be amenable to analytic solution, we can inves-

tigate it using numerical simulation. In this case, we approximate the evolution of the system

numerically using Beeman integration.⁵ According to the simulation, trajectories started near

an aractor tend to approach it quickly and directly. However, for initial conditions a cer-

tain distance away from each of the magnets, the pendulum’s motion becomes quite complex.

Typical system trajectories wander wildly all over the plane, looping erratically around the

magnets, before eventually seling down near an aractor (see Fig. 3.7). In the short term,

the pendulum’s behaviour seems chaotic. Its behaviour cannot, however, be truly chaotic be-

cause it converges to a simple point aractor rather than a chaotic aractor. is short-term

chaos-like behaviour is called transient chaos (Tél and Gruiz 2006, 224).

More interesting than the trajectories themselves, at least for our purposes, are the mag-

⁴For details consult, e.g., Peitgen, Jürgens, and Saupe (2004, 714) and Tél and Gruiz (2006, 9). Aguirre, Viana,and Sanjuán (2009) gives a discussion of the history of the Wada property and additional references.

⁵Simulation by the author.


Figure 3.7: A characteristic trajectory of a magnetic pendulum swinging over three mag-nets. Here we are looking down on the pendulum from above. e pendulum exhibitstransient chaos as it spirals wildly around the three magnets before seling down nearan aractor.

netic pendulum’s spatial basins of araction. We can use a computer to get a visual rep-

resentation of these basins by considering a grid of initial conditions, evolving each initial

condition forward in time, and then colouring in the corresponding pixel with one of three

colours according to the aractor it approaches. If we take a system with three aractors, for

example, and associate one of the colours red, green, and blue with each aractor, we will get

a picture with red, green, and blue patches corresponding to regions where the pendulum ap-

proaches each of these aractors. is approach gives results which agree qualitatively with

the mathematical literature (e.g., Tél and Gruiz 2006, Plates III-VI). If the magnets are very

weak, or if the damping forces due to friction are very strong, then the basins of araction

tend to be regularly shaped and to have simple boundaries (see Fig. 3.8a). As the coefficient of

friction decreases, the boundaries become much more complex (see Fig. 3.8b). And when the

damping forces are very weak, some of the solid regions from the first scenario remain, but

far from the origin the basins become very convoluted indeed (see Fig. 3.8c). In cases where

the pendulummoves quite wildly, these basin boundaries are more than just convoluted: they


(a) α = 0.5 (b) α = 0.1

(c) α = 0.05

Figure 3.8: Spatial basins of araction for magnetic pendula with varying coefficients offriction. In each case, sources were located at (0.5, 0.0), (−0.5, −0.5), and (−0.5, 0.5),with the pendulum height h = 0.28, w0 := 1.0, Beeman integration time-step dt = 0.03,and image boundaries −3.0 < x < 3.0, −3.0 < y < 3.0. Color brightness representsspeed of convergence.

are fractals.

Our simulation gives some evidence for the fractal Wada structure of these basin bound-

aries. Although this complicated system is not perfectly self similar like the Koch curve, it

does show fine structure upon magnification. As we examine what appears to be a simple

boundary between two regions more closely, we see more complicated boundaries between

boundaries, between boundaries. Figures 3.9a—3.9c show progressive zooms on the whorl


(a) −1.164 < x < 0.360, −2.142 < y <−0.696

(b) −0.417 < x < −0.181, −1.727 < y <−1.530

(c) −0.304 < x < −0.283, −1.647 < y <−1.631

Figure 3.9: Progressive zooms on the whorl above the central red portion of Fig. 3.8c.Boundaries rounded to three decimal places.

over the central red area of Fig. 3.8c. As we zoom in, more and more detail becomes visible,

just as we would expect in a fractal. Fig. 3.9b shows faint alternating bands of red, green,

and blue at a tiny scale. When enlarged further, these bands give way to speckled clouds of

intermingled blue, green, and red points (see Fig. 3.9c). ese isolated basin points should

not be taken literally as features of the system, but as an indication that we are hiing the

“resolution” of our simulation, and that rounding errors and other computational artefacts are

influencing our results. Still, the intermingling of these points suggests that we have not en-


countered a simple boundary, and that the fractal Wada structure continues on down through

all levels of magnification.

ese images are compelling, but not definitive. In the absence of proof, one might object

that we cannot be sure that the magnetic pendulum’s basins of araction are Wada fractals.

is is a fair point, but nothing in my argument hinges on the magnetic pendulum in par-

ticular. I chose the magnetic pendulum as an example because it is easy to visualize and it

already appears in the philosophical literature on chaos (Smith 1998b, 58). ere are, how-

ever, simpler dynamical systems that are provably Wada, and which could be used to make

my point (Aguirre, Viana, and Sanjuán 2009, 341).

Before closing, I should pause to address some possible worries about the many idealiza-

tions in the models I have considered. One might worry that the fractal basin boundaries

could disappear if, in Ernan McMullin’s words, we “de-idealize” these idealized features of

our models (McMullin 1985, 261). In response, I would note first that virtually all models in-

volve idealizations. ere is widespread agreement that to demand a perfect model would be

an unrealistic and perhaps impossible standard (e.g., Teller 2001; Judd and Smith 2004). e

models I have used make idealizations similar to many used in scientific practice; in fact, these

models are actually drawn from scientific practice. is shows that the received view’s whole-

sale rejection of the mere possibility of fractals in nature was much too strong. Furthermore,

the features I have relied on—for example, the continuity of space and homoclinic points—are

quite general, and would likely be found in any de-idealized models as well. However, a fuller

investigation of this point is outside the scope of this chapter.

To conclude this section, the two-well particle and the magnetic pendulum models have

fractal spatial basins of araction. ese basins were one-dimensional in the case of the

two-well particle, and two-dimensional in the case of the magnetic pendulum. If either of

these models were to be instantiated in nature, then fractal geometry would be required to

characterize an important spatial aspect of their behaviour.

3.7. CONCLUSION 98

3.7 ConclusionI’ve argued here that we need to rethink the received philosophical wisdom concerning

the relation between fractals and nature. First, the two main objections to natural fractals, the

constructivist objection and the atomic objection, can be challenged on several fronts. e

constructivist objection assumes a strict finitist metaphysics, and so applies just as much to

the real numbers and infinity as it does to fractals. As such, this objection entails the rejection

or reinterpretation of most of modern science along with fractals. One reason for rejecting

fractals was that they cannot be completely characterized by a finite set of more traditional

shapes; but neither can an ellipse, aer all, be completely characterized by a finite set of

straight lines. e atomic argument coherently outlaws atomic fractals, but it outlaws only

atomic fractals. If to be a “geometry of nature” is to be a set of tools and techniques for char-

acterizing parts of nature, then I argued that we should consider whether fractal geometry

could characterize properties of natural systems other than their atomic structure. I argued,

with reference to three models of increasingly realistic systems, that the tools of fractal ge-

ometry are in fact oen necessary to characterize important regions of a dynamical system’s

phase space. I then showed that a dynamical system’s spatial basins of araction, defined

as the cross-section of their phase-space basins of araction with zero momentum, can have

fractal boundaries. is makes it plausible that fully spatial aspects of nature could also be

fractal.

In this paper I have relied on imperfect evidence in the form of numerical simulations

and idealized models, and so I can only make tentative conclusions about the relationship

between fractals and the natural world. At the very least, a particle ontology of maer is

consistent with scientifically interesting fractal regions of space. Indeed, if we take dynamical

models seriously, such fractal-bounded regions of space may even exist. Whether fractal

spatial basins of araction exist is an empirical question, but since they appear in scientific

models, I believe it is possible—and perhaps even likely—that fractal geometry is a geometry

of nature.

.. 4Chaos and Observational

Equivalence:Manifest Isomorphism andε-congruence Reconsidered

And when ε is chosen at the finite limit of our measurementaccuracy, the Newtonian mechanical and a Markov process modelof a Sinai billiard are observationally indistinguishable, as theyare ε-congruent.

Patrick Suppes and Jose Acacio de Barros, Photons, Billiards andChaos, p. 200.

4.1 Introduction

4.1.1 Baground and OutlineObservational equivalence is a fraught topic in the philosophy of science, and there is lile

agreement as to when—if at all—it holds between scientificmodels. If the observational equiv-

alence of distinct models could be reduced to a provable mathematical relation, this would be

an important development in the epistemology of science. Charloe Werndl has recently

argued that two such relations, to be introduced shortly, could be useful in mathematical def-

99


initions of observational equivalence (Werndl 2009b). In this chapter I will consider whether

these two relations can be considered purely mathematical definitions of observational equiv-

alence, and argue that this is unlikely.

e definitions to be considered apply to deterministic chaotic dynamical systems and to

stochastic processes, and so I will begin with a brief introduction to the relevant mathemat-

ics. Next, Werndl’s two proposed definitions will be considered in turn. e first is based

on manifest isomorphism and is, to my knowledge, original to Werndl (2009b). e second

is based on ε-congruence, which was introduced in Ornstein and Weiss (1991). I will present

two main arguments for the position that neither of these relations is an acceptable condition

for observational equivalence. First, I will present counterexamples which demonstrate that

each definition can hold between models which are easily distinguishable, and so each pro-

posal is insufficient as stated. Second, I will consider whether these counterexamples could

be avoided by making the definitions more restrictive. I will argue that, while this is the case,

these restrictions will take the form of physical hypotheses about the systems in question.

is undermines the idea of a purely mathematical definition of observational equivalence,

and limits any application of such modified definitions to a given theoretical context. Accord-

ingly, I suggest that the prospects for a non-contextual, provable, and purely mathematical

definition of observational equivalence are dim. In closing, I will suggest that a strength of

Werndl’s analysis is that it can deepen our understanding of observational equivalence by

clarifying this distinction between the provable and unprovable elements of our scientific

judgements.

4.1.2 Deterministic Dynamical SystemsIn the broadest terms, dynamical systems are mathematical representations of how values

change over time.¹ Oen these values are given physical interpretations, and the dynamical

system can be said to model some part of the real world. e framework here is that of ergodic

theory, and as such a measure-preserving dynamical system is a quadruple (M , ΣM , µ, T). M

¹For an accessible introduction to measurable dynamical systems, see Silva (2008).


is a non-empty set, called the phase space, whose members represent the possible states of

the system. ΣM is a σ-algebra on M , which is a set of subsets of M that must be nonempty,

closed under complementation, closed under countable unions, and must contain ∅ and M .

µ is the probability measure, a function whose domain is the elements of ΣM and whose

codomain is [0, 1]. Intuitively, µ gives the “size” of the regions of M contained in ΣM , and

we make the controversial but common assumption here that µ(m) is the probability that the

system will be found in a region m of M . µ is defined such that µ(M) = 1, and is closed under

countable additions. Any set assigned measure zero is called a null set, but note that a set of

measure zero need not be small in any intuitive sense.

e transformationT moves points and regions of M to other points and regions of M , and

is interpreted as the dynamical system’s evolution operator. e present chapter is concerned

with discrete-time dynamical systems, and so if a system’s state is initially m ∈ M , then

T(m) is the new state of the system aer one time step. is process can be iterated, and

Tn(m) denotes the state of the system aer n time steps. Repeated iteration of T generates a

sequence of points called an orbit or trajectory of the system. e types of dynamical systems

considered here are both invertible and measure-preserving, and so µ(T−1(m)) = µ(T(m))

for all m ∈ ΣM .

A deterministic system (M,ΣM , µ,T) is called Bernoulli if it meets two conditions. First,

and most relevantly for our purposes here, there must exist a partition α of M such that

µ(Tnαi ∩ Tmα j) = µ(Tnαi)µ(T mα j), where −∞ < m, n < ∞ and αi and α j range over

all the atoms of α. Second, the T iα must generate the full σ-algebra of M (Ornstein 1970,

337). Seing aside this second condition, which does not bear directly on the arguments of

this chapter, note that the first condition states that the past and future states of a Bernoulli

system exhibit a kind of statistical independence. In a well-defined sense Bernoulli systems

are the most unpredictable of all deterministic systems (Berkovitz, Frigg, and Kronz 2006,

677). e KS-entropy of a Bernoulli system is defined as the following sum over its transition


probabilities pi:

−n∑

i=1

pi ln pi (4.1)

Ornstein famously demonstrated that Bernoulli systems are isomorphic if and only if they

have the same KS-entropy, and this result will be useful in a later section (Ornstein 1970).

4.1.3 Stoastic ProcessesStochastic processes have irreducibly probabilistic state transitions, and so their evolu-

tions cannot be predicted with certainty even given precise knowledge of their dynamics and

current state. Despite stochasticity’s intuitive connection to ‘chancy’ or indeterministic pro-

cesses, the fact that a physical system can be modelled by a stochastic process must not be

taken as proof that it is indeterministic in some ontological sense.² Epistemic limitations,

for example, will sometimes require us to use stochastic models even when we believe the

systems under study are deterministic.

To define a stochastic process, let M = m1,m2, ...,mn be a set of states identified as pos-

sible outcomes. In analogy to the way we constructed a deterministic system, letΩ be a phase

space, ΣΩ be a σ-algebra, and ν be a probability measure. Intuitively, Ω represents all pos-

sible sequences of outcomes, and each ω ∈ Ω represents one possible sequence of outcomes.

Let us also define a set of functions Zt from Ω to M , which we call random variables. Zt is

interpreted as the outcome of ω at time t. Since we are concerned with discrete processes,

the time index will range over the integers. Our focus here is on stationary stochastic pro-

cesses, for which the transition probabilities are constant through time. Puing these pieces

together, a stochastic process is a one-parameter family of such random variables:

Zt ; t ∈ Z (4.2)

²is connection is oen made in the physics literature, eg: “Observe, however, that the system is stillconsidered deterministic: only themodel becomes stochastic … this is quite different from the common approachthat assumes the system must be stochastic too” (Judd and Smith 2004, 232).


where, again, the values given by Z0, Z1, and so on represent outcomes of the process at times

t = 0, 1, . . . . As long as the transition probability from at least one state to another has a

probability that is strictly between 0 and 1 we call the stochastic process non-trivial (Doob

1953, 46–47).

ere are several different types of stochastic behaviour, and in this chapter we will be

concerned particularly with Bernoulli and Markov processes. A Markov process’ transition

probabilities depend only on its current state, and on none of its past states. Bernoulli pro-

cesses, on the other hand, have ‘no memory,’ and their transition probabilities are state in-

dependent. As they are stochastic, Bernoulli processes are conceptually distinct from the

deterministic Bernoulli systems defined in the previous section.

4.1.4 Finite-valued observation functionIn practice, our observations are always limited to some finite precision, and we can rep-

resent this mathematically with an observation function Φ. Let the observation space MO

represent the values we can actually observe in practice. MO is oen generated by applying a

partition to the full phase space M , which divides M up into non-overlapping regions called

atoms (see figure 4.1). Each atom of the partition αi is associated with an observable value,

and the observation function Φ takes each point of the phase space M to the value of its cor-

responding atom. is is oen referred to as a ‘coarse-graining’ of the phase space. e size

of the atoms will depend on factors like the resolution of our instruments, and upgrading our

instruments would correspond to using a ‘finer-grained’ partition, but as long as we impose a

finite partition we can never know the system’s precise state. Here we assume MO has finitely

many elements, and Φ is called a finite-valued observation function.

4.1.5 Deterministic Representation of a Stoastic Processe last piece of technical apparatus needed is called the deterministic representation of

a stochastic process. is is a method for ‘replacing’ any stochastic process Zt ; t ∈ Z from

(Ω,ΣΩ, ν) to (M ,ΣM) with a deterministic system. Recall that each ω ∈ Ω represents an in-

4.2. TWO MATHEMATICAL DEFINITIONS OF OBSERVATIONAL EQUIVALENCE 104

Figure 4.1: An example of a finite partition dividing M into the five atoms α1 through α5.An observation function Φ would then associate an observable value with each αi .

finite sequence of possible outcomes of the stochastic process, which is called the realization

of ω. e idea is to set up a deterministic le-shi system whose phase space consists of all

possible realizations, and gives the same probabilities as the stochastic process. Let the phase

space M of the deterministic system be the set of all bi-infinite sequences (...m−1m0m1...)

with each mi a member of the outcome space M . Let the transformation T : M → M be the

le shi, which bumps each element over one place by moving mi to mi−1. Only the 0ᵗʰ ele-

ment m0 can be observed, and this is represented mathematically by the observation function

Φ : M → M ,Φ(m) = m0. As we keep applying the le shi T , different elements of the se-

quence will be moved to m0 and become visible when we observe the system through Φ. e

finite realizations of Zt ; t ∈ Z are cylinder sets, and the probability assigned to these realiza-

tions forms a pre-measure which may be extended to the measure µ onΣM . (M,ΣM , µ,T,Φ)

as so constructed is a deterministic system which reproduces the given realizations and prob-

abilities of the stochastic process Zt ; t ∈ Z, and is called its deterministic representation

(Werndl 2009b, 236).

4.2 Two Mathematical Definitions of Observational

Equivalence

4.2.1 Manifest IsomorphismIf two probabilistic models assigned (nearly) the same probabilities to the same outcomes,

we might want to call them observationally equivalent; conversely, if the models assigned


very different probabilities to the same outcomes, it would be unintuitive to call them obser-

vationally equivalent. Charloe Werndl has suggested that in some cases manifest isomor-

phism, a special case of general measure-theoretic isomorphism, could provide a rigorous

justification of this kind of probabilistic, evidence-based observational equivalence (Werndl

2009b, 234). Aer outlining manifest isomorphism itself, I will present counterexamples to

show that models can be manifestly isomorphic but not observationally equivalent, and ob-

servationally equivalent without being manifestly isomorphic. A strengthened version of

manifest isomorphism might avoid these difficulties, but I will argue that this strength comes

at the cost of introducing contextual and defeasible assumptions. Such a strengthened pro-

posal could help us gain a beer understanding of observational equivalence, but would not

be a purely mathematical and provable relation.

Two measure-preserving systems (M1,ΣM1 , µ1,T1) and (M2,ΣM2 , µ2,T2) are isomorphic

if there is an invertible measure-preserving map φ between them which takes orbits of T1 to

orbits of T2 ‘almost everywhere’—that is, except perhaps for a set of points of measure zero

(Ornstein andWeiss 1991, 15–16). e intuitive idea is that the systems are isomorphic if each

orbit of T1 has a corresponding orbit of T2 with identical probabilistic features, and the map

φ tells us how to translate between them. To make the “almost everywhere” requirement

explicit, we consider two subsets of M1 and M2, M1 and M2 respectively, which differ from

the full sets by a set of measure zero, and then demand that φ take orbits of T1 to orbits of T2

everywhere on M1 and M2. e systems are called manifestly isomorphic if identical subsets

M1 and M2 can be found.

If M1 and M2 are identical, then the two systems inhabit the same phase space, which

means that they have the same possible outcomes. Since the systems are isomorphic, each

trajectory in one system will have an analogue in the other, and corresponding bundles of

trajectories will have the same probabilities. is sounds like the intuitive position, outlined

above, that two systems could be observationally equivalent if they assigned the same prob-

abilities to the same outcomes.


Along these lines, Werndl proposes the following definition of observational equivalence

between deterministic and stochastic models. Suppose we want to determine whether a de-

terministic system (M,ΣM , µ,T) and a stochastic process Zt ; t ∈ Z are observationally

equivalent. Recall our assumption that we can only view the deterministic system through

the finite-valued observation function Φ, which coarse-grains the phase space into a finite

number of observable values. If our deterministic system is of the right type, specifically

if it is totally ergodic, then applying a finite-valued observation function produces a non-

trivial stochastic process (Werndl 2009b, 235).³ In effect, when we coarse-grain the phase

space with Φ we can no longer predict the system’s next state with certainty, and this yields

the non-trivial stochastic process Φ(T t); t ∈ Z. Since manifest isomorphism is defined for

deterministic systems, we consider the deterministic representation of these two stochastic

processes. If the deterministic representation of the derived stochastic process Φ(T t); t ∈ Z

is manifestly isomorphic to the deterministic representation of the original stochastic process

Zt ; t ∈ Z, then they have the same set of possible outcomes, and all trajectories in one have

probabilistically equivalent analogues in the other. According to the proposal being enter-

tained, this is exactly what it means for the process Zt ; t ∈ Z and the system (M,ΣM , µ,T)

(observed with Φ) to be observationally equivalent.

Formally, Werndl defines manifest isomorphic observational equivalence of a determinis-

tic system (M,ΣM , µ,T) and a stochastic process Zt ; t ∈ Z as follows:

e stationary stochastic process Zt ; t ∈ Z and the measure-preserving de-

terministic system (M,ΣM , µ,T), observed withΦ, are observationally equivalent

if and only if the deterministic representation of Φ(T t); t ∈ Z is manifestly iso-

morphic to the deterministic representation of Zt ; t ∈ Z. (Werndl 2009b, 236)

Aminor problemwithmanifest isomorphism is that the “if and only i” wording inWerndl’s

³A measure-preserving transformation T is totally ergodic if Tn is ergodic for all integers n > 0 (Silva 2008,101). For more on ergodicity and its connection to randomness and chaos in deterministic systems, see Berkovitz,Frigg, and Kronz (2006) and Werndl (2009a)).


DeterministicSystem S : (M1,ΣM1 , µ1,T1)

Observe with Φ

StochasticProcess

Φ(T t1); t ∈ Z

P : Zt ; t ∈ Z

GenerateDeterministicRepresentation

DeterministicSystem (M1,ΣM1 , µ1,T1,Φ)

''OOOOOOOOOOOOOO(M2,ΣM2 , µ2,T2,Φ)

wwpppppppppppppp

Manifestly isomorphic? If so,S observed with Φ and P areobservationally equivalent.

Figure 4.2: Schematic representation of Werndl’s proposed method of determining obser-vational equivalence of deterministic and stochastic models.

definition rules out many model pairs that yield arbitrarily similar probabilistic predictions,

and so might actually be observationally equivalent. Consider a deterministic Bernoulli sys-

tem S1 and a stochastic Bernoulli process P1 with the same two outcomes, A and B. Let S1

be shorthand for the system (M,ΣM , µ,T) where M is the set of all bi-infinite sequences of

A and B, let ΣM be the σ-algebra generated by the appropriate cylinder sets on M , and let µ

be the appropriate probability measure on ΣM . Similarly, let P1 be shorthand for Zt ; t ∈ Z.

Let the probabilities of the outcomes (A, B) in any individual trial be as follows:

S1 : (0.5, 0.5) (4.3)

P1 : (0.50001, 0.49999) (4.4)

Although not identical, intuitively, in many experimental situations the evolutions of S1 and

P1 would seem to be observationally equivalent. However, since S1 and P1 have different KS-

entropies, they cannot be manifestly isomorphic, and thus cannot be observationally equiva-

lent according to the proposal under consideration. All this shows, of course, is that manifest


isomorphismwill not do as a necessary condition for observational equivalence between these

sorts of models. Since it might well be the case that it was never intended as such, we can

avoid this problem by taking manifest isomorphism as a sufficient condition.

Yet, this modified proposal is also not sufficient for observational equivalence. Consider

S2 and P2, which are again two-outcome deterministic and stochastic Bernoulli models re-

spectively, but this time set their probabilities as follows:

S2 : (0.1, 0.9) (4.5)

P2 : (0.9, 0.1) (4.6)

In order to satisfy the requirements for the proposed definition of observational equivalence,

we need a stationary stochastic process Zt ; t ∈ Z; a measure-preserving deterministic sys-

tem (M,ΣM , µ,T) observed with Φ; and the deterministic representation of Φ(T t); t ∈ Z to

be manifestly isomorphic to the deterministic representation of Zt ; t ∈ Z.

e first condition is met by P2 = Zt ; t ∈ Z. e second condition is met by S2 observed

with Φ, since Werndl’s proposition guarantees that this gives a stationary stochastic process

Φ(T t); t ∈ Z. For the third condition we need to show that the deterministic representations

of Φ(T t); t ∈ Z and Zt ; t ∈ Z are manifestly isomorphic. By construction these models are

isomorphic since they are Bernoulli and have the same KS-entropy. eir phase spaces also

both consist of the set of all bi-infinite sequences of A and B. Since they are isomorphic and

share the same phase space, these two deterministic representations are manifestly isomor-

phic and therefore observationally equivalent according to the proposal under consideration.

Yet, also by construction, S2 and P2 are intuitively not observationally equivalent since

they assign very different probabilities to the same sequences of outcomes. S2 and P2 there-

fore stand in different relations to any set of evidence. In fact, by altering the probabilities

of S2 and P2, we can devise manifestly isomorphic models which assign arbitrarily different

probabilities to any given observation. If observational equivalence for probabilistic models


is supposed to be something like assigning the same (or similar) probabilities to the same

outcome sequences, then manifest isomorphism is not well suited to the task.

A likely response to this counterexample would be to add the condition that two systems

are observationally equivalent if the isomorphism map is the identity function φ(x) = x.⁴

Since φ takes orbits of one system to orbits of the other almost everywhere, if φ is the iden-

tity function then almost all trajectories of the two deterministic representations are identi-

cal—and identity should certainly be sufficient for indistinguishability.

However, even if we do accept this modified definition, the inference from manifest iso-

morphism to observational equivalence requires non-mathematical assumptions, and soman-

ifest isomorphism cannot be a purely mathematical definition of observational equivalence.

For example, whether a deterministic model S and stochastic model P are manifestly iso-

morphic will depend in general on the specific finite-valued observation function Φ which

is applied to S. But there are no mathematical facts which compel the choice of a particular

Φ, and the selection of an appropriate Φ will be based on contextual and defeasible factors

such as our confidence in our data, or physical theories about our measurement apparatus

and the system under consideration. If we choose a strange or inappropriate Φ, such as the

function Φ6 which takes all elements of S’s phase space to the value 6, then S will be man-

ifestly isomorphic to systems it is certainly not observationally equivalent to. Of course Φ6

will be excluded from serious consideration in most circumstances, but this rejection is not a

mathematical necessity. Rather, the decision to reject Φ6 will be based on physical theories

and beliefs about the system S, such as our expectation that it will deliver values other than 6.

What this demonstrates is that the question of whether a manifestly isomorphic pair S and P

are truly observationally equivalent depends on whether an appropriate Φ has been applied,

but the appropriateness of a givenΦwill vary depending on the circumstances, and there may

not be a single correct—or at least undisputed—way of resolving this maer. While the mani-

fest isomorphism of two models may be provable, the further inference to their observational

⁴anks to Charloe Werndl for pointing this out to me.


equivalence will be based on contextual and defeasible factors.

To summarize, a purely mathematical definition of observational equivalence based on

manifest isomorphism is in trouble if we accept the intuition that observational equivalence

for probabilistic models should be something like assigning the same or similar probabilities

to the same outcomes. Since manifestly isomorphic models can have radically different prob-

ability distributions, this definition picks out many wrong systems, and since non-manifestly

isomorphic models can have arbitrarily similar probability distributions, it excludes many

right ones. ese problems can be avoided by restricting the claim to sufficiency and adding

the requirement that φ(x) = x. However, while this modified manifest isomorphism is a

provable relation, the inference to observational equivalence will only be as reliable as the

non-mathematical assumptions supporting it.

4.2.2 ε-CongruenceIn 1991 the mathematicians Ornstein and Weiss introduced ε-congruence, which was pre-

sented as a well-defined notion of observational equivalence (Ornstein and Weiss 1991, 23).

ε-congruence has been an influential concept, and some have argued that, since it entails

observational equivalence, it has important implications for the metaphysical thesis of deter-

minism (Suppes 1993; Suppes and de Barros 1996). In this section I will begin by outlining

ε-congruence and the arguments purporting to establish it as observational equivalence. I

will then argue that this view is untenable, since two dynamical systems can be ε-congruent

yet observationally distinguishable. ε-congruence plus some extra conditions may be more

feasible as a definition of observational equivalence, but, as with manifest isomorphism, these

additional conditions will generally be fallible physical hypotheses, and the inference to ob-

servational equivalence will no longer be deductively certain.

Take two deterministic measure-preserving dynamical systems, associated with transfor-

mations T1 and T2 respectively, that act on the same phase space M . We introduce a metric,

which is a function that gives the distance between any two points in M . Recall that T1 and

T2 are isomorphic if there is a measure-preserving map φ which takes orbits of T1 to orbits


of T2 almost everywhere. ε-congruence requires that two systems be isomorphic but also

puts restrictions on their geometrical and statistical properties. Ornstein and Weiss give the

following definition of ε-congruence for two systems inhabiting the same phase space M :

We say that two measure-preserving [transformations] … on the same com-

pact metric space M are [ε]-congruent if they are isomorphic and the map φ from

M to M that implements the isomorphism moves the points in M by < [ε] except

for a set of points in M of measure < [ε].⁵ (Ornstein and Weiss 1991, 22–3)

More generally, let f t and f t be flows on abstract measure spaces X and X , and g and g

be functions from X and X respectively to a metric space. en we say that ( f t , X, g) and

( f t , X , g) are ε-congruent if there is an invertible measure-preserving function φ such that

φ f t(x) = f tφ(x) almost everywhere, and, leing d denote distance in the metric space, we

have d(g(x), g(φx)) < ε everywhere except for in a set of measure less than ε (Ornstein and

Weiss 1991, 23).

Given the time-average interpretation of probability, the second part of this definition

stipulates, roughly, that corresponding orbits are allowed to have at most distance ε between

them ‘most of the time,’ but ‘ε of the time’ they are allowed to be farther apart. ε is a parameter

that ranges between zero and one, and the smaller ε is the more alike the two flowsmust be. If

we set ε small but not too small, then we get an interesting notion of close-but-not-too-close:

the corresponding trajectories of the two system are within some small distance ε of each

other most of the time, but a proportionately small ε of the time they are allowed to differ by

more.

Ornstein and Weiss thought this looked suggestive. ey wrote:

If we agree that we cannot distinguish points in M that have distance < [ε],

and if we are willing to ignore events of probability less than [ε] (experimental

error), then [ε]-congruent flows are indistinguishable. (Ornstein andWeiss 1991,

⁵Notation changed for consistency.


23)

ε-congruence does seem promising as amathematical definition of observational equivalence.

For example, two Bernoulli systems with radically different probability distributions—such as

those that gave rise to the counterexamples in section 2.1—could scarcely be ε-congruent for

any small value of ε, since for the most part their trajectories will not be close to each other.

ε-congruence has aracted philosophical aention due to a theorem, proved by Ornstein

and Weiss, which establishes that any deterministic Bernoulli system is, for all ε > 0, ε-

congruent to some (generally ε-dependent) stochastic process Ornstein andWeiss 1991, 39. If

to be ε-congruent is to be observationally equivalent, and if we accept Ornstein and Weiss’s

interpretation of ε-congruence, then this theorem entails that there are classical deterministic

models which are indistinguishable from stochastic models at all observation levels.⁶

e ε-congruence theorem concerns deterministic Bernoulli systems and stochastic semi-

Markov processes. A Bernoulli system, recall, is a deterministic measure-preserving system

whose behaviour is chaotic and extremely unpredictable. A semi-Markov process is a stochas-

tic process which remains in one of a finite number of states for a period of time and then

jumps to a new state, where both the time between jumps and the transition probabilities are

state dependent. Since the process is stochastic, the result of this jump cannot be predicted

with certainty. Ornstein and Weiss’s theorem proves that every deterministic Bernoulli sys-

tem on amanifold M is, for every ε > 0, ε-congruent to some stochastic semi-Markov process

on M , where this stochastic process will in general depend on the value of ε (Ornstein and

Weiss 1991, 39).

We might expect deterministic and stochastic models to behave very differently, yet the

ε-congruence result appears to imply arbitrarily similar behaviour in some cases. is is

surprising, to say the least, and Ornstein and Weiss suggested the following interpretation:

is may mean that there is no philosophical distinction between processes

⁶Ornstein and Weiss’s result has been philosophically influential, but other authors have argued for similarconclusions (e.g. Werndl (2011)).


governed by roulee wheels and processes governed by Newton’s laws … we are

comparing, in a strong sense, Newton’s laws and coin flipping. (Ornstein and

Weiss 1991, 39)

Ornstein and Weiss did not develop this idea any further, but several philosophers have ex-

plored its implications. Patrick Suppes accepted that ε-congruence implies observational

equivalence, and argued that this leads to a strong form of underdetermination wherein any

thesis concerning the deterministic or indeterministic nature of the world must necessarily

“transcend experience” (Suppes 1993; Suppes and de Barros 1996). In a reply to Suppes, John

Winnie conceded that while the ε-congruence results show that deterministic Bernoulli and

stochastic Markov models are observationally equivalent, there are inductive reasons for pre-

ferring the deterministic model, since it “outstrips any single Markov model in its conceptual

and predictive power” (Winnie 1998, 317). Entering this debate here would take us far afield,

but both authors accept as a starting premise that ε-congruence implies observational equiv-

alence.

It is surprising, given the potentially far-reaching ramifications of Suppes and Winnie’s

debate, that very lile aention has been given to the question of whether ε-congruence is

in fact a sufficient condition for observational equivalence. In the remainder of this section I

will consider the two main justifications of this claim in the literature, the first from Ornstein

and Weiss’s original paper, and a more recent and thorough treatment by Charloe Werndl.

I will argue that these justifications are problematic, and that ε-congruence, like manifest

isomorphism, is susceptible to counterexamples which show that it cannot be sufficient for

observational equivalence. I will close by suggesting that ε-congruence can be made more

adequate by imposing additional conditions; however, as with manifest isomorphism, these

conditions will in general be based on defeasible physical hypotheses.

To begin, Ornstein and Weiss’s interpretation of ε-congruence risks inappropriately con-

flating an experiment’s precision with its accuracy. eir interpretation, recall, is that if we

cannot distinguish measurements within ε of each other, and if we ignore events of proba-


bility less than ε, then ε-congruence is observational equivalence. However, using the same

parameter ε to quantify both precision and accuracy is difficult to motivate since we expect

these factors to vary independently, and sometimes to differ quite greatly. If we use only one

variable to account for both types of inexactitude, we seem commied to accepting greater

uncertainty in each measurement if we eliminate more measurements as outliers. is will

oen be unwarranted. If we use a precision instrument in a noisy environment, eachmeasure-

ment may be quite exact even if we choose to disregard a great many measurements as due to

external noise factors. A delicate acoustical experiment, for example, will detect a great deal

of outliers if it is located in a bowling alley; but even if we ignore a large proportion ε of its

results, it would be unwarranted to consider each individual measurement correspondingly

inexact. In cases where precision and accuracy differ, Ornstein and Weiss’s interpretation of

ε-congruence seems not to apply.

Werndl’s explication of ε-congruence overcomes this difficulty by making ε dependent

on two other quantities. First, she says, let ε1 be the minimum distance at which states of

the deterministic system can be distinguished. en, note that “in practice, for sufficiently

small ε2, one will not be able to observe differences in probabilities of less than ε2 ” (Werndl

2009b, 238). Presumably ε1 and ε2 will be determined on an experiment-by-experiment basis,

depending on the situation at hand. Now let ε be smaller than ε1 and ε2. en, claims

Werndl, two models “give the same predictions at observation level ε” if their solutions can

be put into one-to-one correspondence in such a way that at each time point they are less than

ε apart, except for a set whose probability is smaller than ε. In other words, ε-congruence is

indistinguishability at observation level ε.

Werndl’s finer-grained approach is an improvement, but problems still arise when we try

to set ε based one ε1 and ε2. If we set ε greater than both ε1 and ε2, the ε-congruence re-

quirement will be less restrictive than the most restrictive condition imposed by the details of

the experiment. is can result in systems being ε-congruent, and thus mislabelled as obser-

vationally equivalent, even if they are quite clearly discriminable. Werndl prudently guards


against this, and advises us instead to choose ε smaller than both ε1 and ε2. Any variations

between the two models must then occur below both thresholds of detectability. However, ε

is more restrictive than the least restrictive condition imposed by the actual experiment, and

consequently this condition will fail to identify some intuitively indistinguishable systems as

observationally equivalent. Werndl’s conservative choice of εwill avoidmisclassifying distin-

guishable systems as observationally equivalent, but only at the cost of denying ε-congruence

the status of a necessary condition for observational equivalence.

It might be objected that picking on ε’s relation to ε1 and ε2 is unfair, since arguably these

finer-grained quantities underlie the mathematical discussion, but using a coarser-grained ε

makes the proofs easier. Furthermore, in the specific case of the Orenstein-Weiss theorem,

seing ε = min(ε1, ε2) may actually be an acceptable heuristic.⁷ is is fine and well, but

the present concern is interpreting this move in the light of a general notion of observational

equivalence. If ε-congruence based on Werndl’s finer-grained ε1 and ε2 is not necessary

for observational equivalence, then in order to be generally applicable it must at least be

sufficient. Unfortunately, as I will now argue, this is not the case.

ε-congruence cannot be a sufficient condition for observational equivalence because the

set of points where two systems differ by more than ε (hereaer called the ε-set, for brevity’s

sake) is restricted only in its measure, and not in its distribution. is means that two ε-

congruent models can have ε-sets that differ in empirically meaningful ways. If the trajecto-

ries of two systems always remain within ε of each other they will be indistinguishable, and

so observationally equivalent. If two systems do have an ε-set, then their trajectories differ at

some points by more than ε, but we can still call them observationally equivalent if we write

these variations off as random error. But if the trajectories of two systems differ at extremely

regular intervals by extremely large amounts, then it is hard to see how they could be obser-

vationally equivalent. Two audio recordings of gently hissing white noise may reasonably be

regarded as indistinguishable even if they are not identical. However, if one recording also

⁷anks to Charloe Werndl for stressing this point.


Figure 4.3: e bump function B(x) = e−1

1−x2 .

includes the distinct and regular ticking of a clock while the other does not, any claim of their

observational equivalence becomes very suspect.

is intuitive argument can be made more precise, and I will now construct two sys-

tems S1 and S2 that are distinguishable despite meeting all the technical requirements for

ε-congruence. Let M be the section of the Cartesian plane (0, 1] × (0, 1] with opposite edges

identified, let ΣM be all Lebesgue-measurable subsets of M , and let µ be the Lebesgue mea-

sure. For simplicity, S1 is a very boring system whose trajectories are straight lines moving

constantly to the right. e trajectories of S2 also move constantly rightward, but are per-

mied to deviate smoothly from straight lines. Let all the trajectories of S2 be the same but

shied up or down on the y-axis, so the phase space of S2 is filled with a stack of similar

trajectories. Let S1 be the system (M,ΣM , µ,T1) with horizontal trajectories given by the

following transformation T1:

T1(x , y) = ((x + τ) mod 1, y) (4.7)

where τ is a parameter that determines the ‘speed’ with which trajectories of S1 move across

the phase space. Let P(x) be a perturbing function. Many options present themselves, but

here wewill use a bump function, a continuous curvewith continuous derivatives of all orders.

A standard bump function is defined as follows:

B(x) =

e−

11−x2 if |x | < 1

0 otherwise.(4.8)


Figure 4.4: Graph of P(x) when ε = 0.4 and ε1 = 0.5. P(x) is below the doed liney = 0.4 most of the time, but climbs above the dashed threshold of detectability y = 0.5on an interval of width strictly less than 0.4.

In appearance B(x) is reminiscent of a Gaussian curve, except it smoothly approaches and

meets the line y = 0 at x = ±1 (see fig. 4.3). Here we will use the custom bump function

P(x). Since ε1 and ε2 can be quite different, to generate a counterexample we need two

systems whose ε-set has measure smaller than the minimum of ε1 and ε2, but whose largest

deviations are greater than the detectability threshold ε1. Following Werndl, we choose a

conservative value for ε, and set ε = min(ε1, ε2). P(x) is then defined as follows:

P(x) =

ε(1 − ε) + ε1e

(1− ε2

ε2−4(x−0.5)2

)if |x − 0.5| < ε2 ,

ε(1 − ε) otherwise.(4.9)

P(x) provides a ‘bump’ of width ε and height ε(1− ε)+ ε1 centred around x = 0.5. Outside

this interval the curve connects smoothly to a horizontal line of height ε(1− ε). We can gen-

erate a system S2, given by the quadruple (M,ΣM , µ,T2), by perturbing the transformation

operator of S1 with P(x) as follows:

T2(x , y) = ((x + τ) mod 1, (y − P (x) + P ((x + τ) mod 1)) mod 1) (4.10)

Trajectories of S1 are mapped to trajectories of S2 by the following φ:

φ(x , y) = (x , (y + P(x)) mod 1) (4.11)


Conversely, φ−1 takes trajectories of S2 back to S1:

φ−1(x , y) = (x , (y − P(x)) mod 1) (4.12)

e determinants of the Jacobians of φ(x , y) and its inverse are 1 and do not depend on x,

y, ε1, or ε2, so φ(x , y) and its inverse are measure-preserving at all points and for all values

of ε1 and ε2. S1 and S2 are therefore isomorphic.

To establish ε-congruence, we need to show that φ(x , y) moves trajectories by less than

ε, except for in a region of measure less than ε, where ε = min(ε1, ε2). Since the situation

will be the same for all trajectories modulo a vertical translation in the phase space, consider

the trajectory of S1 that moves uniformly across the phase space along the line y = 0. e

corresponding trajectory of S2 spends most of its time moving along the horizontal line y =

ε(1 − ε), and so lies within ε of S1. Within the bump of width ε, the trajectory increases to

a height of ε(1 − ε) + ε1, which is greater than ε1, and so represents a detectable deviation

from the trajectory of S1. Since the bump began slightly below y = ε, the portion of the bump

above this line, which is this trajectory’s contribution to the ε-set, will be of width strictly less

than ε. Similar considerations apply for all trajectories, and so the total ε-set has measure

less than ε. erefore, no maer which of ε1 and ε2 is smaller, S1 and S2 are isomorphic,

inhabit the same phase space, and the measure-preserving map φ(x , y) that takes trajectories

of S1 to trajectories of S2 moves orbits by less than ε except for a set of measure less than ε.

S1 and S2 are therefore ε-congruent, but have a regular and detectable difference.

e function P(x) is not terribly exciting as it stands, but it can be modified to generate

a more pathological perturbation P∗(x) (see fig 4.5). First, the size of the bump can be made

arbitrarily large by increasing the coefficient of the exponential term. ε1 was chosen here

because it ensures detectability and seems to fit with the general spirit of the proposal, but

the systems will remain ε-congruent under a perturbation by an arbitrarily large bump. Sec-

ond, while P(x) consists of only one bump, the number of bumps in P∗(x) can be made an


Figure 4.5: Graph of P∗(x) with five peaks, again with ε = 0.4 (doed line) and ε1 = 0.5(dashed line).

arbitrarily large number n by compressing the original function P(x) to a width of 1/n and

repeating it n times. Since the width of the observable portion of the original bump is less

than ε, the width of the observable portions of each shrunken bump will be less than ε/n,

and the combined width of all n observable bumps will still be less than ε. If we perturb S1

with such a P∗(x), the result will be an ε-congruent S2 with arbitrarily many arbitrarily large

deviations.⁸

erefore no maer which of ε1 or ε2 is smaller, two systems can be ε-congruent and

yet differ in empirically meaningful, observationally detectable ways. Since in both cases the

trajectories of S2 are those of S1 perturbed by a regular function, the trajectories of S2 will

vary measurably from those of S1 at regular intervals. Since both the number and size of the

bumps in the perturbing function P∗(x) can be made arbitrarily large, the number and size of

the detectable variations between S1 and S2 can be made arbitrarily large. It is highly counter

intuitive to say that, in the absence of any other considerations, two mathematical models

could be observationally equivalent when they diverge detectably and systematically. Some

contextual or theoretical explanation would need to be given for why these observations

differ systematically yet still count as observationally equivalent, but any explanation will

necessarily go beyond the mathematics. e conclusion is that ε-congruence alone cannot

⁸Although not proved here, this result should generalize to an arbitrary S1, further problematizing any con-textless interpretation of the ε-congruence of two mathematical models.


be sufficient for observational equivalence, since it will, at least in some cases, need to be

supplemented with a physical theory, model, or hypothesis about the systems in question.

Perhaps for a sufficiently small ε we can expect not to observe any members of the ε-set,

and so it can be safely ignored. However, this cannot be guaranteed. In many cases, and

particularly in particle physics, the number of measurements performed may be quite enor-

mous, making it highly probable that a member of the ε-set will be observed. Furthermore,

these improbable outlying events may be important. is is, in fact, oen the point, and many

experiments are designed specifically to detect such low-frequency events.

Perhaps ε-congruence could be strengthened with the requirement that the ε-set be dis-

tributed randomly in some sense, to match our expectations about experimental noise. In-

deed, since the systems considered byOrnstein andWeiss are Bernoulli, and therefore strongly

chaotic, it already seems implausible, although not impossible, that the outliers in the ε-set

will be distributed regularly. is may be feasible, but there are two possible problems. First,

an explicit randomness requirement on the ε-set would complicate the mathematics, and

there is no guarantee that any of the interesting results about deterministic and stochastic

processes—which, recall, are what sparked philosophical interest in the subject in the first

place—would still hold with this modified definition. Second, there are many different defini-

tions of randomness, and the justification of any choice would likely have to take into account

the theoretical characteristics of the system and the experimental context at hand. is would

introduce a large amount of context sensitivity into the definition, once again eliminating its

purely mathematical and general nature.

Given these considerations, the prospects for using ε-congruence as a purely mathemat-

ical definition of observational equivalence seem dim. Strict ε-congruence permits of coun-

terexamples, and resolving these counterexamples will introduce non-mathematical consid-

erations. e mere ε-congruence of two models therefore does not entail their observational

equivalence. For ε-congruence to be an adequate criterion of observational equivalence, it

must be supplemented by hypotheses about the physical systems under study, the assump-

4.3. CONCLUSION 121

tions embedded in our models, and the kinds of data discrepancies we are willing to aribute

to chance error events.

4.3 ConclusionDeterministic and stochastic models can be manifestly isomorphic and ε-congruent, but

these relations alone are not adequate formalizations of observational equivalence since the

strict mathematical requirements of each can be met by distinguishable systems. Nor can

either relation easily be made into purely mathematical sufficient conditions, since natural

aempts to strengthen them will introduce non-mathematical physical hypotheses. us,

it seems that whether two models are observationally equivalent or not will be a context-

sensitive judgement based on physical hypotheses, and neither manifest isomorphism nor

ε-congruence can deal with these sorts of experimental vagaries in a strict, axiomatic way.

Plausibly, any purely algorithmic approach to observational equivalence will risk ignoring

contextual subtleties.

is only goes to show that Werndl’s purely mathematical definitions are incomplete,

not that they are wrong, and indeed I believe that manifest isomorphism and ε-congruence

can guide us towards a more nuanced understanding of observational equivalence between

certain types of mathematical models. By making it clear which parts of our arguments for

observational equivalence are provable, Werndl’s definitions can likewise help us to deter-

mine which parts are not. Armed with this knowledge, anyone wishing to advance, or to

dispute, an argument for observational equivalence on the basis of manifest isomorphism

or ε-congruence can do so in a more productive way. If these arguments also contain non-

mathematical components, then our judgements of observational equivalence will only be as

strong as the theoretical assumptions underpinning them; but this seems appropriate for a

scientific concept.

.. 5Epilogue

In this dissertation I examined several philosophical disputes concerning the foundations of

chaos theory, and provided a re-evaluation of the place of chaos and fractals in science and

the philosophy of science. Overall, while I was skeptical about the direct application of chaos

and fractals in some particular circumstances, I found that oen the prevailing philosophical

wisdom was unduly negative towards them. I hope, therefore, that this dissertation will do

something to rehabilitate the reputation of chaos theory, and especially of fractals, in the

philosophical community. Let me briefly recap the main conclusions of each chapter, and

close by pointing out some questions that could not be answered in this dissertation, and

some promising directions for future research.

I began in Chapter 1 by considering the question of whether chaotic models can be con-

sidered approximately true. Several authors have argued that traditional accounts of approx-

imate truth face difficulties when applied to chaotic models, and have proposed accounts

meant to overcome these problems. I considered two case studies in detail to show that

chaotic models represent and misrepresent their target systems in complicated and unpre-

dictable ways. I then considered three claims that chaotic models are approximately true,

each based on a different account of approximate truth: Peter Smith’s geometrical-modelling

account, Jeffrey Koperski’s continuum-mechanics account, and a Popperian verisimilitude

122

123

account. I argued that none of these accounts of approximate truth is adequate for the evalu-

ation of chaotic models: the first two are inadequate because they are essentially incomplete,

and the third cannot serve as an analysis of approximate truth on pain of circularity, because it

presupposes a separate account of approximate truth. Furthermore, I argued that completing

the first two accounts is a formidable challenge. Given this pessimistic analysis, I proposed

that oentimes focussing on big questions like approximate truth can actually be counter-

productive. Instead, I proposed that it is more fruitful to study in detail the merits of chaotic

models than to try to evaluate broad and vague honourifics like “approximate truth.”

In Chapter 2 I examined instrumental fractal models of non-fractal physical objects in

more detail. Philosophers have tended to be very skeptical of such models as a whole, and

have argued that non-fractal models should always be pragmatically and philosophically

preferable. I challenged this consensus in two steps. First, I addressed the objections to frac-

tal models, and argued that they could be overcome. Second, I provided a positive argument,

based on a case study from the engineering literature, that in some cases fractal models of

non-fractal physical objects can out-perform non-fractal models. I concluded that fractal rep-

resentations of non-fractal physical objects can be instrumentally useful in various scientific

applications.

In Chapter 3, I consideredwhether fractals could be realized in physical space, and I argued

that classical mechanics entails that there could be scientifically relevant fractal regions of

space. I first argued that the philosophical arguments for the impossibility of spatial fractals

establish only that fractals cannot be made out of atoms, since any fractal-like paern cannot

continue below the atomic scale. I then demonstrated, with the help of both mathematical

analysis and computer simulations of several chaotic models, that there are reasons to believe

that some parts of the physical world could be fractals. Although fractal geometry is clearly

unsuitable for describing a great many things, if chaos theory or some similar theory is true

then there are (or could be) actual things in the world which have truly fractal shapes.

In Chapter 4 I analysed two accounts of observational indistnguishability that pertain

124

specifically to chaotic models. I looked at Charloe Werndl’s manifest isomorphism-based

account, and Ornstein and Weiss’s notion of ε-congruence. I argued, for two related rea-

sons, that neither can function as a purely mathematical definition of observational equiv-

alence, and that contextual non-mathematical scientific judgement will play a role in any

determination of observational equivalence. First, I presented counterexamples to each of

these definitions; second, I argued that overcoming these counterexamples will introduce

non-mathematical premises about the systems in question. Accordingly, the prospects for a

broadly applicable and purely mathematical definition of observational equivalence are un-

promising. Despite this critique, I argued that Werndl’s proposals are valuable because they

help separate the provable mathematical elements of our judgements from the equally nec-

essary but less-rigorous contextual judgements, and that recognizing the distinction between

provable and unprovable elements can bring clarity to arguments for observational equiva-

lence.

ere are, unfortunately, many questions that could not be answered in this dissertation,

and I would like to point out some topics for future research.

One question le open by this dissertation is whether a satisfactory notion of approximate

truth for chaotic models can be articulated. In Chapter 1 I argued that extant accounts of

approximate truth for chaotic models are incomplete. My proposal was to set approximate

truth aside, but one could also take up the challenge to amend the proposals considered, or to

advance a completely new account of approximate truth. More could also be said to elaborate

my own approach, and possibly to develop it into a fuller rival account.

More work could be done to investigate the fractal regions of space I identified in Chapter

3. For example, I suspect they may have some interesting consequences for metaphysics and

ontology. We generally think of fractals as mathematical objects, aer all, which makes some

of their unintuitive properties interesting from a merely mathematical perspective. But if I

am right, and a compelling case can be made for the physical existence of fractals, then this

means that such unintuitive properties can be physically instantiated. If, furthermore, these

125

fractal regions of space count as physical objects—which seems at least plausible, given some

accounts of objecthood—then this could have some interesting consequences for ontology. I

suspect specifically that Ned Markosian’s “maximal continuity” account of what it is for an

object to be simple—roughly, to have no internal parts—could be fruitfully applied to fractal

physical objects (Markosian 1998, 2000, 2004). However, this line of thought must be le to

future work.

Much more could also be said about chaos, observational indistinguishability, and inde-

terminism. For example, in Chapter 4 I argued that several proposed definitions of obser-

vational indistinguishability are inadequate. is means that any arguments based on those

definitions have questionable foundations, and one avenue for future work would therefore

be to reconsider such arguments in light of my analysis. Patrick Suppes, in particular, has

taken ε-congruence to entail observational indistinguishability in several papers, and drawn

far-reaching conclusions on this basis (Suppes 1993; Suppes and de Barros 1996). It would be

worthwhile to see how my analysis affects his arguments and others along the same lines.

ere are also questions about chaos that go beyond the topics touched on in this dis-

sertation. For example, since fractals contain a form of “infinite intricacy” not found in

non-fractals, it could be worth investigating whether fractal representations pose challenges

for accounts of misrepresentation, idealization, and abstraction such as that proposed by

Chakravary (2001, 327). I also did not address the question of whether and how chaotic

models pose challenges to traditional accounts of scientific explanation, or if, as Peter Smith

has argued, chaotic explanations are “very much business as usual” (Smith 1998b, 131). And,

although quantum chaos was briefly mentioned (see Section 1.3.2), I have also not addressed

the question ofwhethermy research has implications for quantum chaos, nor have I addressed

any questions about the nature of quantum chaos and its relation to classical chaos. ese

and other questions must be le for future work.

In the end, we have seen that chaos theory and fractals are still fruitful topics for philo-

sophical research, and that a close study of chaos and fractals can still bring us new knowledge

126

about the world’s possible structure and behaviour. It is true that chaos’s early proponents

may have over-stated its case; as a sub-branch of classical dynamics, chaos theory could never

cause us to completely re-imagine the structure of space, time, and objects in the same ways

that general relativity or quantum mechanics did. But we should not let the over-enthusiasm

of chaos’s early proponents distract us from the ways it deepened the richness of classical

mechanics. Perhaps the most important lesson we can learn from chaos is that oen ordi-

nary things can behave in extraordinary ways. As scientists, as philosophers of science, and

indeed just as human beings, sometimes we would do well to give the ordinary things around

us a closer look.

Appendix: Some Details of FractalGeometry

is section contains an extended discussion of fractal geometry, essentially an expanded

version of Section 0.3, to complement the philosophical arguments in Chapters 2 and 3. First,

I will present the Koch curve, a standard fractal example, and discuss some of its noteworthy

properties. is will leadme to consider several notions of dimension, in particular topological,

similarity, and Hausdorff dimensions, and their particular applications to fractal geometry.

A.1 e Ko CurveeKoch curve is an exemplary fractal, and it is simple to approximatewith a construction

procedure. e intuitive idea here is that the fractal itself is what would result if we repeated

the construction procedure a countably infinite number of times. When constructing the Koch

cure, we begin with a straight line of unit length (figure A.1a). e construction procedure

is to take each straight line segment—of which, to begin, there is only one—and to divide it

into three equal segments, remove the middle third segment, and replace it with a tent shape

whose sides are the same length as the piece we just removed (figure A.1b). Alternatively,

we could think of placing an equilateral triangle over the middle third segment, and then

removing the boom of the triangle. e resulting figure, K1, represents the first iteration

of a construction process that leads to a fractal. Now we repeat the operation, this time

performing it on each of the straight lines in K1, which gives us the more complicated shape

K2, the second iteration of the process (figure A.1c). is process can be repeated as oen as

127

A.1. THE KOCH CURVE 128

(a) K0 (b) K1

(c) K2 (d) K5

Figure A.1: A series of curves converging to the Koch curve. Figure A.1a is a straight lineK0, corresponding to the 0ᵗʰ iteration. Figure A.1b shows the line K1 aer one iteration,figure A.1c shows the result K2 aer two iterations, and figure A.1d shows the Koch curveK5 aer five iterations.

desired. Each iteration will deliver a more complicated shape with more kinks in the line. In

the infinite limit of iterations we get the Koch curve.

Since discussions of fractals oen limit themselves to this sort of intuitive presentation,

I note as an aside here that the claim that this construction process converges to a unique

curve can be made precise. As a brief sketch, if we represent each step of the process by

a continuous and piecewise affine curve gk : [0, 1] → R2, the sequence (gk) can be shown

to be a Cauchy sequence, which, by the completeness of the space of functions from [0, 1]

to R2, converges uniformly to a function g. Since the functions gk are continuous, g too is

continuous. is is, of course, not a constructive proof, but I sketch it here to emphasize that

my superficial overview has a rigorous mathematical underpinning.¹

e Koch curve has some unusual properties. First, although it remains within a bounded

region of the plane, it is infinitely long. To see this, consider what happens when we begin

with a line of length 1 and repeatedly apply the construction step. Aer one iteration, we

have four lines of length 13 , and so the total curve is of length 4

3 . In general, at each step we

replace each line segment with four lines 13 as long, and so the total length of the figure is

¹For details of the proof see Edgar (1992), especially pp. 60—4.

A.2. SIMILARITY DIMENSION 129

(43

)naer n iterations. Since 4

3 > 1, this value diverges in the infinite limit:

limn→∞

(43

)n= ∞, (A.1)

and so the total length of the Koch curve is infinite. However, the area under the curve

remains finite. To see this, note that with each iteration we add the area bounded by smaller

and smaller equilateral triangles. Specifically, if we consider the starting straight line our 0th

iteration, the nth iteration adds 4n−1 equilateral triangles of base length(13

)n. Since the area

of an equilateral triangle of base length ` is equal to√34 `

2, the area an added at iteration n is:

an =

√3

4

4n−1

32n (A.2)

It can be verified using standard methods (e.g. the root test, or the ratio test) that the value

an approaches zero as n approaches infinity. Since each term represents the additional area

added under the curve at the nᵗʰ construction step this means that the total area under the

Koch curve, given by the following sum, converges:

∞∑1

an =∞∑1

√3

4

4n−1

32n (A.3)

A computer approximation gives a total value of around 0.195. Several Koch curves can be

joined together to form a closed loop, and then we have an infinitely long curve surrounding

a finite bounded area of the plane. A curious object indeed!

A.2 Similarity Dimensione notion of dimension is crucial to fractal geometry. Textbook authors will oen appeal

to the intuition that our ordinary notion of dimension, which is close to topological dimen-

sion, seems ill-suited to describe fractals (e.g. Devaney 1992, 185). We understand topological

dimension well enough for familiar subsets of Rn: a point is zero dimensional, a line is one


dimensional, and so on.² Fractals, however, oen seem ill-classified by topological dimension.

e Koch curve, for example, is a continuous curve and so is topologically one dimensional.

However, because of its infinite intricacy, and particularly because it manages to cram an

infinite amount of length into a finite region of the plane, it seems ‘bigger’ than a regular

curve. What this demonstrates, Devaney concludes, is the need for a more refined notion of

dimension to captures these intuitions.

ere are, in fact, many other notions of dimension which can be useful in fractal geome-

try, but here I will focus on the similarity dimension and Hausdorff dimension. is restriction

has both pragmatic benefits and a theoretical justification, since many of these definitions are

complex and in many cases are postulated to be equivalent or nearly equivalent (Farmer, O,

and Yorke 1983).

e similarity dimension provides a helpful way of characterizing self-similar shapes. Re-

call that we call a shape self-similar if it can be divided into smaller pieces that, when magni-

fied, yield a reproduction of the whole.³ e similarity dimension D is defined as the ratio of

the logarithms of the number of pieces the shape is broken into to the magnification factor

required to regain the initial shape (Devaney 1992, 188):

D =log(number of pieces)

log(magnification factor)(A.4)

More concretely, take the Koch curve. If we chop it along the bends introduced in its first

iteration, we get four pieces which, when magnified by a factor of 3, yield a replica of the

whole curve (see figure 0.4). us, for the Koch curve:

DKoch curve =log 4log 3

' 1.262 (A.5)

Note that we will get the same value D no maer which smaller self-similar portion of the

²ere are actually several types of topological dimension, the definitions of which are quite involved, butthis intuitive gloss suffices for present purposes (cf. Edgar 1992, Ch. 3).

³For a more technical discussion of self-similarity, see Edgar (1992, 185).


curve we use. We could just as well have divided the curve up at the joints introduced in the

second iteration, in which case we would have 16 smaller replicas of length 1/9, which would

need to be magnified by a factor of 9 to recover the original figure. Since log(16) = 2 log(4)

and log(9) = 2 log(3), when we divide on by the other we get the same answer as we did in

equation A.5.

One justification for calling the similarity dimension (and the Hausdorff dimension) a

“dimension” is that, despite its different conceptual groundings, it gives the same value as the

topological dimension for conventional shapes. A square, for instance, is a highly self-similar

shape, and so has a similarity dimension. If we divide a square into four equal parts, we get

four smaller squares, each of which, when magnified by a factor of two, yields the original

square. Applying equation A.4 and calculating its similarity dimension, we get:

DSquare =log 4log 2

=2 log 2log 2

= 2 (A.6)

Which is the same as its topological dimension. It is easy to see that a straight line will have

similarity dimension 1, a cube 3, and so on, and indeed in general “well-behaved” self-similar

shapes will have equal similarity and topological dimensions.

e similarity dimension is easy to calculate and has intuitive appeal, but it is limited

in that it cannot be applied to shapes that are not strictly self similar, and that, at least as I

have formulated it here, it is restricted to subsets of Rn. A more general notion is given by

the Hausdorff dimension, which applies to shapes that are not strictly self similar, and works

in general metric spaces. e Hausdorff dimension is, however, more difficult to calculate

than the similarity dimension, and its definition is more involved. e reader is referred to

Appendix A.3 for a fuller discussion. e important points for our purposes are that the

Hausdorff dimension is defined for shapes that are not strictly self-similar, and that it works

in general metric spaces.

A.3. THE HAUSDORFF-BESICOVITCH DIMENSION 132

A.3 e Hausdorff-Besicovit DimensionHere I will sketch a definition of the Hausdorff-Besicovitch dimension, one of the standard

notions of fractal dimension. is presentation draws heavily on Edgar (1992, Ch. 6).

First, we define a countable cover. A family A of subsets of S is called a countable cover

of a set F iff:

F ⊆∪A∈A

A (A.7)

and A is a countable, possibly even finite, family of sets. e diameter of a subset A of a

metric space S with metric ρ is given by:

diamA = supρ (x , y) : x , y ∈ A (A.8)

diamA can be thought of as giving the “largest distance” between any two members of the set

A. If we let ε be small but greater than zero, then A is an ε-cover of a set F if and only if A

is a cover of F and diamA ≤ ε for all A ∈ A.

We now define the following function:

H sε (F) = inf

∑A∈A

(diamA)s (A.9)

As ε shrinks, H sε (F) gets larger. If we take the limit as ε approaches zero, we get a quantity

called the s-dimensional Hausdorff outer measure of F :

H s = limε→0H sε (F) = sup

ε>0H sε (F) (A.10)

As s increases, the value ofH s either remains constant or decreases. In fact, for most values

of s H s will either be zero or infinity. However, it can be shown that for a given set F there


is a unique critical value s0 ∈ [0,∞] such that:

H s = ∞ ∀s < s0

H s = 0 ∀s > s0(A.11)

is critical value s0 is called the Hausdorff dimension of the set F, wrien s0 = dimF.

To recap this definition, first we defined a cover of a set F, which is a family of sets A

whose union contains F. We then defined an ε-cover, which is a cover whose elements each

have a diameter smaller than ε. We then defined H sε (F), which is the smallest possible sum

over the diameters of ε-covers of F raised to the power of s. e Hausdorff measure H s

is defined as limit of H sε (F) as ε approaches zero; in other words, as the diameters of the

elements of the cover of F become vanishingly small, we raise each of these diameters to the

power of s and add them all together. e Hausdorff dimension of a set F is the critical value

s0, below whichH s is infinite and above whichH s is zero. (It is possible thatH s is zero or

infinite for all s, in which case dimF = 0 or∞ respectively.)

e Hausdorff dimension is difficult to calculate in most cases, but to aid in understanding

it might be helpful to intuitively consider how we might compute the Hausdorff dimension

of the line segment F = [0, 1]. First we need to define a cover of F. Many examples present

themselves: the set [−∞,∞] obviously covers F, but for simplicity we will restrict ourselves

to disjoint sets that divide F evenly into n segments of length 1/n. (Strictly the Hausdorff

measure requires us to consider all possible covers, so this derivation is not rigorous.) So,

for example, the two line segments [0, 1/2], (1/2, 1] form a cover of F. If we let ε be 1/n, then

H sε (F) will be equal to n(1/n)s = n1−s. H s is defined as the limit of H s

ε (F) as ε approaches

zero, and so as n approaches infinity:

limn→∞

n1−s (A.12)

For s < 1, the value given by equation A.12 grows without bound, and so the limit approaches


infinity. For s > 1, the value of equation A.12 shrinks, and so the limit approaches zero. But

for the critical value s = 1, equation A.12 gives the value of one for all n. erefore the

critical value s0 = 1, and, given the caveat about the cover restrictions above, the Hausdorff

dimension of the line segment [0, 1] is 1.

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Index

ε-congruence, 110–121as observational equivalence, 112, 120definition, 111

Belousov-Zhabotinsky, see Reaction, Belousov-Zhabotinsky

Bernoulli property, 101Bump function, 116

Chakravary, Anjan, 45Coin flipping, 113Constructivism, 79

Dimensionfractal, of surfaces, 67Hausdorff, 131similarity, 9, 130topological, 9, 130

Dynamical systems, 27, 100Bernoulli, 112continuous, 29discrete, 30iterative, 3, 83

Ergodic theory, 100

Fractals, 31as models, 61, 70–71definitions, 10–11, 77in phase space, 83–87in physical space, 87–97pre-, see Prefractals

Functionbump, see Bump function

Goldblum, JeffShirtless, 1

Instrumentalism, 59

Isomorphism, 105manifest, 105, 106

Kadanoff, Leo, 64Kicked rotator, 32

map, 33quantum, 35standard map, 33

Koch curve, 77as a limit of a sequence of functions, 80construction, 7, 127properties, 128

Magnetic pendulum, 90–97McMullin, Ernan, 97Measure, probability, 101

Observation function, 103Oregonator, 40

Particle in a double well, forced, 84, 88Phase space, 28, 83

aractor, 29, 83basins of araction, 83orbit, see trajectoryreconstruction, 39trajectory, 28

Poincaré map, 85Prefractals, 62, 71–73Properties, point-like, 40

coarse-graining, 41fixed-volume definition, 41limit definition, 41

Reaction, Belousov-Zhabotinsky, 38–40Recursion, see Recursion

SDIC, 5, 28

145

INDEX 146

Self-similarity, 8statistical, 8

Sensitive dependence on initial conditions, seeSDIC

Shenker, Orly, 78Standard map, see Kicked rotatorStochastic processes, 102

semi-Markov, 112Suppes, Patrick, 113

Tribology, 65–69classical, 66fractal, 66

Truth, approximate, 42–54Continuum-mechanics account, 48GM-theory account, 43

Verisimilitude, 52and approximate truth, 54

Wada property, 92Werndl, Charloe, 105, 114Winnie, John, 113

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