WHAT SPLENDOUR, IT ALL COHERES
Notes on complexity, computation, information, and the origins of order.
1"Creation can be simple."
Poundstone, The Recursive Universe (1985)
2“Would it not be extraordinary, however, if these laws could by themselves contrive to generate rich and beautiful patterns?”
Ball, The Self-Made Tapestry (1999), p. 4
3“The formative process is the supreme process, indeed the only one, alike in nature and art.”
Goethe (1790), in Whyte (ed.), Aspects of Form (1951), p. 233
4"A cellular automata machine is a universe synthesizer."
Toffoli & Margolus, Cellular Automata Machines (1987), p. 1
5"Cellular automata are stylized, synthetic universes defined by simple rules much like those of a board game. They have their own kind of matter which whirls around in a space and a time of their own. One can think of an astounding variety of them. One can actually construct them, and watch them evolve. As inexperienced creators, we are not likely to get a very interesting universe on our first try."
Toffoli & Margolus, Cellular Automata Machines (1987), p. 1
6"I did what is in a sense one of the most elementary imaginable computer experiments: I took a sequence of simple programs and then systematically ran them to see how they behaved. And what I found — to my great surprise — was that despite the simplicity of their rules, the behavior of the programs was often far from simple. Indeed, even some of the very simplest programs that I looked at had behavior that was as complex as anything I had ever seen."
Wolfram, A New Kind of Science (2002), p. 2
7“Apparent randomness can arise from very simple deterministic systems.”
Mitchell, Complexity: A Guided Tour (2009)
8"Although the underlying dynamics describing this system is very simple, and entirely deterministic, there is an enormous variety, and complexity, of emergent particle-particle interactions. Suppose that we had been shown such a space-time pattern but were told nothing whatsoever about its origin. How would we make sense of its dynamics? It would take a tremendous leap of intuition to fathom the utter simplicity of the real dynamics."
Ilachinski, Cellular Automata: A Discrete Universe (2001), p. 17–18
9"What we are thus seeing is rather dramatic evidence of the phenomenon of self-organization, in which an initially random state evolves to a state exhibiting long-range correlations and whose overall space-time patterns contain manifestly nonlocal structure."
Ilachinski, Cellular Automata: A Discrete Universe (2001), p. 53–54
10“An apparent global ordered collective dynamics need not stem from an intrinsically global cooperative dynamics, but may instead be nothing more than an emergent high-level construct stemming from an essentially non-cooperative low-level dynamics.”
Ilachinski, Cellular Automata: A Discrete Universe (2001)
11“You put in featureless, indiscriminate energy, and the out-of-equilibrium system uses it to organize itself into patterns that can astonish.”
Ball, The Self-Made Tapestry (1999), p. 52
12"In a cellular automaton, there are no 'particles' — only bits. Yet when we watch the screen, we irresistibly see objects that move, collide, bounce, and scatter. The question of what constitutes an 'object' in a CA, and what it means for that object to 'move,' is more subtle than it first appears. Identity is not given by the substrate but by the pattern."
Toffoli & Margolus, Cellular Automata Machines (1987), p. 101
13“There is better reason than ever for believing that the structure and complexity of our world is inherent in our physical laws and not in some special, unknowable microstate. The universe is a recursively defined geometric object.”
Poundstone, The Recursive Universe (1985)
14"Complexity generically increases with randomness up until a phase transition is reached, beyond which further increases in randomness decrease complexity."
Ilachinski, Cellular Automata: A Discrete Universe (2001), p. 57
15“A completely ordered universe, however, would be dead. Chaos is necessary for life.”
Crutchfield, “Between Order and Chaos,” Nature Physics 8 (2012), p. 22
16“The mere fact of adopting a certain type of modelling, e.g., choosing to describe the situation in discrete time, has already a profound influence on the outcome of the analysis.”
Collet & Eckmann, Iterated Maps on the Interval as Dynamical Systems (1980), p. 1
17"In principle, evolution, life, and intelligence can take place within a world governed by a very simple cellular-automaton rule."
Toffoli & Margolus, Cellular Automata Machines (1987), p. 209
18“No matter how impressive the products of an algorithm, the underlying process always consists of nothing but a set of individually mindless steps succeeding each other without the help of any intelligent supervision.”
Dennett, Darwin’s Dangerous Idea (1995), p. 59
Below a certain threshold of complexity, every automaton can only produce things simpler than itself. Above it, something changes.
19"In a cellular automaton, objects that may be interpreted as passive data and objects that may be interpreted as computing devices are both assembled out of the same kind of structural elements and subject to the same fine-grained laws; computation and construction are just two possible modes of activity."
Toffoli & Margolus, Cellular Automata Machines (1987), p. 9
20“Any system is a channel that communicates its past to its future through its present.”
Crutchfield, “Between Order and Chaos,” Nature Physics 8 (2012), p. 18
21"Complication, on its lower levels, is probably degenerative, that is, every automaton that can produce other automata will only be able to produce less complicated ones. There is, however, a certain minimum level where this degenerative characteristic ceases to be universal. At this point automata which can reproduce themselves, or even construct higher entities, become possible."
von Neumann, Theory of Self-Reproducing Automata (1966), p. 80
22"It is a fundamental fact of the theory of automata that any automaton which is powerful enough to be a universal constructor must also be powerful enough to include a universal computing machine."
von Neumann, Theory of Self-Reproducing Automata (1966), p. 125
23"The secret of self-reproducing programs — and, as we shall see, of self-reproducing molecules — is that one string functions in two ways: first as program, and second as data."
Hofstadter, Gödel, Escher, Bach (1979), p. 495
24"Life — like all computationally universal systems — defines the most efficient simulation of its own behavior."
Ilachinski, Cellular Automata: A Discrete Universe (2001), p. 21
25"Information theory claims that every observation obscures at least as much information as it reveals."
Poundstone, The Recursive Universe (1985)
26“Inside any sufficiently large random broth, we expect just by chance, there will be some of these self-replicating creatures.”
Griffeath & Moore (eds.), New Constructions in Cellular Automata (2003), p. 2
27“We prove a strict hierarchy: globally universal CA form a strict subset of universally self-replicating CA, which form a strict subset of locally universal CA.”
Cotler, Hongler & Hudcova, “Self-Replication and Universality in Cellular Automata” (2025)
28“The existence of self-reproducing machines is only a very special case of a much wider phenomenon, the theory of which might conceivably be applied to situations of quite a different kind from those occurring in biological simulation.”
Myhill, in Burks (ed.), Essays on Cellular Automata (1970), p. 206
What can be communicated? What can be known?
29"The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem."
Shannon, A Mathematical Theory of Communication (1948), p. 3
30"Rarely does it happen in mathematics that a new discipline achieves the character of a mature and developed scientific theory in the first investigation devoted to it."
Khinchin, Mathematical Foundations of Information Theory (1957), p. 3
31"Of course, information existed before Shannon, just as objects had inertia before Newton. But before Shannon, there was precious little sense of information as an idea, a measurable quantity, an object fitted out for hard science. Before Shannon, information was a telegram, a photograph, a paragraph, a song. After Shannon, information was entirely abstracted into bits."
Soni & Goodman, A Mind at Play (2017), p. 10
32"Information is a measure of one's freedom of choice when one selects a message."
Weaver, in Shannon & Weaver, The Mathematical Theory of Communication (1949), p. 100
33"This means that when we write English half of what we write is determined by the structure of the language and half is chosen freely."
Shannon, A Mathematical Theory of Communication (1948), p. 26
35"If a source can produce only one particular message its entropy is zero, and no channel is required. A computing machine set up to calculate the successive digits of π produces a definite sequence with no chance element. No channel is required to 'transmit' this to another point. One could construct a second machine to compute the same sequence at the point."
Shannon, A Mathematical Theory of Communication (1948), p. 31
36“One bit of information — one atom of knowledge, the smallest unit that can exist — is 0.693 of the thermodynamic unit.”
Poundstone, The Recursive Universe (1985)
37“The exact arrangement of molecules in the water is a secret, and the laws of thermodynamics are the rules for keeping the secret. No matter how clever your experiments, you will never learn the complete microstate.”
Poundstone, The Recursive Universe (1985)
38"I don't think I was ever motivated by the notion of winning prizes, although I have a couple of dozen of them in the other room. I was more motivated by curiosity. Never by the desire for financial gain. I just wondered how things were put together. Or what laws or rules govern a situation, or if there are theorems about what one can't or can do. Mainly because I wanted to know myself."
Shannon, Kyoto Prize lecture, in Soni & Goodman, A Mind at Play (2017), p. 337
39"Maybe it is too much to presume that the character of an age bears some stamp of the character of its founders; but it would be pleasant to think that so much of what is essential to ours was conceived in the spirit of play."
Soni & Goodman, A Mind at Play (2017), p. 14
40“In 1943, Alan Turing visits Bell Labs as part of a British-American intelligence liaison, and he and Shannon meet over tea in the cafeteria. The two men share a fascination with the idea of thinking machines.”
Soni & Goodman, A Mind at Play (2017)
41"The specter of information is haunting sciences."
Zurek (ed.), Complexity, Entropy, and the Physics of Information (1990), p. vii
42"Shannon issued his famous warning in a 1956 editorial: information theory had become a bandwagon, and workers in many fields were applying its concepts far beyond their proven domain."
Kline, The Cybernetics Moment (2015), p. 102
43"Words that come to mean everything may finally mean nothing; yet their very emptiness may allow them to be filled with a mesmerizing glamour."
Roszak, quoted in Kline, The Cybernetics Moment (2015), p. 225
44"The collapse of cybernetics as a unifying science in the United States — signaling the end of the cybernetics moment — occurred precisely when information was becoming the keyword of our time."
Kline, The Cybernetics Moment (2015), p. 7
45"John McCarthy recalled, 'one of the reasons for inventing the term artificial intelligence was to escape association with cybernetics.'"
Kline, The Cybernetics Moment (2015), p. 181
46"The concept of information developed in this theory at first seems disappointing and bizarre — disappointing because it has nothing to do with meaning, and bizarre because it deals not with a single message but rather with the statistical character of a whole ensemble of messages. I think, however, that these should be only temporary reactions; and that one should say, at the end, that this analysis has so penetratingly cleared the air that one is now, perhaps for the first time, ready for a real theory of meaning."
Weaver, in Shannon & Weaver, The Mathematical Theory of Communication (1949), p. 116
47"The meaning of a message can be defined as its selective function on the range of the recipient's states of conditional readiness."
MacKay, Information, Mechanism and Meaning (1969), p. 24
48"A noise generator can be regarded as a source of completely original but meaningless information. A logical computer per contra provides completely unoriginal but rigorously meaningful information."
MacKay, Information, Mechanism and Meaning (1969), p. 137
49"It seems important to rid ourselves of the current heresy that only what can be expressed precisely in words has precise meaning. The selective function of an utterance for an individual may obviously be perfectly well-defined, although he can say nothing well-defined about it."
MacKay, Information, Mechanism and Meaning (1969), p. 110
50"There can be a moral element in conceptual blindness."
MacKay, Information, Mechanism and Meaning (1969), p. 114
51"The freedom we attribute to each other is not a matter of convention, but a matter of fact."
MacKay, Information, Mechanism and Meaning (1969), p. 155
52"The value of a message appears to reside not in its information (its absolutely unpredictable parts), nor in its obvious redundancy (verbatim repetitions, unequal digit frequencies), but rather in what might be called its buried redundancy — parts predictable only with difficulty, things the receiver could in principle have figured out without being told, but only at considerable cost in money, time, or computation."
Bennett, in Pines (ed.), Emerging Syntheses in Science (1988), p. 222
53"In general, when sufficient physical insight is lacking and one makes more or less arbitrary probabilistic assumptions about the data and proceeds to make logical deductions, the results must be considered irrelevant to the task at hand, which is to learn from the data."
Rissanen, Stochastic Complexity in Statistical Inquiry (1989), p. 3
54"It is my experience from talks I have given on the subject that a person well indoctrinated in traditional statistical thinking may have greater difficulty in understanding the new concepts than one who is not so blessed."
Rissanen, Stochastic Complexity in Statistical Inquiry (1989), Preface
55“Kolmogorov complexity and Shannon entropy both increase monotonically with disorder, making them measures of randomness rather than complexity. When we call a random pattern ‘not complex,’ we are implicitly assigning it to an ensemble, and it is the ensemble that is simple.”
Grassberger, “Toward a Quantitative Theory of Self-Generated Complexity” (1986)
56"A random variable is neither a variable nor random."
Green & Swets, Signal Detection Theory and Psychophysics (1966), p. 358
57“Information like liquids ‘has volume but no shape’: the amount of information is measurable by a scalar. Just as the time necessary for conveying the liquid content of a large container through a pipe is determined by the ratio of the volume of the liquid to the cross-sectional area of the pipe, the transmission rate equals the ratio of two numbers, one depending on the source, the other on the channel.”
Csíszár & Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems (1981)
58"It from bit. Otherwise put, every it — every particle, every field of force, even the spacetime continuum itself — derives its function, its meaning, its very existence entirely — even if in some contexts indirectly — from the apparatus-elicited answers to yes-or-no questions, binary choices, bits."
Wheeler, in Zurek (ed.), Complexity, Entropy, and the Physics of Information (1990), p. 5
59“Build physics, with its false face of continuity, on bits of information!”
Wheeler, in Zurek (ed.), Complexity, Entropy, and the Physics of Information (1990)
Where does complexity live? Not in order, not in chaos, but somewhere between.
60"Ordered systems are too frozen to coordinate complex behavior; chaotic systems are too wild to coordinate complex behavior; systems near the edge of chaos are most likely to coordinate complex behavior."
Kauffman, The Origins of Order (1993), p. 134
61"Our experiment produced very different results, and we suggest that the interpretation of the original results is not correct."
Mitchell, Hraber & Crutchfield, "Revisiting the Edge of Chaos" (1993), p. 1
62"A completely ordered universe, however, would be dead. Chaos is necessary for life."
Crutchfield, "Between Order and Chaos," Nature Physics (2011), p. 23
63"Patterns live on the edge, in a fertile borderland between these extremes, where small changes can have large effects. Pattern appears when competing forces banish uniformity but cannot quite induce chaos. It sounds like a dangerous place to be, but it is where we have always lived."
Ball, The Self-Made Tapestry (1999)
64“Determinism and randomness are not opposites but essential and unavoidable twins.”
Crutchfield, “Between Order and Chaos,” Nature Physics (2011)
65“The crucial lesson: maximal randomness (fair coin) has zero structural complexity, and perfect periodicity has zero randomness but nonzero structural complexity. The two quantities are genuinely independent.”
Crutchfield, “Between Order and Chaos,” Nature Physics (2011)
66"Scaling is the observation; the renormalisation group is the explanation."
Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (1992)
67“Universality lurks just below the surface of almost any system complex enough to be interesting.”
Moore & Mertens, The Nature of Computation (2011)
68"An ice cube floating in a glass of water, at just the melting point, poses a genuine conceptual problem in theoretical physics."
Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (1992)
69"The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious."
Tauber, Critical Dynamics (2014), p. 5
70"The physics lies in the fluctuations."
Frieden, Physics from Fisher Information (1998), p. 3
71“The phenomenon of critical opalescence is one of the most visually dramatic manifestations of critical phenomena. Normally transparent fluids become milky white near the critical point, because density fluctuations occur on all length scales, including those comparable to the wavelength of visible light.”
Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (1992)
72“Quantitative theories of physics are possible because macroscale phenomena are often independent of microscopic details.”
Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity (2021), p. 353
73"We conclude from this fact that undecidability and computational irreducibility are not good measures for physical complexity."
Israeli & Goldenfeld, "Computational Irreducibility and the Predictability of Complex Physical Systems" (2006), p. 17
74“The choice of macroscopic states is not a detail — it is the entire problem.”
Crutchfield & Shalizi, “Thermodynamic Depth of Causal States” (1999)
75"It was a sort of investigation where you got a good lead, and certainly you had to pursue that; and before you reached the end of that lead up opened another, and this was if anything even more fascinating. One kept on going in several stages; one beautiful lead opening up after the other, and every one much too good to abandon."
Onsager, quoted in Domb, The Critical Point (1996), p. 130
76“Imagine poor Ising’s disappointment on deriving this result!”
Binney et al., The Theory of Critical Phenomena (1992), p. 63
77"Without broken symmetry, we would have no way of making a measurement — no rigid rods, no stable configurations."
Anderson, Basic Notions of Condensed Matter Physics (1984)
78“Symmetry-breaking is a paradox only if you confuse the symmetry of the underlying laws with the symmetry of the things that those laws govern.”
Stewart & Golubitsky, Fearful Symmetry (1992), p. 14
79“Symmetry is not a number, not a shape, not a property: it is a transformation — an action — that leaves the thing apparently unchanged.”
Stewart & Golubitsky, Fearful Symmetry (1992), p. 28
80“The Geometer God constructed the universe using symmetry as a guide: then She broke it.”
Stewart & Golubitsky, Fearful Symmetry (1992), p. 127
81“If the grand unified theories are right, there were symmetries in the infant universe that no longer exist. The universe was like a marble perfectly balanced on the tip of a flagpole.”
Poundstone, The Recursive Universe (1985)
82"These statistical laws resulting from the very presence of a large number of particles forming the body cannot in any way be reduced to purely mechanical laws."
Landau & Lifshitz, Statistical Physics, Part 1 (3rd ed., 1980), p. 1
83"According to the results of statistics, the universe ought to be in a state of complete statistical equilibrium. Everyday experience shows us, however, that the properties of Nature bear no resemblance to those of an equilibrium system; and astronomical results show that the same is true throughout the vast region of the Universe accessible to our observation."
Landau & Lifshitz, Statistical Physics, Part 1 (3rd ed., 1980), p. 30–31
84"Anomalous dimensions are the fingerprints of the microscopic world, persisting at macroscopic scales through the collective action of fluctuations at all intermediate scales."
Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (1992)
85“There is something spooky about entropy and the second law of thermodynamics. Entropy seems to be the only quantity in physics that requires a particular direction for time.”
Poundstone, The Recursive Universe (1985)
86“The presence of chaos in a system implies that perfect prediction is impossible not only in practice but also in principle, since we can never know x(0) to infinitely many decimal places.”
Mitchell, Complexity: A Guided Tour (2009)
87“A chaotic map possesses three ingredients: unpredictability, indecomposability, and an element of regularity.”
Devaney, An Introduction to Chaotic Dynamical Systems (1989), p. 50
88“If F has a periodic point of period 3, then F has periodic points of all periods.”
Devaney, An Introduction to Chaotic Dynamical Systems (1989), p. 62
89“The period-doubling route to chaos is one of the most commonly observed transitions from simple to complicated behavior in parameterized families of maps. It is also universal: the same constants arise for all families.”
Devaney, An Introduction to Chaotic Dynamical Systems (1989), p. 130
90“The two sets (periodic and aperiodic) are intimately intertwined: between any two values of μ for which the orbit is periodic, there is one for which it is aperiodic, and vice versa.”
Collet & Eckmann, Iterated Maps on the Interval as Dynamical Systems (1980), p. 30
91"Sudden cardiac death is a problem in topology."
Winfree, When Time Breaks Down (1987), p. 171
92"The singularity is a state, not merely a stimulus."
Winfree, When Time Breaks Down (1987), p. 55
93"The rotor is a singularity incarnate in a physical medium."
Winfree, When Time Breaks Down (1987), p. 119
94“The theory of mutually entraining oscillators has a potential importance comparable to the theory of general cooperative phenomena which has been one of the central subjects of statistical and solid-state physics.”
Kuramoto, Chemical Oscillations, Waves, and Turbulence (1984), p. 60
95“The critical state is a global attractor of the dynamics.”
Aschwanden, Self-Organized Criticality in Astrophysics (2011), p. 40
Pattern and form. The hurricane or the jigsaw?
96"The great question of developmental theory is this: Do the self-organizing structures of living organisms paint themselves into three-dimensional pictures by joining together structurally, bit by bit in building-block or jigsaw-puzzle fashion, in sufficient strength to overcome the hurricane, or does development at some stage make use of the hurricane and produce order out of chaos in the Boltzmann-statistical manner?"
Harrison, Kinetic Theory of Living Pattern (1993), p. 24
97"Anyone can feel this wind when hearing the remarkably evocative music played. Only a fully trained musician can feel it when looking silently at the printed score. When one contemplates an electron micrograph, and then a diagram of chemical changes in biochemical cycles, one should feel first the storm which raises the question and then the great determined flows through it which shape the organism."
Harrison, Kinetic Theory of Living Pattern (1993), p. 30
98"The terms in equations have functions in the organism, as groups of amino acids do. Here is an alpha-helix; it helps to make the protein span a membrane. Here is a cubic term; it helps the organism to make stripes."
Harrison, Kinetic Theory of Living Pattern (1993), p. 264
99"Scientific theory is an art in which the surface on which to elaborate the details is not available a priori, but is itself a product of the enterprise."
Harrison, Kinetic Theory of Living Pattern (1993), p. 322
100"Diffusion is usually considered a stabilising process which is why this was such a novel concept."
Murray, Mathematical Biology II (2003), on Turing's reaction-diffusion instability
101“It took until 1990 for the first conclusive experimental evidence of Turing patterns to appear.”
Epstein & Pojman, An Introduction to Nonlinear Chemical Dynamics (1998), p. 299
102"From a theoretical point of view, since we are not near equilibrium there is no a priori reason to suppose that we have a Gibbs ensemble or a free energy functional whose minima yield the patterns obtained under given external conditions."
Cross & Hohenberg, "Pattern Formation Outside of Equilibrium," Rev. Mod. Phys. 65 (1993), p. 855
103"The most interesting aspect of mollusk shell patterns is that they are generated one row at a time, during growth, so that the two-dimensional pattern on the shell surface represents a space-time record of a one-dimensional solution of the model equations."
Cross & Hohenberg, "Pattern Formation Outside of Equilibrium," Rev. Mod. Phys. 65 (1993), p. 1060
104"We provide the first quantitative evidence, via the local information dynamics framework, for the long-held conjecture that in cellular automata, domains are information storage, particles are information transfer, and particle collisions are information modification."
Lizier, Prokopenko & Zomaya, "Local Information Dynamics" (2012)
105"Perhaps we should turn the pattern formation question around and ask: 'What patterns cannot be formed by such simple mechanisms?'"
Murray, Mathematical Biology II (2003)
106“The fundamental importance of pattern and form in biology is self-evident. Whatever pattern we observe in the animal world, it is almost certain that the process that produced it is still unknown. The genes themselves cannot create the pattern.”
Murray, Mathematical Biology II (2003)
107“Classification schemes are the holy grail in the theory of cellular automata.”
Hadeler & Müller, Cellular Automata: Analysis and Applications (2017), p. 111
108“There is no general method to directly link the definition of a cellular automaton with its global, long-term behavior.”
Hadeler & Müller, Cellular Automata: Analysis and Applications (2017), p. 374
109"As on the sea's wrinkled surface, it is the pattern that remains, not its individual components."
Ball, The Self-Made Tapestry (1999)
110"If life were started from scratch a thousand times over, it would every time alight on these fundamental structures eventually."
Ball, The Self-Made Tapestry (1999)
111"Here also order seems to spring from chaos, and this is perhaps even more remarkable as the crystal has no metabolic machinery, and cannot capture and store energy."
Bonner, Morphogenesis: An Essay on Development (1952), p. 26
112"Why does nature appear to use only a few fundamental forms in so many different contexts? Why does the branching of trees resemble that of arteries and rivers? Why do crystal grains look like soap bubbles and the plates of a tortoise shell?"
Stevens, quoted in Adam, Mathematics in Nature (2003), p. 4
113"Instability is not merely the breakdown of order — it is the mechanism by which nature spontaneously generates spatial structure."
Adam, Mathematics in Nature (2003)
114"No river, regardless of size, runs straight for more than about ten times its average width."
Adam, Mathematics in Nature (2003)
115“The symmetry of a pattern formed by a symmetry-breaking force does not always reflect the symmetry of that force. If you heat a shallow pan of oil, it will develop roughly hexagonal circulation cells. The system was initially uniform, and the symmetry-breaking force was also applied uniformly — yet suddenly this uniformity is lost. Where has this sixfold pattern come from?”
Ball, The Self-Made Tapestry (1999)
116“The diverse range of pelt patterns and markings can be explained with the same basic mechanism. William of Ockham would have reminded us that if we can account not only for the leopard’s spots but also for the zebra’s stripes and the giraffe’s dapples with the same theoretical model, that is surely more satisfactory.”
Ball, The Self-Made Tapestry (1999)
117“There is no way we could detect the ‘presence’ of this geometric principle by decoding the genetic information in the bee’s DNA. We could see that they make use of it only by watching the organism as a whole go about her job.”
Ball, The Self-Made Tapestry (1999)
118“It is as though individuals in a jabbering crowd were able to converse with one another from opposite sides of a room.”
Ball, The Self-Made Tapestry (1999)
119“Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed. They are no exceptions to the rule that God always geometrizes.”
D’Arcy Thompson, On Growth and Form (1917/1942), p. 10
120“The beauty of life is, therefore, geometrical beauty of a type that Plato would have much appreciated.”
J. D. Bernal, The Origin of Life, epigraph in Winfree, The Geometry of Biological Time
121“The bee makes no claim to skill in mathematics; but the mathematician has much to learn from the bee’s honeycomb.”
D’Arcy Thompson, On Growth and Form (1917/1942), p. 525
122“The engineer, who had been busy designing a new and powerful crane, saw in a moment that the arrangement of the bony trabeculae was nothing more nor less than a diagram of the lines of stress, or directions of tension and compression, in the loaded structure: in short, that Nature was strengthening the bone in precisely the manner and direction in which strength was required; and he is said to have cried out, ‘That’s my crane!’”
D’Arcy Thompson, On Growth and Form (1917/1942), p. 977
123“An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be a much harder thing to understand than the most widespread and remote departures from such a condition.”
D’Arcy Thompson, On Growth and Form (1917/1942), p. 1027
124"The developing organism is at all times a going concern, at all times a functioning whole, and not merely a set of parts being assembled. It is self-forming, and the 'information' which guides its development is not a blueprint external to the process but is embedded in the process itself."
Berrill, in Whyte (ed.), Aspects of Form (1951), p. 48
125“Equilibrium is a dull place to be. Nothing happens there. If the Universe were itself at thermodynamic equilibrium, it would be a lifeless place pervaded by a uniform, dim glow of just a few degrees above absolute zero. Just about every phenomenon that interests us is an out-of-equilibrium process — life, to mention one.”
Ball, The Self-Made Tapestry (1999)
126"A perfect crystal is, from the point of view of growth, a dead crystal."
F. C. Frank, in Whyte (ed.), Aspects of Form (1951), p. 19
127"Fractal geometry will make you see everything differently. There is danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpets, bricks, and much else besides."
Barnsley, quoted in Flake, The Computational Beauty of Nature (1998), p. 93
128“When two waves in an excitable medium collide, they annihilate one another, because each wave leaves behind it a refractory region in which the system cannot be excited. This behavior is exactly the opposite of what occurs with linear waves such as sound or light, which pass through one another.”
Epstein & Pojman, An Introduction to Nonlinear Chemical Dynamics (1998), p. 124
129“Between any two entrained regions, no matter how close together in frequency ratio, there is always another entrained region — a chemical analog of the number-theoretic result that between any two rational numbers there is always another rational.”
Epstein & Pojman, An Introduction to Nonlinear Chemical Dynamics (1998), p. 240
130“The BZ reaction in a Petri dish provides some of the most beautiful and instructive demonstrations in all of chemistry.”
Epstein & Pojman, An Introduction to Nonlinear Chemical Dynamics (1998), p. 116
What does it mean for a physical system to compute?
131"A shockingly wide variety of physical systems — from ordinary physical objects to biological cells to brains to the universe as a whole — have come to be regarded by researchers and scholars in multiple disciplines as 'computational.' The notion of computation may seem to have lost its distinctiveness."
Anderson & Piccinini, The Physical Signature of Computation (2024), p. 21–22
132"The pansculpturalist would declare that a block of marble is every sculpture that could be chiseled from it because they can interpret the block that way, denying any real distinction between an imagined surface within the block of marble and a real physical surface rendered from the chiseling of the block to expose it."
Anderson & Piccinini, The Physical Signature of Computation (2024), p. 96
133"For any given sequence of physical microstates visited in the arbitrary time evolution of any sufficiently complex system (e.g., a rock), there exists an assignment of physical microstates to computational states such that the time evolution of the microstates is consistent with the system having performed the computation f. This is not very impressive."
Anderson & Piccinini, The Physical Signature of Computation (2024), p. 144
134"It would be absurd to maintain that all there is to sunshine, rain, snow, or hail is that it is the output of a computation. Computational outputs may be taken to represent sunshine, rain, snow, or hail, but they make nothing illuminated, wet, or cold."
Anderson & Piccinini, The Physical Signature of Computation (2024), p. 190
135“Whatever policy learned inside of this generated environment will achieve a perfect score most of the time, but will obviously fail when unleashed into the harsh reality of the actual world, underperforming even a random policy.”
Ha & Schmidhuber, “Recurrent World Models Facilitate Policy Evolution” (2018), p. 6
136"We believe that the boundary between easy and hard problems, between the computable and the uncomputable, is not an artifact of a particular mathematical formalism. We believe it is carved into the fabric of the universe."
Moore & Mertens, The Nature of Computation (2011), p. 18
137"If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in 'creative leaps,' no fundamental gap between solving a problem and recognizing the solution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss."
Moore & Mertens, The Nature of Computation (2011)
138"Gödel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved."
Hofstadter, Gödel, Escher, Bach (1979), p. 27
139"It is an inherent property of intelligence that it can jump out of the task which it is performing, and survey what it has done; it is always looking for, and often finding, patterns."
Hofstadter, Gödel, Escher, Bach (1979), p. 46
140"The fascinating thing is that any such system digs its own hole; the system's own richness brings about its own downfall."
Hofstadter, Gödel, Escher, Bach (1979), p. 454
141"I think; therefore I have no access to the level where I sum."
Hofstadter, Gödel, Escher, Bach (1979)
142"The self comes into being at the moment it has the power to reflect itself."
Hofstadter, Gödel, Escher, Bach (1979), p. 698
143"'Know thyself' seems to be an impossible command for a computer program. If a model of computation is so weak that it cannot ask questions about programs, then it will skirt the abyss, but at the cost of being too stupid to solve interesting problems."
Flake, The Computational Beauty of Nature (1998), p. 47
144"Computational mechanics — in its focus on letting the process speak for itself through (possibly impoverished) measurements — follows the spirit that motivated one approach to experimentally testing dynamical systems theory."
Shalizi & Crutchfield, "Computational Mechanics" (2001), p. 22
145“The Church-Turing thesis is one of the most remarkable convergences in the history of ideas. A dozen different formalisms were proposed for capturing the notion of ‘effective computation.’ Every single one turned out to define exactly the same class of computable functions.”
Moore & Mertens, The Nature of Computation (2011)
146“The universality of NP-completeness is one of the most remarkable phenomena in all of mathematics. Thousands of problems, arising from the most diverse corners of science, engineering, and mathematics, turn out to be equivalent to each other.”
Moore & Mertens, The Nature of Computation (2011)
147“Zero-knowledge proofs are among the most paradoxical objects in all of mathematics. They allow you to convince someone that a statement is true without conveying any information beyond the truth of the statement itself.”
Moore & Mertens, The Nature of Computation (2011)
148“IP = PSPACE tells us that conversation is extraordinarily powerful: a single conversation with an untrusted genius is as powerful as an exponential amount of trusted computation.”
Moore & Mertens, The Nature of Computation (2011)
149“Randomness and hardness are two sides of the same coin: the existence of hard problems lets us generate pseudorandomness, and pseudorandomness lets us remove the need for true randomness.”
Moore & Mertens, The Nature of Computation (2011)
150“The geometry of the solution space of a random formula undergoes a series of remarkable transitions as the density of clauses increases. At low densities, the solutions form a single giant connected cluster. At higher densities, this cluster shatters into an exponential number of smaller clusters.”
Moore & Mertens, The Nature of Computation (2011)
151“In the frozen phase, the landscape of solutions is like an archipelago of tiny islands in an exponentially vast ocean, and no local algorithm can island-hop its way to satisfaction.”
Moore & Mertens, The Nature of Computation (2011)
152“The theory of computation is not just about computers — it is about the nature of the physical universe, and what it allows us to know.”
Moore & Mertens, The Nature of Computation (2011)
153“If a computation is encoded as an attractor of a dynamical system, then small perturbations are automatically corrected as the system relaxes back to the attractor. The system is self-repairing.”
Pines (ed.), Emerging Syntheses in Science (1988), p. 168
154“Griffeath and Hickerson construct a finite Game of Life initial seed — exactly 2,392 cells — whose growing crystal has an asymptotic density of (3 − √5)/90, an irrational number.”
Griffeath & Moore (eds.), New Constructions in Cellular Automata (2003)
155“The right strategy is to formalize pattern, then define complexity as the amount of pattern, then define organization as the growth of complexity, and finally define self-organization as self-generated growth of complexity.”
Shalizi, Causal Architecture, Complexity and Self-Organization (PhD thesis, 2001)
156"An epsilon-machine reconstruction algorithm takes in data and gives back a representation of causal patterns, suitable for use in prediction or intervention, ‘untouched by human hands.’ Such an algorithm is a phenomenological engine."
Shalizi, Causal Architecture, Complexity and Self-Organization (PhD thesis, 2001), p. 126
How do you measure complexity? The question is harder than it looks.
157“The term ‘complexity’ has been used without qualification by so many authors, both scientific and nonscientific, that it has been almost stripped of its meaning.”
Feldman & Crutchfield, “Measures of Statistical Complexity: Why?” (1998)
158"If we think of complexity as a physical property of an object (such as mass or entropy) then there is a puzzle. Objects can be copied. A bull is a complex object. Are seven bulls seven times as complex as one bull? Can complexity proliferate so cheaply?"
Lloyd & Pagels, "Complexity as Thermodynamic Depth" (1988), p. 187
159"It took billions of years for the earth to evolve one bull; but one bull and a few compliant cows will produce seven bulls relatively speedily."
Lloyd & Pagels, "Complexity as Thermodynamic Depth" (1988), p. 187
160"The algorithmic definition of complexity is really a definition of randomness (a profound one)."
Lloyd & Pagels, "Complexity as Thermodynamic Depth" (1988), p. 188
161"Take a glass and smash it with a hammer. Keep on smashing, and you will in short order create a mass of shards that contains an amount of mutual information greater than that between all the DNA in the human body. Since a bowl of glass dust ought not to be more complex than human genes, mutual information ought not to be considered a measure of complexity."
Lloyd & Pagels reporting Bennett, "Complexity as Thermodynamic Depth" (1988), p. 200
162"A cat following a bird with its eyes need not do floating-point arithmetic to know where to pounce."
Lloyd & Pagels, "Complexity as Thermodynamic Depth" (1988), p. 189
163“Periodic behavior has low information content; purely random behavior has high information content; but both are simple. The interesting behavior lies between these extremes, where a process is an amalgam of regular and stochastic components.”
Crutchfield & Young, “Inferring Statistical Complexity” (1989), p. 105
164"The genetic complexity of an organism is proportional to the amount of genetic information tried out and discarded by the process of natural selection on the ancestors of the organism."
Lloyd & Pagels, "Complexity as Thermodynamic Depth" (1988), p. 209
165“Statistical mechanics handles randomness and disorder well, but it lacks a coherent, principled way to describe, quantify, and detect the many kinds of structure nature exhibits.”
Shalizi & Crutchfield, “Computational Mechanics” (2001)
166“Epsilon-machines are simultaneously maximally predictive, minimally complex, unique, and minimally stochastic.”
Shalizi & Crutchfield, “Computational Mechanics” (2001)
167“Ockham’s razor is a consequence, not an assumption.”
Crutchfield, “Between Order and Chaos,” Nature Physics (2011)
168“One is ‘swimming around in Occam’s pool’ of possible state representations, and epsilon-machines sit at the shallowest point.”
Crutchfield & Shalizi, “Thermodynamic Depth of Causal States” (1999)
169"Complexity, on this view, must be a function of the process that generates configurations; we need movies, not snapshots. But this should not be distressing to physicists: we, of all people, should be very suspicious if pattern appeared without a causal history to back it up."
Shalizi, Causal Architecture, Complexity and Self-Organization (PhD thesis, 2001), p. 113
170“‘Organized’ does not equal ‘ordered’ — order means low entropy, while organization lies between order and randomness.”
Shalizi et al., “Automatic Filters for the Detection of Coherent Structure” (2005)
171"To call something emergent is therefore not to say anything about the property at all, but merely to make a confession of scientific and mathematical incompetence."
Shalizi, Causal Architecture, Complexity and Self-Organization (PhD thesis, 2001), p. 115
172"We are not trying to explain everything we can measure; we are trying to find what's intrinsically important in our measurements. Emergence is anti-regression."
Shalizi, Causal Architecture, Complexity and Self-Organization (PhD thesis, 2001), p. 116
173"No unbiased estimator for entropy or mutual information exists."
Paninski, "Estimation of Entropy and Mutual Information" (2003)
Order for free. The heresy that won't die.
174"If we take selection as the sole source of order, it is because we have come to suppose that without selection there could be only chaos."
Kauffman, The Origins of Order (1993), p. xiii–xviii
175"In sufficiently complex systems, selection cannot avoid the order exhibited by most members of the ensemble."
Kauffman, The Origins of Order (1993), p. 45–46
176"Life is an expected, emergent collective property of complex systems of polymer catalysts."
Kauffman, The Origins of Order (1993), p. 287
177"The fundamental order lies deeper, the routes to life are broader."
Kauffman, The Origins of Order (1993), p. 341
178"To suppose, as I do, that such an intellectual task may one day be achieved is, among other things, to suspect with quiet passion that below the particular teeming molecular traffic in each cell lie fundamental principles of order any life would reexpress."
Kauffman, The Origins of Order (1993), p. 362
179“It is a fundamental question whether metabolic stability and epigenesis require the genetic regulatory circuits to be precisely constructed.”
Kauffman, “Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets” (1969), p. 438
180“The formation of a connected web of metabolic transformations arises almost inevitably in a sufficiently complex system of organic molecules and polymer catalysts. From the outset, life possessed a certain inalienable holism.”
Kauffman, The Origins of Order (1993)
181“Cell types are the attractors of the underlying genomic regulatory network.”
Kauffman, The Origins of Order (1993)
182"The usual biologists' explanations are fundamentally hierarchical. You don't try to get back to the boss and change his mind. The essence of a Turing model is the existence of feedback loops in which everything interacts back on everything else and there is no boss."
Harrison, Kinetic Theory of Living Pattern (1993), p. 175
183“Contrary to intuition, morphogenesis may be deeply robust. Organisms, rather than being tinkered-together contraptions, may exhibit a nearly inevitable and stable order.”
Kauffman, The Origins of Order (1993)
184"Unfortunately, this is a statement that is 'obviously true' but turns out to be false."
Durrett & Limic, "Rigorous Results for the NK Model" (2003), p. 1716
185“The beautiful fractal patterns of rules like 60, 90, and 150 are infinitely fragile. They exist only at a measure-zero point in parameter space and have no robustness whatsoever to stochastic perturbations.”
Edlund & Jacobi, “Renormalization of Cellular Automata and Self-Similarity” (2011)
186“The very capacity to evolve may itself be subject to evolution and may have its own lawful properties.”
Kauffman, The Origins of Order (1993)
187"No matter how strong selection may be, adaptive processes cannot climb higher peaks than afforded by the fitness landscape. This limitation cannot be overcome by stronger selection."
Kauffman, The Origins of Order (1993), p. 63–64
188"Evolution is not just 'chance caught on the wing.' It is not just a tinkering of the ad hoc, of bricolage, of contraption. It is emergent order honored and honed by selection."
Kauffman, The Origins of Order (1993), p. 312
189"If I were to give an award for the single best idea anyone has ever had, I'd give it to Darwin, ahead of Newton and Einstein and everyone else. In a single stroke, the idea of evolution by natural selection unifies the realm of life, meaning, and purpose with the realm of space and time, cause and effect, mechanism and physical law."
Dennett, Darwin's Dangerous Idea (1995), p. 21
190"No matter how impressive the products of an algorithm, the underlying process always consists of nothing but a set of individually mindless steps succeeding each other without the help of any intelligent supervision."
Dennett, Darwin's Dangerous Idea (1995), p. 59
191"The actual is, as a matter of brute historical fact, a Vanishingly small subset of the possible."
Dennett, Darwin's Dangerous Idea (1995), p. 104
192“Searching through the Library of Mendel for the best genomes is rather like searching for a particular book in the Library of Babel, but with one important difference: there is a non-miraculous, indeed algorithmic, process that has been running for billions of years.”
Dennett, Darwin’s Dangerous Idea (1995), p. 222
193"Should it be that much harder for an algorithmic process to write an ode to a nightingale or a poem as lovely as a tree? Surely Orgel's Second Rule is correct: Evolution is cleverer than you are."
Dennett, Darwin's Dangerous Idea (1995), p. 451
194"A scholar is just a library's way of making another library."
Dennett, Darwin's Dangerous Idea (1995), p. 346
195“The fitness-based approach is no better than random search, demonstrating that the objective function, rather than helping, has actually been misleading search away from the true answer.”
Lehman & Stanley, “Exploiting Open-Endedness to Solve Problems Through the Search for Novelty” (2008), p. 6
196“Rather than measuring novelty explicitly, nature is guided by a single, fundamental constraint: survive long enough to reproduce. Surprisingly, this simple constraint produces both complexity and diversity in a continual process unparalleled by any algorithm to date.”
Brant & Stanley, “Minimal Criterion Coevolution” (2017), p. 67
197“Unrestricted adaptability requires that the adaptive system be able initially to generate any of the programs of some universal computer.”
Holland, in Burks (ed.), Essays on Cellular Automata (1970), p. 299
Matter that dances.
198"When I first read a biology textbook, it was like reading a thriller. Every page brought a new shock. As a physicist, I was used to studying matter that obeys precise mathematical laws. But cells are matter that dances."
Alon, An Introduction to Systems Biology (2006)
199"Although cells evolved to function and did not evolve to be comprehensible, simplifying principles make biological design understandable to us."
Alon, An Introduction to Systems Biology (2006)
200“I am happy to still have a sense of wonder when reading the chapters. The wonder comes because networks of thousands of interacting components are generally incomprehensible, yet simplifying principles can still be found.”
Alon, An Introduction to Systems Biology (2006)
201“Evolution converges again and again to the same network motifs in transcription networks. This suggests that motifs are selected because they confer an advantage relative to other circuit designs.”
Alon, An Introduction to Systems Biology (2006)
202“Development is the remarkable process in which a single cell, an egg, becomes a multicellular organism. During development, cells must assume different fates in a spatially organized manner.”
Alon, An Introduction to Systems Biology (2006)
203"There is a great feeling of satisfaction, that no one can deny, in being able to take a living phenomenon and apply the ruler and balance to it."
Bonner, Morphogenesis: An Essay on Development (1952), p. 57
204“Neither development nor crystallization are downhill processes, yet each will occur spontaneously. They are machines, as Aristotle thought of them, but unlike clocks or mechanical devices, they do not decrease their thermodynamic order, but increase it by bleeding the environment.”
Bonner, Morphogenesis: An Essay on Development (1952), p. 27
205"The principal function of the higher centres, which must pilot a vulnerable animal through a changing and hostile world, a world where chaos and cosmos are interlaced and superimposed, where anything may happen, but nothing happens twice, is to deal with uncertainty."
W. Grey Walter, in Whyte (ed.), Aspects of Form (1951), p. 179
206"The question is not whether matter has form, but whether form has causal efficacy."
Toulmin, in Whyte (ed.), Aspects of Form (1951), p. 121
207"There is no gruesome vitalistic fascination about viruses; they have the same gruesome mechanistic fascination as nitro-glycerine or a soufflé."
Pirie, in Whyte (ed.), Aspects of Form (1951), p. 7
208"Of all regular tessellations that tile the plane, hexagons enclose the most area for a given perimeter — a fact bees have exploited for millions of years, and that Thomas Hales proved rigorously only in 1999."
Adam, Mathematics in Nature (2003)
209"The deep ideas of computation are intimately related to the deep ideas of life and intelligence."
Mitchell, Complexity: A Guided Tour (2009), p. xi
210“This is a story about dynamics: about change, flow, and rhythm, mostly in things that are alive. The subject matter being dynamics, we are embarked upon a study of temporal morphology, of shapes not in space so much as in time.”
Winfree, The Geometry of Biological Time (2nd ed., 2001)
211“One can visualize the necessity of some such irregularity by trying to construct a smooth surface of data points over the square, subject to the constraint that along the boundary it must rise up one floor like a parking garage rampway. Think how this is arranged in the garage: It isn’t.”
Winfree, The Geometry of Biological Time (2nd ed., 2001)
212“The phase singularity itself is not an instability, but just an alternative — and fatal — mode of operation in normal tissue.”
Winfree, The Geometry of Biological Time (2nd ed., 2001)
213“A lot of behavioral physiology is temporally organized in periodic patterns. If I had to decide what impresses me as the single most conspicuous feature of natural ecosystems, I would say that it is the daily and seasonal periodism.”
Winfree, The Geometry of Biological Time (2nd ed., 2001)
214"Lenia: a continuous generalization of cellular automata in which space, time, and state are all made continuous. A handful of parameters — a kernel peaks vector, a growth center, a growth width — generate over 400 species organized into 18 families. The parameter space maps exactly to Wolfram's four classes: desert, savannah, forest, and river — where the river is where the lifeforms swim."
Chan, "Lenia — Biology of Artificial Life" (2019)
215“The Nagel-Schreckenberg traffic model: the addition of a single stochastic step to deterministic acceleration and deceleration rules produces spontaneous traffic jam formation from homogeneous initial conditions.”
Boccara, Modeling Complex Systems (2nd ed., 2010)
216“Even without any background mutation, this state transition occurs with roughly the same frequency. It is not simply random bit-flipping that causes self-replicators to arise.”
Agmon et al., “Computational Life” (2024), p. 6
217"Anyone who accepts small risks of losing everything will lose everything."
Poundstone, Fortune's Formula (2005), p. 297
Departures, echoes.
218"The formative process is the supreme process, indeed the only one, alike in nature and art."
Goethe (1790), quoted in Whyte (ed.), Aspects of Form (1951), p. 233
219“The word complex comes from the Latin root plectere: to weave, entwine.”
Mitchell, Complexity: A Guided Tour (2009)
220"The great beauty and power of a mathematical theory or model lies in the separation of the relevant from the irrelevant."
Pierce, Symbols, Signals and Noise (1961), p. 45
221"Mathematically, white Gaussian noise is the least predictable. To a human being, however, all white Gaussian noise sounds alike."
Pierce, Symbols, Signals and Noise (1961), p. 251
222"It is conceivable that there are processes which organize themselves into conditions so complex that no human being can grasp them. They would be so organized, in other words, that they would look very like noise."
Shalizi, Causal Architecture, Complexity and Self-Organization (PhD thesis, 2001), p. 119
223"No one really knows what entropy really is."
von Neumann, quoted in Poundstone, Fortune's Formula (2005), p. 58
224"Every room is its own recording studio."
Hofstadter, Gödel, Escher, Bach (1979)
225"The mind is the effect, not the cause, of the brain's being the way it is."
Dennett, Darwin's Dangerous Idea (1995), p. 370
226"Reductionism, roughly speaking, is the view that everything in this world is really something else, and that the something else is always in the end unedifying. So lucidly formulated, one can see that this is a luminously true and certain idea."
Gellner, Legitimation of Belief (1974), p. 107
227"Shannon and Thorp had built a machine that beat a casino at its own game, and then more or less lost interest in using it."
Soni & Goodman, A Mind at Play (2017)
228"Even in wartime, Shannon carved out space for the kind of undirected thinking that would produce his greatest breakthroughs."
Soni & Goodman, A Mind at Play (2017)
229“Good problems and mushrooms of certain kinds have something in common: they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near.”
Polya, epigraph in Winfree, The Geometry of Biological Time (2nd ed., 2001)
230“Curiosity reward should be proportional to predictor improvement, not predictor error.”
Schmidhuber, “Developmental Robotics, Optimal Artificial Curiosity” (2006)
231“Collapse the distinction between learning and reward by making the reward signal identical to the rate of learning itself.”
Storck, Hochreiter & Schmidhuber, “Reinforcement Driven Information Acquisition” (1995)
232"Animals prove that this kind of learning is possible, and set a lower bound on how well it can be achieved: anything a sea slug, a lorikeet, or a tenured professor can do, a learning algorithm can do."
Shalizi, Causal Architecture, Complexity and Self-Organization (PhD thesis, 2001), p. 125
233“The ‘law of series’ refers to a popular, intuitively felt tendency of rare events to occur in clusters rather than being spread uniformly in time. While it has been commonly dismissed as a purely psychological effect of selective attention, in many cases it has a sound mathematical basis rooted in the ergodic theory of dynamical systems.”
Downarowicz, Entropy in Dynamical Systems (2011), p. 135
234“For the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.”
D’Arcy Thompson, On Growth and Form (1917/1942), p. 1097
235"Would it not be extraordinary, however, if these laws could by themselves contrive to generate rich and beautiful patterns?"
Ball, The Self-Made Tapestry (1999)
236"This world is sacred."
Dennett, Darwin's Dangerous Idea (1995), p. 520
Where to dig next.
237"What are artificial organisms actually computing? Apply Crutchfield's epsilon-machine reconstruction to Lenia species. Nobody has measured this."
Open question.
238"Does the "edge of chaos" survive renormalization? If you coarse-grain a critical CA, does it stay critical?"
Open question.
239"What makes one rule space rich and another barren? We have examples of fertile parameter spaces but no theory of fertility."
Open question.
240"Can MacKay's "selective function" be formalized in terms of epsilon-machines? Shannon measures surprise. Crutchfield measures structure. Nobody measures meaning."
Open question.
241"Which features of Lenia species are dynamically selected, and which are topologically forced?"
Open question.
242"Plot real systems on the thermodynamic-depth vs. statistical-complexity plane. Where does life fall? Where does turbulence? Where does language?"
Open question.
243"What if the edge is special not because computation is optimal there, but because measurement is? A system at criticality has maximal susceptibility — an observer learns the most per measurement."
Open question.
244"Life extracts maximal structure from minimal entropy production. Is there a variational principle? Do living systems maximize thermodynamic depth per unit negentropy consumed?"
Open question.