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Links between Entropy, Complexity, and the
Technological Singularity
Theodore Modis
Address reprint requests to:
Theodore Modis
Growth Dynamics
Via Selva 8
6900 Massagno
Lugano, Switzerland.
Tel. 41-91-9212054, E-mail: tmodis@yahoo.com
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Abstract
Entropy always increases monotonically in a closed system but complexity increases at
first and then decreases as equilibrium is approached. Commonsense information-related
definitions for entropy and complexity demonstrate that complexity behaves like the time
derivative of entropy, which is proposed here as a new definition for complexity. A 20-year
old study had attempted to quantify complexity (in arbitrary units) for the entire Universe in
terms of 28 milestones, breaks in historical perspective, and had concluded that complexity
will soon begin decreasing. That conclusion is now corroborated by other researchers. In
addition, the exponential runaway technology trend advocated by supporters of the
singularity hypothesis—which was in part based on the trend of the very 28 milestones
mentioned above—would have anticipated five new such milestones by now, but none have
been observed. The conclusions of the 20-year old study remain valid: we are at the
maximum of complexity and we should expect the next two milestones at around 2033 and
2078.
Keywords: entropy; complexity; singularity; logistic growth; S-curve
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1. Introduction
This work was triggered by the author’s invitation to speak at the international
symposium on Social singularity in the 21st century: At the crossroads of history in Prague, CZ on
September 18, 2021 (InstituteH21, 2021.) They asked him for an update of his 20-year old
work on the evolution of complexity and change in our lives (Modis, 2002; Modis, 2003) and
its impact on the possibility of an approaching technological singularity. The author has
previously published three related updates (Modis, 2006; Modis, 2012; Modis, 2020.)
During the last ten years there has been much literature published on the subjects of
complexity and singularity. One notable example is the work of theoretical physicist Sean M.
Carroll whose bestselling book The Big Picture: On the Origins of Life, Meaning, and the Universe
Itself argues that complexity is related to entropy and that ―complexity is about to begin
declining‖ (Carroll, 2016). The idea that complexity first increases and then decreases as
entropy increases in closed systems had been previously suggested by several researchers
(Huberman et al., 1986; Grassberger, 1989; Li, 1991; Gell-Mann, 1994; Carroll, 2010; Carroll,
2016). In the same direction Kauffman had coined the term ―complexity catastrophe‖ to
explain the low complexity of an overly connected network similar to that of a sparsely
connected network (Kauffman, 1995). But in a more recent publication, Carroll together
with Aaronson and Ouellette demonstrated quantitatively the phenomenon of decreasing
complexity when approaching equilibrium by calculating the complexity and the entropy in a
cup of coffee that is undergoing the mixing of coffee and cream (Aaronson et al., 2014).
These publications provided fertile ground for the work presented here. Two short videos
by Sean Carroll popularize these ideas in YouTube for the layperson (Carroll, 2021).
Entropy and complexity are subjects that have enjoyed enormous attention in the
scientific literature. Their treatment in the next section is very brief and relates only to their
connection to the concept of a technological singularity. With information-related definitions
for entropy and complexity, a simple mathematical relationship between them is established
in light of which the author reinstates his 20-year old conclusion, namely that we should
expect a decreasing complexity in the future instead of an approaching technological
singularity. This conclusion has been corroborated by Magee and Devezas who studied
shorter-timescale technologically-driven or simply human-driven profound societal changes
(Magee et al., 2011).
2. Entropy and Complexity
2.1 Entropy
There are many definitions of entropy. The concept was first developed by Rudolf
Clausius, a German physicist in the mid-nineteenth century (Clausius, 1867). The
classical thermodynamic entropy is defined in terms of the energy (heat) and the
temperature of a system. Boltzmann’s definition involves the number of different ways
the atoms or molecules of a thermodynamic system can be arranged; his celebrated
formula for entropy has been carved on his gravestone (Allen et al., 2017). The
definition of Gibbs involves the energy and the probability that it occurs for all
microstates of the system (Klein, 1990). There is also the quantum-mechanical entropy
defined by von Neumann (Zyczkowski et al., 2006). All these definitions of entropy are
related to each other but they are not relevant here.
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In this paper we will concentrate on the fact that entropy is ―a measure of the
number of different ways a set of objects can be arranged‖ or ―a measure of disorder‖
(Martin et al., 2013), even though entropy isn’t always disorder (Styer, 2019).1 With
disorder defined as the number of possible configurations, a messy or disordered room
has higher entropy than a tidy room. The number of possible configurations of the
items in a messy or disordered room is higher than the number of possible
configurations in a tidy room, where the items ―inhabit a small set of possible places –
the books on the bookshelf, the clothes in the dresses, and so on‖ (Martin et al., 2013).
―The concepts of entropy and disorder are inherently linked‖ (Martin et al., 2013).
When entropy is high disorder is generally high and vice versa. Entropy always increases
in a closed system in accordance with the 2nd law of thermodynamics, which stipulates
that the entropy S will always increase: ΔS > 0. Entropy may locally decrease, but it will
increase elsewhere in the system by at least the same amount so that in a closed system
entropy (and also disorder) will generally increase.
There is a link between entropy and information. The higher the number of
possible configurations in a system, the more information is needed to describe the
system, i.e. the higher its information content will be. In information theory Shannon
has defined entropy as a measure of the information content in a message (Shannon,
1948). This is the amount of information an observer could expect to obtain from a
given message. A highly ordered, low-entropy state contains less information compared
to a highly disordered, high-entropy state. Let’s go back to the tidy-room example. If
they tell us a living-room is tidy (ordered), the information content of the message is
limited. Probably there is a sofa with pillows on it, there is an easy chair, a television
against the wall, chairs around a table, etc. But if they tell us that the living room is
utterly disordered, the information content of the message is much higher, because it
may include oddball situations like pillows on the floor, the television upside down, dirty
dishes on the table, chairs scattered around, etc. The more disordered the living room,
the greater the information content of the message we are given.
For the rest of this paper we will define entropy as information content.
On a larger scale entropy began increasing at the beginning of the Universe with
the Big Bang, when the Universe is thought to have been a smooth, hot, rapidly
expanding plasma and rather orderly; a state with low entropy and low information
content. Entropy will reach a maximum at the end of the Universe, which in a prevailing
view will be a state of heat death, after black holes have evaporated and the acceleration
of the Universe has dispersed all energy and particles uniformly everywhere (Carroll,
2010). The information content of this final state of maximal disorder (everything being
everywhere), namely the knowledge of the precise position and velocity of every particle
in it will also reach a maximum.
Entropy’s trajectory grew rapidly during early Universe. As the Universe expansion
accelerated, entropy’s growth accelerated. Its trajectory followed a rapidly rising
exponential-like growth pattern. At the other end, heat death, entropy will grow slowly
to asymptotically reach the ceiling of its final maximum (Patel, 2019). It will most likely
happen along another exponential-like pattern. It follows that the overall trajectory of
1 In recent times there has been criticism of the long-standing association of disorder with entropy. The
interested reader can go in more depth on this subject by consulting such publications as Floyd, 2007;
Lambert, 2002; Low, 1988; Styer 2020; and Wright, 1970.
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entropy will trace some kind of an S-shaped curve with an inflection point somewhere
around the middle.
2.2 Complexity
There are also many definitions for complexity. In fact, John Horgan in his essay in
his June 1995 Scientific American editorial entitled ―From complexity to perplexity‖, has
mentioned a list of 31 definitions of complexity (Hogan, 1995). Among them notable is
the Kolmogorov complexity, which defines it as a measure of the computational
resources needed to specify the object (Kolmogorov, 1963; Kolmogorov 1998). Also,
the Effective complexity, defined by Murray Gell-Mann and Seth Lloyd as a measure of
the amount of non-random information in a system (Gell-Mann et al., 1996).
But in this paper, and for the sake of consistency with the previous section, we will
use the following information-related definition for complexity: the capacity of a system
to incorporate information at a given time. Complexity is more like a snapshot while
entropy is more like a sum. Informally, complexity reflects the amount of information
needed to describe everything ―interesting‖ about the system at a given point in time
(―interesting‖ information is non-random information.) More intuitively, complexity
reflects how easy it is to describe the human system; the higher the complexity, the
more difficult it is to describe.2
In a closed system, entropy and complexity increase together initially, in other
words the greater the disorder the more difficult it is to describe the system. But things
change later on. Toward the end, as entropy approaches its final maximum where there
is also maximal disorder, complexity diminishes. Maximal disorder is simple to describe.
By the time entropy reaches its final ceiling the information content has become
maximal but also not ―interesting‖ because it has become 100% random information.
The degradation of the information content into non-interesting random information
begins when entropy reaches the inflection point of its trajectory, i.e. when the rate of
growth becomes maximal. At that point complexity goes over a maximum and begins
decreasing. Aaronson et al. have likened complexity to ―interestingness.‖ They have
demonstrated that it declines as entropy reaches a ceiling with the example of a cup of
coffee with cream (Aaronson et al., 2014). In the beginning when the cream rests calmly
on top of the coffee, the entropy of the system is small (there is also order) and the
complexity is also small because the situation is very easy to describe. At the end of the
stirring when coffee and cream are completely mixed together, entropy is maximal
(there is also maximum disorder because everything is everywhere) but the situation is
again easy to describe, so the complexity is low again. Around the middle of the mixing
process when entropy (and also disorder) is growing fastest the complexity of the
system is maximal.
Another example is the Universe itself. The very early Universe near the Big Bang
was a low-entropy and easy to describe state (low complexity.) But the high-entropy
state of the end will also be easy to describe because everything will be uniformly
distributed everywhere. Complexity was low at the beginning of the Universe and will
be low again at the end. It becomes maximal—most difficult to describe—around the
middle, the inflection point of entropy’s trajectory, when entropy’s rate of change is
2 This echoes Rosen’s epistemological account of complexity: “To say that a system is complex … is to say
that we can describe the same system in a variety of distinct ways …” (Rosen, 2000).
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maximal (see milestone numbers 27, 28 in next section.) Complexity follows a bell-
shaped curve similar to the time derivative of a logistic function.
2.3 A new relationship between entropy and complexity
With the above-mentioned information-related definitions for entropy and
complexity for a closed system, namely:
Entropy: the information content
(or a measure of the amount of disorder)
Complexity: the capacity to incorporate information at a given time
(or a measure of how difficult it is to describe at a given time)
we see that entropy results from the accumulation of complexity, or alternatively, that
complexity is the time derivative of entropy. Entropy traces out an S-shaped curve while
complexity traces a bell-shaped curve. The ―interestingness‖ of entropy’s information
content diminishes during the second half of the growth process and so does the
complexity of the system. At the end there is purely random information everywhere
and zero capacity to incorporate ―interesting‖ information.
In this case—i.e. with the chosen definitions—a new relationship between entropy
and complexity can be written as:
𝐶
＝
𝑑𝑆
𝑑𝑡 (1)
or
𝑆= 𝐶
‧
𝑑𝑡 (2)
The patterns of the trajectories followed by entropy and complexity may turn out not to be
exactly the classical logistic patterns, which are symmetric around the midpoint. But in the
coffee-and-cream study mentioned earlier, and with the particular quantitative definitions
the investigators used, they found indeed complexity to trace a symmetric bell-shaped curve
while entropy approached a ceiling asymptotically, see Figure 2 in (Aaronson et al., 2014).
3. Forecasting Complexity
In his 2002 article the author attempted to quantify the evolution of complexity in the
Universe in terms of 28 ―canonical‖ milestones—events of maximum importance, breaks in
historical perspective—based on data he collected from thirteen different sources (Modis
2002; Modis 2003). In his book The Singularity Is Near Kurzweil presented the data behind
these 28 milestones in different ways demonstrating the rapid rate of change in our lives, see
four figures on pp 17-20 of his book. Together with other runaway trends Kurzweil arrived
at the conclusion that there is an approaching technological singularity (Kurzweil, 2005).
These 28 ―canonical‖ milestones generally consist of clusters of events. They are
reproduced here in Appendix A. The importance of each milestone was assumed to be
proportional to the amount of complexity it brought multiplied by the length of the
following stasis until the next milestone. Consequently the increase in complexity ΔCi
associated with milestone i of importance I is:
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Δ𝐶𝑖=𝐼
Δ𝑇𝑖 (3)
where ΔTi is the time period between milestone i and milestone i+1.
Under the assumption that milestones of maximum importance were also milestones of
comparable (see equal) importance, values for complexity were obtained for 27 milestones in
relative terms (i.e. with arbitrary units) as being inversely proportional to the time difference
from one milestone to the next one.
In view of the discussion in Section 2.3 the accumulation of this complexity—i.e. the
integral—should be akin (if not equal) to the system’s entropy. The evolution of the world
seen by these 28 milestones is a non-equilibrium open system and for such systems Grandy
has demonstrated that it is the time derivative of entropy rather than entropy itself, which
plays the major role governing the ongoing macroscopic processes (Grandy, 2004).
Below are reproduced some results from the author’s work of twenty years ago. Figure 1
shows the ―primordial‖ S-curve, a logistic fit (thick gray line) to the cumulative complexity
values, which should be akin (if not equal) to the entropy of the system. Figure 2 shows
complexity per milestone and the fitted curve here (thick gray line) is the bell-shaped logistic
life cycle, i.e. the derivative of the logistic function.
Figure 1. A logistic fit (thick gray line) and an exponential fit (thin black line) to the
cumulated complexity values of 27 milestones. The graph at the bottom has a logarithmic
vertical scale. The red line is on the 28th milestone and coincides with the center of the
logistic.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0 5 10 15 20 25 30 35 40
Arbitrary units
Canonical milestone number
Cumulative change
S-curve fit
Exponential fit
Amount of change
1E-11
1E-10
1E-09
1E-08
0.000000
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
0 5 10 15 20 25 30 35 40
Canonical milestone number
Cumulative Complexity (Entropy)
Data
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Figure 2. A logistic life-cycle fit (thick gray line) and an exponential fit (thin black line) to the
complexity values of 27 milestones. The error bars reflect the spread on the values of the
milestones in the particular cluster. The little open circles forecast the position of future
milestones according to a logistic and to an exponential extrapolation. The graph at the
bottom has a logarithmic vertical scale. The red line is on the 28th milestone and coincides
with the center of the logistic.
The red line indicates the 28th milestone for which a complexity value cannot be
assigned yet not knowing the 29th milestone. The penetration level of the fitted logistic curve
at this time (1990) is 50.1%.
We also see in these two figures an exponential fit to the data (thin black line), which
would be compatible with the hypothesis of an approaching singularity. The two fits seem to
describe the data comparably well with exception the most recent data point, which is
overestimated by the exponential fit, something more obvious in Figure 2.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 5 10 15 20 25 30 35 40
Arbitrary units
Canonical milesto ne number
Amount of change between successive milestones
Logistic life cycle
Exponential fit
Amou nt of change
1E-10
0.000000001
0.00000001
0.0000001
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
0 5 10 15 20 25 30 35 40
Arbitrary units
Canonical milestone number
Amount of change between successive milestones
Logistic fit
Exponential fit
ln(Amount of change)
Complexity
Data
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The little open circles in Figure 2 forecast complexity values for future milestones
according to a logistic and to an exponential extrapolation. Since complexity was calculated
as being inversely proportional to the time to the next milestone, the forecasted complexity
of future milestones—be it with a logistic or an exponential fit—can be translated to dates
using Equation (3). Table 1 gives time estimates for the next five milestones according to the
two forecasting methods.
Table 1. Milestone Forecasts
Milestone
Logistic
fit
Exponential
fit
Number
Complexity*
Year
Complexity*
Year
29
0.0223
2033
0.1540
2009
30
0.0146
2078
0.3247
2015
31
0.0081
2146
0.6846
2018
32
0.0041
2270
1.4435
2020
33
0.0020
2515
3.0436
2021
* In arbitrary units
4. Discussion
Twenty years after the authors original work, his conclusion that complexity and change in our
lives will soon begin decreasing is corroborated. First by the work of other scientists who not only
claim that complexity in a closed system must eventually decrease, but have also demonstrated with
quantitative calculations that it does so symmetrically (Aaronson et al., 2014; Carroll, 2016). And
second by the mere fact that no milestones of paramount importance—breaks in historical
perspective—have been observed, while five of them had been expected during these twenty years
according to the exponential rate of growth advocated by supporters of the singularity hypothesis.
The relationship between entropy and complexity as expressed by Equations (1) and (2) is a direct
consequence of the definitions used in Section 2.3, but its validity could be more general despite the
fact that the relationship between entropy and complexity is not always one-to-one, as Wentian Li has
demonstrated (Li, 1991). As we said earlier the various definitions of entropy are related to each other
and so are most of the definitions of complexity. Seeing complexity as the derivative of entropy may
have widespread appeal and utility on an intuitive level. After all, complexity reaches a maximum value
when entropy grows the fastest. Grandy has amply demonstrated the importance of the role played by
the derivative of entropy (Grandy, 2004).
In any case complexity, as determined by the 28 milestones, has reached a maximum and now
begins on the declining slope of its bell-shaped pattern. It is a direct consequence of having described
the accumulation of entropy by a natural-growth (logistic) pattern, which so far seems to hold as there
haven’t been any ―milestones‖ in the last 25 years. There have been many small ones but nothing like
the Internet, DNA, or nuclear energy. The idea that our world’s complexity will decrease in the future
may seem difficult to accept but such a unimodal pattern (namely low at the beginning and at the end
but high in between, not unlike the normal—Gaussian—distribution3) is commonplace in everyday life.
It is associated with a reversal appearing at extremes. We say for example, that too much of a good
thing is not good. We saw that too much disorder is easy to describe in the examples of coffee and
3 The Gaussian and the derivative of the logistic function, the so-called life cycle are very similar (Modis,
2006).
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cream, and in the evolution of the entire Universe. Also, I mentioned how Kauffman points out that an
overly connected network is as dysfunctional as a sparsely connected network. John Casti in his book
X-Events defines complexity as ―the number of independent decisions a decision-maker can make at
any given time‖ (Casti, 2012). Thus, if a decision-maker has only few decisions in his or her set of
possibilities, he/she faces low complexity. The complexity will increase as the number of possibilities
increases. But I believe—Casti does not say this—that if the decision-maker faces millions of
possibilities, life in fact will become simpler rather than more complicated because the situation will
trigger alternative ways to make decisions (e.g. random choices). Life may not be as simple as having
only one choice, but it will be simpler than having to choose among 20 or 30 possibilities, each of
which requires individual attention.
Because the time frame considered by this analysis is vast and the crowding of milestones in recent
times is extremely dense functions such as logistics and exponentials cannot describe the growth
process adequately. There are processes for which our Euclidean (linear) conception of time does not
accommodate an appropriate description. That’s why for this analysis, a better-suited time variable was
chosen: the sequential milestone number, which is a logistic time scale.
We are obviously dealing with an ―anthropic‖ Universe here since we are overlooking how
complexity has been evolving in other parts of the Universe. Still, the author believes that such an
analysis carries more weight than just the elegance and simplicity of its formulation. John Wheeler has
argued that the very validity of the laws of physics depends on the existence of consciousness.4 In a
way, the human point of view is all that counts! In astronomy/cosmology this is referred to as the
Anthropic Principle (Bostrom, 2010), which in its weak form basically states that one sapient life form
(humans) looks back to the past from its point of view (Penrose, 1989).
One may object to including such cosmic events as the Big Bang and the formation of galaxies in
the same set of milestones as the invention of agriculture, or the internet. But if we dropped the first
two milestones and repeated our analysis beginning with the 3rd milestone cluster (the formation of our
solar system and the earth, oldest rocks, and origin of life on earth), then the fitted curves would
change only imperceptibly. But at the same time, there would now be rough corroboration of the
conclusion that complexity and entropy are presently around their midpoints: the sun is close to its
midlife (is thought to be 4.6 billion years old and expected to go out in 5.5 billion years from now.)
But we could restrict further our data set to those milestones that have to do only with humans.
The reader’s attention is drawn to the fact that the trends in Figures 1 and 2 remain purely exponential
(straight line on the lower graphs with the logarithmic vertical scales) with extremely low values for
most of the range. The trends begin deviating from exponential only very recently, namely after
milestone No. 23, i.e. after the fall of Rome, and zero and decimals invented. So even if we dropped all
pre-human milestones, we wouldn’t obtain a significantly different fit.
One of the thirteen data sets used to distill the 28 ―canonical‖ milestones of Figures 1 and 2 has
been provided by Nobel Laureate, Paul D. Boyer. In his contribution he had anticipated two future
milestones without specifying their timing. Boyer’s 1st future milestone was ―Human activities devastate
species and the environment,‖ and the 2nd was ―Humans disappear; geological forces and evolution
continue.‖ The logistic-fit time estimates for the two next milestones from Table 1 are 2033 and 2078
respectively. It is likely that there are bona fide scientists who would agree more with Boyer’s future
milestones and these time estimates rather than with an approaching technological singularity.
Alternatively, and on a more positive and realistic tone the next two milestones could well be along
the lines:
4 John Wheeler was a renowned American theoretical physicist best known for first using the term "black
hole" in 1967.
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2033. A cluster of achievements in AI, robotics, nanotechnology, bioengineering, NASA’s
scheduled human mission to Mars, etc. could qualify as one milestone in the same way
modern physics, radio, electricity, automobile, and airplane had done at the turn of the
twentieth century (milestone No. 26).
2078. Teleportation or creation of life, two fields that have been attracting attention of
researchers for some time now.
In his publication of 2002 the author had concluded that ―we are sitting on top of the world‖ from
the point of view that we are experiencing complexity and change at their maximum and that they will
begin decreasing soon. Twenty years later there is no reason to revise that conclusion.
Author statement
I would like to thank Alain Debecker and Athanasios G. Konstandopoulos for fruitful discussions.
12
Appendix A
The 28 ―canonical‖ milestones generally represent an average of clustered events not all of which are
mentioned in this table. That is why some events, e.g. the asteroid collision, may appear dated
somewhat off. Highlighted in bold are in the most outstanding event in the cluster. The dates given are
expressed in number of years before year 2000.
No. Milestone Date
1. Big Bang and associated processes 1.55 x 1010
2. Origin of Milky Way, first stars 1.0 x 1010
3. Origin of life on Earth, formation of the solar system and the Earth, oldest rocks 4.0 x 109
4. First eukaryotes, invention of sex (by microorganisms), atmospheric oxygen, 2.1 x 109
oldest photosynthetic plants, plate tectonics established
5. First multicellular life (sponges, seaweeds, protozoans) 1.0 x 109
6. Cambrian explosion, invertebrates, vertebrates, plants colonize land, 4.3 x 108
first trees, reptiles, insects, amphibians
7. First mammals, first birds, first dinosaurs, first use of tools 2.1 x 108
8. First flowering plants, oldest angiosperm fossil 1.3 x 108
9. Asteroid collision, first primates, mass extinction, (including dinosaurs) 5.5 x 107
10. First hominids, first humanoids 2.85 x 107
11. First orangutans, origin of proconsul 1.66 x 107
12. Chimpanzees and humans diverge, earliest hominid bipedalism 5.1 x 106
13. First stone tools, first humans, Ice Age, Homo erectus, origin of spoken language 2.2 x 106
14. Emergence of
Homo sapiens
5.55 x 105
15. Domestication of fire, Homo heidelbergensis 3.25 x 105
16. Differentiation of human DNA types 2.0 x 105
17. Emergence of “modern humans,” earliest burial of the dead 1.06 x 105
18. Rock art, protowriting 3.58 x 104
19. Invention of agriculture 1.92 x 104
20. Techniques for starting fire, first cities 1.1 x 104
21. Development of the wheel, writing 4907
22. Democracy, city-states, the Greeks, Buddha 2437
23. Zero and decimals invented, Rome falls, Moslem conquest 1440
24. Renaissance (printing presss), discovery of New World, the scientific method 539
25. Industrial revolution (steam engine), political revolutions (France, USA) 223
26. Modern physics, radio, electricity, automobile, airplane 100
27. DNA structure described, transistor invented, nuclear energy, 50
World War II, Cold War, Sputnik
28. Internet, human genome sequenced 5
13
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Biography
Theodore Modis is a physicist, strategist, futurist, and international consultant. He is
author/co-author to over one hundred articles in scientific and business journals and ten
books. He has on occasion taught at Columbia University, the University of Geneva, at
business schools INSEAD and IMD, and at the leadership school DUXX, in Monterrey,
Mexico. He is the founder of Growth Dynamics, an organization specializing in strategic
forecasting and management consulting: http://www.growth-dynamics.com
