Role of Entropy in Biological Systems
Eduardo Alfredo Zevallos-Giampietri, M. D.
Carlos Barrionuevo, M.D.
Departamento de Patología, Instituto de Enfermedades Neoplasicas, Lima, Perú
Vishnu S. Shukla, M. D.
Phoenix Cancer Help Group, Green Acres, ITC, Bangalore, India
Address correspondence:
Eduardo A. Zevallos Giampietri, M.D.
Clínica San Marcos & Instituto Diagnóstico Cayetano Heredia
Jr. A. B. Leguía 604
Tarapoto, San Martín
Perú
Telefax: + 51 42 523838
E-mail: edzevallos@yahoo.com
Running head: Entropy, Biology, Health, Disease, Life Evolution.
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Abstract
In this article the authors use an extended concept of Entropy to understand
biological systems. Entropy is considered as a universal function that can be
applied in different scenarios, from the Planck level to the Cosmological level.
Establishing the universal importance of Entropy is crucial to understand the
behavior of biological systems. Hence, concepts such as life, evolution,
reproduction, health and disease then acquired a profound significance.
Interpretation of data depends on the relativistic position of the observer, either
internal or external to the system. Entropy is highest (Smax) or lowest (Smin) for an
external or internal observer, respectively. This is corresponding to the “outside
view” and “inside view” that emerged in the context of quantum mechanics. If this
essential fact is not taken into account then fallacies are inevitable, and
subsequently, weird interpretations and twisted believes take place in any types of
scenarios. In addition, the authors explain the concept of intrinsic or physiological
time based on Smax/Smin associated to cybernetics concepts.
Key words: biology, death, disease, dissipation, evolution, health, information, life,
maximum entropy, second law, thermodynamics.
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Introduction
Medicine is not an independent discipline, that it can be placed into the context of
well-established physical laws and scientific principles. Particularly, the Second
Law of Thermodynamics (1), Popper’s falsification principle (2), and Cybernetics
are very suitable for this purpose. Here, Entropy is treated as a broad physical
principle. Entropy is one of the most controversial concepts ever described.
Clausius, who discovered this property of nature in 1865, used this term (in
German “entrepein”, and from Greek entrope, change) to denote conversion of
energy. All types of energy potentially can be interconverted among themselves.
The discovery of Clausius was based on the Carnot´s principle that no machine
can be perfectly efficient. In the process of energies interconversions part of the
total energy cannot be reused. Despite the conservation of energy, this apparently
unrecoverable share of energy is integrated into the rest of the universe in the form
of heat. Meanwhile, and simultaneously, the temperature of the universe
decreases as the universe expands. The Second Law is like a security device for
the First Law, and it reassures that equilibrium (maximum entropy) will be reached
(Third Law). In fact Clausius concisely summarized the laws of thermodynamics,
as “the energy of the universe is constant, while the entropy of the universe tends
to be maximal”. The Second Law of Thermodynamics is one of the most
fundamental principles; it says that when a type of energy converts into another
inside a system, there must be always some amount of energy that is transformed
into heat, and it dissipates into the rest of this universe. This may reflect a basic
mechanism that guarantees consistency and historical conservation in the
universe. Therefore, as there is equivalence between energy and mass through the
Special Theory of Relativity, then all the mass in the universe is dissipating.
Unrecoverable heat can be cautiously considered to as “useless” energy. On the
other hand, the energy that produces an effective work can be considered as
“useful” energy (at the very moment that such work is being done). Entropy is a
measurement of the interrelations inside the system, which determines the
outcome of heat and work. Entropy is a measurement of the uncertainty of
destination of energy. Heat represents that part of information that is lost; while
work represents the information that is preserved (3).
Entropy can be conceptualized as a broad principle in the development of systems.
Clausius’ (thermodynamic) Entropy, Boltzmann’s (probabilistic) Entropy, and
Shannon’s (information) Entropy are all related, and they provide a basic
framework to understand Entropy at different levels, including Planck, Quantum,
Newtonian, Relativistic, and Cosmological. Since the macroscopic physical “reality”
emerges from the quantum level, therefore Heisenberg’s Uncertainty Principle (4)
can be applied to the study of systems, and ultimately it may be the basis of
evolution. A crucial relationship between Shannon’s Entropy and Cybernetics was
established by Ashby’s Law of Requisite Variety: “The law of Requisite Variety
says that R’s capacity as a regulator cannot exceed R’s capacity as a channel of
communication…the law of Requisite variety can be shown in exact relation to
Shannon’s Theorem 10, which says that if noise appears in a message, the
amount of information that can be removed by a correction channel is limited to the
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amount of information that can be carried by that channel…Thus the use of a
regulator to achieve homeostasis and the use of a correction channel to suppress
noise are homologous” (5). We think that all types of definitions in the context of
complex systems must be avoided, because definitions inevitably fall into
anthropomorphic or reductionist fallacies.
Medicine is not a science, as it lacks a theory, and so far cannot be expressed in
logical true values. This weakness is justified among physicians by appealing to
“inevitable” subjective factors. However, there is no intellectual effort in medical
circles to explain as to on what these subjective factors consist of, or how do they
arise. For the same reason the health activity, including facilities and providers, are
close circles and have “invented” themselves usually non-scientific terminology and
doubtful formalities. Moreover, the attempts of finding “hidden philosophies” by
physicians have been so far worthless, because they are restricted to the rules
imposed in such close circles. The health system, in this sense, projects this
inadequate approach towards conventional medical schools. Therefore,
physicians are mostly unaware of this lack of “philosophy”, or they consider with
resignation that this deficiency is unsolvable. So-called basic sciences such as
Biochemistry and Physiology are also unable to unveil such “hidden philosophies”
since they are also influenced by the prevailing anthropomorphic orientation.
Contrarily and paradoxically, non-conventional medicine has more scientific basis
because it has a dynamical non-linear scope, as it usually focuses on body and
mind in a comprehensive manner (holistic). Health and disease, to the best of our
knowledge, have not been fully associated with physics or mathematics. The
question of internal observer and external observer also come in play with Yoga,
as well as subjectivity and objectivity of modern medicine (6). Therefore, we go
beyond and conjecture a model of health and disease based on Entropy.
In this article Entropy is placed into a biological perspective, and its importance is
extended to health and disease. This is important for the perspective of Entropy as
a probabilistic measurement in diagnostic process (manuscript in preparation).
Biological systems
Thermodynamically, a system can be described by its state, which includes the
overall properties of the system such as pressure, volume, and temperature as well
as composition. It is not a coincidence that Thermodynamics is also known as
Statistical Mechanics, since the principles can be obtained based on physics as
well as in a probabilistic manner (3).
According to the interaction with the environment systems can be separated as
follows:
1. Systems in equilibrium with the medium
ƒ=0
Si = Sext (> 0)
- “Energy” only flows
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- Organic polymerization not significant
2. Systems far from equilibrium with the medium
ƒ>0
Si < Sext (> 0)
- “Energy” produces a function (heat, mechanical and chemical work)
- Organic polymerization significant (proteins, nucleic acids, etc).
Where,
ƒ is the function of the system,
Si the internal Entropy, and
Sext the external (environmental) Entropy.
Essentially, this is an arbitrary thermodynamical division of systems to highlight the
behavior of Entropy at both extremes: close and open systems. Ultimately, in all
systems inside the universe there must be some sorts of matter and/or energy flux,
thus stringently and broadly speaking all systems are truly open. Therefore,
classical thermodynamics, dealing with “isolated” systems, is unrealistic. Biological
systems are at extreme far from equilibrium status, which renders them a false
sense of dislodgement from the media. The anthropic principle may be rooted in
this fallacy. In open systems, such as biological, the informational or working
energy can be replenished allowing the flow of “new” energy to produce more
informational or working energy. In consequence, it is wrong to say that the
Entropy of an open system must always increase.
We shall generalize the concept of Entropy as “the development of the elements of
the systems”. This is not a strict definition. Notice that terms such as energy and
time are not used. Besides, the term element has to be taken in its broadest
meaning, since the elements determine the function, as well as vice verse the
function determines the elements. For the same reason we do not used a term
such as “interaction” (of elements), precisely to avoid separation of elements from
the function. At a quantum level, similarly, there is no difference between particles
and waves. Moreover, it is likely that at the Planck-quantum interface this
“smearing effect” is even blurrier. Entropy deals with the development towards
equilibrium of the overall elements and sets of elements. Time is not primordial,
since equilibrium inevitably has to be reach. This concept regards Entropy as
“development”, therefore, implies a dynamical and measurable process similar to
Information. Because of this “development”, Entropy has also a connotation of
measurement of freedom. One bit of information means the probability between
two states, but also the same bit allows choosing between one of the two states.
This freedom is rooted in quantum mechanics as uncertainty and non-determinism,
and reflected in the macroscopic world as evolution.
"While some parts of the universe may operate like machines, these are closed
systems, and closed systems, at best, form only a small part of the physical
universe. Most phenomena of interest to us are, in fact, open systems, exchanging
energy or matter (and one might add, information) with their environment. Surely
biological and social systems are open, which means that the attempt to
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understand them in mechanistic terms is doomed to failure” (7). Biological systems
function at their own level of “relatively Smax” (eigen, proper, intrinsic or
physiological entropy). This is an extension of Jaynes’s Principle of Maximum
Entropy (8), which for an external observer appears to be the most frequent state
of a system, meaning more Entropy, and therefore more uncertainty. However
different combinations can produce the same state or result (9 – 12). Jaynes’
principle of Maximum Entropy has the advantage that maximizes the uncertainty of
the observer; therefore this observer cannot have more information than is actually
available. Though, this intrinsic or “eigen” entropy is lower compared to that of the
environment. An inherent condition of biological systems is that energy flows
through their constituents (cells, organelles, molecules, etc.). The role of the
metabolic work is, in fact, the reduction of the internal Entropy compared with that
of the environment. This is essentially the principle of what is life.
Time, after all, is an interim term, and its measurement is relativistic (13). Each
system in the universe carries its own intrinsic time (eigen time) (14), likely implicit
in the system’s own intrinsic Entropy (eigen Entropy). Systems are at dynamical
status between two extremes. On one side they tend towards their own Smax (to
say the maximum “allowed” eigen entropy). Meanwhile, on the other side, they try
to avert the surrounding Entropy (Suniverse). To persist as such, the system has to
stay away from both extremes. If it reaches the exact status of Smax, the system
collapses. However, if the system cannot sustain the tendency towards Smax, then
it starts to merge with the rest of the universe, a process known as dissipation (7).
At both extreme situations systems cease to exist. All systems tend to their Smax,
and so do the biological systems. Since biological systems are extremely complex,
this dynamical “struggle” between the Smax and Suniverse is not well understood. The
more complex is a system, the more words and more complex models are
necessary to explain the system; therefore it is easier to produce paradoxes and
fallacies. In fact, a full explanation of a complex system can be accomplished only
reproducing exactly the same system. “Explicit definitions for central concepts
concerning complexly organized systems are often not just impossible to provide,
but can be quite misleading. The reason is simple: Explicit definitions place the
defined term on only one side of the definition, so that all explicitly defined
concepts are in principle eliminable” (15). Our reasoning is akin to Ashby’s concept
of regulation of very large systems; in fact this scientist says, “Then he [Sir Ronald
Fisher] showed that any given extraction of information had a maximum, and the
statistician’s duty was simply to get near the maximum- beyond that no man could
go. Similarly before Shannon’s work it was though that any channel, with a little
more skill, could be modified to carry a little more information. He showed that the
engineer’s duty is to get reasonably near the maximum, for beyond it no-one can
go. The law of Requisite Variety enforces a similar strategy on the would-be
regulator and controller: he should try to get near his maximum - beyond that he
cannot go.” (5). Moreover, complex systems naturally produce statements lacking
supporting theorems inside them (16, 17). This fact may be related to the
unavoidable increasing Entropy inside systems, which can be also considered as
error accumulation.
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The “struggle” Smax/Suniverse determines all the complex biological functions, which
in turn maintain integrity and function. This is indeed the thermodynamic basis of
life. Life frequently is referred as to something ethereous. Life, indeed, is the
overall thermodynamical expression of high carbon-based complex systems that
are called organisms, including the human species. Biological systems are
governed by the same universal thermodynamical principles that apply equally to
every system of this universe. Similarly to any other systems inside this universe,
biological systems are constantly changing, as well as transforming themselves
into non-informational energy (heat). This is essentially the core of the dissipation
process (7). Not only biological systems have a dissipative structure, but also
everything inside the universe, and possibly the universe itself. A universal
“dissipation principle” can be conjectured, which perhaps is in correspondence with
Thermodynamics and Information theory. Gravitation itself might be a macroaspect of Smax, acting as a physical tendency to reduce the Entropy of the systems
(18). Suniverse, on the other hand, may be driven by the dark energy (or by the
boundary of the universe), as a physical tendency towards dissipation or
“antigravitation”. Very speculatively speaking, in analogy to the elusive “graviton”,
there may be “particles” associated to Information (to say the “informon”), and even
associated to observation (to say the “observon”, or the “voyeuron”). Neutrinos
and/or neutralinos may play an important role in the process of information.
In biological systems, an increased of Entropy is somewhat undesirable or
unpleasant, but unavoidable. For instance, when we are hungry, an unpleasant
sensation, it signifies that our body is tending to high Entropy and dissipating. The
food is made up of elements that are either at lower or not too much higher Entropy
compare to that of our own body’s Entropy. Indeed, metabolism has a functional
specificity to decrease Entropy, since enzymes are highly efficient factories of low
Entropy (19). Metabolism, potentially, can be translated into informatics. Enzymes
are essentially polymers. Polymers (macromolecules) have “self-assembly”
property according to their sequence and/or interaction with the media. This
property is a stochastic process that can be traced back to origin of planet Earth. A
similar extension of this principle is that of Information Gathering and Using System
(IGUS), which in the case of proteins the amino-acid sequence makes
measurements while is growing to “attain a high unique stable native form that
promotes the updating of the information content”, which means that proteins tend
to be at equilibrium or at their Smax. Through IGUS proteins are comparable to a
Maxwell Demon-like activity as they reduce the Entropy, but different than the
Demon they cannot violate the Second Law (20). Melkikh uses a similar cybernetic
approach regarding DNA and quantum behavior (21); however, we think he falls
into a reductionist trap because DNA is not necessarily the primordial polymer that
characterizes the function “life”. It would have been more interesting if this author
performed a probabilistic analysis relating DNA, proteins and species. We
conjecture that “packing” or “quantatizing” is a basic universal principle, which
counteracts variability. In fact, as proposed by Everett, further elaborated by
Schmidt and Zeh (22), for the outcome of our “reality” there is “no collapse” of the
wave of the universe, but instead branching, and consequently development of
multiple realities (multiverse). The more stable or robust configurations prevail (but
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we cannot ascertain which are the more stable universes). We see such
“branching” of the wave as an averaging process that reduces variability. In fact
this averaging can be in compliance with Cybernetics principles such Ashby’s Law
of Requisite Variety (5), and the Principle of Selective Retention or “More stable
configurations are less easily eliminated than less stable ones” (23). A relation
between Cybernetics and Chaos Theory can be traced back to the Ashby’s
concept of Markovian machine and his analysis of Stability when he says, “The
stable region is a set of sates such that once the representative point has entered
a state in the set it can never leave this set…A state of equilibrium is simply the
region shrunk to a single sate. Just as, in the determinate system, all machines
started in a basin will come to a state of equilibrium, if one exists, so too do the
Markovian; and the state of equilibrium is sometimes called an absorbing
state…S is now well known, a system around a state of equilibrium behaves as if
‘goal-seeking’, the state being the goal…Thus, the objective properties of getting
success by trial and error are shown when a Markovian machine moves to a state
of equilibrium” (5). Here we can see the analogy between Ashby’s “stable region”
concept and that of “attractor” of Chaos Theory.
On the other hand, pleasant sensations are associated with the reduction of
Entropy from a relatively previous state of increased Entropy. Retaking the
previous example, an unpleasant sensation such as hunger signifies increasing
Entropy. Eating reduces the pre-increased Entropy, and as a pleasant sensation
the amount of pleasure is directly proportional to the previous degree of preincreased Entropy. However, if we are not hungry eating is not pleasant, or it is
even unpleasant. Perhaps in anorexia nervosa there is a corruption of this entropic
mechanism at hypothalamic level. The behavior of animals, including human
species, can be governed by the same principles. For instance, sexual intercourse
renders great pleasure because it is intimately linked to reproduction, which is one
of the most remarkable way to reduced Entropy, therefore this mechanism of
gratification ensures perpetuation of the species. If Entropy is not already
increased the system is thermodynamically indifferent, as there is nothing to
decrease and there are no disturbances to constrain (24). Therefore, Entropy may
play an important role in the internal homeostasis.
A fetus is a very pictorial example of a biological system. It cannot be at equilibrium
because it is constantly growing (dividing cells, accommodating cells and making
tissues). It acquires energy from the mother’s blood through the placenta. This
energy is carried by nutrients that cross the placenta into the baby’s blood flow.
Thermodynamically speaking, to increase its efficiency, the system “baby” has to
be far from equilibrium, and then energy can flow more efficiently inside the system
“baby”. Energy is used to decrease and maintain the condition “far from
equilibrium”. This condition means that the system “baby” has less Entropy than
the system “mother”. In this model it is relatively simple to elucidate how this
property “far from equilibrium” initially happens, because the origin of the system is
two halves of a cell (oocyte and spermatozoid) that readily join forming a complete
cell. However, where is the real origin of the individual? What is the underlying law
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that conducts this process? The answer comes by itself: Entropy. The baby
represents an intense entropic shortcut to maintain the species.
The Smax of the universe is the final equilibrium, and its Suniverse is at its own
boundary. At Smax any system would achieve the maximum equilibrium; therefore,
asymmetry vanishes and symmetry arises, but with an established symmetry
information stops, then the system ceases as such. In biology, maybe this
corresponds to aging process until death. After death a biological systems tends to
become again chaotic and ultimately dissipates into the entropy of the
environment, and subsequently merges with the S universe (its “new” attractor). In
cosmology massive stars after dying are shrunk into black holes (25). In both
cases, for death systems and black-hole converted systems the fate is merging
with the entropy of the universe, which mean re-entering in “the game” breaking
the symmetry, and asymmetry arises again. Meaning that the Smax of the universe
is that for an internal observer, and the Suniverse of the universe is that for an
external observer, and perhaps ultimately both are the same. The principle of “0”
and “1”, neither can exists without the other. Collier has hypothesized biological
information based on hierarchical levels of information depending on the molecular
complexity; however, this author claims that there is ambiguity between syntactic
information (understood as the information encoded by the parts) and functional
information (15). Perhaps the answer to this “gap” is the asymmetric character of
Entropy.
Observer’s position
The functionality of biological systems is reliant on Smax and Suniverse, therefore
maximum efficiency is comparable to an actual Entropy level of the organism, and
this Entropy level identifies the system itself at any present instant. As an ultimate
argument it may be possible that no system is equal to another and not even to
itself at any moment. This conjecture is rooted in the Heisenberg’s Uncertainty
Principle. Nonetheless, at least, macroscopically systems apparently maintain
certain integrity and identity through a dynamic interaction with the media, which
means that they are constantly constraining disturbances. At this point we can infer
again a relation with Ashby’s Law of Requisite Variety, in the sense that systems
can overcome a disturbance of the media only if they have sufficient flexibility or
variety to counteract such disturbance. Meanwhile, the internal tendency towards
Smax keeps enough internal variety inside the systems, and therefore maximizes
their capacity to beat the disturbances of the media. However, the internal Smax is
always less than the external Suniverse, and this status is achieved through
controllers. The controllers must be replicas or “aliquots” of the system (26), which
are able to detect variations of the actual internal entropy related to the Smax of the
system. Suniverse minus Smax may be a constant, and perhaps similar inside same
species or genres, and even possibly proportional among living organism and nonbiological system. Dewar has proposed a similar theory using Jaynes’ principle
and probabilistic thermodynamics (27). This researcher’s formalization starts with
stationary-type macroscopic approach to support conservation of mass and energy
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inside the system, but we think this is questionable because ignores the previous
history of systems, and does not consider the quantum phases of systems.
Naturally speaking systems does not just show up from nothing. Thereafter, he
uses Lagragians to reaffirm the proper stability of the system, which essentially is
similar to our rationale that the Principle of Least Action as an internal observer’s
point of view. Then, the same author maximizes the entropy using the Jayne’s
Principle, thus, in this way he turns to an external observer’s position. Here, we
have to remark that the maximum entropy obtained through the Jayne’s Principle is
equivalent to the most uncertain interaction likelihood for an external observer,
which in terms of Shannon’s Entropy is the most probable state rendering the most
information to an external observer. Despite Dewar mathematically demonstrates
that systems are at “maximum entropy production”, this is not surprising, as this is
the same as to say that systems function at a maximum currently allowable entropy
or “actual Smax”. Consequently, the inner and outer flux of energy may also be the
“maximum” allowable for such specific “actual Smax” (however, the “actual Smax” is
not equal to the “attractor Smax”). Here a further explanation is needed; the “actual
Smax” reflects the integrity and functionality of the system, which is related to the
identity of the system itself. Therefore, the energy that flows through the system,
strictly speaking, cannot be called either maximum or minimum, as it is just the
amount of energy related to such Smax, and no more and no less. The fundamental
difference compare with our conjecture is that Dewar is limited to macroscopic
systems, thus he “discards all of the irrelevant microscopic information from Pг”
(phase state probabilities). He does not determine the scale of microscopic
probabilities; therefore, what is “irrelevant” might not be so “irrelevant”. His
argument might be somewhat skewed. For him constrains are highlighted, but
constrains are related to degrees of freedom. For instance, we do not know if his
theory can be applied to the quantum scale. Besides, it might be naive that inner
conditions are just preserved by equal inner and outer flux of energy, otherwise all
systems would be eternal, and obviously this is not the case. Systems must have
regulatory mechanisms (controllers), which keep the level of entropy into a
permissible range. However, these controllers even if highly efficient are not
infallible, as entropy always increases (error accumulation) inside the systems,
and, the regulatory mechanisms themselves also ensue entropy.
Concepts such as “minimum entropy production” and “maximum efficiency” can be
redefined through these principles. In fact, the separation of “physiological time”
from “metabolism rate” (14), and the apparently impossibility of finding a link
between both, is because of this preconceived division between structure and
function. Ultimately, elements and function of the systems are so intimately
interrelated that they are the same. Similarly, matter and energy have common
identity through Special Relativity, so do particles and waves in Quantum
Mechanics. Observer’s constrains and setting of degrees of freedom may also
hamper the understanding that no-separation is an inherent condition. Science has
to be extended to a vast array of dimensional spacetime. Separation of structure
from function is a flaw that arises from being local and reductionist. The powerful
Principle of Least Action and its extensions can be also embedded into Entropy
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and Information (28-30). Therefore, understanding organisms through
thermodynamics/informatics, i.e. Entropy, is the best “natural” way to do so.
In Entropy/Information, if the relative position of the observer (internal or external)
is not taken into account then misleading and twisted interpretations are produced
in every scenario. Thermodynamically speaking for an external observer the only
important quantity is work, which in Information is analogous to data (bits).
Contrarily, for an internal observer heat is the most important, because it increases
the Entropy and capacity (freedom) to produce such work. Of course, not all heat
can be converted into work, and a fraction of the heat indefectibly diffuses to the
media. This share of energy lost as heat can be seen as an erasing procedure that
reassures that more energy (data input) can enter into the system to generate
more work (data output or information). Therefore Entropy is the overall function
that brings possibilities and flexibility to systems. From an external observer point
of view, and at the beginning of an observation, the Entropy of a system is highest
(Smax) and the potential amount of data (information) is also the highest.
Conversely, for an internal observer (inside the system) the Entropy and the
amount of data (information) is the lowest (Smin), since the system itself is the
“normal” status of such observer, and, moreover, the observer is part of the
system. Therefore, any observation made by humans is always contaminated by
bias.
Particularly in Many Worlds Interpretation (MWI) (Everrett’s theory) of Quantum
Mechanics, again it arises the argument of the position of the observer. As a matter
of fact, Tegmark says, “Everett’s brilliant insight was that the MWI does explain
why we perceive randomness even though the Schrödinger’s equation itself is
completely causal. To avoid linguistic confusion, it is crucial that we distinguish
between
 the outside view of the world (the way mathematical thinks of it, i.e., as an
evolving wavefunction), and
 the inside view, the way it is perceived from the subjective frog perspective
of an observer in it.” …
”It is in this sense that the MWI predicts apparent randomness from the inside view
while maintaining strict causality from the outside view-point…”The reader must
choose between two tenable but diametrically opposite paradigms regarding
physical reality and the status of mathematics:
 PARADIGM 1: The outside view (the mathematical structure) is physically
real, and the inside view and all the human language we use to describe it is
merely a useful approximation for describing our subjective perceptions”.
 PARADIGM 2: The subjective perceived inside view is physically real, and
the outside view and all its mathematical language is merely a useful
approximation.” (31)
Bear in mind that Tegmark’s inside and outside views are, to say, from a
conceptual quantum perspective. Instead, and compounding the issue, ours
internal and external observers are from a broad thermodynamics/informatics
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perspective, not only conceptual, but also active, as the observers are actively
performing measurements. Of course, if such observers were place at a quantum
scale, then we would face similar Tegmark’s dissertation. We prefer Tegmark’s
Paradigm 1, as it is more precise and likely not interfered by semantics (or
semiotics). Since Entropy/Information, according to Bekenstein, has a physical
location, then it is more likely that physical reality evolves from Entropy/Information,
and be explained by theories “living” at the locus of Entropy/Information.
Bekenstein noticed this similarity and proposed the identity of the area of the event
horizon with entropy, in 1972. Bekenstein’s insight is a milestone in theoretical
physics. Additionally to the interconversion matter-energy by Special Relativity,
both matter and energy can be then expressed in another state such is Entropy
(32)
We think that well-known paradoxes (i.e. quantum suicide, quantum holocaust,
quantum immortality, etc), arising from MWI as an extension of the Schrödinger’s
cat, are consequence of the lack of consideration of the position of the observers,
including the one (like us) who is reading or thinking about these paradoxes.
Briefly, overall, the setting of such paradoxes is that a “predominant quantum
system” (or mechanism) is confronted with “predominant macroscopic systems”,
and, moreover, the outcome is realized at a macroscopic scale (by human mind).
This assertion does not dichotomize what is “quantum” from “macroscopic”,
because we are conscious that the macroscopic realm arises from an underlying
quantum realm. The “abracadabra” of these paradoxes is when different scale
measurements are mixed, this blurs (as some sort of smoke curtain) the
awareness of the external observer (whoever is doing the experiment or even
reading the paradox itself). Basically, the “macroscopic realm” has much less
degrees of freedom compared with those of the “predominant quantum realm”.
Conversely, once a quantum event is placed and observed into the “macroscopic
realm” the degrees of freedom of such quantum event are “reduced”, as they are
aligned with those of the “macroscopic realm”. Therefore the uncertainty of the
quantum event is apparently also reduced. Another “abracadabra” is that the
“predominant macroscopic” and “the predominantly quantum” system are
apparently treated as completely isolated, for instance by placing the Schrödinger’s
cat inside of a box, it gives the impression that they do not interact with the outside
world (stressing that there is no interaction with the observer situated inside the
experiment itself, but outside the box). In the Schrödinger’s cat situation the poor
animal is death and alive at once, as the result of some sort of “propagation” of
uncertainty coming from the “predominantly quantum system” (we do not see how
this is possible!). In the case of the “quantum suicide” there is not only such a
“propagation” of uncertainty from the quantum system, but also, in the context of
the MWI, the “predominantly macroscopic system” (i.e. the suicide-prospective
scientist) acquires uncertainty, which keeps him forever alive (unable to die), while
his assistance witnesses the scientist death. In these though experiments there is
another catch, even if there is no apparently observation of the systems in
question, this does not means that the uncertainty is not already reduced to match
the permissible degrees of freedom of the “macroscopic realm”. “Even if the many
worlds interpretation is correct, the measure (given in MWI by the squared norm of
12
the wavefunction) of the surviving copies of the physicist will decrease by 50% with
each run of the experiment. This is equivalent to a single-world situation in which
one starts off many copies of the physicist, and the number of surviving copies is
decreased by 50% with each run. Therefore, the quantum nature of the experiment
provides no benefit to the physicist; in terms of his life expectancy or rational
decision making, or even in terms of his trying to decide whether the many-worlds
interpretation is correct, the many-worlds interpretation gives results that are the
same as that of single-world interpretation” (33). This means that same results are
obtained using the unitary collapse of wave Copenhagen interpretation and the
“non-collapsed” MWI. In fact Tegmark says,
 “What Everett does NOT postulate:
At certain magic instances, the world undergoes some sort of
metaphysical “split” into two branches that subsequently never interact.”
In essence, what Everett says is that there are superpositions of possible
outcomes, and inside each one the observer sees only his respective outcome.
The same author continues, “According to the MWI, there is, was and always will
be only one wavefunction, and only decoherence calculations, not postulates, can
tell us when it is a good approximation to treat two terms as non-interacting…When
confronted with experimental questions, adherents of the first four [Copenhagen,
Many worlds, Bohm, Consistent histories] will all agree on the following cookbook
prescription for how to compute the right answer, which will term the ‘shut-up-and
calculate’ recipe:
Use the Schrödinger equation in all your calculations. To compute the probability
for what you personally will perceive in the end, simple convert to probabilities in
the traditional way at the instant when you become mentally aware of the outcome.
In practice you can convert to probabilities much earlier, as soon as the
superposition becomes ‘macroscopic’, and you can determine when this occurs by
a standard decoherence calculation” (34). The same author, in another publication,
says, “Everett’s viewpoint become known as the ‘many worlds’ or, perhaps more
appropriately, ‘many minds’ interpretation of quantum mechanics because each of
one’s superposed mental states perceives its own world. This viewpoint simplifies
the underlying theory by removing the collapse postulate, implying that there is no
new undiscovered physics that makes these superposition go away…Even though
the wave function technically never collapses in the Everett view, it is generally
agreed that decoherence produces an effect that looks like a collapse and smells
like a collapse.” (35). Decoherence was first demonstrated by Zeh in 1970. Briefly,
it consist on the own “measurement” or “observation” or “scanning” that elements
of an environment makes on the objects, as a result of such interaction objects are
turn on reality. To say, for example a pen on a desk comes to reality or
materializes because is scanned by the environment. The environment is
technically the rest of the universe. In the Schrödinger’s cat experiment, therefore,
because of decoherence we do not expect that the cat or the gun or the box would
split into other worlds just because they are in proximity to a “predominant quantum
system”. Despite Tegmark postulate the end result would be the same using the
collapse-wave theory (Copenhagen) or the “non-collapse” MWI theory, according
to Page there might be differences in different cosmological scenarios with different
number of observers, which means different realities (36). This might be possible if
13
our universe is one of multiple waves, and just one of them is what we call the
Schrödinger’s equation. The question then is where and how these damned
equations appears?
Tegmark’s paradigms should not be arbitrarily mixed; each one has their own
scope. Consequently, we consider that applying the so-called “Principal Principle”
(37) to analyzed “quantum suicide” (38) does not have too much sense. We
consider that this “principle” is subjective, semiotic, anthropomorphic, and it is a
form of counterfactual internalism. Despite Everett’s theory has been also called
“Many Minds Interpretation”, we think this designation can be misused with an
anthropomorphic connotation. However, nothing prevents from the possibility of
worlds or universes without conscious minds. Indeed, decoherence happens
without the presence of brains. Actually, the “minds” part of this theory may
produce misinterpretations at the macroscopic level. Thus, according to Papineau
“Peter J. Lewis seems to have no good reason for ascribing the death-defying
implication to the many minds theory” (39)
Health and disease
If a biological system at any instant matches its perfect Smax it collapses. This
makes plausible that the biological system encounters a state of total equilibrium of
all its elements, at all degree of freedoms, and of course, instantaneously. Under
this hypothetical status, at Smax, the biological system halts because no more
events are possible. This means that the biological system is unable to produce
useful energy, thus it cannot produce work so it cannot produce more information.
In Chaos Theory terms, the biological system falls into its attractor (which is Smax),
and in turn, it falls to the “infinitesimal infinite” or zero (40, 41). We are not quite
aware of any known pathological condition associated with perfect equilibrium at
Smax. No research has been done on this particular issue. Smax-like conditions may
occur at ultrastructural levels, or as “hyperfunctional” levels. Disseminated
intravascular coagulation may follow such thermodynamic mechanism; in fact at
some point, most or all coagulation factors can be interacting at a Smax-like status.
Atherosclerosis might be also some kind of Smax condition. Well, if a biological
system reaches a deleterious Smax status, what is next? Since it cannot sustain its
integrity and its function any more, it would be absorbed by the S universe. However,
pathological conditions may primarily follow the opposite thermodynamic route,
which is the tendency to merge with the Suniverse. We can say that diseases start as
by increasing Entropy at ultrastructural sub-cellular level in an organ. For instance
cancer and autoimmune diseases may follow this entropic pattern. The increasing
Entropy disseminates into a cell, then from cell to cell, then to the whole organ,
then to the system, and finally to the whole body. Clinical manifestations such as
pain, fever, malaise, etc. may be associated with increasing Entropy. When the
entire biological system reaches certain high Entropy level, being unable to sustain
function and integrity, it dies. Another possibility is that pathological conditions
represent alternating phases of high/low Entropy, resembling different directions in
relation to space state attractors (different Lyapunov exponents) (40). The immune
14
system can be better understood through Chaos Theory, because it is complex,
self-assembled, interactive, fractal, evolutive, and capable of memory. There is a
balance between Th1 and Th2 immune responses, thus the immune system can
be conceptualized as a self-referential network revolving “strange attractors”, which
in turn are very important in tumorigenesis and cancer treatment (42). Brú, et al,
have found that malignant tumors show a clustering fractal dimensional growth at
their contour, which renders this microenvironment more acidic. Space
competence with the host is apparently the main factor in the growth of tumors.
Brú’s results contradict the classical oncology tumoral kinetics, and, moreover,
challenge the foundations of chemotherapy and radiotherapy; besides, he
conjectured that neutrophils by resistance to acidosis and competing for space
may play an important role to inhibit tumoral expansion (43, 44). In fact, the same
group has reported the cure of a terminal hepatocellullar carcinoma by inducing
patient’s neutrophilia with granulocyte colony-stimulating factor (45).
Understanding boundaries may be extremely important, as all systems follow the
holographic principle as the expression of the Information (Entropy) “living” at
boundaries (46 - 49). Goldberger, by the analysis of heart rate plotted against time
has conceptualized a similar model of heart disease. Healthy hearts exhibit rates of
nonlinear dynamics and possible chaotic behavior, which reflects the physiological
flexibility and adaptability. Contrarily, in heart disease rates became periodic or
totally random (50). Possibly the periodic rate indicates abnormal tendency to Smax,
in the sense that the functional pathways are being reduced by an equilibrium
imposed by the generic attractor of the system.
Life and evolution
Looking for “the cause of death” in an autopsy appears a bit odd as
thermodynamically speaking death is a whole process, besides the entire body
dies. Death is a dynamical process, therefore cannot be localized. The conjecture
here is why doctors, then, do not look for “the cause of life”? Seriously, it will be
more beneficial.
The basis for what is life is at the core of the “struggle” between the Smax and
Suniverse inside the biological system. Entropy, as a universal principle with different
aspects, renders integrity and functionality to all systems, including biological
systems. Integrity and functionality are sides of the same coin, ultimately. This
approach has many points in common with the universal law “Tao” of Lao-Tzu (400
B.C.) (51). The Yang and the Ying are comparable to Smax and Suniverse, and
similarly to the Second Law of Thermodynamics and Entropy, this can be extended
to the whole universe. “Ying exists within Yang; Yang exist within Ying”, it is an
expression that implies binarity and, therefore, it is related to the principle of 0 and
1. Lao-Tzu in the opening of his book “Tao Te Jing” says “The Tao that can be told
is not the eternal Tao, The name that can be named is not the eternal name”, but
we assume that this uncertainty of Tao depends on its extreme complexity. Entropy
might be dual with the “eternal” Tao. Similarly, at dimensional extremes Entropy
also falls into uncertainty. According to Schrödinger’s famous book “What is life”
15
(52), life basically covers two aspects “order to order” and “disorder to order”.
“Order to order” is akin to reproduction or replication of the organism. In biological
systems usually this is through DNA, which is essentially a macromolecule
(polymer) with high binary code capacity. The immense probability of base
combinations makes DNA extremely uncertain, therefore likely having high
Shannon’s entropy. Knowing the sequences of DNA may have reduced almost
insignificantly such uncertainty, since the informational power of this polymer
depends vastly more on its functional expressions than that of its sequence.
“Disorder into order”, this means that biological systems can exchange “energy”
and matter with the media. In animals the source is basically matter exchange. We
uptake matter with relatively low Entropy, and subsequently the programmed
enzymatic metabolism reduces even more the Entropy of this uptake.
Metaphorically, animals bite the bits of vegetables, in the sense that animals
nourish with the information taken from the vegetables. Salthe has evoked a similar
concept of transference-degradation of energy between “gradients and
consumers”, where the consumers are also “gradients” (53). As an extension of
this hypothesis, he has proposed that users of energy do not increase, locally, the
overall Entropy of the system; however, they do shape the distribution of Entropy.
In this context, Evolution can be conceptualized as introduction of new levels of
consumers to optimize the management of Entropy, which ultimately restrain the
tendency for dissipation. Salthe, metaphorically, concludes, “Evolution, then, is the
Universe’s devious route to its own negation” (54). “The second law implies that
the free energy of an isolated system is successively degraded by diabatic
processes over time, leading to entropy production…The formulations from
classical thermodynamics can be applied to non-equilibrium systems which are not
isolated (e.g., Prigogine 1962)… For these systems, the Second law then takes the
form of a continuity equation, in which the overall change of entropy of the system
dS/dt is determined from the local increase in entropy within the system dSI/dt and
the entropy flux convergence dSE/dt (i.e., the net flux of entropy across the system
boundary): dS/dt = dSI/dt + dSE/dt…A non-equilibrium system can maintain a state
of low entropy by “discarding” high entropy fluxes out of the system” (55). A
complicated mathematical and statistical form of non-equilibrium Entropy has been
proposed (56).
Production of vitamin D, reduction of melanin in skin, and reduction of retinoic acid
in retina are few of the reactions in animals that involve direct usage of solar
radiation. Contrarily, in vegetables the solar radiation is the main substrate source
through the process of photosynthesis. Another flaw of anthropomorphism is that
DNA is the starting source or origin of life. Instead, DNA is just another biological
polymer, and as any other polymer in nature it also has Entropy, and, therefore
carries information. Besides, Schrödinger’s dichotomization of “what is life” is
arbitrary, since DNA is a molecule produced by the biological metabolism.
Therefore, ultimately life can be summarized as just “disorder to order”, and
reproduction is an entropic shortcut in this process to reassure preservation of the
species. It should be clear that in thermodynamics and informatics, terms such
“order” and “disorder” must be avoided, because this entails anthropomorphic
biases (1). Therefore the question as to “what is life?” can be answer as
16
maintaining lower Entropy than that of the environment (Suniverse), and higher
entropy than that of Smax, by exchanging matter and/or “energy” with the
environment. Aging process can be thermodynamically explained as a
progressively increase Entropy (“error accumulation), and, therefore dissipation of
the system. Interestingly, Gladyshev has formulated a similar model using
concepts of Gibbs and Helmholtz functions, which are placed in the context of the
so-called “law of temporal hierarchies” (Gladyshev’s law) and the so-called
“principle of the stabilization of chemical substances” (Gladyshev’s principle) (57).
This model is essentially dual with the Smax/Suniverse presented here, since
quantities such as internal energy, enthalpy, Gibbs free energy and Helmholtz free
energy (“thermodynamic potentials”), as well as Entropy can be obtained based on
statistics arguments. Briefly, in these terms Entropy can be expressed as
S = U – F/T, where U is internal energy, F is Helmholtz free energy, and T the
absolute temperature, which means, that when F is minimized then S is
maximized. Entropy can be also expressed as S = (U + PV) - G/T, where U is
internal energy, P is pressure, V is volume, G is Gibbs free energy, and T is the
absolute temperature; additionally, PV is work (W) , and (U + PV) is enthalpy.
Similarly, this means that when G is minimized then S is correspondingly
maximized. According to Gladyshev’s model for biological systems G and F are
minimized through a hierarchic gradient sequence of environment/system, which
apparently is intimately related with the “law of temporal hierarchies”. Specifically,
this law establishes that “a biological system consist of the given organism’s cells,
the organism itself, and the population formed by these organisms (i.e., fragment of
the hierarchic sequence of biological structures). Identifying the average life-span
(life time) of structures makes it possible to assert that the average life-span (t) of a
cell (cel) in the organism is much less than the average life span of the organism
(org), which, in its turn, is much less than the life-span of the population (pop):
<<tcel <<torg << tpop <<…”
This assertion is a natural fact. For instance, in the intestine, of course, a cell lives
less than villi, the villi lives less than the intestine itself, the intestine less than the
animal, the animal less than all the animals, all the animals less than the whole
ecosystem, the whole ecosystem less than the planet, the planet less than the
galaxy, the galaxy less than the universe, etc. Therefore, by itself this “law”
apparently does not have too many consequences if it is not place into the context
of a higher physical law or principle. This apparently time hierarchy is perhaps the
consequence of not accounting a relativistic time frame. Therefore, this may be
indirectly an anthropomorphic bias, because this assessment is done through an
apparently external observer only. Moreover, these “temporary hierarchies” may
vanish if they are considered as eigen (physiological) times, which together with
the metabolic rate can be unified using a broad concept of Entropy, which
combines the external and internal observations. The tendency towards
minimization of G and F, in biological system, can be part of the universal tendency
of systems towards Smax. Thus, this so-called “law of temporal hierarchies” is very
likely a consequence of Smax. Regarding the so-called “principle of the stabilization
of chemical substances” (Gladyshev’s principle) it can be deduced that the G of
simple molecules such as H2, N2, O2, CO2 and H2O is much more compared to
17
the G of macromolecules. In other words, macromolecules (polymers) have more
Entropy than that of simple molecules, but correspondingly less Entropy compared
to that of the environment (rest of the universe). This assertion is another
consequence of the Smax/Suniverse model presented here. Subsequently, because
macromolecules are more uncertain for an external observer they carry more
information than simple molecules. Igamberdiev’s concept of “internal quantum
sate” (IQS) is essentially similar to state phase invariant(s). IQS behaves as
cellular automata, and it is “concatenated within the 3D space as a molecular
computer (MC).” Enzymes are MCs operators, while error corrections are effected
by RNA (short term) and DNA (long term). The error corrections do not affect the
IQS (58). In the context of Smax/Suniverse model, IQS can be the thermodynamic
status that depends on the instant interactions of all invariants. These invariants
interaction is in turn focused on fulfilling constrains imposed by the media. Besides,
such media can be internally related to Smax, and externally related to Suniverse.
Living organism by evolution are thermodynamically open (to matter and energy)
systems far from equilibrium. Otherwise life would not be possible. Here, the key is
“far from equilibrium” to explain life. How living organisms acquired such property?
Is this exclusively of living organism? By answering the second question the first is
also solved. Chemical reactions naturally can occur “far from equilibrium”. Three to
four billions years ago possibly a crack in the clay at the beach of the primitive sea,
that eventually sealed and concentrated raw chemical elements, which in turn
underwent innumerable chaotic (Brownian movements) probabilities, can explain
this property of life. Therefore, when the odds of a natural phenomenon or event
(i.e. a macromolecule or biological polymer such a protein or nucleic acid) are
statistically or probabilistically analyzed, a reductionist analysis is totally naive. This
means just breaking down the biological polymer into the number and different
types of moieties, followed just by a conventional probabilistic calculation of the
odds of reassembling it exactly as the initial macromolecule. This is not only terribly
primitive, but also misleading. It is like breaking down a cathedral into bricks,
counting the bricks, and making a statistical calculation how the bricks can be
reassembled by themselves into a cathedral again. Obviously, this approach is
wrong. But there is also another “little” difference, a cathedral is usually built in few
years, instead a natural event had already taken about four billions years to
happen, and, moreover, with the advantage that energy constantly flowed through
the system. Besides, systems cannot be built from nothing. Nothing can be
completely placed into a definition. After all, when a pizza can be “defined” as a
pizza? By the recipe, by the ingredients solely, by the ingredients together, before
introducing into the oven, during the cooking, when is out of the oven, or while is
being eaten or later? As the case of a pizza, every natural phenomenon implies a
continuum process. The term “life” does not have a definition, and similarly to time
and energy it is also a formality. Any definition of life, unfortunately, falls into the
anthropomorphic semantic rhetoric as a “property” of living organisms…and living
organisms live as they posses this “property” of life; however as such this
“property” remains obscure. On this regard we partially agree with Nasif. However,
this author involuntarily also falls into this anthropomorphic trap, because he tries
to push a definition of life, which can be considered as a reductionist attempt.
18
Dynamical complex systems are irreducible and impredicative, therefore defined
terms cannot be clearly differentiated from defining terms. The term life is just
provisional, as a summarized conceptualization of many complex functions.
Contrarily, I conceptualized but not define life. Nasif appeals a great deal to energy
disregarding the concept of Entropy, and also dichotomizes considerably energy
and matter. Besides, for this author the quality of life resides at the cytoplasm of
cells related to chemical reductions. We consider life as a whole process that
cannot be related to any particular locus, molecule or process, otherwise it will fall
again into anthropomorphic terms. In this context the same author says, “…
complex self-replication pattern can occur in inert self-replicating templates, for
example in prions, viruses, autocatalytic proteins and ribozymes, and all of these
structures cannot be considered as living dissipative thermodynamic systems
because the postmanipulation of energy for self-replication occurs spontaneously,
not autonomously as it occurs in living dissipative thermodynamic systems” (59).
What does he mean by “spontaneously” and “autonomously”? If we could ask a
virus, a prion, or even a peptide if it is dead or alive, we are sure that any of them
will answer that they are alive, and, in fact, from their own perspective (as internal
observers) we may be inert. A cell, an organ, and even a whole plant or animal,
including human species are not autonomous. Moreover, nothing can occur
“spontaneously”. Olalde has proposed another complex model of life and
extended to health status. According to this researcher’s philosophy life is a
function of the triad “intelligence”, “energy”, and “organization”, while health is the
“survival potential” as the result of the interaction of this hypothetical triad (60). In
this theory, however, the core of this triad remains elusive. As per personal
communication with Olalde, he also thinks that Entropy plays a pivotal role in
health and disease. Perhaps “intelligence” and “organization”, in this model, could
be expressed in terms of Entropy/Information, and may be explained by
coding/decoding-forwards/backwards algorithms (similar to artificial intelligence).
On the other hand, energy can be carried by Entropy, and, therefore it can be
cancelled. Igamberdiev`s model of living systems has many similarities to our
Smax/Suniverse model, as both models establish a relationship with thermodynamics,
chaos theory, evolution and self-organization. Admittedly, Igamberdiev`s model is
mathematically much more elaborated. Moreover, this model elegantly explains
evolution as memory kept inside of a reflective mathematical loop and series.
Then, this author emphasizes the intricate interrelation of DNA with structure and
function. “The reflecting control in genome is realized by tools (molecular
addresses) organizing combinatorial events. Thus, the molecular addresses
establish the set of rules for language game corresponding to such hierarchical
organization…During this strategy the error-correction is realized, and this takes
place in the potential field. We can suppose that the whole organism possesses
the ability to forecast the splicing result before is actualized, i.e. it can realize errorcorrection in the potential field by eliminating wrong potential possibilities, by
implicating error-correcting codes. This means that living systems realize
computation at quantum level, the process maintaining their dynamic stability at the
macroscopic time level” (58). Huang et al had studied about 12600 genome
expressions in the cellular differentiation process of neutrophils, and by means of
dynamical approach they have found that a subset of 2773 genes converge in
19
state space to a stable attractor associated with phenotype. According to these
researches, “Thus formal network architecture considerations as experimental
observations of cell fate behavior also support the idea that the genome-scale
regulatory network can act as an integrated entity and give rise to coherent, higherorder dynamic patterns, such as stable high-dimensional attractors” (61).
Igamberdiev`s conclusion is fascinating, since it can be related to quantum
entanglement and “instant communication” (62 - 64). An EPR (Einstein-PodolskyRosen)-pair and entanglement mechanism must occur inside biological systems,
reassuring their stability, and consequently involved in health status. Another
possible aspect of disease could be an imbalance between the physical state of
the biological system and the “macroscopic time level”, to be discussed in a future
article. Melkikh has proposed a quantum model computation of biological evolution
(21), which despite some criticisms done before, is very similar to the core of
Igamberdiev’s and our models, as he uses quantum mechanics, as well as
reduction of Entropy through a “parametric or force control” operating as a
“quantum demon”. However, this author does not further elaborate the emergence
of such “parametric or force control”, which according to our model it is most likely
equivalent to intrinsic or eigen Smax of the system. In addition, we explain in the
context of Entropy/Information and Chaos how this Smax may arise.
In summary, biological systems can be placed into the Universal
Entropy/Information. Biological systems can be understood, in this context, by
means of well-established concepts such as Smax (Jayne’s Maximal Entropy
Principle), Smin (based on Least Action Principle), which are relativistic according to
the observer’s position (internal or external). Function and structure are dual, and
eventually they merge. In the case of Biological Systems these principles can
explain the overall role of metabolism. Terms such as life, death, evolution, health
and disease, then acquire a profound and dynamical significance when are framed
by Entropy/Information.
Acknowledgements
The authors are indebted to Mrs. María Olinda Tello-Rodriguez for her invaluable
secretarial support.
20
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