The
Matrix movies present an interesting metaphor for reality.
In the Matrix, there is an artificial world of illusionary mental states. If
you seek out and are saved by Morpheus, then you go back to the “real” world,
which unfortunately is a world at war. Metaphorically speaking, the world of
humans and machines is at war because determinism has taken over society and spirit
has died. Human had treated machines as soul-less robots, and now the machines
use humans as batteries and live in the dark confines of meaningless world. The
Matrix is required to keep most humans from realizing that they are just
batteries. The machines don’t think that the humans have souls or individual
rights, they just find that they get more productive batteries if the humans
think that they are living in world pre-collapse.

Is this theory of the world inconsistent with the
facts of this world? Not really, but I think that the Matrix movies are a great
starting point for understanding Plato’s parable of the cave or Aristotle’s
concept of eudaimonia (sometimes translated as happiness and sometimes as flourishing.)
As Aristotle put it,

__you don’t even know what you really want__. You think you want a new car, a powerful job, or lots of sex. But according to him, what you really want is eudaimonia. The problem is that most of us don’t know what eudaimonia really is. In the movie, Morpheus and Neo realize that there is a Matrix and have figured out how to escape from it. To me, the Matrix we live in today is the philosophy of consumerism, utilitarianism, pluralism, and determinism. It is the “air we breathe” in the U.S. and in Europe. We are constantly bombarded with ads for products telling us that the goal of life is to be happy and that we should satisfy our desires/wants/cravings because satisfying those desires/wants/cravings is what’s actually good for us.
One of the problems we face today is that there’s a
disconnect between those things that brings us pleasure and those things that brings
us growth. Pleasure and growth are not the same, and the amount of overlap between
the two is quickly shrinking. Tens of thousands of years ago, what brought us
pleasure was the nearly the same as what brought us growth, and it is growth
that we should be ultimately are aiming for. Thousands of years ago, eating and
having reproductive sex actually helped grow societies. Now we can can’t say
the same. So for Aristotle, the problem is first how to recognize good
pleasures from bad pleasures, and then ultimately figure out how to make bad
pleasures undesirable. Aristotle thought that this could be done through the
use of reason and through the help of a mentor/coach who had already learned or
had already been taught how to make the bad (i.e. what does not lead to growth)
unpleasant. The other problem of our age (also discussed throughout the Matrix)
is the lack of spirit in a world of empiricism, determinism and cultural
relativism. In the first movie, Neo had to overcome the problem of pleasure vs.
eudaimonia (i.e. escape the Matrix in order to help the rebel society flourish),
but then he had to go beyond that. In the later movies, he had to bring spirit
to a world of machines and determinism.

The goal of this blog has been to supply rational
answers to the following questions: (1) why does pleasure not always equate
with growth? (2) why is the world not deterministic? (3) what is the purpose of
life? The goal of this post is to try to put the nail in the coffin in the
theory of determinism and reductionism.

To do so, I’ll argue that there is a digital structure co-existent with, but not subservient to the analog world of physical space. By digital structure, I don’t mean zeroes and ones; I mean discrete countable symmetries between particles in the physical world. Examples of digital (i.e. discrete and countable) objects are the symmetries of a cube, the letters of DNA, and the 0,1’s of computer systems. In contrast to the discrete symmetries of a cube, there are also analog (i.e. continuous) symmetries of the circle. What’s fascinating is that there are ways of going between the infinite-sized symmetry groups (such as the symmetry of a circle or of a sphere) and countable, finite symmetry groups. (This is due to the fact that every Lie group…symmetry group of objects like circles and spheres…has an associated Lie algebra with a finite number of elements. For example, the Lie algebra of the circle has one element. The Lie algebra of the sphere has three elements, and the Lie algebra of the 4-D sphere has 8 elements.)

So, here I’m using the term digital to mean “discrete,
finite and countable” and I’m using the term analog to mean “infinite and
continuous.” The digital world and the analog world co-exist in this world, but
they are not the same thing. The analog world is the world of particles moving
according to the analog laws of physics. I’m using the word ‘analog’ here because
the forces of gravity and E&M are differentially continuous (and probably
the strong nuclear force as well.) What are digital (or finitely countable) are
the exchange symmetries between similar particles, and this digital structure
has the capability of growing. The digital structure actually grows out of the
differential equations of the analog world; it’s the symmetries in the
differential equations that are finite and countable, and that can increase
with time. In their case of living creatures, the symmetries can
self-replicate. The exchange symmetries are what we mean when we say that the entropy
increases in the universe. For small-sized sets of differential equations, the
number of symmetry operators is small and constant with time. For example, when
studying the symmetries of the differential equations that describe the
gravitational motion of the Sun and the Earth, there are only a few types of
symmetries (such as symmetry with respect to time translation, symmetry with
respect to time reflection, symmetry with respect to spatial reflection through
the plane of motion.) The algebra of these symmetries is complete and
consistent. However, when you put together the full set of differential
equations required to describe the motion of the components of a bacteria and
its environment (such as proteins, gases, sunlight, DNA, cell walls, etc…), the
number of symmetries (including exchange symmetries) is extraordinarily large,
and because of irreversible processes, the number of symmetries increases with
time. The algebra of the symmetries is incomplete and the motion of particles
is unpredictable (i.e. statements in the algebra may be true but can’t be
proven.) In other words, living creatures are extremely large, self-replicating
symmetry groups. The symmetry group of a human being is much larger than the symmetry
group of a bacteria. Stated another way, there is a digital structure that
drives the movement through analog space-time.

If this is the first time with some of these
concepts, I suggest reading an
earlier post in which I go through the concept of
entropy generation and how the symmetries of differential equations can
increase with time. In that post, I discuss the concept of “Non-linear,
Non-equilibrium of degree five” (in which the algebra of the symmetry operators
of a set of differential equations follows Gödel’s incompleteness theorem
rather than his completeness theorem. An example of a complete algebra is the propositional
logic, which is an algebra which follows Gödel’s
completeness theorem. An example of an incomplete algebra is arithmetic in the natural numbers.)
When an algebra follows Gödel’s incompleteness theorem, then there are true
statements that can’t be proven.

When the set of differential equations of the
universe has an underlying algebra of the group of symmetry operators that
obeys Gödel’s incompleteness theorem, then we reach the point in which there
may be true statements in the algebra of the symmetry group that can never be
proven. In arithmetic in the natural numbers, there are true statements that
can’t be proven using only arithmetic in the natural numbers. What this seems
to imply is that the future state of the universe might have certain
symmetries, but we can’t prove that this will be so. We don’t know the future
symmetries of the universe and therefore we don’t know the future entropy of
the universe. The entropy of the universe is the number of symmetry operators
in the symmetry group of relations between similar particles in the universe. (This
was shown by Joe Rosen. I suggest reading Chapters
5&7. Or download
the following paper. In addition, he has a more
recent book that you can download: Ch4 and Ch11.3 are
good starting points.) To paraphrase Rosen, “

__The symmetry group of the past is a subgroup of the symmetry group of the future__.” And “If the symmetry group of the future is larger than the symmetry group of the past (while containing the symmetry group of the past as a subgroup), then this implies irreversibility, i.e. that the laws of physics are time-reversal asymmetric.” This means that if the entropy of an isolated system increases, then the laws of physics are time-reversal asymmetric, which is what we have found for the weak nuclear force. Because I’m discussing a lot of mathematics and physics, this post might sound reductionist and deterministic, but it’s not either of these things. But before continuing, I’d like to go into more detail into what I mean by determinism and reductionism.
First, what do I mean by our deterministic, analog
world? I mean what Plato and Gnostic
Christians call the material world; it’s what we see and can experience with
our senses, even though we all have different sense organs. By this world, I
mean the mass/energy, and I mean the differential equations that occur over
differentially continuous space and time. I mean the continuous (analog) world
of physics as taught to me and many others in college. Because of relativity,
we can’t agree on certain aspects of the world (like 4-D vector space-time as
well as 4-D apparent mass and momentum), but we can agree on total energy. When
you just study the analog world and accept Laplace’s idea that the future and
past state of the universe can be determined if you knew the position and
velocity of every particle in the universe, then you are necessarily led down the
following path (as described by Plato in the Theaetetus
dialogue): (1) Empiricism,
(2) Relative values (i.e. no absolute values or morality), and (3) the
inability to communicate. (Yes! Most physicists forget that the inability to
communicate is a logical conclusion of living in a purely analog world…most
physicists haven’t read Plato or many post-modernist philosophers to understand
that the “desert
of the real” that would be a logical conclusion if
we lived in a purely deterministic world.)

Most of today’s physicists have forgotten about our
underlying digital structure, and are only focusing on the analog forces of
nature acting between small numbers of particles. As stated before, when I say
digital structure, I don’t mean that space-time is discrete (like Wolfram argues at times.) Space-time is
most likely differentially continuous (or else we couldn’t take Noether’s
Theorem to derive conservation of energy-momentum.) The digital structure that
I’m talking about is not the same as the discrete 0,1’s of the Matrix. By
digital structure, I mean the finite structures or elements of the Lie algebras
or finite symmetry groups. This underlying digital structure is the countable structure
of the system, such as the finite number of exchange relationship between the
particles of the world.

__What’s digital is the symmetry group of the universe.__Physicists today would like to find the smallest Lie group that explains the fundamental force (i.e. Grand Unified Theory.) And more power to them, but the problem is that many of these physicists often think to themselves “Once we know the symmetry group of the fundamental forces, then that’s it.” This is not a healthy mindset.

While if these Grand Unified Theories (GUTs) turn out to be true, it might seem like a proof of Plato’s idea of the abstract logical forms. But when you look deeper, it’s clear that these GUTs aim to wipe out one of the main elements in Plato’s world of the abstract: ethics. In the supposedly, time-reversal symmetric world of many GUT theories, there is no ethics, no morality, and no goal to life. What I find so fascinating is that some physicists will talk about a time-symmetric universe and then in the next breathe, they say that’s it’s unethical to use fossil fuels. The disconnect in the world today is so apparent it’s funny. You can’t have ethics in a time-reversal symmetric universe. You can only have ethics if there is actually a goal to life. You can’t get mad about fossil fuels unless you can prove that fossil fuels are bad. And for something to be bad, there must be something that is good, and that good must be common to all living creatures. This is the foundations of ethics, and it requires a time-asymmetric universe. In Plato’s abstract universe, in addition to mathematical truths, there are also things like beauty and ethics, which according to him don’t change with time or space.

As stated earlier, many physicists think they are
living in an analog world where (1) space-time is continuous, (2) the entropy
of the universe is constant, and (3) there is no such thing as ethics or
morality. Many physicists think that entropy is a made-up term due to “coarse graining.” Worse, many
physicists talk about “spontaneous symmetry breaking of the natural laws.” They
believe that the laws of the universe were more symmetric in the past than the
laws of physics are today. So, not only
do these physicists think that entropy is constant, but they somehow think that
symmetry breaking occurred in the universe without causing entropy to decrease.
This violates Rosen’s principle that the symmetry group of the future contains
the symmetry group of the past as a subset. If there was spontaneous symmetry
breaking and no entropy generation due to irreversible collisions, then the
entropy of universe would actually decrease. But this is not what we see when
we run experiments. In every case, the entropy of an isolated system increases
or remains constant with time.

This is what we mean by the Second Law of
Thermodynamics. The future is more symmetric than the past, or exactly as
symmetric as the past if we ever reach equilibrium. If we only include
gravitational and E&M forces in the differential equations of motion of
particles, then the number of symmetries will remain constant. If we include the
weak nuclear force into the differential equations of motion, then we have a
source of time asymmetry, which can convert inequivalent microstates of a
system into equivalent microstates of the system. Due to these time-asymmetric,
irreversible processes, the number of exchange symmetries between equivalent
microstates increase with time.

Let me state this again: The number of symmetry operators in the symmetry group of the universe is increasing with time. Not only is the symmetry group of the universe increasing with time, but as stated before, the symmetry group of the future contains the symmetry group of the past as a subgroup. The symmetry group of the universe is so large than I can’t even comprehend the number of symmetry operators in the group. The symmetry group of the universe includes the symmetry group of the fundamental forces (which doesn’t appear to change with space-time) and includes the symmetry group of space-time itself (which doesn’t appear to change with space-time.) But in addition, the symmetry group of the universe also includes the exchange symmetry between particles, i.e. the exchange symmetries between equivalent microstates of the system. The exchange symmetries are discrete symmetries, unlike the continuous Lie symmetries of a circle or of an n-dimensional sphere. The exchange symmetries are similar to the permutation group S(n). This is what we mean by entropy, i.e. the symmetry and relationship between particles. This symmetry can only increase with space-time.

Let me state this again: The number of symmetry operators in the symmetry group of the universe is increasing with time. Not only is the symmetry group of the universe increasing with time, but as stated before, the symmetry group of the future contains the symmetry group of the past as a subgroup. The symmetry group of the universe is so large than I can’t even comprehend the number of symmetry operators in the group. The symmetry group of the universe includes the symmetry group of the fundamental forces (which doesn’t appear to change with space-time) and includes the symmetry group of space-time itself (which doesn’t appear to change with space-time.) But in addition, the symmetry group of the universe also includes the exchange symmetry between particles, i.e. the exchange symmetries between equivalent microstates of the system. The exchange symmetries are discrete symmetries, unlike the continuous Lie symmetries of a circle or of an n-dimensional sphere. The exchange symmetries are similar to the permutation group S(n). This is what we mean by entropy, i.e. the symmetry and relationship between particles. This symmetry can only increase with space-time.

As discussed in the
last post, if you could measure the surface area of the
universe, then you could see that space-time only goes forward. There is no going
backwards. The surface area of the universe is increasing with time, and the
surface area of the universe is proportional to the entropy of the universe
which is proportional to the logarithm of the number of symmetry operators in
the universe.

Entropy is a property of a system. There is no such thing as the entropy of particle. It appears that the weak nuclear force is the cause of the increase of symmetry operators between particles because it is the only time-asymmetric of the four known forces of nature and because it only acts on fermions. (This may be why bosonic states of superfluids or superconductors don’t have irreversible entropy generation.) How does the weak nuclear force do this? It randomizes outcomes after collisions between particles who interact via the weak nuclear force. In other words, it destroys knowledge of the final state of particles, given the position-velocity of the particles before a collision. As our knowledge of the position-velocity of the particles decreases with time, the symmetry group between the particles increases with time. (For example, when we mix nitrogen in half a box with oxygen in half a box, we have specific knowledge about where to find certain types of particles. When we remove the divide between the two halves of the box, we slowly lose information about the particles, but the symmetry group of the overall universe has increased because inequivalent microstates turn into equivalent microstates.) What’s interesting is that the value of entropy of a system is invariant (i.e. not relative) to the velocity of the observer with respect to the system. In part this is due to the fact that the entropy is a system property, not a property of any specific particle. Moving to a different rest frame can’t change the exchange symmetries between particles, or as I stated in the last post, moving to a different rest frame can’t change the overall surface area of the universe.

So, let me step back and use a metaphor to describe what I mean by the “symmetry group of the Universe”:

Entropy is a property of a system. There is no such thing as the entropy of particle. It appears that the weak nuclear force is the cause of the increase of symmetry operators between particles because it is the only time-asymmetric of the four known forces of nature and because it only acts on fermions. (This may be why bosonic states of superfluids or superconductors don’t have irreversible entropy generation.) How does the weak nuclear force do this? It randomizes outcomes after collisions between particles who interact via the weak nuclear force. In other words, it destroys knowledge of the final state of particles, given the position-velocity of the particles before a collision. As our knowledge of the position-velocity of the particles decreases with time, the symmetry group between the particles increases with time. (For example, when we mix nitrogen in half a box with oxygen in half a box, we have specific knowledge about where to find certain types of particles. When we remove the divide between the two halves of the box, we slowly lose information about the particles, but the symmetry group of the overall universe has increased because inequivalent microstates turn into equivalent microstates.) What’s interesting is that the value of entropy of a system is invariant (i.e. not relative) to the velocity of the observer with respect to the system. In part this is due to the fact that the entropy is a system property, not a property of any specific particle. Moving to a different rest frame can’t change the exchange symmetries between particles, or as I stated in the last post, moving to a different rest frame can’t change the overall surface area of the universe.

So, let me step back and use a metaphor to describe what I mean by the “symmetry group of the Universe”:

The symmetry group of the universe is like a book
that can only grow in size as we move into the future. In other words, it’s
like a book that has the capability of growing. The language of the book looks
like it’s written in an alphabet called the simple groups.
Think of the groups as the letters in the alphabet. The smallest size non-cyclic,
simple group is the group A5, which has 60 symmetry operators. When the set of
differential equations of the universe contain A5 as an underlying symmetry
group, we get our first taste of the first unique letter in this alphabet. When
the set of differential equations in the universe contain even more symmetry
operators, then we can get a taste for even larger simple groups. There are an
infinite number of cyclic simple groups, but only a finite number of sporadic simple
groups, just as there are only a finite number of letters in any given
language. The largest sporadic simple group is called The Monster,
whose size is larger than I’m willing to write out here. The symmetry group of
the relationship between the particles in the Universe is increasing in time,
and the number of symmetry operators in the future can’t be computed even if we
know the symmetry group of Universe right now. The length of what I call the
Book (i.e. the symmetry group of the universe) is increasing and there’s no way
to predict the end length of the Book from knowing its stating length. There’s also
no way to know what the Book will say at any given point in time in the future.
The crazy thing about this alphabet is
that the alphabet contains the capability of self-replication, i.e. encoded in what
I’ll call “paragraphs” of the book is the ability for the overall length of the
book to increase through self-replication of those “paragraphs.” The
irreversibility of self-replication is not the only way that the length of the
Book increases, but it is one way that the Book can increase in length.

**Once something is written in the Book, it can’t be deleted because the symmetry group of the universe of the future contains the symmetry group of the past.**
The fact that there are self-replicating structures
in the digital world is due to underlying symmetries in the differential
equations themselves. The “analog” differential equations contain within
themselves the very structures can self-replicate. But you can only find these
self-replicating sets of symmetry operators when you put together the
differential equations for large sets of particles and when you include the
weak nuclear force in the differential equations. If you only put a few
particles together, you don’t get enough symmetry groups to see simple groups
like A5, let alone The Monster Group.
I’ll put it this way: physicists rarely work with enough sets of
particles to see the self-replication of the simple groups. Biologists see
self-replication and think that the self-replication is due to DNA. But the DNA
we see in living creatures is only a metaphor for the underlying alphabet of
simple groups. The DNA is a visible reminder that there is underlying digital
structure to the universe, but the DNA is not the same thing as the simple
groups. There’s a problems with thinking that DNA itself is the replicator: the
DNA needs to have other material and to be part of larger feedback loops in
order to replicate. The feedback loops occur in a system of differential
equations. There is an underlying set of symmetries in the differential
equations that allow the self-replication to occur

So, when Richard Dawkins states that a human being
is just a means for DNA to make more DNA, he is partially correct and partially
incorrect. While the symmetries (abstract) and DNA (physical) are both digital (i.e.
discrete and finite), DNA can be destroyed, but symmetry groups can’t be
destroyed because the symmetry group of the future contains the symmetry group
of the past. The Book can only increases in length. In this sense, thinking
that DNA is the self-replicator is a little bit misguided…though it’s thinking
in the right path. DNA growth and replication is an outward sign of the inward
truth: growth and replication of the
underlying feedback loops and underlying symmetries of the differential
equations of the universe.

The purpose of life is not to replicate any
particular set of genes. The purpose of life is to grow life as a means of
increasing the exchange symmetries of the universe. Without reference to a
common good or without the use of a common alphabet of simple groups, there is no
possibility for language or communication. What Richard Dawkins forgets is that
if we take his idea that the purpose of the gene is to make more copies, then
we are still living in a reductionist universe with no ethics and no reference
to the Good, the Truth, and The Beautiful. It’s a world with no communication
and therefore it’s not the real world that he’s describing. The reason that we
can communicate is that there really is a discrete, countable structure
co-exist with the continuous, analog world of the four forces of nature. In
general, humans and other species are ethical. We all think that murder is
wrong. Most of the time, our disagreements are about what constitutes a murder
and not whether murder is wrong. Where does this ethics come from? In a
time-reversel-symmetric universe, there could be no ethics and no way to refer
to the Good, the Truth, and The Beautiful.

So, when I say “grow, baby, grow”, what I’m saying
is that you should live your life so as to grow life because growing life is
the best way to increase the symmetry group of the universe as quickly as
possible. Sure, you could burn some coal/oil/gas/biomass right now in air and
generate a lot of entropy (exchange symmetries), but this is not useful in the
long run. The trick is figuring out how to burn coal/oil/gas/biomass such that
you can derive useful work, so that you can collect more coal/oil/gas/biomass
and build more power plants. In the end, this will increase the symmetry group
of the universe faster than just burn some coal/oil/gas/biomass in the air. Therefore,
the “Good” is

__the capability to self-replicate quickly in a way that does not harm or prevent other life forms from being able to self-replicate__. This definition of the “Good” does not change with time or space, and is the basis of ethics for all life forms.
Restated, the purpose of life is to grow life. You
are in control of your actions because your actions take place both in the
analog and digital world. In the analog world, everything looks deterministic,
but in the digital world, entropy increases as rational agents self-replicate,
causing the symmetry group of the universe to increase. In the digital world (i.e.
the finite, countable relationships between particles), the universe is not
deterministic. There’s no way to predict the future consequences of an action.
There’s history, and life-forms must learn from history in order to help them
determine how best to act so as to grow life as fast as possible.

Knowing only the fundamental forces of nature is
like studying the alphabet of a language but never reading a book written in
that language. It would be silly to learn the movement of chess pieces but
never play a game of chess, just as it would be silly to study the letters in
an alphabet but never a book in the language. What’s important is the book, but
in order to read the book, you have to know the language it’s written in. The
alphabet is apparently the ‘finite simple groups’ (note that these are the
equivalent of prime numbers in group theory.) This is what I mean (and probably
what Plato/Socrates meant) by the underlying digital form of the universe. As
Aristotle would put it, the abstract world is the relationship between the
parts of this world, i.e. Plato’s theory of forms is correct in some sense
(though not completely correct because there is no abstract concept called a
table or a horse.)

There is purpose, but the purpose is completely of
this world. Purpose (i.e. the Truth, the Good, and the Beautiful) come from the
relationship between the particles of the universe. Purpose comes from striving
for symmetry. Purpose is not inherent in the particles themselves. It’s the
fact that there is an underlying structure to the universe that we can even
have this discussion right now. In a purely deterministic universe, there is no
such thing as language and all communication is impossible. It’s only the fact
that we have a common reference, such as the alphabet of simple groups that
life itself is able to communicate. In other words, purpose is a part of the
nature world, and when we think that we live in a deterministic universe, we
blind ourselves to this truth and to the purpose of life.

In summary, remember that (1) the symmetry group of the
universe can only increase; (2) there’s no rule about how it can increase; (3) what
you do effects how the book will be written; (4) the book of the universe is
not predetermined; (5) the book of the universe can’t be destroyed by anything;
(6) the universe will not end in a Big Crunch that destroys everything; (6) what
you write in the Book is written forever; (7) if irreversible processes
continue, then the universe can continue to expand; (8) the symmetry groups of
the past can’t be destroyed; (9) the digital self-replicating structures have
free will when you study the digital structure of the world; and (10) it’s only
when you focus on the analog world and deny the digital world of relationships
between particles that you could think that you don’t have free will.

In conclusion, I’d like
to go back to the prior discussion of The Matrix. Neo’s job in the last of the
three movies was to return spirit to a world with no soul, to return love to
the world of deterministic machines, and finally to return ethics to a world of
merely sense perceptions. Throughout the ages, this has been the job of the
philosopher. It’s what Socrates,
Plato
and Aristotle
did thousands of years ago in response to the determinism of Heraclitus.
It’s what Kant
was trying to do in responding to the determinism of Newton
and Hume.
It’s what physicist/biologist/philosopher Stuart Kaufmann
and physicist/linguist/philosopher Douglas
Hofstadter are doing today in response to the determinism and
reductionism of today’s physics and the post-modernism, relativism, and
pluralism of today’s philosophies. The question is: can we learn enough about
the laws and particles of the world in order to prove that determinism and
reductionism are wrong? In other words, can a mathematician take the ideas I’ve
discussed above, and then put them into a proof that refutes determinism? (Just
as Gödel did to prove that arithmetic on the natural numbers is incomplete, but
consistent. As side note, I’d like to point out that Gödel was a Platonist even
though he regularly attended meetings in Vienna with the logical
positivists.)
Until we have more mathematicians who can put these anti-deterministic
ideas into mathematical proofs, it is likely that many people will continue to
believe in ethical
non cognitivism (i.e. determinism and a world with no
ethics or absolute morality) or ethical reductionism (i.e. that God or something not of this
universe is the source of all ethics and morality.) My hope is that this post
will help convince people that ethics and the good are a part of the universe
itself, or at least I hope that this post will induce you to study Plato’s
refutation of determinism by showing that empiricism and determinism lead to
the inability to communicate (Lecture
4 of the following CD course.) My hope is that
people can take the ideas in this post and prove mathematically the concept of
self-replicating of the underlying symmetries in the differential equations or prove
the incompleteness of the algebra of the underlying symmetry group of those
differential equations. Then we will have placed the nail in the coffin of
determinism and reductionism.

p.s. To those who dare this mathematical task, best
of luck. This is no easy task. I’ll leave you with some possible words of
wisdom from Louis H Kauffman “Self-reference is the inﬁnite in ﬁnite guise!” Remember that Gödel was able to prove the
incompleteness of arithmetic over the natural numbers by showing that any
statement in arithmetic could be converted into a natural number…i.e. every
statement (such as 2+2=5) had a corresponding Gödel number (such as 2

^{w }∙ 3^{x }∙ 5^{w }∙ 7^{y}∙ 9^{z}where w,x,y,z are the numbers assigned to symbols in the arithmetic statements.) In other words, arithmetic over the natural numbers is self-referential. Every natural number can be broken down into its prime factorization, and hence every natural number can be converted into a statement in arithmetic. The question yet to be proved mathematically is the following: can you put together the correct differential equations of motion of particles in a system, and come up with a group for the symmetry operators, such that the algebra of the symmetry group is incomplete?
Gödel was able to do
his proof on the natural numbers because it was countable, discrete and able to
break down into prime factorization. His proof would not have worked with the
integers (i.e. negative natural numbers, zero and positive natural numbers)
because he could no longer use his prime factorization ‘trick.’ The questions
in my mind are: (1) Since the time-irreversibility of the weak nuclear force
destroys the time reflection symmetry of the differential equations, is this
someway analogous to converting the real numbers or the integers into the
positive natural?

(2) Can we turn any symmetry group into a statement
about ‘group multiplication’ in the same way that Gödel turned any natural number into a statement about
arithmetic over the natural numbers? I think this is possible because every
symmetry group can be factorized into simple groups multiplied by other simple
groups, and once decomposed into simple groups there is no further
decomposition possible. The decomposition of any symmetry group is unique.

For example, the
group Z(6) (i.e. the cyclic group of size 6) is equal to Z(2) x Z(3). You
essentially prime factorize any group. But it’s a little bit more complicated
than prime factorization of natural numbers because Z(9) ≠ Z(3) x Z(3) and
because you can have multiple groups with the same number of symmetry
operators. This is at least the start of what somebody more knowledgeable than me
in group theory can hopefully finish.

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