Tuesday, August 28, 2012

There is an underlying digital structure to our analog world: A refutation of reductionism and determinism

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.
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”: 
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 2w ∙ 3x ∙ 5w ∙ 7y ∙ 9z   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.

1 comment:

  1. This comment has been removed by a blog administrator.