Notes on a Cosmology – Part 24a, Epilogue

In the next few posts, I plan to wrap up the Notes on a Cosmology series and draw some general conclusions. The idea I want to convey is very difficult to put into words not because it is a very complicated or novel idea but because there are so many ways to miscommunicate it.

Let’s begin the project of summarizing the series by looking at one of the emerging technologies of our time: Bitcoin. The principles of Bitcoin are defined as follows:

• 21 million coins
• No censorship: Nobody should be able to prevent valid txs from being confirmed.
• Open-Source: Bitcoin source code should always be open for anyone to read, modify, copy, share.
• Permissionless: No arbitrary gatekeepers should ever prevent anybody from being part of the network (user, node, miner, etc).
• Pseudonymous: No ID should be required to own, use Bitcoin.
• Fungible: All coins are equal and should be equally spendable.
• Irreversible Transactions: Confirmed blocks should be set in stone. Blockchain History should be immutable.

This may seem to be a topic far removed from the lofty ontology of the Holy Trinity and the other ideas I have covered throughout the series. But it is not so far removed. Let’s re-word the Bitcoin principles slightly (while retaining their essential properties):

• Fixed, finite pie - no one can pad their pocket or create additional value for themselves, ex nihilo
• Non-obstruction – everyone can engage the system without the possibility of being censored
• Reverse-engineering is allowed – no one owns a patent on the system or can act as its central director
• Permissionless – similar to non-obstruction; you don’t have to ask permission to participate in the system
• Secrecy is possible within the system (if you don’t divulge your key, no one can access the associated funds)
• Homogeneity – every part of the system follows the same rules as every other part of the system
• Irreversibility – no take-backs

If you think about it, these properties describe another system that is very familiar to all of us, whether tech-savvy or not. This system is called the world. Look around you, and ask:

• Is there anyone who can wave a magic wand and bring gold bars into being and make themselves rich, thusly? No. The physical pie is a fixed, finite pie.
• Is there anyone who can obstruct you from using your own body? Sure, you can be put in jail or even in restraints. But this does not obstruct you from using your own body within those constraints, it merely encloses the extents in which your body can exist, while you remain free as ever to use your body. You could be drugged but this entails some loss of ordinary consciousness, and it is the ordinary world that I mean to examine.
• Is there anything preventing the rules of Nature from being reverse-engineered and copied verbatim? No. You are free to reverse-engineer the very fabric of reality itself and copy it, verbatim, if you are able.
• Must you ask anyone’s permission in order to be alive? Sure, someone might kill you if you do not do what they want. But that is not the same as having to ask permission, it just means that your existence is not unconditional and permanent.

As far as we know, it is possible to have secrets in the physical world; that is to say, no one has proved that the physical world is a holographic construct in which all facts are accessible (with sufficient energy) to every observer within that construct. The laws of the physical world are homogeneous – the speed of light on Arcturus is the same as the speed of light on Earth. Finally, all physical processes are irreversible.

It is, at once, remarkable and unremarkable that the principles of a currency that was invented to be ideal, in some sense, would happen to correlate with properties of our world. The correlation is remarkable because physics appears to be a very different kind of thing than a digital currency. But it is unremarkable in that we are physical beings – what we consider to be useful and relevant is inescapably shaped by our physical mode of being.

My purpose in mentioning the correlation between the principles of the world’s largest cryptocurrency and some of the properties of the physical world is this: even though computational systems and material systems appear to be very different, our belief in this “computational dualism” is purely our own prejudice. We see the physical world as being very different from the artificial world only because we are so used to the physical world.

I have recently done a deep-dive into the subject of artificial neural net (ANN) architectures. One of the things I have realized as a result of this study is that the relationship between discrete and continuous computation is not well understood; at least, there is some persistent confusion in the various specializations. If we were to identify one property that makes artificial neural nets so useful and flexible, it would have to be that they are end-to-end differentiable. The backpropagation algorithm is possible because differential equations work equally well going from the input to the output or vice-versa. Differential equations merely specify the differential relationships between two or more variables; they do not specify which variable “causes” or “drives” which. For this reason, we can use an iterative training method that allows us to alternately forward-propagate and back-propagate the values in any set of equations that is end-to-end differentiable (including ANNs).

The physical world has a property very similar to end-to-end differentiability – our best physical theories describe the world using complex-valued functions; we can use analytical continuation to extend these functions beyond the reach of observation. The logic of analytic continuation is this: supposing the world continues to behave at very large and very small scales in a manner that is consistent with the way it behaves at scales we can observe, then it must behave so-and-so. Note that such reasoning is not a substitute for experiment. We only resort to this kind of indirect reasoning for those scales where experiment is simply not possible with current technology. The key point is this: analytic functions are everywhere differentiable. This means that every aspect of the physical world that is described by our best theories is amenable to back-propagation!

In Part 16 of this series, I described the quantum monad. Key to understanding the idea of the quantum monad is understanding the relationship between continuous information and discrete information. Mathematics leads us to think of the discrete and the continuous as two, irreconcilably separate domains. But the physical world exhibits a unity between discreteness and continuity. Our best theories of physics describe a continuous world, a world that is everywhere differentiable – yet discrete signals abound within this world. Speech, the written word and hand gestures are all familiar examples of discrete signals. Every digital electronic computer is built using analog circuitry and yet the electronic digital computer is a physical system that almost perfectly implements idealized, discrete computation. What this tells us is that discreteness can be thought of as a matter of staying far away from boundary conditions. In digital electronics, the boundary condition is called the non-allowed region – the circuit is simply not allowed to remain at voltage levels that are not clearly “high” or clearly “low”. Any circuit that stays in this region is exhibiting undefined behavior. Note that such conditions commonly arise in real electronic circuits as the result of high-impedance states but a logic failure is likely if these high-impedance states are used as input to logic gates.

Another, less known form of electronic computation that enjoyed some popularity in the pre-PC era is called stochastic computation. The basic principle of stochastic computation is to binarize a signal, while treating the value of that signal as a ratio of its levels – if a signal is ‘1’ 70% of the time and ‘0’ 30% of the time, then the value of the signal is 0.7. Note that the mathematics that describes the signal values in a stochastic system is continuous, not discrete. The resolution of measurement is, at any given time, finite but this is a limitation that is very well-understood since physical theories must always take into account measurement error. The point is that we can build a discrete system and use continuous mathematics to correctly describe its behavior, just as we can build continuous systems and implement discrete-symbol systems within them.

An article just published in Nature has calculated that the efficiency gains of artificial neural networks built with analog circuitry are around 100-fold over the GPUs that are commonly used. Biological brains – including the human brain – are analog computers, not digital computers. We can see that there is a deep relationship between analog, discrete, noise-tolerance, power-consumption and stability. This relationship is one of tradeoffs, not a black-and-white choice between discrete or analog.

The theory of quantum computation predicts (and experiment confirms) that computations using qubits are able to harness modes of computation not available to classical computers. But here’s the punchline: the mathematics of quantum systems straddles the very same divide as the discrete-analog distinction. Is it a wave? (Continuous?) Is it a particle? (Discrete?) The correct answer: it is both. This is true of all real information and all real computation. There is no exception – all digital computers are actually just analog systems that we interpret discretely by staying far away from the boundary conditions.

The theory of computational complexity tells us how hard it is to compute the solutions to different kinds of mathematical problems. Certain problems are easy to solve. Others are very difficult. Some are provably impossible. But the domain of computational complexity theory is restricted to symbolic computation (discrete computation) – the answer to hard mathematical problems can sometimes be found very directly with analog methods. The circuits used to implement artificial neural networks are a good example of this disconnect. Digital multiplication is an O(n²) operation which basically means that the number of steps required to calculate the multiplication grows as the square of the number of digits in the numbers to be multiplied. But an analog multiplier is very simple and its time complexity is O(1) – it returns a result with a single, fixed delay regardless of the number of digits in the multiplication. The limitation is in the precision of the multiplier. The only way to get more digits of precision is to measure the output of the multiplier more precisely and experience shows that the cost of measurement grows geometrically with each additional bit of precision.

Deep Learning is just the first baby step towards something much bigger. Some people are confidently predicting that we will soon build quantum neural networks but it is difficult to imagine how we are going to do that when we are not yet even harnessing the power of analog electronic neural networks or optical neural network technologies. We know that the mathematics of classical systems (such as analog computers and digital computers) is just a special case of the mathematics of quantum systems. Despite the computational speedup that quantum computers can deliver over classical computers, there is no well-defined problem that a QC can solve that no analog or digital computer cannot solve, given enough time and resources. The difference is one of degree, not of kind. The power of quantum computers is not the result of some kind of quantum pixie-dust.

From the information theoretic point-of-view, quantum systems are nothing more or less than continuous systems that admit to discrete conventions, like the non-allowed regions in a digital logic circuit. The mathematical implications of these kinds of continuous systems are subtle and easy to miss from a purely physical perspective. When a physicist has a theory of physics, he takes this to mean that the entire evolution of the physical system is, in some sense, predictable or determined by the parameters of the theory itself (even if the theory is probabilistic, as quantum theory is). But if that physical system can compute, then its long-run evolution might be undecidable. Specifically, the long-run evolution of any physical computer that can simulate a Turing machine is undecidable, otherwise, we could build such a physical machine and use it to solve the halting problem[1].

I have often encountered confusion about the underlying basis of quantum computation, a confusion that I think it is important to dispel. A quantum computer cannot exist in two, mutually exclusive states at the same time. This is true by definition because what we mean by “mutually exclusive states” is that nothing can be in both of those states, at the same time. Quantum experiment only breaks our intuitive notions of the microscopic world, a world that we envision as consisting of tiny classical marbles hurtling unimpeded through empty space and colliding with one another like microscopic billiard balls. This intuitive notion is untenable and is thoroughly contradicted by laboratory experiment.

In Part 16, I laid out the cosmological idea of the quantum monad. The quantum monad is the idea of abolishing the veil of mystery that surrounds quantum phenomena. The problem with quantum physics is not that it is quantum; the problem with quantum physics is how we talk about it. It is one thing to have wonder and awe at the intricate patterns of the natural world. It is another thing to speak of the natural world in frankly magical language, something that frequently happens in popular discussions of quantum physics. If we mean to do science, then we must dispense with non-rational and non-causal thinking.

Let’s return to the two-slit experiment. If we perform this experiment in a shallow pool of water perturbed by a single wave source, we will find that the interference patterns produced exactly correspond to the quantum experiments. Light was thought to be a wave by many early modern physicists. So, there is no surprise here. The surprise arises when we perform the experiment with a single slit – in this case, the pattern formed by the light is not like that which we would see in a shallow pool of perturbed water. Instead, we see the light striking the back-screen in a particle-like pattern. Some early physicists (including Isaac Newton) thought that light was a particle like any other. The surprise of quantum experiment is that light behaves as either a wave or a particle. But light never behaves as both at once. It either exhibits wave-like properties, or it exhibits particle-like properties.

Of course, quantum particles exhibit many other counter-intuitive properties. If we perform the two-slit experiment by emitting a single photon (or electron) at a time, we will see the same wave-pattern on the back-screen as if we had emitted the photons or electrons all at once. Quantum theory explains this phenomenon by extending the wave-equation through both space and time. Of course, classical fluids and gases do not behave this way. Water waves interfere as they do because the countless water molecules are interfering with each other, all at once.

The analogy I assert is this: quantum systems are to classical systems as analog computation is to digital computation. This analogy is a loose one. The idea is this – discrete computation is just a convention, the only computers we can actually build are analog systems. Discrete computers get their discreteness from staying well clear of boundary conditions – nothing more. Similarly, quantum theory says that all classical systems are really just quantum systems with so many copies (nearby quantum particles in similar states) that they exhibit classical behavior.

I opened this post by discussing the principles of Bitcoin in order to make the following point: it is conceivable that all the properties we associate with the material world are what they are for computational reasons. This idea puts the Simulation Hypothesis in a new light – rather than being a reflection of a generic existential angst or nihilism, perhaps the idea that the world is a simulation has a solid basis in laws that must regulate any information system. If this is the case, then we can see every particular fact of the material world in light of a symbolic computation. I will expand on this idea in more depth in an upcoming post.

Next: Part 24b, Epilogue cont'd

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[1] – As a pedantic point, any finite physical machine is fully computable because it has only finite memory resources; however, we can make a limiting argument to counter this… we mean by “physical computer that can simulate a Turing machine” any physical computer whose indefinite operation is merely a matter of adding more of a homogeneous physical resource, such as loading more and more tape into a mechanical Turing device.