02 Apr

The Paradox of Natural Laws and Its Resolution

In an earlier post, I described the problem of computing in nature that scientific laws employ mathematical formulae, but it is not clear how these formulae are being calculated in nature. The reasons for this are historical and date back to Newton’s formulation of the three laws of motion. While Newton had produced a mechanics, he had not himself envisioned machines. He was only trying to describe celestial and terrestrial motion, and his laws were later used to create machines. As a result, the components of reality in Newton’s mechanics (particles and properties) are unrelated to the components of a Turing Machine that can calculate the formulae. This post discusses how the separation of motion and computation leads to a paradox in which the computer that calculates the natural laws for a single finite universe must be infinite in space or time.

The Problem of Natural Computation

When the material reality is different from the machine that computes, it appears that we have two kinds of machines—(1) the computer that computes the laws, and (2) the material object that causes physical changes as a result of that computation. This separation between the computational and the material mechanism now leads to the computational problem mentioned above: If nature is only the material mechanisms that we can measure, then where is the computer necessary to make this mechanism work? How does nature work without a universal computer that computes the equations for every single particle in the universe? And how big that computer must be?

All known laws of nature are difficult to solve for any but the simplest systems. The computational complexity for solving an N-body problem in Newton’s mechanics is O(N2) which means that as the number of objects in the universe increase as N, the space or time required to compute the solution increases as N2. Assuming that the space in the universe is proportional to the number of particles, the size of the computer that can compute the universe (in real-time) would be significantly bigger. 

Thus, if there are N particles in the universe, a computer that can compute the dynamics for all the particles in a single universe would have to span N universes with N particles each. Of course, each of these universes requires N more universes to compute its own state, in order to compute the dynamical states of the original universe. This quickly explodes into infinite universes required to compute the states of a single universe!

We can conclude that this can never be the right solution—i.e. the mechanical laws and the computational scheme that computes these laws cannot be two separate mechanisms. Rather, the natural law has to be the machine which computes itself, without the need for a computer. Every object in the universe, therefore, has to be described as a self-computing machine—i.e. that which computes itself.

The Dream of Perpetual Motion Machines

Any self-computation, however, leads to a problem of infinite recursion making the computation impossible. A self-computing machine is a perpetual motion machine. We can imagine that such a machine is comprised of two parts that compute each other. The two parts of such a computer are analogous to M. C. Escher’s sketch of two hands drawing each other. In computing theory, such a problem is called self-recursion and in layman’s terms, it is the chicken and egg problem. For the first hand to draw the second hand, the first hand must be fully functional because just a part of the hand cannot function as a hand. But, the first hand can only be fully functional if the second has completed its task of drawing out the first hand. In other words, the first hand can work only if the second hand has finished its job. But the second-hand doesn’t exist unless the first hand has finished its job. Both hands are therefore deadlocked. This deadlock means that the time to draw the hands is infinite.

The picture that M. C. Escher drew can never exist in a physical world because it entails the ability to bootstrap a fully-functional hand out of nothing so that it can draw the second hand. If this were possible, then the second hand would never be required.

The Impossibility of Mathematical Laws

And yet, people have dreamed of perpetual motion machines for a very long time. That such machines are physically impossible is not hard to see. And yet, the problem doesn’t disappear by just denying the possibility of a man-made machine because, eventually, we can see that the universe is also one such machine. Why is the universal machine working perpetually when the man-made machines cannot work perpetually? The answer to this question, as we have seen above, is that for one universe to work, an infinite number of computing universes are required.

Therefore, either we must have infinite universes (for a single universe to exist computationally) or we must have a perpetual motion machine. Without these alternatives, nothing can possibly exist. Since we exist, and neither perpetual motion nor infinite universe seems very plausible, we have a paradox.

We are caught between a rock and a hard place. If we separate the material and computation mechanisms, we require infinite space. If we combine the material and computation mechanism, we require infinite time. Neither option appears to be adequate to explain how nature actually works from a computational standpoint—regardless of the laws of nature we use.

The Notion of a Hierarchical Machine

In the following paragraphs, I will try to sketch a conceptual solution to this problem which I will call a Hierarchical Machine. The idea is simple: my machine must be designed in such a way that even a partially drawn hand can function effectively—i.e. it can draw the other hand. If a fully drawn hand is not necessary to draw the other hand, then two hands can draw each other. Such a pair of hands (which draw each other) to successively greater levels of detail will produce the fully detailed hand.

The above figure illustrates this. The machine starts with a Right-Hand Outline which then draws the Left-Hand Outline. The trick is that the Left-Hand Outline must be drawn in such a way that the Left Hand is now capable of producing greater levels of details into the Right Hand. The result of the Right Hand drawing out the Left Hand is that the Left Hand can now draw a slightly more detailed version of the Right Hand, which then produces a slightly more detailed picture of the Left Hand.

Once both hands are fully drawn, then the process has to be reversed—i.e. instead of drawing the other hand, the hands start erasing each other in exactly the opposite path in which the hands were traced. For example, the fully drawn Left Hand can now erase some details from the Right Hand which tells the Right Hand to erase some more details in the Left Hand and so forth. Until, of course, we reach the top of the hierarchy again, and the process is reversed again, and it continues indefinitely.

Hierarchical Machines and Semantics

I call this is a Hierarchical Machine because it grows through hierarchical stages, thereby building a tree of parent and child nodes. The machine also collapses in exactly the reverse order in which it was created. Therefore, at each stage, the machine can create the next stage or collapse back into the previous stage. 

The “outline” or “detail” in question are concepts. The “outline” is an abstract concept, while the “detail” is a contingent concept. An abstract concept is capable of producing a detailed concept, and the detailed concept is capable of collapsing back into the abstract concepts. This “emergence” and “collapse”, when conducted cyclically, constitutes a Self-Computing Perpetual Motion Machine. The machine constructs itself from a rough outline, and then collapses back into the outline.

This scheme resolves the paradox of infinite space and time in the current notion of natural laws where the motion of matter has no relation to the structure of the computing machine. For example, in Newton’s physics, the matter is a moving particle governed by forces. However, the computer—as a Turing Machine—is a moving device that reads and writes onto a tape. When the moving particle in physics is not the moving device that reads and writes then clearly we require two machines, and that in turn leads to an infinite time or space paradox. The above resolution to this paradox suggests a dialectical model of change in which two hands draw and erase each other.

In other words, we replace two machines—computational and mechanical—with a single machine that has two interlocked parts. Each part defines the other part. I have earlier discussed how semantics requires us to think in terms of these dialectical opposites because all concepts are described through mutual oppositions (e.g. hot vs. cold). Hence, there isn’t a unidirectional “force” pushing matter from the past to the future. Rather, the past to future change involves an interaction of mutually complementary conceptual entities, which are at once mutually-defining and mutually-opposing. The constructs of logic described in earlier posts constitute a computer architecture.

Enfolded and Unfolded Order

David Bohm used an evocative terminology to describe the quantum problem and its solution. According to his thesis, reality has two kinds of orders—Implicate and Explicate. The implicate order, according to Bohm, is a “hidden” or “deeper” reality from which an explicate order or the “visible” world is created. Bohm wanted to solve the problem of quantum theory and believed that by separating these two kinds of orders he would be able to explain how the manifest phenomena emerge from an unmanifest reality. Bohm’s key realization was that we never measure the unmanifest part, although we may be able to describe this reality mathematically or conceptually.

It is well-known that David Bohm had extensive interactions with J. Krishnamurthy, and Bohm even attributed the idea of Implicate Order partially to some discussions with him.

“We had many discussions, you see. I think partly through these discussions, although not entirely, I came to this idea of the Implicate Order. He used to greatly encourage me in that direction. I may have had the idea before in a very germ form.” [D. Bohm’s Site]

However, Bohm still thought of the Implicate Order as a physical reality governed by an algebra that is in turn based on Aristotelian logic, Set theory, and Number theory. The novelty was that he believed that reality is an algebra which gets “projected” into a geometry, quite like we can formulate a reality using an algebraic formula and then “project” it into space as a line or curve. In that sense Bohm did not sufficiently break into new ground in order to maintain continuity.

In the enfolded [or implicate] order, space and time are no longer the dominant factors determining the relationships of dependence or independence of different elements. Rather, an entirely different sort of basic connection of elements is possible, from which our ordinary notions of space and time, along with those of separately existent material particles, are abstracted as forms derived from the deeper order. These ordinary notions in fact appear in what is called the “explicate” or “unfolded” order, which is a special and distinguished form contained within the general totality of all the implicate orders. [Bohm, David (1980), Wholeness and the Implicate Order, London: Routledge]

My presentation of Enfolded and Unfolded Order substantially differs from Bohm’s in one key respect, namely, that in my presentation both enfolded and unfolded realities exist in the same space although the space is a tree rather than a box. Therefore, there can be “higher” or “deeper” locations in space, just as there can be “lower” or “visible” locations.

The higher location exists as the outline of a picture. The lower location exists as the details inside that picture. The higher location is therefore the “whole” while the lower location is the “parts”. The parts manifest after the whole has been created, and the parts can dissolve into the whole. The higher nodes of the tree, therefore, are the Implicate or the Enfolded Order, while the lower nodes of the tree are the Explicate or the Unfolded Order. These orders are like trunks and leaves of a tree.

The Relation to Mystic Philosophy

In Vedic mystic traditions, nature is described as a “mother” who produces “children” within herself. The “mother” is the “big picture” of reality, and the child is the “detailed picture” of the same reality which the “mother” creates within herself. The trigger for this creation and dissolution is the “father” of the universe who represents time. Time periodically creates the universe—which is a euphemism for drawing out the details in the picture. Time, similarly, annihilates the universe—which is a euphemism for erasing the details in the picture. The details are created when two opposites within nature increase in lock-step, thus drawing out a full tree: the world of meaning as opposites is created. The details are annihilated in nature when the opposites within nature are erased by each other.

The “mother” and “father” are, however, eternal. The mother is the three modes of nature called sattva-guna, rajo-guna, and tamo-guna, of which the last two are mutually opposed. Before the Implicate Order is manifest, all the modes exist in “balance”. When the Explicate Order is manifest, then the balance is disturbed and the universe is manifested from a “seed” into a “tree”. The interaction between the “mother” and the “father” has been described as their sexual union.

While these words are often used euphemistically, there are not just metaphors or myths. They are accurate scientific concepts that we can described in logic and mathematics which deals directly with concepts. This age-old description can also be a scientific model for the nature of reality.