The Game of Life

The regularities in the motion of astronomical bodies such as the sun, the moon, and the planets suggested that they were governed by fixed laws rather than being subject to the arbitrary whims and caprices of gods and demons.

At first, the existence of such laws became apparent only in astronomy (or astrology, which was regarded as much the same).

The behavior of things on earth is so complicated and subject to so many influences that early civilizations were unable to discern any clear patterns or laws governing these phenomena.

Gradually, however, new laws were discovered in areas other than astronomy, and this led to the idea of scientific determinism: There must be a complete set of laws that, given the state of the universe at a specific time, would specify how the universe would develop from that time forward.

These laws should hold everywhere and at all times; otherwise they wouldn’t be laws. There could be no exceptions or miracles. Gods or demons couldn’t intervene in the running of the universe.

At the time that scientific determinism was first proposed, Newton’s laws of motion and gravity were the only laws known. We have described how these laws were extended by Einstein in his general theory of relativity, and how other laws were discovered to govern other aspects of the universe.

The laws of nature tell us how the universe behaves, but they don’t answer the why? questions that we posed at the start of this book: Why is there something rather than nothing?

Why do we exist?

Why this particular set of laws and not some other?

Some would claim the answer to these questions is that there is a God who chose to create the universe that way. It is reasonable to ask who or what created the universe, but if the answer is God, then the question has merely been deflected to that of who created God. In this view it is accepted that some entity exists that needs no creator, and that entity is called God.

This is known as the first-cause argument for the existence of God. We claim, however, that it is possible to answer these questions purely within the realm of science, and without invoking any divine beings.

According to the idea of model-dependent realism introduced in Chapter 3, our brains interpret the input from our sensory organs by making a model of the outside world. We form mental concepts of our home, trees, other people, the electricity that flows from wall sockets, atoms, molecules, and other universes.

These mental concepts are the only reality we can know. There is no modelindependent test of reality. It follows that a well-constructed model creates a reality of its own.

An example that can help us think about issues of reality and creation is the Game of Life, invented in 1970 by a young mathematician at Cambridge named John Conway.

The word “game” in the Game of Life is a misleading term.

There are no winners and losers. In fact, there are no players.

The Game of Life is not really a game but a set of laws that govern a 2D universe.

It is a deterministic universe. Once you set up a starting configuration, or initial condition, the laws determine what happens in the future.

The world Conway envisioned is a square array, like a chessboard, but extending infinitely in all directions.

Each square can be in one of 2 states:

1. Alive (shown in green)
2. Dead (shown in black)

Each square has eight neighbors: the up, down, left, and right neighbors and four diagonal neighbors.

Time in this world is not continuous but moves forward in discrete steps. Given any arrangement of dead and live squares, the number of live neighbors determine what happens next according to the following laws:

1. A live square with two or three live neighbors survives (survival).
2. A dead square with exactly three live neighbors becomes a live cell (birth).
3. In all other cases a cell dies or remains dead. In the case that a live square has zero or one neighbor, it is said to die of loneliness; if it has more than three neighbors, it is said to die of overcrowding.

That’s all there is to it: Given any initial condition, these laws generate generation after generation.

An isolated living square or two adjacent live squares die in the next generation because they don’t have enough neighbors. Three live squares along a diagonal live a bit longer.

After the first time step the end squares die, leaving just the middle square, which dies in the following generation.

Any diagonal line of squares “evaporates” in just this manner. But if three live squares are placed horizontally in a row, again the center has two neighbors and survives while the two end squares die, but in this case the cells just above and below the center cell experience a birth.

The row therefore turns into a column. Similarly, the next generation the column back turns into a row, and so forth. Such oscillating configurations are called blinkers.

If three live squares are placed in the shape of an L, a new behavior occurs. In the next generation the square cradled by the L will give birth, leading to a 2 × 2 block. The block belongs to a pattern type called the still life because it will pass from generation to generation unaltered.

Many types of patterns exist that morph in the early generations but soon turn into a still life, or die, or return to their original form and then repeat the process.

There are also patterns called gliders, which morph into other shapes and, after a few generations, return to their original form, but in a position one square down along the diagonal.

If you watch these develop over time, they appear to crawl along the array. When these gliders collide, curious behaviors can occur, depending on each glider’s shape at the moment of collision.

What makes this universe interesting is that although the fundamental “physics” of this universe is simple, the “chemistry” can be complicated. That is, composite objects exist on different scales.

At the smallest scale, the fundamental physics tells us that there are just live and dead squares. On a larger scale, there are gliders, blinkers, and still-life blocks.

At a still larger scale there are even more complex objects, such as glider guns: stationary patterns that periodically give birth to new gliders that leave the nest and stream down the diagonal.

If you observed the Game of Life universe for a while on any particular scale, you could deduce laws governing the objects on that scale.

For example, on the scale of objects just a few squares across you might have laws such as “Blocks never move,” “Gliders move diagonally,” and various laws for what happens when objects collide. You could create an entire physics on any level of composite objects.

The laws would entail entities and concepts that have no place among the original laws.

For example, there are no concepts such as “collide” or “move” in the original laws.

Those describe merely the life and death of individual stationary squares. As in our universe, in the Game of Life your reality depends on the model you employ.

Conway and his students created this world because they wanted to know if a universe with fundamental rules as simple as the ones they defined could contain objects complex enough to replicate.

In the Game of Life world, do composite objects exist that, after merely following the laws of that world for some generations, will spawn others of their kind? Not only were Conway and his students able to demonstrate that this is possible, but they even showed that such an object would be, in a sense, intelligent!

What do we mean by that? To be precise, they showed that the huge conglomerations of squares that self-replicate are “universal Turing machines.”

For our purposes that means that for any calculation a computer in our physical world can in principle carry out, if the machine were fed the appropriate input—that is, supplied the appropriate Game of Life world environment—then some generations later the machine would be in a state from which an output could be read that would correspond to the result of that computer calculation.

To get a taste for how that works, consider what happens when gliders are shot at a simple 2 × 2 block of live squares. If the gliders approach in just the right way, the block, which had been stationary, will move toward or away from the source of the gliders.

In this way, the block can simulate a computer memory. In fact, all the basic functions of a modern computer, such as AND and OR gates, can also be created from gliders. In this manner, just as electrical signals are employed in a physical computer, streams of gliders can be employed to send and process information.

In the Game of Life, as in our world, self-reproducing patterns are complex objects.

One estimate, based on the earlier work of mathematician John von Neumann, places the minimum size of a selfreplicating pattern in the Game of Life at 10 trillion squares—roughly the number of molecules in a single human cell.

One can define living beings as complex systems of limited size that are stable and that reproduce themselves.

The objects described above satisfy the reproduction condition but are probably not stable: A small disturbance from outside would probably wreck the delicate mechanism.

However, it is easy to imagine that slightly more complicated laws would allow complex systems with all the attributes of life. Imagine a entity of that type, an object in a Conway-type world.

Such an object would respond to environmental stimuli, and hence appear to make decisions. Would such life be aware of itself? Would it be self-conscious? This is a question on which opinion is sharply divided.

Some people claim that self-awareness is something unique to humans. It gives them free will, the ability to choose between different courses of action.

How can one tell if a being has free will? If one encounters an alien, how can one tell if it is just a robot or it has a mind of its own? The behavior of a robot would be completely determined, unlike that of a being with free will. Thus one could in principle detect a robot as a being whose actions can be predicted.

Chapter 2 said this may be impossibly difficult if the being is large and complex.

We cannot even solve exactly the equations for three or more particles interacting with each other.

Since an alien the size of a human would contain about a thousand trillion trillion particles even if the alien were a robot, it would be impossible to solve the equations and predict what it would do. We would therefore have to say that any complex being has free will—not as a fundamental feature, but as an effective theory, an admission of our inability to do the calculations that would enable us to predict its actions.

The example of Conway’s Game of Life shows that even a very simple set of laws can produce complex features similar to those of intelligent life. There must be many sets of laws with this property.

What picks out the fundamental laws (as opposed to the apparent laws) that govern our universe? As in Conway’s universe, the laws of our universe determine the evolution of the system, given the state at any one time.

In Conway’s world we are the creators—we choose the initial state of the universe by specifying objects and their positions at the start of the game.

In a physical universe, the counterparts of objects such as gliders in the Game of Life are isolated bodies of matter. Any set of laws that describes a continuous world such as our own will have a concept of energy, which is a conserved quantity, meaning it doesn’t change in time.

The energy of empty space will be a constant, independent of both time and position. One can subtract out this constant vacuum energy by measuring the energy of any volume of space relative to that of the same volume of empty space, so we may as well call the constant zero. One requirement any law of nature must satisfy is that it dictates that the energy of an isolated body surrounded by empty space is positive, which means that one has to do work to assemble the body.

That’s because if the energy of an isolated body were negative, it could be created in a state of motion so that its negative energy was exactly balanced by the positive energy due to its motion. If that were true, there would be no reason that bodies could not appear anywhere and everywhere.

Empty space would therefore be unstable. But if it costs energy to create an isolated body, such instability cannot happen, because, as we’ve said, the energy of the universe must remain constant.

That is what it takes to make the universe locally stable—to make it so that things don’t just appear everywhere from nothing.

If the total energy of the universe must always remain zero, and it costs energy to create a body, how can a whole universe be created from nothing? That is why there must be a law like gravity.

Because gravity is attractive, gravitational energy is negative: One has to do work to separate a gravitationally bound system, such as the earth and moon.

This negative energy can balance the positive energy needed to create matter, but it’s not quite that simple. The negative gravitational energy of the earth, for example, is less than a billionth of the positive energy of the matter particles the earth is made of.

A body such as a star will have more negative gravitational energy, and the smaller it is (the closer the different parts of it are to each other), the greater this negative gravitational energy will be.

But before it can become greater than the positive energy of the matter, the star will collapse to a black hole, and black holes have positive energy. That’s why empty space is stable. Bodies such as stars or black holes cannot just appear out of nothing. But a whole universe can.

Because gravity shapes space and time, it allows space-time to be locally stable but globally unstable.

On the scale of the entire universe, the positive energy of the matter can be balanced by the negative gravitational energy, and so there is no restriction on the creation of whole universes.

Because there is a law like gravity, the universe can and will create itself from nothing in the manner described in Chapter 6.

Spontaneous creation is the reason there is something rather than nothing, why the universe exists, why we exist. It is not necessary to invoke God to light the blue touch paper and set the universe going.

Why are the fundamental laws as we have described them?

The ultimate theory must be consistent and must predict finite results for quantities that we can measure. We’ve seen that there must be a law like gravity, and we saw in Chapter 5 that for a theory of gravity to predict finite quantities, the theory must have what is called supersymmetry between the forces of nature and the matter on which they act. M-theory is the most general supersymmetric theory of gravity.

This is why M-theory is the only candidate for a complete theory of the universe.

If it is finite—and this has yet to be proved—it will be a model of a universe that creates itself. We must be part of this universe, because there is no other consistent model.

M-theory is the unified theory Einstein was hoping to find.

The fact that we human beings—who are ourselves mere collections of fundamental particles of nature—have been able to come this close to an understanding of the laws governing us and our universe is a great triumph. But perhaps the true miracle is that abstract considerations of logic lead to a unique theory that predicts and describes a vast universe full of the amazing variety that we see.

If the theory is confirmed by observation, it will be the successful conclusion of a search going back more than 3,000 years. We will have found the grand design.