Saints & Simulators 9: #ChaosAndEmergence

Ninth in an ongoing series about the places where science and religion meet. The previous episode is here.

The paired opposite to reductionism is called emergentism, and in recent years it has begun to gain an increasing number of advocates. In summary, it means that the whole is more than the sum of its parts. Unexpected behaviors and properties can emerge, even from simple well-understood parts, at high enough levels of organization.

One major impetus behind the popularization of the concept of emergence was the discovery of a new branch of mathematics called chaos theory, and an associated and radically new approach to mathematical shapes called fractal geometry. Entire volumes have been written introducing the core concepts of chaos theory, but it all centers around emergent mathematical phenomena; complex and unexpected patterns that arise spontaneously from simple equations.

Several sources converged to create chaos theory. One was the work of an eccentric group of early-twentieth-century mathematicians and logicians—Georg Cantor, Gaston Julia, Pierre Fatou, and Waclaw Sierpinski being among the best known—who created a bizarre group of mathematical shapes by repeatedly applying the same simple rules over and over again. For instance, Sierpinski created a fractal called the “gasket” by taking a triangle, removing an upside down triangle from the middle, thus leaving three smaller triangles, and then repeating the same operation on each of the new, smaller triangles. He then calculated that if the same process were repeated infinitely, a crazy mathematical monstrosity would emerge, something stretching over the entire area of the original shape, but with zero calculable area of its own, a piece of nothing wrapped in a something, with a surprising elegance and inherent beauty.

Fractals, so-called because they are neither truly two-dimensional like a sheet of paper, nor one dimensional like a line, but somehow possess some fractional dimensionality somewhere in between, were first thought to be nothing more than the feverish ravings of some intellectuals’ overheated brains. Soon, however, it was discovered that fractals are all around us. The meandering of a river is a fractal shape, as is the branching structure of a tree and most other plants. Ferns, among the oldest of all terrestrial life forms, are nearly perfect fractals. The pattern of veins in the body is a fractal, and so are the lungs. The Sierpinski gasket has been found hidden inside Pascal’s Triangle (a venerable mathematical tool central to the science of statistics) and more poetically, has even been found naturally inscribed on the shell of a mollusk with the lyrical name false melon volute. When it comes to the natural world, it turns out that the magic of simple rules leading to complex patterns is ubiquitous, perhaps because of the efficiencies it grants organisms in the game of evolution. (Nor did it turn out that these shapes have been entirely unknown throughout human history, given, as theorist Ron Eglash has demonstrated, that fractal patterns are a core part of the favored aesthetic and principles of design used in traditional societies and ancient civilizations across the continent of Africa.)

Another source for chaos theory was the discovery that simple equations could yield fantastically complicated results, if only you fed the results of those equations back into themselves. One such set of equations was intended by mathematician Edward Lorenz, working on an early computer in 1961, to model the ever-changing patterns of the weather. The only problem was that his simple, easy-to-understand set of equations worked only too well once he began feeding their results back into them. They began outputting data that swung up and down in unpredictable ways, just like real weather data, such as the temperature. Even odder, when the results of these equations were plotted on a graph, they produced beautiful, unexpected, fractal shapes. By 1971, such shapes had become known, in a term coined by physicist David Ruelle and mathematician Floris Takens, as “strange attractors.”

Perhaps the best known of all fractals is a divine beast known as the Mandelbrot Set. An extension of the work of Julia and Fatou, it was mathematician Benoit Mandelbrot’s attempt to find a single shape that contained all possible Julia sets within it. He accomplished this with the help of a single equation, expressing a process so simple that even a non-mathematician can understand it. Take a complex number (which is basically a pair of numbers, indicating a point on a graph, just like in high-school geometry). Square that number. Add the original number. Square it again. Add the original number. And so forth, and so on, forever. Does the result get infinitely big? If so, color the original point white. Does it stay small forever? If so, color it black.

Oddly, this simple process, fully describable in a single paragraph, plots out a very strange and complex object if you graph it, an object sometimes called the “most complex in all mathematics.” It is not a circle, a square, a triangle, or any classical shape. It is a fractal, but not a Sierpinski gasket, nor a fern shape, nor any other easily described fractal shape. Instead it is a completely unexpected, irregular blob, sometimes described as a “bug on fire.”

The real magic begins, however, when you magnify this shape, because upon magnification, all sorts of amazing things are hidden inside: little copies of itself, swirling curves like seahorse tails, jagged tendrils, and what looks, for all the world, like baroque ornamentation. Pieces of it are so beautiful that one might well imagine them deliberately created by a human artist, and yet they are composed only of pure mathematics. In addition, they have a property no human artist could duplicate. No matter how much you magnify them, there is always new detail to see. The shape goes on to infinitely tiny levels of detail, often mirroring the larger levels, but never exactly duplicating them, often unexpected, always new, without end.

The mere contemplation of the existence of this shape might count as a mystical experience for some, but what, the reader might well ask, does this have to do with artificial intelligence? The relevant claim is this: The top-down, reductionist, programming-centric approach to creating artificial intelligence is all wrong, because intelligence is an emergent phenomenon. It may come from simpler parts, but it cannot be understood by reference to them, and it cannot be built to plan following a set of instructions, as if it were a cabinet from IKEA.

This is not to say that emergentist theorists disbelieve in the possibility of artificial intelligence, but they they disagree about the way in which it should be built. It is well-established that the neural net of the brain is not laid out with the Boolean (true-or-false) logic of the circuitry of a computer. Rather, it is a diffuse network of fuzzy connections that can be incrementally strengthened or weakened. Similarly, things such as thoughts and memories are not contained at discrete storage locations, such as on a computer drive, but rather spread out across the entire net in counterintuitive ways.

Some of the ways emergentists have proposed creating artificial intelligence include building or simulating artificial neural nets, or using quantum computers, which take advantage of wave-particle duality and superimposition to perform fuzzy logic. Others reject the entire idea of shortcuts to emulating human intelligence, in favor of simply duplicating the entire fine structure of the human brain in virtual form—something not possible today, but perhaps in the future.

As starkly different as the reductionist and emergentist approaches to artificial intelligence are, however, they both take one thing for granted: that we live in a purely physical universe, and therefore that minds can be duplicated or simulated simply by reproducing their physical structure and characteristics. The rejection of that assumption, through the endorsement of a dualist, body-and-soul universe, is therefore the one sure way to decisively rule out the possibility of the world being a simulation. Or is it?

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