In no way is this article meant to be complete. It is more in the way of musings—a crude introduction—so that we can see where to go.
Is the universe a computer? Asked differently: is the universe discrete and finite? Are all physical manifestations the result of simple interactions on a very small, well-connected, discrete grid of blocks, where each block is allowed to assume only a finite number of states?
I like to think so—an opinion I state immediately, so that you can see where my biases lay.
Says Manfred Requardt:
There exists a certain suspicion in parts of the scientific community that nature may be “discrete” on the Planck scale. The point of view held by the majority is however, at least as far as we can see, that quantum theory as we know it holds sway down to arbitrarily small scales as an all-embracing general principle, being applied to a sequence of increasingly fine grained effective field theories all the way down up to, say, string field theory. But even on that fundamental level one starts from strings moving in a continuous background. It is then argued that “discreteness” enters somehow through the backdoor via “quantisation”.
The hunch is that even string theory is a manifestation of something deeper, and discrete. Requardt learned his suspicions at the knees of Konrad Zuse (which, we’re informed, is pronounced “Tsoosay”, which does not sound like the Greek deity).
Away back in 1967 Zuse proposed, essentially, that the universe is a computer, much like a cellular automaton. Think of it like a three? four? more?-dimension set of building blocks, all packed together. The blocks themselves are now called hodons. We merely assume that the hodons are all similarly sized and shaped, incidentally.
We have all heard of John Conway’s Game of Life. Well, Zuse surmised, the universe is like that, only more so.
Conway’s two-dimensional blocks were allowed only two states, and the laws governing their behavior were limited. Allow a larger (but still finite) number of states and blocks, increase the allowable rules, and rich behavior emerges. Zuse—and many others since—showed how, for example, particle interactions might be manifestations of automata.
Zuse mapped out a progression of mechanics, from classical, which is analog, to quantum, which is a hybrid of analog and digital, to what he called “calculating space”, which is entirely digital. A similar progression exists for mathematics.
Classical mathematics assumed a continuum, from which is derived the subject of analysis, which becomes differential equations in quantum mechanics, and finally difference equations and logical operators in calculating space.
Roughly before the twentieth century, mathematics was synonymous with physics. Or, that is, advances in mathematics usually were propelled by needs in physics. But then came Cantor and others, and with them an uncountably infinite amount of baggage, and math soon became “math for math’s sake.”
The push towards finiteness is, in a sense, a way of restoring mathematics to its historical role of providing physical understanding. Various infinities and continuums are less useful because what can’t be constructed is suspect.
Familiar example: Zeno’s “paradoxes” are now solved by recourse to limits, but disappear utterly under finiteness. Rather, they are shown to contain false premises. Take the dichotomy paradox, for example. From Aristotle, “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”
If the distance between start and finish is (arbitrarily) 1, then you must first go to point 1/2, then 1/4, then 1/8, and so on ad infinitum: you’ll never arrive! This argument tacitly assumes a continuum premise. Replace it with discreteness, and the paradox vanishes.
The distance of start to finish is comprised of a finite, discrete number of steps. There may not even be a half way point! Suppose the distance is 3 units; then, with your first step, you may either end the walk (step to block 1), or go only 1/3 of the way (step to block 2).
There are, as with all ideas, objections to finiteness. But they are of an odd sort. “Where’s Pythagoras’s theorem?” ask critics. “Where’s geometry?” Well, they’re not there, for simple reason that triangles cannot be constructed discretely. Most geometrical objects are jagged, or, as we might now say, pixelated. Once more the computer analogy is useful.
You can read Requardt’s paper for his derivation of a “differentials” in a discrete world. Discrete versions are no longer the simple creatures of calculus; they do not obey, for example, the chain rule. They are harder to write down.
Polarizer, and poster child for self-esteem, Stephen Wolfram’s A New Kind of Science may also be read, but cautiously.
Again, this is just a tease. More to follow.