Finitism, Physics, Cellular Automata: The Universe as Logic

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.

12 Comments

  1. What does finiteness have to do with the universe being a computer? I think it is a computer, but there are analog computers, as well as discrete, and in my experience the two will arrive at the same answer–with due caution about assuring the two means are, in fact, solving the same problem. The unique results of quantum mechanics, on the other hand, arise not from discreteness, but from the failure of operators to commute. Am I wrong?

    Not even cautiously do I read Wolfram again. I got about 250 pages into A New Kind of Science and gave up. He seemed to be suggesting that we do computations, and if we recognize some physics in the result, then explore that path more thoroughly. It’s like trying to organize serendipity.

    At any size scale where I have an interest, the Planck length is so small it looks like a continuum. Someone would need to convince me of the utility of a new way of doing mathematics to get me on-track. Zeno’s paradoxes will not do.

  2. Bollocks. You don’t need discreetness to debunk Zeno.

    If you regard both time and space as continuum, and if you think that there are infinitesimals everywhere, then the paradox vanishes.

    This is the case if you think about numbers. For instance, if discreetness would be required to solve Zeno’s paradox, how on earth is possible to exist any number beyond 1? Like 2? Put Zeno’s paradox on the table about numbers, and you’ll be able to prove that you’ll never reach 2 if you start counting every single number from 1 to 2.

    Or, IOW, if discreetness is the problem, then how on earth is it possible to solve Zeno’s paradox using continuous maths’ tools?

    So you see, it doesn’t follow that for the Zeno’s problem you require discreetness.

    And then there’s the Bell Inequality experiment: there’s nothing beneath quantum randomness. There’s no discreetness. Only randomness. How’s that for a computer, huh?

  3. Zeno’s paradox isn’t much of a paradox. There are an infinite number of points in an ineterval. So what! It takes no time to cross a point.

  4. How could one possibly imagine refuting quintessential arguments such as “Bollocks” and “So what!”.
    For those of us who haven’t yet reached that level of logical sophistication, I suggest reading Joseph Mazur’s “Zeno’s paradox Unravelng the ancient mystery behind the science of space and time”.

  5. Is the universe a computer? Asked differently: is the universe discrete and finite?
    .
    Well such a big question suffers from a lack of definition what Universe is .
    If I take the usual definition – universe is space , time and energy then I have an adequate description of this being via General Relativity .
    As the General Relativity links space-time (metrics) to energy , I have everything and the answer to the question is NO .
    It is NO because of mundane considerations about Einstein equations which are partial differential equations in a space-time continuum .
    It works pretty well and explains about everything that concerns the Universe as we can observe it .
    So it could be the end of story .
    However this representation runs in a problem at origin (of time) .
    If one admits (what most people do) that the age of Universe is finite then the space-time was infinitely small and infinitely hot at origin .
    But being so small , clearly it was governed by this other (also continuous) theory which is Quantum Mechanics .
    String theory which we can’t resume in a post is a theory that takes over from QM and GR in those extreme conditions .
    .
    Let’s be clear about the word “quantum” . Neither space nor time is “quantified” .
    Space-time plays in QM exactly the same role as in GR – it is a continuum .
    The characteristics of QM is , like Kevin has rightly observed , that the operators don’t commute .
    It is trivial to show the Heisenberg’s Uncertainty principle from this fact .
    You add to this another postulate and it is INDEED a postulate and not a theorem that the square of the wave function is a probability of presence and you have the whole QM .
    The “quantisation” or “discretisation” of some variables is then just a result in some cases .
    I said “some” because by no means is everything quantified in QM , continuous spectra can exist too .
    .
    So when one takes a lucid look at what we really have , then we see that “discreteness” is just a very partial , particular property of some specific cases and doesn’t deserve vast extrapolations to become some fundamental or even governing property of everything there is aka Universe .
    As I have already written somewhere here , the wavelength of a photon is damn continuous all the way and there is no reason that it should not be a small or as large as you wish .
    So even in QM the answer is clearly NO .

  6. P.S
    And I must add this .
    When I hear “discreteness” (no I don’t grab my gun :)) I immediately think Lorentz invariance .
    Discrete space-time horribly violates Lorentz invariance . Always .
    And that kills any physical theory without any possibility of redemption .
    Abandon hope all ye enter here .

  7. Tom,

    Agreed. Lorentz invariance is no more under discreetness. The question becomes: do we abandon discreetness—accept an infinitely (which kind of infinity?) graded continuum—and keep Lorentz invariance, or abandon Lorentz invariance except as an approximate phenomenon and accept discreetness with all of its concomitant niceties? It is beyond my competence to say which is correct, though I believe discreetness worthy of investigation.

  8. William

    The one point I can’t see is WHY would one want abandon Lorentz invariance or the obvious evidence that a photon’s wavelength can be as small or as big as you want and run into LETHAL difficulties with no compelling reason .
    For me it looks like this movie where a small scrawny cowboy in a bar keeps annoying a 2m , 150 kg big guy minding his business and drinking a beer . Untill he says “You are looking for trouble stranger and you just found it .” and then proceeds to teach him the wrong of his ways by dispersing him all over the walls .
    .
    A century shockfull of experimental and theoretical evidence that Mother Nature put at our disposal is telling us in the frame of a very consistent picture that the Lorentz invariance is exactly respected in all cases .
    So while I could understand that due to the finiteness and discreteness of our brains , we could have some anthropocentrical emotional preference for finite discrete things , Mother Nature clearly doesn’t care about our preferences .
    In other words what is the worthiness of investigating a concept that is extremely unlikely to be true and for which there is not a trace of experimental evidence ?
    This could certainly be a kind of artistic endeavour (e.g “l’art pour l’art”) , perhaps even amusing or intellectually challenging but it would have apparently little to no relevance to the physical Universe we live in .

  9. Contrary to what is often stated, a fundamental spacetime discreteness need not contradict Lorentz invariance.
    Dowker F., Joe Henson J. & Rafael D. Sorkin: Quantum Gravity Phenomenology, Lorentz Invariance and Discreteness

    .
    This is an old story and this paper is wrong . However neither am I enough versed in the string theory to follow all arguments nor , I think , is this blog a place to enter in technicalities .
    But this class of “theories” (deformed or double special relativity) has for consequence that the speed of light is energy dependent .
    Since the above preprint was put on ArXiv , all theories based on the variability of the speed of light have been effectively killed by observation .
    I would also add that Perimeter Institute is famous for employing some of the most prominents crackpots (f.ex L.Smolin) so most papers from there are generally not read by the HEP community . This , of course , is no rational argument 🙂

    Also see : http://motls.blogspot.com/2006/12/lorentz-violation-and-deformed-special.html
    I recommend to read this article as t is the clearest and most understandable analysis of this question that I know in Internet .

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