Wednesday, 16 May 2012

A part is more complex than the whole

Here's two apparently unrelated hypotheses for you:

1. The universe contains infinite space and an infinite quantity of stuff.

2. The many worlds interpretation of quantum mechanics is true.

Both of these appear at first glance to be astonishingly wasteful, positing infinities upon infinities of things we can never observe and which can never affect us. As such, they appear to be more complex than ideas which assume that space is finite or that there is only one quantum world.

It would seem that Occam's razor weighs against them.

In this post, I will argue that in fact the reverse is true.

Imagine I want to describe to you all the pictures that could ever exist. It would seem that would take a long time. Indeed it would, so let's constrain ourselves somewhat so that at least we are dealing with a finite quantity.

Let's talk about all images 256 pixels high by 256 pixels across, with each pixel having an 8-bit colour describing one of 256 possible colours. That's a staggering number of possible images (256 to the power of 256 squared, which is a number 157,827 digits long), but at least it's finite.

But guess what? I've just described them to you. You now have all the information you need to go and generate all those images I'm talking about, one by one. It may take you longer than the lifetime of the universe to actually inspect each one, but all the information is there in the preceeding paragraph.

If, on the other hand, I want to describe a particular picture to you, then I have to tell you the colour of 256 squared pixels. It turns out this would take me 8 * 256 * 256 bits of information. This means I'd have to send you 524,288 ones and zeros (let's leave image compression out of it!). This, paradoxically, is more information than I would need to send you to describe the complete set of possible images, which is simply the paragraph before last.

The argument could also be made with the real number co-ordinates of a vector drawing. Picture the familiar Cartesian plane that we know from high school. The x-axis is horizontal, the y-axis is vertical. We could draw a picture on this plane by specifying a number of arcs, curves and line segments. This is essentially how some vector graphics programs such as Adobe Illustrator work. Every time we add to the picture, we're adding information to the file representing the picture.

And yet, in another sense, all we're doing is setting out a more and more specific subset of the points that already exist on the plane that's already there. The Cartesian plane, before we do anything to it, already contains an infinite number of points. A drawing on the Cartesian plane is just a representation of a subset of the points, and yet contains more information. Again, a part is more complex than the whole.

The phenomenon can also be described intuitively without mathematics. Consider a block of marble. That "contains" an innumerable multitude of possible sculptures within it. A sculptor adds information and thus complexity by removing chips of stone. If we imagine the block of marble is infinite in size, we have to specify even less information because we don't have to give its height, width or depth.

In the same way, an infinite universe is simpler than a finite universe, because to describe a particular finite universe you'd have to specify the initial conditions of every proton, neutron, electron, photon, neutrino, muon... etc. You get the picture.

An infinite universe, on the other hand, need only specify the laws, and this is because no matter what finite configuration of particles you're looking for, that configuration will be found somewhere within the space (as long as the distribution of particles is effectively random). It's just like the digital image - no matter what configuration of pixels you're looking for, it will be found somewhere within the space of possible pictures.

The many worlds interpretation of quantum mechanics is similar. It's essentially an explanation for how certain subatomic events really do seem to be genuinely random. I'm not going to get into it in depth, but just apply what I'm saying to it. If this universe is the only one, then every time a quantum observation is made, it would seem that new information is either being created from nothing, or that we are for the first time revealing some information that was there since the dawn of time but which has previously not been discovered.

If this universe is the only "quantum world", then in order to provide a complete description of reality we need to describe not only the initial state of the particles and the physical laws which govern them, but also the results of every single quantum interaction since the beginning of time. Now this is what seems astonishingly wasteful to me!

The alternative Many Worlds Interpretation is much simpler. There are an infinite number of universes. Anything that can happen does happen (with a frequency proportional to what we perceive as probability). We just happen to live in one of those quantum worlds. In order to give a complete description of reality, you just explain this as well as specifying the physical laws. You quite possibly don't need to give the initial state because the initial state is in all probability the result of quantum fluctuations in the first place. So all possible initial states exist in some quantum world or other.

This seems much simpler to me -- in fact I find it beautifully elegant -- and so I believe it is probably true. It only seems wasteful to us because we have the unfortunate egotistical tendency to believe that all of reality exists for our benefit, so anything that we cannot see must be wasted.

We still cling to this attitude even though we now understand that we are not at the centre of the universe but standing on just one planet of several orbiting a star, which is one of billions in a galaxy, which is just one of billions of other galaxies which we can observe.

All of this for us? Parsimony is, in some cases at least, on the side of the infinite.

Note: This is my take on an argument given by Max Tegmark.


  1. My two thoughts about an infinite universe:

    First, there is no evidence that it is infinite, according to all current observations. Time towards the past is certainly not without limits, there was the Big Bang after all (and if there was pre-BB stuff, this must have been of a very different nature). Towards the future, the universe might go on forever, but the conditions change drastically. If the expansion keeps on accelerating, configurations that you see now, with young stars, clusters of galaxies etc., will not exist anymore a hundred billion years from now.

    Spatially, it is extremely unlikely that the universe is infinite. Inflation tells us that the universe is flat, which is what we observe. This would indeed imply an infinite spatial extend. But flatness is only necessary within a certain range, to the high degree that inflationary models tell us. That means that before inflation, the universe could have had any geometry within that range, and flatness (= infiniteness) is only one single, unlikely, possibility out of that range.

    Second, cosmologists describe the universe in a statistical sense. Nobody can predict the exact formation of the Earth, the solar system, or the Milky Way. We can only predict the distribution of planets, stars, and galaxies. The description of such models need a certain number of parameters, e.g. initial conditions, the matter density, the curvature, etc. These numbers can describe both finite and infinite universes, none is simpler than the other.

    1. Hi energie_sombre,

      There is neither evidence that it is finite or infinite, but it certainly seems to be flat to within our ability to measure it, which as you say might imply an infinite size if it is flat (or open). It is interesting that a flat universe would have precisely zero energy, which might hint at a reason for how it could be precisely flat.

      But when you discount the possibility that it is perfectly flat as being too improbable, you're forgetting the possibility that it is open. Even if we rule out a flat universe entirely, there's still a 50% chance it's open, which would mean infinite.

      >The description of such models need a certain number of parameters, e.g. initial conditions, the matter density, the curvature, etc.<

      I see you've missed the point of my post. An infinite universe is simpler because you don't need to specify initial conditions, as all possible initial conditions will be found in that infinity.

  2. You're right of course about the open universe being infinite.

    But also in this case, you need to know the initial conditions. You still want to know what e.g. the distribution of matter is, or how rare certain events are.

    1. Yes, you need to define all the parameters that define the universe, including matter density, rarity of events, physical laws etc.

      What you don't need to specify is the specific configuration of particles that exists at the moments just after the big bang, because in an infinite universe all such configurations will exist somewhere.

      Conversely, a finite universe does need this specific configuration and this is an absolutely enormous amount of information, far more than anything else that defines the universe.

      The infinite universe is simpler since you don't need this information to define it.

    2. I don't know what you mean by 'specific configuration of particles'. No cosmological model does provide this, since those processes are of statistical nature (in particular in the early universe, which is dominated by quantum fluctuations if you accept inflationary scenarios), so you only describe the statistical properties of random fields.

      You can quantify the amount of information by the entropy density, and with that there is basically no difference between finite and infinite models.

    3. Excellent point. Taking quantum mechanics into consideration, it is probably not fair to say that the universe is simpler if it is infinite, but only if you allow the many worlds interpretation.

      If you do not assume the MWI, then this initial configuration of particles is replaced by the information which would encode the outcome of every random quantum event leading up to today - a massive amount of data that's not required in an infinite universe because all possibilities are realised somewhere.

      On the other hand, if you do assume the MWI, the universe is not simpler because it is infinite in spatial extent, but because it has infinite instances corresponding to all the different ways quantum events could have turned out.

      Again, infinity reduces complexity.