Science of Discworld III

background tells us so. If space were finite, then traces of that finitude would show up in the statistical properties of the cosmicbackground and the various frequencies of radiation that make it up.
    This is a curious argument. Only a year or so ago, some mathematicians used certain statistical features of the cosmic microwave background to deduce that not only is the universe finite, but that it is shaped a bit like a football. 8 There is a paucity of very long-wavelength radiation, and the best reason for not finding it is that the universe is too small to accommodate such wavelengths. Just as a guitar string a metre long cannot support a vibration with a wavelength of 100 metres – there isn’t room to fit the wave into the available space.
    The main other item of evidence is of a very different nature – not an observation as such, but an observation about how we interpret observations. Cosmologists who analyse the microwave background to work out the shape and size of the universe habitually report their findings in the form ‘there is a probability of one in a thousand that such and such a shape and size could be consistent with the data’. Meaning that with 99.9 per cent probability we rule out that size and shape. Tegmark tells us that one way to interpret this is that at most one Hubble volume in a thousand, of that size and shape, would exhibit the observed data. ‘The lesson is that the multiverse theory can be tested and falsified even when we cannot see the other universes. The key is to predict what the ensemble of parallel universes is and to specify a probability distribution over that ensemble.’
    This is a remarkable argument. Fatally, it confuses actual Hubble volumes with potential ones. For example, if the size and shape under consideration is ‘a football about 27plex metres across’ – a fair guess for our own Hubble volume – then the ‘one in a thousand’ probability is a calculation based on a potential array of one thousand footballs of that size. These are not part of a single infinite universe: they are distinct conceptual ‘points’ in a phase space of bigfootballs. If you lived in such a football and made such observations, then you’d expect to get the observed data on about one occasion in a thousand.
    There is nothing in this statement that compels us to infer the actual existence of those thousand footballs – let alone to embed the lot in a single, bigger space, which is what we are being asked to do. in effect, Tegmark is asking us to accept a general principle: that whenever you have a phase space (statisticians would say a sample space) with a well-defined probability distribution, then everything in that phase space must be real.
    This is plain wrong.
    A simple example shows why. Suppose that you toss a coin a hundred times. You get a series of tosses something like HHTTTHH … TTHH. The phase space of all possible such tosses contains precisely 2 100 such sequences. Assuming the coin is fair, there is a sensible way to assign a probability to each such sequence – namely the chance of getting it is one in 2 100 . And you can test that ‘distribution’ of probabilities in various indirect ways. For instance, you can carry out a million experiments, each yielding a series of 100 tosses, and count what proportion has 50 heads and 50 tails, or 49 heads and 51 tails, whatever. Such an experiment is entirely feasible.
    If Tegmark’s principle is right, it now tells us that the entire phase space of coin-tossing sequences really does exist . Not as a mathematical concept, but as physical reality.
    However, coins do not toss themselves. Someone has to toss them.
    If you could toss 100 coins every second, it would take about 24plex years to generate 2 100 experiments. That is roughly 100 trillion times the age of the universe. Coins have been in existence for only a few thousand years. The phase space of all sequences of 100 coin tosses is not real. It exists only as potential.
    Since Tegmark’s principle doesn’t work for coins, it makes no sense to suppose that it works for universes.
    The evidence advanced in favour of level 4 parallel worlds is eventhinner. It amounts to a mystical appeal to Eugene Wigner’s famous remark about ‘the unusual effectiveness of mathematics’ as a description of physical reality. In effect, Tegmark tells us that if we can imagine something, then it has to exist.
    We can imagine a purple hippopotamus riding a bicycle along