Max Tegmark has a provocative paper up on the arXiv claiming that if there exists an external physical reality independent of humans, then that external physical reality is a mathematical structure. There’s too much in it to analyse in one blog post, so for now I’ll just comment on one section. Tegmark argues that if external physical reality is a mathematical structure (this statement he labels the Mathematical Universe Hypothesis — MUH), then physics no longer needs to explain initial conditions.
Just why initial conditions are undesirable is an assertion that itself deserves explanation. Tegmark explains the usual stance of physicists:
The traditional view of these matters is eloquently summarized by e.g. [3, 35] as splitting our quantitative description of the world into two domains, “laws of physics” and “initial conditions”. The former we understand and hail as the purview of physics, the latter we lack understanding for and merely take as an input to our calculations.
As Houtappel, van Dam and Wigner write in , initial conditions “are complicated and no accurate regularity has been discovered in them”. Physical laws explain the regular, so initial conditions are the things that fall outside the purview of physics. Since physicists like to think they can explain everything, this is a problem.
Taking a typical reductionist tack, Tegmark then explains how our increasing knowledge of the so-called fundamental laws of physics has led to the borderline between initial conditions and physical laws shifting at the expense of initial conditions. He claims that the MUH would complete this historical trend: “The MUH leaves no room for ‘initial conditions’, eliminating them altogether.” What he means by eliminating physical conditions, however, is somewhat confusing. He illustrates his point with the following diagram:
As examples of the shifting boundary between initial conditions and physical laws, he describes various scientific revolutions in which what had been accepted as fundamental laws were reclassified as initial conditions. Thus Ptolemy thought that the circularity of orbits was a fundamental laws, but Newton reclassified that as an initial condition — gravitational orbits can be non-circular, so circularity is not a regularity that can be accounted for by a general physical law. Thus too, he claims, the string theory landscape, together with inflation, reclassifies what we now think of as fundamental physical laws (the weak force, the electromagnetic force, etc.) as initial conditions.
I say that this is a confusing formulation of his point, because this way, it would seem that we are conceding that more aspects of our universe are initial conditions rather than physical laws. In what sense, then, can the borderline between intial conditions and physical laws be shifting “at the expense of initial conditions”? In what sense can we be eliminating physical conditions when we move to the ultimate Theory of Everything of (say) the string theory landscape? If, as originally formulated, initial conditions are the part of the universe that physics cannot explain, then wouldn’t the physical-laws-as-initial-conditions theory mean that we are essentially wringing our hands and conceding that physics cannot go any further?
Let’s look more closely at why Tegmark thinks that a landscape theory eliminates initial conditions. It does so in a somewhat linguistically paradoxical fashion, so that one suspects that the tension between eliminating initial conditions and turning all laws into intial conditions stems from that inherent paradox. A landscape theory eliminates initial conditions by postulating that a range of different physical laws persists in different parts of the universe. Each part is communicationally isolated from the others via inflation, so we would never encounter signs of another part of the universe where, say, the fine structure constant has a different value from the one we love and know to 12 significant figures. The impossibility of detecting them notwithstanding, a landscape theory would claim that variants of the physical laws we know hold in other parts of the universe. Naively, one would think that this seems to only add to the problem: instead of eliminating initial conditions, we are now saying that many possible initial conditions exist! But that’s exactly the point, and it’s where the issue is badly formulated: The problem was not that any initial conditions at all exist, but why one set of initial conditions, as opposed to other sets, happens to exist in our universe. By saying that they all exist, we don’t have to tackle that problem, since (we then claim, somewhat controversially) ascribing an existent status to all of them means we aren’t discriminating between any of the possible sets of initial conditions. Thus the landscape theory would eliminate the need to explain the existence of a particular set of initial conditions as opposed to other sets. This is what Tegmark means by the MUH “leaving no room for ‘initial conditions'”.
This ties into Tegmark’s description of physics as progressively reclassifying fundamental physical laws as initial conditions. He is essentially saying that physics is progressively removing the need for an explanation for why certain physical laws pertain as opposed to others. In this view of physics, we can think of theoretical physics as an exercise in answering questions of the form “Why does law X pertain as opposed to some other law?” As we move to more “general” physical laws, we see that law X is only a special consequence of law Y, and law Y is only a special consequence of law Z, and so on. Before a given law is described as a consequence of some other more general laws, that law cannot be explained within physics — it must be accepted, as it were, as something analogous to a physical initial condition. This is how I think Tegmark ends up saying that the MUH eliminates initial conditions, rather than that it eliminates the need to explain why one particular set of initial conditions pertains. It’s because he is treating the fundamental laws as initial conditions of a sort — as things that cannot, at this time, be explained within physics. And it is true (maybe not, actually, but I don’t want to get into that argument) that the MUH eliminates laws-acting-as-initial-conditions by showing that those laws are actually consequences of some more encompassing set of laws.
Perhaps I’m cooking up a trivial linguistic error out of nothing, but it struck me as exceedingly sloppy prose to speak of progress as simultaneously:
1) eliminating initial conditions, and
2) reclassifying more laws as initial conditions.
Not to mention first making a firm distinction between laws and initial conditions, and then going on to treat laws as though they were initial conditions to be eliminated.