Sep. 7th, 2010

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To some, science is all about models.

Exactly what is the purpose of science? It depends on whom you ask. Some might say that it aims to find the ultimate reality of things (to the best of our ability). Others might say that this is, in fact, a ludicrous idea. All that we can do is to construct the best models possible to predict and describe reality. I think this is a useful way to think of it.

You may have heard the joke about the physicist who was asked to help a dairy farmer optimise milk yield, worked on his calculations for a few weeks, and came back confident that he had found a solution: …First, let’s assume that we have a perfectly spherical cow in a vacuum…. Well, rather than poke too much fun at this fictional physicist, I actually think that (if his math was right) this is actually a very good model. The reason why I think so is that it’s immediately clear that, if the calculations based on the model are valid at all (and checking that is what empirical science is for), there’s a domain in which it is useful, but we are not tempted to extend the analogy beyond its proper domain. In a slightly more realistic example, we could calculate the features of a head-on collision between cars and consider them as simple lumps of material, and use a model that we originally devised to figure out the collision of lumps of clay. The model is useful (in that we can calculate some features of inelastic collisions), but we are not tempted to extend the car–clay analogy beyond the domain in which the model is useful: We know that cars behave very unlike lumps of clay in many ways (just as we know that cows sometimes act decidedly un-spherically, e.g. in their mode of locomotion).

So far, so good. What these models have in common is that they use examples from common experience (we are all passingly familiar with the ideas of cows, cars, spheres, and lumps of clay) to explain other phenomena that take place more or less in the realm of common experience. However, things can get very different when we move to radically different scales. Consider, for instance, when we use models (analogies) to explore and explain large-scale features of space, or small-scale features of particles on the atomic or subatomic level.¹


Consider the atom.

You may know that the atom has a nucleus and a bunch of electron. So, it looks like a lump of spheres (positively charged protons and electrically neutral neutrons) comprising the nucleus, orbited by a bunch of negatively charged electrons, equal in number to the protons. It all looks rather like a solar system with the electrons standing in for planets, orbiting their star, the nucleus. And this is a very good model that is not only evocative to us laymen, but has helped scientists figure out all sorts of things about how matter works. We can even elaborate the model to say that electrons spin about their axis, just like planets do; and they can inhabit different orbits—and change orbits—just like planets and satellites can.

Unfortunately, virtually nothing I said in the preceding paragraph is really true in any fundamental sense about what atoms are really like. It’s true that Niels Bohr’s model of the atom is rather like that, and that it is indeed helpful—but that’s not what the atom is like. It is here that the common sense familiarity of the model deceives us laymen into overextending the analogy to domains where it is no longer useful and valid. For instance, when I say that an electron can change its orbit, you may imagine something like a man-made satellite orbiting the Earth at 30,000 km, whose orbit decays gradually to 20,000 km. But an electron behaves nothing like that: It can only ever exist in certain orbits, and when an electron goes from one energy state to another, it goes there immediately. It is physically incapable of being in between. It’s as though our satellite suddenly teleported from its higher to its lower orbit, but even weirder because the satellite not only doesn’t, but in fact cannot ever inhabit an altitude of, say, 15,000 km.

Nor is it true that electrons and other subatomic particles spin in the sense that planets do. It’s certainly true that they have certain properties, called spin, and that if you use the same mathematics to work out their consequences as you would usually apply to simple spinning objects, you get good results. In this sense, a spinning sphere is a good model for a spinning subatomic particle. But I gather that there are at least some subatomic particles that have the rather curious property that they have to go through two full revolutions to get back where they started.

In fact, when I said that the atom looks like a lump of spheres, et cetera, even that isn’t fundamentally true. Everything you perceive with your sense of sight consists of objects with considerable spatial extent, and your sight is a complicated function of the fact that photons of varying energy levels bounce off them. You can’t do this with atomic nuclei, let alone electrons, because they are too small. Shoot a photon at an atom and you will change the atom. (The fact that quarks are said to have colour is pure whimsy. Consider the fact that one type of quark is “charm”, and only one of them is formally labelled “strange”.)


One of my favourite ideas—not an idea of my own, mind!—is due to John Gribbin, author of In Search of Schrödinger’s Cat: Quantum Physics and Reality. In mentioning peculiar things like the above, where electrons ‘orbit’ nuclei and electrons ‘spin’, he suggests that you may cleanse your mind of misleading connotations by reminding yourself that these properties are not really the ones you’re familiar with at all. Instead, he suggests (if I recall correctly) not that spinning electrons orbit nuclei, but rather that gyring electrons gimbal slithy toves—I’m sure I’m getting this slightly wrong, but I am very sure that I am using the right words from Lewis Carroll’s Jabberwocky.

`Twas brillig, and the slithy toves
  Did gyre and gimble in the wabe:
All mimsy were the borogoves,
  And the mome raths outgrabe.

The key observation to keep in mind is that, to paraphrase Richard Dawkins, humans have evolved to have an intuitive grasp of things of middling size moving at middling speed across the African savannah. They make sense to us on a visceral level. The large and the small—stars and galaxies and clusters, or atoms and electrons and quarks; the slow and the fast—evolution and geological change and stellar evolution, or photons and virtual particles and elemental force exchange: We did not evolve to comprehend them, because comprehension thereof had no adaptive value for Pleiocene primates (well, and the data were not available to them). We can only describe and predict them using science and mathematical models, and we can only make them seem comprehensible by constructing models and analogies that relate them to things that we do seem to understand.

But every so often we encounter natural phenomena that just don’t seem to make sense. The quantum-mechanical particles that simultaneously travel multiple paths and probabilistically interfere with themselves appear to contradict all common sense. But perhaps this is only because we attempt to think of them as little balls moving in much the way that little balls do, only faster and on smaller scales. It may be mathematically sensible to say that an electron travels infinitely many paths at once, with differential probability—which is clearly contradictory nonsense. Right? But in reality, there’s no such thing as an electron simultaneously travelling multiple paths; it’s just outgrabing rather mimsily. And the only reason why we have to model it as though our middling-scale phenomenon of probability were at issue is that we don’t have the ability to appreciate the gyring and gimbling of the slithy toves.


Now, if you enjoyed reading that, you’ll enjoy Gribbin’s In Search of Schrödinger’s Cat even more. Go forth and read it.

¹ It is a fact, sometimes held up as remarkable, that on a logarithmic scale of size from elemental particles to the universe as a whole, we’re somewhere in the middle. Once or twice, I have even heard people mention this as though it were imbued with mystical significance, in a sort of muddled anthropic principle:

On the smallest scales, elemental particles are too simple to do anything very interesting; on the largest scales, the universe is just a highly dilute space with some fluffy lumps called galaxy clusters floating around. In the middle, where we are, is where the interesting stuff happens: Large enough to combine the elemental effects into highly intricate patterns, but not so large that the patterns average out.

Such thinking is pretty fluffy and dilute—of course (the weak anthropic principle) we are necessarily on a level where “interesting” stuff happens, as reflecting brains would not occur on levels where they cannot, but I also think that this is an artefact of thinking that could apply to any phenomenon logically intermediate between other levels, so long as it’s the intermediate level that the observer is interested in.

On the smallest scales, cellular respiration just deals with chemical reactions too elemental to be very interesting; on the largest scales, muscle fibres just aggregate in big lumps that do nothing more than produce boringly linear forces of contraction. In the middle, where the individual cells are, is where the interesting stuff of semipermeable membranes, ionic drives, mitosis, protein synthesis and folding, and all that happens: Large enough to combine the elemental chemical reactions into highly intricate patterns, but not so large that the patterns average out.

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Petter Häggholm

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