The blogosphere, even the tiny little bit of it that I know anything about, has a habit of summoning up strange coincidences between things so, following EM Forster’s maxim “only connect”, I thought I’d spend a lazy saturday lunchtime trying to draw a couple of them together.

A few days ago I posted what was intended to be a fun little item about the wave-particle duality in quantum mechanics. Basically, what I was trying to say is that there’s no real problem about thinking of an electron as behaving sometimes like a wave and sometimes like a particle because, in reality (whatever that is), it is neither. “Particle” and “wave” are useful abstractions but they are not in an exact one-to-one correspondence with natural phenomena.

Before going on I should point out that the vast majority of physicists are well away of the distinction between, say, the “theoretical” electron and whatever the “real thing” is. We physicists tend to live in theory space rather than in the real world, so we tend to teach physics by developing the formal mathematical properties of the “electron” (or “electric field”) or whatever, and working out what experimental consequences these entail in certain situations. Generally speaking, the theory works so well in practice that we often *talk *about the theoretical electron that exists in the realm of mathematics and the electron-in-itself as if they are one and the same thing. As long as this is just a pragmatic shorthand, it’s fine. However, I think we need to be careful to keep this sort of language under control. Pushing theoretical ideas out into the ontological domain is a dangerous game. Physics – especially quantum physics – is best understood as a branch of epistemology. *What is known?* is safer ground than *what is there*?

Anyway, my little piece sparked a number of interesting comments on Reddit, including a thread that went along the lines “of course an electron is neither a particle nor a wave, it’s actually a spin-1/2 projective representation of the Lorentz Group on a Hilbert space”. That description, involving more sophisticated mathematical concepts than those involved in bog-standard quantum mechanics, undoubtedly provides a more complete account of natural phenomena associated with the electrons and electrical fields, but I’ll stick to my guns and maintain that it still introduces a deep confusion to assert that the electron “is” something mathematical, whether that’s a “spin-1/2 projective representation” or a complex function or anything else. That’s saying something *physical* is a mathematical. Both entities have some sort of existence, of course, but not the same sort, and the one cannot “be” the other. “Certain aspects of an electron’s behaviour can be described by certain mathematical structures” is as I’m prepared to go.

Pushing deeper than quantum mechanics, into the realm of quantum field theory, there was the following contribution:

The electron field is a quantum field as described in quantum field theories. A quantum field covers all space time and in each point the quantum field is in some state, it could be the ground state or it could be an excitation above the ground state. The excitations of the electron field are the so-called electrons. The mathematical object that describes the electron field possesses, amongst others, certain properties that deal with transformations of the space-time coordinates. If, when performing a transformation of the space-time coordinates, the mathematical object changes in such a way that is compatible with the physics of the quantum field, then one says that the mathematical object of the field (also called field) is represented by a spin 1/2 (in the electron case) representation of a certain group of transformations (the Poincaré group, in this example).I understand your quibbling, it seems natural to think that “spin 1/2″ is a property of the mathematical tool to describe something, not the something itself. If you press on with that distinction however, you should be utterly puzzled of why physics should follow, step by step, the path led by mathematics.

For example, one speaks about the ¨invariance under the local action of the group SU(3)” as a fundamental property of the fields that feel the nuclear strong force. This has two implications, the mathematical object that represents quarks must have 3 ¨strong¨ degrees of freedom (the so-called color) and there must be 3^{2}-1 = 8 carriers of the force (the gluons) because the group of transformations in a SU(N) group has N^{2}-1 generators. And this is precisely what is observed.

So an extremely abstract mathematical principle correctly accounts for the dynamics of an inmensely large quantity of phenomena. Why does then physics follow the derivations of mathematics if its true nature is somewhat different?

No doubt this line of reasoning is why so many theoretical physicists seem to adopt a view of the world that regards mathematical theories as being, as it were, “built into” nature rather than being things we humans invented to describe nature. This is a form of Platonic realism.

I’m no expert on matters philosophical, but I’d say that I find this stance very difficult to understand, although I am prepared to go part of the way. I used to work in a Mathematics department many years ago and one of the questions that came up at coffee time occasionally was “Is mathematics invented or discovered?”. In my experience, pure mathematicians always answered “discovered” while others (especially astronomers, said “invented”). For what it’s worth, I think mathematics is a bit of both. Of course we can *invent* mathematical objects, endow them with certain attributes and proscribe rules for manipulating them and combining them with other entities. However, once invented anything that is worked out from them is “discovered”. In fact, one could argue that all mathematical theorems etc arising within such a system are simply tautological expressions of the rules you started with.

Of course physicists use mathematics to construct models that describe natural phenomena. Here the process is different from mathematical discovery as what we’re trying to do is work out which, if any, of the possible theories is actually the one that accounts best for whatever empirical data we have. While it’s true that this programme requires us to accept that there are natural phenomena that can be described in mathematical terms, I do not accept that it requires us to accept that nature “is” mathematical. It requires that there be some sort of law governing some of aspects of nature’s behaviour but not that such laws account for everything.

Of course, mathematical ideas have been extremely successful in helping physicists build new physical descriptions of reality. On the other hand, however, there is a great deal of mathematical formalism that is is *not* useful in this way. Physicists have had to *selec*t those mathematical object that we can use to represent natural phenomena, like selecting words from a dictionary. The fact that we can assemble a sentence using words from the Oxford English Dictionary that conveys some information about something we see doesn’t not mean that what we see “is” English. A whole load of grammatically correct sentences can be constructed that don’t make any sense in terms of observable reality, just as there is a great deal of mathematics that is internally self-consistent but makes no contact with physics.

Moreover, to the person whose quote I commented on above, I’d agree that the properties of the SU(3) gauge group have indeed accounted for many phenomena associated with the strong interaction, which is why the standard model of particle physics contains 8 gluons and quarks carrying a three-fold colour charge as described by quantum chromodynamics. Leaving aside the fact that QCD is such a terribly difficult theory to work with – in practice it involves nightmarish lattice calculations on a scale to make even the most diehard enthusiast cringe – what I would ask is whether this description in any case sufficient for us to assert that it describes “true nature”? Many physicists will no doubt disagree with me, but I don’t think so. It’s a map, not the territory.

So why am I boring you all with this rambling dissertation? Well, it brings me to my other post – about Stephen Hawking’s comments about God. I don’t want to go over that issue again – frankly, I was bored with it before I’d finished writing my own blog post – but it does relate to the bee that I often find in my bonnet about the tendency of many modern theoretical physicists to assign the wrong category of existence to their mathematical ideas. The prime example that springs to my mind is the *multiverse*. I can tolerate certain versions of the multiverse idea, in fact. What I can’t swallow, however is the identification of the possible landscape of string theory vacua – essentially a huge set of possible solutions of a complicated set of mathematical equations – with a *realised* set of “parallel universes”. That particular ontological step just seems absurd to me.

I’m just about done, but one more thing I’d say to finish with is concerns the (admittedly overused) metaphor of maps and territories. Maps are undoubtedly useful in helping us find our way around, but we have to remember that there are always things that aren’t on the map at all. If we rely too heavily on one, we might miss something of great interest that the cartographer didn’t think important. Likewise if we fool ourselves into thinking our descriptions of nature are so complete that they “are” all that nature is, then we might miss the road to a better understanding.