You may be wondering about the title: “Theoretical Atlas”. Both words have a double meaning here.

First, Atlas: originally, this was the name of a Titan in Greek mythology, who was condemned by Zeus to stand at the Western edge of the world and hold up the sky on his shoulders forever. The Western edge of the Greek world – the Mediterranean – is indeed where the Atlas mountains are found, in the Maghreb. Also named for him is the Atlantic Ocean (and, therefore, Atlantis, a continent once speculated to be located somewhere in it). You can see a picture of the Atlas mountains in the banner at the top of this blog’s main page.

So one meaning comes from a notion that tends to crop up fairly often when one talks about the project of finding a quantum theory of gravity. This is the prospect of a complete unified theory of physics, a Theory of Everything (TOE), or some such name. People peering into the mist of our limited knowledge sometimes seem to see prospects of a single theory that unifies every aspect of the physical world in one single model – all forms of matter, energy, forces, gravity, etc. The name “M-theory” is popular in some circles for this idea – an as-yet undiscovered theory which might go beyond what string theory can do today. Other prospects have been proposed, but the image I have is of a single, immensely powerful theory, holding up the entire world on the strength of its explanatory power – a theoretical Atlas holding up this enormous burden.

But this great Atlas of a theory has never been written down – alas. For myself, I’m quite skeptical if it even could be: why should there be a short, pithy idea that encodes the whole huge, complex, endlessly surprising universe? Even if we had a theory which accounted for all particles and forces in nature, would that be a theory of everything? The point of a theory, after all, is to help us understand things: we’d still need, at the very least, a theory to explain how chemistry emerges from physics, what life is and how it can come into being – all just to account for even our most basic experience. Then there are whole areas of the world that open up from there. So this great single Atlas of an idea that accounts for the entire world of experience is, as they say, just a theory. It’s a (merely) theoretical Atlas.

(Of course, this use of the phrase “just a theory”, often used to dismiss the insights of Darwin, and much less prominently used any other way, is simply wrong. The meaning of “theory” depends on context, but it always means something more than a mere guess. Still, as I said before, I’m not going to worry TOO much about being wrong now and then – and the more accurate hypothetical Atlas just didn’t sound as good.)

The other meaning of the word “atlas” has to do with maps. The other element of the banner above mentions the Bellman’s map from The Hunting of the Snark. It had no markings on it at all – “purely conventional signs”. But mathematics is all about using purely conventional signs as a reference point in describing the features of the world. The Bellman’s map showed no land – only sea – and so it left out not only the conventional reference points, but also anything definite to refer to.

A “theory” can be seen as a way of taking some standard, pre-existing structure, and trying to “map” it onto the features we see in the real world. In a way, a literal map is an example of a theory: it imposes a regular grid of coordinates on some convoluted shape, which is itself a model of some territory off elsewhere in the world. It’s an artificial imposition – but it allows us to find our way around. Assuming it’s accurate enough, and we know how to read it.

In the case of a literal atlas, we have a collection of – usually flat, generally rectangular – drawings of the surface of a sphere (more or less). Each one is a little bit distorted, because the Earth isn’t flat (no, no, I know – that’s just a theory – but I think it’s accurate enough). In the study of manifolds, these are called “charts” – each one is a map from some open subset of Rn to a subset of the manifold. Generally – and, for instance, on the surface of the Earth – one chart won’t be enough. You need several charts, and an understanding of how they fit together. The collection of charts is an atlas, and one imagines a big book filled with these charts, each one imposing a rectilinear grid of coordinates onto some underlying terrain. “Transition maps” tell you how they fit together to cover the whole surface.

So the other meaning of theoretical atlas is the notion that we may need many theories to properly account for the world. Each one may describe some part of it fairly well – maybe with a bit of distortion, but certainly not so much that it doesn’t help to find our way around. None by itself explains everything – but given enough, and some knowledge of how to manage the transition between the domain of one theory and the domain of another, they can tell us a lot. This is my image of what our researches into physics, and the world in general, are aiming at: an atlas of theories that covers everything.

Mind you, I realize that such an atlas, like the other kind of Atlas, is purely theoretical.

Oh, all right: hypothetical.

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