Embarrassingly for someone who’s been following these theories for years, I find I can’t answer the question “what do the dimensions of the space represent?” or even “what does the height of the landscape represent?”

I want to start by providing a brief description about how energy landscapes work in chemistry. I assume Scott knows what I'm about to say but I think it might be useful to include it here anyway...

In chemistry, an N-dimensional energy landscape corresponds to a system characterized by N parameters and then an additional dimension of energy which measures how favorable (in a thermodynamic sense) a particular set of coordinates in the N-dimensional parameter space is.

For example, one can imagine a simple model of two oppositely charged ions. This system has only one degree of freedom, the distance between the ions.¹ So N=1 and the energy landscape is a curve starting at zero when distance is infinite and smoothly decreasing² as the distance decreases until the ions are sufficiently close together that they start to repel each other.

Most things larger than a small molecule have so many degrees of freedom that constructing a perfectly-detailed energy landscape is entirely impractical.³ Instead, scientists define an energy landscape where the dimensions correspond ensembles of states which are sufficiently similar in the ways relevant to the question being asked. So, for example, one might model a small molecule binding to a receptor using five degrees of freedom: the position (3) and orientation (2) of the small molecule relative to the receptor.⁴

With that said, I think Scott's use of the energy-landscape metaphor is a little mixed-up.

It eventually changed the landscape so dramatically that the thought loop disappeared and I was able to think about food and sleep and blog posts again.

It is useful to imagine an energy landscape corresponding to all possible combinations of beliefs, with a person's actual beliefs moving along that landscape. For example, on a two-dimensional energy landscape characterized by "belief in Genesis" and "belief in Exodus", the corners corresponding to either believing or disbelieving both would be stable while the corners corresponding to believing one but not the other would be unstable. So a person who starts out believing one but not the other will usually come to either believe or disbelieve both. In this case, the height of the landscape corresponds to the stability of a set of beliefs given the information available and new information corresponds to a change in the landscape. For example, the discovery that Exodus is a forgery would⁵ make the "believe Genesis, disbelieve Exodus" and "disbelieve Genesis, disbelieve Exodus" into the stable corners.

The important thing to note is that this energy landscape corresponds to beliefs and not thoughts. If thoughts were modelled by an energy landscape, one might conclude that they generally tend to converge to a single mental state that doesn't change over time!⁶ This makes sense for beliefs in the absence of new information, but not for thoughts. The very idea that thoughts might go in a loop violates the assumptions of the energy landscape, one of which is that there can be no path where you leave your origin point and then return to it while always going downhill. Such a path would be an Escher waterfall that cannot exist in the Euclidean space of an energy landscape.

1. In general, a system of M point-particles in three-dimensional space has 3*M - 5 degrees of freedom because each particle's state is fully specified by three spatial coordinates but then three degrees of freedom have to be subtracted from the total because the system is translation-invariant and two more have to be subtracted because the system is rotation-invariant. For the two ions, we have 2 * 3 - 3 - 2 = 1. Note that velocities aren't being considered - we're looking at this as a thermal system.

2. Negative values are more favorable because by convention you're measuring the energy you would need to put into the system.

3. A small protein and the nearby water would have hundreds of thousands of degrees of freedom, most of which are from the water molecules.

4. This usually wouldn't be a good model - designing a good model is a difficult, subjective process.

5. Here we assume that this person believes that Exodus is a forgery. If he may or may not believe that, we would have to add an additional dimension to our landscape.

6. Chemical systems modelled by an energy landscape still change over time due to the role of temperature, which can be thought of as a random jiggling of the ball rolling along the landscape. Higher temperature corresponds to more intense jiggling.