One of the big problems that emerges from all the proceeding discussions of perception is how we are able to perceive *space*. Space has been considered as a mathematical concept (in terms of Euclidean geometry), as a psychological concept (a construction of the mind) but never really as a biological, ecological concept. This first chapter about space perception is focused on one mathematical conception, some of it's implications, and one specific attempt to deal with those implications (Berkeley's *New Theory of Vision*).

Space perception has to work with whatever space is, so a theory of space perception has to work with whatever your theory of space is. Mathematically, space has been considered as Euclidean - continuous, isotropic (no preferred direction), homogeneous, and infinite. (ADW note: there are lots of kind of geometrical systems, and they vary in how many of these kinds of symmetries they require. Affine geometries give up isotropism, for example, while topology relaxes all symmetries. So a mathematical conception of space for perception to work with does not have to be Euclidean, it just mostly has been considered as such. Identifying the correct geometry for perception is actually an empirical question, e.g. Todd et al, 2001, and there are plenty of options that may be weird enough.)

The Euclidean conception of space poses some problems for visual space perception to solve. In essence, vision inhabits Flatland, and faces all the challenges of coming to grips with Spaceland detailed in Abbott's famous book. In the first story, a 2D square living in Flatland tries but fails to convince a 1D inhabitant of Lineland that the 2nd dimension exists; in the next two stories. a visitor from 3D Spaceland tries to convince the square that the 3rd dimension exists, which only happens with the 'miracle' of the square being lifted into Spaceland. This convinces the square, who goes on to reason that there may be a 4th dimension; he cannot convince the sphere from Spaceland that this reasoning works, however.

The relevant moral of the story is that there is nothing essential (inherent) to Flatland than can provide access to Spaceland, so you cannot get from one to the other using logic or analysis on Flatland data. The implication is that, in order to perceive space, Flatland vision needs help from a Spaceland inhabitant, which might come in one of two guises:

- You could enrich 2D vision with a source of knowledge about the rules of Spaceland geometry. This is the Kantian style solution, but is an example of an
*unrepayable loan of intelligence* - You could ground 2D vision with a perceptual system that necessarily inhabits 3D Spaceland, such as
*touch*: this is Berkeley's solution, because he was worried about the loan, and which Turvey spends the rest of the Lecture on.