The previous two lectures have covered aspects of the problem of space perception. We perceive objects, and these inhabit space - they are at some distance from us and each other, etc. So what is space, and how do we come to know about it? We've tried using touch (a sense that lives in 3D) to provide the necessary clues (this was Berkeley, and the story of Flatland). This doesn't work because touch isn't a perfectly reliable source of space information, it's a perceptual system like vision and faces the same kinds of problems. We've also tried to just make space a necessary feature; not an object of perception, but a mode of perception (this was Kant, and the power of Euclidean geometry). This doesn't work, because there are perfectly coherent non-Euclidean geometries, that all provide different answers when measuring the same space. As soon as there is more than one geometry, choosing one becomes part of the problem and you cannot simply assume it.
Given this, how might you go about selecting the right geometrical description to use as the basis for your experience of space? Turvey discusses one major attempt to show how this might happen, specifically Helmholtz's account of how to derive a geometry (a notion of space) from non-spatial local signs.
This Lecture involves quite a bit of mathematical detail, which I am not going to get into here because it's slightly besides the point. The overall goal, however, is to take some sensory raw material that isn't intrinsically about space (because it can't be), and work to turn it into an experience that is about space. This is going to be the start of the move to talk about sensations and perception, the organising principle still at the heart of all non-ecological discussions of how we come to experience the world. More on this in the next lecture.
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