Lecture 9: The Space Enigmas II: Kant, the Nature of Geometry, and the Geometry of Nature (Turvey, 2019, Lectures on Perception)

The first space enigma was the fact that vision lives in the two dimensions of Flatland, but produces an experience of three dimensional Spaceland. You can't logic or experience your way from Flatland to Spaceland (as described in the famous book). Berkeley tried to solve this problem by providing a guide, in the form of the Spaceland-dwelling body, but this fell apart and the only remaining suggestion was an unrepayable loan of intelligence from God. 

Another way to consider this problem that leads to another proposal is what Turvey calls 'the outness problem'. This is the annoying fact that sensations on the retina are experienced as things out there, in the world. This makes space a necessary precursor to perceptual experience: however the outness problem is solved, a notion of space is required to drive the search for a solution. Kant is the main person who worked to establish how space might be baked into perception; 'Space, therefore, is not an object of perception...but something very different, namely, a mode of perceiving objects' (Turvey, 2019, pg 124). Spoiler alert: it doesn't work, for interesting reasons that feed into the development of the ecological analysis. 

How can we come to know what is true or false? There are two basic ways. An analytic claim is one that can be evaluated as necessarily true or false a priori, without any sort of empirical investigation, just the application of logic (e.g. 'an orange is a fruit'). A synthetic claim brings two or more pieces of information together and must be empirically evaluated to see if it is contingently true or false, so a posteriori. The notion of space as a necessary precursor to perception requires something different, specifically a synthetic a priori analysis, and this was where Kant came in. 

He begins with four metaphysical claims about the nature of space:

1. Space is presupposed by sensory experience, not derived from it (spatial terms like 'next to' don't emerge from the experience of two things next to each other, that experience is the way it is because of a notion of space)
2. Space is a necessary precondition to perception of objects (because you can imagine no objects in space but you cannot imagine no space)
3. Space is a single thing; it's not made up of 'the space here + the space over there'
4. Space is not a concept; we consider it as an infinite thing, and in order for it to be a concept we would have to be able to conceptualise that infinity. 

He concludes with a transcendental argument proposing a synthetic a priori claim: perception of space is necessarily Euclidean. The idea here is that geometry is the set of formal propositions about space, and mathematical propositions are always a priori type claims (they entail a necessity that isn't derived from experience). However, geometry is also synthetic: it brings together a set of information (axioms) in order to be a complete claim. At the time Kant wrote, geometry meant Euclidean geometry, a set of five axioms from which you can derive everything you need to characterise space. 

The practical upshot is that Euclidean geometry is the kind of claim that can be the basis for a notion of space that fulfils the metaphysical requirements. Space is not a thing, it simply is Euclidean geometry, and given that this geometry had been the only coherent geometry for 2000 years, it seemed like we have a pure-reason solution for what space is and how it forms the basis of experience. 

Non-Euclidean Geometry

This entire argument falls apart, however, because it turns out there is more than one internally coherent and complete geometry (set of axioms about space). The 5th axiom asserts that there is one and only one way for two lines to be parallel to one another, but it turns out that if you relax this claim in different ways you get different geometries that are inconsistent with Euclidean geometry but are still internally coherent. Geometry is no longer an a priori fact, and which geometry best describes a given space is an empirical (a posteriori) question. 

As soon as geometry became an empirical game, the key purity of it fell away. The various axioms are about abstract, idealised forms such as points, lines and planes, none of which exist in the real world, so there were now limits on the axioms imposed by the details of the world and the resolution of the measuring system. Then people started trying to empirically establish how many dimensions space has and found that fractional dimensionality was an option and everywhere: fractal geometry had arrived. Sizes, shapes, and distances were suddenly not necessarily definite things, but functions of how they were measured, and perception is an act of measurement using wide varieties of embodied systems. 

The practical upshot is that it no longer makes any sense to talk of 'space' as if it is a single, coherent thing, because there are multiple coherent ways of describing it and which one applies to a given question is an empirical question. Finding which one applies to the question of the perception of space is also an empirical question, and what space is is now a negotiable thing for an organism. 

Reflection

One of the key themes emerging over these chapters is that the history of the study of perception is a story of attempting to derive experience from theories of physics and geometry that simply are not up to the task, and instead of noticing that the theories are inadequate, trying to find solutions that still live within the bounds of that theory (often because the theory was overly revered). The recent (last 150 years or so) history of science has involved a lot of theory proliferation: physics and geometry have both produced a large number of more complex theories that are more up to the challenge of explaining the facts at hand (by facing up to what the data was telling them instead of treating the inadequate theory as some magnificent truth). Psychology has access to these more interesting theories, but has yet to fully embrace any of them, and as a result is still flailing around making the same mistakes. The promise of the ecological approach is that it is going to built on the basis of these more interesting theories, and that is why we're going to succeed.