Geometrical Knowledge

Geometrical Knowledge

How do we come to have geometrical knowledge? How do we acquire geometrical concepts? The question has been around since Plato and his theory is still probably the best—we have such knowledge innately. But this doesn’t answer the question of what triggers the innate knowledge (it isn’t there fully formed from the start): what in our experience enables the innate knowledge to enter our consciousness in explicit articulate form? It can’t be Socratic interrogation because someone has to have the knowledge without such interrogation, and obviously most people don’t learn what a triangle or circle is by explicit verbal instruction. Is it by seeing shapes, or touching them? But how do we see and touch shapes—how does our experience come to have this kind of geometrical content? You might suppose that triangles and circles are present in the stimulus as it reaches the sense organs, ready to picked off like apples from a tree, but this is not the case. The pattern of light that impinges on the retina is notoriously fragmented, fluctuating, and fleeting; yet the concepts of geometry are fixed immutable forms, as Plato emphasized. The proximal stimulus is remarkably impoverished compared to the representational powers that exist in the mind; no mere copying process could produce the latter from the former. You can’t make a Euclidian purse from a retinal sow’s ear. In fact, as we now appreciate, the visual system makes elaborate computations that lead from the mess on the retina to a stable, intelligible, three-dimensional sensory world—a world of shaped objects in space. We see an objective world of determinate forms based on a stimulus that is anything but, thanks to our sophisticated visual processing mechanisms. Vision science investigates these mechanisms and it has made impressive progress. Can we conjoin this with Plato’s innateness hypothesis to answer the question of the origin of geometrical knowledge? It sounds like a promising idea, but we need to be more specific about the eliciting stimulus that triggers the innate schematism: what is it exactly that causes geometrical understanding to form in the mind? There is an innate structure in the mind-brain of unknown character, and somehow the course of experience elicits it; but what is it about the perceptual system that makes it suitable for performing this role? What does it do that leads to the formation of geometrical understanding? I suggest the following: it works by means of perceptual constancies. As is well known, the visual system (and tactile) corrects for variations in the proximal stimulus that result from changes of distance, illumination, perspective, and so on. It extracts distal invariants from proximal variations, thus producing a world of objective forms from an array of idiosyncratic subjective impingements.[1] These forms—shapes, among other qualities—constitute the geometry of the human visual world. They are generated by a process of correction applied to the proximal stimulus. Thus, we perceive cubes and spheres and other forms that are not subject to the vagaries of the proximal stimulus. The visual system attributes these forms to distal objects. This is not to be understood as a process of conscious rational thought but as an automatic unconscious computational mechanism (unreasoning animals have it too). Perceptual constancies are primitive properties of visual systems, which can occur in creatures that lack rational thought. However, in the case of humans rational thought does coexist with perceptual constancy, and an innate potential for geometrical understanding comes with the genes; in humans, geometrical knowledge can flower. We have the conception of geometrical forms as fixed determinate shapes with specific properties by virtue of a combination of innate schematism, rational thought, and perceptual constancy—with the last of these doing the heavy lifting. This is the capacity that triggers the formation of geometrical understanding, given the right innate structure. For it deals in perceptual invariants, objective shapes, a repertoire of determinate geometric forms—not in flickering patterns of light striking the retina. There is no geometry worthy of the name in the patches of light that cause nerve endings to fire in the retina, but by the time the visual system has done its work a fully formed geometry has been attributed to the distal environment. It only remains for this visual geometry to interact with the system of representations we call “innate ideas” and we have the roots of geometrical understanding—concepts of triangles, circles, etc. Perceptual constancies are vital to this process of cognitive production. It isn’t that distal objects somehow emit shapes that end up on the retina and can be copied by the brain to yield shape concepts; the shapes have to be generated by the visual system from clues contained in the proximal stimulus. Perceptual representational content is endogenously generated from perceptual primitives and then attributed to the environment; geometrical knowledge, as we ordinarily conceive it, results from the interaction between this system and the innate endowment we bring with us into the world. It is as if the innate geometry lies dormant in us until stimulated by the operations of the perceptual systems as they impose constancies on the data of sense. This capacity arises early in all perceiving creatures, so human children are primed to output geometrical competence before the school years begin: they are then ready for formal geometrical instruction. They know what triangles and circles are well before they are taught their mathematical properties. They have been having experiences as of triangles and circles from an early age—forms subject to perceptual constancies—so they are acquainted with geometrical figures; they just need to attach these perceptual primitives to their innate system of “ideas”. The triggering is perceptual, but not in the way the old empiricists supposed, i.e., by a copying procedure; rather, perceptual constancies play the eliciting role. Not that any of this is easy to understand—the nature of the innate schematism, the act of triggering—but it is the general character of the formation of geometrical knowledge. In the case of language an innate schematism is triggered by verbal sensory input, and the child comes to understand what words and sentences are (a basic level of grammatical knowledge); in the case of geometry the same kind of thing happens, only here the innate schematism is triggered by a perceptual system that is built around constancy effects. When a ball, say, moves away from the perceiver, retaining its apparent size as the retinal image shrinks, this embodies the idea of a geometrical form that persists across space and time; it is an invariant in a world of variation. It is the primitive prototype of Plato’s eternal geometric forms existing in a flux of fleeting impressions. Constancies imply permanence amid change, and that is precisely what Platonic forms are supposed to be. Geometric objects are constancies idealized.[2]

[1] For a thorough discussion of perceptual constancy, see Tyler Burge, Origins of Objectivity (2010) and Perception: First Form of Mind(2022).

[2] Our knowledge of color is not dissimilar. There is also color constancy and this affects our conception of color: we don’t think of colors as varying with every change of illumination, etc. We think of colors as invariant under variation of proximal stimulus, and hence as possessing a certain sort of objectivity. We can then study colors as stable qualities with a fixed nature. If colors and shapes were conceived as varying with the proximal stimulus, we could hardly study them at all—they would be too fleeting and lacking in any fixed nature. They would not be colors and shapes as we know them (there would be no such thing, for us, as the color red or the shape square).