The schema, it seems, is a kind of mental image. But what is mental imagery? There has been considerable debate in the literature over whether mental images even exist as actual images, many insisting that they do not see mental images as pictures in their mind. Surely this is a question best left to introspection. When you instruct yourself for example to imagine a table in your mind’s eye, what, if anything do you see? Well, in the first place I see absolutely nothing in the usual sense: the space before my eyes remains an empty void, with nothing in it. And yet at the same time I do see a totally invisible mental image of a table in that same empty space, or at least in a space that is somehow superimposed on my normal visual space. I can both see the mental image, in an invisible ghostly way, and at the same time I don’t see anything at all. Furthermore, what little I see of the imagined table does not always have a specific location, nor a specific size or scale, nor a specific viewing angle, nor a specific color or furniture style. The “image” of the table (if it can be called such) often appears either fleeting and unstable in location, scale, and orientation, or it appears totally abstract, non-spatial, as if expressed only in some symbolic non-spatial code, like a node in a neural network model that is labeled “table”. It is this fleeting evanescence and instability of the mental image that allows so many to deny its very existence as a spatially extended image in our imagination.
But although the mental image of a table can remain totally unspecified with respect to location, orientation, and scale, it is clearly possible to imagine a specific table at a specific location, orientation, and scale, and we can even select a color, and furniture style for our imagined table. The mental image remains perfectly invisible, we would still swear there is nothing there in the empty space before us. And yet at the same time we can see the imagined table right there in that empty space, with greater or lesser vividness and detail, even though its appearance in that space seems coincidental and inconsequential, like reflections in a pane of glass superimposed on the world seen through the glass. An artist or sculptor routinely sketches their mental image as if copying from a real image, demonstrating that there is some kind of information present in the mental image, and that information can be clearly spatially extended, like an actual three-dimensional image of a scene. It is even possible to locate the imagined table at a specific location in the space before us, outlining the spatial limits of its top and sides and legs with our hands, as if polishing the invisible surfaces of the imagined table. I call this exercise morphomimesis, miming the morphology of an imagined object with a wave of your palms, and thus revealing its explicit three-dimensional spatial structure. Although we can only mime two parts of the image at a time with two palms, the image itself can remain fixed and stable in space during the morphomimesis, demonstrating that it is possible to have a fully specified mental image that has the property of spatial extension across a specific region of space, even though it remains completely invisible in that space.
The fact that it is possible to form a mental image with a specific location and specific dimensions, and to mime its morphology with your palms, is proof that mental images can exist as stable three-dimensional structures, and that they can carry a specific information content. And the mental image can be formulated to have a specific location and spatial extent, even if it is not usually specified so precisely, but often remains in an indeterminate state. The fleeting evanescence and instability of many mental images should not be viewed as counter-evidence for their existence as images, but is merely evidence of a fleeting and unstable imaging system, one that is capable of representing multiple possibilities all superimposed, much like a quantum particle that can exist in multiple states simultaneously. Like a quantum particle, the mechanism or principle underlying the mental image can apparently flip or morph continuously into different forms, unless it is held to a stable state by an act of will.
So let us examine the mental image medium to see what mental images are composed of, how they present themselves to consciousness. Picture, if you will, a square, of the geometrical variety, that is, composed of Euclidean lines that span four Euclidean points marking the four corners, to define a square segment of a Euclidean plane, a surface that is perfectly flat and thin. It is possible to imagine such a square of any size, I can rotate it in my mind to any orientation in three dimensions, and I can trace out with my fingertips exactly where I am imagining the square at any moment. In other words, what I see in my mind’s eye is an image, very much like the images I see with normal vision, except that the mental image is totally and completely invisible.
Mental imagery compels us to acknowledge two different types of seeing: One is the regular type of seeing, as when viewing a colored cardboard square whose edges are defined by a visible transition in color and/or brightness across the edge, and the other is a kind of invisible seeing in which imagined objects are completely invisible, and yet we can “see” them as spatially extended structures that can occupy specific volumes of visual space. Michotte (Michotte, 1963; Kanizsa, 1979; Michotte, Thine`s & Crabbe´, 1991) has called this kind of vision “amodal” perception, because it is perception in the absence of a particular visual modality, such as color or brightness. We see a square in our imagination, but it is in a kind of invisible outline form, like a figure in a geometry text, without color or substance, just a shape.
But it is also possible to imagine a color with your mental image. I can just as easily imagine a red square, or a green one, and I can see my imagined square change color on my command, with a specific square region being painted out with the specific color that I choose to imagine, all the while remaining totally invisible in the sense that I’d be willing to swear in a court of law that I do not see a colored square before me, even though I can locate with precision the edges and corners of the colored square in my imagination that I don’t see. This ability to conjure into existence any simple geometrical structure I might choose, and to imagine it at any location and orientation and scale I might choose, and even paint it with imagined color, and yet to remain acutely aware of the distinction between reality and my imagination, is both the foundational origin of mathematical thought, and at the same time, it reveals the most basic operational principle behind human intelligence. We think primarily in pictures, the words only follow after the mental image is formed. And the words lose their meanings if they become disconnected from the images that they represent in the symbolic code of language.
The Perceptual Origins of Mathematics 