mathematics is visual.
thankfully blind people use the visual system as well.
Mental imagery experiments, using fMRI techniques, have shown that much of the visual system, down to the primary visual cortex, is active when we create mental imagery without visual input. The brain’s vi- sual system is also active when we dream (Hobson, 1988, 1994). Moreover, con- genitally blind people, most of whom have the visual system of the brain intact, can perform visual imagery experiments perfectly well, with basically the same results as sighted subjects, though a hit slower (Manner &. Zaback, 1976; Car- penter &. Eisenberg, 1978; Zimler &. Keenan, 19831 Kerr, 1983). In short, one should not think of the visual system as operating purely on visual input. Thus, it makes neurological sense that structures in the visual system can be used for conceptual purposes, even by the congenitally blind.
Moreover, the visual system is linked to the motor system, via the prefrontal cortex (Rizzolatti, Fadiga, Gallese, &. Fogassi, 1996; Gallese, Fadiga, Fogassi, &. Rizzolatti, 1996). Via this connection, motor schemas can be used to trace out image schemas with the hands and other parts of the body. For example, you can use your hands to trace out a seen or imagined container, and correspondingly you can visualize the structure of something whose shape you trace out with your hands in the dark. Thus, congenitally blind people can get “visual” image- schematic information from touch. Image schemas are kinesthetic, going be- yond mere seeing alone, even though they use neural structures in the visual system. They can serve general conceptual purposes and arc especially well suited for a role in mathematical thought.
There are many image schemas that characterize concepts important for math- ematics: centrality, contact, closeness, balance, straightness, and many, many more. Image schemas and their logic.-; are essential to mathematical reasoning.
Motor Control and Mathematical Ideas
One might think that nothing could be further from mathematical ideas than motor control, the neural system that governs how we move our bodies. But certain recent discoveries about the relation between motor control and the human conceptual system suggest that our neural motor-control systems may be centrally involved in mathematical thought. Those discoveries have been made in the field of structured connectionist neural modeling.
Building on work by David Bailey (1997), Srini Narayanan (1997) has observed that neural motor-control programs all have the same superstructure:
• Readiness: Before you can perform a bodily action, certain conditions of readiness have to be met (e.g., you may have to reorient your body, stop doing something else, rest for a moment, and so on).
• Starting up: You have to do whatever is involved in beginning the
process (e.g., to lift a cup, you first have to reach for it and grasp it).
• The main procss.~: Then you begin the main process.
• Possible interruption and resumption: While you engage in the main
process, you have an option to stop, and if you do stop, you may or may
• lteratio11 or continuing: When you have done the main process, you can
repeat or continue it.
• Purpose: If the action was done to achieve some purpose, you check to
see if you have succeeded.
• Completion: You then do what is needed to complete the action.
• Final .’~tate: At this point, you are in the final state, where there arc re-
sults and consequences of the action.
This might look superficially like a flow diagram used in classical computer science. But Narayanan’s model of motor-control systems differs in many sig- nificant respects: It operates in real time, is highly resource- and context- dcpendcnt, has no central controller or clock, and can operate concurrently with other processes, accepting information from them and providing information ~o them. According to the model, these arc all necessary properties for the smooth function of a neural motor-control system.
Where Mathematics Comes From.