Cognition, Brains and Riemann


Joselle DiNunzio Kehoe in Plus magazine:

By the nineteenth century mathematicians struggled with the meaning and implications of their ideas and tried to shore up the foundation of mathematics, fearing perhaps that the weightlessness of the purely abstract could threaten the integrity of their discipline. They reflected on, and argued about, the meaning of their work — what it could address, and how. What was a function? Was “infinity” simply shorthand for an unending process, or was it something? In 1829 the legendary mathematician Carl Friedrich Gauss wrote, “mathematics has for its object all extensive quantities.” Other kinds of quantities could be considered “only to the extent that they depend on the extensive.” By extensive quantities Gauss meant lines that are described by length, surfaces, solid bodies and angles, as well as time and number. The non-extensive quantities he allowed included speed, density, hardness, height, the depth and strength of tones, the depth and strength of light, and probability. But he also provided an important qualification, “One quantity in itself cannot be the object of a mathematical investigation: mathematics considers quantities only in their relation to one another.” Here, quantities and their measures are considered together, and they can each be thought of as magnitudes.

Gauss's student Bernhard Riemann brought a definitive clarification to the meaning of measure. He acknowledged in the introduction to his famous lecture On the hypotheses which lie at the bases of geometry that this was influenced, not only by Gauss, but also by ideas of the philosopher John Friedrich Herbart, who pioneered early studies of perception and learning. Herbart's work played a significant role in debates centered on how the mind brings structure to sensation.

Like cognitive scientists today, Herbart broke down the world of appearances into the subjective impressions that build it. He rejected the idea that space was the thing that contained the physical world. For him spatial forms were mental images derived from relationships among any number of things we experience. They arise in our conception of time (the future being ahead of us and the past behind us), as well as number, and are applied to all aspects of the physical world. Herbart accepted that any perceived object could be thought of as a collection of properties bound together.