The equivalence of geometry with algebra, of visible shapes with abstract formal operations, had escaped the Greeks in spite of their fascination with number and form. But this insight was the clue that illumined Descartes' search for the nature of mind as mechanism. Just as geometrical figures are generated by algebraic operations, so are the shapes and colors we experience in vision, for instance, generated by logical operations represented by physical events in the brain. The language of experience is as conventional as the language of numbers or words. In the meaning of each symbol we find nothing necessary in itself, but only whatever the mind has agreed to.
Cognition is like an interpreted formal system, one pressed into representational service.
Other interpretations could be possible for a given system, and other systems could map the same territory. This faith is underwritten by, and ultimately refers to, the fact that organisms in possession of such a system survive. We are here because we are just what we are, and the world is just as it is in our experience for the same reason. Descartes sets out first to reduce physics to the mathematics of space. For him, extension in space is the only objective quality of matter, and space is in turn reducible to number through the coordinate system.