# Objects in Space

“*A mental creation that evolved to study objects in the world*”^{1 }─ this sounds to me like a pretty good description of mathematics ─ or at any rate of mathematics prior to the late 19th century. It highlights the two most important features of mathematics:

**(1) **that it is a (human) *creation* and,

**(2) **that it is not a *free *creation ─ it cannot be if its aim is to study ‘objects in the *world’*.

**(2)** in effect means that mathematics is, or rather was, subject to serious constraints. What sort of constraints? If we examine the two earliest branches of mathematics, arithmetic and geometry, we see that the principal constraints are *discreteness *and *distancing*. *Discreteness* because arithmetic sees the world as made up of lots of little bits that *can be counted* ─ if everything was mixed up with everything else like the ingredients in a cake arithmetic would not give sensible results (and would not be needed). Secondly, experience tells us that whatever is happening *here* is not simultaneously happening *over there*, i.e. that objects and events are separated by something, thus geometry, metric spaces, causality, anything that depends on spatial separation.

Viewed thus, the two earliest and (arguably) still the two most important branches of mathematics depend either on *separateness *or *separability *: in the first case, that of arithmetic, the focus is on the *objects *themselves, in the second, the focus is more on what lies *between *and *around *the objects. Practically all pre-Renaissance mathematics was in effect the study

**(1)** *of particular objects separated by empty space *or

**(2)** *of particular objects such as points and curves embedded in empty space.*

No hint of movement so far ─ as we know the Greeks, though they initiated both statics and hydrostatics baulked at the creation of dynamics, the study of objects in motion and, by implication, of the unseen forces that give rise to motion.

^{1} From Lakoff & Núñez, *Where does Mathematics come from?*