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Objects in Space

January 23, 2019
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A mental creation that evolved to study objects in the world1 ─ 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?

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