| The question is which n-dimensional space? The concept that we call a "point" gives us an (n-dimensional) "affine space". Affine spaces don't require any sort of coordinate system, axes, origin, etc. which makes them quite general. For example, consider the sleepers of a railway track, or the crossing-points of a chain-link fence, or the hands on a clock face, or the electrical potential at various positions around a circuit, or a date, or the temperatures of various objects, etc. It makes no sense to "add" or "multiply" points; but we can find the difference between two points. The result will actually be a vector (in the examples above: a distance and direction along the track; a distance and direction along the 2D plane of the fence; an angle; a voltage; a duration; and a temperature difference). We can add such vectors to our points; if we add the vector (pointA - pointB) to pointB, we get pointA! This relationship between points and vectors leads to the concept of a "torsor" https://math.ucr.edu/home/baez/torsors.html Vectors live in an (n-dimensional) "vector space", which requires more concepts than an affine space; e.g. a notion of "zero", a notion of "size", a notion of "direction", etc. This is less general, but lets us do operations like adding and scaling vectors, as well as the various notions of multiplication defined here. Some vector spaces arise naturally, e.g. taking the angle between clock hands gives us a natural zero (the difference between identical positions) and a natural size (a full turn), although whether positive/negative indicates clockwise/anticlockwise is still arbitrary. Other times we will "impose" some arbitrary coordinate system on an affine space, since vector operations are so useful; often ignoring the space's affine nature entirely! That way we can treat "point" interchangeably with "vector from the origin"; even though most of the fancy things we want to do are only defined for the latter (e.g. taking dot-products, comparing cosine similarity, etc.) For example, ere are some arbitrary coordinates that we impose on affine spaces every day: - The top of a clock is 12:00 - The Greenwich meridian - Grounding/earthing electrical circuits - Celsius and Fahrenheit - 0AD, New Year's Day, the Unix epoch |