What is a Dimension ?
This text is about the mathematical and geometrical foundation of the term. It starts with simple definitions, but waters will get muddier, when we examine how the word gets actually used. During this first part I limit the number of dimensions to the visible.
If you look up "dimensio" in the Latin dictionary, it reads measure or measurement. That might be first surprising, because you could measure anything. But like the word meter (which means to measure too), it refers in the most used sense to the unit, humans perhaps measure for the longest time: length. Even today dimension is sometimes used in the sense of size.
The dictionary will tell you, that "mensio" also means to measure and "di" is two - since you can measure along an axis in two directions. That's why axis and earth axis is also referred to as ''dimensio''. Truly, you can imagine an infinite amount of axis through space, but three are necessary to define and travel space. Left-right, front-back and up-down. This is so basic, that it seems we don't have to talk about it.
That is not at least due to a book that shaped the mindset of our culture for millennia. Long before Russel or Newton it set the standard, of what definitions, axioms, postulates and theorems were and how to write a math book or how to teach math at an university level. Its content was cited during the middle ages by book number, paragraph and sentence - pretty much like the bible. These are the "Elements" by Euclid, written just one generation after Plato in the 3rd century B.C.
Book 1, Definition 1: A dot is what can not be parted. Already here we see a shimmer of the idea of an atom, and a more practical approach, than modern definitions, that say a dot has no dimension. In case you did (like me during the first reading) miss the subtle difference, the old Greeks make physical reality the basis of their reasoning, we today on other hand use abstractions. If you paint a dot, visible or not - it will have of course hight, length and width (aka dimensions). Even if I use a nano-pen to set a dot with the size of one atom or smaller, the previous statement stays true. A point with no such dimensions can only live in our mind. This distinction is important and people who can not distinguish between their imagination and the physical world are called ideologs and are hard to reason with.
Book 1, Definition 2: A line has length but no width. Suddenly Euclid gives up seeing dots as atoms or he just does not care about what he can not measure (remember dimensio ment that what can be measured).
Book 1, Definition 5: An area has length and width. This will be followed by all the wonderful maneuvers you can do with lines, triangles and angles as well as number theory (greatest common divisor, primes and alike).
Book 11, Definition 1: A body has length, width and depth.
Still : the observable object defines the mathematical idea here.
We could also take the opposite approach and construct the basic shapes, starting with an ideal dot. When we move the dot along one dimension - the trace would be a line. Move that line along another dimension and get an rectangle - moved along the third we end up with a cube. Please note, that during that process we define our world by preconceived ideas. It might not be harmful on this occasion, but being able to recognise it, is sign of awareness.
Euclid pursues the similar view in definition 3 and 4 by stating, that not only the end points of a line are dots, but also that the line itself is composed of dots. That raises the question: How can something that has length be composed of something which has none . To Euclid that is not an issue, since he never claimed that a dot has no length. David Hilbert in his now 100 years old attempt to make the Euclidian system mathematically watertight, avoids this question.
Euclid also realised that he needed real numbers (numbers with digits after a point) in order to make his system work. If you have a line between position 1 and 2 of your axis (dimension) you need a lot of dots between them. Here again you can observe the difference of the learned man of old and new. The former are more consumed by practical ideas of proportion and angles, which can be used for instance for architecture. The notion that there are a lot of points seems good enough to them.
Today we get taught that no matter how much you zoom into a line there will be always an infinite number of dots - as if infinite times nothing gives you something. If you apply this property to all dimensions, you get a picture of what is called nowadays complete space.
Infinity is not only imaginable in the very small, but also in the very big, which also was not a big concern to Euclid. He just states in the last definition of Book 1: Parallel are lines, which lay in the same plain, and if you would prolong the indefinitely, would never meet. There are geometries where they do, but they are called none - Euclidian.
The most famous ones are called projective because they can be imagined by projecting an image on a bent surface (the dimensions are not straight lines). A practical example would be our earth. Because the surface of our home is not flat but a kinda - sphere, a rectangle will never have angles that are precisely ninety degrees. Seemingly parallel lines cross here on the back side of the earth. If you make a perfect rectangle, it can touch the surface only in one point. House-builder do not care about that, because for one: there is topology (the surface of the earth has hills and valleys) and secondly: the earth is so big, that there are no practical differences.
That also illustrates another important concept of modern math: the manyfold. Yes its sounds fancy, because the name is derived from the notions that there might be a lot more dimensions. But the basic statement here is: I am a human with a limited horizon of observation. And within it, things that look like an Euclidean space, I will treat as if it were an Euclidean space.
There is a new field (<120 years) where mathematicians got very creative with the term space and dimension. It is named topology - not to be confused with the former mentioned meaning of the the word. In topology spaces can be shaped and limited as people are able to imagine. They can even introduce their own definition of distance. Sure, there is still a thing called "distance function induced by Euclidean norm" which is math speak for "that what can be measured with a ruler" (our original meaning of dimensio). But in modern topology any function that obeys three rules is allowed to be called "distance":
1. The distance from a point to itself is zero.
2. The distance between two points is equal in both directions.
3. If you move from A to B and then C it must not be shorter than directly A to C.
In fact, each "point" of our custom made "space" might be a function and our distance function is just an equation calculating the similarity between two equations. This is clearly very far removed from the original meaning of space and its axis. Why do mathematicians do this to us?
You may not believe this, but they are humans too. They struggle to orient themselves between the many ideas they produce. And to do so, they use terms with strong emotional connotation, because we know from neuroscience, that in order to be memorized, a fact has to be attached to some emotion: the stronger - the better. Please read the works of Antonio Damasio on that.
If you would read the mathematical literature (or the closely related theoretical computer science) you would be astounded how often they use the word magic, oracle and other terms that don't sound like the primal occupation of an mathematician. These terms provoke strong feelings and are therefore able to transport meaning. Ironically: the more abstract and removed from everyday life mathematical concepts become, the greater the need to connect them with a strong word, that carries their meaning.
Moving and orienting ourself in space is connected with brain circuits we share with simple animals that display only very base level feelings. For that reason spacial terms get an instinctual meaningful connotation - instantly and always.
I think one task of a philosophers is it, to not be carried away by such impulses and to assure that meaning and usage of words are consistent and foster self reflection.