**This ended up being more complex than I originally intended to**
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I haven't read much of the posts before posting this, so this might have already been said.
First let's start by defining how we define speed and length. At first, length was just length. Some scientist said that this amount of distance would be a meter and so the meter was born. We had an "official meter ruler" which is what we defined all other measurements by and many copies of this meter was made. This system proved to have many problems. First is in the field of Probability & Statistics. I've you've studied this, then we know that everything has a probability between 0 and 1. 0 meaning that it is impossible to happen, and 1 that it will surely happen. In the field of Engineering, nothing is 0 or 1. This means that we will never make all of these "official meter ruler" copies exactly the same. Some of these are bound to be incorrect by minimal amounts, and then these create more copies which can create more errors. The second mayor error in such a system was in the field of Thermodynamics. We know that the temperature in a volume is a 3 dimensional gradient which involves many factors. We know that objects expand as they heat up and they contract as they lose heat. Problem is that different parts of this meter will not lose/gain heat at the same rate. So even though we know how to adapt properties of expansion via temperature to this problem, it will never be 100% accurate.
These are the two most basic problems to the old definition of a meter, along with others such as varying rotational speed of the Earth and radiation. Our meter wasn't always the same, we needed a way to define a meter to always be the same.
In the early 20th century, scientists found out that the speed of light is always constant in a vacuum. We now had something which was 100% constant under certain conditions. We then redefined the meter to be a fraction of the distance the light traveled in one year. (about 3x10^8)
Now to define time.
Time can or cannot be the 4th dimension. Same goes with a point, lines and volumes. We just define them as the 1st, 2nd, 3rd and 4th dimensions for ease of study, so the question is, can time be a dimension?
The answer is simply, it depends.
We define the first dimension as a single point in space. (Don't try to imagine this, it is impossible for our minds to see a 1D world)
The second dimension is two points and everything that is in between these two points on the straight line that joins them. The same goes for any additional point you may want to place in this universe. The only exception is that you can only have distance and one angle between any two different points. This can be an (X,Y) plane using two perpendicular distances or (r,θ) being the distance and an angle between two points or polar coordinates.
The third dimension is any combination of 3 points of more by which the distance between 2 points is defined by either (X,Y,Z) three perpendicular distances, (X,Y,θ) two perpendicular distances and an angle, or cylindrical coordinates and (r,θ,ρ) which is one distance and two angles, or spherical coordinates.
Describing the next set of dimensions is very complicated. (We won't assume time is the 4th) You may recall from Calculus III that a point is just a set of numbers.
2-D objects are the following:
f(x) = y -> line
3-D objects are the following:
f(x,y) = z -> surface
f(x,y,z) = w -> closed surface
We could then define f(x,y,z,w) = ζ as probably the best function set which we can most easily understand that belongs to the 4th spatial dimension. The spatial dimensions can go on and on, but it is impossible for our brains to picture any dimension that is not 2D or 3D. The numbering of dimensions is just a relation to the number of spatial axis which we are studying.
Time being described as the fourth dimension is just to simplify everything to those who have not yet taken advanced studies. As we've seen in all dimensions, they revolve around and axis around a common point, or a variable. For time, we can use two variables like distance and speed to define time. But isn't speed the amount of distance traveled in a set amount of time? Yes.
We define F = ma. a = dv/dt
F = m dv/dt
(F/m)dt = dv
∫(F/m)dt = ∫dv
∫(F/m)dt = v
Velocity is the integrand of the amount of force per unit of mass with respect to time. We can measure force and mass at specific time intervals. But another problem, we used time. The problem with time is that we use it for everything, but it's hard to understand. Mathematically it's very simple, but Physically it's a paradox. We use it to explain nearly everything, but in order to explain something you need to use variables which are not itself.
None.
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