Some time ago I was asked about dimensions by a friend's son. The following is an extract from the very non-scientific response I gave him. Perhaps it might be a help.

Perhaps the simplest way to think about the dimensions of any given space is to think in terms of the number of co-ordinates that would have to be given in order to specify one particular point. First let us consider a straight line drawn on a sheet of paper; if this line is 10cm long, with 0 at the left and 10 at the right, then the single co-ordinate 0.9cm will designate a unique point just short of the 1cm mark. Nothing further is needed, so we can say that the line has one dimension. In the case of a surface area the situation is only a little more complex. We already have a straight line, marked off in 1cm intervals from the left. Raise a vertical (i.e. towards the top of the paper) line from the 0 point of the first line and mark it in 1cm intervals from the bottom. If it helps you to do so, complete the square and fill in the grid of 1cm squares. Call the left side of this square A and the bottom edge B. Clearly, a unique point on this surface can be designated by using two co-ordinates, as for example, A7 - B4, or to be more specific about the location, A7.5 - B3.2. Two co-ordinates are needed to specify a unique point on our surface, so it is deemed to have two dimensions. Let us now imagine we have a very large number of small blocks, each measuring 1cm by 1cm by 1cm. One hundred of these little cubes can be placed on the surface of our square. If we place another nine layers, each of 100 cubes, on top of the first, we will have a larger cube made up of 1,000 small cubes. In order to specify any one particular small cube within the larger one we will have to use the two co-ordinates from our earlier surface, and also raise a vertical (this time perpendicular to the surface of the paper) line on which to mark a third set of co-ordinates, similar to the other two. If we call this new line C, any small cube can be identified by giving three co-ordinates, as for example, A9 - B3 - C5. If we wanted to pinpoint the small cube in the bottom, left hand, front corner, its co-ordinates would be A0 - B0 - C0. Notice that in the case of the verbal description and the co-ordinates, the designation of the position requires three elements. The cubes are therefore said to have three dimensions.

Whilst the definition of dimensions set out above is perfectly valid in terms of theoretical calculations, there are serious problems when trying to apply these definitions on a practical basis. Let us start with the concept of a two dimensional surface. No one would deny that a surface has only two dimensions. Any point on the surface can be designated by two co-ordinates, as we have seen. However, it also has to be accepted that in order to be a surface, in the real world, it must be the surface of something. Remove the something and you remove the surface. Furthermore, any activity undertaken on that surface requires a third dimension. Any mark made on that surface, for example, must either rest on the surface, in which case it is in a third dimension, or it must soak into the fabric underlying the surface, which also requires a third dimension. The something of which the surface is the surface.

Draw a straight line on a sheet of paper. We considered a straight line as having only one dimension, but in fact it has width as well as length. Imagine reducing the width of that line; how far do you have to reduce it in order to claim that it has no dimension of width? The reality is that the dimension of width does not vanish until the whole of the line has gone. Logic tells us that the same applies to the third dimension; the only way to remove the depth of the line is to remove the entire line.

This is probably where the concept of infinity will creep into the discussion. Valuable as reduction to infinity might be to the theoretician, it could have serious consequences in the real world.

Then there was Einstein!!! Einstein had this disconcerting habit of pointing out that nothing was as simple as Newton would have had us believe. Three dimensions were no longer enough, suddenly we needed a forth. We established earlier that three co-ordinates were needed in order to specify a particular point in three dimensions. Let us now apply that to a particular example. At the junction of High Street and Church Street there is a multi-story car park. You arrange to meet a friend on the third floor of that car park. The three dimensional reference gives all you need for the meeting to take place, or does it? Suppose your friend arrived at 10am and you did not turn up until 5pm. Either you would have the most patient friend in the world, or the meeting would not take place. A forth co-ordinate is needed for the success of the arrangement. Before Einstein, no-one would have suggested that there was any significant connection between time and space, other than that time was the measure by which we could keep track of the changes that took place in space. Einstein, however, maintained that not only were matter and energy interchangeable (E =mc² took its bow on the scientific stage), but so were time and space. Time had to be considered as a dimension, and this 4D "thing" became spacetime.

BTW, Senga14, if you are looking for 100% proof of anything, science is probably not the place to look.