First off to my mind the word "race" is racist because it implies we're in some kind of competition and a linear one at that.
Numbers are events, not things, everything is in flux, so maybe the distinction isn't that important.
At any rate, numbers are hierarchically ordered, in a line. The so called real numbers are what is known as linearly ordered which amounts to the same thing.
Dimensions create partial orders, trees, which are not linearly hierarchical because two branches can be on the same level.
It is true that these can always be collapsed into a linear order by applying a suitable metric, but one looses continuity in the process.
And so we come to space and time, numbers in more than one dimension and continuous movement throughout these dimensions.
Continuity is a choice, as Zeno of Elia pointed out and was doubtless known before.
One chooses the approach and the outcome according to these choices.
What does this have to do with "race"?
There are at least two theories of "race" that I know of that are based on finite orderings.
The first is the three so called "races": Negroid, Caucasoid and Mongoloid. In this Whites are a subgroup of Caucasoid, as are Asian Indians and Semitic People, who are Indo-European, but not "White".
The other is the four so called "races" by skin colour: Black, White, Red, Yellow. In this it is hard to fit some people like Semitics, Asian Indians or Aboriginal Australians.
However, the significance of both of these theories is the numbers, three and four.
All of us have either three or four types of cones in our eyes with which we perceive colour or hue and one type of rod with which we perceive intensity.
Three is the minimum number of colours or labels needed to partition any circle.
Four is the minimum number of colours of labels needed to partition any surface.
But these partitions hold that the boundaries of the partitions are finite.
In fact if the boundaries are infinite, even though they may contain only finitely many points, then the number of colours or labels needed is as many as the number of partitions, so we can partition the world into any number of "races" we choose.
"Race" is arbitrary, given that it has an infinite boundary, and we can choose as many or as few "racial" categories as we want, it simply depends what we put our attention on.
Many people still want to put their attention on "Black" and "White".
These "racial" categories don't work for me because I am a complex mix, but if others wish to use them, that is there prerogative. However it has no reality outside of their own heads and those who agree with them.
Some reality stems from the number of cones, and some justification stems from the three and four colour theorems, but the reality is that both people and groups of people have infinite boundaries, infinite subtlety and though a person may contain a finite number of cells and a group of people a finite number of people, their depth of presence extends for ever.
There is nothing wrong with forming groups either wilfully or by observation, but at the end of the day it depends what we put our attention on, and like all numbers, is an event, constantly in flux, constantly changing.