It would make sense to specify dates in either of two ways: smallest-to-largest or largest-to-smallest.
In England specifically (and Europe more broadly, I understand) people have opted for the former, writing dates as Day Month Year (e.g., 10 July 2014).
Americans, on the other hand, use the apparently-crazy Month Day Year system, which is neither smallest-to-largest, nor largest-to-smallest. Why?
The answer is that Americans actually use the largest-to-smallest method, but shift the year to the end, like we do when alphabetizing personal names by surname.
In a phone book or online directory, I would be listed as “Tillman, Micah” rather than “Micah Tillman.” Similarly, in common usage, Americans write “July 10, 2014” rather than “2014 July 10.” We shift the year to the end. In both “Tillman, Micah” and “July 10, 2014,” the added comma indicates the shift to the end of what belongs at the front.
Why do Americans do this? I suspect it’s a compromise between theoretic purity and pragmatism. The largest-to-smallest system has theoretic purity on its side, but in everyday life, we usually know what year it is, and just want to know what month and day it is. So, we start our dates “in the middle,” and move the year to the end.
(This, please note, is an explanation of the justificatory rationale for our system, not of the historical genesis of our system.)
I would argue, furthermore, that we come to know most things by moving from largest to smallest, genus to species (first learning to distinguish animate from inanimate objects, then learning to distinguish animate reptiles from animate mammals, then to distinguish animate mammal dogs from animate mammal cats, etc.). Therefore, the largest-to-smallest arrangement style of specifying dates fits the way we come to know everything better than the smallest-to-largest style.
What’s inconsistent about American usage, however, is the fact that we write our names in the opposite direction of our dates, moving from personal name to family name, rather than family name to personal name, as in many Asian cultures. We name people in the individual-to-species direction (as it were) rather than in the species-to-individual direction. Only in alphabetized lists, it would seem, do we acknowledge the utility of the largest-to-smallest approach to specification (a utility perhaps indicated by the very term “specification”!).