A Roman mile is indeed defined a thousand Roman paces. However, a Roman pace is considered five Roman feet. This developed into the English mile, 5000 English feet. The English mile was changed by parliamentary decree to 5280 ft. so that it could be divided into furlongs, which are 660 ft., as they were an important unit of measure at the time.
Something I've always wondered: is a pace, stride, Roman pace, etc. measured from heel to heel? This would lead to consistent measurement. I was always under the impression that they were measured from heel to toe, which would mean that one would always have to subtract the length of a foot from every subsequent unit of measure, meaning that only the first pace is three feet, the first stride about four, the first Roman pace about five.