Archaeology uses many numerical methods grounded in mathematics: photogrammetry, the search for numerical patterns—or, conversely, proof that such patterns are illusory. In another area, however, mathematics is used to build logical models with which to test lines of reasoning. This has been particularly true in the archaeology and paleontology of human settlement, where the "wave of advance" model has become widely renowned.
Invasions and settled populations
For many years, the settlement of the world was described as a succession of invasions. The different cultures identified by paleontologists were explained in terms of invasions and the replacement of populations. In Eurasia, these invaders came first from the eastern steppes; then, at the end of the Roman Empire, came the Germanic and northern peoples, followed by Attila's Huns and the Mongols of Genghis Khan and Tamerlane. Europeans likewise invaded the Americas from the 15th century onward. Even today, some people fear a "great replacement." These sudden population movements were attributed to the invaders' military superiority—for example, to the Mongols' invention of the stirrup or the conquistadors' invention of firearms.
The "invasion theory," however, faced several obstacles. The earliest populations were settled. Paleolithic hunter-gatherers moved from one hunting and foraging ground to another throughout the year, but followed the same route year after year. The first farmers, meanwhile, remained tied to their fields. The further back in time one went, the less plausible the invasion hypothesis became.
