Cryptohistory is full of accounts of putative pre-Columbian contacts between New and Old World Civilizations; their frequency in lay literature and on the web is dependent much more on how interesting the encounter would be than on how likely it is to have happened based on the evidence.
For example: Zunis are a linguistic isolate with a bizarrely high frequency of Type B blood. They must be the descendants of lost Japanese pilgrims! Or - it's possible for a late iron age Norwegian to sail west from South America across the Pacific, approximating Inca technology - therefore, the Pacific could have been settled by South Americans! We know better today, half a century later, from DNA and linguistic evidence. But did Polynesians come to South America? The discovery of conquistadors' texts that said there were already chickens there suggested this, but so far DNA analysis has been inconclusive. For my money, the only decent evidence so far for trans-Pacific contact prior to Europeans is the linguistic and technology evidence in raft-building by the Chumash on California's central and southern coasts.
But all of these contact theories are relatively recent, in terms of prehistory. The major gene distributions and language families that we see today are the echoes of earlier migrations, the most famous of which is the crossing of the Bering Sea land bridge by Siberians to populate the Americas, 20,000 years ago or more. But we know that sometimes paleolithic people crossed water too: the first Australians were able to cross channels that were still tens of miles wide during the last ice age at least 40,000 and perhaps as long as 70,000 years ago. In Japan, at least by 30,000 years ago, people were able to make two > 50 km open water jumps across the Tsushima Strait. Consequently there's one other place where I'm curious as to why there isn't more discussion of why a water crossing didn't seem to have happened - the Strait of Gibraltar.
What is now the Sahara was much wetter as the Earth's climate shifted away from its most recent glacial maximum, and only now are we starting to understand the diversity of the people that lived there. One site in particular (the Tenerian culture at Gobero in Niger), only discovered in 2000, has burials dating to 7500 BCE. These were pastoralists, and it's clear that a greener Sahara could have supported a much denser population of pastoralists then than now. We invariably assume that Europe was colonized from its southeast, through the Middle East. Why could "green Saharans" not have made it across the Strait of Gibraltar to Iberia?
We know for certain that neolithic Afro-Asiatic speakers (the Guanches) settled the Canaries. Of course the interesting possibility is a connection to the Basques. It's possible that some Tenerians did colonize Iberia, but that water crossing is an effective enough barrier to gene flow that the genes coming through the Middle East into Europe swamped the Tenerian genes. A quick test would be an analysis of Basque mitochondrial DNA against Guanche and Gobero samples; if it has been done at this point I'm not aware of it.
Optimism and longevity.
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