First, I have seen a great deal of difference between "archaeologists" who call themselves Book of Mormon scholars and those who call themselves Mesoamericanists. The three I mentioned are in the latter category. I have personally heard one of them say "my faith and my work have nothing to do with each other. There is no proof linking the Book of Mormon to the ancient Maya." He said this publicly at a conference at Yale. My guess is that these sort of men are not typically sent out to speak to congregations. So you might actually get a good answer from them if you ask without directly challenging their faith.
As far as I know from scientific research, all paleontological evidence demonstrates that horses went extinct in the New World about 12,000 years ago. There were people here at that early time during the Ice Ages, but this is many thousands of years before there were Jews anywhere. Moreover there is no evidence of the use of iron anywhere in North, Central, or South America before the arrival of Europeans. The earliest example I know of is at L'anse Aux Meadow, Newfoundland, a small Viking colony that dates to ca. AD 1000. I'm not sure if the iron was local or imported, but the site is located near an iron bog. The wheel was known in Mesoamerica and perhaps elsewhere, but was little used. Without a large beast of burden (there were no oxen or domesticated cattle, either), wheeled carts or chariots are useless. These too, did not exist. In Veracruz, ca. AD 500-1000, the wheel was used to make small pull toys. In the Andes, there *was* a beast of burden--the llama--but it can only carry about 90 lbs so was too small to pull a cart. There was no cement, and I think somewhere in the Book of Mormon it is mentioned. A kind of "concrete" was used at Teotihuacan and other places. It is very hard, but does not use cement. Most cultigens mentioned in the Bible and the New Testament were not found here until they were brought to the New World by colonists in the 16th and later centuries. The Americas did have maize. In South America there were potatoes. In parts of both N and S America, the grass family Chenopodium was domesticated: it is now fashionable to eat quin~oa, a seed from Bolivia. But this is not mentioned to my knowledge in the Book of Mormon.
You mention synagogues. Does the Book of Mormon use this word? I wonder why it wouldn't just use the word "temple" or "house of assembly" or "house of prayer" or even "school." These are the terms that jews actually use for places of worship (a "temple" is really different from the others and hasn't truly existed since AD 70). In any rate XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX When we travel, my husband and I like to see what they look like over the centuries. They all have this in common: a central hall for congregation and prayer. At the center of this is the bimah, a table or lectern from which the Torah is read. In one wall of the hall there always is an aron kodesh, a niche or box where the scrolls are kept. Most also have a room for study called a beit midrash. Lots have secondary rooms for all sorts of purposes. The oldest one I've personally seen or even read about is in Aswan, Egypt, on Elephantine Island (Yebu). It dates to about 400-650 BC or so. In any case, I personally don't see any important similarities between the temples and synagogues of Mesoamerica and Jewish ones--even this very early example. I guess they all have walls and roofs and a place that serves like a stage for a priest. But that can be said for any religious building. In particular, however, Mesoamerican temples are often built on pyramids and generally have very small inner spaces; they are no places for people to congregate inside. Only priests and kings and the like go in them. Jewish, Christian, Muslim, Hindu, Buddhist, etc. "temples" generally are places for people to gather inside a building, not outside of it (but they too can have a hidden sanctum sanctorum; ours undoubtedly come from Egyptian examples dating back at least 2,500 years before Christianity and at least 800 years before even the earliest roots of Judaism). Also, Jewish examples aren't built on pyramids that symbolize holy mountains. Mesoamerican temples are. Finally, *no* Mesoamerican temple has a central bimah or an aron kodesh. Without those two features, you cannot have a beit knesset (house of congregation).
DNA evidence is "incomplete," by which I mean a few Book of Mormon scholars have indicated that we would need to collect it from every skeleton and every living Native American before they would accept the results. Nonetheless, what evidence we have--and it is substantial--is that all living Native Americans (including the Maya and the descendants of the Olmec) are relatively closely related and come from northeast Siberia/Asia. Inuits (Eskimos) are somewhat different, but they are definitely the last to arrive. There is no evidence of a Middle East connection.
Linguists, too, have long looked at Native American languages. Most see a very confusing and complicated web of hundreds (maybe thousands) of languages that are not related close enough to link together, except in scores of distinct language families. One linguist has argued that there are basically three language superfamilies: Inuit, Athabaskan, and all the rest (which he calls "Amerind"). Not one--not a single, credible, modern scholar anywhere--has posited a connection to Hebrew, Aramaic, or any other Semitic language. I should say that a mere 2000 years (since Jesus' arrival in the New World) should be detectable by linguist analysis. Generally it is after 5,000 or 8,000 years of separation that things get difficult. And 5000 years ago, there were no Jews!
So, as far as I see it, there is no *archaeological* material evidence connecting the New World to the Old before about AD 1000 (footnote: a Roman ceramic head was found at a site in Mexico, but probably was planted there), and--with the exception of the Vikings--most evidence dates to after 1492. There is no *paleontological* research supporting the existence of some Old World animals here after the Ice Ages and before the 16th century. There is no *biological* evidence supporting a genetic connection between current or ancient Native Americans and the Old World, except that their ancestors came from northeastern Siberia. There is no *linguistic* evidence for a connection between any New World language and any Old World one.
Finally, you raised the Mound Builder controversy, which was still raging during the life of Joseph Smith. Let me speak about such ideas more generally, far beyond the LDS. Many people have asserted that the pyramids of Mexico, Central America and South America, and the mounds of North America, were not built by native populations. Some, yes, have argued for ancient lost tribes of Israel. Others have argued for the Egyptians, or the Chinese, or black Africans (Olmec heads being the principal "data"), or by people from the lost continents of Atlantis or Mu, or even by space Aliens (there is a ridiculous tv show about this now). What all of these ideas share is one common thread: THEY ARE ALL RACIST.
The basic notion shared by all these racist perspectives is that native Americans are too dumb, ignorant, degraded, subhuman--whatever--to build the works that we see today. Since they are incapable, we must look elsewhere. Book of Mormon followers look to the Lost Tribes. Afro-centrist scholars look to subsaharan Africa. Con-men like Erick von Daniken look to outer space and make millions.
This inherently racist notion is actually still reflected in American institutions. If you want to learn about Greek or Roman culture, you go to a museum of fine art. If you want to learn about native Americans, you go to a natural history museum. Such museums teach you about rocks, insects, animals, and--as a logical extension--native Americans.
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