Absence of evidence is not evidence of absence.
So it has been said. Again...and again...and again...
Like most wise sayings that are actually wise (which, I would note, is a very small minority of allegedly wise sayings), this one is true only to a limited extent.
I used to think that this was an absolute. You could never hold a lack of evidence for a claim as evidence that the claim wasn't true. It was, instead, necessary to build up evidence that countered the claim.
However, about 15 years ago, as I was finishing my bachelors degree in anthropology, one friend of mine who had been raise Mormon began to get more involved with his church, and another two converted to Mormonism, and all three routinely wanted to talk with me about their church's alternative history of the Americas*. I would point out that the archaeological data did not support their church's claims, they would counter with data that FARMS (an apologetics group masquerading as a historical and archaeological research organization associated with BYU) had taken out of context to support the churches claims. The FARMS data was invariably poor, and when I would point this out, and direct them back to there being no actual evidence to support the church's conclusions, all would then repeat back to me a phrase that, in another context, I had taught them: The absence of evidence is not evidence of absence.
At first, this stopped me cold. It was a sentiment with which I agreed, how could I argue against it. And yet, I knew that there was something wrong with its use here.
This line is a favorite of people who hold to a dubious position unsopported by evidence - whether you are talking about religious beliefs, political ideals, positions on social issues, belief in paranormal claims, etc. If you press someone hard enough regarding the lack of supporting evidence for their belief, you are likely to be told that the lack of evidence doesn't provide evidence that the belief is false.
But if you give the matter some momentary thought, it becomes clear that there are, in fact, cases where a lack of evidence for a proposition does serve to forward an argument that a proposition is false.
Let's compare and contrast, shall we...
I have written before that I am a supporter of the pre-Clovis hypothesis, that is, the idea that people were in North American prior to the appearance of Clovis tools ca. 11,000 to 12,000 years ago. Although some evidence to support this view (all of it interesting, none of it overwhelming) has been trickling in in recent years, I supported this hypothesis prior to that based on some fairly basic arguments (follow the link if you want more information). Now, the early peoples of North America of whom we are aware were, generally, highly mobile hunter-gatherers who left what are typically referred to as ephemeral traces of their activities and movements. So, their sites are few and far between, and often we know them more by isolated artifact finds than full sites. There is every reason to believe that their predecessors would have had a similar settlement and land use pattern, meaning that there sites would be rare, would often be in difficult to find places (buried under alluvium, or drowned in the since-risen coastal waters), and might be ambiguous as to age or constituents (many tools aren't time diagnostic, and datable materials has to be found in order to figure out the age of a site, so there are many sites that we encounter that simply can not be dated). Given these circumstances, it is likely that pre-Clovis sites would be few and far between and might not be recognized for what they are when found. In this sort of situation, the fact that we had no known pre-Clovis sites meant that there was no material support for the pre-Clovis hypothesis, but it in no way precluded the existence of such sites.
Contrast this with the various claims of vast Old World-derived empires being established int he Americas. These types of claims are not limited to the Mormon church, there is no shortage of people who believe that the Egyptians, Sumerians, Celts, Knights Templar, etc. etc. established a strong presence in the Americas**. In such cases, the remains of the various cities, plantations, etc. that have been suggested would be pretty damn hard to miss, and would bear features that would make them easy to distinguish from indigenous settlements (in other words, if you think Cahokia or Mesa Verde or any other Native American site is evidence of Old World influence, you have demonstrated a definite ignorance of Native American culture technology AND Old World practices and material culture). Moreover, given the way that the native economies of the Americas functioned, it is beyond unreasonable to expect that artifacts associated with these Old Worlders would not be spread far and wide in Native American sites, and be especially common in the sites nearest to the Old World outposts. In other words, not only should the Old World-derived sites be obvious on the landscape, but even if, for some reason, they weren't, like being able to detect a planet by it's gravitational pull on other objects, we should be able to see evidence of a major Old World incursion by its effects on the Native Populations***.
So, in this latter case, the fact that we have no evidence of Old World empires being established in the Americas is, in fact, pretty strong evidence that they never existed. Any attempt to explain how they could have existed without having the huge impacts that they would invariably have had always results in either misconstrued claims about the actual archaeological record, or else special pleading.
*For the record, I usually didn't start these conversations. That I didn't buy their church's teachings didn't change the fact that these three were all really good people, and that I was happy to have them as friends. Even when we had these conversations, they were usually good-humored and friendly.
**Note, this is different from claims that people from outside of the Americas may have occasionally visited the Americas or established a weak presence (such as the Viking settlement known in Canada). Small and/or brief presences might leave little physical evidence, and there is always the possibility that such will be found (though it is unlikely). I'm talking here about the various common claims that various European, Asian, or African powers established cities, large colonies, or other large and long-lasting settlements in the Americas.
***For example, the Spanish missions in the American Southwest and Florida are much smaller-scale than anything proposed by the Mormon church, and no smaller than those things proposed by proponents of the notion that the Knights Templar ended up here, and they had huge impacts on the native populations of their surroundings, impacts that, 250 years later, we are still strugglign to comprehend the enormity of.