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Firstly, it is important to be aware that our understanding of the civilisations and culture of Pre-Columbian America is far from complete. Secondly, I don't propose to attempt to cover every civilisation, so this will - at best - be only a partial answer.
However, with those caveats:
As far as I am aware, we have no evidence that any pre-Columbian cultures or civilisations knew the true shape of the Earth. All the evidence we do have supports the idea that they all supported a flat-Earth model. What is more, there seems to be a considerable degree of overlap in the cosmologies of the major civilisations.
Aztec:
The Aztecs believed that the Earth had the approximate shape of a giant disk. That disk was divided into four cardinal directions. Above the Earth were 13 heavens, while below the Earth were the 'nine Hells of Mictlan'.
Maya:
The Mayans also believed that 13 heavens were stacked in layers above the Earth. The Earth itself was flat and four-cornered, and carried on the back of a giant reptile (probably a crocodile) which was, in turn,floating on the ocean. Below the Earth were nine underworlds once again.
Olmec, Zapotec, Mextec:
In the paper, Cosmology in Mesoamerica, Keith Jordan argues that:
While each distinct culture generated its own cosmology, all such models of the nature of the cosmos are united by common features marking them as distinctively Mesoamerican.
Among those 'common features' seem to be the idea of 13 heavens above a flat Earth, with nine underworlds below.
Our current understanding is that this represents part of a continuity of belief which extended from the Olmec civilisation (c 1500 BCE - c 400 BCE) until the arrival of the Spanish, although there remains some debate about just how much the beliefs of the Olmec influenced later civilisations. See, for example, The gods and symbols of ancient Mexico and the Maya by Miller & Taube (Thames & Hudson, 1993).
The Olmec civilisation is the oldest of these groups, and evidence for their world view in this regard was found on a square jade plate excavated at Ahuelican in State of Guerrero:
"On its small surface were recorded the three regions of the Mesoamerican universe. At the top, the sky with thirteen estrata surrounding a glyph in form of X that for the Olmecs and Mayas represented the sky. Immediately below appears a quincunx with four elements similar to seeds representing the directions of the universe or the solstitial extremes in the horizon. In the center was carved a fruiting maize plant with three cobs. This plant arises of a design that seems to be an architectural structure that stands on a hill. The base of this hill leads to the underworld, to the primeval water indicated by three oval elements and one rectangular."
For more on our current understanding of the relationships between the various early Mesoamerican groups, see Jeffrey P Blomster: Complexity, interaction, and epistemology: Mixtecs, Zapotecs, and Olmecs in Early Formative Mesoamerica, Ancient Mesoamerica, Vol 21, No 1 (Spring 2010), pp 135-149.
Inca:
The Inca called their land "Tahuantinsuyu" which (according to the course notes for Lecture 4 of Astronomy 161 at Ohio State) means "The Four Quarters of the Earth". This suggests that they had similar views to those of the Aztec and the Maya. In his paper Animals and Astronomy in the Quechua Universe, Gary Urton observed:
"... the celestial River is believed to carry into the sky the actual water which flows through the Vilcanota River. As the Vilcanota River flows from the southeast to the northwest, it carries terrestrial water to the edge of the earth. The water then flows into the mar, the cosmic sea, which completely encircles the earth.
Again, this suggests that the Inca also believed that the Earth was flat and four-cornered.
Upvote:-3
The assumption is that non-Europeans somehow needed to prove that they didn't believe something that we know Europeans at some point came up with. This assumption is made based on the idea that European knowledge is the relative standard on which to judge the rest of the world. But if other cultures never thought the earth was flat why would they ever go about disproving it? What is the actual basis to believe the earth is flat? Because maps are flat? You're talking about humans who built massive structures which you simply cannot do without an understanding of weight, measurement, and the effect of gravity (pyramids > apple falling). Why would such builders think they themselves lived on a structure with no support for the weight of mountains and oceans?
Have you ever seen a drop of water hit the ground and turn into a perfect rectangle? triangle? hexagon? No. That's because these shapes are not naturally occurring. In nature, objects tend to be three dimensional and so the laws of physics are acting on all sides of the object. A rain drop is an elongated sphere or oval only due to its motion. When hits a flat surface it becomes a disc. Therefore, it is only logical to start with the assumption and premise that all natural objects in three dimensional space tend to be round/spherical and to represent these objects on a flat surface would more logically be done as "disc" or oval if one was to be accurate whereas maps tend to be rectangular because of the medium and machines used to print them.
Even those who built pyramids or ziggurats could see more of the ground below from higher elevations. They were making star charts. They could do that but not compare the sight range from the ground vs at the top of the pyramid? They knew that the sky itself was curved. Look at every depiction of the goddess Nut. Ancient Africans knew the sky was curved and that's a much harder thing to prove than the earth being curved. But if the sky is curved it is only logical that the earth would be as well.
See pictures of Nut where her body is sometimes flat directly overhead but curved at the "horizons". Assuming the earth is flat leads to 2 unnatural suppositions. 1... that the land is infinite in all directions. 2. that at some point the land simply ends and somehow everything within the bounds of it is contained. If the ancients had star charts it means that they knew the whole sky was either moving or that they were. And which one is the farthest logical jump? They would have known their own rotation based on the position of the stars which were accurate enough for humans to use for navigation on land and sea. So let's give them more credit.
Upvote:-2
I like your reasoning Jonathan Sinclair. So many people are Eurocentric but fail to see their bias.
One commenter who stated 'flat earth' was the common notion in the Americas ended his ideas with the observation that the Inca felt: "The water then flows into the mar, the cosmic sea, which completely encircles the earth." This would imply a round earth not a flat earth. Of course without extensive writing and what books (Aztec and Maya) in existence were mostly destroyed by the Spanish 'civilization', all we can do is conjecture.
But even with the belief in 13 cosmic heavens, etc. it would in not be a leap to think these were abstract ideas (much as Christians now hold of heaven and earth) and that the earth was solid and spherical. Clearly observation their of the moon and planets demonstrated these were not flat. Why would native astronomers think the earth was any different?