[Ratboy]

Guys,
I believe this is a case of being an expert at framing the question.  Science is truly based on a house of cards. Carl Sagan criticized most of the Greeks we idolize for the fact that they did mostly deductive reasoning, without the back up of inductive reasoning.
The purpose of science is to acknowledge that what is believed to be the best theories to explain how things work are built on a house of cards and to shake that house.
For most of the hard sciences, there are very few collapses of widely used principles, and I like to use the belief of phlogiston in the late 1700s as an example.  People studied it for decades until Lavasier (spelling is wrong I know) offered an element called oxygen as an alternative that better explained what is going on. The phlogiston house of cards collapsed and a new house based on burning through oxidization was built.  Geocentric round earth models are another example because they were pretty good at explaining what we see and the earliest attempts of building heliocentric models were not any better.  As the houses were shaken the geocentric house collapsed. 
So we can say that because all science is based on the belief that this is all just a bunch of theories we can believe whatever stupid thing we want to, or we can accept that what we currently believe might be changed in the future.  But collapses in the hard science area are not very frequent in the last 100 years or so and we should probably just use the best models we have.
So if you want to be arrogant and say that you don't care about people living south of the equator, you can invent some wacky theory that can sort of explain what you see if you do not look closely.  For example, it is just fluke that you get equal amounts of daylight and night in a year because of how the sun circles above where you are.  If you acknowledge that this happens for everyone everywhere in the world, the flat earth model cannot be built on any house of cards that does not immediate collapse.  So a better house of cards is either the geocentric or the heliocentric house. So much lack of progress in science (Newton and Liebniz for example) comes from not recognizing the foundations are houses of cards. 
In the work world, people who believe stupid things (bridges stand up because of UA and you cannot calculate how to make them stronger) will not get hired for designing bridges.  The people who believe that they can design a bridge based on current scientific theories gets the job.  So we can believe in a flat earth because a better model might be wrong, but what is the point of that?

[InquisitiveREer]

So I just did some google searches and found some sites that talk about the ancient Greeks and circles. Hear they are:

https://nrich.maths.org/2561 , http://www.storyofmathematics.com/greek.html , https://www.smithsonianmag.com/travel/world-full-circles-180954529/

[xenotolerance]

Okay, we've started with Geometry. So that's interesting and I really like how this is going so far. Thanks to everyone who is participating, this is pretty much exactly what I had in mind for the thread, getting into first principles.

I'd like to respond to this in particular:

Here are some of the fundamental elements the models and maths of the Ancient Greeks assume:

- That perfect circles can exist
- That one could zoom into a circle forever and see a curve
- That any length of space can be divided into infinitely smaller parts
- That the space can be infinitely long
- Time can likewise be infinitely divided, or infinitely long

This is what is meant by "continuous universe." The math further takes such elements and runs with them. Continuous this, continuous that. None of it is justified. The Ancient Greeks performed no experiments before coming up with those ideas. It is merely their idea of a "perfect" universe.

Making conclusions from this continuous universe model, such as the sun would never set on a plane, relies on many axioms of the continuous universe model being true. Since the Ancient Greeks never really performed experiments to justify their ideas, that calls any such calculation into question.

emphasis mine

In a discrete universe, the sun still never sets on a plane.

I don't mean to agree with Tom's logic or argument about geometry / a continuous universe; it's just, even if true, it doesn't actually lead to his conclusion or support his overall position. So, from his first two posts where he denies that circles and the circle constant can even exist, and refers to sunsets, his logic doesn't work. In this case, taking out the bottom level of the house doesn't matter. A better fundamental topic here might be the law of similar triangles.

[Ratboy]

So if circles have never been proven to exist, does this mean that the path the sun orbits around a flat earth cannot be a circle or how does any of this lead me to conclude that what I see every day should be interpreted to mean that the sun is circling (or not circling if circles do not exist) around (and can I say around if round is not proven to exist?) the North Pole?  That is, if my eyes are tricking me into believing that a round earth is rotating, how does knowing that this is based on unproven assumptions help me conclude that a better model is one that defies what I actually see? I think a better stance would be to proceed with since we cannot believe anything we should just stay in bed all day.