I've looked around for discussions on this, and they all seem to have gotten off track pretty quickly. I'm hoping this one doesn't.
The crux of my argument is that if the world is flat, there are a lot of people who would have to know, and who would have to be keeping it a secret. Leaving aside the question of why this conspiracy would get started in the first place, I want to know why each of these groups of people would find it in their interest to willingly lie to the public about this.
1. Geoscientists. As you can see from pictures like (A) below, geoscientists certainly rely on the assumption of a round earth. From their papers, they seem to operate quite well under these assumptions, and perform experiments which would immediately falsify either FE or RE (you'd never be able to detect seismic waves as shown in that picture in FE, right?). Therefore, there must be a rather large number of geoscientists who know the earth is flat, yet continue to practice their sham science and, I suppose, just straight up create false data from nothing. Why? What do they have to gain from being part of a field whose fundamental assumption they know to be false? Add under this heading Meteorologists as well, whose mapping of weather patterns in the Southern hemisphere must be completely incorrect (although they do a pretty good job with those radar maps despite apparently having no idea how the earth is shaped...) and Oceanographers as well, whose detailed maps of sea currents, etc., must be fatally flawed since they don't even know how far it is from Australia to South America.
2. Airlines. If the world actually looks like (B), then all southern-hemisphere flights must be longer than they're actually billed. Airline pilots and those creating their routes must know how far they're actually going, even if they try to keep it a secret from their passengers for some reason. There are, after all, nonstop flights from Australia to South America all the time, for instance. They'd have to either fly over the North Pole (which would be very hard to justify) or take an extremely long path across the enormous Pacific Ocean (which would be extremely inefficient). Either way, there have to be a great many people in the airline business that know. Why do they keep quiet?
3. Sailors. From the voyages of James Cook (C) to the Vendee Globe yacht race (D) to basically every commercial freighter that sails in the Southern Hemisphere, the distances these people travel and the way they navigate is based on a round earth. It would be impossible for the Vendee Globe sailors to go around the perimeter of the disk earth as quickly as they do, so is that whole thing a sham? If so, why even host it? Why should those sailors, along with all the commercial sailors, keep this quiet? As a subcategory here, include cartographers, who even hundreds of years ago were drawing world maps that look entirely incompatible with (B).
4. Physicists. A lot of physics, even stuff that doesn't seem to be at first, is predicated on the idea of gravity working pretty much the way Isaac Newton said it does. If you're saying Newtonian gravity is completely wrong, then the Cavendish experiment is a hoax, along with General Relativity. There are many papers that confirm the findings of Newton and Einstein, so these all must be fraudulent and at least some of those physicists must know that. Why don't they go public with this? I've seen that there are answers to this on the wiki, so in the interest of not re-hashing old arguments, I'm dropping this point for now.5. Smaller governments. To the extent that I've seen any justification for a vast RE conspiracy, it's that larger governments, like the US, want to brainwash the masses or something. But what about other countries that have geoscientists and airline pilots and commercial sailors -- in other words, countries that have the ability to figure this out? Do they have the same incentive as the big guys to keep this under wraps? What about countries 100, 200, 500 years ago, and all the ones in between? How have all these governments agreed that maintaining this sweeping RE deception is beneficial?
6. Larger governments. As above, they certainly know. This is the group usually focused on in these discussions, so I'd like to de-emphasize it somewhat, but I still would like to know why in the world they would want to perpetrate this massive hoax. When has any other government project gone on for so long, so successfully, and even through massive regime changes? And whatever their ultimate goals are, is pouring the trillions of dollars it must take to fake NASA, pay off whoever needs paying off, and secure the Antarctic Ice Wall *really* the best way these countries have of acheiving those goals?
Also, not only is there the direct cost of all that falsifying, but the additional cost of having a public that's not scientifically literate. If the Earth is actually flat, surely there are consequences of that which could be better exploited if it didn't need to be kept a secret?
I'm purposely de-emphasizing NASA and other space agencies as well -- I don't want this to get sidetracked into a discussion solely about them, that ground has been well-trod many times. It probably leaves out many other groups of people that would also have to know the shape of the Earth, no matter what it is. The bottom line is that the conspiracy seems too enormous to be sustainable.
---Pictures---
(A)
(B)
(C)
(D)
(see
here for more information on the Vendee Globe race.)