Absence Of Evidence Is Not Evidence Of Absence

"Absence of evidence is not evidence of absence!"
-- Carl Sagan, Astronomer

"Absence of evidence is not evidence of absence!"
-- Donald Rumsfeld, Military strategist

The above is a favorite (and sometimes annoying ;-) catch phrase often repeated by many, many, many people (apparently including BradAppleton, for whatever that may be worth). Simply put, it means that if we don't know that something exists, it doesn't mean that it doesn't; It only means we don't know one way or the other, we just haven't been made aware of it yet so it's not part of our knowledge.
If you look for "X" and don't find it, does that prove that there is no "X"? No.

But the more you look in places where X "ought to be" in ways and at times that X "should be likely to be there," the more confidence you can have that there is no "X".

Like, I flip a coin 5 times and get "heads" every time (no "tails"). Is it a "heads only" coin? (...having a "head" printed on both sides.) Well, say I flip it 500 times and get "heads" every time. Now I'm pretty convinced that it is "heads only."

Or you could just look at both sides of the coin. OTOH, if you find no evidence of a "tails" on either side, does that mean the tails isn't there? Hmm...
Certainly - but you have some evidence here. You know that there is a "tails" on the other side of the coin, that does not come up when you flip it. And if you've done a good job of looking in likely places where you might find X, but still don't find it, there is no question that increases one's confidence in it's scarcity.

On the other hand, if I am the only person doing the coin flipping in the above example (and especially if I flip it with same hand every time in a similar fashion) then from the information provided, it's certainly quite possible that the problem is not with the coin, but with the way I'm flipping it time after time after time!

Unless I've been switching hands or had several others flip the coin as well with the same results, I'm dismissing the possibility that the problem may be me and not the coin. And in doing so, I've committed precisely the kind of error in judgment that is addressed by the saying AbsenceOfEvidenceIsNotEvidenceOfAbsence. I didn't consider the problem might be elsewhere, so I jumped to the conclusion it was the thing and not me.

Some people do the same with a great many things. "I've never successfully used (single or multiple) inheritance without always running into problems, therefore the problem must be that inheritance is bad (and not simply that I used it badly), despite the fact that others talk of using it successfully. They must just be fooling themselves." Or replace "inheritance" with "O-O" or "C++" or "pair programming" or "parallel development" or anything else that is the subject of many popular raging debates on both sides of the continuum.

The problem here, as with so much of the discussion, is with the word MUST. But evidence is not about proof or certainty or necessity, it is simply something that can contribute to belief. Failure to have ever used multiple inheritance successfully is EVIDENCE that it's bad, but it is in no way conclusive. This is the fundamental mistake that one finds throughout discourse about evidence -- the belief or assumption that there can only be evidence for claims that are true. But there is in fact a vast amount of evidence for claims that are in fact false.

Every quote or saying has a context that goes with it, and AbsenceOfEvidenceIsNotEvidenceOfAbsence is certainly no exception. It has to do with the flip-side of the all too frequently recurring case of assuming facts not in evidence. It's no less safe to assume those non-evident facts are true than it is to assume they are false. It has to do with the cases where people weren't aware of some possibility or piece of information and therefore jumped to the conclusion that it wasn't there. If we don't see or understand something, let's at least consider that maybe our vision or thinking needs some adjusting before rushing to declare it completely absent or unworkable.

It's all too easy to jump to that wrong conclusion when it's a case of blaming something other ourselves. It's so much easier to point the finger at "O-O" or C++ or inheritance or parallel development, etc., than it is to point the finger in the other direction (just remember when you point your finger the other way, all the other fingers on that hand are pointing back at you ;-)

The context here has to do with lack of awareness. When people dismiss that possibility simply because they weren't aware of it, or haven't made the attempt to explore it, or haven't looked beyond their own personal usage/viewpoint -- then they are all too frequently jumping to conclusions based on lack of information rather than the presence of it. It may be that the conclusion was correct after all, but that doesn't make it any less premature at that time, before it was adequately explored.
Psychological experiments by Ward Edwards and others have shown that humans are generally very conservative when cumulating evidence toward probabilistic results. For example, they generally estimate the probability that the coin is biased as much lower than Bayes' Theorem would indicate.

This is not evidence, of course, that when a person does draw a correct conclusion before we're ready, that they are in fact merely closer to veridical estimation than we are. However, Edwards' results do call into question our objections when there is some evidence, but we aren't personally ready to generalize.

On the other hand, I'm not aware of any research addressing people's ability to instantly [over-]generalize from zero or one cases. We're a weird species ... -- RonJeffries

An infinite number of lines go through one point, two points determine a line, three points landing on the same line is amazing. I can't imagine why this applies to people, but I notice people (myself included) immediately drawing a conclusion from a sample population of 1 - all different conclusions, of course. If they see two samples, then they announce the conclusion as "certain" or obvious. Three samples make it irrevocable. -- Alistair
For a skewed but related notion, see CoinTossingBof.
"1000 to 1 shots come up 9 times out of 10" - From TerryPratchett's DiscWorld novels.

It's "1,000,000 to 1 shots". And it has to be one million. If it's more likely than that, do something to make it worse (like blindfold yourself and stand on one leg when shooting the dragon - see "Guards, guards").

The only problem is, it's not true. In fact, million-to-one chances come up ten times out of ten. The trick is knowing which million-to-one chance is going to come up...
This page is full of StrawMan arguments, so I'm listing it in the FallaciousArguments.

Which arguments? How about the misstatement at the top, that AOEINEOA "means that if we don't know that something exists, it doesn't mean that it doesn't" -- it doesn't mean that at all. The latter is true, but AOEINEOA is false.

The "simply put" is simply mistaken; it doesn't mean that at all. The paraphrase is certainly true, and expresses the fallaciousness of argumentum ad ignorantiam -- absence of a proof is not proof of absence. But AOEINEOA says something different entirely; it is quite wrong, and indicates a misunderstanding of evidence and empirical science. Absence of evidence certainly is no proof of absence -- there aren't any empirical proofs in the mathematical sense -- but it is evidence of absence. That is, it does not falsify absence, and in fact gives reason to suspect absence. Just how good evidence of absence it is depends on how hard evidence of presence was sought. If we try really really hard to find evidence of life on Mars, using every technique we can think of, and yet we fail to do so, this is strong evidence that there is no life on Mars.

It is evidence of absence, true, but very, very weak evidence. To become a proof, one must exhaustively search all possible locations and establish absence in each. This is only rarely feasible, though it does happen. In those cases, AOEINEOA is a fallacy. Far more often, though, absence is established in some vanishing fraction of the possible locations of some phenomenon and that is held up as a proof. So the vast majority of times this saying is trotted out, the point holds. Compare "evidence of existence", where one good piece of evidence establishes the point beyond all doubt.

This misses 99% of the point I made, as it talks repeatedly about proof, which is not the subject at hand. A fine example is William Safire's claim that absence of evidence of WMD's in Iraq isn't evidence of absence -- he's just plain wrong, and the evidence isn't "very very weak", in fact it's very strong, because of the effort to find evidence and the eagerness of the searchers to find it. That I can't find any elephants in my living room is overwhelming evidence that there are none because they ought to be easy to find.

[Not quite the same. You can argue that you checked all possible locations of elephants in your living room, with no exceptions whatsoever, due to the size ratio. The most you can say about a country is that (for the sake of the argument) "all reasonable locations have been checked", or some such. Countries are too big to literally eliminate 100% of the possible locations, unlike your living room.]

I don't want to argue that I checked all possible locations because that's not my point. We're talking about evidence, not proof. Talk about "all", "100%", and "possible" is always talk about proof. Evidence is simply something that contributes to belief. Absence of positive evidence of elephants is a reason to believe there are no elephants, because of the obviousness of elephants. In the absence of reason to believe there are elephants in your living room, I have reason to believe there are no elephants there.

[And even in your living room, it's not the absence of reports of elephants that is evidence of something, it's the positive fact that you reported that you searched and found nothing that provides evidence. If on the other hand we just said "we haven't heard anything at all about elephants in that living room (but we don't know if anyone even checked)", that would be true "absence of evidence".]

This is just naysaying. Absence of reports of elephants is evidence of something, because of the meaning of the word evidence. I already made this point -- that this is a matter of empirical science; this is a fundamental issue in philosophy of science, and is discussed in any good text on the subject. And I already wrote "Just how good evidence of absence it is depends on how hard evidence of presence was sought", so you're repeating what I said with your talk of reports, but with less clarity. To add to that, it isn't just how hard evidence of presence was sought, but all sorts of other factors concerning the nature elephants, living rooms, or whatever. Absence of evidence of men with 25 penises is evidence of absence of men with 25 penises, even if no one has ever thought to look for one -- because if there were one, we would have been likely to hear of it. Evidence, unlike proof, is statistical in nature, and it is cognitive and subjective -- "a thing or things helpful in forming a conclusion or judgment", says the dictionary. Unfortunately your discussion, and much of the rest of the discussion of this subject, blithely ignores the meaning of the word. And there's no excuse for it, as I already wrote "[absence of evidence] does not falsify absence, and in fact gives reason to suspect absence". Reason to suspect absence is evidence of absence, by the meaning of the word evidence. People seem to want the notion of evidence to be stronger than it is. They don't like the fact that there can be evidence for claims that aren't true. But absence of evidence for P is evidence of (reason to believe) the absence of P, whatever P is, which is why we generally disbelieve things for which there is no evidence, rather than being purely agnostic about them.
The case for evidence of absence depends upon whether or not evidence of any kind exists. If none exists, then absence of evidence is neither evidence of absence or of existence.
A good example proving this statement is the usual discussion I get into around NonFunctionalRequirements in software specifications. Many people have no idea what an NFR is, and even when they know what it is, have no idea how to elicit them. However, NFRs always exist and will usually be discovered when it's most expensive and difficult to meet them. -- PhilStubbington
Absence of evidence certainly is evidence of absence. It's simply a matter of deciding whether it's very weak evidence or something stronger. Clearly it is not proof. -- DavidPlumpton

Absence of evidence is only "evidence" of absence if one has gone about making a reasonable attempt to gather such evidence (most likely beyond simply their own immediately recollectable experience). In that case, one has actually made the effort to look and found no evidence -- they have found evidence of the absence of evidence. Otherwise, there is no known absence of evidence, only ignorance of evidence or ignorance of the absence of evidence.

The statement at the top of this page is talking about the cases where someone simply "off the cuff" says something like "well, I've never seen it" or "I haven't seen it in my experience." That's a far cry different from making a concerted effort across numerous sources too look for it and then coming up empty handed. It's little more than an illegitimate excuse to keep a closed-mind closed.

The real problem being addressed by the (perhaps oxymoronic, or just plain moronic :-) phrase at the top of this page is that IgnoranceOfEvidenceIsNotAbsenceOfEvidence. The phrase "AbsenceOfEvidenceIsNotEvidenceOfAbsence" may in fact be logically incorrect according to the rules of empirical science. However, its intended meaning (from both the original source and others on this Wiki that use it) is indeed what the top of this page says: Lack of knowledge/experience/awareness of something (which is a polite way of saying "ignorance") does not imply lack of existness/correctness. Ignorance of evidence is not the same as the "absence" of evidence, and therefore Ignorance of evidence is not evidence of absence.

Even if absence of evidence really is evidence of absence, ignorance of evidence is neither one of those things and shouldn't be mistaken to imply either one. Ignorance of evidence is evidence of ignorance (no, more and no less). -- BradAppleton
However, absence of evidence is certainly not evidence of existence. If one wants to prove something, he must provide some justification.
The straw man arguments point above is the best comment here. The Absence of Evidence argument is NOT argument FOR a phenomenon per se, it only refutes the argument the phenomenon cannot exist (unless you can circumscribe all possible occurrences, which is extremely rare). Occam's Razor is useful but not infallible. Buying a lottery ticket a hundred thousand times does not prove a lottery cannot be won, it at best proves the statistical unlikelihood. Annoying to some, but a critical point in fields in which evidence may be either hard to come by or even methodically suppressed.
This page is quote mining Carl Sagan. Carl Sagan didn't support that proposition, he criticized it. Here's those words in context:

* appeal to ignorance -- the claim that whatever has not been proved false must be true, and vice versa (e.g. There is no compelling evidence that UFOs are not visiting the Earth; therefore UFOs exist -- and there is intelligent life elsewhere in the universe. Or: There may be seventy kazillion other worlds, but not one is known to have the moral advancement of the Earth, so we're still central to the Universe.) This impatience with ambiguity can be criticized in the phrase: absence of evidence is not evidence of absence."

--Carl Sagan, "The Fine Art of Baloney Detection" (collected in "The Demon-Haunted World: Science as a Candle in the Dark")

Absence of evidence is always evidence of absence.

• No... that isn't true unless you both searched for evidence and failed to find it where it should have been found. If you never looked for evidence, or if you wouldn't have found it even if you looked (bad measurement tools for the problem), then the absence of evidence isn't evidence of anything. And that's presuming honesty. If you allow dishonesty and incentives towards burying a truth, absence of evidence might also mean that someone destroyed the evidence - not that an accusation should be leveled unless one can find evidence for the intentional destruction of evidence.

A more defensible statement would be: Absence of proof is not proof of absence.

• Well, that's true insofar as 'proof' is stronger than 'evidence', but isn't nearly so useful.

Let H be a hypothesis that something is true, and E be an observation of evidence supporting H. The probability of H given E is P(H|E), and the probability of H given not-E is P(H|~E).

If P(H|E) > P(H) then conversely P(H|~E) < P(H). By Bayes' theorem, E increases our estimate of P(H) and ~E lowers our estimate of P(H).

Absence of evidence is indeed evidence of absence, always, though that evidence may be strong or very weak. How strong it is depends on how likely E is given H. If you don't search for evidence and don't observe any, that's obviously much weaker than if you had searched.
I can accept the Bayesian formulation above, but there is a stronger form which may be more correct and less dependent on the choice of lower bound of P(H):

Absence of evidence is not evidence at all. This results in no conclusion at all.

If P(H) is chosen low enough then for some bound epsilon they are the same. But some may not pick P(H) well, or they may pick it differently. Is P(god exists) high or low -- discuss ( but not here make a new page :) )

-- MarcGrundfest
Just chiming in my two cents. Not trying to prove or disprove anyone. Simply trying to understand better. What is wrong with the following example: early humans did not have evidence/proof that Earth is a spheroid. Hence they lived on a non-spheroid planet? Say flat piece of rock? Modern humans did find evidence/proof that Earth is a spheroid. Hence we believe/know that we live on a round planet. Hence sometime in the past, say, 1000 years, Earth switched from flat piece of rock to a spheroid? What about some primitive tribes nowadays that have no evidence/proof to what planet are they on. Does it mean we share with them a flat and round planet at the same time? Help!!!!!

Putting aside the vast factual problems regarding the alleged beliefs of `early humans`, I would point out that said humans also did not have evidence/proof that the Earth is a flat piece of rock.

{If you extrapolate what it looks like up close, you'd say it's generally "flat".}

That is unlikely. Though I'm sure some loud cranks did make such an extrapolation.

{I don't see any reason why some other shape would be a more popular guess than "flat" (or a variation of). What other wrong guesses would you expect based on "looking around outside"? Some may indeed guess that Earth is "round like the moon and sun". Complicating that is that they may interpret them as disks, not spheres, "confirming" flatness. But without further evidence we couldn't say for sure and are just speculating. And human imagination tends to be all over the map:}

No guess is necessary. The curvature of the Earth has been known for a long time, and is readily observable to those who look, think, and travel. Sailors can easily see how the Earth curves away in all directions. Some may have thought it curved like a turtle's shell. Yet, for thousands of years it has been known and taught (with the logic and abundant evidence) that the Earth is spherical, and even a decent estimate for its diameter (Erastothenes ~200BCE).

Yes, the Greeks figured it out by studying shadows and comparing notes, but they are the exception. And ship masts were generally not tall enough to make the "ship effect" clear until fairly recently.
• The "ship effect" isn't necessary - the same thing happens for land masses, mountains, etc. And I don't believe your claim about the Greek being the exception. They just happen to be where `Western Civilization` traces its understanding. The `popular` idea today seems to be that the flat earth hypothesis was popular when Columbus sailed.

{Can sailors really see that the Earth curves or is this just our eyes not being able to focus at long distances. If the earth was flat could sailors even see things at a far distance anyway? I think not - things would go way out of focus and disappear.}
• Things might be fuzzy, but you'd certainly be able to see large structures - mountains, for example - at distances proportional to their size. This hypothesis is easily justified - climb a mountain and you will see large structures at a greater distance, therefore it is not the distance alone that is hiding the structures. Instead, if you sail away from a mountain, you'll see it sink below your view of the ocean - the curvature blocks the view. Same if you walk away from it through plains, but forest and hills might confuse you.

On a clear day, tall ships appear to "sink" at a distance, as if half is under water. However, it may be hard to interpret what is going on, similar to heat mirages in the desert where the surface appears to act mirror-like.

Yes, you'd certainly need to think to interpret. And a lot of people will never take the time to think about it. But people from thousands of years ago seem to be no less intelligent than people today, and just as willing or unwilling to think about things.

• Yes. Extending the reach and intelligence (information processing, knowledge gathering, external memory) of humans is a worthy goal for technology. Yet, when people search the Internet, they still seek to confirm their opinions rather than judge them. They still are subject to cherry-picking and a hundred cognitive biases. Perhaps we can augment humans in these roles, too, by automatically measuring consistency in a MindMap or similar, software for presenting sound arguments (up to premises, with confidences). But it is unclear how popular would be software that rejects your ideas as fallacious - I expect it would be preaching to the small crowd of rationalists.
• Yes, the internet is a DoubleEdgedSword - similar to how some people read something in a book and they think it is true (or read it in a newspaper)

But there are many different conclusions to draw. A "round Earth" is only one. And who's to say that water wouldn't have similar mirages as deserts (or hot days) for items near the horizon?

Indeed, one might not conclude `sphere` from noting that the Earth curves away. One might instead think `curved, like the back of a turtle` and think there are still edges out there (lacking evidence for either). But there is no evidence to support a flat Earth. Granted, many cultures believed in flat Earth, but that isn't the result of extrapolating from evidence. (Nor are prejudice, superstition, religion, cold fusion, the benefits of OOP... humans like to fit evidence into their world views, rather than vice versa.)

I meant more along the line of if heat mirages can blur the bottom of objects at a distance, then it's not unrealistic to assume some property of the ocean may be similar optics-wise. (Maybe "mirage" is the wrong word for it.)

Perhaps you should look at the ocean, or take a cruise, rather than hand-wave about what assumptions might not be unrealistic. Unless you're dealing with fog or polluted haze, you can generally get crisp lines off the ocean.

I don't see how crispness level would make a significant difference. The ancient man couldn't explain heat mirages such that crispness makes little difference. If you can't explain a non-crisp phenomenon, the addition of crispness won't change that. Both phenomenon are still similar enough to suggest similar sources.

What is similar about these phenomenon? You seem to be stretching. Perhaps you can find some evidence that ancient man was likely confused by the possibility that a horizon is somehow like a heat mirage, rather than speculating on the matter. Mere speculation makes a poor counterpoint.

Further, even if one does conclude the ocean can be "curved", it could merely be lumpiness or local curvature. "Water mountains", for example. And more, ancients tended to sail close to the coast, lacking advanced navigation. Thus, the opportunity to study ships in open waters was limited.

It doesn't take open water. You can observe the Earth's curvature anywhere with a clear view of a flat horizon. It's just easier to achieve such views on the ocean or from a high point with large visibility in all directions. Climbing does effectively counter the notion that our eyes simply can't see large, distant objects.

I've never seen clear evidence of such, and I've been to Pike's Peak and Mauna Kea on a clear day. Without measurable reference points, flat and not flat are hard to distinguish (except in the extreme), even from a passenger jet.

Flat is relative, of course, but even rolling hills will appear on the curve of the horizon from a high enough vantage point. The fact that a horizon exists at all is a measurable reference point. The fact that you can see more distant objects (i.e. have a larger horizon) when you climb or fly a jet is a measurable reference point. You've seen the evidence; you've only failed to understand and apply it.

I am not understanding you. I saw no clear signs of a round Earth. The horizon is not perceived as arched to the naked eye, and without repetitious uniformity, one cannot judge the proportions of distant objects to a useful degree. Stereo vision only has a usable range of a few thousand feet. Without sufficient indication of the scale of distant objects, a flat Earth and a non-flat Earth would not be different enough to make a clear determination (and ancient man had no luxury of simulations). Further, it's hard to know why you are not seeing what you are not seeing. A distant mountain not seen because the Earth is round is just as invisible as one hidden by distant fog or haze.

Arched? No, on a flat surface the horizon appears as a flat circle to the naked eye - just as it would to a Lego man standing on a soccer ball. I casually walked outside and looked at the ocean to verify this. And you can distinguish `hidden by haze` with `hidden by Earth's curvature` by waiting for a clear day or moving to a higher vantage point. If you did not "see" the signs, go to an eye doctor. Seriously. Stereo vision isn't necessary to these observations, but you do need to observe and think. If you lack understanding, or cannot "recognize" the signs, or did not bother thinking about them - even AFTER knowing what to look for - that is just your own idiocy, which is entirely unsurprising and quite typical of you, TopMind.

I guess I am just blind, stupid, and dumb. Bummer. -t

If only I could believe that. I suspect it's a lie. You're too arrogant to ever seriously consider your particular brand of idiocy.

Whatever, Mr. Columbus. You keep flip-flopping on what roundness looks like from mountain.

I have not. I did presume you know what a horizon looks like, such that I wouldn't need to describe it in much detail. But that was clearly expecting too much of you. ("arched"?) Whatever. You aren't a worthy partner for further argument.

The feeling's mutual.

Indeed so. For you, a worthy peer would be another hypocritical, arrogant ignoramus who thumbs his nose at academia and substitutes education and investigation for speculation.

I challenge somebody to tell me what in this pic clearly demonstrates the Earth is round. (The horizon is curved a bit, but it could be the optics.) If you don't like this example, you are welcome to present another. -t

Take multiple pictures, not one. If you take pics in every direction and stitch them together, you'll have a very clear circle for the horizon (so no, that is NOT a trick of optics, you moron). If you move and take pics, you'll see objects moving in and out of view that were previously hidden `over` the horizon - i.e. by the curvature of the Earth. Your parenthetical comment is exactly the sort of speculation I called you on before.

Ancient man wouldn't have the luxury of moving so readily.

Walking distances are entirely sufficient. Standing at ground level, horizon for a man is about 3 miles. This easily rises to 7 miles if they climb a moderate hill or tree. Even seven miles is much smaller than the trade routes, nomadic routes, hunting routes, of even a typical farming route of ancient man. Can you stop claiming stupid things? I mean, you weren't born stupid - for you, it's obviously by choice and habit.

• YOU are the one claiming stupid things. One would have to make careful sketches, and then compare them after walking several more miles. Further, unless the inspected objects are directly ahead or behind the travel route, their shape will change because one is looking at them from a different angle, making it a "non-controlled" test. In other words, the visual changes caused by change in horizontal angle overrides or "pollutes" any changes due to distance. Yes, it's possible to take such efforts and control for viewing angle etc, but only an extremely small percentage of the population would take on such an endeavor and find the right circumstances.
• I'm not suggesting someone could precisely measure the diameter of Earth by walking and taking notes. The evidence I speak of is recognizing that the Earth is curving away no matter which direction they walk in, and that `haze` doesn't nearly account for why distant mountains are hidden. That doesn't take careful sketches or controlled tests. It doesn't even require an above-average memory. You seek invalid excuses to ignore valid evidence. Very typical of you.
• How does one know a mountain exists if it's hidden? A tall mountain may also look like a shorter mountain, but unless you measure, you don't know actual size.
• One can know a mountain exists by seeing it before walking out of view; it isn't as though the mountain is going to move on you. And if you're confused whether the mountain is shrinking due to distance vs. curvature - simply remember some features near the base, middle, and top of the mountain (plant some if necessary). You don't need to know size.

```         A -------- B -------- C:
```
• Suppose you are at "C" (colon is eyeballs) looking back at mountain A whose top appears just barely above mountain B. You know mt. A is much higher than mt. B because you climed both. However, because the Earth is round, mt. B appears lower than we would expect. But one could perhaps also speculate that mt. B is on a subtle bump such that it looks shorter than it really is when one is on it. In other words, the low-lands surrounding B may not be that low after all. Vegetation etc. may suggest otherwise, but climate varies per region, and any experienced traveler will know this. In practice mountains and terrain are difficult to get an accurate 3D mental map of just by looking because one does not travel in strait lines among mountains. They are not "clean" cardboard cutouts like one would find in a 2nd-grade model. I'm not saying one could not surmise roundness with enough observations, but the evidence is baked into many other factors. The observant Greeks found a less subtler approach: shadows.
• It seems you've lost track of the problem, TopMind. The goal was evidence that the Earth is round (or at least not flat), NOT an accurate estimate of diameter. Greeks spent at least 300 years observing that the Earth was round before Erastothenes figured out a good way to estimate its diameter (with shadows). Accurate maps and mental models and all that rubbish are relevant only if you confuse the problems of `surmising roundness` with `computing diameter`. To surmise roundness, it is sufficient to observe that stuff is over the horizon in every direction, no matter where you stand.
• Analyzing shadows at different latitudes is one way to detect that the earth is round. It can ALSO be used to measure diameter with some extra steps. And your mountain-top observation claim is bogus because there are too many variables influencing one's view, as already described. Removing these variables requires both positional luck and a lot of planning. It may be possible, but not superior to the shadow approach.
• Studying shadows was not, historically, used to determine that the Earth is round. And your argument about variables for mountain-top observation is analogous to arguing you can't surmise the curve of a woman's body because clothing might interfere. One doesn't need to remove the variables (or the clothes), or rely on positional planning and getting lucky. You only to estimate and account for the variables, or choose a better sample. A distinct tree, rock outcropping, or city wall would serve just as well as a mountain. It does help to control for a few variables. Bays, seas, and dry lakebeds offer the best control over height of the observer, for example. But such control isn't essential to surmise roundness. It's sufficient to note we have a horizon in every direction no matter which hill or mountain we climb, even if we don't know our exact height. Anyhow, it's way past time to SetTheBozoBit for you. Congrats on ruining another page by filling more than half of it with your foolishness and fallacy and off-topic pictures, TopMind.
• A tree? You are kidding, right? How about UFO's, do they work?, you fucking mental case.
• Yes, a tree - but a distinct one, as I said. Unlike flying objects, trees don't move on you.
• Even if one could find a place with the right conditions to pull it off, one would have no direct way to know if a "flat" surface, such as a dry lake bed, matches the curve (or lack of) of the Earth. You know you are testing the dry lake bed, but that's not necessarily the same as testing Earth in general. It's the same issue with potential subtle long-period elevation variations described above. If one sampled many, perhaps they could begin to rule that out, but it's no longer a casual observation, but a concerted and costly plan (by their standards).
• It is not so costly if the observations are carried out in parallel with other tasks that call for travel. In any case, the evidence of curvature is there. And, if it weren't for your double standards, you should be arguing that a similar effort would be required to surmise the Earth is flat - minus the good evidence for it.
• No, the evidence is NOT there. You are wrong. It can be teased out with a lot of effort to the point where the Greek shadow experiment seems like the simpler path. And it's easier to explain to your buddies.
• The evidence of the Earth's roundness is abundant, and visible to the naked eye. It doesn't take much effort to see it, but it does take some thinking to understand it. The Greek shadow experiment was to find the diameter, not to learn the Earth is round.
• The Greek shadow experiment can be used for both. And I disagree observation of the horizon is an easy way to tell.
• The horizon provides plenty evidence for the sharp-eyed rational travelers. Hint: the knowledge of Earth's roundness came at the same time as early math and logic were invented by the Greeks, around three centuries earlier than Erastothenes and the shadow experiment. Perhaps you cannot recognize the evidence because your competence with logic is UnskilledAndUnawareOfIt. The shadow experiment is not easy to perform. It requires making detailed, time-coordinated observations at hundreds of miles distance, and also requires a working knowledge of trigonometry.
• I haven't seen any clear evidence that any ancients concluded the Earth was round by observing the horizon from land areas. (Their historical evidence is pole shadows at diff latitudes, lunar eclipse shadows, star positions, and later sea observations[1].). You are welcome to present such evidence. Nor have you presented a step-by-step observation and logic process that rules out the mentioned caveats. It appears you just ignore them without any detail comment: "Your caveats don't matter because you are stupid" (Paraphrased). I don't see details; I don't see numbered steps; I only see insults. Kill me with details; kill me with logic; kill me with StepwiseRefinement; not insults. -t
• Your caveats aren't well enough justified or demonstrated relevant to deserve serious treatment; for example, I have no idea what you planned to do by trying to observe the heights of two mountains (as opposed to one mountain against a horizon). I have not argued that the Greeks made their conclusion from observing the land. (I have argued that such evidence is available even on land.) I wouldn't kill you even if you begged for it, but I'd happily sentence you to life without Internet access.
• I agree that some evidence can be gained from land-only horizon observations. But that by itself is not enough to make a fairly definitive claim unless the observations were numerous and rigorous. You implied that land-horizon-only observations provided clear and easy evidence. -t
• I said that land-horizon-only observation is sufficient to `surmise` roundness. You were the first to use the word; perhaps you should it look up in a dictionary. For a definitive claim, I agree that I'd want some high-quality evidence, too, such as star charts and sea horizons. But sea horizons are also available from land - bays and peninsulas, for example.
• "Surmise" is a fuzzy word. We should have avoided it. (If I used it first, I apologize.) If you mean horizon observations are sufficient to generate a hypothesis that the Earth is round, we are in agreement. But your strong wording implied that horizon observations alone were enough to make roundness the top candidate model. -t
• Make enough horizon observations, from enough locations, and roundness will be the only falsifiable hypothesis left. The Earth is round, after all, and all valid paths towards truth will converge. But land-horizon-only certainly wouldn't be ideal or the most efficient.
• Well, I generally agree with that. You initially made it sounds like "a snap" to see obvious roundness.
• It's "a snap" to see evidence of roundness. It's all around us. If you get confused about evidence vs. proof, you're really really stupid.
• One has to be familiar with the objects observed. How do I know the bottom is "chopped off" if I don't know what the bottom looks like?
• Get familiar.
• From multiple angles/distances? You are MovingGoalPosts.
• My very first response to your challenge told you to take more pictures, more angles and positions. Did you miss it? Your challenge was the attempt to move goal posts. I spoke of travelers and horizons of 3-20 miles, within capability of a sharp human eye on a clear day. You gave me single still picture with a narrow snapshot of a horizon from a plane with a horizon of at least a hundred miles.
• This didn't start out being about airplanes and cameras, but about what "ancient man" may observe. The airplane shot was merely to make it clear we were "up high" in the example.
• Ancient man can walk, turn his head, gather more evidence from better places. Hence, multiple angles and distances isn't MovingGoalPosts. Asking for a demonstration from a single picture IS MovingGoalPosts. You're the bad guy here.
• Sure, with enough time and travel under one's belt. You implied it was obviously and casually available.
• I have said that evidence of curvature is obviously and casually available. Don't confuse evidence with proof. Don't confuse evidence of curvature with evidence of spherical. Communication requires making distinctions, and recognizing when other people make them.
• It's not casually available. One has to first find a way to build a reasonably accurate 3D model in their head of several mountains. On the Silk Road people mostly followed the silk road rather than dance around the landscape like a Discovery Channel billionaire. Life was hard and risky back then: you focused on your goal and didn't take big risks most of the time because there was no 911. Most travel routes were between the larger mountains, not over the top such that your view of the distance is largely obscured.
• The evidence is casually available - it's visible to the naked eye, unlike fingerprints, and even of high quality in many places humans travel (plains, beaches, bays, peninsulas, coasts and seas). And stars or shadows, of course, though those take a lot more travel. Accurate mental maps aren't necessary to see the evidence, but even if they were, they aren't difficult to form if you've traveled a path a few times (and most travel was routine, cyclic, even in ancient times). You're HandWaving invalid excuses into existence. It reflects on your intelligence.
• I don't change my mind based on insult levels. Personal insults are a commodity on the web, and thus of no use or value. You are HandWaving about the allegedly simplicity. Most mountainous routes wind and twist. One mostly uses landmarks, not a big-picture mental 3D map to figure out where they are. It's pattern matching, not absolute mapping. Even if you have a map, the actual route rarely "feels" like the shape on the map. It's natural to mentally exaggerate the length of a rough, rocky part of the path, for example, because it "feels" longer due to the difficulty. The mind uses lots of little UsefulLies like this. The mental vision of extra length is a filler for terrain that takes longer due to texture.
• Landmarks and pattern matching are all that is necessary to see the evidence. Which is why I keep saying: we don't need accurate 3D maps. If you want to argue that 3D maps are difficult to form in most mountainous regions, I really don't care - it's irrelevant to any point I've made. For my arguments, a mountain would serve as a landmark, or as a platform to see large distant landmarks (like other mountains) at the edges and foothills of a mountainous region. And I know you're just as immune to insults as you are to logic, which is why I feel free to send them your way without feeling even slightly guilty about it.
• I told you how to present true logic if you have true logic. You ignore that pattern and just keep making claims and insults. I cannot read your flucking mind. If logic is in there, it's still hiding. Serialize the shit into discrete numbered steps.

And one does NOT see a "curve" from a plane via human eye. Whether the photo curve is an artifact or not irrelevant because we are not testing cameras here.

Sure they do. Humans have a much broader peripheral view than the photo offers. They also have flexibility to turn their heads a little. It's easy, with the naked eye, to see that the horizon is circular - i.e. that there is a horizon in every direction, and thus there is a curve of the horizon. From a plane, of course, it will be a very big circle. (If you get high enough, the horizon becomes half the Earth.)
• Perhaps, but people generally need reference objects to compare, and are easily swayed by context (aka optical illusions). Anyhow, without some kind of controlled test, this is all just ArgumentFromAuthority.
• Ah, I see. Any empirical evidence TopMind doesn't like is ArgumentFromAuthority. And unjustified speculation (e.g. about optical illusions) is sufficient cause to ignore evidence. These properties combine well with selective blindness to make a truly impenetrable shell around Top's mind. Well, I don't expect rational reasoning from the #1 most active WikiZen, so I'm no more disappointed than I expected to be.
• You haven't talked about empirical evidence, only "I know it when I see it."
• Evidence doesn't depend on you recognizing or acknowledging it. If someone left fingerprints at the scene of a crime, and you think fingerprints are optical illusions and dismiss them, they're still evidence to people who aren't idiots like yourself. Plenty evidence of the curvature of the Earth is readily accessible to those who know to look for it.
• Your sanity is an optical illusion.
• Your stupidity is plainly self-evident.
• Projection. Go take your pills, now.
• Again, you look for excuses to never look too closely at yourself. Pathetic.
• Projection.
• WishfulThinking. Pathetic WishfulThinking. If you really think I'm projecting, provide evidence.
• Do I have your permission to smash your thick skull in and take the bloody neurons down to the lab?
• No. Shall I accept this as admission that you do not already possess evidence to claim I'm projecting?
• Just like all the accusations you made about me that assumes you know my internal motivations. Touche.
• I've not assumed any internal motivations. I don't know why you've developed the habits of stupidity, though I think you're really foolish for doing so. I've only noted your evident stupidity. (It won't be evident to you, of course, since `know thyself` is a subject at which you evidently fail spectacularly.) Your claim that I assumed internal motivations is another fine example of your stupidity.
• You accused me of lying. I think you are a stupid, stubborn, vague, idiot, and you think I am a stupid, stubborn, vague, idiot. Let's not argue about who is stupider any more. I won't get anywhere.
• I accused you of claiming something that you lack evidence to claim. AssumeStupidityNotMalice. Are you suggesting you're a liar instead? or also?
• I don't know what you are talking about, but I don't care anymore. You haven't described in a clear way how to remove the other variables mentioned.

As far as your speculation comment, I don't know what you are talking about. Improve your writing skills to avoid being obtuse.

So either you don't know when you're speculating, or you don't know the meaning of speculation. Hmm.

No wonder this page is in the rant mode. Wow! What a load of crap! Who cares about all this nonsense! The point was to use a simple model to discuss math.

Anyway. People WITH a brain. That one is for you. You secretly plant an Easter egg in John's house. Then you ask him "John can you find an Easter egg in your house?" John looks for Easter egg in his house and can't find it. John comes out and says "I don't have any evidence that the Easter egg is in the house." Question. What does that mean? If absence of evidence is evidence of absence then Easter egg is not in the house, as far as John is concerned? But you, and other witnesses, do know that Easter egg IS in the house. So what is it?

John meant just what he said. He did not say: "I have proven there is no Easter egg in the house." John does have reason to believe there is not an Easter egg (with confidence proportional to the quality and coverage of his search), but he also has reason to suspect you might have hidden one (based on your challenge). I note that your own information might be wrong - the wily witness standing behind you may have secretly moved or removed the Easter egg in the time between you planting it and John searching for it. If I were John, I'd also be tempted to move the Easter egg then lie about not finding it.

In any case, if you're going to complain about nonsense and crap, you probably should stop contributing to it by asking FalseDichotomy questions. Go learn about hidden Markov models and Kalman filters and Bayesian probability if you're interested in some cognitive and computational aspects of working with limited data in an open world.

Sadly, this page has degenerated into RantMode. Therefore, it goes into that category until it can be cleaned up at a later date.
FOOTNOTES

[1] I don't consider the "half ship" or "half island" observations sufficient evidence by itself of a round Earth because one cannot rule out large but subtle waves that raise up to block the bottom portions of distant views. One could surmise that such large "lumps" dissipate closer to shore such that they are not detectable at shore. Without the math of fluid dynamics or positional sea-bottom measurements, ancients couldn't prove or disprove the "subtle lump" theory of why part of the view may be blocked. While I consider "half view" to be evidence that the Earth is round, it is not high-certainty evidence by itself, due to the caveats. It's a weight on the evidence scale, but by itself does not tip the scale. It appears the Greeks added up multiple bits of evidence, especially in terms of astronomical observations. (Tides are actually such "subtle lumps", but their wave length is few magnitudes larger than the target speculative phenomena.) -t

Speculating about possibilities that `one cannot rule out` is still your first resort. But in science and logic, we don't do that. It would be difficult to `rule out` godly interference of our perception, for example. Instead, we ask you to provide some evidence to rule it in. You seem to think you know something about science, but your argument (like most of your arguments) evidences nothing positive about your competence with it.

• I don't think you are understanding: there are (at least) two plausible explanations of the "chopped island": 1) round Earth, 2) large subtle waves. Why should #2 just be ruled out? The ancient human would need a reason to rule it out, no?
• There is a third plausible (to ancient humans) explanation of the "chopped island": 3) godly interference of perception. Why did you rule #3 out as plausible? The ancient human would need a reason to rule it out, right? Sigh. Your entire concept of logic and science is FuBar, TopMind. I need evidence that large, subtle waves are causing the problem before I treat it as a possibility - OccamsRazor. But if I did need to rule it out, I'd ask why the boat isn't rising and falling with these proposed `subtle waves`. If I'm at the peek of the wave, I should see the whole island, right?.
• I am not sure what you point you are trying to make by bringing in the supernatural. "God did it" can explain anything and everything if we allow it.
• Your argument for the subtle wave theory was made without evidence. You called it `plausible` without evidence. That puts it in the same category as `god did it`. If you want to provide some evidence, go for it.
• Neither model has evidence for it at this point (per scenario).
• Why not?
• I agreed with the "multiple sources of evidence" stance. That's not the original premise, though.
• How does that answer my question?
• Perhaps you should flesh out "why not?".
• Why do the models not have evidence for them at this point?
• We are considering only horizon-based observations. Is this what you meant? Not astronomical observations. Your original "are you blind?" statements did not assume astronomical observations. If we rule out astronomical and shadow observations, then we have pretty much virgin models for virgin observations.
• I did not assume astronomical observations (it wasn't a necessary assumption for me), but I also do not rule them out (indeed, they're another fine source of evidence). Why would you?
• Because that was not the original premise.
• I suspect you meant to say that there is no evidence to distinguish the two models, which is quite different than saying neither model has evidence (which I found baffling). Presumably, astronomical evidence could distinguish them. But there's no reason to bother with more than one model until you have evidence the first model can't explain. (Models aren't about truth or explanation; they're about prediction; these just happen to coincide a lot, modulo quantum theory and a few others.) Pick one model and run with it.
• For now, let's put astronomical and shadow observations away in the bottom drawer. You seem to be saying the bulge model is somehow worse or inferior to the sphere model, and used a supernatural reference to dismiss the bulge model. I don't see any connection to the supernatural here, nor an OccamsRazor problem.
• [What exactly is the bulge model, where is it described? I searched this page for bulge theory and bulge model and there isn't much referencing exactly what the theory is.]
• My `supernatural reference` regarded your argument for the bulge model, not the model itself: "I don't consider BLAH because one cannot rule out FOO". Unless you have evidence to support FOO, it's a violation of OccamsRazor - i.e. you're introducing an unnecessary assumption. It's just another way of asking "What if" without actually checking. Who knows whether the Earth is really round - what if we could see in more than three dimensions? what then? In Science, there is a reason the incumbent model wins by default. Mind and textbook overhaul economics. So pick one model to be incumbent - it doesn't matter which - and start from there. Stop saying `what if FOO` unless you're ready to search for FOO.
• Re: "I don't consider BLAH because one cannot rule out FOO". A better representation would be: "I don't consider BLAH sufficient evidence to make a conclusion because one has to first rule out FOO". And, the incumbent model was "Earth is flat".
• No, just include the "sufficient evidence to make a conclusion" as part of BLAH. Anyhow, if you start with "Earth is flat" you'll quickly move to your bulge theory or something else based on observing the sea horizon, then discard bulge theory for the land horizon, QED. At least if you were a rational thinker who cared about that subject, which most people aren't.
• Please clarify your revised FOO statement. As far as "most people", most THINK they themselves are rational. The human ego is very powerful and blinding. If you truly think it's sure-shot logic, then use numbered semi-formal logic statements with each step's source clearly described. Otherwise, few will just outright believe you. A "safe" default would be that the Earth's shape is "unknown". Wikipedia suggests that prevailing opinion of the day was "flat". In other words, when the subject came up in ancient writings, flat seems to be the most common. And when it was debated back then, it was usually flat versus something else (such as flat versus cylinder). Superficially, most seem to agree that the Earth "looks" flat. It's the model that best explains the vast majority of what we see. -t
• Just because you have this fascinating idea in your mind of what logic looks like doesn't mean anyone else agrees with you. And, yes, prevailing opinion was that the Earth is `flat` right up until Greeks invented the earliest versions of classical logic. Funny coincidence, that.
• The Greeks used at least three different kinds of evidence together. Multiple kinds of evidence backing the same model makes a much stronger case. If you don't want to serialize your logic, then go ahead and be vague and obtuse. You don't have any logic, just puffery. That's why you can't serialize it. Talky Flunky. Fail. No wonder you believe such weird shit: your tolerance for low-grade sloppy logic presentation is nearly infinite. Your evidence and logic is essentially encrypted with the fuzzshit algorithm such that it's indistinguishable from repetitious and rude gibberish.
• If my tolerance of low grade sloppy logic was as you say, I'd probably be more accepting of your arguments. People who use logic regularly learn to recognize it without having it spelled out for them. Anyhow, I agree that the Greeks use multiple kinds of evidence - all of which, note, were readily visible in our environments to the naked eye with just a little travel. If that was supposed to be a counterpoint to my position, I'm really not seeing it.
• If "regular" English was sufficient to encode and present logic and clarity, we wouldn't need programming languages and legal terms/conventions. (Programming languages are designed mostly to be read by humans, not machines.) If you think your writing is clear-cut logic, then we have a new disagreement. Plus, wouldn't it be easier and quicker to encode your logic more formally and clearly up front rather than debate around in circles using "regular" English? That doesn't sound like a rational choice. That's almost like saying you'll light all buildings you walk past on fire so that you don't have to wear a jacket in the cold.
• I agree that regular English is often unclear. But if you aren't familiar enough with both English and the subject to play word games, notice loopholes, ask for precise clarifications, and recognize both logic and fallacy, then you simply aren't a worthy peer or debate partner. Anyhow, your request might get more respect if it wasn't sheer hypocrisy.
• I do not "play word games" for the sake of word games. If it looks like I'm toying with language, it's usually an attempt to illustrate that a term or English is insufficient or imprecise. If you insist your writing style is "good enough" for technical debates, then we will never get anywhere. It's that simple. As far as "hypocrisy", I generally do not make absolute statements, unlike you (because I learned that ProgrammingIsInTheMind).
• I did not suggest you play word games. Indeed, I implied you aren't competent enough to do so.
• And you don't seem competent enough to serialize your (alleged) logic.
• [How would publishing it in parts make it any clearer?]
• 1) Easier to reference (name or number) parts in a finer and clearer granularity when talking about them, 2) The rules applied and the parts these rules are applied to are clearer (less ambiguous). 3) Statements are shorter, which generally reduces ambiguity.
• [1) I don't see the connection. 2) (Look up the definition of serialize.) 3) What's wrong with ordinals? 4) The rules are rarely mentioned. 5) (They are considered well-known.) 5) Sometimes shorter is less clear. 6) Still plenty of room for ambiguity. 7) (Did that really help you?)]
• As far as ordinals, if you can find a way to reference something by name or ID, that's fine. But it's usually easier to find something in a list if the labeling is in ID order (alpha or numeric) rather than random. I'm not sure you mean by “the rules are rarely mentioned”. Perhaps that's part of the problem. And yes, sometimes shorter can be less clear, but in general there's less need for deep StepwiseRefinement (clarification requests) if the statement is kept short. Why are you arguing against it? I told you this is the best way to communicate alleged “clear-cut logic” with me. If you don't wish to communicate with me, just say so and move on. When it comes to driving directions, some prefer written steps and some prefer maps. Why argue over what format they “should” prefer? If you wish to communicate location accurately with THEM, give them their preferred format, or just honestly say, “It's more effort than I wish to devote to better communication”.
• [Why am I arguing against your proposal? Because it's clear that even you don't think that it would be an improvement, otherwise you would already be doing it.]
• Top is a hypocrite. He does believe it would be an improvement. He just holds himself to a lower standard than he asks of other people.
• One generally cannot use such a tool for demonstrating absence of evidence or absence of logic behind a claim. If you find a claim of mine that is subject to such a tool, please do point it out. -t
• You just made such a claim. I think you make them far more often than you realize.
• [What do you mean by subject to such a tool? Are all "positive" arguments subject? If so, you have the very claim you are making here. If not, we need some sort of criteria to make that decision. (Just in case you haven't figured it out yet, I think the situations where this tool is useful to be rare. Mostly confined to learning about logic, reasoning about logic, and machine reasoning.)]
• I don't know what the hell you are talking about. Too many pronouns and ambiguous noun references. What the fuck is "here" and "that" referencing? You were the first to imply your argument was solid logic, so you get to go first. You suggested that I was stupid for not seeing your allegedly obvious and clear logic. Thus, you made the first claim about having presented logic. Stop trying to deflect. List it.
• [The only pronouns I used were "you" and "I". I had thought you knew who those referred to. "here" is a noun refering to this "place", i.e. this section of this page of this wiki.... In particular, I was specifying which claim of yours that I thought might be subject to being "serialized". "that" is an adjective. It tells you which decision I was referring to, the decision about whether or not to use your tool. Really, this is all stuff you should have learned in grade school (AmericanCulturalAssumption). In any case, I need to be convinced that "serializing" helps. I don't believe that "serializing" does help. You claim it so much better, yet even you don't use it. That leads me to believe that you also don't really believe it's better. (BTW, why does it matter who goes first? If you really think that it makes things clearer, shouldn't you be doing it regardless?)]
• I asked you first. And, I don't want to argue about arguing, but about the topic. I don't recall claiming I presented objective logic related to this topic. Thus, there's no claim to test. YOU made a claim. You are deflecting. The definition of "idiot" is somebody who keeps doing the same thing over and over yet expects a different result. Your communication technique has failed in multiple topics. Time to try something different. If you don't want to serialize your claims of logic, just say so and we end this topic. I'm not going to suffer your horrible documentation style anymore. --top

• I'm not sure what the point of this basketball analogy is. Its relationship with the material above it is hazy.
• [A basketball is a sphere and can be proven a sphere. Can you provide evidence that it is a bulge? what is the bulge theory? Since a basketball can be proven as a sphere, so can the earth - it's just a bigger basketball sphere. If indeed the earth is a bulge, or is flat, then there is possibility of a basketball also being flat or a bulge. Do we have any evidence that basketballs are spheres? How would we go about proving it? what logic would we use? The same logic and proof could be used on the Earth since both Earth and basketball are exactly the same shape... It's less of an analogy and more of a DIRECT almost exact COMPARISON.]
• That's an after-the-fact comparison. The ancients had to take it step-by-step: create models and test the models against actual observations. In practice, it appears the Greeks did NOT rely heavily on horizon-based observations. This fact of history tends to back my assertion that horizon-based clues were rather subtle to the ancients' travel capabilities and patterns, at least in comparison to their other evidence-gathering options. The actions of history did the test already (not your after-the-fact speculation) and multi-faceted won in this particular instance of the dice-toss of the occurrences of our universe.
• Of course the Greeks used multiple evidence sources. They aren't stupid. But that doesn't support your own assertion, TopMind.
• Yes it does. The actual methods were tested by those who didn't have prior knowledge. You have no equivalent "live" test, only speculation.

• In any case, the bulge model IS inferior to the sphere model, even if we only consider horizons. After all, the sphere model can describe why structures disappear behind horizons even on land, whereas the bulge theory only works for ships. So if we started with bulge theory, we'd quickly need to discard it (even if sticking to horizons), whereas if we started with round theory we'd never get back to seriously considering bulge theory (lacking evidence to get past the hand-wavy what-if level). Thus we progress in science.
• As described above, land is a much messier "lab" than sea. You seem to be assuming hindsight. You haven't given a good reason to discard bulge theory so quickly.
• Land horizons have plenty of good `lab` areas. You try to argue "there are some bad areas therefore they are all bad areas", which is fallacy. If you look across a bay, you can easily prove that there is no bulge (by seeing the other coast) yet that there is curvature (by not seeing the base of a structure or mountain beyond the coast). There are many places you can make such observations.
• Please clarify the bay statement. How do they know that the part after the far shore and before the mountain doesn't "naturally" slope down relative to land near the shore? Observing stream flow? Perhaps, but that's "additional" observations that requires coordinating multiple evidence sources, which I already agreed may be sufficient.
• They walk over there, or at least closer, then look around a bit. Easy enough.
• One cannot necessarily tell gradual slopes just by looking around via the naked eye, especially if there are a lot of trees around. It generally takes surveyor-like equipment or techniques, or at least lots of training and feedback.
• Perhaps. If we picked our sample poorly there might still be some ambiguity. OTOH, we are unlikely to see so-gradual-as-to-be-ambiguous downhill slopes towards a mountain. And the trees themselves would also be useful for evidence - i.e. if we can only see the top half of the tree rather than its base. Speculating on possible excuses why each point-sample of evidence is by itself insufficient doesn't really undermine an inductive argument.
• They are not "excuses", dammit! You exaggerate the obviousness of such. Anyhow it's at a point of anecdotal versus anecdotal evidence such that continuing this line is probably not going to be fruitful. The reader can do their own field work to test these.
• You exaggerate the probability of these `non-obvious` cases occurring in the first place.

• As far as the waves... (moved to [2])
• Moved, since it isn't very relevant.

And of course lots of evidence adds together. You can't conclude `sphere` from a point sample, after all. Anyone with competence in math knows that. Even the shadow experiment doesn't distinguish `sphere` from `cylinder` or `torus` - Erastothenes was starting with a 300 year old understanding that the Earth is round.

Agreed.

[2] As far as the waves, we cannot rule them out anymore than we can rule out a round Earth. They are two different models that would need further testing. One is not automatically "better" than the other without further evidence. The wave "lump" could be in a fixed position, by the way (such as land causing a "push-away" effect, similar to a big water droplet curving "up" in the middle). They would have to test multiple islands from multiple points to rule it out. A stationary lump seems silly with our modern knowledge, but it's not something an ancient person could rule out initially. Their "logic" would be built on what they can observe and their world's "givens", not our givens. Their logic would be built on top of what THEY can observe and test, not our modern science books. A logic "test" based on modern givens is not applicable to our case. As far as OccamsRazor, how is "simpler" being measured? If your logic is clear-cut and without holes, then write in out semi-formal numbered steps.

I think you only know the paraphrased version of OccamsRazor. And I agree that there are many hypotheses ancients might need to rule out, but the problem with your thinking is the `initially` (in `rule out initially`). We don't need to consider and reject every possible hypothesis up front. We can wait for evidence to distinguish one hypothesis from another, and we can certainly wait for a good reason to consider a hypothesis. Regarding the `big water droplets curving up` model - you must think the ancient people are idiots. The prevailing theory of the time is that water rolls downhill, and that bodies of still water (even as little as in a cup) will sit flat if undisturbed. The tides were already known to move with the moon, and waves with the wind. They'll have never seen evidence of these large, still waves in their bays or anywhere else.

Well, they've never (yet) seen evidence of roundness either. Water droplets on wax "violate" the flat-water model on a small scale such that a similar phenom perhaps could be available on a larger scale. The flat-water model is a working model, not a given. I'm not sure what you mean by your bay comment. If the distance between shores in a bay is too small, it wouldn't show up in either model, so it's a wash. -t

Your idea about `small scale` water tension... doesn't scale even to a teacup of water. In any case, sailors would notice pretty quickly if they got to sail downhill to reach a port.
• a tea cup of water causes the water to be concave due to adhesion between the cup and the water (cup sucks up edges of water) but this is just a diversion or nitpick (ArguingForTheSakeOfArguing). Sometimes it is convex though... depends on the cup material vs liquid material.
• (http://en.wikipedia.org/wiki/Meniscus)
And 3-foot spheres don't have detectable gravity either. They are both pretty "out of the blue" (assuming we are not considering non-horizon-related evidence at this point). As far as going "down-hill", it's a pretty shallow slope. Plus, the bulges could be caused by bunched gravity localities, meaning it's not really "down". Gravity bulges are a real phenom, by the way, due to density variations, but not at a visible level:

http://news.nationalgeographic.com/news/2011/04/110406-new-map-earth-gravity-geoid-goce-esa-nasa-science/

A flat Earth where the density of the dirt under the oceans was significantly less [corrected] than that under land could create the "water on wax" kind of effect at a large scale. No god needed. But this is merely a side issue, other than illustrating it doesn't require supernatural forces.

Perhaps, but the land-horizon observations would then provide evidence of flatness. So would starcharts. And you'd be able to see further from the top of a bulge. And you'd certainly notice the gravity sinking ships that are weighted for coastal travel. A scenario will have evidence for it. You seem to want to hand-wave scenarios into existence without providing evidence to justify them - i.e. you introduce entities and assumptions unnecessarily. Reasoning: you're doing it wrong.

Re: "Perhaps, but the land-horizon observations would then provide evidence of flatness."

The visibility limit at the horizon is just a bit longer than the distance needed to see the phenom. Almost every photo I've seen of the phenom looks like the object is about 90% obscured by haze. The top of one bulge would allow one to just barely see the top of the next bulge (assuming they are equal in size). When you get close enough to an island, you do see water on the "other side" of it. Whether this is standard perspective or "the next bulge" is hard to tell without reference points or modeling. The geometry of perspective was not really developed until the renascence, and would require a sextant or the like to check against the paper math. Plus, we are not assuming complex math models here, but an "ordinary but observant" ancient person.

Haze depends much on weather. On clear days - not uncommon - I sometimes see "ship effect" easily with unaugmented vision. But it is true I couldn't distinguish flatness from a very big sphere (much larger than Earth).

• Perhaps you get out a lot more than me. I've only seen the effect on large freighters, and there was a lot of haze between me and the ships, which were pretty far away. Ships were not very tall back then and would get lost in the haze I observed.
• Anyhow, it's difficult to envision what the bulge model view would look like beyond distant land masses without actual tests or simulations. The beyond-island bulge may be too far away to make the view stand out.
• While you might find it counter-intuitive, the ship effect is easier to observe with smaller ships - the lower horizon means that ships are closer to the island before the effect occurs. A larger ship must be much further away before `half` of it has disappeared, for example. Conversely, someone on a high deck of a big ship will have a longer horizon and therefore must be further away before `half` of the island has disappeared. Haze tends to be proportional to distance, so haze becomes a bigger problem with bigger ships.
• I've never seen it with short vessels. Plus, 10-foot waves are not uncommon, and such could be blocking one's view instead.
• Calm days on seas are common, so sailors and people living on or near docks would experience plenty of them. Short waves are easy to account for.
• [What if the ships really are sinking, and they sink into another parallel universe. Then they come back out of the parallel universe later when no one is looking !!! !1111!!!1111 AnythingIsPossible!!11!111 (!!!one!!!one!!!111)]

Re: "And you'd certainly notice the gravity sinking ships that are weighted for coastal travel. "

I should have said the density was lighter under the water, not heavier. I'll make the corrections. Anyhow, the water top is at the equilibrium point either way. The gravity level at the surface would be the same. The bulge is relative.

Might want to work on your physics, there.

• No, I think that's a law of physics, or at least a property of it. It's almost like "local roundness": there's no "sliding". Floating is caused by relative density, not absolute gravity. But it may be moot anyhow because the ancients are not using modern physics to test their models. Even under a round Earth, they don't know how gravity "works"; it's just "there". The hypothesis is that "water bulges up away from land". The "cause" of the bulging is no more necessary than the "cause" of Earth's roundness. The ancients are at the stage of working out the shape of things, not the causes of the shapes.

• I agree it's moot. Similarly, your argument about detecting gravity off a 3 foot sphere in the backyard (or flat shapes for that matter) was moot. We don't care why the Earth is round or flat, only whether. Regarding the physics: floating is caused by relative weight, not mass (density), and since water doesn't compress the boat's weight will change relative to the displacement of water. And I think you are right about the bulge being with lower gravity, but that doesn't mean gravity at the surface is the same, only that the potential energy of a unit water at the surface is the same.

• [You should undo your correction Top. You would get a bulge over the higher density spots (assuming a uniform thickness for this "Earth". This happens because gravity is pointing somewhat towards the higher density areas instead of perpendicular to the surface of the "Earth". Floating is caused by relative density, not weight (although that works out the same if you have a fixed volume).]

• Not density, unless you mean weight density. There would be no buoyancy if you were submerged in a tub of water at freefall. And I think Top's correction was right at least insofar as where the bulges lie (he has the wrong equilibrium). Any given unit of water will flow always to a lower potential energy - PE = m*g*h. Generally g is constant so we flow downhill (down h). But if gravity varies, then we can also flow downgrav, and we will do so until there is nothing to be gained (g balances h).

• [Floating is not caused by relative weight. Do you really think that if you heat water at the bottom of a tank that it rises to the top because it now weighs less? Yes the bouyancy force is directly proportional to g and in 0g its magnitude is 0, but for any g > 0, it suffices to know the relative density to know if something will float.]
• Hmm, true. I wasn't accounting for the increased weight of the water. Though, you'd still risk sinking a ship by moving it into higher gravity, due to concerns of structural integrity.
• [Yes, it would be pretty easy to differentiate his theory from a round Earth theory, since the gravity changes would have to be pretty big to make a noticeable bulge. You would have things falling faster, stuff breaking on a calm ocean that wouldn't break on land, etc.]

• [Regarding Top's correction, you might want to notice that the very Earth we live on is an example of his original scenario. If you draw a plane through the Earth, it would be an area of heavy density surrounded by less dense areas. Alternatively, you can do the math.]
• That really isn't Top's scenario. Start with a flat Earth, and manipulate gravity under the seas so we have hilly water. That's top's scenario.

• [Yes, that's Top's scenario; he even specified how (by placing more mass under the water). The thing is, that his scenario (when taken to an extreme) is also descriptive of the Earth in space. Space maps to the low density land, and the Earth maps to the high density seabed. Now if you took a situation closer to what he was expecting, where the land density isn't essentially zero, you won't end up with the bulge being shaped like a hemisphere, but you will end up with a bulge.]
• The analogy doesn't work. It only seems to because, in your example, the Earth is the only element that reflects light. You need to ask: where would water naturally `flow` if placed on a zero-grav plane? What would be its maximum height or bulge? Effectively infinite, I would think, though depending on how far you moved the water from Earth's gravity well.
• [Well, assuming that the water is stationary to start with, it would form a sphere centered on the mass center of the water. If it's initially moving things get more complicated. However, we aren't dealing with a zero-gravity situation. Even though the plane has zero mass, there is still the gravity provided by the Earth. In this situation, the gravity is parallel to the plane, and the water will be pulled towards the Earth.]

• It may depend on how much water compresses on a large scale. I assumed it did enough to affect the results, but I only play a fluid physicist on TV. -t

• Water doesn't compress appreciably on any scale. Density of water depends on temperature and salinity more than pressure. The non-compression of water is, btw, the basis for hydraulic technologies.

Re: "i.e. you introduce entities and assumptions unnecessarily"

Please clarify. A round Earth is not necessarily simpler or less assumptions. Again, at least the bulge phenom can be observed in small-scale models, unlike the gravity of a backyard sphere. (Except maybe a polished magnet, but that's only for metals.)

I wouldn't assume round Earth without any evidence, either. You'll need to find evidence for each assumption you introduce - i.e. so you introduce them necessarily.

True. But both models require assumptions.

That goes without saying. Models are assumptions.

You seem to be implying that bulge theory has "extra" or "unnecessary" assumptions? Please clarify.

Entertaining bulge theory is itself an extra or unnecessary assumption, until you have evidence to distinguish it from the preferred theory.

"Preferred"?

Yes. I could also say "incumbent".

Another possibility to consider is that light does not travel in a strait line. Example occurrence:

"The super-thick atmosphere [of Venus] bends light so severely that the horizon appears to curve upward, allowing you to see all the way around the planet [if not for haze]."

http://tvtropes.org/pmwiki/pmwiki.php/UsefulNotes/Venus

This is the opposite direction needed to simulate the appearance of a round Earth, but it illustrates the point that the path of light could be a factor also.

No, that's the right direction, though it would need to be a weaker curvature. Light would bend up into the night sky, i.e. away from the horizon. Interesting thought.

Venus' atmosphere appears to make light bend down. That would make the surface appear concave. In other words, it's counter-intuitive.

Ah, then I retract my statement. I don't think there is much intuitive about optics.

There is if you draw an imaginary little planet and an imaginary little eyeball cross-section (remembering that images are projected upside down). Actually, use a pin-hole camera instead of an eyeball to keep the model simpler. I wonder if different gases have different bend properties (or different mixtures at different levels)? I believe the Venus phenomenon is caused largely by density differences between the very surface, and air just above the surface. Otherwise, photons wouldn't know or "care" what is "up" or "down".