John responds to me:
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I've been doing it for quite some time. It's called "indirect proof."
But that isn't what he is talking about. In fact, what you've done isn't really proving a negative. It is a round about way of proving a positive, or of proving something about a positive-- that positive being the number you choose to test.
No, it is, indeed, proving a negative. There is no largest prime. The result is not that the individual number before us isn't the largest prime (though that, too, would be proving a negative) but that there is no prime anywhere no matter how you try to construct one that is the largest.
The problem with "proving a negative" is not that it cannot be done but that the techniques available to do so require the ability to look at every single possibility.
That isn't always possible.
I can prove the negative that something doesn't exist in a particular room. Depending upon the specifics, it can be because the object in question is too large to fit in the room or that the object is definitively somewhere else or by doing an exhaustive search of all the objects in the room and determining that none of them are the object in question.
The problem with something like proving god doesn't exist is that the definition of "god" is not very specific and the arena which needs to be examined is too large for a brute force method such as an exhaustive search as well as being too vaguely defined to be amenable to abstract methods.
It isn't that you can't prove a negative. You can. It's that many things aren't amenable to the techniques involved. On the flip side, not all positives are amenable to proof, either.
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Throughout the operation you have values, data, and other such things with which to work.
Didn't I say that? I'm sure I did. Yep...right here:
"However, it requires well-defined objects behaving in well-defined ways. If you deviate from that, such as the methods of science which rely upon observation rather than definition, then things become more difficult."
The fact that you have data to work with doesn't mean you aren't proving a negative. The fact that there is a positive means there is a negative, too. By the fact that an object is X, that means it is not Y.
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What is meant by 'proving a negative' is different. Consider: Someone tells you that a giant lives in the mountains, but this person offers no proof but instead challenges you to disprove the existence of the giant.
No, it's still the same thing. It's just that the scenarios are dissimilar. Someone tells you that there is a largest prime but this person offers no proof but instead challenges you to disprove the existence of the largest prime.
Quite simple: Assume there is a largest prime, lead yourself to a contradiction, and thus logically conclude the negative: There is no largest prime.
The only difference between your situation and mine is that my scenario has well-defined objects behaving in well-defined ways. If you refuse to define what a "giant" is and how it behaves, not only does it become impossible to prove that there is no giant, but also it becomes impossible to prove that there is one.
Until we can get a concrete definition, one cannot prove anything.
Take, for example, the proof of the four-color theorem. For those who don't know, the four-color theorem states that any map drawn on a contiguous plane surface (that is...a plain sheet of paper) can be colored using only four colors such that no two areas that share an edge have the same color. You might be able to get away with fewer, but you don't need more than four.
Well, how do we prove that? For the longest time, we could handily show that for any map anybody had ever drawn, it could be colored using at most four colors. It's because we could come up with a concrete definition (this particular map), we could come up with an actual result (yes, it can be colored using no more than four colors). But, we couldn't come up with any progress on the larger question because we couldn't define what a "map" was in any concrete method.
Eventually, through a lot of work, a method was developed that could generate every possible map you could draw. It was noticed that some maps are equivalent. For example, suppose you have a map that consists of a circle within a circle such that the two don't touch. Well, that is the same as a map that consists of just a circle: The color of the inner circle can be anything because it is completely surrounded by a single color. Therefore, we can remove that inner circle completely and not affect the number of colors needed to color the rest of the map.
The technique allows us to get a much more concrete definition of "map." By using the greater comprehension, we can generate an exhaustive list of all possible maps. We can then go through every single one and determine if it can be colored using only four (this is where the computer came in which caused the controversy...did anybody bother to check if the computer did it right? Yes, they did.) If it turns out that the entire set of all possible maps can be colored using only four colors, then we have proven that there is no map that requires five.
By proving the positive, each map can be colored using four colors, we also prove a negative, each map requires no more than four colors to color.
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If the giant does exist then he will leave evidence and can be proven to exist. This is proving a positive.
Indeed. But if the definition of "giant" requires a necessary result that does not appear, then the giant can be proven not to exist. This is proving a negative.
In fact, that's exactly what science does: Prove negatives. Nothing in science can ever be proven true. What is done all the time, however, is prove things not to be true.
Newtonian mechanics, for example, necessarily requires a linear universe. The universe, however, is relative and not linear, therefore we prove Newtonian mechanics to be not true. Oh, it's very close to being true for many instances, but there is always an error between what the theory claims and what we actually observe. It may require extremely sensitive equipment to actually measure that discrepancy, but that doesn't mean the discrepancy isn't there.
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The giant could also be proven to be something other than a giant-- like a hoax or the result of the consumption of a peculiar fungus. This also is proving a positive.
Actually, it's also proving a negative: The evidence is not a giant. If a giant and only a giant results in X and what we have before us is Y, then we prove that it is
not from a giant.
But again, it requires well-defined objects behaving in well-defined ways. When things become fuzzy, the ability to prove anything, positive or negative, becomes much more difficult.
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You have proven that something other than a giant can account for the alleged sighting of a giant. But you still haven't proven the non-existnce of the giant, but only that you have no evidence for the giant and do have evidence for other explanations.
Only if the definition of "giant" is so vague as to be incapable of providing a definitive result.
What, precisely, is a "giant"? How does it behave? What actions can be ascribed to a giant? What actions must be excluded from the definition? Are there any that are unique to a giant?
Until these questions are answered, not only can we not prove the non-existence of the giant, we also can't prove the existence of the giant. We are left with an ill-formed premise behaving in an ill-formed way. No wonder we are at a loss to make any sort of acceptable statement.
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A believer in the giant could always say 'that was a hoax but the giant is real.
That's where the logical part of burden of proof comes along. The claimant is the one that needs to show evidence.
And yes, that means those that say there is no giant. But, since this started with the person making the claim that there is a giant, it is his burden of proof to show that it is there. If my response is that there is no giant there, then it is my burden to justify my claim.
I can easily do that by defining what a "giant" is and then showing that the mountains do not contain any evidence that is necessarily required for there to be a giant.
If the other person then claims that my definition of "giant" is insufficient, then he needs to provide a better one. No proof in either direction can be provided unless a specific definition of "giant" is forth-coming.
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Where is the proof that he isn't?' Here is the problem. If the giant does not exist there is no evidence, and there never will be.
Not a problem if the definition of "giant" allows there to be a scenaro that necessarily results in a certain situation. If that situation does not exist, then the absence of evidence is, indeed, evidence of absence.
The definition of ice requires the existence of a solid. If we can determine that there are no solids anywhere in the system, then we can necessarily conclude that there is no ice. The absence of solids is evidence of the absence of ice.
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If the giant does not exist, he leaves no trace.
Then the definition of "giant" is so vague that not only can we not show it doesn't exist, we also can't show that it does.
If the giant leaves no trace, then there is also no evidence for the existence.
Of course, a difference that makes no difference is no difference. If the evidence that we have is perfectly consistent with both the presence and the absence of the giant, then we logically conclude that there is no giant since there is no difference between the two.
Note, this assumes that we have complete information. Again, the problem is not the act of proving the negative. The problem is the lack of definitive information. If the objects are not well-defined and we do not have a concrete definition of how they behave, then we become susceptible to overlooking something.
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There is no evidence to examine, and no information with which to construct a proof or a disproof. You can't prove, or disprove, a negative, because there isn't any evidence with which to work. So we are stuck with proving, or disproving, positives.
You've just stated my point: With no information to go off of, you can't say anything either way. We aren't stuck with proving positives. We don't even have the ability to prove a positive.
Nothing in science is ever proven true.
Instead, science concerns itself with proving things false. The function of every experiment is to try and disprove the theory because a successful experiment doesn't prove the theory true, it merely shows it to be accurate. It may be, like Newtonian physics, that our instruments are sensitive enough to detect the error.
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Rrhain
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