I don't see the difference between positive and negative. Both are based on several impossible to proof assumptions.
Well, eh...
It is a bit more complicated than that. The base case is proof of existence. In an empirical system, we can prove general existence, because that can be proven by specific appearance. "I can prove horses exist, because look here - Horse!". We cannot prove general non-existence. "I can prove unicorns don't exist, because look here - No unicorn!" doesn't work. The unicorn might have its day off and be down on the pier playing the slot machines for all we know.
In that way, it is easy to prove that there is (specific) damage to something - "Look here, cracked board". It is not possible to prove that there is (generally) no damage to something - "Well, this is ok, and this is ok, and all seems ok, but, well...". That is, unless you have enumerated all the possible cases of what constitutes damage and have a test for each end every one of them. Which is the equivalent of having enumerated each and every possible hiding place for the unicorn, and checking all of them. In the general empirical world, stuff is not enumerable.
But this is still all proof of existence. When it comes to proof of causation, it gets a bit more complicated. Strictly speaking, we can prove that a particular something does
not cause a
specific effect. That is general falsification, Popperism, call it what you like. You have a theory about the world ("All red objects fall sideways because of the redshift effect"), you logically deduct a specific testable consequence of this ("If I drop this red marble, it will fall into the wall on my left"), and then you test it. If the consequence holds, you really cannot be sure that it really means your theory holds - there could, for instance, be a hidden electromagnet behind the wall - but if it doesn't hold, and the marble falls straight down, then you've proven that your theory is wrong. So, we can prove a theory false, but not true. This is down to the specific/general thing again - We're looking for a general explanation for all of these cases of falling marbles, and there is a multitude of possible general explanations whilst we're checking a specific one. But that also mean that that
specific one can be struck of the chart when we see that it fails.
Science never proves. Science always disproves, and when it doesn't, we've just made it a little more likely that our theory is true. Likely, not proven.
So, strictly speaking, we cannot generally prove that a vibration motor causes damage to a board. We can experiment - We can mount the motor, run a typical usage test, and find a broken solder joint. We can fix it, rerun the test, and find another broken solder joint. We can repeat this ad infinitum and never have actually
proven that it is the vibration that breaks the board. It could have different causes - a small earthquake the first time, an unseen lightning strike the second, a bad joint to start off with the third...It could also be a systematic but unmeasured cause - It could, for instance, be a telekinetic pulse that your brain emits every time you think the thought "now I shall start the vibration test". However, in practice, we tend to accept induction as a practical proof method - Even though a million marbles falling straight down does not prove that the next one will do the same, it is rational to assume that it will, even at the lack of a proper logically sound proof. We have a specific effect we're looking for, and we've seen it so many times that we inductively assume the theory of gravitation is correct. Or the theory that vibration causes board damage, as was the case. Particularly when we have supporting theory in mechanics (transfer of vibrational energy) that would explain the mode of causation
However, if proving the positive is
slightly logically dodgy, proving the negative is several magnitudes worse. And it goes back, again, to proof of existence. For the positive, we need to see a specific effect. Any specific effect. For the negative, we need to
not see
any of the multitudes of possible effects. That is, a general theory for the causation of a general effect.
In the first case, we could be sure of the outcome (
That is cracked, therefore something was broken, therefore the experiment was a success), although we couldn't be logically certain of the cause. Here, we can't even be sure of that - We think nothing is broken, because we can't see anything broken, but we can't be sure because that is a general statement - so we can't even, logically speaking, be sure whether our experiment came out a success or a failure. Where we could have a fairly good induction going rather quickly in the positive case, here we can't.
So: It is impossible to prove a negative. And while it is also true that strictly speaking it is impossible to prove a positive, too, that does not mean that those two impossibilities are on equal footing. Proving a negative is
a lot more impossible than proving a positive.
All this, of course, in the empirical realm. If we go into formal analytic subjects like mathematics, then it is a different thing.
