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I think this is along the lines of Rummy's "unknown unknowns"
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Alison P wrote: the absence of a particular bug
on this computer at that very moment in time.
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Actually I don't agree.
Here is a simple example to illustrate what I mean, if I forget to clear a variable when required in a function then I fix that bug and retest, I have established 'the absence of that bug' on more than just that computer at that very moment in time. That particular bug has been fixed on all computers all the time.
Although I fully accept that there could be any number of other bugs that may appear on other computers at other times, I do believe that by fixing then testing I have proven beyond reasonable doubt that I have truely fixed that particular bug.
If I didn't believe that I think I would have to give up programming and become a baker or something that requied less logic. 
Ali
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OK, if you actually locate a bug, yes.
My statement mostly applies to passing tests - yet another reason for automated tests 8where applicable).
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That may very well be so, but that's not an argument to skip the test altogether.
It's a fact that you can only find your lost carkeys if you look for them. How many keys you're about to find, remains to bee seen.
Bastard Programmer from Hell 
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I 100% agree with that. Testing is required... Whether formal (dependant on the need, by specialized software testers). Or super informal (by the developer himself). Or informal (by another developer). Why I mention two informal's here is because it is better for someone else to test your code, but if left no choice, which is in many cases the case the former would have to do.
"Program testing can be used to show the presence of bugs, but never to show their absence."
<< please vote!! >>
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A short (real) story:
A dev thought he was above having bugs. His colleagues managed to prove he had a critical bug, no less.
When he was shown the bug he reacted as such "I will not let reality confuse me", got up and went back to his cube.
Alberto Bar-Noy
---------------
“The city’s central computer told you? R2D2, you know better than to trust a strange computer!”
(C3PO)
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hopefully he was then fired for being unprofessional and being a general twit?
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Yep... Oh now there are 2 of "them" 
To Err is human but to make a mess you need a computer
Alberto Bar-Noy
---------------
“The city’s central computer told you? R2D2, you know better than to trust a strange computer!”
(C3PO)
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