On one view, the purpose of voting is to gather evidence of the will of the people. Recounting and even revoting, when done properly, then seem plausible as ways to gather more and better evidence. On the other view, elections give everyone an opportunity to support their candidate, and voting is somewhat like signing a contract. In a rental car contract, you have an opportunity to get insurance, but, if you don't initial the correct box, you cannot come back after an accident and say that you intended to get insurance. Analogously, if the ballot was not too obscure, each voter had a fair chance to support their candidate, so they cannot come back later to correct mistakes, such as not fully pushing out a chad or marking the spot for Buchanan when they wanted Gore.
Most people adopt both views of voting to some extent, even though they conflict.
This incoherence within our attitudes towards voting is part of what makes it so difficult to know what should be done in Florida.