Science

Why is it said that scientific results must be replicable? Is this also possible or should that also be the same for mathematics, history, arts or other natural or social sciences?

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One difference between the sciences and other disciplines is that scientists publish experimental and observational results. Since other scientists rely on these results, they want some way of checking that they are genuine. However, this can't be done simply by scrutinizing their publication, as you might assess the value of a historical argument or piece of literary critcism. Hence the demand that the results be replicable. (This is why it is so highly charged in scientific circles to say that so-and-so's results 'are not replicable'--the suggestion is that so-and-so is incompetent or even fraudulent.)

As David says, replication in science is a way of checking that a result is genuine. We can distinguishing two different senses in which a result may fail to be genuine. One is that it was made up. Replication is a good way of detecting (and discouraging) fraud. Here there is a parallel in the study of history. If one historian makes a claim about what has been found in a document in an archive, other historians may want to check that this is what the document really said.

But in science there is also another sense of 'genuine' that gives another reason for wanting replication. Scientific results are usually not just reports of what the meter said. They are often causal claims, like the claim that a certain drug reduces cholesterol. That claim may be based on an experiment where people on the drug ended up with lower cholesterol than people not on the drug. But that doesn't prove the claimed result, that the drug really does lower cholesterol. It might be that there is some other difference between the two groups of people that is the real cause of the difference in cholesterol levels and the drug had nothing to do with it. But if the experiment is replicated in diverse contexts, all with the same effect, that increases the probability that the causal claim is correct, that the drug really works.

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