There is only one way to make sure of the validity of a chain of logical reasoning. This is to put it in the form in which it is most easily testable: we break it up into many small steps, each easy to check by anybody who has learnt the mathematical or logical technique of transforming sentences. If after this anybody still raises doubts then we can only beg him to point out an error in the steps of the proof, or to think the matter over again. In the case of the empirical sciences, the situation is much the same. Any empirical scientific statement can be presented (by describing experimental arrangements, etc.) in such a way that anyone who has learned the relevant technique can test it. If, as a result, he rejects the statement, then it will not satisfy us if he tells us all about his feelings of doubt or about his feelings of conviction as to his perceptions. What he must do is to formulate an assertion which contradicts our own, and give us his instructions for testing it. If he fails to do this we can only ask him to take another and perhaps a more careful look at our experiment, and think again.
An assertion which owing to its logical form is not testable can at best operate, within science, as stimulus: it can suggest a problem. In the field of logic and mathematics, this may be exemplified by Fermat’s problem, and in the field of natural history, say, by reports about sea-serpents. In such cases science does not say that the reports are unfounded; that Fermat was in error or that all the records of observed sea-serpents are lies. Instead, it suspends judgment. [81]
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This echoes Mill’s explanation of how to achieve real understanding.