In most empirical sciences, the penumbra is at first prominent, and becomes less important and thinner as the accuracy of physical measurement is increased. In mechanics, for example, the penumbra is at first like a thick obscuring veil at the stage where we measure forces only by our muscular sensations, and gradually is attenuated, as the precision of measurements increases. But with the arithmetical concept of an individual identifiable object it is the exact reverse; a crude point of view does not suspect the existence of the penumbra at all, and we discover it only by highly refining our methods. Doubtless arithmetic owes its early development to this property. We may now go still further. Operations themselves are, of course, derived from experience, and would be expected also to have a nebulous edge of uncertainty. We have to ask such questions as whether the _operations_ of arithmetic are clean-cut things. Is the operation of multiplying 2 objects by 2 a definite operation, with no enveloping haze? All our physical experience convinces us that if there is a penumbra about the concept of operations of this sort it is so tenuous as to be negligible, at least for the present; but the question affords an interesting topic for speculation. We also have to ask whether mental operations may similarly be enveloped in a haze. EXPLANATIONS AND MECHANISMS Perhaps the climax of our task of interpreting and correlating nature is reached when we are able to find an explanation of phenomena; with the finding of the explanation we are inclined to feel that our understanding of the situation is complete. We now have to ask what is the nature of the explanation which we set as the goal of our efforts. The answer is not easy to give, and there may be difference of opinion about it. We shall get the best answer to this, as to so many other questions, by adopting the operational point of view, and examining what we do in giving an explanation. I believe that examination will show that the essence of an explanation consists in reducing a situation to elements with which we are so familiar that we accept them as a matter of course, so that our curiosity rests.[5] "Reducing a situation to elements" means, from the operational point of view, discovering familiar correlations between the phenomena of which the situation is composed. [Footnote 5: The ultimate elements of explanation are analogous to the axioms of formal mathematics.]