It is a general consequence of the approximate character of all measurement that no empirical science can ever make exact statements. This was fairly obvious in the case of mechanics, but it required a Gauss[4] to convince us that the geometry in which we are interested as physicists is an empirical subject, and that one cannot say that actual space is Euclidean, but only that actual space approaches to ideal Euclidean space within a certain degree of approximation. I believe that we are compelled to go still further, and recognize that arithmetic, so far as it purports to deal with actual physical objects, is also affected with the same penumbra of uncertainty as all other empirical science. A typical statement of empirical arithmetic is that 2 objects plus 2 objects makes 4 objects. This statement acquires physical meaning only in terms of certain physical operations, and these operations must be performed in time. [Footnote 4: C. F. Gauss, Gesammelte Werke, especially vols. IV and VIII.] Now the penumbra gets into this situation through the concept of object. If the statement of arithmetic is to be an exact statement in the mathematical sense the "object" must be a definite clean-cut thing, which preserves its identity in time with no penumbra. But this sort of thing is never experienced, and as far as we know does not correspond exactly to anything in experience. It is of course true that in most experience the penumbra is so very thin and snug-fitting that it requires special effort to recognize its presence at all; but scrutiny, I believe, shows that it is always there. If our experience had been restricted to phenomena in a vacuum, and the objects we were trying to count had been spheres of a gas which expand and interpenetrate, it is obvious that the concept of "object" as a thing with identity would have been much more difficult to form. Or, if our objects are tumblers of water, we discover when our observation reaches a certain stage of refinement that the amount of water is continually changing by evaporation and condensation, and we are bothered by the question whether the object is still the same after it has waxed and waned. Coming to solids, we eventually discover that even solids evaporate, or condense gases on them, and we see that an object with identity is an abstraction corresponding exactly to nothing in nature. Of course the penumbra of uncertainty which surrounds our arithmetical statements because of this property of physical objects is so exceedingly tenuous that practically we are not aware of its existence, and expect never to find undiscovered phenomena within the penumbra. But in principle we must recognize its presence, and must further recognize that _all_ empirical science must be of this character.