However, between complete knowledge (having run each of the infinite tests) and no knowledge (having not so much as existed in the same reality as the object) there lie degrees of identity. Through these degrees do humans - and other intelligences - understand identity. The degree in question is dependent on the tests, conscious or unconscious, conducted on the object. For example, if one has seen two identical pictures, and performed no other tests on them, then, seeing the first picture an hour later, one would not know which of the two original pictures it was - their degrees of understanding of each picture’s identity would be the same, and these degrees of understanding of identity would be less than the degree of identity at which these pictures differ. To this person, the two pictures would be the same - they would have the same identity.
This idea, though, extended, seems at first to show a problem with this theory of identity. Imagine two perfectly identical objects - cubes of metal. These cubes are precisely the same in every way. The atoms are arranged precisely the same inside each cube, they are completely indistinguishable to empirical tests, and so have the same identity - when each is taken individually. However, these objects do not exist in a vacuum. As large a factor as internal structure is in identity, it is only a degree of that identity. Other factors are considerable too, and two objects are not completely identical unless all these factors are the same - in which case they are the same object. One of these factors is location. Two different things can not occupy the same space at the same time. To be sure, in the case of the cubes, they have a very high degree of common identity - but they are different in other degrees. These degrees are all reliant on tests that can be performed, and there is a simple test to tell them apart: (assuming they’re both on Earth) place a compass on each - the angle of the needle will differ slightly. Alternatively, and more accurately, a GPS would do as well. Theoretically, in a completely empty space, these cubes could be told apart - would have a degree of difference in identity - by relation to one another. Not spatially, for that would be meaningless, but directionally.In any case, one might say, location and direction are momentary aspects of identity. They are constantly changing. What if these identical two cubes were placed in a black box through which no energy, in the form of vibrations or otherwise, could be transmitted out, and shaken like a pair of dice before being removed? How could they possibly be told apart? Each cube still has a separate identity, but there are no empirical ways to differentiate which of the cubes coming out of the box pairs up with one going in. Does this prove that identities are destroyed and reforged inside the box? That would be absurd. There are still the same two cubes. Clearly there is a historical degree to identity as well, and like all other degrees, an object’s identity can not be fully known unless the historical degree is. Earlier, though, it was claimed that all of identity is empirical, based on interactions. How can history be discovered through tests when every molecule in two cubes is the same? The answer is simple. Three pieces of data are needed, all of which can be determined empirically. First, complete models of the physical degree of identity of each cube must be made. Second, the locations of the cubes within the black box before and after they had been shaken. Third, the precise way in which the box was moved. Some would say that it’s silly to have to know the second and third points in order to know the identities of the cubes. They may be right, and in fact the identities can be known without that - but only to a degree, just like having only glanced at two pictures and not being able, later, to tell them apart. However, knowing all three parts of the data, and with proper use of knowledge of physics, the precise paths of the cubes through the black box during the shaking can be determined - even without any information being allowed to move out of the box. This figuring is empirical, if not in a strictly literal sense - it measures interactions between a perfect theoretical representation of the cubes and of the world, rather than between the real things. However, they can still comfortably be called a test of the object. The results of these calculations can be matched up with the locations of the two indistinguishable cubes after shaking, therefore both distinguishing them and providing a continuity of identity with the cubes placed in the box.
By a great extension of this idea, all historical identity of an object can be found empirically - if the entire rest of the universe surrounding it is. That is, of course, impossible in reality - but no less impossible than knowing the precise position and velocity of each particle in the object, which is required to completely understand the physical degree of identity of anything. However, like the physical identity, the historical identity can easily be approximated - which is how humans keep track of the continuity of identities of objects they encounter. All identity is known in degrees.
As has been shown, all identity can be reduced to either empirical or virtually empirical interactions with reality. It is determined by tests - formal or informal, in the case of humans - and can only be known to the degree of which these tests are conducted. Due to this, only theoretically is it possible to fully comprehend the identity of anything. The philosophical implications of this are a matter which will not be covered here. Similarly is the greatest assumption made by this theory: a deterministic universe. This theory, so far, disregards both quantum randomness and the idea of souls. For the former, remember that unknown information due to quantum events can be completely consistent with understanding of a limited degree of identity. For the latter, that is a problem that, as was already true in the time of Nietzsche, has been disproved a hundred times and yet still needs a brave philosopher to successfully contradict it.