The principles of mathematics are as clearly postulates.  In Euclidean geometry we assume definitions of ‘points,’ ‘lines,’ ‘surfaces,’ etc., which are never found in nature, but form the most convenient abstractions for measuring things.  Both ‘space’ and ‘time,’ as defined for mathematical purposes, are ideal constructions drawn from empirical ‘space’ (extension) and ‘time’ (succession) feelings, and purged of the subjective variations of these experiences.  Nevertheless, geometry forms the handiest system for applying to experience and calculating shapes and motions.  But, ideally, other systems might be used.  The ‘metageometries’ have constructed other ideal ‘spaces’ out of postulates differing from Euclid’s, though when applied to real space their greater complexity destroys their value.  The postulatory character of the arithmetical unit is quite as clear; for, in application, we always have to agree as to what is to count as ‘one’; if we agree to count apples, and count the two halves of an apple as each equalling one, we are said to be ‘wrong,’ though, if we were dividing the apple among two applicants, it would be quite right to treat each half as ‘one’ share.  Again, though one penny added to another makes two, one drop of water added to another makes one, or a dozen, according as it is dropped.  Common sense, therefore, admits that we may reckon variously, and that arithmetic does not apply to all things.

Again, it is impossible to concede any meaning even to the central ’law of thought’ itself—­the Law of Identity (’A is A’)—­except as a postulate.  Outside of Formal Logic and lunatic asylums no one wishes to assert that ‘A is A.’  All significant assertion takes the form ‘A is B.’  But A and B are different, and, indeed, no two ‘A’s’ are ever quite the same.  Hence, when we assert either the ‘identity’ of ‘A’ in two contexts, or that of ‘A’ and ‘B,’ in ‘A is B,’ we are clearly ignoring differences which really exist—­i.e., we postulate that in spite of these differences A and B will for our purposes behave as if they were one (’identical’).  And we should realize that this postulate is of our making, and involves a risk.  It may be that experience refuses to confirm it, and convicts us instead of a ‘mistaken identity.’  In short, every identity we reason from is made by our postulating an irrelevance of differences.

There is thus, perhaps, no fundamental procedure of thought in which we cannot trace some deliberately adopted attitude.  We distinguish between ‘ourselves’ and the ‘external’ world, perhaps because we have more control over our thoughts and limbs, and less, or none, over sticks and stones and mountains; fundamental as it is, it is a distinction within experience, and is not given ready-made, but elaborated in the course of our dealings with it.  Similarly, in accordance with its varying degrees of vividness, continuity, and value, experience itself gets sorted into ‘realities,’ ‘dreams,’ and ‘hallucinations.’  In short, when the processes of discriminating between ‘dreams’ and ‘reality’ are considered, all these distinctions will ultimately be found to be judgments of value.