He is dealing with mere ciphers, symbols or abstractions, and consequently the mathematician has everything his own way, and so long as he abides by ciphers, symbols and abstractions we have no wish to interfere. He can carry his egalitarian principle right through the English weights and measures, _viâ_ the decimal system, into geometry. He is speaking of pure abstractions, arbitrarily supposed to be identical, and if it amuses him to postulate equality as their characteristic, nobody cares. They are his abstractions, his ciphers, he can postulate what he likes about them. We have the feeling that it does not matter. It is only when the mathematician, who, as a rule, is a hopeless psychologist, begins to apply [p63] the lifeless notions he has learnt in his study, to the world of activity and reality; it is only when he begins to speak of things that are not his own abstractions--things that really have an existence known to us--that we immediately begin to feel that he is taking liberties with reality. For instance, if he say that a certain 2,000 pear leaves are equal to another 2,000 pear leaves, we who know that no two leaves have ever been known to be exactly the same, straightway call him to order and say: “No, sir, abide by your abstractions! That statement of yours is not true.” Likewise, if he say that a certain 2,000 cows are equal to another 2,000 cows, we feel that he is either taking too much for granted, or else that he should try to enlist our confidence by specifying the precise weight and individual qualities of each cow in each set, before inviting us to acquiesce in his assertion. And even if the two sets of 2,000 cows weighed exactly the same amount and were of the same race, we should still feel that there were differences in the quality and supply of milk in each set, as also in the vitality of the respective cows in each set, etc., which ought to be taken into account and which it would hardly be possible to estimate with perfect accuracy. But let us think of things which have less individual divergence from the common type. Let us think of screws, bolts, plates, chain links, etc. After these have been made with the utmost care by means of machines capable of almost mathematical precision, and when once they have been accurately weighed and found equal both [p64] as regards size and ponderability, you would think that you had groups of things or individual things as between which you would be justified in postulating the attribute equality. But if you should ask anyone accustomed to dealing with such things, he would tell you that one bolt in ten or in twenty usually splits, that one screw in a hundred or in a thousand usually strips, and that one plate in fifty usually cracks. Thus here and there, even when enormous pains have been taken to attain uniformity, marked differences become apparent. What about those differences that are not sufficiently marked to be noticeable until some considerable time has elapsed?