=The Association of Contiguity.= This is the most elementary of the three classes of association, and concerns the spatial and temporal order in which impressions come to us. Two impressions come at the same time or in succession, and when one of them is remembered, the other is likely to be remembered also. We see a man and hear his name; when we remember the man’s face, we may remember his name also. Hume maintains that this association of succession or coexistence is given with the impressions themselves. It is the order of the _outer impressions_. We perceive the order of the outer impressions with the same certainty that we perceive the contents of the impressions. _This is the only certainty we have about “matters of fact,”_――a certainty of the exact order of our immediate outer impressions. We know the order in which our impressions do occur, but, as we shall see, when we argue from this that our impressions must recur in the same order we are involved in a fallacy. Any order may recur. The fact that the sun rises in the east to-day does not make certain that it will rise in the east to-morrow. It is only a matter of probability, however many times repeated. There is no certain science of “matters of fact.” =The Association of Resemblance.= This is a clear and distinct association which is given with the impressions. When we have an impression, we see intuitively its similarity or difference to other impressions, and the degrees of likeness and unlikeness. The face of one man reminds us of another man, or we contrast it with a brute’s face. _This association concerns only inner impressions_, while the association of contiguity concerns outer impressions. This has to do with the “relation of ideas,” while the association of contiguity has to do with “matters of fact.” =1. Mathematics.= But there is this difference between the association of resemblance and that of contiguity――upon resemblance is founded a demonstrative science. This is mathematics――the sole demonstrative science. The subject-matter of mathematics consists of the possible relations between the contents of our ideas――the possible relations between our inner impressions. These relations are intuitively known by us, and out of them we get a science of complete certainty. We make a comparison between the magnitudes in the contents of ideas, and we analyze their regularity. This is mathematics, and it is a perfectly legitimate science. Because it confines itself to the relations between ideas, and has nothing to do with “matters of fact,” it can be a demonstrative science. All mathematical knowledge is restricted to the study and verification of ideas, and has therefore nothing to do with the external world.