A further point must be made: Sigma pQ, where the Q's are interpreted as abstract numbers, is a summary of concrete money payments, each of which has a causal explanation, and each of which has effected a concrete exchange. Mathematically, PT is equal to [Greek: S] pQ, just as 3 times 4 is equal to 2 times 6. But from the standpoint of the theory of causation, a vast difference is made. Three children four feet high equal in aggregate height two men six feet high. But the assertion of equality between the three children and the two men represents a high degree of abstraction, and need not be significant for any given purpose. Similarly, the restatement of [Greek: S] pQ as PT. One might restate [Greek: S] pQ as PT, defining P as the _sum_ (instead of the average) of the p's, and T as the weighted average (instead of the sum) of the Q's. Such a substitution would be equally legitimate, mathematically, and the equation, MV = PT equally true. [Greek: S] pQ might be factorized in an indefinite number of ways. But it is important to note that in PT, as defined by Professor Fisher,[144] we are at three removes from the concrete exchanges in which actual concrete causation is focused: we have first taken, for each commodity, an average, for a period, say a year, of the concrete prices paid for a unit of that commodity, and multiplied that average by the abstract number of units of that commodity sold in that year; we have then summed up all these products into a giant aggregate, in which we have mingled hopelessly a mass of concrete causes which actually affected the particular prices; then, finally, we have factorized this giant composite into two numbers which have no concrete reality, namely, an average of the averages of the prices, and a sum of the abstract numbers of the sums of the goods of each kind sold in a given year--a sum which exists only as a pure number, and which, consequently, is unlikely to be a causal factor! It may turn out that there is reason for all this, but if a _causal_ theory is the object for which the equation of exchange is designed, a strong presumption against its usefulness is raised. Both P and T are so highly abstract that it is improbable that any significant statements can be made of either of them. As concepts gain in generality and abstractness, they lose in content; as they gain in "extension" they lose (as a rule) in "intension." On the other side of the equation, we also look in vain for a truly concrete factor. V, the average velocity of money for the year, is highly abstract. It is a mathematical summary of a host of complex activities of men. Professor Fisher thinks that V obeys fairly simple laws, as we shall later see, but at least that point must be demonstrated. Even M is not concrete. At a given moment, the money in circulation is a concrete quantity, but the average for the year is abstract, and cannot claim to be a direct causal factor, with one uniform tendency.