The Measurement Chimera
In chapter 1 of his groundbreaking treatise, The Theory of Money and Credit, Ludwig von Mises explains what money is: a universally, or at least commonly, used medium of exchange. In chapter 2, Mises stresses what money is not. Contrary to the common fallacy, it is not a measure of value.
According to Mises, the notion of money as a measure of value is an artifact of the value theory of the "older political economy." By this he means the "classical economics" of Adam Smith, David Ricardo, and John Stuart Mill.
The classical economists by and large believed that the value of a good was an objective attribute of the good itself. Economic actors, according to classical theory, only exchanged goods if the respective values of the goods were equal (a fallacy that goes back to Aristotle).
And how do economic actors determine if an objective attribute of one thing is equal to the same objective attribute of another thing? Well, how do you determine equality between other objective attributes, like length, weight, volume, temperature, etc.? You measure, of course! And assuming value is an objective quantitative attribute, it would seem that the best unit of its measurement would be the money unit.
However, the classical economists were entirely backwards in their value theory. Therefore, their conception of money as a measure of value (derived, as it was, from their value theory) was equally backwards. Classical value theory was finally supplanted by what Mises calls "modern value theory" in the late 19th century. By this, Mises means the subjective-marginal-utility theory of value.
According to modern value theory, then, value is derived from utility. Valuation is a matter of preferring one good over another, according to the goods' respective utilities.
To prefer one good over another is to give the goods a rank order. Therefore "ordinal numbers" (first, second, etc.) can be applied to the valuation of goods. For example, you can say that, in order of your preference, a plum is first, an apple is second, and an orange is third.
But preferring is not measuring. Therefore, "cardinal numbers" (1, 2, 3¼, 4.5, etc.) cannot be applied to the valuation of goods.
While, after the advent of modern value theory, most economists accepted that valuation is not objective, and thus not cardinal, they just could not let go of cardinality altogether. Cardinality is necessary for the use of measurement and mathematics, and according to the prejudice of many thinkers, "science is measurement."
Value could not be cardinal, because it is a subjective preference based on utility. But maybe, thought some, utility itself could be thought of as cardinal!
Even one of the greatest pioneers of modern value theory (and Mises's teacher) Eugen von Böhm-Bawerk (1851–1914) tried to bring cardinality back in in this manner. However, instead of formulating measurable utility, Böhm-Bawerk tried to formulate measurable satisfaction. (The two notions are closely related; the "utility" of a good is the good's "causal relevance" for the satisfaction of a desire.)
Irving Fisher (1867–1947), on the other hand, did try to conceive of a way to measure utility itself.
In rebutting Böhm-Bawerk and Fisher in The Theory of Money and Credit (1912), Mises never quite explicitly spells out why utility, value, and satisfaction cannot be measured. For the most part, he just states that the law of marginal utility precludes it, and assumes his target audience (other economists) should be able to see why that is so at once.
However, Mises later summed up the argument nicely in Economic Calculation in the Socialist Commonwealth (1920):
Marginal utility does not posit any unit of value, since it is obvious that the value of two units of a given stock is necessarily greater than, but less than double, the value of a single unit.
Böhm-Bawerk's Attempt to Measure Satisfaction
Let us explore Böhm-Bawerk's attempt to find a measure for satisfaction, and see how, according to Mises, it does not "measure" up.
Böhm-Bawerk's argument runs as follows:
Let us say you would rather have eight plums than one apple, and you would rather have one apple than seven plums. We can then say that the satisfaction afforded by the consumption of an apple is more than seven times, but less than eight times, as great as the satisfaction afforded by the consumption of a plum. Voilà, cardinality!
Now let us see why this runs contrary to the very law of marginal utility that Böhm-Bawerk himself did so much to advance.
Böhm-Bawerk's claim is based on the supposition that the satisfaction provided by one apple is over seven times greater than the satisfaction provided by one plum.
But that is only true if you assume the satisfaction provided by seven plums is exactly seven times the satisfaction provided by one plum.
But the only reason one might assume that would be if one assumed that each plum provides exactly one seventh of the satisfaction of the whole collection of plums. And that would mean the satisfaction provided by each plum is equal.
But according to the law of marginal utility, the expected satisfaction provided by each successive unit of a good diminishes. So, the satisfaction provided by each successive plum cannot be equal to the satisfaction provided by the previous plum.
Therefore, even if you assume satisfaction can be measured at all, the satisfaction provided by seven plums cannot be equal to seven times the satisfaction provided by a single plum.
And thus there is no basis for concluding that the satisfaction provided by one apple is greater than seven times the satisfaction provided by one plum.
Fisher's Attempt to Measure Utility
Irving Fisher tried to discover a unit for the measurement of utility. And he purported to do so in a way that took the law of diminishing marginal utility into account.
His argument ran as follows:
Say you have 100 loaves of bread at your disposal in a given year. The marginal utility of 1 loaf, given that you have 100, is greater than the marginal utility of 1 loaf if you have 150.
Now let's say in that same year, you also have B gallons of oil.
Furthermore, let us call β (beta) the increment of B that is equal to the marginal utility of 1 loaf if you have 100 loaves.
Now let us say in the case in which you have 150 loaves, you have the same amount of oil (B). And let us say that, with 150 loaves, the marginal utility of a single loaf is equal to the marginal utility of half of β.
Since 1 loaf out of 150 gets you the marginal utility of half the oil that 1 loaf out of 100 gets, we can say that the marginal utility of 1 loaf out of 150 is half that of 1 loaf out of 100. Voilà, cardinality!
From that point Fisher proceeds to try to deduce a unit for measuring utility: the util.
At first glance, Fisher's attempt may seem more sophisticated than Böhm-Bawerk's. But it is really just more convoluted, and every bit as wrong (and for the exact same reason).
His conclusion only follows if it is supposed that the marginal utility of a given amount of oil (β) is twice the marginal utility of half that amount (β/2).
But that assumes that the marginal utility of a first increment of oil is equal to the marginal utility of a second equal increment of oil.
But just as with Böhm-Bawerk's plum argument, that flies in the face of the law of marginal utility. The marginal utility of a second increment of oil must be less than the marginal utility of the first increment.
Therefore there is no basis for saying that the marginal utility of a given amount of oil is twice the marginal utility of half that amount. And thus there is no basis for saying that the marginal utility of 1 loaf out of 150 is half that of 1 loaf out of 100.
Besides, if one is going to operate under the fallacious assumption that the marginal utility of a good necessarily rises and falls in strict proportion with its amount, why bother introducing the comparison good (oil) in the first place?
Under the same fallacious assumption, one could simply say at the outset that the marginal utility of 100 loaves of bread is necessarily 2/3 of the marginal utility of 150 loaves and be done with it!
Later in chapter 2, Mises also counters Joseph Schumpeter's attempt to quantify satisfaction. He does so by pointing out that Schumpeter assumes that valuation must be preceded by some prior measuring process. But simple reflection demonstrates that we are perfectly capable of looking at an apple and an orange and simply selecting one based on a direct comparison of the two choices. We do not need to consider any intermediary quantities and then decide based on an arithmetic comparison of those two quantities.
In section 2 of the chapter, Mises argues that, since value cannot be quantified, neither can values be summed up to infer the "total value" of a collection of goods.
In section 3, Mises brings money back into the picture. Money is not a measure of value, because valuation is a process of prioritization, not of measurement.
When a man buys a newspaper for 25¢, he is not really demonstrating that 25¢ is the "measure" of its value to him. He is demonstrating the he values the newspaper over 25¢. Furthermore he values the newspaper over 24¢, 23¢, etc. And he may even value the newspaper over 26¢ as well; although of course he'd rather pay 25¢ than 26¢. And at some point, there is a certain number of cents above which he values the money over the newspaper.
Strictly speaking, when a newspaper is purchased for 25¢, 25¢ is the newspaper's price, not its value.
Money, however, does introduce arithmetic into economic affairs in an important way. While money does not measure value, money prices can quantitatively express value in a somewhat commensurable way. This makes economic calculation (which is the hallmark of the market economy) possible.
One of Mises's greatest contributions was to purge many of the vestigial fallacies infecting modern economics that were carried over from pre–Marginal Revolution days. Pointing out why value, utility, and satisfaction can never be measured was one great instance of this.
 Murray N. Rothbard, "It All Began, As Usual, with the Greeks" (excerpted from An Austrian Perspective on the History of Economic Thought), Section 1.8
 This being the original motto of the Econometric Society. See Murray N. Rothbard; "The Mantle of Science."