"What about the theorem - that Rothbard dismissed - which claims that utility-maximizing individuals equalize the marginal utilities of goods consumed divided by their prices? Doesn't this show that neoclassicals believe in cardinal utility? No, it does not; statements made in technical jargon often sound absurd if you forget the underlying definitions. A utility function just uses numbers to summarize ordinal rankings; it doesn't commit us to belief in cardinal utility. Deriving the marginal utility of individual goods from this function commits us to nothing extra.[10]"

Deriving the marginal utility from ordinal utility functions commits us to nothing extra. Is that true? Is cardinality defined by being non-ordinal? ie. is marginal operations, taking derivatives (for the moment ignoring discrete vs continious and just focusing on ordinal vs cardinal) possible in ordinal rankings? can you derive a derivative, but then continue to say yes yes this derivative is from an ordinal ranking. Is he trying to have his cake and eat it too?

Could someone point me to a mathematics text (introductory as possible) that explains the requirements of a scale before these operations can be conducted?

Cardinality is absolute, while ordinality is relative. But dealing with ordinality per se requires discrete values. Discrete values, in turn, make standard calculus impossible. One can approximate standard calculus with discrete values by using finite differences to simulate derivatives and finite sums to simulate integrals. But this won't tell you much with strict ordinality. Taking the derivative of the 4th ordinal will simply tell you that it's in between the 3rd and 5th ordinals. Integrating the 3rd, 4th, and 5th ordinals will give you a result of 3 ordinals.

On another note, neoclassical economists use multivariate utility functions. They do this to compare the utility of two or more goods (typically only two). Using the simplest case of a two-variable utility function, where each variable represents the quantity of a different good, a three-dimensional space would be needed to graph this function. In theory, this function maps quantities of the two goods to rankings in ordinal-utility space. Because of this, there's no need for the mapping to be linear - it only needs to be monotonic.

However, monotonicity allows for successive values of the variables to map to the same function value. This is important as it allows for the notion of indifference (i.e. bundles of goods that provide an equal utility mapping). Indifference relaxes the ordinality requirement of utility - strict ordinality requires that each bundle of goods be either more or less preferred to every other. From the standpoint of human action, however, for a person to be indifferent with respect to two different bundles of goods means that he won't act to obtain one over the other, but that doesn't mean he can't try to obtain one of them. Upon doing so, he'll presumably not try to obtain the other - which seems to mean that the bundle not obtained now has a different position in his ordering of preferences. This suggests that not only are individuals' preference rankings variable over time, but that they also change when acted upon. Neoclassical economics doesn't seem to account for this at all, but I could be wrong.

1. Just to be clear, I do not think preferences can be cardinal.

2. However, I think the paper referred to in the link above just doesn't know enough math to be relevant.

As an analogy, the positive integers are discrete and therefore a function into them canot be differentited etc. And yet, many results about them, for example, the prime number theorem, rely on their being embedded in the complex plane, and functions of complex numbers then differentiated etc, with results about the positive integers.

Similarly, I see no reason why ordinal functions cannot be mapped into cardinal ones, represented, as the paper calls it.

3. Also, I think it possible for someone to be indifferent to two things. And just because we don't know, because he hasn't acted on it, so what? If we are discussing theory, it is theoretically possible. i don't get the arguments taht claim otherwise.

Indifference has no bearing on praxeology, as the latter focuses on action. In other words, indifference may be a thymological category, but not a praxeological one.

1. Can you please explain what thymological means, and what thymology is?

2. I don't see why, just because we cannot read minds, we have to blind ourselves to the existence of thoughts. A person may be indifferent in his mind, and this may have theoretical consequences that cannot be ignored, it seems to me.

We go much further by discussing an ERE, which is a fictional thing, and yet it is studied and gives insight into the real world.

Ever so much more so real thoughts, which should not be ignored in a theoretical discussion, if their existence has comsequences for the theory of economics.

2. Let us not confuse the object being studied with the methods used to study it.

Say we define arithmetic as the study of the rational numbers. This does not stop us from using real and complex numbers [and for that matter functions], in studying them, even though all those things are not rational numbers.

Going a step further, we use formal rules of logic to study arithmetic, and no one says that since the rules of logic are not numbers, we cannot use them.

Going a step further, we use normal human discourse while studying math, meaning English words and the concepts they represent, even though they are not rational numbers.

The rule is, or should be, all's fair in the quest for the truth. If considering thoughts gives us economic insight, as Caplan for example claims it does, then by all means.

Put another way, if the study of thymology leads to insights into praxeology, then we should go right ahead.

"Deriving the marginal utility from ordinal utility functions commits us to nothing extra."

I don't see the point of computing a marginal utility in terms of cardinal quantities OR differences in ordinal rankings. If a unit of one good advances things from state #7 to state #5 in my rankings, it is not twice as a good as a subsequent unit that advances things from state #5 to state #4 (i.e. an improvement of only one rank). And if I use cardinal numbers then I can (using arbitrary monotonic transformations) make the computed numbers take on any ratio I please so it appears that the first unit was 100x or 0.01x as valuable as the second consumed unit.

This whole discussion is absurd. No one prefers one friend 2.63 times as much as another friend. There is no scientific evidence for cardinal utilities or that the brain's neural patterns are storing floating-point numbers or can be represented as such. Your brain has only limited capacity; don't fill it with this garbage.

2. Praxeology concerns action and only action, does it not?

Thinking is an action, isn't it?

But, as it pertains to utility, one can't be indifferent if one chooses to consume one good over another as such an act demonstrates preferences, goals, values, etc. Even if someone thinks, "I don't care which ... that one" it's praxeologically not true that that person was indifferent to which he chose for the sake of maximizing utility. It demonstrates that this certain good had the added value, to whatever such value he may have already assigned to it in his mind, to the actor of ending the choice by making the decision of choosing this good over the other good.

If I had a cake and ate it, it can be concluded that I do not have it anymore. HHH

Go ahead and use complex numbers as a means to improve your understanding of human values and praxeology and economics.

Then report your findings back here. P.S. for extra credit, try using surreal and hyperrreal numbers and tensors too.

Tell us what economic insights Caplan uncovered that Mises missed due to his aversion to drawing graphs involving made-up units and made-up functions subject to monotonic transformations.

Since everything is fair game, why not use praxeology to understand physics problems. For example, maybe a ball perched on an incline rolls down the slope to remove it's felt uneasiness. The Earth's gravitational field is a capital good because it produces these lower-order goods (tugs downward) over time. It's like bartering: the ball is exchanging potential energy for kinetic energy.