Mises Versus Rothbard On Mathematics and Economics

Within the community of adherent to the "Austrian school" of economic thought, there exists certain members of the community who are willing to accept only the Rothbardian conception of things. Unfortunately for all of us, these folks believe that Rothbard is the only post-Mises economist whose theories accurately capture, modernize, and improve upon the Misesian praxeological tradition. They will not tolerate any other point of view. They adhere to this point of view with a rabid dogmatism better befitting a religious fervor than an intellectual pursuit.

They also happen to be among the most vocal members of the Austrian school community. Consequently, there is a widespread idea that Rothbard's views are the same as or are improvements upon Mises' views, and that all other notions within the school are either wrong, "confused" (to use a term common among a particularly noteworthy Austrian school adherent), or bad in some way.

Now, it's one thing to believe the above, state the position, and defend it intellectually and civilly. It's quite another thing to shout down all opposition, to publicly berate those holding opposing viewpoints, and to be so intolerant to questioning that the rest of us get turned off to the conversation long before any truth presents itself.

In the following blog post, I will highlight one example of Rothbard's views being a rather paler send-up of Misesian praxeology, to the point of being - in my opinion - extremely mistaken, wrong, confused, and bad. In doing so, my hope is to break down some of the religious reverence some hold over Rothbard's views.

Why? Because no intellectual pursuit can ever hope to be advanced if the assumption is that it was fully and perfectly developed in 1959. If Rothbard was correct about everything and it's unacceptable to question him, then we are no longer talking about a study or a discipline, we're talking about a religion or a cult. (Yep, I said it. A cult. Pretty rich, if you ask me.)

Rothbard On Mathematics In Economics
Murray Rothbard, supposedly the great palladin of Austrian school economics and its last great innovator, rejected all forms of mathematics in economics. However, in saying so, I'm not really doing justice to complete and utter disdain Rothbard held for not just mathematical economic models, but even using mathematical notation to express logical ideas.

In Man, Economy, and State, Rothbard writes (footnotes eliminated for brevity):
The suggestion has been made that, since praxeology and economics are logical chains of reasoning based on a few universally known prem­ises, to be really scientific it should be elaborated according to the symbolic notations of mathematical logic. This represents a curious misconception of the role of mathematical logic, or “logistics.” In the first place, it is the great quality of verbal propositions that each one is meaningful. On the other hand, algebraic and logical symbols, as used in logistics, are not in themselves meaningful. Praxeology asserts the action axiom as true, and from this (together with a few empirical axioms—such as the existence of a variety of resources and individuals) are deduced, by the rules of logical inference, all the propositions of economics, each one of which is verbal and meaningful. If the logistic array of symbols were used, each proposition would not be meaningful. Logistics, therefore, is far more suited to the physical sciences, where, in contrast to the science of human action, the con­clusions rather than the axioms are known. In the physical sciences, the premises are only hypothetical, and logical deductions are made from them. In these cases, there is no purpose in having meaningful propositions at each step of the way, and therefore symbolic and mathematical language is more useful.
It is difficult to see where in this paragraph Rothbard made any correct statement about mathematical logic.

First, there is Rothbard's claim that the notation of formalized, mathematical logic is not in itself meaningful, whereas verbal statements are inherently (i.e. "in themselves") meaningful. For Rothbard to simply leave these claims standing without proving them is to have failed at the requirements of a praxeologist. One cannot simply state that one's own method is inherently meaningful while the method of one's contemporaries is not. One must prove it, or at least provide a compelling argument why this is so.

So, I will counter with the following argument:
  • Rothbard claims that all verbal propositions are inherently meaningful.
  • If I can establish one verbal proposition that is not inherently meaninful, I will have demonstrated that Rothbard was wrong.
  • The following proposition is not inherently meaningful: If this sentence is true, Rothbard was an Objectivist.[Note: This is an example of Curry's Paradox. For further discussion regarding the meaninglessness of paradoxes, see this Stationary Waves post.]
  • Therefore, Rothbard was wrong.
The reader may find it particularly pedantic to disprove Rothbard in this way, but I find it important to use this particular line of reasoning, since it is the only one Rothbard himself thought valid.

As to the allegation that the symbols of mathematical logic "are not in themselves meaningful," one could make the same claim about letters of the alphabet. That is, in the philosophy of language, we generally accept that words have specific definitions. That Om means "peace" in Hindi and "mango" in Bangla is a fact that essentially boils down to the fact that one community of people agrees that the associated noise means one thing, while the other community agrees that it means another thing. That is, the sound Om is not in itself meaningful, and is in fact quite meaningless in English.

But Rothbard isn't merely suggesting that mathematical logic clouds the issue. In fact, he states in the above quote that "If the logistic array of symbols were used, each proposition would not be meaningful." In Rothbard's view, merely using mathematical logic notation to express an idea at all is "not meaningful."

We may as well argue that the English language is the only "meaningful" way to engage in dialectics. Who would ever make such a claim?

To this absurdity, Rothbard offers only one defense:
Simply to develop economics verbally, then to translate into logistic symbols, and finally to retranslate the propositions back into English, makes no sense and violates the fundamental scientific principle of Occam’s razor, which calls for the greatest possible simplicity in sci­ence and the avoidance of unnecessary multiplication of entities or processes.
Rothbard calls the expression of ideas in a language other than verbal exposition a violation of Occam's razor (which, by the way, is hardly a law). By that line of reasoning, Human Action violates Occam's razor by virtue of its being a re-worked English-language version of Mises' Nationalokonomie, especially if read by a native English speaker who must translate the German text into English-based thoughts in order to understand them. Indeed, Man, Economy, and State itself began as an update to Human Action and could likewise be dismissed as a violation of Occam's razor.

But this is ridiculous!

Mises On Mathematics in Economics
The question arises, is this really praxeology in the Misesian tradition? Did Mises feel this way about mathematical economics? Some say yes. Let's take a look at what Mises did say.

In Human Action, he writes:
The mathematical method must be rejected not only on account of its barrenness. It is an entirely vicious method, starting from false assumptions and leading to fallacious inferences. Its syllogisms are not only sterile; they divert the mind from the study of the real problems and distort the relations between the various phenomena.
This is from a section of the book entitled "Logical Catallactics Versus Mathematical Catallactics." While this is a damning passage indeed, right away we notice that Mises is not attacking the use of the mathematical expression of ideas, but rather the mathematical method as it pertains to solving economic problems.

In other words, Mises would never childishly attack mathematical expression as being "inherently meaningless." Mises had great respect for mathematics and science. He simply felt that logical catallactics were a more appropriate methodology than mathematical catallactics.

And please note: Mises makes no reference to verbal logic. This is important because the use of mathematical logic is still an example of logical catallactics. This fact must be underscored several times over. The particular language used to express logical ideas does not impact the validity or "meaningfulness" of the logic itself. Whether we notate our dialectics using the Queen's English, mathematical notation, or street Bangla, we may freely engage in logical catallactics so long as we follow a logical chain.

Mises found three methodological problems with mathematical catallactics. The first problem was the use of econometrics as a tool of economic theory. As Mises put it:
Statistics is a method for the presentation of historical facts concerning prices and other relevant data of human action. It is not economics and cannot produce economic theorems and theories.
He further states that "The arrangement of various price data in groups and the computation of averages are guided by theoretical deliberations which are logically and temporally antecedent." In other words, a theory must be developed prior to the application of a statistical model.

This, however, is a far cry from condemning the use of mathematical symbols at all, as Rothbard did.

Mises' second objection to mathematical economic theory was rather complex, but I shall try to summarize it succinctly. First, Mises remarks that supply and demand equations implicitly assume the existence of money to arrive at a particular price or cost. One objection to this is that it ignores the case of non-monetary transactions and barter. But this first objection is merely a lead-in to a greater problem with mathematical economics, which is the (mathematical) assumption of (mathematical) continuity:
On the other hand prices--if this term is applicable at all to exchange ratios determined by barter--are the enumeration of quantities of various goods against which the "seller" can exchange a definite supply. The goods which are referred to in such "prices" are not the same to which the "costs" refer. A comparison of such prices in kind and costs in kind is not feasible.
Note that again this is not a problem with the notation of mathematical logic. Continuity is an assumption of calculus functions. The problem is not the symbolism used, but rather the theoretical assumptions.

Mises' third objection to mathematical economics is unique only to those mathematical economists who prefer to explore mathematical equations outside the particular context of economic theory:
The characteristic mark of this third group is that they are openly and consciously intent upon solving catallactic problems without any reference to the market process. Their ideal is to construct an economic theory according to the pattern of mechanics. They again and again resort to analogies with classical mechanics which in their opinion is the unique and absolute model of scientific inquiry. There is no need to explain again why this analogy is superficial and misleading and in what respects purposive human action radically differs from motion, the subject matter of mechanics. It is enough to stress one point, viz., the practical significance of the differential equations in both fields.
Thus, we see a large discrepancy between the views of Rothbard and those of Mises with respect to mathematical economics. While both economists were critical of mathematical analysis in the context of economics, Mises' objections were methodological, while Rothbard felt that mathematical notation itself is inherently meaningless.

It is important to stress this difference because Mises' views were epistemological in nature. He felt that mathematical models were inappropriate for tackling the abstract problems of economics.  Rothbard did not even address this point, but rather simply dismissed mathematical logic notation because he himself felt it was meaningless.

It is possible to come to the correct conclusion for the wrong reasons. Rothbard correctly adhered to a logical catallactic methodology, as well he should have. Yet, he quite incorrectly chose this methodology only because he himself didn't understand the meaning of mathematical symbols.

It is akin to claiming that the world is round based on the notion that "flat is a meaningless concept."

There are two important takeaways from this discussion:
  1. This is one point of many on which Rothbard's views are wildly and importantly different from Mises' views. That is, Rothbard and Mises were not engaged in the same reasoning. Rothbard's work is not a "continuation" of Mises' work, nor is it in any way an "improvement" upon it.
  2. This is one point of many on which Rothbard was not just wrong, but ridiculously wrong.
This second point is especially important because if we do not note important shortcomings in the theoretical works of famous Austrian school economists, the praxeological tradition will evaporate into the haze of cultish dogma.


  1. What makes things worse is that Rothbard was a trained mathematician.

    1. To me, it's downright puzzling, and highlights an underlying problem with Rothbardian analysis. It's accurate, but only in a manner of speaking. It's highly rhetorical. One would expect that if he were going to place such a great emphasis on logical analysis, there would be more logical proofs involved. Instead, "Man, Economy, and State" reads more like a series of possibilities, contingent on Rothbard's use of language.

  2. Is it possible we just need more words?

  3. Mathematically Perfected Economy disproves any other proposition with a mathematical THEOREM.
    MPE is in fact a set of undeniable/irrefutable principles.