Von Mises Confused About Formal Reasoning, Praxeology

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Robert Vienneau
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In his book, Human Action, Ludwig von Mises defines 'praxeology' as the science of human action. He says that it is a subject of formal reasoning. Human action is conscious action in which the actor attempts to decrease felt uneasiness. 'Catallactics' is a subset of praxeology treating market exchanges. But von Mises is quite confused.

Von Mises does not know. At one point, he says formal reasoning does enlarge our reasoning. And he takes geometry as an example:

"All geometrical theorems are already implied in the axioms. The concept of a rectangular triangle already implies the theorem of Pythagoras. This theorem is a tautology, its deduction results in an analytic judgment. Nonetheless nobody would contend that geometry in general and the theorem of Pythagoras in particular do not enlarge our knowledge. Cognition from purely deductive reasoning is also creative and opens for our mind access to previously barred spheres. The significant task of aprioristic reasoning is on the one hand to bring into relief all that is implied in the categories, concepts, and premises and, on the other hand, to show what they do not imply. It is its vocation to render manifest and obvious what was hidden and unknown before." -- Von Mises, Human Action, Chapter II, Section 3. The a priori and reality

Von Mises makes a distinction between class and case probability. Class probabilities can be characterized by relative frequencies. Think of craps or blackjack. Case probabilities deal with uncertainity, where events cannot be repeated in identical circumstances. A horse race starts to be uncertain. But what about the prospects of a general European war in the next twenty years? Of course, von Mises does not reference Frank Knight or John Maynard Keynes.

Class probability can be formalized by Kolmogorov's axioms. But, strangely enough, von Mises claims that formal reasoning here cannot enlarge our knowledge:

"We know, for instance, that there are ninety tickets in a lottery and that five of them will be drawn. Thus we know all about the behavior of the whole class of tickets. But with regard to the singular tickets we do not know anything but that they are elements of this class of tickets...

...For this defective knowledge the calculus of probability provides a presentation in symbols of the mathematical terminology. It neither expands nor deepens nor complements our knowledge. It translates it into mathematical language. Its calculations repeat in algebraic formulas what we knew beforehand. They do not lead to results that would tell us anything about the actual singular events. And, of course, they do not add anything to our knowledge concerning the behavior of the whole class, as this knowledge was already perfect- or was considered perfect-at the very outset of our consideration of the matter.

..It is a serious mistake to believe that the calculus of probability provides the gambler with any information which could remove or lessen the risk of gambling. It is, contrary to popular fallacies, quite useless for the gambler, as is any other mode of logical or mathematical reasoning." -- Von Mises, Human Action, Chapter VI, Section 3. Class probability

I guess von Mises did not get his brother Richard, the more intelligent one, to review his work. I gather that von Mises was not exposed to David Hilbert's notion that axiomatic reasoning in mathematics is not about physical things in the external world, albeit he does have an observation about Einstein that suggests maybe he was.

In discussing this question, von Mises makes an elementary mistake in reasoning:

"Logic and mathematics deal with an ideal system of thought. The and implications of their system are coexistent and interdependent. We may say as well that they are synchronous or that they are out of time. A perfect mind could grasp them all in one thought. Man's inability to accomplish this makes thinking itself an action, proceeding step by step from the less satisfactory state of insufficient cognition to the more satisfactory state of better insight. But the temporal order in which knowledge is acquired must not be confused with the logical simultaneity of all parts of this aprioristic deductive system. Within this system the notions of anteriority and consequence are metaphorical only. They do not refer to the system, but to our action in grasping it. The system itself implies neither the category of time nor that of causality. There is functional correspondence between elements, but there is neither cause nor effect.

What distinguishes the praxeological system from the logical system epistemologically is precisely that it implies the categories both of time and of causality. The praxeological system too is aprioristic and deductive. As a system it is out of time. But change is one of its elements. The notions of sooner and later and of cause and effect are among its constituents. Anteriority and consequence are essential concepts of praxeological reasoning. So is the irreversibility of events. In the frame of the praxeological system any reference to functionaI correspondence is no less metaphorical and misleading than is the reference to anteriority and consequence in the frame of the logical system." -- Von Mises, Human Action, Chapter V., Section 1. The temporal character of praxeology

I more or less agree with von Mises that an axiomatic system does not exist in time, despite the fact that mortals, in stepping through deductions in such a system, use up time. But these claims' are distinct from the idea that such a system cannot characterize events in time in some sense. In fact, temporal logic exists. I suppose I could stand to know more about C. A. R. Hoare's Communicating Sequential Processes (CSP). In one of my papers, I illustrated a trace of an algorithm.

I think the answer to this question to be negative. I think I am agreeing with dominant trends in logic and philosophy for a century and a half. Von Mises, though, answers this question in the positive:

"The human mind is not a tabula rasa on which the external events write their own history. It is equipped with a set of tools for grasping reality... the logical structure of human reason.

praxeology ... is human because it claims for its theorems, within the sphere precisely defined in the underlying assumptions, universal validity for all human action

the logical structure of mind is uniform with all men of all races, ages, and countries -- Von Mises, Human Action, Chapter II, Section 2. The formal and aprioristic character of praxeology.

Is this logic build into all minds propositial logic? Predicate logic? Modal logic? Three-valued logic? Von Mises does not say. He does say that praxeology need not take into account the findings of psychology.

Maybe the issue is with von Mises' understanding of logic and formal reasoning. He dismisses the call for axioms and says his 'theorems' are drawing out the concepts implicit in the categories of human action. To me, this suggests an approach drawing on Immanuel Kant. Of course, von Mises does not reference Kant.

Anyways, let us look at some examples of what von Mises says are theorems:

"Thinking is to deliberate beforehand over future action and to reflect afterward upon past action. Thinking and acting are inseparable. Every action is always based on a definite idea about causal relations. He who thinks a causal relation thinks a theorem. Action without thinking, practice without theory are unimaginable. The reasoning may be faulty and the theory incorrect; but thinking and theorizing are not lacking in any action. On the other hand thinking is always thinking of a potential action. Even he who thinks of a pure theory assumes that the theory is correct, i.e., that action complying with its content would resuit in an effect to be expected from its teachings. It is of no relevance for logic whether such action is feasible or not." -- Von Mises, Human Action, Chapter IX, Section 1. Human Reason

The above sounds implausible to me. Not everybody acting, including me, can always articulate a proposition about why they are doing what they are doing. I suppose one could read von Mises as narrowing the concept of human action, contrary to his intent.

Here is another example:

"In the concept of money all the theorems of monetary theory are already implied. The quantity theory docs not add to our knowledge anything which is not virtually contained in the concept of money. It transforms, develops, and unfolds; it only analyzes and is therefore tautological like the theorem of Pythagoras in relation to the con- cept of the rectangular triangle." -- Von Mises, Human Action, Chapter II, Section 3. The a priori and reality

To my mind the properties of money, as a means of exchange, a unit of account, and a store of value, do not seem to be of the nature to easily support formal reasoning. Various assets have some degree of moneyness in various communities.

Von Mises nowhere lays out the basis of his formal reasoning in one compact place. He makes up his principles as he goes. For example, consider the tradeoff between labor and leisure. Here is some sentences about his assumptions:

"The disutility of labor is not of a categorial and aprioristic character. We can without contradiction think of a world in which labor does not cause uneasiness, and we can depict the state of affairs prevailing in such a world. But the real world is conditioned by the disutility of labor. Only theorems based on the assumption that labor is a source of uneasiness are applicable for the comprehension of what is going on in this world." -- Von Mises, Human Action, Chapter II, Section 10. The Procedure of Economics

And when it comes to intertemporal tradeoffs, he invents another principle.

"The theorem of time preference must be demonstrated in a double way. First for the case of plain saving in which people must choose between the immediate consumption of a quantity of goods and the later consumption of the same quantity. Second for the case of capitalist saving in which the choice is to be made between the immediate consumption of a quantity of goods and the later consumption either of a greater quantity or of goods which are fit to provide a satisfaction which-except for the difference in time-is valued more highly. The proof has been given for both cases. No other case is thinkable." -- Von Mises, Human Action, Chapter XVIII, Section 2. Time Preference as an Essential Requisite of Action

Von Mises just does not seem to appreciate the requirements for formal reasoning. Here is how some recent authors start to explain the difference between formal and informal reasoning:

Here are two tasks for you to try:

1. Define "chair". That is, give a set of properties so that:
1. everything which is a chair has those properties;
2. everything which has those properties is a chair;
3. everything you might deduce from that set of properties holds for all chairs.
2. Without giving the formal definition, describe the notion of a basis of a vector space.

Task 1 is hard, if not impossible. Dictionaries give definitions of concepts such as chair, but these are concise descriptions designed to capture pre-existing concepts; they do not, and are not designed to, satisfy constraints (a), (b) and (c). Indeed, most of us have probably never looked at a dic- tionary definition of "chair", but our experience with chairs means that we have no problem recognising and using them in everyday life (Vinner, 1976; Vinner, 1991; Edwards & Ward, 2004)

Task 2, on the other hand, is relatively easy. It may be somewhat unsatisfying to give an informal description of a concept such as basis, because we know that in doing so we are glossing over technical subtleties. But most of us do such things regularly in our teaching and in talking to colleagues. When introducing the concept in a lecture, we probably provide a formal definition and also try to give students a sense of the concept by saying something like "as small a set of vectors as possible, from which you can make all the others in the space". Indeed, we move easily between informal statements that describe a concept and a formal definition that, unlike a dictionary definition, prescribes what is (and is not) an instance of that concept. We recognise, however, that there is a huge functional difference between the two types of statement, and that formal definitions serve to structure the world of mathematics precisely because they do satisfy constraints (a), (b) and (c). -- Lara Alcock and Adrian Simpson, Ideas from Mathematics Education

Spinoza's Ethics offers a system laid out with formal reasoning, yet treating a subject that many today think resistant to such a treatment. Von Mises talks a good game, but does not do this.

I have hardly exhausted all the problems in von Mises' unscholarly book. It is fine for a researcher to produce a work against the dominant trends of a discipline. Israel Kirzner can argue that there are some insights here missed by, for example, Lionel Robbins's definition of economics. I, however, find the confusion to outweigh any insight.

• Ludwig von Mises. 1966. Human Action Third revised edition. Chicago: Contemporary Books.