Mathematician: Plain English Often Works Better Than Mathematical Notation

Critics of Austrian economics often say that praxeology lacks rigor. Praxeologists rely on imprecise verbal logic that is difficult to assess. Instead, modern neoclassical economics is to a large extent couched in mathematics. The definitions and axioms of the model used are stated exactly, and then theorems can be proved to follow from them. Isn’t the Austrian school behind the times in not availing itself of the modern tools that mathematics provides?

Austrians respond to this that verbal reasoning can be as exact as mathematical, and that there are advantages to avoiding mathematics in economic theory. In particular, the functional equations of mathematics are inadequate to express causal relations. As Murray Rothbard notes,

Mathematics rests on equations, which portray mutual relationships between two or more “functions.” Of themselves, of course, such mathematical procedures are unimportant, since they do not establish causal relationships. They are of the greatest importance in physics, for example, because that science deals with certain observed regularities of motion by particles of matter that we must regard as unmotivated. These particles move according to certain precisely observable, exact, quantitative laws. Mathematics is indispensable in formulating the laws among these variables and in formulating theoretical explanations for the observed phenomena. In human action, the situation is entirely different, if not diametrically opposite. Whereas in physics, causal relations can only be assumed hypothetically and later approximately verified by referring to precise observable regularities, in praxeology we know the causal force at work. This causal force is human action, motivated, purposeful behavior, directed at certain ends. (Man, Economy, and State, with Power and Market, pp. 323–24)

I’d like to offer support for the first of these contentions, that verbal logic can be as exact as mathematics, from a surprising source. Paul Samuelson was a strong critic of Austrian economics; he said, for example, that Austrian “demonstrated preference” is trivial. Further, he is the economist mainly responsible for making mainstream economics mathematical. But in “Economic Theory and Mathematics—An Appraisal” (American Economic Review, 1952), he says: “In principle, mathematics cannot be worse than prose in economic theory; in principle, it certainly cannot be better than prose. For in deepest logic … the two media are strictly identical.”

An article by one of the greatest living mathematicians, Terence Tao, also supports Rothbard’s views, though I hasten to add that I do not know whether Tao has any opinion about the use of mathematics in economics. The article is called “Taking Advantage of the English Language.”

In the article, Tao says:

Mathematical notation is a wonderfully useful tool, and it can be exciting to learn for the first time the meaning of mysterious and arcane symbols such as $\forall$ , $\exists$ , $\emptyset$ , $\implies$ , etc. However, just because you can write statements in purely mathematical notation doesn’t mean that you necessarily should. In many cases, it is in fact far more informative and readable to use liberal amounts of plain English; if used correctly and thoughtfully, the English language can communicate to the reader on many more levels than a mathematical expression, without sacrificing any precision or rigour. In particular, by subtly modulating the emphasis of one’s text, one can convey valuable contextual cues as to how a statement interacts with the rest of one’s argument.

An example should serve to illustrate this point. Suppose for instance that P and Q are properties that can apply to mathematical objects x and y. The mathematical statement

$P(x) \wedge Q(y)$ ,

which asserts that x satisfies P and y satisfies Q, is a well-formed and precise mathematical statement. But there are many possible ways one could express that mathematical statement in English, for instance:

P(x) and Q(y) are both true.

P(x) is true. Also, Q(y) is true.

P(x) is true. Furthermore, Q(y) is true.

P(x) is true. Therefore, Q(y) is true….

From the viewpoint of formal mathematical logic, each of these English statements is logically equivalent to the mathematical sentence $P(x) \wedge Q(y)$ . However, each of the above English statements also provides additional useful and informative cues for the reader regarding the relative importance, non-triviality, and causal relationship of the component statements P(x) and Q(y), or of the component symbols P, x, Q, and y. For instance, in some of these sentences P(x) and Q(y) are given equal importance (being complementary or somehow in opposition to each other), whereas in others P(x) is only an auxiliary statement whose only purpose is to derive Q(y) (or vice versa), and in yet others, P(x) and Q(y) are deemed to be analogous, even if one is not formally deducible from the other. In some sentences, it is the objects x and y which are indicated to be the primary actors; in other sentences, it is the properties P and Q; and in yet other sentences, it is the combined statements P(x) and Q(y) which are the most central.

Thus we see that English sentences can be considerably more expressive than their formal mathematical counterparts, while still retaining the precision and rigour that mathematical exposition demands. By using such humble English words as “also”, ”but”, “since”, etc., a sentence conveys not only its semantic content, but also how it is going to fit in with the rest of one’s argument (or in the wider theory of the subject), giving the reader more insight as to the overall structure of that argument. The paper may become slightly longer because of this, but this is a small price to pay for readability (which is not the same as brevity!) …

Finally, there is one situation in which it does make sense to use the terse language of mathematical notation rather than a more leisurely English equivalent, and that is when you are performing a tedious and standard formal computation. In those cases, the reader should already know in general terms what is going to happen (especially if you flag the computation as being standard beforehand), and will only be distracted by superfluous explanation or digression.

The situation in which Tao says it is appropriate to use mathematical notion is one that does not apply to Austrian economics, which does not involve formal computations. To the contrary, in Austrian theory, the praxeologist is trying to understand each step of the deductions from the action axiom. (See my “Praxeology and Mathematical Logic” for more details on this point.)