Is Math More Precise Than Words?

5 December 2006 at 10:02 pm Peter G. Klein

| Peter Klein |

Commentator Michael Greinecker suggests below that mathematics, as a language for expressing economic arguments, is more precise than words. Indeed, Samuelson’s landmark Foundations of Economic Analysis (1947) opens with this statement from J. Willard Gibbs: “Mathematics is a language.” Samuelson felt he had to justify his translation of conventional economic analysis into mathematics — a defense hardly needed today!

As Roger Garrison once noted, mathematics is indeed a language, but so is music:

There is no reason for economists to observe a categorical prohibition against either mathematical formulation or musical expression. The relevant question is: What sort of language — music, mathematics, or, say, English — allows economists best to communicate their ideas? Which language serves the economist without imposing constraints of its own upon his subject matter?

Garrison argues for English (or French, German, Spanish, whatever) on the grounds that mathematics cannot express causality, and economics — here Garrison follows Menger and Mises — is essentially a causal science. (I make this point about Menger here.) That is a subject for another day, however.

For now, Michael’s comment reminded me of these remarks by the mathematician Karl Menger, Jr. (son of the economist Carl Menger):

Consider, for example, the statements (2) To a higher price of a good, there corresponds a lower (or at any rate not a higher) demand. (2′) If p denotes the price of, and q the demand for, a good, then q = f(p) and dq/dp = f'(p) <= 0 Those who regard the formula (2′) as more precise or “more mathematical’ than the sentence (2) are under a complete misapprehension. . . . The only difference between (2) and (2′) is this: since (2′) is limited to functions which are differentiable and whose graphs, therefore, have tangents (which from an economic point of view are not more plausible than curvature), the sentence (2) is more general, but it is by no means less precise: it is of the same mathematical precision as (2′).

The passage is quoted by Murray Rothbard in his “Praxeology: The Methodology of Austrian Economics.”

The point is that in defending mathematical formulations as more precise than verbal ones, social scientists are often really referring not to precision, but to generality, which is a different dimension. “More specific” or “less general” does not mean “more precise.” (By analogy, see Roderick Long’s excellent discussion of “precisive” versus “non-precisive” abstractions, a critique of Friedman’s methodology.)

Email

Print

Facebook

Tumblr

LinkedIn

Twitter

Reddit

Related

Entry filed under: - Klein -, Methods/Methodology/Theory of Science.