Is Math More Precise Than Words?

5 December 2006 at 10:02 pm Peter G. Klein 16 comments

| 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.)

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

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16 Comments Add your own

1. Michael Greinecker | 6 December 2006 at 7:05 am

I don’t understand what Rothbard wants to show us with his example. One can define monotonicity without any requirement of differentiability on functions:

(p > p’) => f(p) =Man, Economy, and State, he tries to show that everything follows from one single assumption. When translating his arguments into math, it becomes obvious that he is constantly introducing new assumptions. You can’t do that with formal economics.

I have never seen a single example of a economic theory that is both consistent and cannot be translated to mathematics.
I have however seen many examples of verbal “theorems” that are not theorems at all (the Coase theorem, the rotten kid theorem…)
2. Joseph Mahoney | 6 December 2006 at 10:18 am

Dear Peter,

A book worth a look is by Philip Davis and Reuben Hersh titled: THE MATHEMATICAL EXPERIENCE. Davis and Hersh write that: “the line between complete and incomplete proof is always somewhat fuzzy, and often controversial” (1981: 34; cf. p. 40).

Real proofs “are established by ‘consensus of the qualified'” and are “not checkable … by any mathematician not privy to the gestalt, the mode of thought in the particular field … It may take generations to detect an error” (1981: 354).

Mathematical knowledge is “fallible, corrigible, tentative, and evolving, as is every other kind of human knowledge” (1981: 406)

The main point is that Economists and Strategic Management Scholars (like you, Nicolai and myself) need not be overly concerned that we have no bedrock of ultimate Truth. Standards of mathematical proof also change.

Another book by mathematician Morris Kline, titled: MATHEMATICS: THE LOSS OF CERTAINTY says much the same: “There is no rigorous definition of rigor. A proof is accepted if it obtains the endorsement of the leading specialists of the time and employs the principles that are fashionable at the moment. But no standard is universally acceptable today” (1980: pp. 6 and 315)

These quotes can be found in D. McCloskey, THE RHETORIC OF ECONOMICS. Thus, there could easily be a book written on the “Rhetoric of Mathematics.”
3. Michael Greinecker | 6 December 2006 at 11:02 am

Most of mathematics today can be checked by a computer, so I somewhat disagree with this conclusion.

Are there any important recent example of how “fallible” mathematics is? Even mistakes humans make, don’t matter much to the discussion. This is why linguists separate performance and competence.

“Thus, there could easily be a book written on the “Rhetoric of Mathematics.””

There is one: Imre Lakatos, Proofs and Refutations
4. Peter Klein | 6 December 2006 at 2:55 pm

Joe, thanks for the cites. Michael, your formulation of the challenge — “I have never seen a single example of a economic theory that is both consistent and cannot be translated to mathematics” — begs the question. If by “consistent” you mean consistent in the mathematical sense, then of course statements that cannot be translated into mathematics cannot be mathematically consistent. If by “consistent” you mean “true,” “valid,” “reasonable,” “useful,” “accurate,” etc., then you have to define criteria for truth or usefulness independent of the language in which the argument is expressed.

As for the more general issue under discussion, well, there is a huge literature on this. For now, if you want an economic proposition that many find true and useful, but that cannot be expressed mathematically, how about:

“Prices are determined by supply and demand.”

The concept of (causal) determination is not easily expressed mathematically. The standard supply-and-demand models, from the Econ 101 version to Arrow-Debreu and beyond, say simply that if the price is equal to some specified price, then the quantity supplied equals the quantity demand, and not otherwise. We can define an equilibrium price mathematically, but such models do not explain what “caused” it.

Of course, the usual response from mathematical social scientists is that causality is not an operationally meaningful concept. It is a non-scientific (or pre-scientific), “philosophical,” or even worse, “literary” concept. Again, that begs the question of what constitutes a scientific explanation.

Along these lines, readers might find the Cohn article cited in this post of interest.
5. srp | 1 January 2007 at 3:00 am

I know this is way late (I just saw this post), but for crying out loud–Karl Menger got it wrong in his verbal argument. The Law of Demand does NOT say that demand is lower (or no higher) when price is higher. It says that the QUANTITY DEMANDED is lower (or no higher) when price is higher. Note that the mathematical setup automatically captures this critical distincition. So, yes, you can do very well with verbal arguments if you are extremely punctilitious and have very clear and distinct ideas, but most people most of the time are much more likely to commit fallacies of equivocation and miss internal contradictions in their ideas if they eschew mathematical or graphical tools.

Reading Israel Kirzner’s work I was struck by how much his verbal arguments rely on the reader already knowing the pictures for supply and demand, indiference curves, etc. That’s smart on his part.

As for causality and math versus causality and words: Come on. Saying the word “cause” doesn’t capture anything about the mechanism by which prices are set. And “causal” processes can be described in formal terms with dynamic models, at least if you’re not a strict Humean skeptic.

It is a fair critique of the simple supply-demand math that it doesn’t include the stories about how shortages and gluts lead to price dynamics, but that isn’t a problem with math. That’s a problem with the theory. (You can add price-adjustment mechanisms, but I think it is safe to say that individual optimizing models of price dynamics are not fully satisfactory.)
6. Peter Klein | 1 January 2007 at 12:11 pm

Steven, I don’t think you’ve caught Karl Jr. in an elementary error. As I recall (though I don’t have the source in front of me) he uses “demand” for quantity demanded and “demand schedule” for demand. He knows the difference.

And yes, you’re right, simply invoking the word “cause” doesn’t make a causal argument. But Menger Sr. does, in fact, offer a causal explanation for price determination. It’s quite distinct from, e.g., Walras’s. And he and his followers do have a theory of price dynamics. It’s only since the post-WWII formalization of the discipline that we’ve lost these notions, IMHO.
7. srp | 2 January 2007 at 5:27 am

Peter, the ambiguity in the verbal, as opoosed to the mathematical, definition of the demand function is neatly captured by our exchange..(And would you really want to explain supply and demand to someone without drawing the Marshallian cross?–that seems perverse.) In any case, there is no reason to assume differentiability to specify weakly monotonically decreasing funcitons, as Michael pointed out in the first comment. So the whole quoted passage seems irrelevant.

As for explaining convergence to equilibrium, I’m not familiar with Menger’s theory. If it looks anything like the neo-Austrians’, though, it is pretty rough and doesn’t lead to specific predictions or give much better intuition than the non-mathematical “just-so” stories in the textbooks about gluts and shortages.

Convergence to equilibrium is a pretty hard and complex problem if you actually want to say something about relative speed of convergence in different markets, price dispersion along the way to equilibrium, impact of different information conditions, etc. The idea that we could make much headway on these questions without some kind of formal modeling (at least an agent-based computer model, if nothing else) seems far-fetched to me. I would be delighted to be proved wrong about this, and people are of course welcome to try, but it seems quixotic at best..
8. Peter Klein | 2 January 2007 at 1:29 pm

Hmmm, I think we’re speaking past each other here (an argument in your favor, perhaps? :-) ) The point of the quoted passage is that mathematical statements are not _necessarily_ more precise than verbal ones, though they may be less general. Generality is not the same dimension as precision. (Again, I recommend Roderick Long’s article on abstraction, mentioned at the end of the original post above.)

As to the broader questions of formal methods in the social sciences, well, that is a huge topic with a vast literature. BTW I agree that the kind of dynamics you’re talking about can’t be modeled verbally. We may disagree about the value added of such modeling, however.
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10. Jason Paul | 15 March 2014 at 8:26 am

“The concept of (causal) determination is not easily expressed mathematically”

Actually that is a very easy concept to express in mathematical symbols and there is a very systematic way to do it. It goes:

Price = f(supply,demand)

Whatever the function may be depend on how you believe supply and demand interacts (with the assumption that you’ve already defined supply and demand numerically)

And that goes without saying a mathematical function should only have variables that are numerical in nature. Non numerical variable do not have a place in mathematics.

Thus, we don’t use mathematics to express non-numerical ideas the same way we paint a picture if you intend to sing a song.
11. Peter Klein | 15 March 2014 at 9:58 pm

Um, you haven’t heard of inverse functions?
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16. Chris King | 27 September 2022 at 8:17 pm

It often appears that mathematics seem accurate because formulas are ‘true to themselves.’ Mathematics creates models with rules for the function of the model, that are based on reality, and in best case scenario, tested repeatedly against reality. But formulas can be generalizations, eminently useful but inaccurate nonetheless until a new formula is generated. Both mathematics and spoken language depend on users experience of the terms. They are meaningless unless the formula or the sentence generates experience in the mind of the ‘reader’ that evokes life experience. But it is possible to create fictional world with words that are consistent in their fictional terms. Perhaps the same is true of mathematics. Newton’s perspectives shaped thinking, and still do. And then there was Einstein.